On December 17 1903, the bicycle mechanic Orville Wright completed the first successful flight in a heavier-than-air machine. A flight that lasted a mere 12 seconds, reaching an altitude of 10 feet and landing 120 feet from the starting point. The Wright Flyer was made of wood and canvas, powered by a 12 horsepower internal combustion engine and endowed with the first, yet basic, mechanisms for controlling pitch, yaw and roll. Only 66 years later, Neil Armstrong walked on the moon, and another 12 years later the first fully re-usable space transportation system, the Space Shuttle, made its way into orbit.

Even though the means of providing lift and attitude control in the Wright Flyer and the Space Shuttle were nearly identical, the operational conditions could not be more different. The Space Shuttle re-entered the atmosphere at orbital velocity of 8 km/s (28x the speed of sound), which meant that the Shuttle literally collided with the atmosphere, creating a hypersonic shock wave with gas temperatures close to 12,000°C -temperature levels hotter than the surface of the sun. How was such unprecedented progress – from Wright Flyer to Space Shuttle – possible in a mere 78 years? This blog post chronicles this technological evolution by telling the story of five iconic aircraft.

Orville Wright&flyer1909

The Wright brothers were the first to succesfully fly what we now consider a modern airplane, but as the brothers would adamantly confirm, they did not invent the airplane. Rather, the brothers stood on the shoulders of a century-old keen interest in aeronautical research. The story of the modern airplane goes back to about 100 years before the Wright brothers, to a relatively unknown British scientist, philosopher, engineer and member of parliament, Sir George Cayley. Although Leonardo da Vinci had thought up flying machines 300 years prior to this, his inventions have relatively little in common with modern designs. In 1799 Cayley proposed the first three-part concept that, to this day, represent the fundamental operating principles of flying:

  • A fixed wing for creating lift.
  • A separate mechanism using paddles to provide propulsion.
  • And a cruciform tail for horizontal and vertical stability.

Many of the flying enthusiasts of the 18th century based their designs on the biomimicry of birds, combining lift, propulsive and control functions in a single oversized wing contraption that was insufficient at providing lift, forward propulsion, let alone a means of control. During a decade of intensive study of the aerodynamics of birds and fish from 1799-1810, Cayley constructed a series of rotating airfield apparatuses that tested the lift and drag of different airfoil shapes. In 1852, Cayley published his most famous work “Sir George Cayley’s Governable Parachutes”, which detailed the blueprint of a large glider with almost all of the features we take for granted on a modern aircraft. A prototype of this glider was built in 1853 and flown by Cayley’s coachman, accelerating the prototype off the rooftop of Cayley’s house in Yorkshire.

The distinctive characteristic of the Wright brothers was their incessant persistence and never-ending scepticism of the research conducted by scientists of authority. By single-handedly revising the historic textbook data on airfoils and building all of their inventions themselves, they developed into the most experienced aeronautical engineers of their day. Engineering often requires a certain intuitive knowledge of what works and what doesn’t, typically acquired through first-hand experience, and the Wright brothers had developed this knack in abundance. In this sense, they were best-equipped to refine the concepts of their peers and develop them into something that superseded everything that came before.

One of the most potent signals of British defiance in WWII is the Supermarine Spitfire. In the summer of 1940, during the Battle of Britain, the Spitfire presented the last bulwark between tyranny and democracy. Between July and October 1940, 747 Spitfires were built of which 361 were destroyed and 352 were damaged. Just 34 Spitfires that were built during the summer of 1940 made it through the war unscathed. Unsurprisingly, the Spitfire is one of the most famous airplanes of all time and its aerodynamic beauty of elliptical wings and narrow body make it one of the most iconic aircraft ever built.

Ray Flying Legends 2005-1

The Spitfire was designed by the chief engineer of Supermarine, RJ Mitchell. Before WWII Mitchell led the construction of a series of sea-landing planes that won the Schneider Trophy three times in a row in 1927, 1929 and 1931. The Schneider Trophy was the most important aviation competition between WWI and WWII – initially intended to promote technical advances in civil aviation, it quickly morphed into pure speed contest over a triangular course of around 300 km. As competitions so often do, the Schneider Trophy became an impetus for advancing aeroplane technology, particularly in aerodynamics and engine design. In this regard the Schneider Trophy had a direct impact on many of the best fighters of WWII. The low drag profile and liquid-cooled engine which were pioneered during the Schneider Trophy were all features of the Supermarine Spitfire and the Mustang P-51. The winning airplane in 1931 was the Supermarine S6.B, setting a new airspeed record of 655.8 km/h (407.4 mph). The S6.B was powered by the supercharged Rolls-Royce R engine with 1900 bhp, which presented such insurmountable problems with cooling that surface radiators had to be attached to the buoyancy floats used to land on water. In March 1936, Mitchell evolved the S6.B into the Spitfire with a new Rolls Royce Merlin engine. The Spitfire also featured its radical elliptical wing design which promised to minimise lift-induced drag. Theoretically, an infinitely long wing of constant chord and airfoil section produces no induced drag. A rectangular wing of finite length however produces very strong wingtip vortices and as a result almost all modern wings are tapered towards the tips or fitted with wing tip devices. The advantage of an elliptical planform (tapered but with curved leading and trailing edges) over a tapered trapezoidal planform is that the effective angle of attack of the wing can be kept constant along the entire wingspan. Elliptical wings are probably a remnant of the past as they are much more difficult to manufacture and the benefit over a trapezoidal wing is negligible for the long wing spans of commercial jumbo jets. However, the design will forever live on in one of the most iconic fighters of all time, the Supermarine Spitfire.

Captain Chuck Yeager, an American WWII fighter ace, became the first supersonic pilot in 1947 when the chief test pilot for the Bell Corporation refused to fly the rocket-powered Bell X-1 experimental aircraft without any additional danger pay. The X-1 closely resembled a large bullet with short stubby wings for higher structural efficiency and less drag at higher speeds. The X-1 was strapped to the belly of a B-29 bomber and then dropped at 20,000 feet, at which point Yeager fired his rocket motors propelling the aircraft to Mach 0.85 as it climbed to 40,000 feet. Here Yeager fully opened the throttle, pushing the aircraft into a flow regime for which there was no available wind tunnel data, ultimately reaching a new airspeed record of Mach 1.06. Yeager had just achieved something that had eluded Europe’s aircraft engineers through all of WWII.

Bell X-1 46-062 (in flight)

The limit that the European aircraft designer ran into during the air speed competitions prior to WWII was the sound barrier. The problem of flying faster, or in fact approaching the speed of sound, is that shock waves start to form at certain locations over the aircraft fuselage. A shock wave is a thin front (about 10 micrometers thick) in which molecules are squashed together by such a degree that it is energetically favourable to induce a sudden increase in the fluid’s density, temperature and pressure. As an aircraft approaches the speed of sound, small pockets of sonic or supersonic flow develop on the top surface of the wing due to airflow acceleration over the curved upper skin. These supersonic pockets terminate in a shockwave, drastically slowing the airflow and increasing the fluid pressure. Even in the absence of shock waves the airflow runs into an adverse pressure gradient towards the trailing edge of the wing, slowing the airflow and threatening to separate the boundary layer from the wing. This condition drastically increases the induced drag and reduces lift, which in the worst case can lead to aerodynamic stall. In the presence of a shock wave this scenario is exacerbated by the sudden increase in pressure and drop in airflow velocity across the shock wave. For this precise reason, commercial aircraft are limited to speeds of around Mach 0.87-0.88 as any further increase in speed would induce shock waves over the wings, increasing drag and requiring an unproportional amount of additional engine power.

It was precisely this problem that aircraft designers ran into in the 1930’s and 1940’s. To make their airplanes approach the speed of sound they needed incredible amounts of extra power, which the internal combustion engines of the time could not provide. Quite fittingly this seemingly insurmountable speed limit was dubbed the sound barrier. It was not until the advent of refined jet engines after WWII that the sound barrier was broken. However, exceeding the sound barrier does not mean things get any easier. The ratio of upstream to downstream airflow speed and pressure across a shock wave are simple functions of the upstream Mach number (airspeed / local speed of sound). Unfortunately for aircraft designers, these ratios change with the square of the upstream Mach number, which means that the induced drag becomes worse and worse the further the speed of sound is exceeded. This is why the Concorde needed such powerful engines and why its fuel costs were so exorbitant.

The North American X-15 rocket plane was one of NASA’s most daring experimental aircraft intended to test flight conditions at hypersonic speeds (Mach 5+) at the edge of space. Three X-15s made 199 flights from 1960-1968 and the data collected and knowledge gained directly impacted the design of the Space Shuttle. Initially designed for speeds up to Mach 6 and altitudes up to 250,000 feet, the X-15 ultimately reached a top speed of Mach 6.72 (more than one mile a second) and a maximum altitude of 354,200 feet (beyond the official demarcation line of space). As of this writing, the X-15 still holds the world record for the highest speed recorded by a manned aircraft. Given the awesome power required to overcome the induced drag of flying at these velocities, it is no surprise that the X-15 was not powered by a traditional turbojet engine but rather a full-fledged liquid-propellant rocket engine, gulping down 2,000 pounds of ethyl alcohol and liquid oxygen every 10 seconds.

North American X-15

The X-15 was dropped from a converted B-52 bomber and then made its way on one of two different experimental flight profiles. High-speed flights were conducted at an altitude of a typical commercial jetliner (below 100,000 feet) using conventional aerodynamic control surfaces. For high-altitude flights the X-15 initiated a steep climb at full throttle, followed by engine shut-down once the aircraft left Earth’s atmosphere. What followed was a ballistic coast, carrying the aircraft up to the peak of an arc and then plummeting back to Earth. Beyond Earth’s atmosphere the aerodynamic control surfaces of the X-15 were obviously useless, and so the X-15 relied on small rocket thrusters for control.

The hypersonic speeds beyond the conventional sound barrier discussed previously created a new problem for the X-15. In any medium, sound is transmitted by vibrations of the medium’s molecules. As an aircraft slices through the air, it disturbs the molecules around it which ensues in a pressure wave as molecules bump into adjacent molecules, sequentially passing on the disturbance. Flying faster than the speed of sound means that the aircraft is moving faster than this pressure wave. Put another way, the air molecules are transmitting the information of the disturbance created by the aircraft via a pressure wave that travels at the speed of sound. While the aircraft is creating new disturbances further upstream, Nature can’t keep up with the aircraft. At hypersonic speeds the aircraft is literally smashing into the surrounding stationary air molecules, and the ensuing compression of the air around the aircraft skin leads to fluid temperatures that are above the melting point of steel. Hence, one of the major challenges of the X-15 was guaranteeing structural integrity at these incredibly high temperatures. As a result, the X-15 was constructed from Inconel X, a high-temperature nickel alloy, which is also used in the very hot turbine stages of a jet-engine.

The wedge tail visible at the back of the aircraft was also specifically required to guarantee attitude stability of the aircraft at hypersonic speeds. At lower speeds this thick wedge created considerable amounts of drag, in fact as much as some individual fighter aircraft alone. The area of the tail wedge was around 60% of the entire wing area and additional side panels could be extended out to further increase the overall surface area.

12 April 1981 marked a new era in manned spaceflight: Space Shuttle Columbia lifted off for the first time from Cape Canaveral. The Shuttle capped an incredible fruitful period in aerospace engineering development. The ground work laid by the original Wright flyer, the Spitfire, the X-1 and the X-15 is all part of the technological arc that led to the Shuttle. The fundamentals didn’t change but their orders of magnitude did.

“Like bolting a butterfly onto a bullet” — Story Musgrave, Columbia astronaut, 1996

Story Musgrave’s description of the Space Shuttle is not far off the mark. On the launch pad the Shuttle sat on three solid-rocket boosters producing 37 million horsepower, accelerating the Shuttle beyond the speed of sound in about 30 seconds. Eight minutes and 500,000 gallons of fuel later the Shuttle was travelling at 17,500 mph at the edge of space. The Space Shuttle was not only powerful but possessed a grace that the Wright brothers would have appreciated. After smashing through the atmosphere upon reentry at Mach 28 (8 km/s) the piloting astronaut had to slow the Shuttle down to 200 mph via a series of gliding twists and turns, using the surrounding air as an aerodynamic break.

Shuttle profiles

The ultimate mission of the Shuttle was to serve as a cost-effective means of travelling to space for professional astronauts and civilians. That vision never came to fruition due to the high maintenance costs between flights, and partly due the Challenger and Columbia disasters that shattered all hopes that space travel would become routine.

Perhaps the Space Shuttle is one of humanities greatest inventions because it reminds us that for all its power, grace and genius it is still the brainchild of fallible men.

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After Germany and its allies lost WWI, motor flying became strictly prohibited under the Treaty of Versailles. Creativity often springs from constraints, and so, paradoxically, the ban imposed by the Allies encouraged precisely what they had actually wanted to thwart: the growth of the German aviation industry. As all military flying was prohibited under the Treaty, the innovation in German aviation throughout the 1920’s took an unlikely path via unmotorised gliders built by student associations at universities.

Before and during WWI, Germany had been one of the leading countries in terms of the theoretical development of aviation and the actual construction of novel aircraft. The famous aerodynamicist Ludwig Prandtl and his colleagues developed the theory of the boundary layer which later led to wing theory. The close relationship of research laboratories and industrial magnates, like Fokker and Junkers, meant that many of the novel ideas of the day were tested on actual aircraft during WWI. Part of the reason why Baron von Richthofen, the Red Baron, became the most decorated fighter pilot of his day, was because his equipment was more technologically advanced than that of his opponents; a direct result of a thicker cambered wing that Prandtl had tested in his wind tunnels.

Given this heritage, it comes to no surprise that German students and professors soon found a way around the ban imposed at the Treaty of Versailles. For example, a number of enthusiastic students from the University of Aachen formed the Flugwissenschaftliche Vereinigung Aachen (FVA, Aachen Association for Aeronautical Sciences). These students loved the art and science of flying and intended to continue their passion despite the ban. Theodore von Kármán, of vortex street and Tacoma Narrows bridge fame, was a professor at the Technical University of Aachen at the time and remembers the episode as follows:

One day an FVA member approached me with a bright idea.
“Herr Professor,“ he said. “We would like your help. We wish to build a glider.”
“A glider? Why do you wish to build a glider?”
“For sport.” the student said.
I thought it over. Constructing a glider would be more than sport. It would be an interesting and useful aerodynamic project, quite in keeping with German traditions, but in view of postwar turmoil it could be politically quite risky … On the other hand, motorised flight was specifically outlawed in the Treaty of Versailles, and sport flying was not military flying. So rationalizing in this way, I told the boys to go ahead.

What von Kármán was not aware of at the time was that he was helping to lay the foundation for an important part of the German air force during WWII. The lessons learned in improving glider design would be directly applicable to military aeronautics later on.

Glider development in itself is a topic worth studying. The French sailor Le Bris constructed a functional glider in 1870, but the most famous gliders of the 19th century were all built by Otto Lilienthal. Lilienthal became the first aviator to realise the superiority of curved wings over flat surfaces for providing lift. Lilienthal conducted some rudimentary wing testing to tabulate the air pressure and lift for different wing sections; data which inspired, but was then superseded by the Wright brothers’ experiments using their own wind tunnel. In the USA, Octave Chanute is famous for his work on gliders and for many years he served as a direct mentor to the Wright brothers, who themselves built a number of successful gliders to optimise wing shapes and control mechanisms.

After the first successful motor-powered flight in 1903, interest in gliders largely subsided, but was then revived by collegiate sporting competitions organised by German universities. Oskar Ursinus, the editor of the aeronautics journal Flugsport (Sport Flying), organised an intercollegiate gliding competition in the Rhön mountains, a spot renowned for its strong upwinds. So work began behind closed doors in many university labs and sheds. Von Kármán’s school, the University of Aachen, built a 6 m (20 foot) wing-span glider called the Black Devil, which was the first cantilever monoplane glider to be built at the time. As a result of the cantilever wing construction, the design abandoned any form of wire bracing to stabilise the wing and relied purely on internal wing bracing, as had been pioneered by Junkers in 1915. In this regard, the glider was already more advanced than most of the fighters in WWI that were based on the classical bi-plane or even trip-plane design held together by wires and struts.

The Black Devil sailplane, designed by Wolfgang Klemperer

By early 1920 the Black Devil was ready to compete. At this point the students faced a new logistical challenge — how were they going to transport the glider a 150 miles south through three military zones (British, French and American), when shipping aircraft components was strictly forbidden?

As reckless students they of course operated in secret. The Black Devil was dismantled into its components, packed into a tarpaulin freight car and then driven through the night. Of this episode von Kármán recounts that,

On one occasion during the journey we almost lost the Black Devil to a contingent of Allied troops. Fortunately the engineer and student guard received advance notice of the movement, disengaged the car holding the glider, and silently transferred it to a dark sliding until the troops rode past.

Overall, the trip took six hours and the teams from Stuttgart, Göttingen and Berlin were already in attendance.

Lacking any motorised aircraft to launch the gliders, two rubber ropes were attached to the nose of the glider and then used as a catapult to launch the glider off the edge of a hill. Once in the air, it was the role of the pilot to manoeuvre the plane purely by shifting his/her body weight to balance the glider in the wind. In 1920, Aachen managed to win the competition with a flight time of 2 minutes and 20 seconds. Not a new revolution in glider design, but proving the aerodynamics of their concept plane nevertheless. A year later, an improved version of the Black Devil, the Blue Mouse, flew for 13 minutes, breaking the long-held record by Orville Wright of 9 minutes and 45 seconds. Some great videos of the early flights at the Wasserkuppe in the Rhön mountains exist to this day.

The Blue Mouse glider flying at the Wasserkuppe in the Rhön mountains.

In the following years, von Kármán and his scientific mentor and aerodynamics pioneer Ludwig Prandtl gave a series of seminars on the aerodynamics of gliding, which were attended by students and flying enthusiasts from all over the country. Among the attendees was Willy Messerschmitt, an engineering student at the time, whose fighters and bombers later formed the core of the Nazi air force during WWII. Even established industrialists, German royalty and other university professors attended the talks. As a result of this broad and democratic dissemination of knowledge and the collaborative spirit at the time, innovations began to sprout quickly. One of the main innovations was efficiently using thermal updrafts in combination with topological updrafts to extend the flying time. In 1922, a collaborative design team from the University of Hannover built the Hannover H 1 Vampyre glider, which first extended the gliding record to 3 hours and then to 6 hours in 1923. The Vampyr was one of the first heavier-than-air aircraft to use the stressed-skin “monocoque” design philosophy and is the forerunner of all modern gliders.

Aircraft Glider Vampyr

The Vampyr glider. One of the first aircraft ever to use the stresses skin”monocoque” concept.

The collegiate sporting competitions continued until the early 1930’s. The Nazis soon realised that the technical knowledge gained in these sporting competitions could be used in rebuilding the German air force. Due to the lack of a power unit and limited control surfaces, the student engineers and industrialists had been forced to design efficient lightweight structures and wings that provided the best compromise in terms of lift, drag and attitude control. Most importantly, the gliders proved the superiority of single long cantilevered wings over the limited double- and triple-wing configuration used during WWI. The internal structure of the wing allowed for much lighter construction as the size of the aircraft grew, the parasitic source of drag induced by the wires and struts was eliminated, and the lift to drag ratio was dramatically improved by the long glider wings. Tragically, some pioneers took these concept too far and lost their lives as a result. While the lift efficiency of a wing is increased as the aspect ratio (length to chord ratio) increases, so do the bending stresses at the root of the wing due to lift. As a result, there were a number of incidents where insufficiently stiffened wings literally twisted off the fuselage.

On the importance of glider developments von Kármán reflects that,

I have always thought that the Allies were shortsighted when they banned motor flying in Germany … Experiments with gliders in sport sharpened German thinking in aerodynamics, structural design, and meteorology … In structural design gliders showed how best to distribute weight in a light structure and revealed new facts about vibration. In meteorology we learned from gliders how planes could use the jet stream to increase speed; we uncovered the dangers of hidden turbulence in the air, and in general opened up the study of meteorological influences on aviation. It is interesting to note that glider flying did more to advance the science of aviation than most of the motorised flying in World War I.

We can only speculate how von Kármán must have felt after leaving Germany in the 1930’s, partly due to his Jewish heritage, and witness from afar how the machines he helped to develop wreaked havoc in Europe during WWII.


The quotes in this post are taken from von Kármán’s excellent biography The Wind and Beyond: Theodore von Karman, Pioneer in Aviation and Pathfinder in Space by Theodore von Kármán and Lee Edson.

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On November 8, 1940 newspapers across America opened with the headline “TACOMA NARROWS BRIDGE COLLAPSES”. The headline caught the eye of a prominent engineering professor who, from reading the news story, intuitively realised that a specific aerodynamical phenomenon must have led to the collapse. He was correct, and became publicly famous for what is now known as the von Kármán vortex street.

Theodore von Kármán was one of the most pre-eminent aeronautical engineers of the 20th century. Born and raised in Budapest, Hungary he was a member of a club of 20th century Hungarian scientists, including mathematician and computer scientist John von Neumann and nuclear physicist Edward Teller, who made groundbreaking strides in their respective fields. Von Kármán was a PhD student of Ludwig Prandtl at the University of Göttingen, the leading aerodynamics institute in the world at the time and home to many great German scientists and mathematicians.

Von Karman and JATO Team - GPN-2000-001652 (cropped)

Theodore von Kármán jotting down a plan on a wing before a rocket-powered aircraft test

Although brilliant at mathematics from an early age, von Kármán preferred to boil complex equations down to their essentials, attempting to find simple solutions that would provide the most intuitive physical insight. At the same time, he was a big proponent of using practical experiments to tease out novel phenomena that could then be explained using straightforward mathematics. During WWI he took a leave of absence from his role as professor of aeronautics at the University of Aachen to fulfil his military duties, overseeing the operations of a military research facility in Austria. In this role he developed a helicopter that was to replace hot-air balloons for surveillance of battlefields. Due to his combined expertise in aerodynamics and structural design he became a consultant to the Junkers aircraft and Zeppelin airship companies, helping to design the first all-metal cantilevered wing aircraft, the Junker J-1, and the Zeppelin Los Angeles.

Furthermore, von Kármán developed an unusual expertise in building wind tunnels — a suitable had not originally exist when he first started his professorship in Aachen and was desperately needed for his research. As a result, he became a sought after expert in designing and overseeing the construction of wind tunnels in the USA and Japan. Von Kármán’s broad skill set and unique combination of theoretical and experimental expertise soon placed him on the radar of physicist Robert Millikan who was setting up a new technical university in Pasadena, California, the California Institute of Technology. Millikan believed that the year-round temperate climate would attract all of the major aircraft companies of the bourgeoning aerospace industry to Southern California, and he hired von Kármán to head CalTech’s aerospace institute. Millikan’s wager paid off when companies such as Northrup, Lockheed, Douglas and Consolidated Aircraft (later Convair) all settled in the greater Los Angeles area. Von Kármán thus became a consultant on iconic aircraft such as the Douglas DC-3, the Northrup Flying Wing, and later the rockets developed by NACA (now NASA).

Von Kármán is renowned for many concepts in structural mechanics and aerodynamics, e.g. the non-linear behaviour of cylinder buckling and a mathematical theory describing turbulent boundary layers. His most well-known piece of work, the von Kármán vortex street, tragically, reached public notoriety after it explained the collapse of a suspension bridge over the Puget Sound in 1940.

The von Kármán vortex street is a direct result of boundary layer separation over bluff bodies. Immersed in fluid flow, any body of finite thickness will force the surrounding fluid to flow in curved streamlines around it. Towards the leading edge this causes the flow to speed up in order to balance the centripetal forces created by the curved streamlines. This creates a region of falling fluid pressure, also called a favourable pressure gradient. Further along the body, where the streamlines straighten out, the opposite occurs and the fluid flows into a region of rising pressure, an adverse pressure gradient. The increasing pressure gradient pushes against the flow and causes the slowest parts of the flow, those immediately adjacent to the surface, to reverse direction. At this point the boundary layer has separated from the body and the combination of flow in two directions induces a wake of turbulent vortices (see diagram below).

Boundary layer separation over cylinder

Boundary layer separation over cylinder

The type of flow in the wake depends on the Reynolds number of the flow impinging on the body,

 Re = \frac{\rho V d}{\mu}

where \rho is the density of the fluid, V is the impinging free stream flow velocity, d is a characteristic length of the body, e.g. the diameter for a sphere or cylinder, and \mu is the viscosity or inherent stickiness of the fluid. The Reynolds number essentially takes the ratio of inertial forces \rho V d to viscous forces \mu, and captures the extent of laminar flow (layered flow with little mixing) and turbulent flow (flow with strong mixing via vortices).

Flow around a cylinder for different Reynolds numbers

Flow around a cylinder for different Reynolds numbers

For example, consider the flow past an infinitely long cylinder protruding out of your screen (as shown in the figure above). For very low Reynolds number flow (Re < 10) the inertial forces are negligible and the streamlines connect smoothly behind the cylinder. As the Reynolds number is increased into the range of Re = 10-40 (by, for example, increasing the free stream velocity V), the boundary layer separates symmetrically from either side of the cylinder, and two eddies form that rotate in opposite directions. These eddies remain fixed and do not “peel away” from the cylinder. Behind the vortices the flow from either side rejoins and the size of the wake is limited to a small region behind the cylinder. As the Reynolds number is further increased into the region Re > 40, the symmetric eddy formation is broken and two asymmetric vortices form. Such an instability is known as a symmetry-breaking bifurcation in stability theory and the ensuing asymmetric vortices undergo periodic oscillations by constantly interchanging their position with respect to the cylinder. At a specific critical value of Reynolds number (Re ~ 100) the eddies start to peel away, alternately from either side of the cylinder, and are then washed downstream. As visualised below, this can produce a rather pretty effect…


This condition of alternately shedding vortices from the sides of the cylinder is known as the von Kármán vortex street. At a certain distance from the cylinder the behaviour obviously dissipates, but close to the cylinder the oscillatory shedding can have profound aeroelastic effects on the structure. Aeroelasticity is the study of how fluid flow and structures interact dynamically. For example, there are two very important locations on an aircraft wing:
– the centre of pressure, i.e. an idealised point of the wing where the lift can be assumed to act as a point load
– the shear centre, i.e. the point of any structural cross-section through which a point load must act to cause pure bending and no twisting

The problem is that the centre of pressure and shear centre are very rarely coincident, and so the aerodynamic lift forces will typically not only bend a wing but also cause it to twist. Twisting in a manner that forces the leading edge upwards increases the angle of attack and thereby increases the lift force. This increased lift force produces more twisting, which produces more lift, and so on. This phenomenon is known as divergence and can cause a wing to twist-off the fuselage.

A different, yet equally pernicious, aeroelastic instability can occur as a result of the von Kármán vortex street. Each time an eddy is shed from the cylinder, the symmetry of the flow pattern is broken and a difference in pressure is induced between the two sides of the cylinder. The vortex shedding therefore produces alternating sideways forces that can cause sideways oscillations. If the frequency of these oscillations is the same as the natural frequency of the cylinder, then the cylinder will undergo resonant behaviour and start vibrating uncontrollably.

So, how does this relate to the fated Tacoma Narrows bridge?

Upon completion, the first Tacoma Narrows suspension bridge, costing $6.4 mill and the third longest bridge of its kind, was described as the fanciest single span bridge in the world. With its narrow towers and thin stiffening trusses the bridge was valued for its grace and slenderness. On the morning of November 7, 1940, only a year into its service, the bridge broke apart in a light gale and crashed into the Puget Sound 190 feet below. From its inaugural day on July 1, 1940 something seemed not quite right. The span of the bridge would start to undulate up and down in light breezes, securing the bridge the nickname “Galloping Gertie”. Engineers tried to stabilise the bridge using heavy steel cables fixed to steel blocks on either side of the span. But to no avail, the galloping continued.

On the morning of the collapse, Gertie was bouncing around in its usual manner. As the winds started to intensify to 60 kmh (40 mph) the rhythmic up and down motion of the bridge suddenly morphed into a violent twisting motion spiralling along the deck. At this point the authorities closed the bridge to any further traffic but the bridge continued to writhe like a corkscrew. The twisting became so violent that the sides of the bridge deck separated by 9 m (28 feet) with the bridge deck oriented at 45° to the horizontal. For half an hour the bridge resisted these oscillatory stresses until at one point the deck of the bridge buckled, girders and steel cables broke loose and the bridge collapsed into the Puget Sound.

After the collapse, the Governor of Washington, Clarence Martin, announced that the bridge had been built correctly and that another one would be built using the same basic design. At this point von Kármán started to feel uneasy and he asked technicians at CalTech to build a small rubber replica of the bridge for him. Von Kármán then tested the bridge at home using a small electric fan. As he varied the speed of the fan, the model started to oscillate, and these oscillations grew greater as the rhythm of the air movement induced by the fan was synchronised with the oscillations.

Indeed, Galloping Gertie had been constructed using cylindrical cable stays and these shed vortices in a periodic manner when a cross-wind reached a specific intensity. Because the bridge was also built using a solid sidewall, the vortices impinged immediately onto a solid section of the bridge, inducing resonant vibrations in the bridge structure.

Von Kármán then contacted the governor and wrote a short piece for the Engineering News Record describing his findings. Later, von Kármán served on the committee that investigated the cause of the collapse and to his surprise the civil engineers were not at all enamoured with his explanation. In all of the engineers’ training and previous engineering experience, the design of bridges had been governed by “static forces” of gravity and constant maximum wind load. The effects of “dynamic loads”, which caused bridges to swing from side to side, had been observed but considered to be negligible. Such design flaws, stemming from ignorance rather than the improper application of design principles, are the most harrowing as the mode of failure is entirely unaccounted for. Fortunately, the meetings adjourned with agreements in place to test the new Tacoma Narrows bridge in a wind tunnel at CalTech, a first at the time. As a result of this work, the solid sidewall of the bridge deck was perforated with holes to prevent vortex shedding, and a number of slots were inserted into the bridge deck to prevent differences in pressure between the top and bottom surfaces of the deck.

The one person that did suffer irrefutable damage to his reputation was the insurance agent that initially underwrote the $6 mill insurance policy for the state of Washington. Figuring that something as big as the Tacoma Narrows bridge would never collapse, he pocketed the insurance premium himself without actually setting up a policy, and ended up in jail…

If you would like to learn more about Theodore von Kármán’s life, I highly recommend his autobiography, which I have reviewed here.

The material we covered in the last two posts (skin friction and pressure drag) allows us to consider a fun little problem:

How quickly do the small bubbles of gas rise in a pint of beer?

To answer this question we will use the concept of aerodynamic drag introduced in the last two posts – namely,

  • skin friction drag – frictional forces acting tangential to the flow that arise because of the inherent stickiness (viscosity) of the fluid.
  • pressure drag – the difference between the fluid pressure upstream and downstream of the body, which typically occurs because of boundary layer separation and the induced turbulent wake behind the body.

The most important thing to remember is that both skin friction drag and profile drag are influenced by the shape of the boundary layer.

What is this boundary layer?

As a fluid flows over a body it sticks to the body’s external surface due to the inherent viscosity of the fluid, and therefore a thin region exists close to the surface where the velocity of the fluid increases from zero to the mainstream velocity. This thin region of the flow is known as the boundary layer and the velocity profile in this region is U-shaped as shown in the figure below.

Velocity profile of laminar versus turbulent boundary layer

Velocity profile of laminar versus turbulent boundary layer

As shown in the figure above, the flow in the boundary layer can either be laminar, meaning it flows in stratified layers with no to very little mixing between the layers, or turbulent, meaning there is significant mixing of the flow perpendicular to the surface. Due to the higher degree of momentum transfer between fluid layers in a turbulent boundary layer, the velocity of the flow increases more quickly away from the surface than in a laminar boundary layer. The magnitude of skin friction drag at the surface of the body (y = 0 in the figure above) is given by

 \tau_w = \mu \frac{\mathrm{d}u}{\mathrm{d}y}_w

where  \mathrm{d}u/\mathrm{d}y is the so-called velocity gradient, or how quickly the fluid increases its velocity as we move away from the surface. As this velocity gradient at the surface (y = 0 in the figure above) is much steeper for turbulent flow, this type of flow leads to more skin friction drag than laminar flow does.

Skin friction drag is the dominant form of drag for objects whose surface area is aligned with the flow direction. Such shapes are called streamlined and include aircraft wings at cruise, fish and low-drag sports cars. For these streamlined bodies it is beneficial to maintain laminar flow over as much of the body as possible in order to minimise aerodynamic drag.

Conversely, pressure drag is the difference between the fluid pressure in front of (upstream) and behind (downstream) the moving body. Right at the tip of any moving body, the fluid comes to a standstill relative to the body (i.e. it sticks to the leading point) and as a result obtains its stagnation pressure.

The stagnation pressure is the pressure of a fluid at rest and, for thermodynamic reasons, this is the highest possible pressure the fluid can obtain under a set of pre-defined conditions. This is why from Bernoulli’s law we know that fluid pressure decreases/increases as the fluid accelerates/decelerates, respectively.

At the trailing edge of the body (i.e. immediately behind it) the pressure of the fluid is naturally lower than this stagnation pressure because the fluid is either flowing smoothly at some finite velocity, hence lower pressure, or is greatly disturbed by large-scale eddies. These large-scale eddies occur due to a phenomenon called boundary layer separation.

Boundary layer separation over cylinder

Boundary layer separation over a cylinder


Why does the boundary layer separate?

Any body of finite thickness will force the fluid to flow in curved streamlines around it. Towards the leading edge this causes the flow to speed up in order to balance the centripetal forces created by the curved streamlines. This creates a region of falling fluid pressure, also called a favourable pressure gradient. Further along the body, the streamlines straighten out and the opposite phenomenon occurs – the fluid flows into a region of rising pressure, also known as an adverse pressure gradient. This adverse pressure gradient decelerates the flow and causes the slowest parts of the boundary layer, i.e. those parts closest to the surface, to reverse direction. At this point, the boundary layer “separates” from the body and the combination of flow in two directions induces a wake of turbulent vortices; in essence a region of low-pressure fluid.

The reason why this is detrimental for drag is because we now have a lower pressure region behind the body than in front of it, and this pressure difference results in a force that pushes against the direction of travel. The magnitude of this drag force greatly depends on the location of the boundary layer separation point. The further upstream this point, the higher the pressure drag.

To minimise pressure drag it is beneficial to have a turbulent boundary layer. This is because the higher velocity gradient at the external surface of the body in a turbulent boundary layer means that the fluid has more momentum to “fight” the adverse pressure gradient. This extra momentum pushes the point of separation further downstream. Pressure drag is typically the dominant type of drag for bluff bodies, such as golf balls, whose surface area is predominantly perpendicular to the flow direction.

So to summarise: laminar flow minimises skin-friction drag, but turbulent flow minimises pressure drag.

Given this trade-off between skin friction drag and pressure drag, we are of course interested in the total amount of drag, known as the profile drag. The propensity of a specific shape in inducing profile drag is captured in the dimensionless drag coefficient C_D

 C_D = \frac{D}{1/2 \rho U_0^2A}

where D is the total drag force acting on the body, \rho is the density of the fluid, U_0 is the undisturbed mainstream velocity of the flow, and A represents a characteristic area of the body. For bluff bodies A is typically the frontal area of the body, whereas for aerofoils and hydrofoils A is the product of wing span and mean chord. For a flat plate aligned with the flow direction, A is the total surface area of both sides of the plate.

The denominator of the drag coefficient represents the dynamic pressure of the fluid (1/2 \rho U_0^2) multiplied by the specific area (A) and is therefore equal to a force. As a result, the drag coefficient is the ratio of two forces, and because the units of the denominator and numerator cancel, we call this a dimensionless number that remains constant for two dynamically similar flows. This means C_D is independent of body size, and depends only on its shape. As discussed in the wind tunnel post, this mathematical property is why we can create smaller scaled versions of real aircraft and test them in a wind tunnel.

Skin friction drag versus pressure drag for differently shaped bodies

Looking at the diagram above we can start to develop an appreciation for the relative magnitude of pressure drag and skin friction drag for different bodies. The “worst” shape for boundary layer separation is a plate perpendicular to the flow as shown in the first diagram. In this case, drag is clearly dominated by pressure drag with negligible skin friction drag. The situation is similar for the cylinder shown in the second diagram, but in this case the overall profile drag is smaller due to the greater degree of streamlining.

The degree of boundary layer separation, and therefore the wake of eddies behind the cylinder, depends to a large extent on the surface roughness of the body and the Reynolds number of the flow. The Reynolds number is given by

 R = \frac{\rho U_0 d}{\mu}

where U_0 is the free-stream velocity and d is the characteristic dimension of the body. The reason why the Reynolds number influences boundary layer separation is because it is the dominant factor in influencing the nature, laminar or turbulent, of the boundary layer. The transition from laminar to turbulent boundary layer is different for different problems, but as a general rule of thumb a value of  R = 10^5 can be used.

This influence of Reynolds number can be observed by comparing the second diagram to the bottom diagram. The flow over the cylinder in the bottom diagram has increased by a factor of 100 ( R = 10^7), thereby increasing the extent of turbulent flow and delaying the onset of boundary layer separation (smaller wake). Hence, the drag coefficient of the bottom cylinder is half the drag coefficient of the cylinder in the second diagram ( R = 10^5) even though the diameter has remained unchanged. Remember though that only the drag coefficient has been halved, whereas the overall drag force will naturally be higher for  R = 10^7 because the drag force is a function of  C_D U_0^2 and the velocity U_0 has increased by a factor of 100.

Notice also that the streamlined aircraft wing shown in the third diagram has a much lower drag coefficient. This is because the aircraft wing is essentially a “drawn-out” cylinder of the same “thickness” d as the cylinder in the second diagram, but by streamlining (drawing out) its shape, boundary layer separation occurs much further downstream and the size of the wake is much reduced.

Terminal velocity of rising beer bubbles

The terminal velocity is the speed at which the forces accelerating a body equal those decelerating it. For example, the aerodynamic drag acting on a sky diver is proportional to the square of his/her falling velocity. This means that at some point the sky diver reaches a velocity at which the drag force equals the force of gravity, and the sky diver cannot accelerate any further. Hence, the terminal velocity represents the velocity at which the forces accelerating a body are equal to those decelerating it.

Beer bubbles rising to the surface

Turbulent wake behind a moving sphere. We will model the gas bubbles rising to the top of beer as a sphere moving through a liquid

The net accelerating force of a bubble of air/gas in a liquid is the buoyancy force, i.e. the difference in density between the liquid and the gas. This buoyancy force  F_B force is given by

 F_B = \frac{\pi}{6} d^3 \left( \rho_l-\rho_g \right)g

where  d is the diameter of the spherical gas bubble,  \rho_g is the density of the gas,  \rho_l is the density of the liquid and  g is the gravitational acceleration 9.81 m/s^2. The buoyancy force essentially expresses the force required to displace a sphere volume  \frac{\pi}{6} d^3 given a certain difference in density between the gas and liquid.

At terminal velocity the buoyancy force is balanced by the total drag acting on the gas bubble. Using the equation for the drag coefficient above we know that the total drag  D is

 D = 1/2 C_D \rho_l U_T^2 \left( \frac{\pi}{4} d^2\right)

where  U_T is the terminal velocity and we have replaced  A with the frontal area of the gas bubble  \frac{\pi}{4} d^2 , i.e. the area of a circle. Thus, equating  D and  F_B

 \frac{\pi}{6} d^3 \left( \rho_l-\rho_g \right)g = 1/2 C_D \rho_l U_T^2 \left( \frac{\pi}{4} d^2\right)

and re-arranging for terminal velocity gives us

 U_T^2 = \frac{4d\left(\rho_l-\rho_g\right)g}{3C_D\rho_l}

At this point we can calculate the terminal velocity of a spherical gas bubble driven by buoyancy forces for a certain drag coefficient. The problem now is that the drag coefficient of a sphere is not constant; it changes with the flow velocity. Fortunately, the drag coefficient of a sphere plateaus at around 0.5 for Reynolds numbers  10^3-10^5 (see digram below) and it is reasonable to assume that the flow considered here falls within this range. Some good old engineering judgement at work!

Drag coefficient of a sphere as a function of Reynolds number

Drag coefficient as a function of Reynolds number. The curve flattens out between 10^3 and 10^5.

Hence, for our simplified calculation we will assume a drag coefficient of 0.5, a gas bubble 3 mm in diameter, density of the gas equal to 1.2 kg/m^3 and density of the fluid equal to 989 kg/m^3 (5% volume beer).

Therefore, the terminal velocity of gas bubbles rising in a beer are somewhere in the range of

 U_T^2 = \frac{4 \times 0.003 \times \left(989-1.2\right) \times 9.81}{3 \times 0.5 \times 989} = 0.0790 \ m^s/s^2

and taking the square root

 U_T = 0.281 \ m/s = 28.1 \ cm/s \left( 11 \ inches/s \right)

Given that the viscosity of the fluid is around \mu = 0.001 Ns/m^2 we can now check that we are in the right Reynolds number range:

 R = \frac{\rho_l U_T d}{\mu} = \frac{989 \times 0.281 \times 0.003}{0.001} = 833

which is right at the bottom of R =  10^3-10^5 !

So there you have it: Beer bubbles rise at around a foot per second.

Perhaps the next time you gaze pensively into a glass of beer after a hard day’s work, this little fun-fact will give you something else to think (or smile) about.


This post is based on a fun little problem that Prof. Gary Lock set his undergraduate students at the University of Bath. Prof. Lock was probably the most entertaining and effective lecturer I had during my undergraduate studies and has influenced my own lecturing style. If I can only pass on a fraction of the passion for engineering and teaching that Prof. Lock instilled in me, I consider my job well done.

At the start of the 19th century, after studying the highly cambered thin wings of many different birds, Sir George Cayley designed and built the first modern aerofoil, later used on a hand-launched glider. This biomimetic, highly cambered and thin-walled design remained the predominant aerofoil shape for almost 100 years, mainly due to the fact that the actual mechanisms of lift and drag were not understood scientifically but were explored in an empirical fashion. One of the major problems with these early aerofoil designs was that they experienced a phenomenon now known as boundary layer separation at very low angles of attack. This significantly limited the amount of lift that could be created by the wings and meant that bigger and bigger wings were needed to allow for any progress in terms of aircraft size. Lacking the analytical tools to study this problem, aerodynamicists continued to advocate thin aerofoil sections, as there was plenty of evidence in nature to suggest their efficacy. The problem was considered to be more one of degree, i.e. incrementally iterating the aerofoil shapes found in nature, rather than of type, that is designing an entirely new shape of aerofoil in accord with fundamental physics.

During the pre-WWI era, the misguided notions of designers was compounded by the ever-increasing use of wind-tunnel tests. The wind tunnels used at the time were relatively small and ran at very low flow speeds. This meant that the performance of the aerofoils was being tested under the conditions of laminar flow (smooth flow in layers, no mixing perpendicular to flow direction) rather than the turbulent flow (mixing of flow via small vortices) present over the wing surfaces. Under laminar flow conditions, increasing the thickness of an aerofoil increases the amount of skin-friction drag (as shown in last month’s post), and hence thinner aerofoils were considered to be superior.

The modern plane – born in 1915

The situation in Germany changed dramatically during WWI. In 1915 Hugo Junkers pioneered the first practical all-metal aircraft with a cantilevered wing – essentially the same semi-monocoque wing box design used today. The most popular design up to then was the biplane configuration held together by wires and struts, which introduced considerable amounts of parasitic drag and thereby limited the maximum speed of aircraft. Eliminating these supporting struts and wires meant that the flight loads needed to be carried by other means. Junkers cantilevered a beam from either side of the fuselage, the main spar, at about 25% of the chord of the wing to resist the up and down bending loads produced by lift. Then he fitted a smaller second spar, known as the trailing edge spar, at 75% of the chord to assist the main spar in resisting fore and aft bending induced by the drag on the wing. The two spars were connected by the external wing skin to produce a closed box-section known as the wing box. Finally, a curved piece of metal was fitted to the front of the wing to form the “D”-shaped leading edge, and two pieces of metal were run out to form the trailing edge. This series of three closed sections provided the wing with sufficient torsional rigidity to sustain the twisting loads that arise because the centre of pressure (the point where the lift force can be considered to act) is offset from the shear centre (the point where a vertical load will only cause bending and no twisting). Junker’s ideas were all combined in the world’s first practical all-metal aircraft, the Junker J 1, which although much heavier than other aircraft at the time, developed into the predominant form of construction for the larger and faster aircraft of the coming generation.

Junkers J 1 at Döberitz in 1915

Structures + Aerodynamics = Superior Aircraft

Junkers construction naturally resulted in a much thicker wing due to the room required for internal bracing, and this design provided the impetus for novel aerodynamics research. Junker’s ideas were supported by Ludwig Prandtl who carried out his famous aerodynamics work at the University of Göttingen. As discussed in last month’s post, Prandtl had previously introduced the notion of the boundary layer; namely the existence of a U-shaped velocity profile with a no-flow condition at the surface and an increasing velocity field towards the main stream some distance away from the surface. Prandtl argued that the presence of a boundary layer supported the simplifying assumption that fluid flow can be split into two non-interacting portions; a thin layer close to the surface governed by viscosity (the stickiness of the fluid) and an inviscid mainstream. This allowed Prandtl and his colleagues to make much more accurate predictions of the lift and drag performance of specific wing-shapes and greatly helped in the design of German WWI aircraft. In 1917 Prandtl showed that Junker’s thick and less-cambered aerofoil section produced much more favourable lift characteristics than the classic thinner sections used by Germany’s enemies. Second, the thick aerofoil could be flown at a much higher angle of attack without stalling and hence improved the manoeuvrability of a plane during dog fighting.

Skin Friction versus Pressure Drag

The flow in a boundary layer can be either laminar or turbulent. Laminar flow is orderly and stratified without interchange of fluid particles between individual layers, whereas in turbulent flow there is significant exchange of fluid perpendicular to the flow direction. The type of flow greatly influences the physics of the boundary layer. For example, due to the greater extent of mass interchange, a turbulent boundary layer is thicker than a laminar one and also features a steeper velocity gradient close to the surface, i.e. the flow speed increases more quickly as we move away from the wall.

Velocity profile of laminar versus turbulent boundary layer

Velocity profile of laminar versus turbulent boundary layer. Note how the turbulent flow increases velocity more rapidly away from the wall.

Just like your hand experiences friction when sliding over a surface, so do layers of fluid in the boundary layer, i.e. the slower regions of the flow are holding back the faster regions. This means that the velocity gradient throughout the boundary layer gives rise to internal shear stresses that are akin to friction acting on a surface. This type of friction is aptly called skin-friction drag and is predominant in streamlined flows where the majority of the body’s surface is aligned with the flow. As the velocity gradient at the surface is greater for turbulent than laminar flow, a streamlined body experiences more drag when the boundary layer flow over its surfaces is turbulent. A typical example of a streamlined body is an aircraft wing at cruise, and hence it is no surprise that maintaining laminar flow over aircraft wings is an ongoing research topic.

Over flat surfaces we can suitably ignore any changes in pressure in the flow direction. Under these conditions, the boundary layer remains stable but grows in thickness in the flow direction. This is, of course, an idealised scenario and in real-world applications, such as curved wings, the flow is most likely experiencing an adverse pressure gradient, i.e. the pressure increases in the flow direction. Under these conditions the boundary layer can become unstable and separate from the surface. The boundary layer separation induces a second type of drag, known as pressure drag. This type of drag is predominant for non-streamlined bodies, e.g. a golfball flying through the air or an aircraft wing at a high angle of attack.

So why does the flow separate in the first place?

To answer this question consider fluid flow over a cylinder. Right at the front of the cylinder fluid particles must come to rest. This point is aptly called the stagnation point and is the point of maximum pressure (to conserve energy the pressure needs to fall as fluid velocity increases, and vice versa). Further downstream, the curvature of the cylinder causes the flow lines to curve, and in order to equilibrate the centripetal forces, the flow accelerates and the fluid pressure drops. Hence, an area of accelerating flow and falling pressure occurs between the stagnation point and the poles of the cylinder. Once the flow passes the poles, the curvature of the cylinder is less effective at directing the flow in curved streamlines due to all the open space downstream of the cylinder. Hence, the curvature in the flow reduces and the flow slows down, turning the previously favourable pressure gradient into an adverse pressure gradient of rising pressure.

Boundary layer separation over cylinder

Boundary layer separation over a cylinder (axis out out the page).

To understand boundary layer separation we need to understand how these favourable and adverse pressure gradients influence the shape of the boundary layer. From our discussion on boundary layers, we know that the fluid travels slower the closer we are to the surface due to the retarding action of the no-slip condition at the wall. In a favourable pressure gradient, the falling pressure along the streamlines helps to urge the fluid along, thereby overcoming some of the decelerating effects of the fluid’s viscosity. As a result, the fluid is not decelerated as much close to the wall leading to a fuller U-shaped velocity profile, and the boundary layer grows more slowly.

By analogy, the opposite occurs for an adverse pressure gradient, i.e. the mainstream pressure increases in the flow direction retarding the flow in the boundary layer. So in the case of an adverse pressure gradient the pressure forces reinforce the retarding viscous friction forces close to the surface. As a result, the difference between the flow velocity close to the wall and the mainstream is more pronounced and the boundary layer grows more quickly. If the adverse pressure gradient acts over a sufficiently extended distance, the deceleration in the flow will be sufficient to reverse the direction of flow in the boundary layer. Hence the boundary layer develops a point of inflection, known as the point of boundary layer separation, beyond which a circular flow pattern is established.

For aircraft wings, boundary layer separation can lead to very significant consequences ranging from an increase in pressure drag to a dramatic loss of lift, known as aerodynamic stall. The shape of an aircraft wing is essentially an elongated and perhaps asymmetric version of the cylinder shown above. Hence the airflow over the top convex surface of a wing follows the same basic principles outlined above:

  • There is a point of stagnation at the leading edge.
  • A region of accelerating mainstream flow (favourable pressure gradient) up to the point of maximum thickness.
  • A region of decelerating mainstream flow (adverse pressure gradient) beyond the point of maximum thickness.

These three points are summarised in the schematic diagram below.

Boundary layer separation over the top surface of a wing

Boundary layer separation over the top surface of a wing.

Boundary layer separation is an important issue for aircraft wings as it induces a large wake that completely changes the flow downstream of the point of separation. Skin-friction drag arises due to inherent viscosity of the fluid, i.e. the fluid sticks to the surface of the wing and the associated frictional shear stress exerts a drag force. When a boundary layer separates, a drag force is induced as a result of differences in pressure upstream and downstream of the wing. The overall dimensions of the wake, and therefore the magnitude of pressure drag, depends on the point of separation along the wing. The velocity profiles of turbulent and laminar boundary layers (see image above) show that the velocity of the fluid increases much slower away from the wall for a laminar boundary layer. As a result, the flow in a laminar boundary layer will reverse direction much earlier in the presence of an adverse pressure gradient than the flow in a turbulent boundary layer.

To summarise, we now know that the inherent viscosity of a fluid leads to the presence of a boundary layer that has two possible sources of drag. Skin-friction drag due to the frictional shear stress between the fluid and the surface, and pressure drag due to flow separation and the existence of a downstream wake. As the total drag is the sum of these two effects, the aerodynamicist is faced with a non-trivial compromise:

  •  skin-friction drag is reduced by laminar flow due to a lower shear stress at the wall, but this increases pressure drag when boundary layer separation occurs.
  • pressure drag is reduced by turbulent flow by delaying boundary layer separation, but this increases the skin-friction drag due to higher shear stresses at the wall.

As a result, neither laminar nor turbulent flow can be said to be preferable in general and judgement has to be made regarding the specific application. For a blunt body, such as a cylinder, pressure drag dominates and therefore a turbulent boundary layer is preferable. For more streamlined bodies, such as an aircraft wing at cruise, the overall drag is dominated by skin-friction drag and hence a laminar boundary layer is preferable. Dolphins, for example, have very streamlined bodies to maintain laminar flow. Early golfers, on the other hand, realised that worn rubber golf balls flew further than pristine ones, and this led to the innovation of dimples on golf balls. Fluid flow over golf balls is predominantly laminar due to the relatively low flight speeds. Dimples are therefore nothing more than small imperfections that transform the predominantly laminar flow into a turbulent one that delays the onset of boundary layer separation and therefore reduces pressure drag.

Aerodynamic Stall

The second, and more dramatic effect, of boundary layer separation in aircraft wings is aerodynamic stall. At relatively low angles of attack, for example during cruise, the adverse pressure gradient acting on the top surface of the wing is benign and the boundary layer remains attached over the entire surface. As the angle of attack is increased, however, so does the pressure gradient. At some point the boundary layer will start to separate near the trailing edge of the wing, and this separation point will move further upstream as the angle of attack is increased. If an aerofoil is positioned at a sufficiently large angle of attack, separation will occur very close to the point of maximum thickness of the aerofoil and a large wake will develop behind the point of separation. This wake redistributes the flow over the rest of the aerofoil and thereby significantly impairs the lift generated by the wing. As a result, the lift produced is seriously reduced in a condition known as aerodynamic stall. Due to the high pressure drag induced by the wake, the aircraft can further lose airspeed, pushing the separation point further upstream and creating a deleterious feedback loop where the aircraft literally starts to fall out of the sky in an uncontrolled spiral. To prevent total loss of control, the pilot needs to reattach the boundary as quickly as possible which is achieved by reducing the angle of attack and pointing the nose of the aircraft down to gain speed.

The lift produced by a wing is given by

L = \frac{1}{2}C_L \rho V^2 S

where \rho is the density of the surrounding air, V is the flight velocity, S is the wing area and C_L is the lift coefficient of the aerofoil shape. The lift coefficient of a specific aerofoil shape increases linearly with the angle of attack up to a maximum point C_{Lmax}. The maximum lift coefficient of a typical aerofoil is around 1.4 at an angle of attack of around 16^\circ, which is bounded by the critical angle of attack where the stall condition occurs.

During cruise the angle of attack is relatively small (\approx 2^\circ) as sufficient lift is guaranteed by the high flight velocity V. Furthermore, we actually want to maintain a small angle of attack as this minimises the pressure drag induced by boundary layer separation. At takeoff and landing, however, the flight velocity is much smaller which means that the lift coefficient has to be increased by setting the wings at a more aggressive angle of attack (\approx 15^\circ). The issue is that even with a near maximum lift coefficient of 1.4, large jumbo jets have a hard time achieving the necessary lift force at safe speeds for landing. While it would also be possible to increase the wing area, such a solution would have detrimental effect on the aircraft weight and therefore fuel efficiency.

High-lift Devices

A much more elegant solution are leading-edge slats and trailing-edge flaps. A slat is a thin, curved aerofoil that is fitted to the front of the wing and is intended to induce a secondary airflow through the gap between the slat and the leading edge. The air accelerates through this gap and thereby injects high momentum fluid into the boundary on the upper surface, delaying the onset of flow reversal in the boundary layer. Similarly, one or two curved aerofoils may be placed at the rear of wing in order to invigorate the flow near the trailing edge. In this case the high momentum fluid reinvigorates the flow which has been slowed down by the adverse pressure gradient. The maximum lift coefficient can typically be doubled by these devices and therefore allows big jumbo jets to land and takeoff at relatively low runway speeds.

Leading edge slats and trailing edge flaps on an aircraft wing

The next time you are sitting close to the wings observe how these devices are retracted after take-off and activated before landing. In fact, birds have a similar devices on their wings. The wings of bats are comprised of thin and flexible membranes reinforced by small bones which roughen the membrane surface and help to transition the flow from laminar to turbulent and prevent boundary layer separation. As is so often the case in engineering design, a lot of inspiration can be taken from nature!

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In the early 20th century, a group of German scientists led by Ludwig Prandtl at the University of Göttingen began studying the fundamental nature of fluid flow and subsequently laid the foundations for modern aerodynamics. In 1904, just a year after the first flight by the Wright brothers, Prandtl published the first paper on a new concept, now known as the boundary layer. In the following years, Prandtl worked on supersonic flow and spent most of his time developing the foundations for wing theory, ultimately leading to the famous red triplane flown by Baron von Richthofen, the Red Baron, during WWI.

Prandtl’s key insight in the development of the boundary layer was that as a first-order approximation it is valid to separate any flow over a surface into two regions: a thin boundary layer near the surface where the effects of viscosity cannot be ignored, and a region outside the boundary layer where viscosity is negligible. The nature of the boundary layer that forms close to the surface of a body significantly influences how the fluid and body interact. Hence, an understanding of boundary layers is essential in predicting how much drag an aircraft experiences, and is therefore a mandatory requirement in any first course on aerodynamics.

Boundary layers develop due to the inherent stickiness or viscosity of the fluid. As a fluid flows over a surface, the fluid sticks to the solid boundary which is the so-called “no-slip condition”. As sudden jumps in flow velocity are not possible for flow continuity requirements, there must exist a small region within the fluid, close to the body over which the fluid is flowing, where the flow velocity increases from zero to the mainstream velocity. This region is the so-called boundary layer.

The U-shaped profile of the boundary layer can be visualised by suspending a straight line of dye in water and allowing fluid flow to distort the line of dye (see below). The distance of a distorted dye particle to its original position is proportional to the flow velocity. The fluid is stationary at the wall, increases in velocity moving away from the wall, and then converges to the constant mainstream value u_0 at a distance \delta equal to the thickness of the boundary layer.

Laminar boundary layer scheme

To further investigate the nature of the flow within the boundary layer, let’s split the boundary layer into small regions parallel to the surface and assume a constant fluid velocity within each of these regions (essentially the arrows in the figure above). We have established that the boundary layer is driven by viscosity. Therefore, adjacent regions within the boundary layer that move at slightly different velocities must exert a frictional force on each other. This is analogous to you running your hand over a table-top surface and feeling a frictional force on the palm of your hand. The shear stresses \tau inside the fluid are a function of the viscosity or stickiness of the fluid \mu, and also the velocity gradient du/dy:

 \tau = \mu \frac{\mathrm{d}u}{\mathrm{d}y}

where y is the coordinate measuring the distance from the solid boundary, also called the “wall”.

Prandtl first noted that shearing forces are negligible in mainstream flow due to the low viscosity of most fluids and the near uniformity of flow velocities in the mainstream. In the boundary layer, however, appreciable shear stresses driven by steep velocity gradients will arise.

So the pertinent question is: Do these two regions influence each other or can they be analysed separately?

Prandtl argued that for flow around streamlined bodies, the thickness of the boundary layer is an order of magnitude smaller than the thickness of the mainstream, and therefore the pressure and velocity fields around a streamlined body may analysed disregarding the presence of the boundary layer.

Eliminating the effect of viscosity in the free flow is an enormously helpful simplification in analysing the flow. Prandtl’s assumption allows us to model the mainstream flow using Bernoulli’s equation or the equations of compressible flow that we have discussed before, and this was a major impetus in the rapid development of aerodynamics in the 20th century. Today, the engineer has a suite of advanced computational tools at hand to model the viscid nature of the entire flow. However, the idea of partitioning the flow into an inviscid mainstream and viscid boundary layer is still essential for fundamental insights into basic aerodynamics.

Laminar and turbulent boundary layers

One simple example that nicely demonstrates the physics of boundary layers is the problem of flow over a flat plate.

Development of boundary layer over a flat plate including the transition from a laminar to turbulent boundary layer.

The fluid is streaming in from the left with a free stream velocity U_0 and due to the no-slip condition slows down close to the surface of the plate. Hence, a boundary layer starts to form at the leading edge. As the fluid proceeds further downstream, large shearing stresses and velocity gradients develop within the boundary layer. Proceeding further downstream, more and more fluid is slowed down and therefore the thickness, \delta, of the boundary layer grows. As there is no sharp line splitting the boundary layer from the free-stream, the assumption is typically made that the boundary layer extends to the point where the fluid velocity reaches 99% of the free stream. At all times, and at at any distance x from the leading edge, the thickness of the boundary layer \delta is small compared to x.

Close to the leading edge the flow is entirely laminar, meaning the fluid can be imagined to travel in strata, or lamina, that do not mix. In essence, layers of fluid slide over each other without any interchange of fluid particles between adjacent layers. The flow speed within each imaginary lamina is constant and increases with the distance from the surface. The shear stress within the fluid is therefore entirely a function of the viscosity and the velocity gradients.

Further downstream, the laminar flow becomes unstable and fluid particles start to move perpendicular to the surface as well as parallel to it. Therefore, the previously stratified flow starts to mix up and fluid particles are exchanged between adjacent layers. Due to this seemingly random motion this type of flow is known as turbulent. In a turbulent boundary layer, the thickness \delta increases at a faster rate because of the greater extent of mixing within the main flow. The transverse mixing of the fluid and exchange of momentum between individual layers induces extra shearing forces known as the Reynolds stresses. However, the random irregularities and mixing in turbulent flow cannot occur in the close vicinity of the surface, and therefore a viscous sublayer forms beneath the turbulent boundary layer in which the flow is laminar.

An excellent example contrasting the differences in turbulent and laminar flow is the smoke rising from a cigarette.

Laminar and turbulent flow in smoke

Laminar and turbulent flow in smoke

As smoke rises it transforms from a region of smooth laminar flow to a region of unsteady turbulent flow. The nature of the flow, laminar or turbulent, is captured very efficiently in a single parameter known as the Reynolds number

Re = \frac{\rho U d}{\mu}

where \rho is the density of the fluid, U the local flow velocity, d a characteristic length describing the geometry, and \mu is the viscosity of the fluid.

There exists a critical Reynolds number in the region 2300-4000 for which the flow transitions from laminar to turbulent. For the plate example above, the characteristic length is the distance from the leading edge. Therefore d increases as we proceed downstream, increasing the Reynolds number until at some point the flow transitions from laminar to turbulent. The faster the free stream velocity U, the shorter the distance from the leading edge where this transition occurs.

Velocity profiles

Due to the different degrees of fluid mixing in laminar and turbulent flows, the shape of the two boundary layers is different. The increase in fluid velocity moving away from the surface (y-direction) must be continuous in order to guarantee a unique value of the velocity gradient du/dy. For a discontinuous change in velocity, the velocity gradient du/dy, and therefore the shearing forces  \tau = \mu \frac{\mathrm{d}u}{\mathrm{d}y} would be infinite, which is obviously not feasible in reality. Hence, the velocity increases smoothly from zero at the wall in some form of parabolic distribution. The further we move away from the wall, the smaller the velocity gradient and the retarding action of the shearing stresses decreases.

In the case of laminar flow, the shape of the boundary layer is indeed quite smooth and does not change much over time. For a turbulent boundary layer however, only the average shape of the boundary layer approximates the parabolic profile discussed above. The figure below compares a typical laminar layer with an averaged turbulent layer.

Velocity profile of laminar versus turbulent boundary layer

Velocity profile of laminar versus turbulent boundary layer

In the laminar layer, the kinetic energy of the free flowing fluid is transmitted to the slower moving fluid near the surface purely means by of viscosity, i.e. frictional shear stresses. Hence, an imaginary fluid layer close to the free stream pulls along an adjacent layer close to the wall, and so on. As a result, significant portions of fluid in the laminar boundary layer travel at a reduced velocity. In a turbulent boundary layer, the kinetic energy of the free stream is also transmitted via Reynolds stresses, i.e. momentum exchanges due to the intermingling of fluid particles. This leads to a more rapid rise of the velocity away from the wall and a more uniform fluid velocity throughout the entire boundary layer. Due to the presence of the viscous sublayer in the close vicinity of the wall, the wall shear stress in a turbulent boundary layer is governed by the usual equation  \tau = \mu \frac{\mathrm{d}u}{\mathrm{d}y}. This means that because of the greater velocity gradient at the wall the frictional shear stress in a turbulent boundary is greater than in a purely laminar boundary layer.

Skin Friction drag

Fluids can only exert two types of forces: normal forces due to pressure and tangential forces due to shear stress. Pressure drag is the phenomenon that occurs when a body is oriented perpendicular to the direction of fluid flow. Skin friction drag is the frictional shear force exerted on a body aligned parallel to the flow, and therefore a direct result of the viscous boundary layer.

Due to the greater shear stress at the wall, the skin friction drag is greater for turbulent boundary layers than for laminar ones. Skin friction drag is predominant in streamlined aerodynamic profiles, e.g. fish, airplane wings, or any other shape where most of the surface area is aligned with the flow direction. For these profiles, maintaining a laminar boundary layer is preferable. For example, the crescent lunar shaped tail of many sea mammals or fish has evolved to maintain a relatively constant laminar boundary layer when oscillating the tail from side to side.

One of Prandtl’s PhD students, Paul Blasius, developed an analytical expression for the shape of a laminar boundary layer over a flat plate without a pressure gradient. Blasius’ expression has been verified by experiments many times over and is considered a standard in fluid dynamics. The two important quantities that are of interest to the designer are the boundary layer thickness \delta and the shear stress at the wall \tau_w at a distance x from the leading edge. The boundary layer thickness is given by

 \delta=\frac{5.2 x}{\sqrt{Re_x}}

with Re_x the Reynolds number at a distance x from the leading edge. Due to the presence of x in the numerator and \sqrt{x} in the denominator, the boundary layer thickness scales proportional to x^{1/2}, and hence increases rapidly in the beginning before settling down.

Next, we can use a similar expression to determine the shear stress at the wall. To do this we first define another non dimensional number known as the drag coefficient

C_f=\frac{\tau_w}{1/2 \rho U_f^2}

which is the value of the shear stress at the wall normalised by the dynamic pressure of the free-flow. According to Blasius, the skin-friction drag coefficient is simply governed by the Reynolds number


This simple example reiterates the power of dimensionless numbers we mentioned before when discussing wind tunnel testing. Even though the shear stress at the wall is a dimensional quantity, we have been able to express it merely as a function of two non-dimensional quantities Re and C_f. By combining the two equations above, the shear stress can be written as

\tau_{w}=\frac{0.332 \rho u_f^2}{\sqrt{Re_x}}

and therefore scales proportional to x^{-1/2}, tending to zero as the distance from the leading edge increases. The value of \tau_w is the frictional shear stress at a specific point x from the leading edge. To find the total amount of drag D_{sf} exerted on the plate we need to sum up (integrate) all contributions of \tau_w over the length of the plate

D_{sf} = 0.332 \rho U_f^2 \int_0^L \frac{\mathrm{d}x}{\sqrt{Re_x}}=\frac{0.664 \rho U_f^2 L}{\sqrt{\rho U_f L / \mu}} = \frac{0.664 \rho U_f^2 L}{\sqrt{Re_L}}

where Re_L is now the Reynolds number of the free stream calculated using the total length of the plate L. Similar to the skin friction coefficient C_f we can define a total skin friction drag coefficient \eta_f

\eta_f = \frac{2D_{sf}}{\rho U_f^2 L} = \frac{1.328}{\sqrt{Re_L}}

Hence, C_f can be used to calculate the local amount of shear stress at a point x from the leading edge, whereas \eta_f is used to find the total amount of skin friction drag acting on the surface.

Unfortunately, do to the chaotic nature of turbulent flow, the boundary layer thickness and skin drag coefficient for a turbulent boundary layer cannot be determined as easily in a theoretical manner. Therefore we have to rely on experimental results to define empirical approximations of these quantities. The scientific consensus of the these relations are as follows:

\delta = \frac{0.37 x}{(Re_x)^{0.2}}
\eta_f = \frac{0.074}{(Re_L)^{0.3}}

Therefore the thickness of a turbulent boundary layer grows proportional to x^{4/5} (faster than the x^{1/2} relation for laminar flow) and the total skin friction drag coefficient varies as L^{-1/5} (also faster than the L^{-1/2} relation of laminar flow). Hence, the total skin drag coefficient confirms the qualitative observations we made before that the frictional shear stresses in a turbulent boundary layer are greater than those in a laminar one.

Skin friction drag and wing design

The unfortunate fact for aircraft designers is that turbulent flow is much more common in nature than laminar flow. The tendency for flow to be random rather than layered can be interpreted in a similar way to the second law of thermodynamics. The fact that entropy in a closed system only increases is to say that, if left to its own devices, the state in the system will tend from order to disorder. And so it is with fluid flow.

However, the shape of a wing can be designed in such a manner as to encourage the formation of laminar flow. The P-51 Mustang WWII fighter was the first production aircraft designed to operate with laminar flow over its wings. The problem back then, and to this day, is that laminar flow is incredibly unstable. Protruding rivet heads or splattered insects on the wing surface can easily “trip” a laminar boundary layer into turbulence, and preempt any clever design the engineer concocted. As a result, most of the laminar flow wings that have been designed based on idealised conditions and smooth wing surfaces in a wind tunnel have not led to the sweeping improvements originally imagined.


For many years NASA conducted a series of experiments to design a natural laminar flow (NLF) aircraft. Some of their research suggested the wrapping of a glove around the leading edge of a Boeing 757 just outboard of the engine. The modified shape of this wing promotes laminar flow at the high altitudes and almost sonic flight conditions of a typical jet airliner. To prevent the build up of insect splatter at take-off a sheath of paper was wrapped around the glove which was then torn away at altitude. Even though the range of such an aircraft could be increased by almost 15% this, rather elaborate scheme, never made it into production.

In the mid 1990s NASA fitted active test panels to the wings of two F-16’s in order to test the possibility of achieving laminar flow on swept delta-wings flying at supersonic speed; in NASA’s view a likely wing configuration for future supersonic cruise aircraft. The active test panels essentially consisted of titanium covers perforated with millions of microscopic holes, which were attached to the leading edge and the top surface of the wing. The role of these panels was to suck most of the boundary layer off the top surface through perforations using an internal pumping system. By removing air from the boundary layer its thickness decreased and thereby promoted the stability of the laminar boundary layer over the wing. This Supersonic Laminar Flow (SLFC) project successfully maintained laminar flow over a large portion of the wing during supersonic flight of up to Mach 1.6.

F-16 XL with suction panels to promote laminar flow

F-16 XL with suction panels to promote laminar flow

While these elaborate schemes have not quite found their way into mass production (probably due to their cost, maintenance problems and risk), laminar flow wings are a very viable future technology in terms of reducing greenhouse gases as stipulated by environmental legislation. An important driver in reducing greenhouse gases is maximising the lift-to-drag ratio of the wings, and therefore I would expect research to continue in this field for some time to come.

Despite the growing computer power and increasing sophistication of computational models, any design meant operate in the real world requires some form of experimental validation. The idealist modeller, me included, wants to believe that computer simulation will replace all forms of experimental testing and thereby allow for much faster design cycles. The issue with this is that random imperfections, and most importantly their concurrence, are very hard to account for robustly, especially when operating in nonlinear domains. As a result, the quantity and quality of both computational and experimental validation have increased in lockstep over the few last decades.

In “The Wind and Beyond”, the autobiography of Theodore von Kármán, one of the pre-eminent aerospace engineers and scientists of the 20th century, von Kármán recounts a telling episode regarding the role of wind tunnel testing in the development of the Douglas DC-3, the first American commercial jetliner. Early versions of the DC-3 faced a problem with aerodynamic instabilities that could throw the airplane out of control. A similar problem had been noticed earlier on the Northrop Alpha airplane, which, like the DC-3, featured a wing that was attached to the underside of the fuselage. When two of von Kármán’s assistants, Major Klein and Clark Millikan, subjected a model of the Alpha to high winds in a wind tunnel, the model aircraft started to sway and shake violently. In the following investigation, Klein and Millikan found that the sharp corner at the connection between the wing and fuselage decelerated the air as it flowed past, causing boundary layer separation and a wake of eddies. As these eddies broke away from the trailing edge of the wing, they adversely impacted the flow over the horizontal stabiliser and vertical tail fin at the rear of the aircraft and resulted in uncontrollable vibrations.

Northrop Alpha (Y1C-19)

The Northrop Alpha plane with the Kármán fillet at the wing-fuselage joint

Fortunately, Theodore von Kármán was world-renowned, among other things, for his work on eddies and especially the so-called von Kármán Vortex Street. Von Kármán therefore intuitively realised what had to be done to eliminate the creation of these eddies. Von Kármán and his colleagues fitted a small fairing, a filling if you like, to the connection between the wing and the fuselage to smooth out the eddies. This became one of the textbook examples of how wind tunnel findings could be applied in a practical way to iron out problems with an aircraft. When French engineers learned of the device from von Kármán at a conference a few years later, they were so enamoured that such a simple idea could solve such a big problem that they named the fillet a “Kármán”.

When testing the aerodynamics of aircraft, the wind tunnel is indispensable. The Wright brothers built their own wind tunnel to validate the research data on airfoils that had been recorded throughout the 19th century. One of the most important pieces of equipment in the early days of NACA (now NASA) was a variable-density wind tunnel, which by pressurising the air, allowed realistic operating conditions to be simulated on 1/20th geometrically-scaled models.

Variable Density Tunnel - GPN-2000-001311

NACA variable density wind tunnel

This brings us to an important point: How do you test the aerodynamics of an aircraft in a wind-tunnel?

Do you need to build individual wind-tunnels big enough to fit a particular aircraft? Or can you use a smaller multi-purpose wind tunnel to test small-scale models of the actual aircraft? If this is the case, how representative is the collected data of the actual flying aircraft?

Luckily we can make use of some clever mathematics, known as dimensional analysis, to make our life a little easier. The key idea behind dimensional analysis is to define a set of dimensionless parameters that govern the physical behaviour of the phenomenon being studied, purely by identifying the fundamental dimensions (time, length and mass in aerodynamics) that are at play. This is best illustrated by an example.

The United States developed the atomic bomb during WWII under the greatest security precautions. Even many years after the first test of 1945 in the desert of New Mexico, the total amount of energy released during the explosion remained unknown. The British scientist G.I. Taylor then famously estimated the total amount of energy released by the explosion simply by using available pictures showing the explosion plume at different time stamps after detonation.

Nuclear Explosion Time Frames

Nuclear explosion time frames

By assuming that the shock wave could be modelled as a perfect sphere, Taylor posited that the size of the plume, i.e. the radius R, should depend on the energy E of the explosion, the time t after detonation and the density \rho of the surrounding air.

In dimensional analysis we proceed to define the fundamental units or dimensions that quantify our variables. So in this case:

  • Radius is defined by a distance, and therefore the units are length, i.e. [R] = L.
  • The units of time are, you guessed it, time, i.e. [t] = T.
  • Energy is force times distance, where a force is mass times acceleration, and acceleration is distance divided by time squared i.e. [E] = \left(\frac{ML}{T^2}\right)L = \frac{M L^2}{T^2}.
  • Density is mass divided by volume, where volume is a distance cubed, i.e. [\rho] = \frac{M}{L^3}.

Having determined all our variables in the fundamental dimensions of distance, time and mass, we now attempt to relate the radius of the explosion to the energy, density and time. If we assume that the radius is proportional to these three variables, then dividing the radius by the product of the other three variables must result in a dimensionless number. Hence,

c = \frac{R}{E^x \rho^y t^z}

Or alternatively, all fundamental dimensions in the above fraction must cancel:

\frac{L}{\left(M L^2 / T^2\right)^x \left(M / L^3\right)^y T^z} = \frac{L}{M^{\left(x+y\right)} L^{\left(2x-3y\right)} T^{\left(-2x+z\right)}} = M^{\left(-x-y\right)} L^{\left(1-2x+3y\right)} T^{\left(2x-z\right)}

For all units to disappear we need:
-x-y = 0 \qquad 1-2x+3y=0 \qquad 2x - z =0

and solving this system gives:

x = 1/5 \qquad y = -1/5 \qquad z = 2/5 

Therefore the shock wave radius is given by

R = c E^{1/5} \rho^{-1/5} t^{2/5} 

and by re-arranging

E = k \frac{R^5 \rho}{t^2}

where k = \frac{1}{c^5}.

So, we have an expression that relates the energy of the explosion to the radius, the density of air and time after detonation, which were all available to Taylor from the individual time stamps (these provided a diameter estimate and the time after detonation. The density of the air was known).

In the example above, specific calculations of E also require an estimate of the constant k. In aerodynamics, we are typically interested in quantifying the constant itself using the variables at hand. Hence, by analogy with the above example, we would know the energy, the density, radius and time and then calculate a value for the constant under these conditions. As the constant is dimensionless, it allows us to make an unbiased judgement of the flow conditions for entirely different and unrelated problems.

The most famous dimensionless number in aerodynamics is probably the Reynolds number which quantifies the nature of the flow, i.e. is it laminar (nice and orderly in layers that do not mix), or is it turbulent, or somewhere in between?

In determining aerodynamic forces, two of the important variables we want to understand and quantify are the lift and drag. Particularly, we want to determine how the lift and drag vary with independent parameters such as the flight velocity, wing area and the properties of the surrounding area.

Using a similar method as above, it can be shown that the two primary dimensionless variables are the lift (C_L) and drag coefficients (C_D), which are defined in terms of lift (L), drag (D), flight velocity (U), static fluid density (\rho) and wing area (S).

Lift coefficient:

C_L = \frac{L}{1/2 \rho U^2 S}

Drag coefficient:

C_D = \frac{D}{1/2 \rho U^2 S}

where 1/2 \rho U^2 is known as the dynamic pressure of a fluid in motion. When the dynamic pressure is multiplied by the wing area, S, we are left with units of force which cancel the unit of lift (L) and drag (D), thus making C_L and C_D dimensionless.

As long as the geometry of our vehicle remains the same (scaling up and down at constant ratio of relative dimensions, e.g. length, width, height, wing span, chord etc.), these two parameters are only dependent on two other dimensionless variables: the Reynolds number

Re = \frac{\rho U c}{\mu}

where U and c are characteristic flow velocity and length (usually aerofoil chord or wingspan), and the the Mach Number

M = \frac{U}{U_{sound}} = \frac{U}{\sqrt{\gamma R T}}

which is the ratio of aircraft speed to the local speed of sound.

Let’s recap what we have developed until now. We have two dimensionless parameters, the lift and drag coefficients, which measure the amount of lift and drag an airfoil or flight vehicle creates normalised by the conditions of the surrounding fluid (1/2 \rho U^2) and the geometry of the lifting surface (S). Hence, these dimensionless parameters allow us to make a fair comparison of the performance of different airfoils regardless of their size. Comparing the C_L and C_D of two different airfoils requires that the operating conditions be comparable. They do not have to be exactly the same in terms of air speed, density and temperature but their dimensionless quantities, namely the Mach number and Reynolds number, need to be equal.

As an example consider a prototype aircraft flying at altitude and a scaled version of the same aircraft in a wind tunnel. The model and prototype aircraft have the same geometrical shape and only vary in terms of their absolute dimensions and the operating conditions. If the values of Reynolds number and Mach number of the flow are the same for both, then the flows are called dynamically similar, and as the geometry of the two aircraft are scaled version of each other, it follows that the lift and drag coefficients must be the same too. This concept of dynamic similarity is crucial for wind-tunnel experiments as it allows engineers to create small-scale models of full-sized aircraft and reliably predict their aerodynamic qualities in a wind tunnel.

This of course means that the wind tunnel needs to be operated at entirely different temperatures and pressures than the operating conditions at altitude. As long as the dimensions of the model remain in proportion upon scaling up or down, the model wing area scales with the square of the wing chord, i.e. S is proportional to c^2. We know from the explanation above that for a certain combination of Mach number and Reynolds number the lift and drag coefficients are fixed.

Using the definition of C_L and C_D the lift is given by

L = C_L * (1/2 \rho U^2 S)

and the drag by

D = C_D * (1/2 \rho U^2 S)

The lift and drag created by an aircraft or model under constant Mach number and Reynolds number scale with the wing area or the wing chord squared. Rearranging the equation for the Reynolds number, the wing chord can in fact be shown to be proportional to the operating temperature and pressure of the fluid flow. So by rearranging the Reynolds number equation:

Re = \frac{\rho U c}{\mu} \Rightarrow c = \frac{Re \mu}{\rho U}

and from the fundamental gas equation

\rho = \frac{P}{RT}

and the Mach Number we have

U = M \sqrt{\gamma RT}

such that we can reformulate the chord length as follows

c = \frac{Re \mu RT}{P M \sqrt{\gamma RT}} = \frac{Re \mu \sqrt{RT}}{P M \sqrt{\gamma}}

Hence, the chord of the model is inversely proportional to the fluid pressure and directly proportional to the square of the fluid temperature. Thus, maximising the pressure and reducing the temperature (maximum fluid density) reduces the required size of the model and the overall aerodynamic forces. The was the concept behind NACA’s early variable density tunnel and is still exploited in modern cryogenic wind tunnels.

(Caveat: There is a little bit more maths in this post than usual. I have tried to explain the equations as good as possible using diagrams. In any case, the real treat is at the end of the post where I go through the design of rocket nozzles. However, understanding this design methodology is naturally easier by first reading what comes before.)

One of the most basic equations in fluid dynamics is Bernoulli’s equation: the relationship between pressure and velocity in a moving fluid. It is so fundamental to aerodynamics that it is often cited (incorrectly!) when explaining how aircraft wings create lift. The fact is that Bernoulli’s equation is not a fundamental equation of aerodynamics at all, but a particular case of the conservation of energy applied to a fluid of constant density.

The underlying assumption of constant density is only valid for low-speed flows, but does not hold in the case of high-speed flows where the kinetic energy causes changes in the gas’ density. As the speed of a fluid approaches the speed of sound, the properties of the fluid undergo changes that cannot be modelled accurately using Bernoulli’s equation. This type of flow is known as compressible. As a rule of thumb, the demarcation line for compressibility is around 30% the speed of sound, or around 100 m/s for dry air close to Earth’s surface. This means that air flowing over a normal passenger car can be treated as incompressible, whereas the flow over a modern jumbo jet is not.

The fluid dynamics and thermodynamics of compressible flow are described by five fundamental equations, of which Bernoulli’s equation is a special case under the conditions of constant density. For example, let’s consider an arbitrary control volume of fluid and assume that any flow of this fluid is

  • adiabatic, meaning there is no heat transfer out of or into the control volume.
  • inviscid, meaning no friction is present.
  • at constant energy, meaning no external work (for example by a compressor) is done on the fluid.

This type of flow is known as isentropic (constant entropy), and includes fluid flow over aircraft wings, but not fluid flowing through rotating turbines.

At this point you might be wondering how we can possible increase the speed of a gas without passing it through some machine that adds energy to the flow?

The answer is the fundamental law of conservation of energy. The temperature, pressure and density of a fluid at rest are known as the stagnation temperature, stagnation pressure and stagnation density, respectively. These stagnation values are the highest values that the gas can possibly attain. As the flow velocity of a gas increases, the pressure, temperature and density must fall in order to conserve energy, i.e. some of the internal energy of the gas is converted into kinetic energy. Hence, expansion of a gas leads to an increase in its velocity.

The isentropic flow described above is governed by five fundamental conservation equations that are expressed in terms density (\rho), pressure (p), velocity (v), area (A), mass flow rate (\dot{m}), temperature (T) and entropy (s). This means that at two stations of the flow, 1 and 2, the following expressions must hold:
– Conservation of mass: \dot{m}_1 = \dot{m}_2 \Rightarrow \rho_1 v_1 A_1 = \rho_2 v_2 A_2
– Conservation of linear momentum: \mathrm{d}F = \mathrm{d}m a = \dot{m} \mathrm{d} v \Rightarrow p_1 A_1 - p_2 A_2 = \dot{m} \left( v_2 - v_1\right)
– Conservation of energy: T_1 + \frac{v_1^2}{2 c_p} = T_2 + \frac{v_2^2}{2 c_p} = constant
– Equation of state: p = \rho R T
– Conservation of entropy (in adiabatic and inviscid flow only): s_1 = s_2

where R is the specific universal gas constant (normalised by molar mass) and c_p is the specific heat at constant pressure.

The Speed of Sound

Fundamental to the analysis of supersonic flow is the concept of the speed of sound. Without knowledge of the local speed of sound we cannot gauge where we are on the compressibility spectrum.

As a simple mind experiment, consider the plunger in a plastic syringe. The speed of sound describes the speed at which a pressure wave is transmitted through the air chamber by a small movement of the piston. As a very weak wave is being transmitted, the assumptions made above regarding no heat transfer and inviscid flow are valid here, and any variations in the temperature and pressure are small. Under these conditions it can be shown from only the five conservation equations above that the local speed of sound within the fluid is given by:

a = \sqrt{\gamma R T}

The term \gamma is the heat capacity ratio, i.e. the ratio of the specific heat at constant pressure (c_p) and specific heat at constant volume (c_v), and is independent of temperature and pressure. The specific universal gas constant R, as the name suggests, is also a constant and is given by the difference of the specific heats, R = c_p - c_v. As the above equation shows, the speed of sound of a gas only depends on the temperature. The speed of sound in dry air (R = 287 J/(kg K), \gamma = 1.4) at the freezing point of 0° C (273 Kelvin) is 331 m/s.

Why is the speed of sound purely a function of temperature?

Well, the temperature of a gas is a measure of the gas’ kinetic energy, which essentially describes how much the individual gas molecules are jiggling about. As the air molecules are moving randomly with differing instantaneous speeds and energies at different points in time, the temperature describes the average kinetic energy of the collection of molecules over a period of time. The higher the temperature the more ferocious the molecules are jiggling about and the more often they bump into each other. A pressure wave momentarily disturbs some particles and this extra energy is transferred through the gas by the collisions of molecules with their neighbours. The higher the temperature, the quicker the pressure wave is propagated through the gas due to the higher rate of collisions.

This visualisation is also helpful in explaining why the speed of sound is a special property in fluid dynamics. One possible source of an externally induced pressure wave is the disturbance of an object moving through the fluid. As the object slices through the air it collides with stationary air particles upstream of the direction of motion. This collision induces a pressure wave which is transmitted via the molecular collisions described above. Now imagine what happens when the object is travelling faster than the speed of sound. This means the moving object is creating new disturbances upstream of its direction of motion at a faster rate than the air can propagate the pressure waves through the gas by means of molecular collisions. The rate of pressure wave creation is faster than the rate of pressure wave transmission. Or put more simply, information is created more quickly than it can be transmitted; we have run out of bandwidth. For this reason, the speed of sound marks an important demarcation line in fluid dynamics which, if exceeded, introduces a number of counter-intuitive effects.

Given the importance of the speed of sound, the relative speed of a body with respect to the local speed of sound is described by the Mach Number:

M = \frac{v}{a} = \frac{v}{\sqrt{\gamma R T}}

The Mach number is named after Ernst Mach who conducted many of the first experiments on supersonic flow and captured the first ever photograph of a shock wave (shown below).

Photography of bow shock waves around a brass bullet, 1888

As described previously, when an object moves through a gas, the molecules just ahead of the object are pushed out of the way, creating a pressure pulse that propagates in all directions (imagine a spherical pressure wave) at the speed of sound relative to the fluid. Now let’s imagine a loudspeaker emitting three sound pulses at equal intervals, t_1 = dt, t_2 = 2 dt, t_3 = 3 dt.

If the object is stationary, then the three sound pulses at times dt, 2 dt and 3 dt are concentric (see figure below).

However, if the object starts moving in one direction, the centre of the spheres shift to the side and the sound pulses bunch up in the direction of motion and spread out in the opposite direction. A bystander listening to the sound pulses upstream of the loudspeaker would therefore hear a higher pitched sound than a downstream bystander as the frequency the sound waves reaching him are higher. This is known as the Doppler effect.

If the object now accelerates to the local speed of sound, then the centres of the sound pulse spheres will be travelling just as fast as the sound waves themselves and the spherical waves all touch at one point. This means no sound can travel ahead of the loudspeaker and consequently an observer ahead of the loudspeaker will hear nothing.

Finally, if the loudspeaker travels at a uniform speed greater than the speed of sound, then the loudspeaker will in fact overtake the sound pulses it is creating. In this case, the loudspeaker and the leading edges of the sound waves form a locus known as the Mach cone. An observer standing outside this cone is in a zone of silence and is not aware of the sound waves created by the loudspeaker.

Speed of Sound

S is the starting point of the load speaker which then moves to the right of the screen emitting three sound pulses at times dt, 2dt and 3dt.

The half angle of this cone is known as the Mach angle and is equal to

\sin \mu = \frac{1}{M}

and therefore \mu = 90^\circ when the object is travelling at the speed of sound and \mu decreases with increasing velocity.

As mentioned previously, the temperature, pressure and density of the gas all fall as the flow speed of the gas increases. The relation between Mach number and temperature can be derived directly from the conservation of energy (stated above) and is given by:

\frac{T_t}{T} = 1 + \frac{\gamma-1}{2} M^2

where T_t is the maximum total temperature, also known as stagnation temperature, and T is called the static temperature of the gas moving at velocity M = v/a.

An intuitive way of explaining the relationship between temperature and flow speed is to return to the description of the vibrating gas molecules. Previously we established that the temperature of a gas is a measure of the kinetic energy of the vibrating molecules. Hence, the stagnation temperature is the kinetic energy of the random motion of the air molecules in a stationary gas. However, if the gas is moving in a certain direction at speed then there will be a real net movement of the air molecules. The molecules will still be vibrating about, but at a net movement in a specific direction. If the total energy of the gas is to remain constant (no external work), some of the kinetic energy of the random vibrations must be converted into kinetic energy of directed motion, and hence the energy associated with random vibration, i.e. the temperature, must fall. Therefore, the gas temperature falls as some of the thermal internal energy is converted into kinetic energy.

In a similar fashion, for flow at constant entropy, both the pressure and density of the fluid can be quantified by the Mach number.

\frac{p_t}{p} = \left( 1 + \frac{\gamma-1}{2} M^2\right)^{\frac{\gamma}{\gamma-1}}
\frac{\rho_t}{\rho} = \left( 1 + \frac{\gamma-1}{2} M^2\right)^{\frac{1}{\gamma-1}}

In this regard the Mach number can simply be interpreted as the degree of compressibility of a gas. For small Mach numbers (M< 0.3), the density changes by less than 5% and this is why the assumptions of constant density underlying Bernoulli’s equation are applicable.

An Application: Convergent-divergent Nozzles

In typical engineering applications, compressible flow typically occurs in ducts, e.g. engine intakes, or through the exhaust nozzles of afterburners and rockets. This latter type of flow typically features changes in area. If we consider a differential, i.e. infinitesimally small control volume, where the cross-sectional area changes by dA, then the velocity of the flow must also change by a small amount dv in order to conserve the mass flow rate. Under these conditions we can show that the change in velocity is related to the change in area by the following equation:

\left( M^2 - 1 \right) \frac{dv}{v} = \frac{dA}{A}

Without solving this equation for a specific problem we can reveal some interesting properties of compressible flow:

  • For M < 1, i.e. subsonic flow, -c \frac{dv}{v} = \frac{dA}{A} with c a positive constant. This means that increasing the flow velocity is only possible with a decrease in cross-sectional area and vice versa.
  • For M = 1, i.e. sonic flow 0 = \frac{dA}{A}. As A has to be finite this implies that dA = 0 and therefore the area must be a minimum for sonic flow.
  • For M > 1, i.e. supersonic flow + c \frac{dv}{v} = \frac{dA}{A}. This means that increasing the flow velocity is only possible with an increase in cross-sectional area and vice versa.
Subsonic and supersonic flow in nozzles

Subsonic and supersonic flow in nozzles

Hence, because of the term M^2 - 1, changes in subsonic and supersonic flows are of opposite sign. This means that if we want to expand a gas from subsonic to supersonic speeds, we must first pass the flow through a convergent nozzle to reach Mach 1, and then expand it in a divergent nozzle to reach supersonic speeds. Therefore, at the point of minimum area, known as the throat, the flow must be sonic and, as a result, rocket engines always have large bell-shaped nozzle in order to expand the exhaust gases into supersonic jets.

RS-68 rocket engine test

The flow through such a bell-shaped convergent-divergent nozzle is driven by the pressure difference between the combustion chamber and the nozzle outlet. In the combustion chamber the gas is basically at rest and therefore at stagnation pressure. As it exits the nozzle, the gas is typically moving and therefore at a lower pressure. In order to create supersonic flow, the first important condition is a high enough pressure ratio between the combustion chamber and the throat of the nozzle to guarantee that the flow is sonic at the throat. Without this critical condition at the throat, there can be no supersonic flow in the divergent section of the nozzle.

We can determine this exact pressure ratio for dry air (\gamma = 1.4) from the relationship between pressure and Mach number given above:

\frac{p_t}{p} = \left( 1 + \frac{\gamma-1}{2} 1^2\right)^{\frac{\gamma}{\gamma-1}} = \left(\frac{\gamma+1}{2} M^2\right)^{\frac{\gamma}{\gamma-1}} = 1.893

Therefore, a pressure ratio greater than or equal to 1.893 is required to guarantee sonic flow at the throat. The temperature at this condition would then be:

\frac{T_t}{T} = 1 + \frac{\gamma-1}{2} 1^2 = 1.2

or 1.2 times smaller than the temperature in the combustion chamber (as long as there is no heat loss or work done in the meantime, i.e. isentropic flow).

Shock Waves

The term “shock wave” implies a certain sense of drama; the state of shock after a traumatic event, the shock waves of a revolution, the shock waves of an earthquake, thunder, the cracking of a whip, and so on. In aerodynamics, a shock wave describes a thin front of energy, approximately 10^{-7} m in thickness (that’s 0.1 microns, or 0.0001 mm) across which the state of the gas changes abruptly. The gas density, temperature and pressure all significantly increase across the shock wave. A specific type of shock wave that lends itself nicely to straightforward analysis is called a normal shock wave, as it forms at right angles to the direction of motion. The conservation laws stated at the beginning of this post still hold and these can be used to prove a number of interesting relations that are known as the Prandtl relation and the Rankine equations.

The Prandtl relation provides a means of calculating the speed of the fluid flow after a normal shock, given the flow speed before the shock.

V_1 V_2 = \frac{2a_t^2}{\gamma+1}

where a_t = \sqrt{\gamma R T_t} is the speed of sound at the stagnation temperature of the flow. Because we are assuming no external work or heat transfer across the shock wave, the internal energy of the flow must be conserved across the shock, and therefore the stagnation temperature also does not change across the shock wave. This means that the speed of sound at the stagnation temperature a_t must also be conserved and therefore the Prandtl relation shows that the product of upstream and downstream velocities must always be a constant. Hence, they are inversely proportional.

We can further extend the Prandtl relation to express all flow properties (speed, temperature, pressure and density) in terms of the upstream Mach number M_1, and hence the degree of compressibility before the shock wave. In the Prandtl relation we replace the velocities with their Mach numbers and divide both sides of the equations by a_t^2

\frac{a_1 M_1}{a_t} \frac{a_2 M_2}{a_t} = \frac{2}{\gamma+1}

and because we know the relationship between temperature, stagnation temperature and Mach number from above:

\frac{a}{a_t} = \sqrt{\frac{T}{T_t}} = \left( 1 + \frac{\gamma-1}{2} M^2 \right)^{-1/2}

substitution for states 1 and 2 the Prandtl relation is transformed into:

M_2^2 = \frac{M_1^2 + \frac{2}{\gamma-1}}{\left(\frac{2 \gamma}{\gamma-1}\right) M_1^2 - 1}

This equation looks a bit clumsy but it is actually quite straightforward given that the terms involving \gamma are constants. For clarity a graphical representation of the the equation is shown below.

Change in Mach number across a shock wave

Change in Mach number across a shock wave

It is clear from the figure that for M_1 > 1 we necessarily have M_2 < 2. Therefore a shock wave automatically turns the flow from supersonic to subsonic. In the case of M_1 = 1 we have reached the limiting case of a sound wave for which there is no change in the gas properties. Similar expressions can also be derived for the pressure, temperature and density, which all increase across a shock wave, and these are known as the Rankine equations.

Both the temperature and pressure ratios increase with higher Mach number such that both p_2 and T_2 tend to infinity as M_1 tends to infinity. The density ratio however, does not tend to infinity but approaches an asymptotic value of 6 as M_1 increases. In isentropic flow, the relationship \frac{p_2}{p_1} = \left(\frac{\rho_2}{\rho_1}\right)^\gamma between the pressure ratio p_2 /p_1 and the density ratio \rho_2 / \rho_1 must hold. Given that p_2 tends to infinity with increasing M_1 but \rho_2 does not, this implies that the above relation between pressure ratio and density ratio must be broken with increasing M_1, i.e. the flow can no longer conserve entropy. In fact, in the limiting case of a sound wave, where M_1 = M_2 = 1, there is an infinitesimally weak shock wave and the flow is isentropic with no change in the gas properties. When a shock wave forms as a result of supersonic flow the entropy always increases across the shock.

Pressure and density ratios across a shock wave

Pressure and density ratios across a shock wave

Even though the Rankine equations are valid mathematically for subsonic flow, the predicted fluid properties lead to a decrease in entropy, which contradicts the Second Law of Thermodynamics. Hence, shock waves can only be created in supersonic flow and the pressure, temperature and density always increase across it.

Designing Convergent-divergent Nozzles

With our new-found knowledge on supersonic flow and nozzles we can now begin to intuitively design a convergent-divergent nozzle to be used on a rocket. Consider two reservoirs connected by a convergent-divergent nozzle (see figure below).

Convergent-divergent nozzle schematic and variations of pressure along the length of the nozzle

Convergent-divergent nozzle schematic and variations of pressure along the length of the nozzle

The gas within the upstream reservoir is stagnant at a specific stagnation temperature T_t and pressure P_t. The pressure in the downstream reservoir, called the back pressure P_b, can be regulated using a valve. The pressure at the exit plane of the divergent section of the nozzle is known as the exit pressure P_e, and the pressure at the point of minimum area within the nozzle is known as the throat pressure P_c. Changing the back pressure P_b influences the variation of the pressure throughout the nozzle as shown in the figure above. Depending on the back pressure, eight different conditions are possible at the exit plane.

  1. The no-flow condition: In this case the valve is closed and P_b = P_e = P_c = P_t. This is the trivial condition where nothing interesting happens. No flow, nothing, boring.
  2. Subsonic flow regime: The valve is opened slightly and the flow is entirely subsonic throughout the entire nozzle. The pressure decreases from the stagnant condition in the upstream reservoir to a minimum at the throat, but because the flow does not reach the critical pressure ratio P_t/P_c = 1.893, the flow does not reach Mach 1 at the throat. Hence, the flow cannot accelerate further in the divergent section and slows down again, thereby increasing the pressure. The exit pressure P_e is exactly equal to the back pressure.
  3. Choking condition: The back pressure has now reached a critical condition and is low enough for the flow to reach Mach 1 at the throat. Hence, P_t/P_c = 1.893. However, the exit flow pressure is still equal to the back pressure (P_e = P_b) and therefore the divergent section of the nozzle still acts as a diffuser; the flow does not go supersonic. However, as the flow can not go faster than Mach 1 at the throat, the maximum mass flow rate has been achieved and the nozzle is now choked.
  4. Non-isentropic flow regime: Lowering the back pressure further means that the flow now reaches Mach 1 at the throat and can then accelerate to supersonic speeds within the divergent portion of the nozzle. The flow in the convergent section of the nozzle remains the same as in condition 3) as the nozzle is choked. Due to the supersonic flow, a shock wave forms within the divergent section turning the flow from supersonic into subsonic. Downstream of the shock the divergent nozzle now diffuses the flow further to equalise the back pressure and exit pressure (P_e = P_b). The lower the back pressure is decreased, the further the shock wave travels downstream towards the exit plane, increasing the severity of the shock at the same time. The location of the shock wave within the divergent section will always be such as to equalise the exit and back pressures.
  5. Exit plane shock condition: This is the limiting condition where the shock wave in the divergent portion has moved exactly to the exit plane. At the exit of the nozzle there is an abrupt increase in pressure at the exit plane and therefore the exit plane pressure and back pressure are still the same (P_e = P_b).
  6. Overexpansion flow regime: The back pressure is now low enough that the flow is subsonic throughout the convergent portion of the nozzle, sonic at the throat and supersonic throughout the entire divergent portion. This means that the exit pressure is now lower than the gas pressure (the flow is overexpanded), causing it to suddenly contract once it exits the nozzle. These sudden compressions cause nonisentropic oblique pressure waves which cannot be modelled using the simple 1D flow assumptions we have made here.
  7. Nozzle design condition: At the nozzle design condition the back pressure is low enough to match the pressure of the supersonic flow at the exit plane. Hence, the flow is entirely isentropic within the nozzle and inside the downstream reservoir. As described in a previous post on rocketry, this is the ideal operating condition for a nozzle in terms of efficiency.
  8. Underexpansion flow regime: Contrary to the over expansion regime, the back pressure is now lower than the exit pressure of the supersonic flow, such that the exit flow must expand to equilibrate with the reservoir pressure. In this case, the flow is again governed by oblique pressure waves, which this time expand outward rather than contract inward.

Thus, as we have seen the flow inside and outside of the nozzle is driven by the back pressure and by the requirement of the exit pressure and back pressure to equilibrate once the flow exits the nozzle. In some cases this occurs as a result of shocks inside the nozzle and in others as a result of pressure waves outside. In terms of the structural mechanics of the nozzle, we obviously do not want shock to occur inside the nozzle in case this damages the structural integrity. Ideally, we would want to operate a rocket nozzle at the design condition, but as the atmospheric pressure changes throughout a flight into space, a rocket nozzle is typically overexpanded at take-off and underexpanded in space. To account for this, variable area nozzles and other clever ideas have been proposed to operate as close as possible to the design condition.

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This is the fourth and final part of a series of posts on rocket science. Part I covered the history of rocketry, Part II dealt with the operating principles of rockets and Part III looked at the components that go into the propulsive system.

One of the most important drivers in rocket design is the mass ratio, i.e. the ratio of fuel mass to dry mass of the rocket. The greater the mass ratio the greater the change in velocity (delta-v) the rocket can achieve. You can think of delta-v as the pseudo-currency of rocket science. Manoeuvres into orbit, to the moon or any other point in space are measured by their respective delta-v’s and this in turn defines the required mass ratio of the rocket.

For example, at an altitude of 200 km an object needs to travel at 7.8 km/s to inject into low earth orbit (LEO). Accounting for frictional losses and gravity, the actual requirement rocket scientists need to design for when starting from rest on a launch pad  is just shy of delta-v=10 km/s. Using Tsiolovsky’s rocket equation and assuming a representative average exhaust velocity of 3500 m/s, this translates into a mass ratio of 17.4:

\Delta v = \left|v_e\right| \ln \frac{M_0}{M_f} \Rightarrow \ln \frac{M_0}{M_f} = \frac{10000}{3500}=2.857
\therefore \frac{M_0}{M_f} = e^{2.86} = \underline{17.4}

A mass ratio of 17.4 means that the rocket needs to be 1-17.4^{-1} = 94.3% fuel!

This simple example explains why the mass ratio is a key indicator of a rocket’s structural efficiency. The higher the mass ratio the greater the ratio of delta-v producing propellant to non-delta-v producing structural mass. The simple example also explains why staging is such an effective strategy. Once, a certain amount of fuel within the tanks has been used up, it is beneficial to shed the unnecessary structural mass that was previously used to contain the fuel but is no longer contributing to delta-v.

At the same time we need to ask ourselves how to best minimise the mass of the rocket structure?

So in this post we will turn to my favourite topic of all: Structural design. Let’s dig in…

The role of the rocket structure is to provide some form of load-bearing frame while simultaneously serving as an aerodynamic profile and container for propellant and payload. In order to maximise the mass ratio, the rocket designer wants to minimise the structural mass that is required to safely contain the propellant. There are essentially two ways to achieve this:

  • Using lightweight materials.
  • And/or optimising the geometric design of the structure.

When referring to “lightweight materials” what we mean is that the material has high values of specific stiffness, specific strength and/or specific toughness. In this case “specific” means that the classical engineering properties of elastic modulus (stiffness), yield or ultimate strength, and fracture toughness are weighted by the density of the material. For example, if a design of given dimensions (fixed volume) requires a certain stiffness and strength, and we can achieve these specifications with a material of superior specific properties, then the mass of the structure will be reduced compared to some other material. In the rocket industry the typical materials are aerospace-grade titanium and aluminium alloys as their specific properties are much more favourable than those of other metal alloys such as steel.

However, over the last 30 years there has been a drive towards increasing the proportion of advanced fibre-reinforced plastics in rocket structures. One of the issues with composites is that the polymer matrices that bind the fibres together become rather brittle (think of shattering glass) under the cryogenic temperatures of outer space or when in contact with liquid propellants. The second issue with traditional composites is that they are more flammable; obviously not a good thing when sitting right next to liquid hydrogen and oxygen. Third, it is harder to seal composite rocket tanks and especially bolted joints are prone to leaking. Finally, the high-performance characteristics that are needed for space applications require the use of massive high-pressure, high-temperature ovens (autoclaves) and tight-tolerance moulds which significantly drive up manufacturing costs. For these reasons the use composites is mostly restricted to payload fairings. NASA is currently working hard on their out-of-autoclave technology and automated fibre placement technology, while RocketLabs have announced that they will be designing a carbon-composite rocket too, and I would expect this technology to mature over the next decade.

The load-bearing structure in a rocket is very similar to the fuselage of an airplane and is based on the same design philosophy: semi-monocoque construction. In contrast to early aircraft that used frames of discrete members braced by wires to sustain flight loads and flexible membranes as lift surfaces, the major advantage of semi-monocoque construction is that the functions of aerodynamic profile and load-carrying structure are combined. Hence, the visible cylindrical barrel of a rocket serves to contain the internal fuel as a pressure vessel, sustains the imposed flights loads and also defines the aerodynamic shape of the rocket. Because the external skin is a working part of the structure, this type of construction is known as stressed skin or monocoque. The even distribution of material in a monocoque means that the entire structure is at a more uniform and lower stress state with fewer local stress concentrations that can be hot spots for crack initiation.

Second, curved shell structures, as in a cylindrical rocket barrel, are one of the most efficient forms of construction found in nature, e.g. eggs, sea-shells, nut-shells etc.. In thin-walled curved structures the external loads are reacted internally by a combination of membrane stresses (uniform stretching or compression through the thickness) and bending stresses (linear variation of stresses through the thickness with tension on one side, compression on the other side, zero stress somewhere in the interior of the thickness known as the neutral axis). As a rule of thumb, membrane stresses are more efficient than bending stresses, as all of the material through the thickness is contributing to reacting the external load (no neutral axis) and the stress state is uniform (no stress concentrations).

In general, flat structures such as your typical credit card, will resist tensile and compressive external loads via uniform membrane stresses, and bending via linearly varying stresses through the thickness. The efficiency of curved shells stems from the fact that membrane stresses are induced to react both uniform stretching/compressive forces and bending moments. The presence of a membrane component reduces the peak stress that occurs through the thickness of the shell, and ultimately means that a thinner wall thickness and associated lower component mass will safely resist the externally applied loads. This is important as the bending stiffness of thin-walled structures is typically at least an order of magnitude smaller than the stretching/compressive stiffness (e.g. you can easily bend your credit card, but try stretching it).

Bending and membrane stress states

Alas, as so often in life, there is a compromise. Optimising a structure for one mode of deformation typically makes it more fragile in another. This means that if the structure fails in the deformation mode that it has been optimised for, the ensuing collapse is most-likely sudden and catastrophic.

As described above, reducing the wall-thickness in a monocoque construction greatly helps to reduce the mass of the structure. However, the bending stiffness scales with the cube of the thickness, whereas the membrane stiffness only scales linearly. Hence, in a thin-walled structure we ideally want all deformation to be in a membrane state (uniform squashing or stretching), and curved shell structures help to guarantee this. However, due to the large mismatch between membrane stiffness and bending stiffness in a thin-walled structure, the structure may at some point energetically prefer to bend and will transition to a bending state.

This phenomenon is known as buckling and is the bane of thin-walled construction.

One of the principles of physics is that the deformation of a structure is governed by the proclivity to minimise the strain energy. Hence, a structure can at some point bifurcate into a different deformation shape if this represents a lower energy state. As a little experiment, form a U-shape with your hand, thumb on one side and four fingers on the other. Hold a credit card between your thumb and the four fingers and start to compress it. Initially, the structure reacts this load by compressing internally (membrane deformation) in a flat state, but very soon the credit card will snap one way to form a U-shape (bending deformation).

The reason this works is because compressing the credit card reduces the distance between two edges held by the thumb and four fingers. The credit card can satisfy these new externally imposed constraints either by compressing uniformly, i.e. squashing up, or by maintaining its original length and bending into an arc. At some critical point of compression the bending state is energetically more favourable than the squashed state and the credit card bifurcates. Note that this explanation should also convince you that this form of behaviour is not possible under tension as the bifurcation to a bending state will not return the credit card to its original length.

The advantage of curved monocoques is that their buckling loads are much greater than those flat plates. For example, you can safely stand on a soda can even though it is made out of relatively cheap aluminium. However, once the soda can does buckle all hell breaks loose and the whole thing collapses in one big heap. What is more, curved structures are very susceptible to initial imperfections which drastically reduce the load at which buckling occurs. Flick the side of a soda can to initiate a little dent and stand back on the can to feel the difference.

Imperfection sensitivity of a cylinder. The plot shows the drastic reduction in load that the cylinder can sustain with increasing deformation once the buckling point has been passed.

Imperfection sensitivity of a cylinder. The plot shows the drastic reduction in load (vertical axis) that the perfect cylinder can sustain with increasing deformation (horizontal axis) once the buckling point has been passed. This means that an imperfect (real) shell will never reach the maximum load but diverge to the lower load level straight away.

This problem is exacerbated by the fact that the shape of the tiny initial imperfections, typically of the order of the thickness of the shell, can lead to vastly different failure modes. Thus, the behaviour of the shell is emergent of the initial conditions. In this domain of complexity it is very difficult to make precise repeatable predictions of how the structure will behave. For this reason, curved shells are often called the “prima-donna” of structures and we need to be very careful in how we go about designing them.

A rocket is naturally exposed to compressive forces as a result of gravity and inertia while accelerating. In order to increase the critical buckling loads of the cylindrical rocket shell, the skin is stiffened by internal stiffeners. This type of construction is known as semi-monocoque to describe the discrete discontinuities of the internal stiffeners. A rocket cylinder typically has internal stringers running top to bottom and internal hoops running around the circumference of the cylindrical skin.

Space Shuttle internal structure of propellant tank. Note the circumferential hoops and longitudinal stringers that help, among other things, to increase the buckling load (via Wikimedia Commons).

The purpose of these stringers and hoops is twofold:

  • First, they help to resist compressive loading and therefore remove some of the onus on the thin skin.
  • Second, they break the thin skin into smaller sections which are much harder to buckle. To convince yourself, find an old out-of-date credit card, cut it in half and repeat the previously described experiment.

The cylindrical rocket shell has a second advantage in that it acts as a pressure vessel to contain the pressurised propellants. The internal pressure of the propellants increases the circumference of the rocket shell, and like blowing up a balloon, imparts tensile stretching deformations into the skin which preempt the compressive gravitational and inertial loads. In fact, this pressure-stabilisation effect is so helpful that some old rockets that you see on display in museums, most notoriously the Atlas 2E rocket, need to be pressurised artificially by external air pumps at all times to prevent them from collapsing under their own weight. If you look at the diagram below you can see little diamond-shaped dimples spread all over the skin. These are buckling waveforms.

Atlas 2E Ballistic Missile (via Wikimedia Commons)

Atlas 2E Ballistic Missile with buckling “diamonds” along the entire length of the external rocket skin (via Wikimedia Commons)

NASA Langley Research Center has been, and continues to be, a leader in studying the complex failure behaviour of rocket shells. To find out more, check out the video by some of the researchers that I have worked with who are developing new methods of designing the next generation of composite rocket shells.

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This is the third in a series of posts on rocket science. Part I covered the history of rocketry and Part II dealt with the operating principles of rockets. If you have not checked out the latter post, I highly recommend you read this first before diving into what is to follow.

We have established that designing a powerful rocket means suspending a bunch of highly reactant chemicals above an ultralight means of combustion. In terms of metrics this means that a rocket scientist is looking to

  • Maximise the mass ratio to achieve the highest amounts of delta-v. This translates to carrying the maximum amount of fuel with minimum supporting structure to maximise the achievable change in velocity of the rocket.
  • Maximise the specific impulse of the propellant. The higher the specific impulse of the fuel the greater the exhaust velocity of the hot gases and consequently the greater the momentum thrust of the engine.
  • Optimise the shape of the exhaust nozzle to produce the highest amounts of pressure thrust.
  • Optimise the staging strategy to reach a compromise between the upside of staging in terms of shedding useless mass and the downside of extra technical complexity involved in joining multiple rocket engines (such complexity typically adds mass).
  •  Minimise the dry mass costs of the rocket either by manufacturing simple expendable rockets at scale or by building reusable rockets.

These operational principles set the landscape of what type of rocket we want to design. In designing chemical rockets some of the pertinent questions we need to answer are

  • What propellants to use for the most potent reaction?
  • How to expel and direct the exhaust gases most efficiently?
  • How to minimise the mass of the structure?

Here, we will turn to the propulsive side of things and answer the first of these two questions.


In a chemical rocket an exothermic reaction of typically two different chemicals is used to create high-pressure gases which are then directed through a nozzle and converted into a high-velocity directed jet.

From the Tsiolkovsky rocket equation we know that the momentum thrust depends on the mass flow rate of the propellants and the exhaust velocity,

F_t = \dot{m} v_{exit}

The most common types of propellant are:

  • Monopropellant: a single pressurised gas or liquid fuel that disassociates when a catalyst is introduced. Examples include hydrazine, nitrous oxide and hydrogen peroxide.
  • Hypergolic propellant: two liquids that spontaneously react when combined and release energy without requiring external ignition to start the reaction.
  • Fuel and oxidiser propellant: a combination of two liquids or two solids, a fuel and an oxidiser, that react when ignited. Combinations of solid fuel and liquid oxidiser are also possible as a hybrid propellant system. Typical fuels include liquid hydrogen and kerosene, while liquid oxygen and nitric acid are often used as oxidisers. In liquid propellant rockets the oxidiser and fuel are typically stored separately and mixed upon ignition in the combustion chamber, whereas solid propellant rockets are designed premixed.

Rockets can of course be powered by sources other than chemical reactions. Examples of this are smaller, low performance rockets such as attitude control thruster, that use escaping pressurised fluids to provide thrust. Similarly, a rocket may be powered by heating steam that then escapes through a propelling nozzle. However, the focus here is purely on chemical rockets.

Solid propellants

Solid propellants are made of a mixture of different chemicals that are blended into a liquid, poured into a cast and then cured into a solid. At its simplest, these chemical blends or “composites” are comprised of four different functional ingredients:

  • Solid oxidiser granules.
  • Flakes or powders of exothermic compounds.
  • Polymer binding agent.
  • Additives to stabilise or modify the burn rate.

Gunpowder is an example of a solid propellant that does not use a polymer binding agent to hold the propellant together. Rather the charcoal fuel and potassium nitrate oxidiser are compressed to hold their shape. A popular solid rocket fuel is ammonium perchlorate composite propellant (APCP) which uses a mixture of 70% granular ammonium perchlorate as an oxidiser, with 20% aluminium powder as a fuel, bound together using 10% polybutadiene acrylonitrile (PBAN).

Solid propellant rocket components (via Wikimedia Commons)

Solid propellant rocket components (via Wikimedia Commons URL)

Solid propellant rockets have been used much less frequently than liquid fuel rockets. However, there are some advantages, which can make solid propellants favourable to liquid propellants in some military applications (e.g. intercontinental ballistic missiles, ICBMs). Some of the advantages of solid propellants are that:

  • They are easier to store and handle.
  • They are simpler to operate with.
  • They have less components. There is no need for a separate combustion chamber and turbo pumps to pump the propellants into the combustion chamber. The solid propellant (also called “grain”) is ignited directly in the propellant storage casing.
  • They are much denser than liquid propellants and therefore reduce the fuel tank size (lower mass). Furthermore, solid propellants can be used as a load-bearing component, which further reduces the structural weight of the rocket. The cured solid propellant can readily be encased in a filament-wound composite rocket shell, which has more favourable strength-to-weight properties of the metallic rocket shells typically used for liquid rockets.

Apart from their use as ICBMs, solid rockets are known for their role as boosters. The simplicity and relatively low cost compared with liquid-fuel rockets means that solid rockets are a better choice when large amounts of cheap additional thrust is required. For example, the Space Shuttle used two solid rocket boosters to complement the onboard liquid propellant engines.

The disadvantage of solid propellants is that their specific impulse, and hence the amount of thrust produced per unit mass of fuel, is lower than for liquid propellants. The mass ratio of solid rockets can actually be greater than that of liquid rockets as a result of the more compact design and lower structural mass, but the exhaust velocities are much lower. The combustion process in solid rockets depends on the surface area of the fuel, and as such any air bubbles, cracks or voids in the solid propellant cast need to be prevented. Therefore, quite expensive quality assurance measures such as ultrasonic inspection or x-rays are required to assure the quality of the cast. The second problem with air bubbles in the cast is that the amount of oxidiser is increased (via the oxygen in the air) which results in local temperature hot spots and increased burn rate. Such local imbalances can spiral out of control to produce excessive temperatures and pressures, and ultimately lead to catastrophic failure. Another disadvantage of solid propellants are their binary operation mode. Once the chemical reaction has started and the engines have been ignited, it is very hard to throttle back or control the reaction. The propellant can be arranged in a manner to provide a predetermined thrust profile, but once this has started it is much hard to make adjustments on the fly. Liquid propellant rockets on the other hand use turbo pumps to throttle the propellant flow.

Liquid propellants

Liquid propellants have more favourable specific impulse measures than solid rockets. As such they are more efficient at propelling the rocket for a unit mass of reactant mass. This performance advantage is due to the superior oxidising capabilities of liquid oxidisers. For example, traditional liquid oxidisers such as liquid oxygen or hydrogen peroxide result in higher specific impulse measures than the ammonium perchlorate in solid rockets. Furthermore, as the liquid fuel and oxidiser are pumped into the combustion chamber, a liquid-fuelled rocket can be throttled, stopped and restarted much like a car or a jet engine. In liquid-fuelled rockets the combustion process is restricted to the combustion chamber, such that only this part of the rocket is exposed to the high pressure and temperature loads, whereas in solid-fuelled rockets the propellant tanks themselves are subjected to high pressures. Liquid propellants are also cheaper than solid propellants as they can be sourced from the upper atmosphere and require relatively little refinement compared to the composite manufacturing process of solid propellants. However, the cost of the propellant only accounts for around 10% of the total cost of the rocket and therefore these savings are typically negligible. Incidentally, the high proportion of costs associated with the structural mass of the rocket is why re-usability of rocket stages is such an important factor in reducing the cost of spaceflight.


Schematic of a liquid-fuelled rocket (via Wikimedia Commons)

The main drawback of liquid propellants is the difficulty of storage. Traditional liquid oxidisers are highly reactive and very toxic such that they need to be handled with care and properly insulated from other reactive materials. Second, the most common oxidiser, liquid oxygen, needs to be stored at very low cryogenic temperatures and this increases the complexity of the rocket design. What is more, additional components such as turbopumps and the associated valves and seals are needed that are entirely absent from solid-fuelled rockets.

Modern spaceflight is dominated by two liquid propellant mixtures:

  1. Liquid oxygen (LOX) and kerosene (RP-1): As discussed in the previous post this mix of oxidiser and fuel is predominantly used for lower stages (i.e. to get off the launch pad), due to the higher density of kerosene compared to liquid hydrogen. Kerosene, as a higher density fuel, allows for better ratios of propellant to tankage mass which is favourable for the mass ratio. Second, high density fuels work better in an atmospheric pressure environment. Historically, the Atlas V, Saturn V and Soyuz rockets have used LOX and RP-1 for the first stages and so does the SpaceX Falcon rocket today.
  2. Liquid oxygen and liquid hydrogen: This combination is mostly used for the upper stages that propel a vehicle into orbit. The lower density of the liquid hydrogen requires higher expansion ratios (gas pressure – atmospheric pressure) and therefore works more efficiently at higher altitudes. The Atlas V, Saturn V and modern Delta family or rockets all used this propellant mix for the upper rocket stages.

The choice of propellant mixture for different stages requires certain tradeoffs. Liquid hydrogen provides higher specific impulse than kerosene, but its density is around 7 times lower and therefore liquid hydrogen occupies much more space for the same mass of fuel. As a result, the required volume and associated mass of tankage, fuel pumps and pipes is much greater. Both the the specific impulse of the propellant and tankage mass influence the potential delta-v of the rocket, and hence liquid hydrogen, chemically the more efficient fuel, is not necessarily the best option for all rockets.

Although the exact choice of fuel is not straightforward I will propose two general rules of thumb that explain why kerosene is used for the early stages and liquid hydrogen for the upper stages:

  1. In general, the denser the fuel the heavier the rocket on the launch pad. This means that the rocket needs to provide more thrust to get off the ground and it carries this greater amount of thrust throughout the entire duration of the burn. As fuel is being depleted, the greater thrust of denser fuel rockets means that the rocket reaches orbit earlier and as a result minimises drag losses in the atmosphere.
  2. Liquid hydrogen fuelled rockets generally produce the lightest design and are therefore used on those parts of the spacecraft that actually need to be propelled into orbit or escape Earth’s gravity to venture into deep space.

Engine and Nozzle

In combustive rockets, the chemical reaction between the fuel and oxidiser creates a high temperature, high pressure gas inside the combustion chamber. If the combustion chamber were closed and symmetric, the internal pressure acting on the chamber walls would cause equal force in all directions and the rocket would remain stationary. For anything interesting to happen we must therefore open one end of the combustion chamber to allow the hot gases to escape. As a result of the hot gases pressing against the wall opposite to the opening, a net force in the direction of the closed end is induced.

Rocket thrust

Net thrust produced by rocket (via Wikimedia Commons)

Rocket pioneers, such as Goddard, realised early on that the shape of the nozzle is of crucial importance in creating maximum thrust.  A converging nozzle accelerates the escaping gases by means of the conservation of mass. However, converging nozzles are fundamentally limited to fluid flows of Mach 1, the speed of sound, and this is known as the choke condition. In this case, the nozzle provides relatively little thrust and the rocket is purely propelled by the net force acting on the close combustion chamber wall.

To further accelerate the flow, a divergent nozzle is required at the choke point. A convergent-divergent nozzle can therefore be used to create faster fluid flows. Crucially, the Tsiolkovsky rocket equation (conservation of momentum) indicates that the exit velocity of the hot gases is directly proportional to the amount of thrust produced. A second advantage is that the escaping gases also provide a force in the direction of flight by pushing on the divergent section of the nozzle.

Rocket nozzle expansion

Underexpanded, perfectly expanded, overexpanded and grossly overexpanded de Laval nozzles (via Wikimedia Commons).

The exit static pressure of the exhaust gases, i.e. the pressure of the gases if the exhaust jet was brought to rest, is a function of the pressure created inside the combustion chamber and the ratio of throat area to exit area of the nozzle. If the exit static pressure of the exhaust gases is greater than the surrounding ambient air pressure, the nozzle is known to be underexpanded. On the other hand, if the exit static pressure falls below the ambient pressure then the nozzle is known to be overexpanded. In this case two possible scenarios are possible. The supersonic air flow exiting the nozzle will induce a shock wave at some point along the flow. As the exhaust gas particles travel at speeds greater than the speed of sound, other gas particles upstream cannot “get out of the way” quickly enough before the rest of the flow arrives. Hence, the pressure progressively builds until at some point the properties of the fluid, density, pressure, temperature and velocity, change instantaneously. Thus, across the shock wave the gas pressure of an overexpanded nozzle will instantaneously shift from lower than ambient to exactly ambient pressure. If shock waves, visible by shock diamonds, form outside the nozzle, the nozzle is known as simply overexpanded. However, if the shock waves form inside the nozzle this is known as grossly overexpanded.

In an ideal world a rocket would continuously operate at peak efficiency, the condition where the nozzle is perfectly expanded throughout the entire flight. This can intuitively be explained using the rocket thrust equation introduced in the previous post:

f = \dot{m} v_{exit} + \left(p_{exit} - p_{ambient}\right) A_{exit} = \text{momentum thrust} + \text{pressure thrust}

Peak efficiency of the rocket engine occurs when p_{exit} = p_{ambient} such that the pressure thrust contribution is equal to zero. This is the condition of peak efficiency as the contribution of the momentum thrust is maximised while removing any penalties from over- or underexpanding the nozzle. An underexpanded nozzle means that p_{exit} > p_{ambient}, and while this condition provides extra pressure thrust, v_{exit} is lower and some of the energy that has gone into combusting the gases has not been converted into kinetic energy. In an overexpanded nozzle the pressure differential is negative, p_{exit} < p_{ambient}. In this case, v_{exit} is fully developed but the overexpansion induces a drag force on the rocket. If the nozzle is grossly overexpanded such that a shock wave occurs inside the nozzle, p_{exit} may still be greater than p_{ambient} but the supersonic jet separates from the divergent nozzle prematurely (see diagram below) such that A_{exit} decreases. In outer space p_{ambient} decreases and therefore the thrust created by the nozzle increases. However, A_{exit} is also decreasing as the flow separates earlier from the divergent nozzle. Thus, some of the increased efficiency of reduced ambient pressure is negated.

A perfectly expanded nozzle is only possible using a variable throat area or variable exit area nozzle to counteract the ambient pressure decrease with gaining altitude. As a result, fixed area nozzles become progressively underexpanded as the ambient pressure decreases during flight, and this means most nozzles are grossly overexpanded at takeoff. Some various exotic nozzles such as plug nozzles, stepped nozzles and aerospikes have been proposed to adapt to changes in ambient pressure and increasing thrust at higher altitudes. The extreme scenario obviously occurs once the rocket has left the Earth’s atmosphere. The nozzle is now so grossly overexpanded that the extra weight of the nozzle structure outweighs any performance gained from the divergent section.

Thus we can see that just as in the case of the propellants the design of individual components is not a straightforward matter and requires detailed tradeoffs between different configurations. This is what makes rocket science such a difficult endeavour.

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