Bio-mimetic Drag Reduction – Part 2: Aero- and Hydrodynamics

Part 1 of this blog series outlined the different sensing mechanisms that aquatic animals possess to create spatial images of the flow fields around them. In summary fish were found to possess a network of mechanosensors distributed over their bodies called the lateral line. The lateral line consists of two separate sensory subsystems:

• a system of velocity-sensitive superficial neuromasts that responds to slow, uniform motions and that integrates large scale stimuli at the periphery such as constant currents

and

• a system of acceleration- or pressure-gradient-sensitive canal neuromasts that responds to rapidly changing motions and gives the fish the opportunity to orient towards sources such as prey or optimize swimming speed or tail-flapping frequency.

In this post I will give a brief overview of general hydrodynamic theory and specifically the flow patterns that swimming fish are expected to sense with their network of neuromasts.

When a body moves relative to a fluid, a boundary layer exists close to the wall because of the “no slip” condition, which arises from the inherent stickiness or so-called viscosity of the fluid. Therefore, fluid in direct contact with the wall adheres to the surface while fluid further away is slowed due to the frictional forces arising from viscosity. This results in a thin layer of fluid where the velocity increases in a U-profile from zero at the wall to the free stream velocity some distance d from the surface; defined as the boundary layer thickness (Figure 1).

Fig. 1. Boundary layer close to a surface (1).

Generally speaking fluid flow can be classified as either laminar or turbulent. In laminar flow (derived from “lamina” meaning finite layers) the fluid moves in lamina or layers of finite speed and with no mixing of the fluid perpendicular to the wall i.e. across layers. As the name suggest in turbulent flow everything is a bit more chaotic with active mixing of the fluid and momentum transfer throughout the boundary layer (Figure 2).

Fig. 2. Laminar and Turbulent Boundary Layer (2).

The type of flow depends on the shape of the body, upstream history of the flow, surface roughness and most importantly the Reynolds Number. The Reynold’s number Re is a non-dimensional ration of the inertial forces to the viscous forces arising in the fluid defined by,

where p is the density of the fluid, v the velocity, u the viscosity and D a characteristic dimensions that describes the body under investigation. At certain critical Reynold’s number there is a natural transition from laminar to turbulent flow. For example if we consider the plate in Figure 3 we can observe that a boundary layer forms close to the surface once the flow encounters the leading edge of the plate. Initially the boundary layer thickness is very small but as we proceed along the length of the plate the boundary layer becomes thicker as increasingly more fluid is slowed down by the frictional effects of viscosity. The characteristic dimension for Re in this case is the distance l from the leading edge. This means that close to the leading edge where l is small the flow will be laminar while at a certain distance l_critical the critical value of Re is reached an the flow naturally transitions to turbulent flow.

Fig. 3. Laminar to turbulent transition over flat plate (2).

Now there are two major types of drag: skin friction drag, which is similar to the friction force you feel when you rub your hand over a table-top, and pressure drag, which results from a difference in fluid pressure between the front and rear of the body. As intuitively expected skin friction drag depends on the viscosity (stickiness) of the fluid but also the relative difference in velocity between different layers of fluid. Figure 3 shows that in a turbulent boundary layer the flow velocity increases more rapidly as we move away from the wall compared to a laminar boundary layer. The steeper velocity gradient close to the wall therefore means that skin friction drag is higher for a turbulent boundary layer (Figure 6).

Fig. 4. Boundary layer separation (3).

On the other hand pressure drag is greatly exacerbated by a phenomenon called boundary layer separation. When flow encounters an adverse pressure gradient (i.e. the fluid pressure increases in the flow direction as found after the point of maximum thickness in aerofoils e.g. Figure 5) the flow has to work against the increase in pressure leading to momentum losses and decelerations in flow. As the flow speed in the boundary layer continues to decreases in the direction of the adverse pressure gradient, at some point the slowest moving fluid close to the wall will actually change direction (Figure 4). This is called boundary layer separation and leads to a larger wake of vortices forming behind the body. The fluid pressure in the vortex wake is much lower than in regions of attached flow close to the leading edge and this pressure difference will therefore push the body backwards. As described earlier the flow velocity in a turbulent boundary layer close to the wall is higher than in a turbulent boundary layer.  This initially higher fluid momentum means that flow separation occurs further downstream than for laminar flow, resulting in a narrower wake and thus less pressure drag.

Fig. 5. Boundary layer separation over aerofoil (4).

Fig. 6. Effect of flow type on drag (5)

Therefore, we have two conflicting criteria to minimise drag as depicted in Figure 6:

• Skin friction drag is minimised by laminar flow and greatly worsened by turbulent flow

While

• Pressure drag is minimised by turbulent flow and greatly worsened by laminar flow

However, it is also clear that overall minimum drag is encountered for purely frictional drag with a laminar boundary layer. Now it is often very difficult to maintain a laminar boundary layer due to chaotic flow conditions that occur further upstream or just due to the inherent surface roughness that can “trip” the boundary layer to go turbulent. In actual fact this “tripping” of the boundary layer is utilised in a controlled fashion in a golf ball. The dimples or indentations on a golf ball serve to trip the naturally low Reynold’s number and therefore laminar flow around a golf ball to go turbulent. The delayed boundary layer separation results in a narrower wake, less pressure drag and thus more distance on Tiger’s drive (Figure 7).

Fig. 7. Delay of flow separation by dimples on a golf ball (6).

If we look at a cross-section of a dolphin (Figure 8) we observe that the general shape is very much the same as that of the aerofoil wing-shape in Figure 5. In fact early wing designs were based on anatomical studies on dolphins, trout and tuna by the “father of aerodynamics” Sir Lord Cayley during the late 18^th century. In dolphins the point of maximum thickness occurs at around 45% of its length in order to push the point of flow separation backwards and minimise pressure drag. This design has since inspired the shape of modern boat hulls and submarines such as the USS Albacore launched in 1953 (Figure 9).

Fig. 8. Streamlined “teardrop” shape of dolphin (7).

Fig. 9. USS Albacore based on biomimetic dolphin design (8).

Similar to the plate example of Figure 3 for gliding fish the boundary layer is laminar close to the head and then transitions to turbulent flow further downstream. However, for actively swimming fish the boundary layer is generally highly turbulent due to the unsteadiness created by the undulation motion of the body. Based on the fact that it is very difficult to maintain laminar flow around their bodies, the third and final post of this series will investigate how fish attempt to reduce the naturally higher skin friction drag associated with turbulent flow.

References