Next: References Up: Viscosity effects at high Previous: Skin friction

## Boundary layer separation and pressure drag

In most situations it is inevitable that the boundary layer becomes detached from a solid body. This boundary layer separation results in a large increase in the drag on the body. We can understand this by returning to the flow of a nonviscous fluid around a cylinder. The pressure distribution is the same on the downstream side of the cylinder as on the upstream side; thus, there were no unbalanced forces on the cylinder and therefore no drag (d'Alembert's paradox again). If the flow of a viscous fluid about a body is such that the boundary layer remains attached, then we have almost the same result--we'll just have a small drag due to the skin friction. However, if the boundary layer separates from the cylinder, then the pressure on the downstream side of the cylinder is essentially constant, and equal to the low pressure on the top and bottom points of the cylinder. This pressure is much lower than the large pressure which occurs at the stagnation point on the upstream side of the cylinder, leading to a pressure imbalance and a large pressure drag  on the cylinder. For instance, for a cylinder in a flow with a Reynolds number in the range , the boundary layer separates and the coefficient of drag is , much larger that the coefficient of drag due to skin friction, which we would estimate to be about .

A Reynolds number-independent drag coefficient leads to a drag force . More importantly, the power P required to maintain a constant speed in the presence of this drag is , so that it increases with the cube of the speed. To see the importance of this dependence, suppose that you ride a bicycle at 15 mph, which is a respectable speed. Most of the resistance at this speed is due to aerodynamic drag (there are other sources, such as mechanical friction, rolling friction, and so on, but I don't think they dominate at this speed). Now suppose you want to get to class in half the time in the morning, so you decide to ride at 30 mph. This requires 8 times as much power!

Boundary layers tend to separate from a solid body when there is an increasing fluid pressure in the direction of the flow--this is known as an adverse pressure gradient  in the jargon of fluid mechanics. Increasing the fluid pressure is akin to increasing the potential energy of the fluid, leading to a reduced kinetic energy and a deceleration of the fluid. When this happens the boundary layer thickens (recall that ), leading to a reduced gradient of the velocity profile ( decreases), with a concomitant decrease in the wall shear stress . For a large enough pressure gradient the shear stress can be reduced to zero, and separation often occurs. The fluid is no longer ``pulling'' on the wall, and opposing flow can develop which effectively pushes the boundary layer off of the wall. Separation is bound to occur in a sufficiently large adverse pressure gradient. On the other hand, boundary layers like decreasing pressure gradients, which accelerate the fluid and cause the boundary layer to thin.

Given these considerations, we see that minimizing the pressure drag amounts to preventing or delaying boundary layer separation. Since adverse pressure gradients are the cause of separation, we want to avoid these or at least make the gradients small. Trailing stagnation points are bound to cause problems, so separation can often be delayed by placing the trailing stagnation point at a cusp, so that the fluid leaves the body smoothly. This is known as streamlining , and is the preferred shape for airfoils, cars, and fish! Another way of delaying separation is by forcing the boundary layer to become turbulent. The more efficient mixing which occurs in a turbulent boundary layer reduces the boundary layer thickness and increases the wall shear stress, often preventing the separation which would occur for a laminar boundary layer under the same conditions. You can see that there is a trade-off here--the turbulent boundary layer produces a greater drag due to skin friction, but can often reduce the pressure drag by preventing, or reducing, boundary layer separation. Since the latter is usually dominant at high Reynolds numbers, various schemes have been invented for producing turbulent boundary layers. The dimples on a golf ball, the fuzz on a tennis ball, or the seams on a baseball are good examples of this. Apparently a dimpled golf ball has one-fifth the drag of a smooth golf ball of the same size! Also, airplane wings are often engineered with vortex generators on the upper surface to produce a turbulent boundary layer.

Next: References Up: Viscosity effects at high Previous: Skin friction

Vittorio Celli
Sun Sep 28 22:13:11 EDT 1997