fluid_mechanics

490 Chapter 9 ■ Flow over Immersed Bodies
The pressure gradient
in the external
flow is imposed
throughout the
boundary layer
fluid.
Separation
location
V9.6 Snow drifts
object in an inviscid fluid is zero, but the drag on an object in a fluid with vanishingly small 1but
nonzero2 viscosity is not zero.
The reason for the above paradox can be described in terms of the effect of the pressure
gradient on boundary layer flow. Consider large Reynolds number flow of a real 1viscous2 fluid
past a circular cylinder. As was discussed in Section 9.1.2, we expect the viscous effects to be
confined to thin boundary layers near the surface. This allows the fluid to stick 1V 02 to the
surface—a necessary condition for any fluid, provided m 0. The basic idea of boundary layer
theory is that the boundary layer is thin enough so that it does not greatly disturb the flow outside
the boundary layer. Based on this reasoning, for large Reynolds numbers the flow throughout
most of the flow field would be expected to be as is indicated in Fig. 9.16a, the inviscid
flow field.
The pressure distribution indicated in Fig. 9.16b is imposed on the boundary layer flow along
the surface of the cylinder. In fact, there is negligible pressure variation across the thin boundary
layer so that the pressure within the boundary layer is that given by the inviscid flow field. This
pressure distribution along the cylinder is such that the stationary fluid at the nose of the cylinder
1U fs 0 at u 02 is accelerated to its maximum velocity 1U fs 2U at u 90°2 and then is decelerated
back to zero velocity at the rear of the cylinder 1U fs 0 at u 180°2. This is accomplished
by a balance between pressure and inertia effects; viscous effects are absent for the inviscid
flow outside the boundary layer.
Physically, in the absence of viscous effects, a fluid particle traveling from the front to
the back of the cylinder coasts down the “pressure hill” from u 0 to u 90° 1from point A
to C in Fig. 9.16b2 and then back up the hill to u 180° 1from point C to F2 without any loss
of energy. There is an exchange between kinetic and pressure energy, but there are no energy
losses. The same pressure distribution is imposed on the viscous fluid within the boundary layer.
The decrease in pressure in the direction of flow along the front half of the cylinder is termed
a favorable pressure gradient. The increase in pressure in the direction of flow along the rear
half of the cylinder is termed an adverse pressure gradient.
Consider a fluid particle within the boundary layer indicated in Fig. 9.17a. In its attempt
to flow from A to F it experiences the same pressure distribution as the particles in the free
stream immediately outside the boundary layer—the inviscid flow field pressure. However,
because of the viscous effects involved, the particle in the boundary layer experiences a loss
of energy as it flows along. This loss means that the particle does not have enough energy to
coast all of the way up the pressure hill 1from C to F2 and to reach point F at the rear of the
cylinder. This kinetic energy deficit is seen in the velocity profile detail at point C, shown in
Fig. 9.17a. Because of friction, the boundary layer fluid cannot travel from the front to the
rear of the cylinder. 1This conclusion can also be obtained from the concept that due to viscous
effects the particle at C does not have enough momentum to allow it to coast up the pressure
hill to F.2
The situation is similar to a bicyclist coasting down a hill and up the other side of the valley.
If there were no friction, the rider starting with zero speed could reach the same height from
which he or she started. Clearly friction 1rolling resistance, aerodynamic drag, etc.2 causes a loss
of energy 1and momentum2, making it impossible for the rider to reach the height from which he
or she started without supplying additional energy 1i.e., pedaling2. The fluid within the boundary
layer does not have such an energy supply. Thus, the fluid flows against the increasing pressure as
far as it can, at which point the boundary layer separates from 1lifts off2 the surface. This boundary
layer separation is indicated in Fig. 9.17a as well as the figures in the margin. (See the photograph
at the beginning of Chapters 7, 9, and 11.) Typical velocity profiles at representative locations
along the surface are shown in Fig. 9.17b. At the separation location 1profile D2, the velocity
gradient at the wall and the wall shear stress are zero. Beyond that location 1from D to E2 there is
reverse flow in the boundary layer.
As is indicated in Fig. 9.17c, because of the boundary layer separation, the average pressure
on the rear half of the cylinder is considerably less than that on the front half. Thus, a large pressure
drag is developed, even though 1because of small viscosity2 the viscous shear drag may be
quite small. D’Alembert’s paradox is explained. No matter how small the viscosity, provided it is
not zero, there will be a boundary layer that separates from the surface, giving a drag that is, for
the most part, independent of the value of m.