The drag force required to move a body at a steady speed V through a viscous fluid is a function of the body shape, size L and speed V and the fluid properties and . The usual dimensional analysis leads to a relationship between the drag coefficient CD and the Reynolds number :
where L is the body dimension in the direction of flow and where A is a suitable body area, either the surface area of the body or the area of the body when projected onto a plane normal to the flow (in the case of blunt bodies). An immense variety of bodies has been tested in wind and water tunnels to determine their drag forces and other interesting attributes of the flow surrounding them. For each shape, the result of such tests may be portrayed by a plot of the drag coefficient as a function of the Reynolds number.
To illustrate the relationship between the drag coefficient and the Reynolds number, we will consider three simple body forms: ( i) a sphere of diameter D, ( ii) a circular cylinder of diameter D and length , whose axis is normal to the flow direction, and ( iii) a flat plate aligned with the flow of length D in the flow direction and length in the normal direction. The drag coefficients for these bodies are shown in figure 10.6 as a function of the Reynolds number . (For the sphere, A while for the cylinder A=DL, where L equals the cylinder length. For the flat plate, A=2DL is the surface area of the plate.)
Figure 10.6: The drag coefficients of a sphere, circular cylinder and a flat plate of dimension D plotted as a function of the Reynolds number . (Reprinted, by permission of Macmillan Publishing Company, from Rolf H. Sabersky, Allan J. Acosta and Edward G. Hauptmann, Fluid Flow. A First Course in Fluid Mechanics, 3 ed. Copyright 1989 by Macmillan Publishing Company)
For creeping or Stokes flow, when , we have analytical solutions for the viscous flow. In the case of a sphere, the drag force is given by equation 6.62 , and the drag coefficient is:
while for a cylinder,
These values agree quite closely with the measurements when . In this regime, the flow field is nearly symmetric in the flow direction and approximately equal contributions to the drag are due to the pressure and viscous stress on the body surface.
At higher Reynolds number, , a low speed flow, called the wake, develops immediately behind the body, within which mostly the same fluid circulates in a symmetric pattern. Toward the higher end of this range, the wake becomes unsteady with a well-defined frequency of motion in which portions of the wake fluid detach themselves from the wake in a regular pattern called a vortex street.
For a cylinder, the Strouhal number formed from this frequency is found empirically to depend upon the Reynolds number:
that reaches a steady value of at high Reynolds numbers.
Similar, but less regular, wakes develop behind a sphere. At the upper end of this range of Reynolds number, the component of the drag due to the pressure force on the body surface, called the form drag, accounts for almost all the drag because the other component, due to the viscous shear stress on the surface, becomes relatively smaller with increasing Reynolds number, as it does on a flat plate (see equations 6.89 -- 6.90 ). The pressure in the wake is nearly the same as that far from the body so that the average pressure difference between the front and the rear of the body is approximately and the drag coefficient approaches unity.
At yet higher Reynolds numbers, , the flow and the drag coefficient show little dependence upon Reynolds number. On the front of the body, a thin laminar boundary layer develops that eventually separates from the body, forming the outside edge of the wake region. The wake region becomes turbulent for , remaining constant in size and pressure until about when the boundary layer on the front of the bodye becomes turbulent. When this occurs, the separation point moves toward the rear of the body, and the wake size is greatly reduced. Both the drag (and drag coefficient) are significantly reduced with only a slight increase in Reynolds number. Once the boundary layer has become fully turbulent, further increases in Reynolds number produce little change in the flow properties and drag coefficient.
The marked changes in flow at occur on smooth, bluff bodies that have a separated boundary layer forming a wake region behind the body, where the separation process can be strongly affected by transition to turbulent flow at, or near, the separation point. It is possible to reduce the Reynolds number at which this transition occurs by roughening the body surface, thereby reducing the drag below the smooth surface value over a limited range of Reynolds numbers centered at .
Now let us return to the drag on a flat plate aligned with the flow. In Stokes flow () the drag coefficient is:
This is hardly different from the drag on a circular cylinder, equation 10.72, if we take into account the difference in the reference area A. This reflects the fact that, in Stokes flow, the drag is not much influenced by the body shape. In the laminar boundary layer regime, , the flat plate drag coefficient is that of equation 6.93 :
This flat plate drag coefficient is always lower than that for the blunt shapes, the sphere and the circular cylinder, the more so at the higher Reynolds numbers. Even when the flat plate boundary layer becomes turbulent (), (equation 7.20 and figure 7.5 ) is still considerably less than that for the blunt shapes. This difference in behavior between the flat plate and the circular cylinder or sphere stems from the fact that the form drag of a flat plate is zero and that the entire drag is caused by the tangential shear stress on the surface of the plate. For , the drag coefficient of blunt bodies is of the order of unity because the separated flow produces a wake of ambient pressure behind the body.
Separated flow can be prevented by streamlining a body, i.e., by rounding sharp corners and increasing the streamwise dimension compared to the transverse dimension. A slender, airfoil-shaped body has a drag coefficient much closer to a flat plate than to a circular cylinder. At high Reynolds number, it pays to streamline bodies to take advantage of the reduced drag.
Values of for bodies of various shapes at selected Reynolds number, are given by Blevens.