fluid_mechanics

470 Chapter 9 ■ Flow over Immersed Bodies
SOLUTION
The characteristics of flow past an object depend on the Reynolds
number. For this instance we could pick the characteristic length
to be the height, h, width, b, or length, /, of the car to obtain three
possible Reynolds numbers, Re h Uhn, Re b Ubn, and
Re / U/n. These numbers will be different because of the different
values of h, b, and /. Once we arbitrarily decide on the
length we wish to use as the characteristic length, we must stick
with it for all calculations when using comparisons between
model and prototype.
With the values of kinematic viscosity for air and glycerin obtained
from Tables 1.8 and 1.6 as n air 1.46 10 5 m 2 s and
n glycerin 1.19 10 3 m 2 s, we obtain the following Reynolds
numbers for the flows described.
Reynolds (a) Model in (b) Model in (c) Car in Air
Number Glycerin Air
Re h
0.571 46.6 2.91 10 6
Re b 0.672 54.8 3.42 10 6
Re / 1.68 137.0 8.56 10 6
Clearly, the Reynolds numbers for the three flows are quite
different 1regardless of which characteristic length we choose2.
Based on the previous discussion concerning flow past a flat plate
or flow past a circular cylinder, we would expect that the flow
past the actual car would behave in some way similar to the flows
shown in Figs. 9.5c or 9.6c. That is, we would expect some type
of boundary layer characteristic in which viscous effects would
be confined to relatively thin layers near the surface of the car and
the wake region behind it. Whether the car would act more like a
flat plate or a cylinder would depend on the amount of streamlining
incorporated into the car’s design.
Because of the small Reynolds number involved, the flow past
the model car in glycerin would be dominated by viscous effects, in
some way reminiscent of the flows depicted in Figs. 9.5a or 9.6a.
Similarly, with the moderate Reynolds number involved for the air
flow past the model, a flow with characteristics similar to those indicated
in Figs. 9.5b and 9.6b would be expected. Viscous effects
would be important—not as important as with the glycerin flow, but
more important than with the full-sized car.
It would not be a wise decision to expect the flow past the fullsized
car to be similar to the flow past either of the models. The
same conclusions result regardless of whether we use Re h , Re b , or
Re / . As is indicated in Chapter 7, the flows past the model car and
the full-sized prototype will not be similar unless the Reynolds
numbers for the model and prototype are the same. It is not always
an easy task to ensure this condition. One 1expensive2 solution
is to test full-sized prototypes in very large wind tunnels 1see
Fig. 9.12.
9.2 Boundary Layer Characteristics
Large Reynolds
number flow fields
may be divided into
viscous and inviscid
regions.
V9.3 Laminar
boundary layer
As was discussed in the previous section, it is often possible to treat flow past an object as a combination
of viscous flow in the boundary layer and inviscid flow elsewhere. If the Reynolds number
is large enough, viscous effects are important only in the boundary layer regions near the object
1and in the wake region behind the object2. The boundary layer is needed to allow for the no-slip
boundary condition that requires the fluid to cling to any solid surface that it flows past. Outside of
the boundary layer the velocity gradients normal to the flow are relatively small, and the fluid acts
as if it were inviscid, even though the viscosity is not zero. A necessary condition for this structure
of the flow is that the Reynolds number be large.
9.2.1 Boundary Layer Structure and Thickness on a Flat Plate
There can be a wide variety in the size of a boundary layer and the structure of the flow within it.
Part of this variation is due to the shape of the object on which the boundary layer forms. In this
section we consider the simplest situation, one in which the boundary layer is formed on an infinitely
long flat plate along which flows a viscous, incompressible fluid as is shown in Fig. 9.7. If
the surface were curved 1i.e., a circular cylinder or an airfoil2, the boundary layer structure would
be more complex. Such flows are discussed in Section 9.2.6.
If the Reynolds number is sufficiently large, only the fluid in a relatively thin boundary layer
on the plate will feel the effect of the plate. That is, except in the region next to the plate the flow
velocity will be essentially V Uî, the upstream velocity. For the infinitely long flat plate extending
from x 0 to x , it is not obvious how to define the Reynolds number because there is
no characteristic length. The plate has no thickness and is not of finite length!
For a finite length plate, it is clear that the plate length, /, can be used as the characteristic
length. For an infinitely long plate we use x, the coordinate distance along the plate from the leading
edge, as the characteristic length and define the Reynolds number as Re x Uxn. Thus, for