For the remainder of this chapter, we consider two-dimensional, steady flow. The nondimensionalized form of the x-momentum equation (one of the Navier-Stokes equations) was developed in Section 15.6 and was given by Equation (15.29);
Let us now reduce Equation (15.29) to an approximate form which holds reasonably well within a boundary layer.
Consider the boundary layer along a flat plate of length c as sketched in Figure 17.7. The basic assumption of boundary-layer theory is that a boundary layer is
very thin in comparison with the scale of the body; that is,
Consider the continuity equation for a steady, two-dimensional flow,
d(pu) d(pv) __ dx dy
In terms of the nondimensional variables defined in Section 15.6, Equation (17.16) becomes
d(p’u’) Э(рУ) dx1 dy’
Because u’ varies from 0 at the wall to 1 at the edge of the boundary layer, let us say that u’ is of the order of magnitude equal to 1, symbolized by 0(1). Similarly, p’ = 0(1). Also, since x varies from 0 to c, x’ = 0(1). However, since у varies from 0 to <5, where 8 <£ c, then y’ is of the smaller order of magnitude, denoted by у’ = О(8/с). Without loss of generality, we can assume that c is a unit length. Therefore, y’ = 0(8). Putting these orders of magnitude in Equation (17.17), we have
!в + Ш=0 [17.18]
Hence, from Equation (17.18), clearly v’ must be of an order of magnitude equal to 8; that is, v’ = 0(8). Now examine the order of magnitude of the terms in Equation (15.29). We have
Let us now introduce another assumption of boundary-layer theory, namely, the Reynolds number is large, indeed, large enough such that
Then, Equation (17.19) becomes
From Equation (17.26), we see that dp’/dy’ = 0(8) or smaller, assuming that yM^ = 0(1). Since 8 is very small, this implies that dp’/dy’ is very small. Therefore, from the у-momentum equation specialized to a boundary layer, we have
Equation (17.26a) is important; it states that at a given x station, the pressure is constant through the boundary layer in a direction normal to the surface. This implies that the pressure distribution at the outer edge of the boundary layer is impressed directly to the surface without change. Hence, throughout the boundary layer, p — P(x) = pe (x).
It is interesting to note that if is very large, as in the case of large hypersonic Mach numbers, then from Equation (17.26) dp’/dy’ does not have to be small. For example, if Mqo were large enough such that 1/yM^ = 0(8), then dp’/dy’ could be as large as 0(1), and Equation (17.26) would still be satisfied. Thus, for very large hypersonic Mach numbers, the assumption that p is constant in the normal direction through a boundary layer is not always valid.
Consider the general energy equation given by Equation (15.26). The nondimensional form of this equation for two-dimensional, steady flow is given in Equation (15.33). Inserting e = h — p/p into this equation, subtracting the momentum equation multiplied by velocity, and performing an order-of-magnitude analysis similar to those above, we can obtain the boundary-layer energy equation as
The details are left to you.
In summary, by making the combined assumptions of 8 <SC c and Re > 1/82, the complete Navier-Stokes equations derived in Chapter 15 can be reduced to simpler forms which apply to a boundary layer. These boundary-layer equations are
Note that, as in the case of the Navier-Stokes equations, the boundary-layer equations are nonlinear. However, the boundary-layer equations are simpler, and therefore are more readily solved. Also, since p = p,. (x), the pressure gradient expressed as dp/dx in Equations (17.23) and (17.27) is reexpressed as dpe/dx in Equations (17.29) and
(17.31) . In the above equations, the unknowns are u, v, p, and h p is known from p = pe (x), and д and к are properties of the fluid which vary with temperature. To complete the system, we have
and h=cpT [17.33]
Hence, Equations (17.28), (17.29), and (17.31) to (17.33) are five equations for the five unknowns, и, n, p, T, and h.
The boundary conditions for the above equations are as follows:
At the wall: у = 0 и = 0 v = О T — Tw
At the boundary-layer edge: у —»• oo и —*■ ue T —*■ Te
Note that since the boundary-layer thickness is not known a priori, the boundary condition at the edge of the boundary layer is given at large y, essentially у approaching infinity.