Orders of Magnitude
We expect that the thickness of the boundary layer has a major impact on the problem formulation. Recall the fact that 5 can represent a length smaller than even the thickness of the paper on which this page is written up to perhaps 1 centimeter or more, depending on the distance from the leading edge of the boundary layer. In most cases, 5 is considerably smaller than the length of the aerodynamic surface.
The method used here for deducing the appropriate set of equations is the same as that used in the early 1900s by Ludwig Prandtl to achieve his astonishing insight into viscous boundary-layer flow. We must examine the various terms in the governing equations to determine which ones control the behavior. Some terms must be retained whereas others that have little effect can be ignored. This is accomplished by comparing sizes of terms on the basis of their orders of magnitude, using the boundary-layer thickness as a primary scaling variable. The arguments here closely follow those used by Prandtl.
To illustrate the approach in the simplest possible way, consider the continuity equation, Eq. 8.29:
й +я = °-
It is clear that because u is of the same order of magnitude as the velocity outside the boundary layer (V<»), then it is proper to assume that:
u ~ Vx – which is often expressed mathematically as u = O(V<»), where O is the order symbol. This is read as “u is of the order of V<».” It expresses the fact that the streamwise velocity anywhere in the region of interest is comparable to that outside of the boundary layer. Therefore, the dimensionless velocity is:
u = V ~1 (or, u = O(1)).
Similarly, if we move downstream from the origin a distance x ~ L (much larger than the thickness of the boundary layer), then the dimensionless length in the x-direction is:
x = — ~1 (or, x = O(1)).
Similarly, if we take changes in the —-position, Ax, and changes in the —-velocity, Au also to be comparable in size to L and Vx, respectively, then Ax ~ 1 and Au ~ 1 Therefore, the derivative of u with respect to x that appears in the continuity equation is on the order of magnitude of unity; that is:
du Ax. ( dx
ж – ax ~ Г»= 0<1)}
Now, changes in the z-direction in the boundary layer must be much smaller than L; that is, they are comparable to the dimensional boundary-layer thickness, 5, which is
the appropriate length scale in the boundary layer. It therefore is useful to introduce the dimensionless boundary-layer thickness:
я 8 ..
8 = — << 1,
which helps in scaling properties of the boundary layer normal to the surface. That is, it is appropriate to write:
Az ~8 (or, Az = 0(8)),
which expresses the fact that changes in distance in the boundary layer normal to the surface are much smaller than in the streamwise direction. So that the continuity equation, Eq. 8.29, is satisfied, it is clear that the velocity and changes in the velocity normal to the surface also must be on the order of 8. Then:
w ~ 8 and Aw ~ 8
so that the derivative:
dw _ Aw 1 dz Az ’
as required for continuity. This justifies the previous observation that:
w«V^ (w <<1).
This ordering process now can be applied to the full set of equations, Eqs. 8.2930, so that any negligibly small terms can be identified and eliminated, if possible. The result is:
where the order of each term is written under its position in the equations. As already noted, continuity is properly satisfied. It is important to notice that only the correct choice of scaling of the normal velocity and displacement allows this; otherwise, the two terms would not cancel to satisfy the equation.
Consider the x-momentum equation, Eq. 8.30. Any term that is to be retained must be of the order O(1). Thus, both parts of the convective acceleration on the left must stay; unfortunately, the problem is still nonlinear! The axial pressure gradient is taken to be of the order O(1) and is dependent only on conditions outside of the boundary layer, as we demonstrate herein. The viscous force on the right of Eq. 8.30 is dominated by the large Re number in the denominator. If it is not similar in size to the other terms in Eq. 8.30, then there is no mechanism present to slow the fluid particles down to zero speed at the surface. In other words, it then would not be possible to satisfy the no-slip condition. If viscous effects are not to disappear, it is necessary that:
8 2~r? (832)
then, the derivative д2и / Эх2 represents a negligible term. At last, something drops out! Because the Re number is very large, the small factor 1/Re in a sense “cancels” the very large factor of the order of the inverse of the square of the dimensionless boundary-layer thickness. The result is a product that is of first order, O(1). For emphasis, the viscous term can be retained only in the momentum balance if:
Thus, all of the terms in Eq. 8.32 are of the same order (namely, O(1)), which is consistent with the terms in the continuity equation.
Rewriting Eq. 8.33 in dimensional form:
du du dp d2u
u— + w — = – — + v —. dx dz dx dZ2
This equation often is called the boundary-layer equation. Notice that unlike most of the other problems solved in this chapter, it is not a linear-differential equation. The convective terms involving products of the variables and their derivatives still appear on the left side. Nevertheless, it is considerably simpler than the x-component of the Navier-Stokes equation.
The z-momentum equation shows that the pressure gradient normal to the surface:
I ~ 5^0 (8.35)
is very small because 8 << 1. It is on this basis that we can assume that the pressure through the boundary layer is controlled by conditions outside of the layer. Corrections to the pressure distribution are not much affected by the presence of the layer, as Eq. 8.33 shows. For these reasons, in the limit of a very small boundary – layer thickness, we assume that the pressure gradient normal to the surface is essentially zero. This is a key element of the Prandtl boundary-layer theory, which states that:
The static pressure gradient through a boundary layer is negligibly small.
Thus, if a static-pressure sensor is inserted into a wind-tunnel wall such that it reads the pressure very close to the surface, the reading also accurately represents the static pressure at the edge of the boundary layer. No correction is required to obtain the value of freestream static pressure. Eq. 8.36 is valid provided that the surface on which the boundary layer is growing is flat or nearly so. If the surface has appreciable curvature, then centripetal forces on the fluid particles may become significant; in that case, there must be a pressure gradient, dp/dz, normal to the surface to maintain the particles in equilibrium. Such a correction is estimated easily from the geometry, whenever necessary.
To summarize, the Prandtl boundary-layer equations rewritten in dimensional form are:
du dw _
IX+1Z = 0
du dw 1 dp d2u (8.36)
dx dz p dx dz2
The continuity and x-momentum balance control the flow. The z-momentum equation provides the information that p = p(x). Then, Eq. 8.37 represents two equations in the three unknowns, u, w, p. This difficulty is resolved by treating p = p(x) as a given quantity. In many practical cases of practical importance, p = p(x) is the pressure-distribution term provided by the external-flow (i. e., potential-flow) solution because p does not change in the direction perpendicular to the surface. The inviscid-flow solutions for the pressure distribution on airfoils, wings, and bodies, as discussed in Chapters 5 through 7, provide the required information describing the pressure distribution p = p(x), which is inserted into Eq. 8.37 to give the dp/dx term in the x-momentum equation. As a result, there are two unknowns, u and w, to be determined by the solution of the two boundary-layer equations. The potential flow – pressure field is said to be “impressed” on the boundary layer. The boundary-layer equations, Eq. 8.36, must be solved subject to boundary conditions appropriate for a given physical and geometrical situation.