Finite-Difference Method

Return for a moment to Section 2.17.2 where we introduced some ideas from compu­tation fluid dynamics, and especially review the finite-difference expressions derived there. Recall that we can simulate the partial derivatives with forward, rearward, or central differences. We will use these concepts in the following discussion.

Also consider Figure 18.13, which shows a schematic of a finite-difference grid inside the boundary layer. The grid is shown in the physical x-y space, where it is curvilinear and unequally spaced. However, in the £-77 space, where the calculations are made, the grid takes the form of a rectangular grid with uniform spacing A£ and A rj. In Figure 18.13, the portion of the grid at four different £ (or x ) stations is shown, namely, at (7 — 2), (7 — 1), 7, and (7 + 1).

Consider again the general, transformed boundary-layer equations given by Equations (18.84) and (18.86). Assume that we wish to calculate the boundary layer at station (7 + 1) in Figure 18.13. As discussed in Section 2.17.2, the general philos­ophy of finite-difference approaches is to evaluate the governing partial differential equations at a given grid point by replacing the derivatives by finite-difference quo­tients at that point. Consider, for example, the grid point (7, j) in Figure 18.13. At this point, replace the derivatives in Equations (18.84) and (18.86) by finite-difference expressions of the form:

[18.87]

[18.88]

where в is a parameter which adjusts Equations (18.87)—(18.90) to various finite – difference approaches (to be discussed below). Similar relations for the derivatives of g are employed. When Equations (18.87)-( 18.90) are inserted into Equations (18.84) and (18.86), along with the analogous expressions for g, two algebraic equations are obtained. If в = 0, the only unknowns that appear are fi+ij and gi+ij, which can be obtained directly from the two algebraic equations. This is an explicit approach. Using this approach, the boundary layer properties at grid point (; + 1, j) are solved explicitly in terms of the known properties at points (i, j + 1), (i, j) and O’, j — 1). The boundary-layer solution is a downstream marching procedure; we are calculating the boundary layer profiles at station (; + 1) only after the flow at the previous station (і) has been obtained.

When 0 < в < 1, then fi+ij+i, fi+,j-, g;+u+i, gi+ij, and gi+ij-i

appear as unknowns in Equations (18.84) and (18.86). We have six unknowns and only two equations. Therefore, the finite-difference forms of Equations (18.84) and (18.86) must be evaluated at all the grid points through the boundary layer at station (i + 1) simultaneously, leading to an implicit formulation of the unknowns. In particular, if в = the scheme becomes the well-known Crank-Nicolson implicit procedure, and if в — 1, the scheme is called “fully implicit.” These implicit schemes result in large systems of simultaneous algebraic equations, the coefficients of which constitute block tridiagonal matrices.

Already the reader can sense that implicit solutions are more elaborate than ex­plicit solutions. Indeed, we remind ourselves that the subject of this book is the fundamentals of aerodynamics, and it is beyond our scope to go into great computa­tional fluid dynamic detail. Therefore, we will not elaborate any further. Our purpose here is only to give the flavor of the finite-difference approach to boundary-layer solutions. For more information on explicit and implicit finite-difference methods, see the author’s book Computational Fluid Dynamics: The Basics with Applications (Reference 64).

In summary, a finite-difference solution of a general, nonsimilar boundary-layer proceeds as follows:

1. The solution must be started from a given solution at the leading edge, or at a stagnation point (say station 1 in Figure 18.13). This can be obtained from appropriate self-similar solutions.

2. At station 2, the next downstream station, the finite-difference procedure reflected by Equations (18.87)—(18.90) yields a solution of the flowfield variables across the boundary layer.

3. Once the boundary-layer profiles of и and T are obtained, the skin friction and heat transfer at the wall are determined from

„ = (»£)_

Here, the velocity gradients can be obtained from the known profiles of и and 7 by using one-sided differences (see References 64), such as

‘dus	—3u]	-f – 4w 2 — и з	[18.91]
	1 – W	2 Ay
dT2	-37,	+ 47) – 7,	[18.92]
9>’y	) – W	2 Ay

In Equations (18.91) and (18.92), the subscripts 1,2, and 3 denote the wall point and the next two adjacent grid points above the wall. Of course, due to the specified boundary conditions of no velocity slip and a fixed wall temperature, и і = 0 and T = Tw in Equations (18.91) and (18.92).

4. The above steps are repeated for the next downstream location, say station 3 in Figure 18.13. In this fashion, by repeating applications of these steps, the complete boundary layer is computed, marching downstream from a given initial solution.

An example of results obtained from such finite-difference boundary-layer solu­tions is given in Figures 18.14 and 18.15 obtained by Blottner (Reference 84). These are calculated for flow over an axisymmetric hyperboloid flying at 20,000 ft/s at an altitude of 100,000 ft, with a wall temperature of 1000 K. At these conditions, the boundary layer will involve dissociation, and such chemical reactions were included in the calculations of Reference 84. Chemically reacting boundary layers are not the purview of this book; however, some results of Reference 84 are presented here just to illustrate the finite-difference method. For example, Figure 18.14 gives the calculated velocity and temperature profiles as a station located at x/RN = 50, where Rn is the nose radius. The local values of velocity and temperature at the boundary layer edge are also quoted in Figure 18.14. Considering the surface properties, the variations of Ся and ty as functions of distance from the stagnation point are shown in Figure 18.15. Note the following physical trends illustrated in Figure 18.15.

1. The shear stress is zero at the stagnation point (as is always the case), then it in­creases around the nose, reaches a maximum, and decreases further downstream.

2. The values of Ся are relatively constant near the nose, and then decrease further downstream.

3. Reynolds analogy can be written as

Сц = [18.93]

2s

where 5 is called the “Reynolds analogy factor.” For the flat plate case, we see from Equation (18.50) that л = Pr1. However, clearly from the results of Figure 18.15 we see that s is a variable in the nose region because Ся is relatively constant while су is rapidly increasing. In contrast, for the downstream region, Cf and Ся are essentially equal, and we can state that Reynolds analogy becomes

u/ue or T/Te Figure 18.14 Velocity and temperature profiles across the boundary layer at x/Rn = 50 on an axisymmetric hyperbloid. (Source: Blottner, Reference 84.}

approximately C#/c/ = 1. The point here is that Reynolds analogy is greatly affected by strong pressure gradients in the flow, and hence loses its usefulness as an engineering tool in such cases, at least when Ся and с/ are based on freestream quantities as shown in Figure 18.15.

Figure 18.15 Stanton number and skin friction coefficient (based on freestream properties) along a hyperbloid. (Source: Blottner, Reference 84.]

18.7 Summary

This brings to an end our discussion of laminar boundary layers. Return to the roadmap in Figure 18.1 and remind yourself of the territory we have covered. Some of the important results are summarized below.

For incompressible laminar flow over a flat plate, the boundary-layer equations reduce to the Blasius equation
2/"’ + //" = 0	[18.15]
where /’ = м/м,,. This produces a self-similar solution where /’ = independent of any particular x station along the surface. A numerical solution of Equation (17.48) yields numbers which lead to the following results.
tw 0.664 Local skin friction coefficient: cf = i——————– г = ,_____ 2 Poo V Л-®*	[18.30]
1-328 Integrated friction drag coefficient: Cf= ________ vRec	[18.33]
5.О* Boundary-layer thickness: S = -……………………….. -■ VRe^	[18.33]
	(continued)