Shooting Method

This method is a classic method for the solution of the boundary-layer equations to be discussed in Chapter 17. For the solution of compressible Couette flow, the same philosophy follows as that to be applied to boundary-layer solutions, and that is why we discuss it now. The method involves a double iteration, that is, two minor iterations nested within a major iteration. The scheme is as follows:

1. Assume a value for r in Equation (16.64). A reasonable assumption to start with is the incompressible value, r = p(ue/D). Also, assume that the variation of u(y) is given by the incompressible result from Equation (16.6).

2. Starting at у = 0 with the known boundary condition T = Tw, integrate Equa­tion (16.64) across the flow until у = D. Use any standard numerical technique for ordinary differential equations, such as the well-known Runge-Kutta method (see, e. g., Reference 52). However, to start this numerical integration, because Equation (16.64) is second order, two boundary conditions must be specified at у 0. We only have one physical condition, namely, T = Tw. Therefore, we have to assume a second condition; let us assume a value for the temperature gradient at the wall, i. e., assume a value for (dT/dy)w. A value based on the incompressible flow solution discussed in Section 16.3 would be a reasonable assumption. With the assumed (dT/dy)w and the known T„ at _y = 0, then Equation (16.64) is integrated numerically away from the wall, starting at у = 0 and moving in small increments, Ay in the direction of increasing y. Values of T at each increment in у are produced by the numerical algorithm.

3. Stop the numerical integration when у = D is reached. Check to see if the numerical value of T at у = D equals the specified boundary condition, T = 7). Most likely, it will not because we have had to assume a value for (dT/dy)w in step 2. Hence, return to step 2, assume another value of (dT/dy)w, and repeat the integration. Continue to repeat steps 2 and 3 until convergence is obtained, that is, until a value of (dT/dy)w is found such that, after the numerical integration, T = Tt. at у = D. From the converged temperature profile obtained by repetition of steps 2 and 3, we now have numerical values for Г as a function of у that satisfy both boundary conditions; that is, T = Tw at the lower wall and T = Te at the upper wall. However, do not forget that this converged solution was obtained for the assumed value of r and the assumed velocity profile u(y) in step 1. Therefore, the converged profile for T is not necessarily the correct profile. We must continue further; this time to find the correct value for r.

4. From the converged temperature profile obtained by the repetitive iteration in steps 2 and 3, we can obtain ft. = ц.(у) from Equation (15.3).

5. From the definition of shear stress,

we have — = — [16.65]

dy fi

Recall from the solution of the momentum equation, Equation (16.60), that г is a constant. Using the assumed value of r from step 1, and the values of ft = ft(y) from step 4, numerically integrate Equation (16.65) starting at у = 0 and using the known boundary condition и = 0 at у = 0. Since Equation (16.65) is first order, this single boundary condition is sufficient to initiate the numerical integration. Values of и at each increment in y, Ay, are produced by the numerical algorithm.

6. Stop the numerical integration when у = D is reached. Check to see if the numerical value of и at у = D equals the specified boundary condition, и = ue. Most likely, it will not, because we have had to assume a value of г and и (у) all the way back in step 1, which has carried through to this point in our iterative

solution. Hence, return to step 5, assume another value for r, and repeat the integration of Equation (16.65). Continue to repeat steps 5 and 6 [using the same values of д = д(у) from step 4] until convergence is obtained, that is, until a value of r is found that, after the numerical integration of Equation (16.65), и = ue at у = D. From the converged velocity profile obtained by repetition of steps 5 and 6, we now have numerical values for и as a function of у that satisfy both boundary conditions; that is, и = Oat у = 0 and и = и,, at v = D. However, do not forget that this converged solution was obtained using /j. — fi(y) from step 4, which was obtained using the initially assumed r and u(y) from step 1. Therefore, the converged profile for и obtained here is not necessarily the correct profile. We must continue one big step further.

7. Return to step 2, using the new value of r and the new и (у) obtained from step 6. Repeat steps 2 through 7 until total convergence is obtained. When this double iteration is completed, then the profile for Г = T (y) obtained at the last cycle of step 3, the profile for m = u(y) obtained at the last cycle of step 6, and the value of r obtained at the last cycle of step 7 are all the correct values for the given boundary conditions. The problem is solved!

Looking over the shooting method as described above, we see two minor iterations nested within a major iteration. Steps 2 and 3 constitute the first minor iteration and provide ultimately the temperature profile. Steps 5 and 6 are the second minor iteration and provide ultimately the velocity profile. Steps 2 to 7 constitute the major iteration and ultimately result in the proper value of r.

The shooting method described above for the solution of compressible Couette flow is carried over almost directly for the solution of the boundary-layer equations to be described in Chapter 18. In the same vein, there is another completely different approach to the solution of compressible Couette flow which carries over directly for the solution of the Navier-Stokes equations to be described in Chapter 20. This is the time-dependent, finite-difference method, first discussed in Chapter 13 and applied to the inviscid flow over a supersonic blunt body in Section 13.5. In order to prepare ourselves for Chapter 20, we briefly discuss the application of this method to the solution of compressible Couette flow.