## Corrector Step

Inserting the flow variables obtained above into the governing equations, Equations

(13.59) to (13.62), using rearward differences for the spatial derivatives, predicted

values of the time derivatives at t + At are obtained, for example, [(3p/30,j](t+A,). In turn, these are averaged with the time derivatives from the predictor step to obtain; for example,

Finally, the average time derivative obtained from Equation (13.64) is inserted into Equation (13.63) to yield the corrected value of density at time t + At. The same procedure is used for all the dependent variables, u, v, etc.

Starting from the assumed initial conditions at t — 0, the repeated application of Equation (13.63) along with the above predictor-corrector algorithm at each time step allows the calculation of the flow-field variables and shock shape and location as a function of time. As stated above, after a large number of time steps, the calculated flow-field variables approach a steady state, where [(3p/3r)(j]ave —>■ 0 in Equation

(13.63) . Once again, we emphasize that we are interested in the steady-state answer, and the time-dependent technique is simply a means to that end.

Note that the applications of MacCormack’s technique to both the steady flow calculations described in Section 13.4 and the time-dependent calculations described in the present section are analogous; in the former, we march forward in the spatial coordinate x, starting with known values along with a constant у line, whereas, in the latter, we march forward in time starting with a known flow field at t = 0.

Why do we bother with a time-dependent solution? Is it not an added complication to deal with an extra independent variable t in addition to the spatial variables x and _y? The answers to these questions are as follows. The governing unsteady flow equations given by Equations (13.59) to (13.62) are hyperbolic with respect to time, independent of whether the flow is locally subsonic or supersonic. In Figure 13.9a, some of the grid points are in the subsonic region and others are in the supersonic region. However, the time-dependent solution progresses in the same manner at all these points, independent of the local Mach number. Hence, the time-dependent technique is the only approach known today which allows the uniform calculation of a mixed subsonic-supersonic flow field of arbitrary extent. For this reason, the application of the time-dependent technique, although it adds one additional independent variable, allows the straightforward solution of a flow field which is extremely difficult to solve by a purely steady-state approach.

A much more detailed description of the time-dependent technique is given in Chapter 12 of Reference 21, which you should study before attempting to apply this technique to a specific problem. The intent of our description here has been to give you simply a “feeling” for the philosophy and general approach of the technique.

Some typical results for supersonic blunt-body flow fields are given in Figures 13.12 to 13.15. These results were obtained with a time-dependent solution described in Reference 35. Figures 13.12 and 13.13 illustrate the behavior of a time-dependent solution during its approach to the steady state. In Figure 13.12, the time-dependent motion of the shock wave is shown for a parabolic cylinder in a Mach 4 freestream. The shock labeled 0 At is the initially assumed shock wave at t = 0. At early times, the shock wave rapidly moves away from the body; however, after about

300 time steps, it has slowed considerably, and between 300 and 500 time steps, the shock wave is virtually motionless—it has reached its steady-state shape and location. The time variation of the stagnation point pressure is given in Figure 13.13. Note that the pressure shows strong timewise oscillations at early times, but then it asymptotically approaches a steady value at large times. Again, it is this asymptotic steady state that we want, and the intermediate transient results are just a means to that

end. Concentrating on just the steady-state results, Figure 13.14 gives the pressure distribution (nondimensionalized by stagnation point pressure) over the body surface for the cases of both Мж = 4 and 8. The time-dependent numerical results are shown as the solid curves, whereas the open symbols are from newtonian theory, to be discussed in Chapter 14. Note that the pressure is a maximum at the stagnation point and decreases as a function of distance away from the stagnation point—a variation that we most certainly would expect based on our previous aerodynamic experience. The steady shock shapes and sonic lines are shown in Figure 13.15 for

Figure 13.1 5 Shock shapes and sonic lines, parabolic cylinder. |

Problems

Note: The purpose of the following problem is to provide an exercise in carrying out a unit process for the method of characteristics. A more extensive application to a complete flow field is left to your specific desires. Also, an extensive practical problem utilizing the finite-difference method requires a large number of arithmetic operations and is practical only on a digital computer. You are encouraged to set up such a problem at your leisure. The main purpose of the present chapter is to present the essence of several numerical methods, not to burden the reader with a lot of calculations or the requirement to write an extensive computer program.

1. Consider two points in a supersonic flow. These points are located in a cartesian coordinate system at {x, yi) = (0,0.0684) and (xi, у2) = (0.0121,0), where the units are meters. At point {x, yi): и і = 639 m/s, г 1 = 232.6 m/s, p = 1 atm, T = 288 K. At point {x3, Уі)’- R2 = 680 m/s, i>2 = 0, P2 = 1 atm, T2 = 288 K. Consider point 3 downstream of points 1 and 2 located by the intersection of the C+ characteristic through point 2 and the C_ characteristic through point 1. At point 3, calculate: m3, i>3, p3, and T3. Also, calculate the location of point 3, assuming the characteristics between these points are straight lines.