Radiation, Outflow, and Wall Boundary Conditions

A computational domain is inevitably finite. For open domain problems, this auto­matically creates a set of artificial exterior boundaries. There are two basic reasons why exterior boundary conditions are needed at the artificial boundaries. First, exte­rior boundary conditions must reproduce the effects the outside world exerts on the flow inside the computation domain. Since only the computed results inside the computational domain are known, in general, it is not possible to know the outside influence. For this reason, a computation domain is often taken as large as possi­ble so that all sources are inside. This will minimize any external influence that is unaccounted for.

Another reason for imposing exterior boundary conditions is to avoid the reflec­tion of outgoing disturbances back into the computational domain and thus contam­inate the computed solution. One way to avoid reflection is to construct boundary conditions in such a way to allow the smooth exit of all disturbances.

In this chapter, it will be assumed that there is little external influence and there are no incoming disturbances. Issues of external influence and incoming entropy, vorticity, or acoustic waves will be considered in later chapters.

As discussed previously, the linearized Euler equations support three types of waves. Two types of these waves, namely, the entropy and vorticity waves, are convected downstream by the mean flow. The acoustic waves, however, propagate and radiate out in all directions if the mean flow is subsonic. Thus, at a subsonic inflow region, the outgoing waves consist of acoustic waves alone. In the outflow region, the outgoing disturbances now comprise of all three types of waves. As a result, it is prudent to develop separate inflow and outflow boundary conditions.

Many aeroacoustics problems involve solid surfaces. To specify the presence of these surfaces, the no-slip boundary conditions are imposed if the fluid is viscous. On the other hand, if the fluid is inviscid, the no-through flow boundary condition is to be used. How to impose these physical boundary conditions on a high-order finite difference computation without creating excessive spurious short waves is one of the main topics of this chapter.

Now, consider an exterior problem involving a uniform flow of velocity u0 and sound speed a0 past some arbitrary acoustic, vorticity, and entropy sources as shown in Figure 5.1. It will be assumed that the boundaries of the computation domain are quite far from the sources. From the analysis of Chapter 5, it is clear that outflow

boundary conditions are needed on the right side of the computational domain with outgoing disturbances formed by a superposition of acoustic, entropy, and vorticity waves. On the other hand, at the top and bottom boundaries, as well as the left inflow boundary of the computational domain, the outgoing disturbances are acoustic waves only. Now, the exterior boundaries are far from the sources, the outgoing waves in the regions close to these boundaries are, therefore, given by the asymptotic solutions of the discretized form of the governing partial differential equations. It will now be assumed that the DRP scheme is used for time-marching computation. However, in the resolved wave number range, the DRP scheme and the original partial differential equations have (almost) identical dispersion relations and asymptotic solutions. These solutions are given by Eqs. (5.14), (5.16a, b), and (5.25). A set of radiation and outflow boundary conditions will now be constructed based on these asymptotic solutions.