Characteristic Boundary Condition Concept

The correct number and types of boundary conditions for a system of partial differential equa­tions can be determined by analyzing eigenvalues and the corresponding eigenvectors of the sys­tem in each coordinate direction separately This approach to boundary condition treatment is called characteristic boundary conditions {155]-( 157) since it suggests one-dimensional applica­tion of certain Ricmann invariants at ihc boundaries that can be either solid or open boundaries. For example, in a 3-D duct flow, if the flow is locally subsonic at the exit, wc will be able to com­pute all How variables at the exit based on the information from the interior points except for one variable that we will have to specify at the exit. If the same characteristic boundary condition pro­cedure is applied in the direction normal to the solid wall and if the desired pressure distribution is specified on die wall, this method of iteratively enforcing the boundary conditions at the wall will result in non-zero normal velocities at the wall which can he used to update the wall shape. The general concept follows.

The F. ulcr equations for 3-D compressible unsteady flows expressed in non-conscrva – tivc form and cast in a boundary-conforming, non-orthogonal. curvilinear <£,r).£) coordinate system can be transformed into

о (73)

w here Q = ф p u v w) is the transposed vector of the non-conservative primitive varia­bles. Figcnvalucs of В are

V * °’ J(n2 * V * n;2 ) V’ V’ V’ V*a * П v* + n:2) (74)

where V ■ n ♦ Л • и ♦ n + and the local speed of sound is defined as a —

1 x У *

(Y P / P),/:: If the П-gnd lines arc emanating from the 3-D aerodynamic configuration, we can

have several situations. If 0 < V < a one eigenvalue is negative requiring a

pressure boundary condition to he specified at that surface point.

Similarly, if-a < 0. four eigenvalues will be negative requiring

s 2 ”> 1/2

pressure, velocity ratio m/(m‘ ♦ »+**) . total pressure and total temperature to be speci­

fied at that surface point. This method has been show n to converge quickly for transonic two – dimensional airfoil shape design when using compressible flow Euler equations (I56].(I57J. The method might be applicable to the inverse design of arbitrary 3-D configurations |155| although no such attempts have been reported yet. If the Euler code is executed in a time-accu­rate mode, the specified unsteady solid wall characteristic boundary conditions will provide for a time-accurate motion of the solid boundaries which is highly attractive for the design of "smart” aerodynamic configurations. This concept is not directly applicable to viscous flow codes since velocity components at the solid w all are zero.