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Derivation of the PML Equation
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The PML equation for the linearized Euler equation (linearized over a uniform mean flow in the x direction) may be derived following three basic steps. To make the discussion of each of these steps as simple as possible, two-dimensional problems in Cartesian coordinates will first be considered. Let L be the length scale, a0 (sound speed of the uniform mean flow) be the velocity scale, L/a0 be the time scale, p0 (gas density of mean flow) be the density scale, and p0a0 be the pressure scale. The dimensionless linearized Euler equation may be written in the following form:
where
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p |
M |
1 |
0 |
0 |
0 |
0 |
1 |
0 |
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u |
, a = |
0 |
M |
0 |
1 |
, в = |
0 |
0 |
0 |
0 |
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U= |
v |
0 |
0 |
M |
0 |
0 |
0 |
0 |
1 |
||
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p |
0 |
1 |
0 |
M |
0 |
0 |
1 |
0 |
M is the Mach number of the mean flow.
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The first step is to make a change of variables from (x, y, t) to (X, y, t). The variables are related by
This transformation is necessary to make the PML equation stable. The linearized Euler equation now becomes
M d U d U 2 1 d U
I +- TA I -=■ + a + (1 – M2)1В— = 01 – m2 J dt dx dy
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In Eq. (9.37), I is the identity matrix. Assuming, for the time being, a time dependence of e-i ®?, Eq. (9.37) may be rewritten as [11]
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so that
dX + aB Yy
Eqs. (9.42) and (9.43) are the PML equations for rectangular computational domain as shown in Figure 9.9.
It is worthwhile to point out that the auxiliary variable q is needed only in the PML domains. This is because the spatial derivative dq/dx disappears for the PML equation when ay = 0 and, similarly, 9q/dy disappears when ax = 0. Thus, 9q/dx is required only in a horizontal PML and 9q/9y is required only a vertical PML. In short, there is no need to compute or store q in the Euler computation domain.
