324
10. Compressible Flow
flow at Ma = 1.65 over a bump (see Fig. 10.4) in a channel is shown. The
same grid and under-relaxation parameters are used for both flows. While
in the compressible case the rate of convergence is nearly constant, in the
incompressible case at low Reynolds number it gets lower as the tolerance is
tightened. At very high Mach numbers, the computing time increases almost
linearly with the number of grid points as the grid is refined (the exponent
is about 1.1, compared to about 1.8 in case of incompressible flows). However, as we shall demonstrate in Chap. 11, the convergence of the method for
elliptic problems can be substantially improved using the multigrid method,
making the method very efficient. The compressible version of the method is
suitable for both steady and unsteady flow problems.
compr., Ma=1.65
tncompr.. Re=iOO
Fig. 10.8. Convergence of the pressurecorrection method for laminar flow at Re =
1 100 and for supersonic flow at Ma;, = 1.65
o
500
, , , , over a bump in channel (160 x 80 CV grid);
Iter.
from Lilek (1995)
For ultimate accuracy one should apply grid refinement locally near the
shocks, where the profiles suddenly change slope. The methods of applying
local grid refinement and the criteria about where to refine the grid will be
described in Chap. 11. Also, the blending of CDS and UDS should be applied
locally, only in the vicinity of shocks, and not globally, as in the above applications. The criteria for decision where and how much of UDS to blend with
CDS can be based on a monotonicity requirement on the solution, on total
variation diminishing (TVD, see next section) or other suitable requirements.
10.3 Methods Designed for Compressible Flow
The method described above is a'n adaptation of methods designed for computing incompressible flows to the treatment of compressible flows. It was
10. Compressible Flow
flow at Ma = 1.65 over a bump (see Fig. 10.4) in a channel is shown. The
same grid and under-relaxation parameters are used for both flows. While
in the compressible case the rate of convergence is nearly constant, in the
incompressible case at low Reynolds number it gets lower as the tolerance is
tightened. At very high Mach numbers, the computing time increases almost
linearly with the number of grid points as the grid is refined (the exponent
is about 1.1, compared to about 1.8 in case of incompressible flows). However, as we shall demonstrate in Chap. 11, the convergence of the method for
elliptic problems can be substantially improved using the multigrid method,
making the method very efficient. The compressible version of the method is
suitable for both steady and unsteady flow problems.
compr., Ma=1.65
tncompr.. Re=iOO
Fig. 10.8. Convergence of the pressurecorrection method for laminar flow at Re =
1 100 and for supersonic flow at Ma;, = 1.65
o
500
, , , , over a bump in channel (160 x 80 CV grid);
Iter.
from Lilek (1995)
For ultimate accuracy one should apply grid refinement locally near the
shocks, where the profiles suddenly change slope. The methods of applying
local grid refinement and the criteria about where to refine the grid will be
described in Chap. 11. Also, the blending of CDS and UDS should be applied
locally, only in the vicinity of shocks, and not globally, as in the above applications. The criteria for decision where and how much of UDS to blend with
CDS can be based on a monotonicity requirement on the solution, on total
variation diminishing (TVD, see next section) or other suitable requirements.
10.3 Methods Designed for Compressible Flow
The method described above is a'n adaptation of methods designed for computing incompressible flows to the treatment of compressible flows. It was