10.2 Pressure-Correction Methods for Arbitrary Mach Number
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These differences are reflected in the pressure-correction equation in another way. Because the equation is no longer a pure Poisson equation, the
central coefficient Ap is not the negative of the sum of the neighbor coefficients. Only when divv = 0, is this property obtained.
10.2.2 Boundary Conditions
For incompressible flows the following boundary conditions are usually applied:
0 Prescribed velocity and temperature on inflow boundaries;
0 Zero gradient normal t o the boundary for all scalar quantities and the velocity component parallel to the surface on a symmetry plane; zero velocity
normal to such a surface;
0 No-slip (zero relative velocity) conditions, zero normal stress and prescribed temperature or heat flux on a solid surface;
0 Prescribed gradient (usually zero) of all quantities on an outflow surface.
These boundary conditions also hold for compressible flow and are treated in
the same way as in incompressible flows. However, in compressible flow there
are further boundary conditions:
0 Prescribed total pressure;
0 Prescribed total temperature;
0 Prescribed static pressure on the outflow boundary;
0 At a supersonic outflow boundary, zero gradients of all quantities are usually specified.
The implementation of these boundary conditions is described below.
Prescribed Total Pressure on the Inflow Boundary. The implementation of these boundary conditions will be described for the west boundary
of a two-dimensional domain with the aid of Fig. 10.1.
One possibility is to note that, for isentropic flow of an ideal gas, the total
pressure is defined as:
where p is the static pressure and y = c,/c,. The flow direction must be
prescribed; it is defined by:
For incompressible flows the static pressure can also be prescribed on either the
in- or outflow boundary. As the mass flux is a function of the difference in pressure
between the inflow and outflow, the velocity at the inflow boundary cannot be
prescribed if the pressure is prescribed at both in- and outflow boundaries.
315
These differences are reflected in the pressure-correction equation in another way. Because the equation is no longer a pure Poisson equation, the
central coefficient Ap is not the negative of the sum of the neighbor coefficients. Only when divv = 0, is this property obtained.
10.2.2 Boundary Conditions
For incompressible flows the following boundary conditions are usually applied:
0 Prescribed velocity and temperature on inflow boundaries;
0 Zero gradient normal t o the boundary for all scalar quantities and the velocity component parallel to the surface on a symmetry plane; zero velocity
normal to such a surface;
0 No-slip (zero relative velocity) conditions, zero normal stress and prescribed temperature or heat flux on a solid surface;
0 Prescribed gradient (usually zero) of all quantities on an outflow surface.
These boundary conditions also hold for compressible flow and are treated in
the same way as in incompressible flows. However, in compressible flow there
are further boundary conditions:
0 Prescribed total pressure;
0 Prescribed total temperature;
0 Prescribed static pressure on the outflow boundary;
0 At a supersonic outflow boundary, zero gradients of all quantities are usually specified.
The implementation of these boundary conditions is described below.
Prescribed Total Pressure on the Inflow Boundary. The implementation of these boundary conditions will be described for the west boundary
of a two-dimensional domain with the aid of Fig. 10.1.
One possibility is to note that, for isentropic flow of an ideal gas, the total
pressure is defined as:
where p is the static pressure and y = c,/c,. The flow direction must be
prescribed; it is defined by:
For incompressible flows the static pressure can also be prescribed on either the
in- or outflow boundary. As the mass flux is a function of the difference in pressure
between the inflow and outflow, the velocity at the inflow boundary cannot be
prescribed if the pressure is prescribed at both in- and outflow boundaries.