4.7 Total pressure condition

Sec. 4.6 concluded that the basic outflow conditions — eqn and eqn — are generally unstable for a free boundary with both inflow and outflow.

PICT\relax \special {t4ht=

The conditions are unstable due to pressure fluctuations at the boundary, shown above. Flows oscillates in and out, shown by 3 stages, creating a vortex that travels from left to right:

  1. the pressure gradient, decreasing outward, causes outflow;
  2. the outflow speed increases causing the pressure gradient to change direction;
  3. inflow begins and the speed increases, until the pressure gradient changes direction, returning back to step 1.

The total pressure boundary condition improves the stability of solutions. It is a fixed value type, calculated according to:

 (p for outflow; p = 0 2 p0 ju j=2 for inflow. \relax \special {t4ht=
The specified total pressure, eqn, can be imagined as the fluid pressure under quiescent conditions far from the free boundary, which decreases as the fluid accelerates towards the boundary. Note that Eq. (4.7 ) is written for the incompressible assumption, Eq. (2.47 ), where eqn and eqn are kinematic, i.e. divided by eqn.

PICT\relax \special {t4ht=

The solution using the total pressure condition converges to the flow field shown above. The critical effect of this boundary condition is that, the boundary eqn decreases by eqn as the inflow speed eqn increases. This reduces the pressure gradient driving inflow, which moderates the increase in inflow speed, enabling it to settle to a stable level.

Total pressure for high speed flow

The total pressure condition can be applied to high-speed flow of a compressible gas. The calculation of eqn for inflow is simply replaced with the 1D isentropic flow equation,4

 1 =(1 ) p = p0 1+ -----Ma2 ; 2 \relax \special {t4ht=
where Mach number is eqn and eqn.
4Ascher Shapiro, The dynamics and thermodynamics of compressible fluid flow, Vol. 1, Ch. 4, 1953.

Notes on CFD: General Principles - 4.7 Total pressure condition