Sec. 4.6 concluded that the basic outﬂow conditions — and — are generally unstable for a free boundary with both inﬂow and outﬂow.
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:
- the pressure gradient, decreasing outward, causes outﬂow;
- the outﬂow speed increases causing the pressure gradient to change direction;
- 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:
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 decreases by as the inﬂow speed increases. This reduces the pressure gradient driving inﬂow, which moderates the increase in inﬂow speed, enabling it to settle to a stable level.
The total pressure condition can be applied to high-speed ﬂow of a compressible gas. The calculation of for inﬂow is simply replaced with the 1D isentropic ﬂow equation,4