Encyclopaedia Index

TITLE : Supersonic flow over a diamond profile

BY : Dr S V Zhubrin, CHAM Ltd

DATE : November, 2000

FOR : Demonstration case for PHOENICS 3.3.1


The supersonic flow over a 2D diamond-shaped body is solved here on Cartesian grid by "cut-off" technique of PARSOL.

The case is aimed to demonstrate the PARSOL performance for compressible flow along with the relevant treatments and settings.


The demonstration case considers the supersonic air flow over a diamond-shaped profile placed in a rectangular channel.

In this case the pressure distribution and accompanying velocity field have to be calculated along with auxiliary field of density to get the Mach number contours.

Symmetry of the flow allows to consider the half of the actual flow pattern.


Introductory remarks

PHOENICS allows compressible flows to be handled in the full range from subsonic to supersonic conditions. All other solution options can also be used, eg. turbulence, two phases, chemical reactions etc.

Shock and expansion waves are predicted without recourse to any special techniques, such as shock fitting. The formulations does however result in some smearing of shock waves, which is reduced as the grid is refined.

Euler-equation-based simulations are performed by simply deactivating all viscous terms in the transport equations.

The solution algorithm has been found to work for inviscid supersonic (ie hyperbolic) flows, and viscous supersonic flows, which are strictly speaking elliptic.

Conservation equations

The flow is supposed to be isentropic, ie. the energy equation need not be solved.

The independent variables of the problem are the two components of cartesian coordinate system, namely X and Y.

The main dependent (solved for) variables are:


For isentropic flows the density is given by:


RHOo and Po refer to pressure and density at stagnation conditions and GAMMA stands for specific heat ratio.

This formulation is activated by selection of isentropic option for 'Density' under the 'Properties' of the Main Menu and specifying RHO1C=0.0, RHO1A and RHO1B, as appropriate.

The specification of 'Reference pressure' is also needed due to relative nature of solved pressure.

The pressure-correction equation, then, takes account of the effects of density changes on pressure to secure convergence in compressible-flow problem. The appropriate term, DRH1DP, which for an isentropic formula reduces to 1/(GAMMA*P1) will be added to the input setting automatically.

Inlet conditions

Fixed mass inflow conditions are specified in the usual way, but care must be taken to ensure that the density of domain fluid is selected for inlet value, otherwise the inlet density must be defined consistently with the selected formulation.

Walls and symmetry planes

These are treated in the usual way. The expected pressure-wave reflection is predicted correctly in supersonic flow.

Exit planes

The prescription of static pressure usually suffices as long as the axial Mach number is not much larger than unity. For supersonic outflow, the effect will be localised.


The plots show the distribution of the pressure, velocities, density and Mach number within the channel.

Pictures are as follows :


All model settings have been made in VR-Editor of PHOENICS 3.3.1

The relevant Q1 file can be inspectedby clicking here.