outletMachNumberPressure

Description

The outletMachNumberPressure is a boundary condition that maintains a certain subsonic Mach number at an outlet patch by dynamically adjusting the static outlet pressure. It makes it possible, for example, to simulate the flow in a preturbine engine exhaust manifold, without resolving details of the flow inside the turbine. In general, the flow in a choked nozzle can be non-trivial and expensive to simulate.

This formulation is derived from a simple model of the gas flow through a nozzle with fixed geometry. The nozzle flow is assumed to be quasi-steady, 1D, isentropic and compressible.

This gives the following general relationship between pressure ratio and Mach number in any cross section inside the nozzle:

$$\frac{p_{tot}}{p}=\left[ 1+ \frac{k-1}{2}\;M^2 \right]^{\frac{k}{k-1}}$$

where the constant ratio of heat capacities is $k=c_p/c_v$. The Mach number in the cross section is $M=V/c$, where $c$ is the speed of sound and V is the uniform velocity in the streamwise direction.

Overall pressure difference across the nozzle is

$$r = pBack/p_{tot}$$

When $k=1.4$, the flow in the nozzle throat becomes choked when $r<0.5$ and non-choked otherwise. This implementation is not applicable when $r>=1$ where backflow would occur.

The nozzle model assumption locks the relationship between nozzle cross sectional areas and Mach numbers. For a choked flow it is only the Mach number on the outlet patch, $M_{outlet}$, that needs to be stated in the boundary dictionary.

Care should be taken however to ensure that the entries in the input dictionary and the CFD geometry satisfy the following equation

$$c1\frac{A_{outlet}}{A_1}=\frac{1}{M_{outlet}}\left[\frac{1+\frac{k-1}{2} M_{outlet}^2}{1+\frac{k-1}{2}}\right]^{\frac{k+1}{2(k-1)}}$$

where $c1$ compensate for non-uniform outlet profiles, $A_{outlet}$ is geometrical outlet patch area and $A_1$ is assumed nozzle throat area.

In the non-choked case the outlet patch Mach number is calculated as

$$M_{outlet} = \frac{A_1} {c1\;A_{outlet}} \sqrt{\frac{2}{k-1}\left[r^\frac{2}{k}-r^\frac{k+1}{k} \right]}$$

The accompanying boundary conditions for velocity should be pressureInletOutletVelocity.

References

Fox, R. W., McDonald, A. T., & Mitchell, J. W. (2020).
Fox and McDonald's introduction to fluid mechanics.
John Wiley & Sons.

Usage

The condition requires entries in both the boundary and field files.

Boundary file

<patchName>
{
    type            patch;
    ...
}

Field file

<patchName>
{
    // Mandatory entries
    type            outletMachNumberPressure;
    pBack           <scalar>;
    choked          <bool>;

    // Optional entries
    M               <scalar>;
    c1              <scalar>;
    A1              <scalar>;
    phi             <word>;
    rho             <word>;
    U               <word>;
    relax           <scalar>;

    // Inherited entries
    ...
}

where:

Property	Description	Type	Required	Default
type	Type name: outletMachNumberPressure	word	yes	-
pBack	Pressure downstream of nozzle	bool	yes	-
choked	Defines nozzle conditions	bool	yes	-
M	Outlet Mach number	scalar	no	0
c1	Correction factor for non-uniform profiles	scalar	no	0
A1	Nozzle throat area [m^2]	scalar	no	0
relax	Underrelaxation of static pressure	scalar	no	0
phi	Name of flux field	word	no	phi
rho	Name of density field	word	no	rho
U	Name of velocity field	word	no	U

The inherited entries are elaborated in:

• fixedValueFvPatchField.H

Further information

Tutorial:

• chokedNozzle

Source code:

• outletMachNumberPressure

API:

• Foam::outletMachNumberPressureFvPatchScalarField

History: Introduced in version v2206