# 1 Overview

The buoyantPisoFoam solver is suitable for unsteady, compressible flow and includes the gravitational body force on the fluid. The Richardson number is a dimensionless measure of the importance of buoyancy in a flow.

# 2 Nomenclature

$p \,$ p pressure
$\rho \,$ rho density
$h \,$ h enthalpy
$\mathbf{\Phi} \,$ mass flux, $= \rho \mathbf{ U }\,$
$\phi \,$ phi mass flux through faces, $= \rho \mathbf{ U } . \mathbf{ S }_{ f } \,$
$\alpha_{ t } \,$ alphaEff effective turbulent thermal diffusivity
$\psi \,$ psi compressibility $= \rho / p$
$\mu_{ eff } \,$ muEff effective viscosity $= \mu + \mu_{ t } \,$
$\mu_{ t } \,$ turbulent viscosity
$\mu \,$ mu laminar molecular viscosity
$\rho \mathbf{ R } \,$ divRhoR Reynolds stresses $= \rho \overline{ U_{ i }' U_{ j }' } \,$

# 3 Momentum

The primary Navier Stokes equation is for the transport of momentum with sources and sinks.

$\frac{\partial ( \rho \mathbf{U} ) }{\partial t} + \nabla \cdot ( \rho \mathbf{U} \mathbf{U} ) + \nabla \cdot ( \rho \mathbf{R} ) = -\nabla p + \rho \mathbf{g}$

or in terms of the flux which must be discretised at cell faces (unlike most other quantities which are discretised at cell centres),

 $\frac{\partial ( \rho \mathbf{U} ) }{\partial t} + \nabla \cdot ( \mathbf{\Phi} \mathbf{U} ) + \nabla \cdot ( \rho \mathbf{R} ) = -\nabla p + \rho \mathbf{g}$

 File: OpenFOAM-1.6/applications/solvers/heatTransfer/buoyantPisoFoam/UEqn.H  fvVectorMatrix UEqn ( fvm::ddt(rho, U) + fvm::div(phi, U) + turbulence->divDevRhoR(U) ); UEqn.relax(); solve(UEqn == -fvc::grad(p) - fvc::grad(rho)*gh); 

We note that the mass flux is on hard-wired linear interpolation from cell centres to faces,

 File: OpenFOAM-1.6.x/src/finiteVolume/cfdTools/compressible/compressibleCreatePhi.H surfaceScalarField phi ( IOobject ( "phi", runTime.timeName(), mesh, IOobject::READ_IF_PRESENT, IOobject::AUTO_WRITE ), linearInterpolate(rho*U) & mesh.Sf() ); 

# 4 Mass

Conservation of mass provides a relation between the local rate of change of density and the degree to which the local mass flux is acting as a source or sink.

 $\frac{\partial \rho}{\partial t} + \nabla \cdot \phi = 0$

 File: OpenFOAM-1.6/src/finiteVolume/cfdTools/compressible/rhoEqn.H { solve(fvm::ddt(rho) + fvc::div(phi)); } 

# 5 Energy

Actually the more general concept of enthalpy is used. This is more relevant for gases and chemical reactions because it accounts for internal energy and the work done by the system, or in other words kinetic and potential energy.

 $\frac{\partial ( \rho h ) }{\partial t} + \nabla \cdot ( \phi h ) + \nabla \cdot ( \alpha_{t} \nabla h ) = \frac{Dp}{Dt}$

 File: applications/solvers/heatTransfer/buoyantPisoFoam/hEqn.H { solve ( fvm::ddt(rho, h) + fvm::div(phi, h) - fvm::laplacian(turbulence->alphaEff(), h) == DpDt ); hEqn.relax(); hEqn.solve(); thermo.correct(); } 

# 6 Turbulence

## 6.1 Momentum source

The Reynolds stresses are estimated [1] via the tensor

\begin{align} \mathbf{R} & = \frac{2}{3} \mathbf{I} k - 2 \frac{\mu_{t} }{\rho} \mathrm{dev} \left[ \mathrm{symm} \left( \nabla \mathbf{U} \right) \right ] \\ & = \frac{2}{3} \mathbf{I} k - 2 \frac{\mu_{t}}{\rho} \left( \mathrm{symm} \left( \nabla \mathbf{U} \right) - \frac{1}{3} \mathrm{ tr } \left( \mathrm{symm} \left( \nabla \mathbf{U} \right) \right) \mathbf{I} \right) \\ & = \frac{2}{3} \mathbf{I} k - 2 \frac{\mu_{t}}{\rho} \left( \frac{1}{2} \left( \nabla \mathbf{U} + ( \nabla \mathbf{U} )^{T} \right) - \frac{1}{3} \mathrm{tr} \left( \nabla \mathbf{U} \right) \mathbf{I} \right) \\ \end{align}

because the trace of the symmetric component of a tensor is just the trace of the original tensor,

$\mathrm{tr}( \mathrm{symm} \mathbf{T} ) = \mathrm{tr} ( T )$

 File: OpenFOAM-1.6/src/turbulenceModels/compressible/RAS/kOmegaSST/kOmegaSST.C 273 tmp kOmegaSST::R() const 274 { 275 return tmp 276 ( 277 new volSymmTensorField 278 ( 279 IOobject 280 ( 281 "R", 282 runTime_.timeName(), 283 mesh_, 284 IOobject::NO_READ, 285 IOobject::NO_WRITE 286 ), 287 ((2.0/3.0)*I)*k_ - (mut_/rho_)*dev(twoSymm(fvc::grad(U_))), 288 k_.boundaryField().types() 289 ) 290 ); 291 } 

The divergence of the product appearing in the momentum equations is estimated as

$\nabla . ( \rho \mathbf{R} ) = - \nabla . ( \mu_{eff} \nabla \mathbf{U} ) - \nabla . \left( \mu_{eff} \nabla \left( \nabla ( \mathbf{U}^{T} ) - \frac{2}{3} \mathrm{tr} ( \nabla ( \mathbf{U}^{T} ) ) \mathbf{I} \right) \right)$

 File: OpenFOAM-1.6/src/turbulenceModels/compressible/RAS/kOmegaSST/kOmegaSST.C 314 tmp kOmegaSST::divDevRhoReff(volVectorField& U) const 315 { 316 return 317 ( 318 - fvm::laplacian(muEff(), U) - fvc::div(muEff()*dev2(fvc::grad(U)().T())) 319 ); 320 } 

For reference,

$\mathrm{ grad \ } \mathbf{ U } = \nabla \mathbf{ U } = \left[ \begin{array}{ccc} \dfrac{\partial U_{x}}{\partial x} & \dfrac{\partial U_{y}}{\partial x} & \dfrac{\partial U_{z}}{\partial x} \\ \\ \dfrac{\partial U_{x}}{\partial y} & \dfrac{\partial U_{y}}{\partial y} & \dfrac{\partial U_{z}}{\partial y} \\ \\ \dfrac{\partial U_{x}}{\partial z} & \dfrac{\partial U_{y}}{\partial z} & \dfrac{\partial U_{z}}{\partial z} \\ \end{array} \right]$

# 7 PISO

The non linearity of the convection term of the momentum equation requires an iterative solution technique if we do not wish to be limited to small time steps. But while the momentum equation provides a way to find the velocity in terms of the pressure gradient, we need some way to determine the pressure as we iterate. In the Pressure-Implicit Split-Operator (PISO) approach, we formulate a linear relationship between the pressure gradient and the velocity in order to obtain a prediction of the velocity (which violates mass conservation), then apply the mass conservation requirement in order to obtain a pressure field that is mass conservative so that it can be fed back to obtain a corrected velocity field. At the same time, we correct for face skewness by updating the mass fluxes at cell faces with the new pressure field.

The momentum equation is discretised at $N$ discrete locations to obtain

$A \mathbf{U} - \mathbf{H}$

or

$\left[ \begin{array}{cccc} A_{11} & A_{12} & \ldots & A_{1N} \\ A_{21} & A_{22} & \ldots & A_{2N} \\ \vdots & \vdots & \ddots & \vdots \\ A_{N1} & A_{N2} & \ldots & A_{NN} \\ \end{array} \right] \left[ \begin{array}{c} \mathbf{U}_{1} \\ \mathbf{U}_{2} \\ \vdots \\ \mathbf{U}_{N} \\ \end{array} \right] = \left[ \begin{array}{c} H_{1} \\ H_{2} \\ \vdots \\ H_{N} \\ \end{array} \right]$

$A$ and $\mathbf{H}$ are then reconstituted to make $A$ into a diagonal matrix with the "offcuts" placed into the square matrix $\mathbf{H}$

Multiplying both sides by the inverse of $A$ yields the momentum corrector equation

$\mathbf{U} = \frac{\mathbf{H}}{A} - \frac{1}{A} \nabla p$

Applying mass conservation, we obtain the pressure corrector equation

\begin{align} & \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{U}) = 0 \\ & \frac{\partial \rho}{\partial t} + \mathbf{U} \cdot \nabla \rho + \rho \nabla \cdot \mathbf{U} = 0 \\ & \frac{\partial \rho}{\partial t} + \frac{\mathbf{H}}{A} \cdot \nabla \rho - \frac{1}{A} \nabla p \cdot \nabla \rho + \rho \nabla \cdot \left( \frac{\mathbf{H}}{A} \right) - \rho \nabla \cdot \left( \frac{1}{A} \nabla p \right) = 0 \\ \end{align}

This is solved to provide a pressure field inserted into a flux corrector equation

\begin{align} \phi & = \mathbf{S}_{f} . \mathbf{U}_{f} \\ & = \mathbf{S}_{f} \cdot \left[ \left( \frac{\mathbf{H}}{A} \right)_{f} - \left( \frac{1}{A} \right)_{f} ( \nabla p )_{f} \right] \\ \end{align}

## 7.1 Algorithm

1 Accept $p, \mathbf{U}, \phi$, and require $\rho$
while (runTime.loop())
{
...
}

The "old" values go into the loop, either initialised in createFields.H or provided by previous time step.
2 $\mathrm{LHS} = \frac{\partial ( \rho \mathbf{U} ) }{\partial t} + \nabla . ( \phi \mathbf{U} ) + \nabla . ( \rho \mathbf{R} )$
    fvVectorMatrix UEqn
(
fvm::ddt(rho, U)
+ fvm::div(phi, U)
+ turbulence->divDevRhoR(U)
);

Assemble the LHS of the momentum equation and discretise it into matrix form using the vector calculus schemes in fvSchemes.
3 $\mathrm{RHS} = -\nabla p + \rho \mathbf{g}$
    UEqn.relax();

if (momentumPredictor)
{
solve
(
UEqn
==
fvc::reconstruct
(
fvc::interpolate(rho)*(g & mesh.Sf())