# ChtMultiRegionFoam

ChtMultiRegionFoam

   Solver for steady or transient fluid flow and solid heat conduction, with
conjugate heat transfer between regions, buoyancy effects, turbulence,


## 1 Equations

For each region defined as fluid, the according equation for the fluid is solved and the same is done for each solid region. The regions are coupled by a thermal boundary condition.

### 1.1 Equations Fluid

For each fluid region the compressible Navier Stokes equation are solved.

#### 1.1.1 Mass conservation

The variable-density continuity equation is

 $\frac{\partial \rho}{\partial t} + \frac{\partial {\rho u}_j}{\partial x_j} = 0$ (1)

The source code can be found in src/finiteVolume/cfdTools/compressible/rhoEqn.H:



{
fvScalarMatrix rhoEqn
(
fvm::ddt(rho)
+ fvc::div(phi)
==
fvOptions(rho)
);

fvOptions.constrain(rhoEqn);

rhoEqn.solve();

fvOptions.correct(rho);
}


#### 1.1.2 Momentum conservation

 $\frac{ \partial (\rho {u}_i)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {u}_j u_i \right) = - \frac{\partial p_{rgh}} {\partial{x_i}} - \frac{\partial \rho g_j x_j}{\partial x_i} + \frac{\partial}{\partial x_j} \left( \tau_{ij} + \tau_{t_{ij}} \right)$ (2)

$u$ represent the velocity, $g_i$ the gravitational acceleration, $p_{rgh} = p - \rho g_j x_j$ the pressure minus the hydrostatic pressure and $\tau_{ij}$ and $\tau_{t_{ij}}$ are the viscose and turbulent stresses.

The source code can be found in Ueqn.H:



// Solve the Momentum equation

MRF.correctBoundaryVelocity(U);

tmp<fvVectorMatrix> tUEqn
(
fvm::ddt(rho, U) + fvm::div(phi, U)
+ MRF.DDt(rho, U)
+ turbulence.divDevRhoReff(U)
==
fvOptions(rho, U)
);
fvVectorMatrix& UEqn = tUEqn.ref();

UEqn.relax();

fvOptions.constrain(UEqn);

if (pimple.momentumPredictor())
{
solve
(
UEqn
==
fvc::reconstruct
(
(
)*mesh.magSf()
)
);

fvOptions.correct(U);
K = 0.5*magSqr(U);
}

fvOptions.correct(U);



#### 1.1.3 Energy conservation

The energy equation can be found in: https://cfd.direct/openfoam/energy-equation/

The total energy of a fluid element can be seen as the sum of kinetic energy $k = 0.5 u_i u_i$ and internal energy $e$ . The rate of change of the kinetic energy within a fluid element is the work done on this fluid element by the viscous forces, the pressure and eternal volume forces like the gravity:

 $\frac{ \partial (\rho k)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {u}_j k \right) = - \frac{\partial p u_j} {\partial{x_i} }- \rho g_j u_j + \frac{\partial}{\partial x_j}\left( \tau_{ij} u_i \right)$ (3)

The rate of change of the internal energy $e$ of a fluid element is the heat transferred to this fluid element by diffusion plus the heat source term $r$ plus the heat source by radiation $Rad$  :

 $\frac{ \partial (\rho e)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {u}_j e \right) = - \frac{\partial q_i} {\partial{x_i} } + \rho r + Rad$ (4)

The change rate of the total energy is the sum of the above two equations:

 $\frac{ \partial (\rho e)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {u}_j e \right) + \frac{ \partial (\rho k)}{\partial t} + \frac{\partial}{\partial x_j} \left( \rho {u}_j k \right) = - \frac{\partial q_i} {\partial{x_i} } + \rho r + Rad - \frac{\partial p u_j} {\partial{x_i} }- \rho g_j u_j + \frac{\partial}{\partial x_j}\left( \tau_{ij} u_i \right)$ (5)