Difference between revisions of "ChtMultiRegionFoam"

Revision as of 13:24, 3 November 2018

ChtMultiRegionFoam

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

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
            (
                (
                  - ghf*fvc::snGrad(rho)
                  - fvc::snGrad(p_rgh)
                )*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)

(2)

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

(2)

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)

(2)

@@ Line 113: / Line 113: @@
 The total energy of a fluid element can be seen as the sum of kinetic energy <math> k = 0.5 u_i u_i </math>  and internal energy <math> e </math> .
-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.
+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:
 <table width="70%"><tr><td>
@@ Line 121: / Line 121: @@
 \frac{\partial}{\partial x_j}\left( \tau_{ij} u_i \right)
+</math></center>
+</td><td width="5%">(2)</td></tr></table>
+The rate of change of the internal energy <math> e </math>  of a fluid element is the heat transferred to this fluid element by diffusion plus the
+heat source term <math> r </math>  plus the heat source by radiation <math> Rad </math>  :
+<table width="70%"><tr><td>
+<center><math>
+    \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
+</math></center>
+</td><td width="5%">(2)</td></tr></table>
+The change rate of the total energy is the sum of the above two equations:
+<table width="70%"><tr><td>
+<center><math>
+    \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)
 </math></center>