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Formulation of the Radial Acutation Disk Source

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Old   September 30, 2013, 09:48
Default Formulation of the Radial Acutation Disk Source
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Antar Netra
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Dear Foamers

Greetings!

I am scratching my head to understand a very small piece of code related to fvOption radialActuationSource.

The function calculates the thrust for a wind turbine and then distributes the thrust over the wind turbine disk using a radial function. In the radialActuationSource.H, the description says the following and I am quoting :

Code:
thrust = 2 \rho A U_{o}^2 a (1-a)

where

a = 1- Cp/Ct 
A = disk area
U_o = upstream velocity

Thrust is distributed by a radial function

Thrust(r) = T (C_0 + C_1 r^2 + C_2 r^4)

Example Usage:
actuationDiskSourceCoeffs
{
   fieldName       U;          // name of field to apply source
   diskDir         (-1 0 0);   // disk direction
   Cp              0.1;        // power coefficient
   Ct              0.5;        // thrust coefficient
   diskArea        5.0;        // disk area
   coeffs          (0.1 0.5 0.01); // radial distribution coefficients
   upstreamPoint   (0 0 0);    // upstream point
}
Now the radialActuationSourceTemplates.C with comments regarding the point i do not understand is given next:

Code:
template<class RhoFieldType> 
void Foam::fv::radialActuationDiskSource::
addRadialActuationDiskAxialInertialResistance
(
    vectorField& Usource,
    const labelList& cells,
    const scalarField& Vcells,
    const RhoFieldType& rho,
    const vectorField& U
) const
{
    
    scalar a = 1.0 - Cp_/Ct_;
    scalarField Tr(cells.size());
    const vector uniDiskDir = diskDir_/mag(diskDir_);

    tensor E(tensor::zero);
    E.xx() = uniDiskDir.x();
    E.yy() = uniDiskDir.y();
    E.zz() = uniDiskDir.z();

    const Field<vector> zoneCellCentres(mesh().cellCentres(), cells);
    const Field<scalar> zoneCellVolumes(mesh().cellVolumes(), cells);

    const vector avgCentre = gSum(zoneCellVolumes*zoneCellCentres)/V();
    const scalar maxR = gMax(mag(zoneCellCentres - avgCentre));

   // Question : how do we get this expression. by the variable name 
   //                it seems that we are integrating the radial function
   //                 (C_0 + C_1 r^2 + C_2 r^4). 
   //                
   //                Is that so ?
    scalar intCoeffs =
        radialCoeffs_[0]
      + radialCoeffs_[1]*sqr(maxR)/2.0
      + radialCoeffs_[2]*pow4(maxR)/3.0;

    vector upU = vector(VGREAT, VGREAT, VGREAT);
    scalar upRho = VGREAT;
    if (upstreamCellId_ != -1)
    {
        upU =  U[upstreamCellId_];
        upRho = rho[upstreamCellId_];
    }
    reduce(upU, minOp<vector>());
    reduce(upRho, minOp<scalar>());

    scalar T = 2.0*upRho*diskArea_*mag(upU)*a*(1.0 - a);
    forAll(cells, i)
    {
        scalar r2 = magSqr(mesh().cellCentres()[cells[i]] - avgCentre);

        // Question 2 : Here we are distributing the thrust as per the 
        //                  radial function. Can somebody please explain
        //                  how does the expression below ensures that
        //                  the total thrust T = sum of (Ti)s when this 
        //                  approach is used.

        Tr[i] =
            T
           *(radialCoeffs_[0] + 
              radialCoeffs_[1]*r2 + 
              radialCoeffs_[2]*sqr(r2))
            /intCoeffs;

        Usource[cells[i]] += ((Vcells[cells[i]]/V_)*Tr[i]*E) & upU;
    }

    if (debug)
    {
        Info<< "Source name: " << name() << nl
            << "Average centre: " << avgCentre << nl
            << "Maximum radius: " << maxR << endl;
    }
}
Any help is highly appreciated

Thanks
Antar
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