Scalar Transport

Overview

Transport of a passive scalar T is done with myScalarTransportFoam. This solver is used to obtain a solution to the advection-diffusion equation given a known velocity field U.

    dT           
    -- + Div(UT) - Laplacian(DTT) = Q
    dt           

The diffusion coefficient DT is assumed to be a constant scalar, Q is the source term.

Steps

See here for an example. The following steps assume the velocity field U is already defined on a mesh. myScalarTransportFoam is compiled in $FOAM_USER_APPBIN and is based on $FOAM_SOLVERS/basic/scalarTransportFoam. The code is in windninja/src/ninjafoam/utility/myScalarTransportFoam and includes createFields.H and myScalarTransportFoam.C.

• The scalar field T and the vector field U are read in from xx/T and xx/U, where xx is the current time step. Boundary conditions are also specified here.
• The diffusion coefficient DT and fluid density rho are defined in constant/transportProperties. DT is read in with createFields.H.
• The source term is defined in 0/source. A non-uniform source can be set by the setFields utility. The cells to set are specified in system/setFieldsDict. For example:

defaultFieldValues

(
    volScalarFieldValue source 0
);

regions
(
    boxToCell //distribute source in cells within a bounding box
    {
        box (-1 -1 -1) (1E10 1E10 1600);        
        fieldValues
        (
            volScalarFieldValue source 333.0
        );
    }
);

regions
(
    nearestToCell //distribute source in cell nearest to a point
    {
        points ((334707 4807053 2200));             
        fieldValues
        (         
            volScalarFieldValue source 333.0
        );
    }

);

• A template controlDict for scalarTransportFoam is included in windninja/data/ninjafoam.zip.

Limitations

Currently we assume a single constant diffusion coefficient. A better approach would be to allow different diffusivities in the horizontal and vertical directions and based on atmospheric stability.

dT/dt + Div(UT) - d/dx(Rx(dT/dx)) - d/dy(Ry(dT/dy)) - d/dz(Rz(dT/dz)) = 0

where Rx, Ry, and Rz are the eddy diffusivities. As a first attempt we could calculate the eddy diffusivities based on mixing length theory as:

Rz = lm * dU/dz
lm = 0.4 * z where z is distance to the wall
Rx = Ry = 2*Rz

Stability parameterizations (e.g., Kumar and Sharan 2012) could also be used.