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$ \color{white} \omega$ varying with height or radius

The widely used boundary condition for  40 1 in rotating machinery inlet is turbulentMixingLengthDissipationRateInlet.


        type            compressible::turbulentMixingLengthDissipationRateInlet;
        mixingLength    0.1;
        phi             phi;
        k               k;
        value           $internalField;

For internal flows the parameter mixingLength is recommended to be equal to hydrostatic diameter of the pipe.