of1612+里面的smagorinsky的k equation是怎样推导的

yuan_neu

Reference:
\verbatim
    Smagorinsky, J. (1963).
    General circulation experiments with the primitive equations: I.
    The basic experiment*.
    Monthly weather review, 91(3), 99-164.
\endverbatim

The form of the Smagorinsky model implemented is obtained from the
k-equation model assuming local equilibrium which provides estimates of both
k and epsilon separate from the sub-grid scale viscosity:

\verbatim
    B = 2/3*k*I - 2*nuSgs*dev(D)

where

    D = symm(grad(U));                               
    k from D:B + Ce*k^3/2/delta = 0
    nuSgs = Ck*sqrt(k)*delta
\endverbatim

实在是看不太懂里面这个D:B + Ce*k^3/2/delta = 0是如何得来的

赵一铭

看起来好像方程18：http://dyfluid.com/LES.html
@李东岳
k from D:B + Ce*k^3/2/delta = 0在哪体现的？

yuan_neu

smagorinsky.c

template<class BasicTurbulenceModel>
tmp<volScalarField> Smagorinsky<BasicTurbulenceModel>::k
(
    const tmp<volTensorField>& gradU
) const
{
    volSymmTensorField D(symm(gradU));

    volScalarField a(this->Ce_/this->delta());
    volScalarField b((2.0/3.0)*tr(D));
    volScalarField c(2*Ck_*this->delta()*(dev(D) && D));

    return tmp<volScalarField>
    (
        new volScalarField
        (
            IOobject
            (
                IOobject::groupName("k", this->U_.group()),
                this->runTime_.timeName(),
                this->mesh_
            ),
            sqr((-b + sqrt(sqr(b) + 4*a*c))/(2*a))
        )
    );
}

yuan_neu

@赵一铭
D:B + Ce*k^3/2/delta = 0 is used
to calculate k_SGS in smagorinsky, which is rather different from earlier version.

yuan_neu

Ce*k^3/2/delta is used to calculate epsilon

yuan_neu

nuSgs = Cksqrt(k)delta
k = 2Ck/Cedelta^2*(D&&D),which is used in earlier version
the second term on the rhs
Cek^3/2/delta = 2Ckdeltak^1/2*(D&&D)=2nuSgs(D&&D)

thus
D:B + Cek^3/2/delta = (2/3kI)&&D + 2nuSgs*((D-dev(D))&&D) =(2/3kI)&&D+2/3nuSgs(tr(D)&&D)

in the end k = -nuSgs*tr(D)??? get puzzled