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

• 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

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

• 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？

• smagorinsky.c

template<class BasicTurbulenceModel>
tmp<volScalarField> Smagorinsky<BasicTurbulenceModel>::k
(
) const
{

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))
)
);
}


• @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.

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

• nuSgs = Cksqrt(k)delta
k = 2
Ck/Ce
delta^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