The eddy viscosity model by Vreman (2004)

We compute the eddy viscosity from a velocity field according to SGS-model proposed by Vreman in Physics of Fluids (13/10) from 2004 under the title An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications. An important aspect of this closure formulation is that the SGS-fluxes vanish in (some) cases of a laminar flow (unlike Standard Smagorinsky).

Publication external link: Vreman.pdf.

I got my implementation inspiration from this link @ www.vremanresearch.nl

The function requires a double Cs for the (classic) Smagorinsky constant, a (3D) vector field \mathbf{u} and the molecular viscosity as input and it fills the Eddyviscosity field Evis[].

void eddyviscosity(double Cs, vector u, double molv, scalar Evis){
double d1v1, d2v1, d3v1, d1v2, d2v2, d3v2, d1v3, d2v3, d3v3;
double b11, b12, b13, b22, b23, b33;
double abeta, bbeta;
foreach(){
d1v1 = (u.x[1,0,0] - u.x[-1,0,0])/2/Delta;
d2v1 = (u.x[0,1,0] - u.x[0,-1,0])/2/Delta;
d3v1 = (u.x[0,0,1] - u.x[0,0,-1])/2/Delta;
d1v2 = (u.y[1,0,0] - u.y[-1,0,0])/2/Delta;
d2v2= (u.y[0,1,0] - u.y[0,-1,0])/2/Delta;
d3v2= (u.y[0,0,1] - u.y[0,0,-1])/2/Delta;
d1v3= (u.z[1,0,0] - u.z[-1,0,0])/2/Delta;
d2v3= (u.z[0,1,0] - u.z[0,-1,0])/2/Delta;
d3v3= (u.z[0,0,1] - u.z[0,0,-1])/2/Delta;
b11 = sq(Delta)*(d1v1*d1v1 + d2v1*d2v1 + d3v1*d3v1);
b12 = sq(Delta)*(d1v1*d1v2 + d2v1*d2v2 + d3v1*d3v2);
b13 = sq(Delta)*(d1v1*d1v3 + d2v1*d2v3 + d3v1*d3v3);
b22 = sq(Delta)*(d1v2*d1v2 + d2v2*d2v2 + d3v2*d3v2);
b23 = sq(Delta)*(d1v2*d1v3 + d2v2*d2v3 + d3v2*d3v3);
b33 = sq(Delta)*(d1v3*d1v3 + d2v3*d2v3 + d3v3*d3v3);
abeta = sq(d1v1) + sq(d2v1) + sq(d3v1)+
sq(d1v2) + sq(d2v2) + sq(d3v2)+
sq(d1v3) + sq(d2v3) + sq(d3v3);
bbeta = b11*b22 - sq(b12) + b11*b33 - sq(b13) + b22*b33 - sq(b23);
// Accordig to [1], Evis = 0 for abeta = 0.
Evis[] =  (abeta > 10E-5 && bbeta > (abeta/10E6))? 2.5*sq(Cs)*sqrt(bbeta/(abeta)) + molv: molv;
}
}

Reference

[1] Vreman, A. W. “An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications.” Physics of Fluids (1994-present) 16.10 (2004): 3670-3681.