Smagorinsky model

15.3 Smagorinsky model

Note that the fine scale vortices are not resolved. They filtered out by the

S GS ij

into account. Since the small vortices are not modeled, the subgrid stress are calculated using phenomenological models. The most recent phenomenolog- ical model was proposed by Smagorinsky in 1963. The Smagorinsky model is just the extension of the Boussinesq approach

S GS instead of the turbulent kinematic viscosity

Expression for the subgrid viscosity was obtained by Smagorinsky with the use of idea taken from the Prandtl mixing length theory. According to P randtl, the turbulent kinematic viscosity is proportional to the mixing length squared and the velocity gradient close to the wall

2 dN u x

t Dl j

dy

According to Smagorinsky, the subgrid viscosity is proportional to the mag- nitude of the strain rate tensor S ij and to a certain length l S squared

The length l S is assumed to be proportional to the mesh size

l S DC S

where C S is the constant of Smagorinsky. The Smagorinsky constant was estimated first by Lilly. The main assumption

of the Lilly analysis is the balance between generation

and dissipation of the turbulent kinetic energy

jQ S j 2 Dl t 2 ij S jQ S ij j 3 (15.3) Lilly estimated the strain rate tensor magnitude for Kolmogorov spectrum

"DN

S Q 2 7C " 2=3 4=3

Substitution of the last formula into (15.3) results in:

Assuming additionally that Q S 2 S 3 , the length l S and the Smagorinsky constant are expressed through the Kolmogorov constant C D 1:5:

.7C / 3=4

The Smagorinsky constant 0:17 is derived analytically with a few strong assumptions. The experience shows that numerical results agree with mea- surements much better if a reduced value of the Smagorinsky constant is used. Common values are 0:065 and 0:1.

Advantages and disdvantages of the Smagorinsky model are summarized in the table 19.1.

Laminar flow is not modelled

Low computational costs Constant of Smagorinsky is constant in time and

space

Stable Actually, the constant is chosen arbitrarily depending on the problem under consideration

Good accuracy in ideal

Sensible to grid

conditions

Purely dissipativ Damping of pulsation is too strong

Table 15.2: Advantages and disadvantages of the Smagorinsky model

We finish this section with a very important comment:

Indeed, if the resolution is increased, The LES equation is passed to the original Navier Stokes equations. The LES simulation becomes the DNS simulation if URANS simulation is not consistent when don’t disappear if the resolution is increased twice resolved.

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