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# Computational Fluid Dynamics – Turbulence Modelling – Sub Grid Models

The Smagorinsky-Lilly (Sub-Grid-Scale) SGS model uses ___________

a) Boussinesq hypothesis and Prandtl mixing length model

b) Prandtl mixing length model and k-ε model

c) k-ε model and k-ω model

d) k-ω model and Boussinesq hypothesis

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Answer: aExplanation: The Smagorinsky-Lilly (Sub-Grid-Scale) SGS model is built on the Prandtl mixing length model and models the SGS eddy viscosities. It uses the Boussinesq hypothesis to assume the effects of the SGS eddies.

According to the Smagorinsky-Lilly SGS model, the SGS stresses depend on the ___________

a) Rate of strain of the SGS eddies

b) Rate of strain of the resolved flow

c) Strain of the resolved flow

d) Strain of the SGS eddies

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Answer: bExplanation: To define the effects of the unresolved SGS eddies on the resolved flow, the Smagorinsky-Lilly SGS model uses the Boussinesq hypothesis. So, the SGS stresses depend on the local rate of strain of the resolved flow.

The characteristic length of the SGS eddies is __________

a) half of the filter cut-off width

b) the filter cut-off width

c) twice the filter cut-off width

d) thrice the filter cut-off width

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Answer: bExplanation: The LES filter accepts and rejects eddies based on the filter cut-off width. So, the size of the SGS eddies is determined by the filter cut-off width and the same is used as the characteristic length.

Which of these assumptions is made in the Smagorinsky-Lilly SGS model?

a) The changes in the flow direction are slow in the resolved flow

b) The changes in the cross-stream direction are slow in the resolved flow

c) The changes in the flow direction are slow in the SGS eddies

d) The changes in the cross-stream direction are slow in the SGS eddies

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Answer: aExplanation: The Smagorinsky-Lilly SGS model is valid only if The changes in the flow direction of the resolved flow are very small that the production and dissipation of turbulence are more or less in balance The turbulent structure is isotropic.

What is the relationship between SGS viscosity (μ_{SGS}), density (ρ), characteristic length (Δ) and the average strain rate of the resolved flow (∣S¯¯¯∣ ) in the Smagorinsky-Lilly SGS model?

a) μ_{SGS}=ρ(C)^{2} Δ∣S¯¯¯∣

b) μ_{SGS}=ρC(Δ)^{2}∣S¯¯¯∣2

c) μ_{SGS}=ρ(CΔ)^{2}∣S¯¯¯∣

d) μ_{SGS}=ρ(CΔ)^{2}∣S¯¯¯∣

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Answer: d

Explanation: The equation for SGS viscosity is obtained by the dimensional analysis. It is given by the equation μ_{SGS}=ρ(CΔ)^{2}∣S¯¯¯∣ . Where, C is the constant of SGS viscosity.

What is the velocity scale taken in the Smagorinsky-Lilly SGS model?

a) The ratio of the length scale and the time scale

b) The square of the average strain rate of the resolved flow

c) The product of the length scale and the average strain rate of the resolved flow

d) The square of the length scale

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Answer: c

Explanation: The Smagorinsky-Lilly SGS model assumes a velocity scale equal to the Product of the length scale and the average strain rate of the resolved flow. It is given by the equation Δ×∣S¯¯¯∣ .

In the higher-order SGS model, what is the velocity scale used?

a) The ratio of the SGS turbulent kinetic energy to the SGS eddy viscosity

b) The product of the SGS turbulent kinetic energy and the SGS eddy viscosity

c) The square root of the SGS eddy viscosity

d) The square root of the SGS turbulent kinetic energy

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Answer: dExplanation: The major difference between the Smagorinsky-Lilly SGS model and the higher-order SGS models is the velocity scale used. The higher order models use velocity scale which is equal to the square root of the SGS turbulent kinetic energy.

The Smagorinsky-Lilly SGS model is ___________

a) Dissipative

b) Convective

c) Diffusive

d) Convective and diffusive

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Answer: aExplanation: The Smagorinsky-Lilly SGS model is completely dissipative. The direction of energy flow is from eddies at the resolved scale towards the sub-grid scales (SGS eddies). This is changed in the later models.

The SGS model uses _________ to reduce the sub-grid-scale eddy viscosity near the wall.

a) van Karman’s constant

b) van Driest damping

c) wall function

d) Leonard stresses

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Answer: bExplanation: The purpose of the van Driest damping is to reduce the sub-grid-scale eddy viscosity near the wall in the SGS models. An alternative method is to reduce the eddy viscosity when the Reynolds number becomes small.

___________ creates a problem in the SGS models.

a) Low Reynolds number flows

b) High Reynolds number flows

c) Anisotropic flow near the wall

d) Viscous flow near the wall

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Answer: cExplanation: Near the wall, the flow structure is very anisotropic. Here, regions of low and high-speed fluids are created. This needs a highly anisotropic grid. But, the choice of length scale is restricted by the cut-off width. This poses a problem in the SGS models.