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# Computational Fluid Dynamics – Turbulence Modelling – Spalart Allmaras Model

The Spalart-Allmaras model differs from the RANS equations by ___________

a) four extra transport equations

b) one extra transport equation

c) two extra transport equations

d) three extra transport equations

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Answer: bExplanation: Spalart-Allmaras is a turbulence model to the RANS equations. This has an extra transport equation. This extra transport equation is used to overcome the non-linear problem of the RANS equations.

The transport equation in the Spalart-Allmaras model is for the transport of ___________

a) kinematic eddy viscosity parameter

b) kinematic eddy viscosity

c) dynamic eddy viscosity parameter

d) dynamic eddy viscosity

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Answer: aExplanation: The extra transport equation of the Spalart-Allmaras method describes the transport of the kinematic eddy viscosity parameter through convection, diffusion, dissipation and source terms. This way, it is different from the other turbulence models.

In the Spalart-Allmaras model, the dynamic eddy viscosity in terms of the kinematic eddy viscosity parameter (v) is given by __________ (Note: fν1 is the wall damping function and ρ is the density of flow).

a) ρvfν1

b) (ρv) ⁄ fν1

c) (ρfν1) ⁄ v

d) v ⁄ (ρfν1)

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Answer: aExplanation: Dynamic viscosity is the product of the kinematic viscosity and the density of the flow. As the kinematic eddy viscosity parameter is used here, the wall function comes into the picture. So, the dynamic eddy viscosity μt=ρvfν1

The first wall damping function in the Spalart-Allmaras model is a function of ___________

a) the product of the dynamic eddy viscosity parameter and the dynamic eddy viscosity

b) the ratio of the dynamic eddy viscosity parameter and the dynamic eddy viscosity

c) the product of the kinematic eddy viscosity parameter and the kinematic eddy viscosity

d) the ratio of the kinematic eddy viscosity parameter and the kinematic eddy viscosity

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Answer: dExplanation: The first wall damping function is introduced in the dynamic eddy viscosity. The dynamic eddy viscosity divided by the density of the flow is the kinematic viscosity. So, the function is a function of the ratio of the kinematic eddy viscosity parameter and the kinematic eddy viscosity.

At high Reynolds numbers, the first wall damping function becomes ___________

a) -1

b) 1

c) 0

d) ∞

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Answer: bExplanation: The first wall damping function becomes one when we consider turbulent flows at high Reynolds numbers. This is because, at these Reynolds numbers, the kinematic eddy viscosity parameter value is close to the kinematic eddy viscosity.

Near the wall, the first wall damping function tends to ___________

a) -1

b) 1

c) 0

d) ∞

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Answer: cExplanation: The value of the kinematic eddy viscosity parameter decreases with the Reynolds number. Near the wall, the Reynolds number is very small. So, the kinematic eddy viscosity parameter and the first wall function also tends to zero.

Expand the Reynolds stress term −ρu‘iu‘j¯¯¯¯¯¯¯¯¯ for the Spalart-Allmaras model.

a) −ρu‘iu‘j¯¯¯¯¯¯¯¯¯=ρv¯¯¯fv1(∂Ui∂xi+∂Uj∂xj)

b) −ρu‘iu‘j¯¯¯¯¯¯¯¯¯=ρv¯¯¯fv1(∂Ui∂xj+∂Uj∂xi)

c) −ρu‘iu‘j¯¯¯¯¯¯¯¯¯=2ρv¯¯¯fv1(∂Ui∂xi+∂Uj∂xj)

d) −ρu‘iu‘j¯¯¯¯¯¯¯¯¯=2ρv¯¯¯fv1(∂Ui∂xj+∂Uj∂xi)

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Answer: b

Explanation: The Reynolds stress term is given as

−ρu‘iu‘j¯¯¯¯¯¯¯¯¯=ρt(∂Ui∂xj+∂Uj∂xi)

Converting to Spalart-Allmaras terms,

−ρu‘iu‘j¯¯¯¯¯¯¯¯¯=ρv¯¯¯fv1(∂Ui∂xj+∂Uj∂xi)

The rate of production of the kinematic eddy viscosity parameter is related to ___________

a) rate of dissipation of kinetic energy

b) turbulence frequency

c) vorticity

d) kinetic energy

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Answer: cExplanation: The rate of production of the kinematic eddy viscosity parameter is a term in the transport equation of the Spalart-Allmaras model. This is related to the local mean vorticity by means of the vorticity parameter.

The rate of dissipation of kinematic eddy viscosity parameter is C_{w1}ρ(ν~κy)2fw. What is the length scale used here?

a) κy

b) (κy)^{2}

c) Cw1y

d) yCw1

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Answer: aExplanation: The length scale cannot be computed in the Spalart-Allmaras model. It must be specified separately. The length scale used here is κy. Where, κ is the von Karman’s constant which is equal to 0.4187 and y is the distance from the wall.

The Spalart-Allmaras model is best suited for ___________

a) turbulent jet flows

b) turbulent mixing layers

c) turbulent boundary layers with slight pressure gradients

d) turbulent boundary layers with adverse pressure gradients

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Answer: dExplanation: The Spalart-Allmaras model is suitable for flow near walls. So, it is suitable for turbulent boundary layers. The other models are also suitable for this case. But, they cannot model adverse pressure gradients for which Spalart-Allmaras is the best model.