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# Computational Fluid Dynamics – Turbulence Modelling – K-epsilon Model

What does the name k-ε model signify?

a) The seven extra transport equations used in the model

b) The variation of k and ε with the flow variables

c) The variation of k with ε

d) The two extra transport equations used in the mod

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Answer: dExplanation: k-ε is a turbulence model used to supplement the RANS equations in overcoming its non-linearity. This model uses two additional transport equations which govern the transport of k and ε.

What does k and ε stand for?

a) Turbulent kinetic energy and its dissipation rate per unit mass

b) Turbulent kinetic energy and turbulent diffusivity

c) Turbulent diffusivity and its dissipation rate per unit mass

d) Turbulent kinetic energy and mass transfer

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Answer: aExplanation: In the k-ε model, the two additional equations govern the transport of turbulent kinetic energy (k) and the rate of dissipation of the turbulent kinetic energy (ε). The behaviour of turbulent flow is given in terms of these two properties in this model.

The k-ε model focuses on the mechanism which affects ____________

a) the Reynolds stresses

b) the cross stresses

c) the transport of scalar fluxes

d) the turbulent kinetic energy

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Answer: dExplanation: The basic mixing length model cannot define a flow which involves flow separation or recirculation. So, a better turbulence model is developed in terms of k and ε. This model focuses on the dynamics of the flow and hence its turbulent kinetic energy.

___________ and _____________ are used in the k-ε model in addition to k and ε to formulate the transport equations.

a) Internal thermal energy and turbulent stresses

b) Internal thermal energy and kinetic energy

c) Rate of deformation and turbulent stresses

d) Rate of deformation and kinetic energy

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Answer: cExplanation: While forming the transport equations for k and ε, the rate of deformation term and the turbulent stresses are also used. These are used in their tensor form. Both of them can be expressed in terms of the mean velocity gradients.

If Sij represents the rate of deformation, μ represents the dynamic viscosity and V⃗ , the velocity of the flow, What does the terms div(2μV⃗ Sij) and 2μSij account for?

a) The effect of turbulent stresses

b) The effect of viscous stresses

c) The effect of Reynolds stresses

d) The effect of kinetic energy

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Answer: b

Explanation: The term div(2μV⃗ Sij) represents the transport of kinetic energy due to viscous stresses. The term 2μSij represents the viscous dissipation of kinetic energy. Together, these two terms represent the effect of viscous stresses on kinetic energy.

The terms accounting for turbulence effects contain ____________

a) Reynolds stresses

b) Turbulent kinetic energy

c) Dissipation of turbulent kinetic energy

d) Length scale terms

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Answer: a

Explanation: The terms accounting for turbulence stresses are div(ρV⃗ u‘iu‘j¯¯¯¯¯¯¯¯¯) and ρu‘iu‘j¯¯¯¯¯¯¯¯¯.div(ρV⃗ u‘iu‘j¯¯¯¯¯¯¯¯¯) represents the turbulent transport of kinetic energy by means of Reynolds stresses. ρu‘iu‘j¯¯¯¯¯¯¯¯¯ represents the net decrease of kinetic energy due to deformation work by Reynolds stresses. Both of these terms contain the Reynolds stress term ρu‘iu‘j¯¯¯¯¯¯¯¯¯.

In high Reynolds number turbulent flows _______________ terms dominate.

a) diffusion terms

b) convection terms

c) viscous stress terms

d) turbulent effect terms

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Answer: dExplanation: In high Reynolds number flows, the difference between the length scales will be very high. The large eddies are more energetic. So, the turbulent effect terms are much larger than the viscous stress terms in high Reynolds number flows.

Express the large scale velocity in terms of k and ε.

a) ε^{1/2}

b) (ε/k)^{1/2}

c) k^{1/2}

d) (k/ε)^{1/2}

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Answer: c

Explanation: In the k-ε model, the properties of turbulence can be expressed in terms of the variables k and ε. The velocity scale of the large eddies are given by k^{1/2}. k is the turbulent kinetic energy term.

Express the large scale length in terms of k and ε.

a) (k/ε)^{3/2}

b) k^{3/2}/ε

c) ε/k^{3/2}

d) (ε/k)^{3/2}

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Answer: b

Explanation: The length scale of the large eddies can be given as k^{3/2}/ε. The small scale dissipation rate of the turbulent kinetic energy can be used to represent the large scale length as the rate at which the large eddies extract energy from the mean flow is matched with the rate of energy transfer to small eddies.

Let C_{μ} be a dimensionless constant and ρ be the density of the flow. Express the eddy dynamic viscosity in terms of k and ε.

a) ρC_{μ}k^{2}/ε

b) ρC_{μ} k/ε

c) ρC_{μ} ε/k

d) ρC_{μ} ε^{2}/k

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Answer: a

Explanation: Using dimensional analysis, the turbulent dynamic viscosity can be given as

μ_{t} = C_{μ} ρϑl

Where,

ϑ → Velocity scale of large eddies

l → Length scale of large eddies

Substituting these two in k and ε terms, we get

μ_{t} = C_{μ} ρk2ε Where, C_{μ} is a dimensionless constant which is adjustable. The standard k-ε model uses C_{μ}=0.09.