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# Computational Fluid Dynamics – Turbulence Modelling – Filtering

When is the LES filter commutative?

a) When the filter function is unity

b) When the filter function is quadratic

c) When the filter function is isotropic

d) When the filter function is uniform

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Answer: d

Explanation: The LES filter function is always linear. If a uniform filter function is used, the order of filtering and differentiation can be swapped with respect to time and space coordinates. Thus, the function will be commutative.

How many extra stress terms occur due to the LES filtering operation?

a) No extra terms

b) Four terms

c) Three terms

d) Two terms

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Answer: cExplanation: The LES filtering operation resolves the flow variables into two – the filtered one and the unresolved spatial variations. This resolution leads to three extra stress terms. They are collectively called as Sub-Grid-Scale stresses.

Identify Leonard stresses from the following.

a) ρui¯¯¯¯¯uj¯¯¯¯¯¯¯¯¯¯¯¯¯¯–ρui¯¯¯¯¯uj¯¯¯¯¯

b) ρui¯¯¯¯¯uj¯¯¯¯¯¯¯¯¯¯¯¯¯¯

c) ρui¯¯¯¯¯uj¯¯¯¯¯

d) ρui¯¯¯¯¯uj¯¯¯¯¯¯¯¯¯¯¯¯¯¯–ρui¯¯¯¯¯uj¯¯¯¯¯

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Answer: a

Explanation: The term ρui¯¯¯¯¯uj¯¯¯¯¯¯¯¯¯¯¯¯¯¯–ρui¯¯¯¯¯uj¯¯¯¯¯ is named Leonard stresses. It is named after an American Scientist A. Leonard who first identified an approximate method to compute them from the filtered flow.

The stress term ρu‘iu‘j¯¯¯¯¯¯¯¯¯is called as ____________

a) Filter Reynolds stress

b) LES Reynolds stress

c) Reynolds stress

d) Sub-Reynolds stress

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Answer: b

Explanation: As the term resembles the Reynolds stress term, it is called the LES Reynolds stress term. This is modelled using Sub-Grid-Scale models, a special type of turbulence model used for these stresses.

Which of these terms corresponds to the cross stresses?

a) ρu‘iuj¯¯¯¯¯¯¯¯¯¯¯¯¯¯

b) ρui¯¯¯¯¯u‘j¯¯¯¯¯¯¯¯¯

c) ρui¯¯¯¯¯u‘j¯¯¯¯¯¯¯¯¯+ρu‘iuj¯¯¯¯¯¯¯¯¯¯¯¯¯¯

d) ρui¯¯¯¯¯u‘j¯¯¯¯¯¯¯¯¯–ρu‘iuj¯¯¯¯¯¯¯¯¯¯¯¯¯¯

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Answer: c

Explanation: The cross-stresses among the other SGS stresses occur due to the interaction between two variables. It can be given by the equation ρui¯¯¯¯¯u‘j¯¯¯¯¯¯¯¯¯+ρu‘iuj¯¯¯¯¯¯¯¯¯¯¯¯¯¯. The filtered and the unfiltered variables come together in each of these terms.

The SGS stresses are obtained from ___________

a) Reynolds stress term

b) Convection term

c) Source term

d) Diffusion term

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Answer: aExplanation: Reynolds stress term comes from the Reynolds-Averaged Navier-Stokes (RANS) equations. When this term is again averaged, but based on spatial coordinate this time, the SGS stresses occur.

If a flow variable ϕ can be resolved by spatial LES filtering as Φ and Φ’. Expand the term ρuiuj¯¯¯¯¯¯¯¯¯.

a) ρui¯¯¯¯¯uj¯¯¯¯¯¯¯¯¯¯¯¯¯¯+ρui¯¯¯¯¯u‘j¯¯¯¯¯¯¯¯¯+ρu‘iuj¯¯¯¯¯¯¯¯¯¯¯¯¯¯

b) ρui¯¯¯¯¯uj¯¯¯¯¯¯¯¯¯¯¯¯¯¯+ρu‘iu‘j¯¯¯¯¯¯¯¯¯

c) ρui¯¯¯¯¯u‘j¯¯¯¯¯¯¯¯¯+ρu‘iuj¯¯¯¯¯¯¯¯¯¯¯¯¯¯

d) ρui¯¯¯¯¯uj¯¯¯¯¯¯¯¯¯¯¯¯¯¯+ρui¯¯¯¯¯u‘j¯¯¯¯¯¯¯¯¯+ρu‘iuj¯¯¯¯¯¯¯¯¯¯¯¯¯¯+ρu‘iu‘j¯¯¯¯¯¯¯¯¯

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Answer: d

Explanation: The given term ρuiuj¯¯¯¯¯¯¯¯¯ represent the Reynolds stresses.

Resolving u_{i} and u_{j}, we get u_{i} = ui¯¯¯¯¯+u‘i and uj=uj¯¯¯¯¯+u‘j. Therefore,

ρuiuj¯¯¯¯¯¯¯¯¯=ρ(ui¯¯¯¯¯+ui‘)(uj¯¯¯¯¯+uj‘)¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

ρuiuj¯¯¯¯¯¯¯¯¯=ρui¯¯¯¯¯uj¯¯¯¯¯¯¯¯¯¯¯¯¯¯+ρui¯¯¯¯¯u‘j¯¯¯¯¯¯¯¯¯+ρu‘iuj¯¯¯¯¯¯¯¯¯¯¯¯¯¯+ρu‘iu‘j¯¯¯¯¯¯¯¯¯

These terms are separated into the three components of SGS stresses

Leonard stresses are caused by ____________

a) The SGS eddies

b) The effects at the resolved scale

c) The effects of the unresolved variables

d) The combined effect of the SGS eddies and the resolved scale

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Answer: bExplanation: The Leonard stress terms occur because of the term ρui¯¯¯¯¯uj¯¯¯¯¯¯¯¯¯¯¯¯¯¯. These are both the resolved velocities. So, we can conclude that the cause for Leonard stresses is the effects at the resolved scale.

Cross-stresses are caused by ___________

a) Convection term

b) Resolved flow

c) SGS eddies

d) Interaction of SGS eddies and resolved flow

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Answer: dExplanation: The cross-stress terms are caused by the combination of the filtered function (Φ) and the eddy function (Φ’). They are given by ρui¯¯¯¯¯u‘j¯¯¯¯¯¯¯¯¯+ρu‘iuj¯¯¯¯¯¯¯¯¯¯¯¯¯¯. So, we can say that they are caused by the interaction of SGS eddies and resolved flow.

LES Reynolds stresses are caused by ____________

a) Source term

b) Resolved flow

c) SGS eddies

d) Diffusion term

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Answer: cExplanation: The LES Reynolds stresses are caused by the convective momentum transfer due to the interaction of the SGS stresses among themselves. These SGS stresses are modelled separately as the Reynolds stresses in the RANS equations.