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The correct option is (b) μt∝K^2/ε

Easiest explanation: In the k-ε model, the TURBULENT dynamic VISCOSITY is given in terms of k and ε. The RELATION is given as

μt∝k^2/ε.

Correct choice is (b) Dimensional analysis

Easiest explanation: The ratio of LARGE and small scales of turbulent FLOWS can be given in terms of the Reynolds number. These relations are ESTABLISHED using dimensional analysis. Dimensional analysis is a method which compares the dimensions in both the sides of the equations to set the RELATIONSHIP.

The correct choice is (c) Re^-1/2

Best EXPLANATION: The ratio of time-scales is OBTAINED USING the LENGTH scale RATIOS. It is given by

Time-scale ratio= Re^-1/2.

The correct answer is (B) Reynolds number

The explanation: The BEHAVIOUR of the BOUNDARY conditions depends on the Reynolds number. NEAR the WALL, the k-ε model does not perform well. So, wall functions are used. This again depends on the Reynolds number of the flow only.

Right choice is (c) Turbulence frequency

To explain I would say: The VARIABLE ω means the turbulence frequency. It gives the rate at which the turbulent kinetic ENERGY is converted into turbulent internal THERMAL energy per unit VOLUME and per unit TIME.

The CORRECT answer is (c) Homogeneous ISOTROPIC TURBULENCE

The best I can explain: The homogeneous isotropic turbulence is the simplest model of turbulence problems. It just needs a UNIFORM grid to simulate the flow. However, the number of grids DEPENDS upon the Reynolds number of the flow.

Right CHOICE is (b) Continuity and momentum equations of incompressible turbulent flow

Easiest explanation: The INSTANTANEOUS continuity and Navier-Stokes equations (momentum equations) for an incompressible turbulent flow form the initial point of the DIRECT Numerical Solution method. These equations form a CLOSED set of four equations with the four unknowns (pressure and three COMPONENTS of velocity).

The correct CHOICE is (c) identical quantities

Easiest explanation: This is useful for identical quantities. Identical in the sense, that they have similar properties in some concern. A NUMBER of quantities which have the simultaneous VARIATIONS can be AVERAGED using this method.

Right choice is (c) Velocity-defect law

The BEST EXPLANATION: Far away from the wall the fluid flow is retarded by the wall shear stress. The velocity (U) at such a point is DEFINED as

\(\FRAC{U_{max}-U}{u_\tau} =FUNCTION(\frac{y}{\delta})\)

This is called velocity-defect law.

Correct option is (d) Law of the wall

The explanation is: The law of the wall is the RELATIONSHIP between the mean FLOW velocity and the distance from the wall derived using dimensional ANALYSIS. This gives the relationship between TWO important dimensionless QUANTITIES u^+ and y^+.

The correct choice is (a) inversely PROPORTIONAL

Easy explanation: Wavenumber is the frequency of eddies and the SIZE will be RELATED to the wavelength. So, the size and wavenumber are inversely proportional. The large eddies have LOW wavenumber and the SMALL eddies have high wavenumbers.

The correct CHOICE is (a) κy

To ELABORATE: 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 CORRECT choice is (C) 0

To elaborate: 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.

Right option is (c) Anisotropic flow near the WALL

To explain: 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.

Right OPTION is (a) \(\FRAC{\sqrt{\overline{\vec{V}^{‘}.\vec{V}^{‘}}}}{\sqrt{\vec{V}.\vec{V}}}\)

Easiest explanation: The turbulence intensity which is used to get the K and ε-values for the inlet boundary conditions is

\(T_i=\frac{\sqrt{\overline{\vec{V}^{‘}.\vec{V}^{‘}}}}{\sqrt{\vec{V}.\vec{V}}}\)

Where,

\(\vec{V}\) → VELOCITY vector.

Right OPTION is (c) √k/ω

Explanation: The turbulence large-scale LENGTH is given in terms of

l=\(\frac{k^{3/2}}{\EPSILON}\)

l=\(\frac{k.k^{1/2}}{\epsilon}\)

l=\(\frac{k^{1/2}}{\OMEGA}\).

The correct option is (b) The effect of viscous stresses

Easiest EXPLANATION: The term div(2μ\(\vec{V}\)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.

Correct answer is (d) The two EXTRA transport EQUATIONS used in the model

The best I can explain: 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.

Right choice is (d) Reynolds number

Best explanation: The GRID SIZE and time-step size of the DNS method is BASED on the largest and the SMALLEST length and time SCALES of eddies in a turbulent flow. This, in turn, depends on the Reynolds number of the flow. So, Reynolds number decides the grid size and time-step size here.

Right answer is (a) transient 3-D EQUATIONS

Easiest explanation: The starting set of equations is TAKEN by the DNS system and the transient 3-D solution is done for a sufficiently fine spatial mesh and sufficiently SMALL time-step sizes to resolve even the smallest TURBULENT eddies.

Correct option is (d) the summation of their mean components

The best explanation: CONSIDER two flow variables which can be decomposed as a=A+a’ and B=B+b’. The mean of their summation means

a+b = A+a’+B+b’

But, a’ = 0 and b’ = 0. THEREFORE,

a+b = A+B

Also, A = A and B = B. Hence,

a+b = A+B.

The CORRECT choice is (a) The RATIO of velocity parallel to the wall to the friction velocity

Explanation: u^+ is the dimensionless velocity. It is DEFINED as the ratio of the velocity of fluid particle parallel to the wall at a particular distance from the wall to the friction velocity. Friction velocity is the square root of the ratio of shear stress to the density of fluid.

Correct answer is (c) Spalart-Allmaras

The explanation: Spalart- Allmaras is a model for solving turbulent FLOW especially invented for aerospace PROBLEMS. It GIVES good results for turbulent boundary layer MODELS. It solves the transport equation for turbulent kinematic VISCOSITY.

Right answer is (d) ν^5/4 ε^1/4

To ELABORATE: Kolmogorov argued that the behaviour of the smallest TURBULENT eddies DEPENDS on the rate of dissipation of turbulent energy. But, later studies revealed that only the spectral energy of the smallest turbulent eddies depends on the rate of dissipation of turbulent energy and the RELATIONSHIP is given by

spectral energy ∝ ν^5/4 ε^1/4.

Right answer is (b) Re^-1/3

To EXPLAIN I would SAY: Once the length and time-scale ratios are KNOWN, we can get the VELOCITIES scale ratios using these TWO. It is given by

Velocity-scale ratio = Re^-1/4.

Correct OPTION is (d) increased energy losses

For EXPLANATION I WOULD say: Small-scale eddy motions are dissipative. The energy associated with these motions is converted into thermal internal energy. This leads to increased energy losses associated with the TURBULENT flows.

Correct choice is (b) Time and length scales

To elaborate: High fluctuation in turbulent flows RESULT in a broad range of length and time scales. This MAKES the Direct Numerical SIMULATION (DNS) of turbulent flows difficult. It IMPOSES a limitation to the DNS method.

The correct CHOICE is (c) The product of the length scale and the average strain RATE of the resolved flow

The best I can explain: 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×\(\MID\overline{S}\mid \) .

Right choice is (c) SGS eddies

The EXPLANATION: 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.

Correct answer is (c) Velocity and k-value

Explanation: Near the wall boundary, the velocity of the flow is reduced by the FRICTION of the wall. So, velocity VANISHES. The variable k STANDS for turbulent kinetic energy. As the kinetic energy DEPENDS on the velocity, when velocity vanishes, k also will VANISH.

The CORRECT choice is (a) Reynolds stress calculation and the k-equation

Easiest EXPLANATION: For a Shear Stress TRANSPORT model, the k-equation and the calculation of the Reynolds stresses are the same as used in the STANDARD k-ω model. The equation for the transport of ε is transformed into the ω-equation by using the RELATIONSHIP ε=kω.

Right choice is (a) Non-linear two-equation turbulence models

Easy EXPLANATION: There is a division of the k-ε models which are non-linear. The transport equations are non-linear in these models. The REALIZABLE k-ε model ALSO COMES under this non-linear k-ε models.

Right OPTION is (b) Simple geometry and low Reynolds number

To explain I would say: The DNS method needs highly refined grids to capture the detailed flow. So, it is not suitable for complex geometries as it will MAKE the system more complicated. As the Reynolds number INCREASES, the number of grids and time-steps also will INCREASE. This will lead to higher complexity again.

Right option is (b) \(lim_{V→∞}\frac{⁡1}{V}\int_V \phi dV\)

The best EXPLANATION: Spatial AVERAGING represents the MEAN based on a particular space interval or volume. So, EQUATION \(lim_{V→∞}\frac{⁡1}{V}\int_V \phi dV\) represents spatial averaging.

Correct answer is (b) steady turbulent FLOWS

To elaborate: Time averaging METHOD is USEFUL when we have to decompose the turbulent flow variables into mean and fluctuating COMPONENTS based on time. They are particularly applicable for steady turbulent flows.

The correct option is (a) Reynolds averaging

Easy explanation: The averaging techniques include time averaging, Spatial averaging and ENSEMBLE averaging. These are collectively called the METHODS of averaging. They are USED to simplify the algebra without ACTUALLY DISTURBING the accuracy much.

Right CHOICE is (d) κ^-5/3ε^2/3

To explain: The spectral energy in terms of the WAVENUMBER and the rate of dissipation of turbulent energy is GIVEN by E(κ) ∝ κ^-5/3 ε^2/3. Where the proportionality CONSTANT is 1.5 (obtained experimentally)

The correct answer is (a) BOUSSINESQ hypothesis and Prandtl MIXING length model

To elaborate: 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.

Right option is (c) Log-law layer

For explanation: 30 < y^+ < 500 is the RANGE of y^+ where the value of U^+ varies LOGARITHMICALLY with the y^+ value. It is called the log-law layer. As the value 50 falls in this range, it BELONGS to the log-law layer.