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The correct answer is (c) Three terms

The explanation is: 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.

The correct ANSWER is (d) an extra source TERM

To explain: While transforming the ε-equation into the ω-equation, an extra source term ARISES. It is called the cross-diffusion term. This cross diffusion term is modified using external BLENDING functions.

Correct choice is (C) RANS model

The best explanation: The governing Navier-Stokes equation is averaged using the REYNOLDS AVERAGING techniques and these averaged equations are used in the RANS method. This is the reason why the technique is named the Reynolds-Averaged Navier-Stokes equations method.

Correct choice is (d) the turbulent kinetic energy

To explain I would say: 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.

Correct OPTION is (d) the cube root of the grid CELL volume

Easy explanation: If Δx, Δy and Δz are the grid SIZES in the x, y and z-directions, the cut-off WIDTH should be the cube root of the cell volume (\(\SQRT[3]{\Delta x \Delta y \Delta z}\)). This is because the cut-off width is in a single direction.

Correct option is (a) Reynolds stresses

Explanation: The terms ACCOUNTING for turbulence stresses are \(div(\RHO \vec{V} \overline{u_{i}^{‘} u_{J}^{‘}})\) and \(\rho \overline{u_{i}^{‘} u_{j}^{‘}}.div(ρ\vec{V} \overline{u_{i}^{‘} u_{j}^{‘}})\) represents the turbulent TRANSPORT of kinetic energy by means of Reynolds stresses. \(\rho \overline{u_{i}^{‘} u_{j}^{‘}}\) represents the net decrease of kinetic energy due to deformation work by Reynolds stresses. Both of these terms contain the Reynolds stress term \(\rho \overline{u_{i}^{‘} u_{j}^{‘}}\).

The correct OPTION is (d) ω=\(\frac{k^{1/2}}{l}\)

To explain: The ε and the ω-values should be obtained from k-value in the k-ε and k-ω models. The formulae used to get these values are

ε=\(C_μ\frac{k^{\frac{3}{2}}}{l}\)

ω=\(\frac{k^{\frac{1}{2}}}{l}\)

Where,

l → TURBULENT length scale.

Cμ → A dimensionless constant.

Right OPTION is (c) 1 to 10%

EXPLANATION: The value of turbulence intensity lies between 1% and 10%. The values below 1% are CONSIDERED to be very LESS and the values above 10% are considered to be a high one.

Right option is (d) 11.63

Explanation: Though the layer varies from viscous sub-layer to buffer layer when y^+ CROSSES the value 5, the VARIATION is STILL linear. This linear variation becomes logarithmic variation when y^+ crosses 11.63 to be exact.

Correct choice is (b) the eddy VISCOSITY

To explain I would say: There are two limiters used in the Shear Stress TRANSPORT MODEL. One of these is on the eddy viscosity. This is done to improve the performance of the model when there are adverse pressure GRADIENTS or with wakes. These are the places where the k-ε model fails.

Right OPTION is (d) the realizability constraint without viscoelastic analogy

The explanation is: The non-linear k-ε models were INITIALLY developed based on the analogy between the viscoelastic FLUIDS and the turbulent flows. This analogy is not used by the realizable k-ε MODEL. The realizability constraint rules out this analogy.

Right option is (c) RATE of deformation and turbulent stresses

Explanation: 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.

The correct option is (d) TURBULENT boundary layers with adverse pressure gradients

BEST explanation: 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.

Correct ANSWER is (c) the SPATIAL partial DERIVATIVE of the mean component

Explanation: The mean of the FLOW variable will be EQUAL to the mean variable. The mean of the flow variable’s spatial partial derivative will be equal to the spatial partial derivative of the mean component of that variable.

The correct answer is (d) Law of the wake

The explanation: Velocity defect law is APPLICABLE to the LAYER far away from the WALL. This layer has less viscous effects and inertia forces are dominating here. The velocity-defect law is otherwise CALLED the law of the wake.

The correct option is (c) Buffer layer

Explanation: The layer above the linear sub-layer has EQUALLY important TURBULENT and viscous stresses. Neither of these is dominating nor inconsiderable. A layer in this AREA where both the viscous and turbulent stresses are of equal MAGNITUDE is CALLED the buffer layer.

Right answer is (a) 0.2 to 3.5

Easy explanation: According to the experiments, the turbulent Schmidt number RANGES from 0.2 to 3.5. Only when using the REYNOLDS analogy, the APPROXIMATION becomes a value near UNITY. OTHERWISE, this value varies.

The correct choice is (d) equal to the PRODUCT of kinematic turbulent VISCOSITY and DENSITY of the fluid

Explanation: Dynamic viscosity is the product of kinematic viscosity and density in general. This applies to the turbulent VISCOSITIES also.

μt=ρνt

Where,

νt →Kinematic turbulent viscosity.

μt →Dynamic turbulent viscosity.

ρ →Density.

Right option is (b) INERTIA force and viscous force

Easy EXPLANATION: REYNOLDS number DECIDES whether the flow is laminar or TURBULENT. Reynolds number of flow gives the relative importance of inertia force (convective forces) and viscous forces.

The CORRECT answer is (a) obtained from TURBULENCE intensity

Explanation: At the inlet BOUNDARY conditions, the k and ε values must be specified for the k-ε model. In most of the industrial CFD applications, these values will not be known. So, they are obtained from the turbulence intensity and the characteristic length of the model.

Correct answer is (C) y^+=(y ut)/ν

The explanation: y^+ is the ratio of the product of the distance from the WALL boundary and the SHEAR VELOCITY to the kinematic VISCOSITY. This is given by the equation y^+=y ut/ν. This is why the value of y^+ increases with the distance from the wall.

The correct option is (c) small-SCALE turbulence

The best I can explain: The statistical mechanics APPROACH has led to NEW mathematical formalizations. A LIMITED number of assumptions regarding the statistics of small scale turbulence is made in this RNG k-ε MODEL.

Right choice is (b) \(-\rho \overline{u_{i}^{‘} u_{j}^{‘}} = \rho \overline{v} f_{v1} (\frac{\PARTIAL U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i})\)

Easy explanation: The Reynolds STRESS TERM is GIVEN as

\(-\rho \overline{u_{i}^{‘} u_{j}^{‘}} = \rho_t (\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i})\)

Converting to Spalart-Allmaras terms,

\(-\rho \overline{u_{i}^{‘} u_{j}^{‘}} = \rho \overline{v} f_{v1} (\frac{\partial U_i}{\partial x_j}+\frac{\partial U_j}{\partial x_i})\) .

The correct answer is (c) turbulent Prandtl/Schmidt number

To explain I would say: Mixing LENGTH model uses the turbulent viscosity coefficients. If a RELATIONSHIP can be established between the turbulent viscosity and turbulent DIFFUSIVITY, the model can be USED for turbulent SCALAR fluxes. This relationship is established by the Turbulent Prandtl/Schmidt number.

Right OPTION is (d) 0.07 of layer width

The explanation: Mixing length varies for DIFFERENT turbulent flows. For free turbulent flow of the mixing layer type, the mixing length is 0.07 times of the layer width. Mixing layer turbulent flow occurs due to the interaction of two flows with VARIOUS VELOCITIES.

The CORRECT answer is (c) turbulence

For explanation: The mixing LENGTH model defines the Reynolds stresses in terms of velocity GRADIENTS, mixing length and density of the fluid. Turbulence is a function of the flow. So, if the turbulence changes, the Reynolds stresses should change. This change is accounted by CHANGING the mixing length.

Right answer is (d) \((\FRAC{\gamma}{\PI \Delta^2})^{\frac{3}{2}} exp(-\gamma \frac{\big|\vec x,\vec{x’}\big|^2}{\Delta^2})\)

Easiest explanation: The filter function, in general, is a function of the spatial vector (\(\vec{x}\)), its derivative (\(\vec{x^{‘}}\)) and the cut-off width (Δ). The GAUSSIAN filter has an additional parameter (γ). The typical value of γ is 6. The function is given as

\(G(\vec{x},\vec{x’},\Delta)=(\frac{\gamma}{\pi \Delta^2})^{\frac{3}{2}} exp(-\gamma \frac{\big|\vec x,\vec{x’}\big|^2}{\Delta^2})\)

The correct ANSWER is (d) 10^9

The explanation: The ratio of the largest to the smallest LENGTH scale gives the number of grid points needed. The number of grid points needed in each direction is 10^3. The TOTAL number of grid points needed in all three DIRECTIONS (as the problem is 3-D) is 10^9.

The CORRECT answer is (B) \(\frac{m^3}{s^2}\)

To explain I would say: Spectral energy is the KINETIC energy per unit mass per unit WAVENUMBER. So, the unit of spectral energy is given by

\(\frac{kg m^2}{s^2}×\frac{1}{kg}×m=\frac{m^3}{s^2}\).

Right option is (d) SQUARE root of the RATIO of WALL shear STRESS to density

The explanation: Shear velocity is USED while defining the y^+. It otherwise called the friction velocity. It is the square root of the ratio of wall shear stress to density. It has the same unit as that of normal velocity.

Correct option is (a) In the near-WALL REGION, the k-ε model is transformed into k-ω model

The explanation is: The Shear Stress Transport model USES a TRANSFORMATION of the k-ε model into a k-ω model in the near-wall region. In the region far from the wall, it uses the standard k-ε model as it gives satisfactory results there.

Correct OPTION is (a) free stream

To elaborate: The k-ω model is sensitive to the free stream SPECIFIED values. The value of ω in the free stream is zero. But, if this is SET to zero, the eddy viscosity becomes infinity or indeterminate. So, a small non-zero value is specified and the whole problem becomes DEPENDENT on this non-zero value. The k-ε model does not have this problem.

Correct choice is (b) Cauchy-Schwarz inequality

The explanation is: Other than the non-negativity CONDITION, a REALIZABLE MODEL should also satisfy the Cauchy-Schwarz inequality. According to this, the term \((\overline{u_{i}^{‘} u_{j}^{‘}})^2 ≤ \overline{u_{i}^{‘2} u_{j}^{‘2}}\). This inequality becomes IMPORTANT as the realizable k-ε model is non-linear.

The correct option is (b) velocity scale with mean flow PROPERTIES

Easiest explanation: The LARGE eddies DIRECTLY interact with the mean flow properties and extract energy from them. So, there is a STRONG connection between the mean flow properties and the behaviour of the large eddies. So, the velocity scale is LINKED with the mean flow properties.

Correct option is (b) Top-hat filter

The best I can EXPLAIN: The top-hat or box filter is the one which is preferred for the finite volume methods in CFD PACKAGES. This is because of their simple ELIMINATION of SMALL eddies. The Gaussian and spectral cut-off filter are USED for research purposes.

The correct option is (a) homogeneous turbulent flows

The best I can explain: Spatial averaging finds the average of a quantity BASED on a spatial interval or volume. It is suitable for homogeneous turbulent flows. In homogeneous flows, the PROPERTIES are invariant under the arbitrary TRANSLATION of the coordinate AXES.

The correct answer is (b) Linear sub-layer

To EXPLAIN I would say: In the FLUID layer which is in contact with a smooth WALL, the value of DIMENSIONLESS velocity and dimensionless cross-stream distance tend to be the same. Because of this linear RELATIONSHIP, the layer is named linear sub-layer.