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Correct OPTION is (d) cross-diffusion

To explain I would SAY: In general, the surface vector of the non-orthogonal GRIDS is given as the sum of the vector CONNECTING the owner and the neighbour elements and an additional vector. This leads to an extra term in the diffusion EQUATION called the cross-diffusion or non-orthogonal diffusion.

Correct answer is (c) Flexibility

For explanation I would say: The least-square method offers more flexibility over the other methods to compute gradients for CHOOSING between the order of accuracy USED and the STENCIL on which to WORK. ACCORDING to the user’s ease, these two can be chosen.

The correct CHOICE is (b) ∇ΦC=1/VC∑f~nb(c)Φf \(\vec{S_f}\)

Easy explanation: The FINAL FORM of the Green-Gauss gradient method is GIVEN by

∇ΦC=1/VC∑f~nb(c)Φf \(\vec{S_f}\)

Where,

\(\vec{S_f}\) is known from the geometry of the grids.

Φf for all the faces should be known to compute ∇ΦC.

Right answer is (d) aS=0; aW=0

Explanation: For the CONTROL volumes ADJACENT to the boundary of the global domain, the boundary-side coefficient is SET to zero. Therefore, for the CELL numbered “13”, the SOUTHERN and the western coefficients are zero (aS=0; aW=0).

Correct option is (b) the SYSTEM is linear

Best explanation: Unless the FIELD where we solve the PROBLEM is linear, the solution using the LEAST square method will not be exact. This is because the number of columns in the coefficient matrix will be more than the number of rows.

The CORRECT OPTION is (c) the coefficients of each discretized equation

The explanation: A proper discretization should RESULT in a discretized algebraic equation that reflects the characteristics of the original conservation equation. The coefficients of each discretized equation should satisfy the zero sum and the opposite SIGNS RULES.

The correct choice is (d) Mass of the cells

The best EXPLANATION: The impact decreases as the centroid of the cell moves AWAY from the vertex. So, the INVERSE is used as the WEIGHT here. This is why the method needs an enlarged stencil. But it leads to more accurate approximation.

Correct answer is (a) Fourier series

To EXPLAIN: For a source-less STEADY STATE diffusion problem, the transfer of the FLOW variable (Φ) occurs only by diffusion. So, the transfer of Φ will be in the direction opposite to the increasing Φ. This is GOVERNED by Fourier series.

The correct ANSWER is (c) Unphysical results

Explanation: Consider a PARABOLIC (second-order) PROFILE is used to find the value of a flow VARIABLE at the face in between two centroids. It will lead to a value HIGHER or lower than the values at the centroids which will be unphysical.

Right ANSWER is (a) 1-D steady-state diffusion is the simplest of all transport equations

Explanation: The one-dimensional steady-state diffusion is the simplest preliminary problem in CFD. This cancels out most of the TERMS in the general transport equation. In HEAT FLOW problems it means conduction and in mass-flow problems, it means diffusion.

Right CHOICE is (d) Computational cost

The BEST I can explain: The cost for the advantage of the least-square gradient METHODS is its high computational cost. This high computational cost is due to its separate CALCULATION of the WEIGHTED average.

Correct option is (a) aP and AF are of opposite signs

For explanation I WOULD say: The boundedness property is ENFORCED only when the coefficients aP and aF are of opposite signs. So, the opposite sign rule SAYS that the coefficients of the flow VARIABLES ΦP and ΦF are of opposite signs.

The correct choice is (c) The OPPOSITE SIGNS rule

Best explanation: For of the discretized equations to be bounded, the SUFFICIENT CONDITION is the opposite signs rule. Sufficient condition means that the rule will be ENOUGH to make sure that the equation is bounded.

Correct answer is (a) Gauss divergence theorem

To explain: The general diffusion term is div(Γ gradΦ). INTEGRATING for the finite volume METHOD, it becomes

∫CV div(Γ gradΦ)dV

Applying the Gauss divergence theorem,

∫A\(\vec{n}.\)(Γ gradΦ)dA

This is the boundary BASED integration as the boundaries will be AREAS.

Correct choice is (d) Jacobian matrices

The EXPLANATION is: COMPACT STENCIL uses implicit methods. So, it is easy to CONSTRUCT compact Jacobian matrices. But the large stencil brings more information into the reconstruction and THEREFORE is more accurate.

Right option is (c) \((S_f cos⁡\theta) \VEC{E}\)

The best explanation: Here, a right-angled TRIANGLE is formed by the vectors \(\vec{E_f}, \vec{S_f}\, and\, \vec{T_f}\). The \(\vec{T_f}\) vector is orthogonal to the \(\vec{E_f}\) vector in this CASE. The relation is given by \((\vec{e}.\vec{S_f})\vec{e}=(S_f cos⁡\theta)\vec{e}\).

Right OPTION is (c) ∑F~NB(P)\(\frac{a_F}{a_P}\) = -1

The best I can explain: From the zero SUM rule,

\(a_P+\sum_{(F \sim NB(P))}a_F = 0\)

Divided by aP, the equation BECOMES

\(1+\frac{\sum_{F \sim NB(P)}a_F}{a_P} = 0 \)

\(\frac{\sum_{F \sim NB(P)}a_F}{a_P} =-1\)

This can be written as

\(\sum_{F \sim NB(P)}\frac{a_F}{a_P} =-1\).

Correct OPTION is (d) the grid sizes in the x, y, z-directions

The best I can explain: The COEFFICIENT matrix obtained by APPLYING the least-square to CARTESIAN grids has non-zero elements only in the DIAGONAL. These non-zero elements are Δx, Δy and Δz (the grid sizes in the x, y, z-directions).

Right answer is (c) \(\FRAC{\Gamma_E A_E}{\DELTA x_{PE}}\)

Explanation: FLUX in the EASTERN DIRECTION is given by

\(\Gamma_E A_E\frac{\partial\phi}{\partial x}\Big|_e=\Gamma_E A_E\frac{(\phi _E-\phi _P)}{\delta x_{PE}}\)

\(\Gamma_E A_E\frac{\partial\phi}{\partial x}\Big|_e=\Gamma _e A_E\frac{\phi_E}{\delta x_{PE}}-\Gamma_E a_E\frac{\phi_P}{\delta x_{PE}}\)

Expanding this while forming the general equation, we will get

\(a_E=\frac{\Gamma_E A_E}{\delta x_{PE}}\).

The correct answer is (b) the boundary CONTROL volumes

The best explanation: There is SPECIAL attention needed at the boundary NODES. For control volumes that are adjacent to the domain BOUNDARIES, the equation is MODIFIED so that the boundary values are incorporated into the equation without any problem.

The correct answer is (B) \(\vec{E_f}=(\frac{S_f}{COS ⁡\theta}) \vec{E}\)

Explanation: The relationship is given by \((\frac{S_f}{cos ⁡θ})\vec{e}\). This is CALCULATED in CFD packages as \(\vec{E_f}=\frac{\vec{S_f}.\vec{S_f}}{\vec{e}.\vec{S_f}}\vec{e}\). The derivation is given as

\(\vec{E_f}=(\frac{S_f}{cos⁡\theta})\vec{e}=(\frac{S_f^2}{S_f cos⁡\theta})\vec{e} =\frac{\vec{S_f}.\vec{S_f}}{\vec{e}.\vec{S_f}}\vec{e}\).

Right CHOICE is (c) When the value of ΦF is increased, the value of ΦP DECREASES

Explanation: The value of ΦP varies with the VARIATION of the values of ΦF. So, it will not remain the same. When the value of ΦF is increased, the value of ΦP decreases. This is the statement of the OPPOSITE signs rule taken PHYSICALLY.

Right choice is (a) Linear profile

To explain: The major approximation made while discretizing a GOVERNING equation is “variation of the FLOW VARIABLE is considered to be linear”. The linear interpolation METHOD is used to get the unknown values near the KNOWN values.

Right option is (d) increases, increases

The EXPLANATION: The importance of the TERM INVOLVING ΦF and ΦC DECREASES when the non-orthogonality decreases for the minimum correction APPROACH. For the over-relaxed approach, the importance increases.

The correct choice is (B) aP+∑F~NB(P)aF = 0

The EXPLANATION is: A consistent discretization METHOD should YIELD an equation which incorporates the PROPERTY of the overall domain – conservation. To ensure this, the equation should satisfy the condition aP+∑F~NB(P)aF = 0.

Correct choice is (a) at LEAST first-order accurate

For EXPLANATION: The ACCURACY of the gradient found using this method is at least first-order. This can be proved by expanding the values of the flow variable using the Taylor series. It can be of HIGHER orders ALSO.

Right choice is (C) the direction of the surface vector

Best explanation: The surface vector and the vector joining the owner and the neighbouring elements are not COLLINEAR for non-orthogonal grids. Thus, non-orthogonal grids need SPECIAL attention in the steady-state diffusion equation.

Correct OPTION is (d) aW

For EXPLANATION I would say: For the left boundary node, there is no WESTERN node present. It has only one neighbour on the eastern side. So, the western node coefficient aW is set to zero. ΦW is not zero as we do not know the flow variable at that POINT and cannot ASSUME.

Correct option is (d) only when the SOURCE term is zero

For explanation: When there is a source term, the linear profile of the flow variable MAY fail. The source or sink term may lead to increased or decreased values than that guesses by the linear INTERPOLATION. So, in this case, boundedness is not POSSIBLE.

Correct choice is (C) inverse of the distance between the vertex and the CENTROID of the cells

Easy explanation: Generally, the weight used is based on the DISTANCES. We can choose it to be the inverse of the distance between the vertex and the cell CENTROIDS. Mostly, the weight is a FUNCTION of the distance between the vertex and the cell centroids.

Right choice is (a) \(\int_A(\Gamma A\frac{\PARTIAL \phi}{\partial x})dA+\int_A(\Gamma A \frac{\partial\phi}{\partial y})dA+\int_{\Delta V} S\,dV=0\)

Easy explanation: The general governing equation for a 2-D steady-state diffusion PROBLEM is given by

\(\frac{\partial}{\partial x}(\Gamma\frac{\partial \phi}{\partial x})+\frac{\partial}{\partial y}(\Gamma\frac{\partial \phi}{\partial y})+S=0\)

Here, partial differentiation is USED as the variable φvaries in both x and y directions, but the differentiation is only in the required direction.

Integrating the equation with respect to the control VOLUME,

\(\int_{\delta V}\frac{\partial}{\partial x}(\Gamma\frac{\partial\phi}{\partial x})dV+\int_{\delta V}\frac{\partial}{\partial y}(\Gamma\frac{\partial \phi}{\partial y})dV+\int_{\Delta V} S \,dV=0\)

Applying Gauss Divergence theorem,

\(\int_A(\Gamma A\frac{\partial\phi}{\partial x})dA+\int_A(\Gamma A\frac{\partial\phi}{\partial y})dA+\int_{\Delta V}S \,dV=0\)

This is the semi-discretized form of the equation.

Correct choice is (a) Deferred CORRECTION

To explain: The cross-diffusion term cannot be expressed in TERMS of nodal values. So, deferred correction is used here. Its value is computed USING the current GRADIENT field and this is added as a source term in the right-hand side of the algebraic equation.

The correct answer is (a) The non-ORTHOGONAL correction is kept as small as POSSIBLE

The best explanation: In the minimum correction approach, the decomposition of the surface vector is DONE in a WAY that the non-orthogonal correction is as small as possible, thus making the \(\vec{E_f} \,and\, the\, \vec{T_f}\) orthogonal.

The correct answer is (a) \(\frac{d}{dx}(k\frac{dT}{dx})\)

For explanation I would say: The GENERAL one-dimensional steady-state diffusion EQUATION is:

\(\frac{d}{dx}(\frac{\Gamma d\phi}{dx})+S=0 \)

For heat conduction problem, the diffusion constant is the heat conductivity (Γ=k) and the flow VARIABLE is temperature (Φ=T). As the given problem is source free, S=0. Therefore, the equation becomes

\(\frac{d}{dx}(k\frac{dT}{dx})=0\).

The CORRECT option is (b) \(\frac{d}{dx}(\Gamma\frac{d\phi}{dx})+S=0\)

Best explanation: The TERM div(Γ GRAD Φ) REPRESENTS diffusion in all three directions. One-dimensional diffusion is given by the equation \(\frac{d}{dx}(\Gamma\frac{d\phi}{dx}.\frac{d\phi}{dt})\) is the TRANSIENT term. So, this should not be present in the steady-state equation. Considering all these, the correct equation is

\(\frac{d}{dx}(\Gamma\frac{d\phi}{dx}+S)=0\).

Right answer is (C) The diffusive flux of Φ leaving the exit face minus the diffusive flux of Φ entering the inlet face is EQUAL to the generation of Φ

For EXPLANATION I WOULD SAY: The diffusive flux of Φ leaving the exit face minus the diffusive flux of Φ entering the inlet face is equal to the generation of Φ. It constitutes the balance equation over the control volume. This ensures conservation.