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The correct answer is (C) known VALUES

To elaborate: In SEQUENTIAL methods of solving coupled equations, the VARIABLE for which the system is solved is treated as unknown. All other VARIABLES are treated as known values with some approximations.

Correct answer is (B) correction

Easiest explanation: Corrections are TRANSFERRED from a coarse GRID to a finer one. Correction is basically obtained from the SOLUTION of the system of equations at the coarse grid. This, in terms of ratios of errors, is transferred to the finer grid.

Right choice is (a) iterative solvers

Explanation: When the iterative solvers are USED to solve medium to LARGE SYSTEM of equations, the error is big and accuracy becomes less. This poses a problem with iterative solvers. So, they are SUPPLEMENTED with the multi-grid approach.

Right choice is (c) MOMENTUM equation

Best EXPLANATION: If grid-oriented velocity components are used, non-conservative source or sink terms will appear in Momentum equations. This will affect the conservation of Momentum equation. EULER equations is a SET including all THREE – continuity, momentum, energy equations.

Right answer is (C) Finite Volume Method and Finite Element Method

Easy EXPLANATION: A hybrid method integrating Finite Volume Method and Finite Element Method called the Control Volume based Finite Element Method (CV-FEM) is also used for solving PDEs. In this, shape FUNCTIONS are used to DESCRIBE the variation of the variables over an element.

The CORRECT ANSWER is (a) Face

To explain: A one-to-one RELATIONSHIP is given by the faces of a mesh. One face is SHARED by two elements. It usually contains information about the FLUXES flowing between those elements.

Correct choice is (d) SUMMATION of the residuals in the (k-1)^th level

Easy explanation: While transferring the errors from ONE level to ANOTHER, the RESIDUAL of the k^th level is given by the summation of the residuals of all the terms in the previous (k-1)^th level.

Correct answer is (c) Block-structured grids with overlapping blocks

To explain: Block-structured grids with overlapping blocks are called COMPOSITE or CHIMERA grids. The DISADVANTAGE of these grids is that conservation is not ensured in BOUNDARIES. This is helpful to follow moving BODIES.

Right answer is (a) \(\frac{r^{k+1}}{r^k} \)

The best I can explain: The first STEP in the multi-GRID approach is the restriction step. Here, the process starts with a FINE grid. After a few iterations, the error is transferred to a coarser grid level. Again, some iterations are performed in that step and the process continues. The restriction operator for the error transferred from one step to the higher step is \(\frac{r^{k+1}}{r^k} \).

The CORRECT choice is (a) LU decomposition

Best explanation: The EXTRA non-zero elements in the factorized MATRICES are called fill-ins. In ILU(p), p INDICATES the order of fill-in allowed. So, where there is no fill-in, the PATTERN of combined L and U matrices and the coefficient matrix will be the same.

Correct OPTION is (c) \(\frac{\rho v_r v_\theta}{R}\)

To ELABORATE: The term \(\frac{\rho v_r v_\theta}{r}\) REPRESENTS Coriolis force in the θ – momentum equation. Coriolis force arises due to the motion of the coordinate system taken. The term \(\frac{\rho v_r v_\theta}{r}\) represents the source or SINK of θ -momentum.

Correct option is (a) Cylindrical coordinates

To explain I would say: In the case of flow over a pipe, cylindrical coordinates do the best as the dependent spatial VARIABLES are just 2 which would be 3 if it is Cartesian coordinates. But, in this case, a swirl COMPONENT is included which makes the flow dependent on three dimensions anyway. To keep the complexities less, Cartesian coordinates should be chosen for this problem.

The correct ANSWER is (c) post-processing

For explanation I would SAY: VERTEX connectivity is important for post-processing especially while computing GRADIENTS. Vertex connectivity generally contains information like the elements and faces SHARING that vertex.

Right OPTION is (a) Newton’s method

Best explanation: Newton Raphson is the most widely USED method for solving a non-linear system of equations. It is preferred in most of the CASES as the RATE of CONVERGENCE is more. It converges fast.

The CORRECT answer is (b) LUΦ=b

Best explanation: For the LU decomposition METHOD,

A=LU

Where L and U stand for Lower and Upper triangular matrices respectively.

Substituting in the global matrix,

LUΦ=b.

The CORRECT answer is (d) Secant method

The explanation is: An alternative to Newton’s method is the Secant method. This is much slower than Newton’s method. However, when the DERIVATIVE of the FUNCTION cannot be evaluated, this method is chosen as it does not involve any derivative.

The CORRECT answer is (C) Taylor series

Explanation: Newton’s METHOD USES the first two TERMS of Taylor’s series to linearize the non-linear system. This is further simplified to get the formula to be iterated and get the roots.

Correct ANSWER is (c) Mathematical model of the flow

Explanation: The FIRST step of any numerical solution of a fluid PROBLEM is converting the PHYSICAL flow into a mathematical model. Physical model of the flow is what we have to solve. After generating the mathematical model only steps like discretization and ITERATIVE solution follows.

Correct answer is (d) METRIC TENSORS

To explain: In general, STRESS components are expressed in the CARTESIAN tensors FORM. It is also possible to define them in covariant or contra-variant terms. But they cannot be expressed in metric tensor. Metric tensors are used to transform covariant tensors into contra-variant tensors.

Correct choice is (d) skewness

Explanation: The skewness of a cell is its variation from the OPTIMAL cell size. This is an apt INDICATOR of the quality and suitability of a mesh. Large skewness leads to less ACCURACY.

The correct option is (b) Density

To explain: Covariant or contra-variant bases can be USED only with vectors or TENSORS. They cannot be used with scalars. Here, Velocities STRESS and FORCES all come under either vectors or tensors. Density is a scalar property. So, it cannot be REPRESENTED using covariant or contra-variant bases.

Correct CHOICE is (C) around one

To explain I WOULD say: Ideally, the aspect ratio of each element should be equal to 1. But, this cannot be practically ensured. So, in real, the elements have their aspect ratio around 1. If it is LARGE, it will lead to errors.

Right answer is (a) LINEAR and highly coupled

To explain I would say: There are two ways to SOLVE coupled EQUATIONS – simultaneous and sequential. In the simultaneous methods, equations are solved together for the unknowns. The sequential methods are USED to solve a highly coupled system with linear equations.

Right ANSWER is (c) U-cycle

The explanation: There are three possible TRAVERSAL cycles for ALGEBRAIC multi-grid APPROACHES. They are W-cycle, V-cycle and F-cycle. V-cycle is a direct traverse without any nesting. W-cycle involves nesting. F-cycle is a hybrid cycle of W and V-cycles.

Correct CHOICE is (a) has better convergence and NEEDS less memory

To explain: Gauss-Seidel METHOD USES the latest values at a particular iteration. So, it has better convergence. The same way, as there is no need for STORING the values of previous iterations, they require less memory too.

Correct option is (d) AΦ^i-b

To explain: While solving the system AΦ=b, AΦ-b should be equal to ZERO. Since, the iterative METHOD will not produce such an answer, the DECISION when to stop the iteration RELIES upon a tolerance value. When the residual AΦ^i-b becomes less than the tolerance, iterations are stopped.

Correct choice is (b) COLLOCATED arrangement

Easy EXPLANATION: The collocated arrangement is the SIMPLEST as all the variables share the same control volume. But this requires more INTERPOLATION. When the grid is non-orthogonal, the collocated arrangement is the best as the other ARRANGEMENTS are more difficult.

The correct option is (B) conservation properties of the EQUATION

Explanation: In the Finite Difference Method, while CHANGING from Cartesian to non-orthogonal grids, the terms change and the NUMBER of terms increases. But, the conservation properties of the equation remain the same.

Right CHOICE is (a) is from the owner ELEMENT to the NEIGHBOUR element

To elaborate: The orientation of the FACES is such that the normal vector to the FACE points from the owner element to the neighbour element. Depending on this the sign of flux term changes. Orientation does not depend on fluxes.

The correct answer is (B) cN-1=0

Easiest EXPLANATION: By analysing the equation for PDMA, for the first TWO EQUATIONS,

d1=e1=e2=0

Similarly, for the last two equations,

bN-1=bN=cN=0.

The CORRECT choice is (d) viscosity and source terms

The best explanation: Coupling problem between velocities and pressure gradients is the reason why staggered grids are formed. Staggered grids ensure the coupling between these TWO terms. The VELOCITY components normal to the cell face should LIE between the pressure COORDINATES on either side of that face.

The correct option is (a) qΦ=B0+b1 Φ

Best EXPLANATION: Picard iteration is used with the source TERM to decompose and linearize it. It decomposes to qΦ=b0+b1 Φ. The term b0 is absorbed by the RHS of the system. The term b1 Φ is added to the coefficient matrix.

Right ANSWER is (a) Centrifugal force

For explanation: When the momentum equation is expressed in CYLINDRICAL terms, due to the transformation from Cartesian coordinates to polar coordinates, a centrifugal term arises. This is the term \(\frac{\rho V \theta^2}{r}\). It describes the transfer of θ-momentum into r-momentum due to the change of DIRECTION of angular velocity.

The correct choice is (b) SUITABLE for SIMPLE geometries

To explain I WOULD say: One of the major disadvantages of STRUCTURED grids is that they are not suitable for complex geometries. Those can be modelled using UNSTRUCTURED grids only. The other disadvantage of structured grids is distribution.

Correct option is (d) GRID is very fine

Explanation: When the grid size is very SMALL, whatever the TYPE of discretization METHOD is, the results will be the same. As very fine grids are not practically acceptable, we choose a particular type of discretization method which will be the best fit for the problem.

The correct choice is (B) Incomplete

For explanation I would say: ILU MEANS Incomplete LU DECOMPOSITION method. This does incomplete factorization of the coefficient MATRIX into upper and lower triangular matrices. The L and U matrices have the same number of non-zero elements as in the lower and upper parts of A.

Right OPTION is (c) Not suitable for sparse matrices

Easiest explanation: Iterative methods are chosen when the coefficient matrix is sparse (less non-zero ELEMENTS). They need less storage and less computational COST. They start the SOLUTION from an initial guess and proceed to find the answer from this initial guess.

The correct choice is (b) speed and SECURITY

Explanation: In SOLVING the non-linear system, there are two methods – Newton’s method and GLOBAL method. Newton’s method is faster and the Global method is guaranteed not to DIVERGE. So, there is always a trade-off between speed and security.

The CORRECT option is (d) Iterative solver

Best explanation: For each inner iteration, ONE variable is unknown and all other variables are TREATED as known values. It is ineffective to solve this SYSTEM accurately for one unknown. So, the iterative solvers are preferred to direct solvers in this CASE.

Correct option is (d) Structured grid problems

Explanation: When a structured grid is used for discretization, it RESULTS in a coefficient matrix with its non-zero ELEMENTS aligning along a few diagonals. The NUMBER of non-zero diagonals DEPENDS on the discretization stencil and the dimension of the problem. So, TDMA and PDMA are suitable to solve this kind of banded matrix.