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The correct answer is (a) only for iterative solvers

The best explanation: Only for iterative solvers, STABILITY can be defined. It describes how well the EQUATIONS can be SOLVED using iterative solvers. It needs the solver not to diverge the solution from the exact ANSWERS.

Correct option is (c) Viscous flow over a flat plate

To ELABORATE: A stretched grid has less grid spacing in one side of the domain and stretched in the other side. For viscous flow over a flat plate, the BOUNDARY layer will have much variations of the flow PROPERTIES than the FAR regions. So, for modelling this, stretched grids can be used.

The correct answer is (c) truncation error

For explanation I would SAY: Truncation error ARISES when partial differential equations are APPROXIMATED. Usually, the partial differential equations are approximated using a SERIES expansion of infinite terms. The higher order terms are cut-off while approximating this series. So, it is called truncation error.

The correct choice is (a) Grids generated for a particular zone of the domain of interest

The explanation: While DEALING with a complex PROBLEM, grids are generated separately for DIFFERENT ZONES which are called zonal grids. These are then attached together to FORM grids for the whole domain.

The correct OPTION is (d) Truncation error is proportional to K^th power

The BEST I can explain: When a discretized equation is said to be of the ORDER ‘k’, it means that the last term has (k-1)^th power and the FIRST term of the truncated equation has k^th power. For example, in spatial derivative, the first term of the truncated equation will be proportional to (∆x)^k.

The correct answer is (d) Fourier series

Explanation: Von Neumann stability analysis is otherwise called as Fourier stability analysis. It is a TECHNIQUE used to ANALYSE the stability of LINEAR PARTIAL differential equations. It is named as Fourier stability analysis as it is based on Fourier decomposition of numerical error.

Correct option is (b) both the upstream and downstream nodes equally

To explain: When the Peclet number (CONVECTION) is zero, FLOW direction does not affect the value of flow PROPERTY. Both the upstream and the downstream nodes have equal INFLUENCE on the value at the CURRENT node.

Correct choice is (c) Boundary-fitted grids

The best I can EXPLAIN: The shape of the AIRFOIL is complex. So, generating a grid that FITS the boundary of this airfoil is best. STRETCHED grids are for flat surfaces. Adaptive grids are needed if the flow gradients are not known to the user. Elliptic grids are USEFUL when the transformation is elliptic.

Right choice is (a) flow field gradients

Explanation: Adaptive grids are the one which can AUTOMATICALLY ADAPT their ARRANGEMENT based on the needs of the PROBLEM. This is based on the flow-field gradients. High flow-field gradients INDICATE a sudden variation of properties based on spatial coordinates. So, more grids are automatically generated there.

The correct option is (a) Round-off errors

To EXPLAIN: Round-off errors are introduced because of the round-off RESULTS produced by COMPUTERS for a particular PROBLEM. When these round-off VALUES are used for further calculations, they become significant after a certain time.

The correct choice is (a) the value of flow PROPERTIES at a node is BOUNDED by its boundary values

For explanation: The flow property is said to be bounded if the internal nodal values of the flow property do not cross the minimum and maximum values of the flow properties in the boundaries. Physically the flow properties will not GO beyond the boundary values. This should be guaranteed by the numerical approximations ALSO.

Correct answer is (B) Grids transformed with elliptic EQUATIONS

The BEST explanation: The one-to-one transformation of the coordinates in the physical domain to the coordinates in the computational domain is done USING elliptic equations. It has nothing to do with elliptic nature of the GOVERNING equations.

Right choice is (d) Accuracy and efficiency

The explanation: By automatically generating the grid points, the PLACES without much variation are given FEWER grids and the places with high variation are given more grids. This helps in MAINTAINING a BALANCE between efficiency of the SOLUTION and accuracy of the answers.

Right choice is (c) in the presence of source term

For EXPLANATION I would say: When there is a source or sink INSIDE the domain of interest, the FLOW properties MAY increase or decrease drastically. This may lead to a value which does not lie in the range of the boundary VALUES. So, a system with a source or sink will not be bounded.

Correct OPTION is (a) have the same sign

For explanation: All the coefficients of the discretized FORM of the governing equations should have the same sign for the system to be BOUNDED. In most of the cases, the signs should be all positive. This means that an increase in the flow VARIABLE at one node increases the variables at the neighbouring nodes too.

Right choice is (c) elliptic

Explanation: When the Peclet number is large, CONVECTION DOMINATES the flow of a PROPERTY. Here, the ISOLINES will be elliptic with the CURRENT node at its focus.

Right answer is (a) ξ=x; η=ln⁡(y+1)

To explain: The horizontal coordinate lines are stretched means that the GRIDS are stretched in the y-direction. COORDINATES are stretched when logarithmically SPACED. So, to GET vertically stretched grid, η=ln⁡(y+1). Horizontal coordinates are UNDISTURBED.

Correct option is (C) Central differencing scheme

To explain I would SAY: When the Peclet number is LOW, the central differencing scheme can be USED. This is because, in this case, the flow VARIABLE is affected by both of the neighbouring nodes equally.

Right answer is (c) QUICK scheme

Explanation: Higher ORDER SCHEMES result in unbounded solutions. This means that the solution is erroneous. This happens only when the GRID is so coarse. The problem of boundedness can be OVERCOME by refining the GRIDS. The QUICK scheme is a higher order scheme.

Correct option is (C) \(\frac{\epsilon_{n+1}}{\epsilon_n} = 1.25\)

Explanation: For a SYSTEM to be stable, the error should be decreasing or at least maintained.

Representing this MATHEMATICALLY,

εn+1 ≤ εn

\(\frac{\epsilon_{n+1}}{\epsilon_n} ≤ 1\)

For an unstable system,

\(\frac{\epsilon_{n+1}}{\epsilon_n} > 1\)

This happens in \(\frac{\epsilon_{n+1}}{\epsilon_n} = 1.25\). So, this system is unstable.

Correct choice is (a) Error is amplified in increasing iterations

To ELABORATE: A system is SAID to be unstable if the error increases with the increasing iterations. So, for a system to be STABLE, the error should decrease or at least should be MAINTAINED in further iterations.

The CORRECT answer is (a) unstable

The BEST I can explain: Transportiveness must be taken care of while choosing the DISCRETIZATION scheme. If not, solutions will have unphysical oscillations and BECOME unstable.

Correct option is (c) QUICK

Easy explanation: QUICK SCHEME is one of the higher-order schemes involving QUADRATIC interpolation. While the other quadratic interpolation schemes give rise to CONSERVATION problems, the QUICK scheme ensures the conservation of the quantities. But, it has boundedness problems.

Right option is (d) Diagonal dominance

The explanation is: A system is bounded if the SCARBOROUGH criterion is satisfied. This needs the COEFFICIENT matrix to be diagonally dominant. This is a desirable feature for BOUNDEDNESS of solutions. This depends UPON the coefficients of the neighbouring nodes and the CENTRAL nodes.

The correct option is (d) Peclet number

Easiest explanation: Peclet number defines the TRANSPORTIVENESS of a FLUID flow property. The Peclet number has a direct impact on the ISOLINES of a flow property around a specific NODE.

Right option is (d) First order schemes

The EXPLANATION is: Irrespective of COARSE or fine grids first order schemes always guarantee boundedness without any OVERSHOOTS or UNDERSHOOTS. So, while choosing a coarse grid, it is better to use the first order schemes to get rid of errors.

Correct choice is (c) \(\frac{\PARTIAL\xi}{\partial t}≠0 \)

To EXPLAIN I would SAY: Adaptive grids change with varying time. So, the time rate of change of coordinates will never be equal to zero. This is GIVEN by\(\frac{\partial\xi}{\partial x}≠0 \). They may or may not vary from the PHYSICAL coordinates.

The correct answer is (b) CONVECTIVE, diffusive

Explanation: Peclet number defines how much convection FLUX DOMINATES the DIFFUSION flux of fluid flow. It will decide if the TRANSPORT of flow property is because of convection or diffusion.

\(Peclet\, number=\frac{Convective\, strength}{Diffusive\, strength}\).

Right ANSWER is (C) CIRCULAR

To explain: When the Peclet number is ZERO, the isolines of flow property are circular with the current node at its centre. Here, diffusive strength dominates.

Correct option is (b) convergence

Easy explanation: Convergence of an iterative solution is when TWO consecutive ITERATIONS result in the same solution. If the discretized system does not satisfy boundedness, it is possible that the solution WIGGLES continuously WITHOUT CONVERGING.

Right answer is (d) artificial source or sink

The explanation is: Non-conservative schemes do not have the net fluxes conserved. In the global DOMAIN, the fluxes ENTERING the domain and that leaving the domain are not the same. This will create artificial sources and SINKS which do not actually EXIST in the physical PROBLEM.

The correct choice is (d) Interpolation curves vary at the face

Best explanation: Quadratic interpolation MODELS the physical PROBLEMS using quadratic equations. At the intersection faces, we get two different quadratic equations and their VALUES do not cancel out there. This gives a problem to conservation in the Quadratic interpolation SCHEMES.

The correct option is (a) The coefficient matrix has larger values in the main diagonal

For explanation I would SAY: SCARBOROUGH criterion needs the coefficient matrix to be diagonally dominant. This MEANS that the diagonal ELEMENTS should be larger than the non-diagonal elements for a particular ROW in the coefficient matrix.

The CORRECT answer is (d) at each node

To elaborate: The algebraic equations for the partial differential GOVERNING equations are formed at each node of the domain. For each of these algebraic equations, consistency should be DEFINED. There is no exception for the boundary nodes or the interior nodes.

Right choice is (B) \(P_i=\frac{b_i}{a_i-c_i P_{i-1}};Q_i=\frac{c_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}}\)

Best explanation: As GIVEN,

Φi = PiΦi+1+Qi

Φi-1 = Pi-1Φi+Qi-1

The i^th equation is,

aiΦi = BI Φi+1 + ciΦi-1 + di

aiΦi = bi Φi+1 + CI(Pi-1Φi + Qi-1) + di

aiΦi – ciPi-1Φi = bi Φi+1+ciQi-1+di

Φi(ai-ci Pi-1) = biΦi+1+ci Qi-1+di

\(\Phi_i = \frac{b_i}{a_i-c_i P_{i-1}}\Phi_{i+1} + \frac{c_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}} \)

Therefore,

\(P_i = \frac{b_i}{a_i-c_i P_{i-1}};Q_i=\frac{C_i Q_{i-1}+d_i}{a_i-c_i P_{i-1}}\).

The correct choice is (c) Gauss-Seidel METHOD

The explanation: Gauss-Seidel method is a method of SOLVING the discretized equations. Finite difference method, finite VOLUME method and SPECTRAL element method are all methods of discretization.

Correct OPTION is (b) piece-wise nature

The EXPLANATION: The discretized FLOW FIELD properties are DISCONTINUOUS and they have a piece-wise nature. Until discretized, the variables are continuous.

The correct option is (b) EFFICIENCY

For explanation I WOULD SAY: The tolerance should be a balance between both ACCURACY and efficiency. If the tolerance is too big, iterations will stop soon but the answers will not be accurate. On the other hand, if the tolerance is too small, the number of iterations will be more. This will MAKE the solution inefficient.

The CORRECT choice is (b) EXPLICIT transient schemes

The explanation is: Courant number is used in explicit TIME integration METHODS INVOLVING numerical methods. It relates grid size and time steps of the explicit transient schemes. It cannot be applied for implicit transient schemes.

The correct OPTION is (c) DISCRETE cells

Explanation: Discretization of the physical DOMAIN means DIVIDING the region of interest into discrete elements. This will result in discrete cells on which the rest of the solution process depends.