Explore topic-wise InterviewSolutions in .

The correct OPTION is (b) Using the EXACT solution of the source and diffusion-free problem

To explain I would SAY: To evaluate the errors, a simplified VERSION of the problem where there is no source and diffusion is taken and further VELOCITY and density fields are assumed to be constants. The exact solution of this problem is used for the analysis.

Right option is (b) unphysical behaviour of assumed INTERPOLATION profile

Explanation: Numerical dispersion ERROR arises with all interpolation profiles EXCEPT the UPWIND SCHEME. It is the result of the unphysical behaviour of the assumed interpolation profile. It leads to unphysical dispersion.

Correct choice is (a) \(\TILDE{\phi_f}=\tilde{\phi_c}+\frac{1}{4}\)

Explanation: The relationship between Φf and Φc is

\(\phi_f=\phi_c+\frac{1}{4}(\phi_D-\phi_U)\)

The normalized forms of Φf, Φc, ΦD and ΦU are \(\tilde{\phi_f}, \tilde{\phi_c},\) 1 and 0 respectively. Therefore,

\(\tilde{\phi_f}=\tilde{\phi_c}+\frac{1}{4}.\)

Correct choice is (d) Unstable and unbounded

Easiest explanation: The QUICK scheme is not bounded. It INVOLVES undershoots and overshoots. The main COEFFICIENTS (immediate eastern and western coefficients) are not guaranteed to be positive. The coefficients AEE and aWW are negative. Therefore, the SOLUTION is not stable.

Right answer is (d) is UPWIND biased

For explanation I WOULD SAY: The FROMM SCHEME is upwind biased. It gives more importance to the upwind nodes than the downwind nodes. IT uses two upwind nodes and one downwind node (totally three nodes).

Correct choice is (c) \(\frac{1}{16} (\Delta x)^3 \phi_C^{iv}\)

The BEST I can explain: The order of accuracy is 3. Therefore, (Δ x)^3 should be there in the first term of the TRUNCATION error. The truncation error is OBTAINED using the TAYLOR series. Therefore, this (Δ x)^3comes along with ΦC^iv.

Right choice is (a) Quadratic Upstream INTERPOLATION for Convective Kinetics

The EXPLANATION: QUICK is a higher-order differencing scheme introduced by BRIAN P. Leonard in his paper in the year 1979. It is the abbreviation of Quadratic Upstream Interpolation for Convective Kinetics.

The correct option is (d) r

Explanation: To find the flux limiter,

\(\phi_f=\phi_C+\FRAC{1}{2}\psi(r)(\phi_D-\phi_C)\)

For the second order UPWIND scheme,

\(\phi_f=\frac{3}{2}\phi_C-1\frac{1}{2}\phi_U\)

Equating both,

\(\frac{1}{2}\psi(r)(\phi_D-\phi_C)=\frac{3}{2}\phi_C-\phi_C-\frac{1}{2}\phi_U\)

\(\frac{1}{2}\psi(r)(\phi_D-\phi_C)=\frac{1}{2}(\phi_C-\phi_U)\)

\(\psi(r)=\frac{(\phi_C-\phi_U)}{(\phi_D-\phi_C)}=r\).

Correct option is (b) False diffusion is small

The explanation: The QUICK scheme involves ONE DOWNWIND NODE also. So, there will be a false-diffusion in this method. But this false diffusion VALUE is not big as it is an upwind biased scheme (extra nodes in the upstream than the downstream).

The correct ANSWER is (c) Both NUMERICAL dispersion and numerical diffusion ERRORS arise

To explain I would say: In the analysis for errors, while comparing the exact and numerical solutions, the first-order upwind scheme gives a complex k-value. THEREFORE, it will have both numerical dispersion and numerical diffusion problems.

Correct option is (b) The second-order upwind scheme is always stable

Best explanation: The numerical STABILITY of a scheme can be analysed by using the rate of change of influx. If the DERIVATIVE of the influx with RESPECT to the flow variable is negative, the scheme is stable. For the second-order upwind scheme, this is always negative.

Right answer is (d) Only numerical DISPERSION ERROR arises

The explanation: For the central DIFFERENCING scheme, there is no PROBLEM of numerical diffusion. Only numerical dispersion error arises here. This is because, the k-value is completely imaginary and free from a real part.

The correct option is (b) Increasing the order of interpolation

Explanation: The NUMERICAL DIFFUSION error can be further DIVIDED into TWO – stream-wise and cross-stream numerical diffusions. The stream-wise numerical diffusion can be reduced by using a higher-order interpolation profile.

The correct CHOICE is (d) Changing the DIRECTION of interpolation

The explanation is: The cross-stream numerical diffusion can be decreased by EITHER INTERPOLATING in the direction of the FLOW which means multi-dimensional interpolation profiles or using a one-dimensional higher-order interpolation profile.

Right option is (a) One-dimensional nature of the ASSUMED profiles

The explanation is: One of the types of numerical DIFFUSION errors is the cross-stream numerical diffusion. Cross-stream numerical diffusion is caused by the one-dimensional nature of INTERPOLATION when the GRID is multi-dimensional. This is caused by cross-flow diffusion or false diffusion.

Correct choice is (a) LESS DIFFUSIVE

The EXPLANATION is: The second-order upwind scheme is more ACCURATE (second-order) accurate than the general (first-order) upwind scheme. But, it is less diffusive when COMPARED to the general upwind scheme.

Right option is (c) boundedness problems

The best explanation: One of the TYPES of sources of the ERRORS is numerical dispersion. This is shown out through OSCILLATIONS in the resulting profiles in the presence of large gradients in the profile resulting in an unbounded solution.

The correct answer is (b) A linear profile is obtained between the FAR upwind and the IMMEDIATE downwind nodes

Explanation: A linear profile is obtained by CONNECTING the values of the far upwind node and the immediate downwind node. A profile with the same slope obtained here is created between the immediate upwind and the current node to get the REQUIRED VALUE.

The correct answer is (B) Second-ORDER

Easy explanation: The FIRST term of the TRUNCATION error while implementing the Taylor series in the FROMM scheme is of order TWO. Therefore, the FROMM scheme is second-order accurate using a linear profile.

Right answer is (a) smearing of SHARP gradients

The best explanation: The SOURCES of NUMERICAL errors CAUSED by the convection flux can be divided into two – numerical diffusion and numerical DISPERSION. The numerical diffusion leads to smearing of sharp gradients.

The correct option is (b) 5

Easy explanation: The discretized FORM of a source-free 1-D CONVECTION problem modelled using the QUICK SCHEME INVOLVES the far upstream and the far downstream nodes too. Therefore, it CONTAINS extra terms than the upwind and the second-order upwind schemes. The stencil is

The discretized equation is

aP ΦP+aE ΦE+aW ΦW+aEE ΦEE+aWW ΦWW=0

It contains 5 terms.

Right answer is (d) Peclet number is high

For explanation: When the Peclet number is low in the positive or NEGATIVE direction, the central differencing scheme is valid. If the Peclet number goes beyond a CERTAIN value both in the positive and negative direction, this APPROXIMATION GIVES unphysical answers.

Correct CHOICE is (a) second-order

To explain: As the QUICK scheme is based on a quadratic function, its ACCURACY in terms of Taylor Series truncation ERROR is third-order. This has a higher order of accuracy than the UPWIND and second-order upwind schemes.

The correct choice is (a) asymmetric linear PROFILE

To explain: The second order upwind SCHEME, LIKE the central difference scheme, uses a linear profile. But, unlike the central differencing scheme, it uses an asymmetric linear profile. This is why it GOT the NAME upwind scheme.

Correct option is (c) It gives EQUAL importance to upwind and downwind NODES

To EXPLAIN I would say: The central differencing scheme gives equal importance to the upwind and the downwind nodes. The contribution of all the neighbouring nodes is CONSIDERED for this APPROXIMATION.

The correct CHOICE is (a) CONVECTION term

Explanation: The upwind SCHEME is not suitable for non-DIRECTIONAL phenomena. The diffusion scheme is a non-directional phenomenon. The convection scheme is a directional phenomenon. So, the scheme is suitable for the convection term.

Correct choice is (B) extrapolation

To explain I would say: Second ORDER upwind scheme uses an upwind biased stencil. Therefore, it needs linear extrapolation to guess the VALUES at the faces instead of interpolation. This is where it is DIFFERENT from the central differencing scheme.

The correct ANSWER is (a) -2≤Pe≤2

For explanation: The CENTRAL differencing SCHEME is valid until the Peclet NUMBER REACHES a value of two. So, in the hybrid difference scheme, in the range -2≤Pe≤2, the central differencing scheme is used.

Right CHOICE is (c) Flow direction

For explanation: The upwind scheme reflects the physics of ADVECTION. The cell-face VALUE is dependent on the upwind nodal value. So, we can say it is dependent on the flow direction and it is SUITABLE for directional flows.

The correct choice is (d) First-ORDER

The explanation: The major DISADVANTAGE of the hybrid difference scheme is its low order of accuracy based on the Taylor SERIES truncation term. It is first-order accurate. Yet, it is USEFUL for solving practical FLOW problems.

Right choice is (a) accuracy

Easy explanation: The upwind scheme is less ACCURATE than the CENTRAL difference schemes. But the central difference schemes are oscillatory. They do not give answers which are PHYSICALLY CORRECT. This is the ADVANTAGE of the upwind scheme over the central-difference scheme.

Right answer is (c) second-order

To explain: The central differencing scheme is second-order accurate. This can be PROVED by using the TAYLOR series expansion. This is more accurate when compared to the upwind or the DOWNWIND SCHEMES.

Right ANSWER is (b) It is conservative and transportive

To elaborate: The HYBRID differencing SCHEME is fully conservative. It satisfies the transportiveness CONDITION by using upwind scheme for high Peclet numbers. So, the scheme is conservative and transportive as WELL.

Right answer is (c) central DIFFERENCE and upwind schemes

Easiest explanation: Hybrid differencing scheme was introduced by SPALDING in 1970s. It is the hybrid between upwind and central differencing schemes so that the ADVANTAGES of both of these schemes is utilized.

Right option is (b) bounded and conservative

The BEST I can explain: The upwind scheme does not produce RESULTS which wiggle. So, it is bounded. It uses CONSISTENT expressions to calculate FLUXES through cells. Therefore, it is sure that the FORMULATION is conservative.

The CORRECT choice is (a) Local Peclet number

The explanation: The central differencing SCHEME works WELL with low Peclet numbers and the upwind scheme works well for high Peclet numbers. So, the differencing scheme to be used in this METHOD is chosen using the Peclet number.

Right ANSWER is (a) is higher than 2

Easiest explanation: When the Peclet number goes beyond 2, the central DIFFERENCE approximation fails. The DISCRETIZATION process BECOMES inconsistent here. In this case, an increase in the neighbouring value will lead to a decrease in the value at the central node.

Right CHOICE is (B) Linear interpolation profile

The best I can explain: The central difference SCHEME MATCHES the linear interpolation profile. The general form (GIVEN below) of both are the same.

Φ=k0+k1(x-xC).

The correct ANSWER is (c) STABILITY

The EXPLANATION: The QUICK scheme also POSSESSES conservativeness and transportiveness. QUICK scheme has a higher order of accuracy. But, it is not STABLE. Stability is the advantage of the hybrid scheme over this scheme.