# Computational Fluid Dynamics Questions and Answers – Convection-Diffusion Problems – QUICK Scheme

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This set of Computational Fluid Dynamics Multiple Choice Questions & Answers (MCQs) focuses on “Convection-Diffusion Problems – QUICK Scheme”.

1. What does QUICK stand for?
a) Quadratic Upstream Interpolation for Convective Kinetics
b) Quadratic Upstream Interval for Convective Kinetics
c) Quadratic Upwind Interval for Convective Kinetics
d) Quadratic Upwind Interpolation for Convective Kinetics

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.

2. Which is correct about the QUICK scheme?
a) A two-point upwind biased interpolation
b) A three-point upwind biased interpolation
c) A three-point downwind biased interpolation
d) A two-point downwind biased interpolation

Explanation: QUICK scheme uses a three-point upstream-weighted quadratic interpolation to approximate the cell face values. It uses two immediate neighbours of the face and an extra upstream node (totally, three points).

3. According to the QUICK scheme, the flow variable (φ) is given by ____
(Note: U, D and C represents the upwind, downwind and the central nodes respectively).
a) $$\phi=\phi_U+\frac{(x-x_D)(x-x_C)}{(x_D-x_U)(x_D-x_C)}(\phi_D-\phi_U)+\frac{(x-x_U)(x-x_D)}{(x_C-x_U)(x_C-x_D )}(\phi_C-\phi_U)$$
b) $$\phi=\phi_U+\frac{(x-x_D)(x-x_C)}{(x_D-x_U)(x_D-x_C)}(\phi_D-\phi_U)+\frac{(x-x_C)(x-x_D)}{(x_C-x_U)(x_C-x_D )}(\phi_C-\phi_U)$$
c) $$\phi=\phi_U+\frac{(x-x_U)(x-x_C)}{(x_D-x_U)(x_D-x_C)}(\phi_D-\phi_U)+\frac{(x-x_C)(x-x_D)}{(x_C-x_U )(x_C-x_D)}(\phi_C-\phi_U)$$
d) $$\phi=\phi_U+\frac{(x-x_U )(x-x_C )}{(x_D-x_U)(x_D-x_C)}(\phi_D-\phi_U)+\frac{(x-x_U)(x-x_D)}{(x_C-x_U)(x_C-x_D)}(\phi_C-\phi_U)$$

Explanation: The scheme should reduce to
$$\phi=\begin{cases} \phi_U & if \, x=x_U \\ \phi_C & if \,x=x_C \\ \phi_D & if \,x=x_D \end{cases}$$
Incorporating these into a formula, the formula for quick scheme is
$$\phi=\phi_U+\frac{(x-x_U )(x-x_C )}{(x_D-x_U)(x_D-x_C)}(\phi_D-\phi_U)+\frac{(x-x_U)(x-x_D)}{(x_C-x_U)(x_C-x_D)}(\phi_C-\phi_U)$$ .

4. Consider the stencil.

Assume a uniform grid. What is φe according to the QUICK scheme?
a) $$\phi_e=\frac{\phi_P-\phi_E}{2}-\frac{\phi_E+2\phi_P+\phi_W}{8}$$
b) $$\phi_e=\frac{\phi_P+\phi_E}{2}-\frac{\phi_E+2\phi_P+\phi_W}{8}$$
c) $$\phi_e=\frac{\phi_P+\phi_E}{2}-\frac{\phi_E-2\phi_P+\phi_W}{8}$$
d) $$\phi_e=\frac{\phi_P-\phi_E}{2}-\frac{\phi_E-2\phi_P+\phi_W}{8}$$

Explanation: According to the QUICK scheme,
$$\phi_e=\phi_W+\frac{(x_e-x_W)(x_e-x_P)}{(x_E-x_W)(x_E-x_P)}(\phi_D-\phi_W) + \frac{(x_e-x_W)(x_e-x_E)}{(x_P-x_W)(x_P-x_E)}(\phi_P-\phi_W)$$
For a uniform grid,
xe-xW=3(xe-xP); xE-xW = 4(xe-xP); xE-xP=2(xe-xP);
xE-xE = -(xe-xP);xP-xW=2(xe-xP);
Applying all these,
$$\phi_e=\frac{\phi_P+\phi_E}{2}-\frac{\phi_E-2\phi_P+\phi_W}{8}$$

5. What is the order of accuracy of the QUICK scheme?
a) second-order
b) first-order
c) fourth-order
d) third-order

Explanation: 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.

6. How many terms does the discretized form of source-free 1-D convection problem modelled using the QUICK scheme has?
a) 3
b) 5
c) 2
d) 4

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.

7. What is the first term of the truncation error of the QUICK scheme?
a) $$\frac{1}{16} (\Delta x)^2 \phi_C”’$$
b) $$\frac{1}{16} (\Delta x)^3 \phi_C”’$$
c) $$\frac{1}{16} (\Delta x)^3 \phi_C^{iv}$$
d) $$\frac{1}{16} (\Delta x)^2 \phi_C^{iv}$$

Explanation: 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)3 comes along with ΦCiv.

8. Which of these is correct about the QUICK scheme?
a) Stable and bounded
b) Stable and unbounded
c) Unstable and bounded
d) Unstable and unbounded

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.

9. Consider the stencil.

Assume a uniform grid. Using the QUICK scheme, what is the convective flux at the western face $$\dot{m_w} \phi_w$$ equal to?
a) $$(\frac{3}{4} \phi_P-\frac{1}{8} \phi_W+\frac{3}{8} \phi_E)×max⁡(\dot{m_w},0)-(\frac{3}{4} \phi_W-\frac{1}{8} \phi_{WW}+\frac{3}{8}\phi_C)×max⁡(-\dot{m_w},0)$$
b) $$(\frac{3}{4} \phi_P+\frac{1}{8} \phi_W+\frac{3}{8} \phi_E)×max⁡(\dot{m_w},0)-(\frac{3}{4} \phi_W+\frac{1}{8} \phi_{WW}+\frac{3}{8} \phi_C)×max⁡(-\dot{m_w},0)$$
c) $$(\frac{3}{4} \phi_P-\frac{1}{8} \phi_W-\frac{3}{8} \phi_E)×max⁡(\dot{m_w},0)-(\frac{3}{4} \phi_W-\frac{1}{8} \phi_{WW}-\frac{3}{8} \phi_C)×max⁡(-\dot{m_w},0)$$
d) $$(\frac{3}{4} \phi_P+\frac{1}{8} \phi_W-\frac{3}{8} \phi_E)×max⁡(\dot{m_w},0)-(\frac{3}{4} \phi_W+\frac{1}{8} \phi_{WW}-\frac{3}{8} \phi_C)×max⁡(-\dot{m_w},0)$$

Explanation: Using QUICK scheme, for a flow in the positive x-direction,
$$\phi_e=\frac{\phi_P+\phi_E}{2}-\frac{\phi_E-2\phi_P+\phi_W}{8} = \frac{3}{4} \phi_P+\frac{3}{8} \phi_E-\frac{1}{8} \phi_W$$
In a similar manner, for a flow in the positive x-direction,
$$\phi_w=\frac{3}{4} \phi_P-\frac{1}{8} \phi_W+\frac{3}{8} \phi_E$$
For a flow in the negative x-direction,
$$\phi_w=\frac{3}{4} \phi_W-\frac{1}{8} \phi_{WW}+\frac{3}{8} \phi_C$$
Therefore,
$$\dot{m_w} \phi_w=(\frac{3}{4} \phi_P-\frac{1}{8} \phi_W+\frac{3}{8} \phi_E )×max⁡(\dot{m_w},0)-$$
$$(\frac{3}{4} \phi_W- \frac{1}{8} \phi_{WW}+\frac{3}{8} \phi_C)× max⁡(-\dot{m_w},0)$$

10. Which of these is correct for a QUICK scheme?
a) False diffusion is zero
b) False diffusion is small
c) False diffusion is big
d) False diffusion is infinity