MCQ

Question 1

Explain very briefly the physical meaning of the governing equation. [Convection- Diffusion]

Answer 1

The governing convection-diffusion equation embodies the rate of change of a scalar quantity within a control volume, accounting for the net effect of advective transport, diffusive fluxes, and any sources or sinks.

Question 2

Describe the principle of Finite Volume Discretisation.

Answer 2

Finite Volume Discretisation involves dividing the domain into discrete control volumes over which the integral balances of mass, momentum, or energy are applied, converting partial differential equations into algebraic equations suitable for numerical solving.

Question 3

3. Consider a conservation problem with the following integral conservation equation for a scalar quantity 𝜙: 𝜕 𝜕𝑡 ∫ (𝜌 𝜙) 𝐶𝑉 𝑑𝛺 + ∫ (𝜌 𝜙) 𝑣⃗ ∙ 𝑛⃗⃗ 𝐶𝑆 𝑑𝐴 = ∫ 𝑆 𝐶𝑉 𝑑𝛺 where 𝑆 is a constant. Which of the following statements is correct? [4 Marks]
   a. ∫ ( ) 𝐶𝑆 𝑑𝐴 represents a volume integral and ∫ ( ) 𝐶𝑉 𝑑𝛺 represents a surface integral.
   b. There is no convective transport and there is a uniform and constant source term.
   c. There is no diffusive transport and there is uniform and constant source term.
   d. There is no diffusive transport.

Answer 3

c. There is no diffusive transport and there is uniform and constant source term.

Question 4

4. Why is a pressure velocity coupling required in the solution of the incompressible Navier Stokes equations by a Finite Volume segregated solution method? [4 Marks]
   a. Pressure does not appear in the momentum conservation equation and must be introduced by coupling it with the energy equation.
   b. The energy equation introduces instabilities in the iterative solution process which is resolved by the pressure-velocity coupling.
   c. The mass conservation equation does not include any pressure term and yet it must be solved for pressure. The coupling of the momentum and mass conservation equations introduces pressure in the equation which can then be solved. This is the purpose of the pressure velocity coupling.
   d. If the fluid is incompressible pressure appears as a source term in the mass conservation equation which does not account for transfers between static and dynamic pressure. This is corrected by the pressure velocity coupling which couples the mass and momentum conservation equations.

Answer 4

c. The mass conservation equation does not include any pressure term and yet it must be solved for pressure. The coupling of the momentum and mass conservation equations introduces pressure in the equation which can then be solved. This is the purpose of the pressure velocity coupling.

Question 5

5. Iterative segregated solution methods can test convergence against a criterion defined by 𝑅 𝜙 = ∑ |∑𝑛𝑏 𝑎𝑛𝑏𝜙𝑛𝑏 + 𝑏 − 𝑎𝑃𝜙𝑃 𝑐𝑒𝑙𝑙𝑠 𝑃 |/ ∑ |𝑎𝑃𝜙𝑃 | 𝑐𝑒𝑙𝑙𝑠 𝑃 . What does the 𝑅 𝜙 represent [4 marks]:
   a. 𝑅 𝜙 is the scaled residual which measures the sum, over all cells in the domain, of the absolute imbalance of the governing equation for the transport of 𝜙. In this definition, it is scaled relative to the sum of the contribution from the cell centre values to the governing equation.
   b. 𝑅 𝜙 is the scaled residual which measures the sum, over all cells in the domain, of the absolute imbalance of the governing equation for the transport of 𝜙. In this definition, it is scaled relative to the sum of the contribution from the cell neighbouring values to the governing equation.
   c. 𝑅 𝜙 is the absolute residual which measures the sum, over all cells in the domain, of the absolute imbalance of the governing equation for the transport of 𝜙.
   d. None of these.

Answer 5

a. 𝑅𝜙 is the scaled residual which measures the sum, over all cells in the domain, of the absolute imbalance of the governing equation for the transport of 𝜙. In this definition, it is scaled relative to the sum of the contribution from the cell centre values to the governing equation.

Question 6

6. Which of the following assumptions is needed to derive the pressure correction equation from the momentum equation in the SIMPLE method? [4 marks]
   a. The momentum source terms are unchanged when evaluated from the initial guess and from the corrected value of the transported quantity at each iteration.
   b. There is no diffusive transport.
   c. There is no convective transport.
   d. Velocity does not need to be corrected after the solution of the pressure correction equation.

Answer 6

a. The momentum source terms are unchanged when evaluated from the initial guess and from the corrected value of the transported quantity at each iteration.

Question 7

7. Why is under relaxation often needed in an iterative solution by the SIMPLE method? [4 marks]
   a. The iterative solution increases pressure constantly by adding the pressure correction at each iteration. Under-relaxation avoid pressure increasing asymptotically to infinity.
   b. The pressure correction equation is derived by neglecting the influence of velocity from neighbouring cells which introduces an error in the correction stage. This can potentially create instability from over correction.
   c. The pressure correction equation is derived by neglecting the influence of pressure from neighbouring cells which introduces an error in the correction stage. This can potentially create instability from over correction.
   d. The pressure correction equation is derived by neglecting the influence of pressure and velocity from neighbouring cells which introduces an error in the correction stage. This can potentially create instability from over correction

Answer 7

b. The pressure correction equation is derived by neglecting the influence of velocity from neighbouring cells which introduces an error in the correction stage. This can potentially create instability from over correction.

Question 8

8. Which of the following statements is correct? [2 marks]
   a. The source term of the pressure correction equation becomes zero when the discrete velocity solution satisfies the discrete momentum conservation equation
   b. The source term of the pressure correction equation is zero at the first iteration only.
   c. The discrete pressure correction equation has no source term.
   The source term of the pressure correction equation becomes zero when the discrete velocity solution satisfies the discrete continuity equation.

Answer 8

d. The source term of the pressure correction equation becomes zero when the discrete velocity solution satisfies the discrete continuity equation.

Question 9

9. Consider the mesh which is shown below and is used to simulate a free shear layer. It shows successive refinements in the streamwise direction (to the right of the page) and the shear layer is shown as the grey overlay. Which of the statements below best describe the mesh and its suitability for CFD? [3 marks]

    a. The coarse mesh resolves the shear layer with well in excess of 10 cells in the streamwise direction and is therefore adequate. 
    b. The velocity gradients are largest outside the grey region on the sketch. The Medium mesh resolves this part of the flow with more than 10 cells in the direction perpendicular to the streamwise direction and is therefore adequate. 
    c. The velocity gradients are largest inside the grey region on the sketch. The Medium mesh resolves this part of the flow with approximately 10 cells in the direction perpendicular to the streamwise direction and is therefore adequate.
    d. The shear layer originates in the coarse mesh. The shear layer in this region is characterised by smaller gradients which explains why larger mesh cells are suitable in this part of the flow.

Answer 9

c. The velocity gradients are largest inside the grey region on the sketch. The Medium mesh resolves this part of the flow with approximately 10 cells in the direction perpendicular to the streamwise direction and is therefore adequate.

Question 10

10. Consider the velocity profile in viscous unit shown below for flow over a flat surface at two Reynolds numbers. From this, which of the following statements below best describe meshing requirements in a boundary layer with a zero pressure gradient? [3 marks]

    a. The extent of the viscous sub-layer and buffer layer parts of the universal law of the wall reduces with the Reynolds number so that specifying the 𝑦 + value for the first cell near the surface, the number of prism layers and the growth ratio alone does not ensure that the same number of cells is used to resolve the thickness of boundary layer for all Reynolds numbers.
    b. The extent of the overlap part of the universal law of the wall reduces with the Reynolds number so that specifying the 𝑦 + value for the first cell near the surface, the number of prism layers and the growth ratio alone does not ensure that the same number of cells is used to resolve the thickness of boundary layer for all Reynolds numbers.
    c. The law of the wall is insensitive to the Reynolds number so that the boundary layer mesh does not need to account for the Reynolds number

Answer 10

b. The extent of the overlap part of the universal law of the wall reduces with the Reynolds number so that specifying the 𝑦+ value for the first cell near the surface, the number of prism layers and the growth ratio alone does not ensure that the same number of cells is used to resolve the thickness of boundary layer for all Reynolds numbers.

Question 11

11. MCQ3 Which of the following statements on the 𝑘-𝜔 SST turbulence model is correct [3 marks]:
    a. The model over-predicts the spread rate of round jets.
    b. The model is highly sensitive to free stream values of 𝜔.
    c. At high Reynolds number, the model is highly sensitive to the wall adjacent cell height when 𝑦 + is between 1 and 5 in prediction of wall shear stress.
    d. Curvature correction is required for flow over flat surfaces.

Answer 11

a. The model over-predicts the spread rate of round jets.

Question 12

12. Consider a two dimensional boundary layer over a flat surface aligned with the 𝑥 coordinate of the Cartesian Coordinate System with 𝝏𝒖 𝝏𝒚 < 0. Which of the following statements is correct [3 marks]
    a. The Reynolds stresses have the same sign as the mean velocity gradient.
    b. The Reynolds stresses have the opposite sign to the mean velocity gradient.
    c. The Reynolds stresses are independent of the mean velocity gradient.
    d. The Reynolds stresses are inversely proportional to the mean velocity gradient.

Answer 12

a. The Reynolds stresses have the same sign as the mean velocity gradient.

Question 13

Discuss whether the flow will be unsteady and what type of Navier-Stokes solver should be chosen when modeling water flow in a hydraulic separation jig depicted in Figure 2.

Answer 13

The flow will likely be steady if the boundary conditions and forces are constant over time, and a pressure-based Navier-Stokes solver is suitable due to the incompressible and steady assumption.

Question 14

Describe the discretisation schemes that should be used for modeling the hydraulic separation jig in Figure 2

Answer 14

For the hydraulic separation jig, discretization schemes should account for turbulence modeling and complex geometry; therefore, a combination of second-order upwind for convective terms and second-order central for diffusive terms could be appropriate.

Question 15

Discuss the appropriate boundary conditions to choose when modeling the hydraulic separation jig in Figure 2.

Answer 15

Boundary conditions for the jig should include a pressure inlet or outlet at the screen, no-slip walls at solid boundaries, and symmetry or periodic boundaries if the geometry and flow conditions permit.

Question 16

Discuss how the mesh should be defined for the hydraulic separation jig in Figure 2.

Answer 16

The mesh should be refined in areas of high gradient such as near solid boundaries and around the screen to capture the flow details accurately while ensuring that the computational load is manageable.

Question 17

MCQ1: Why is a pressure–velocity coupling required when solving the incompressible form of the Navier-Stokes equations with a segregated method? Choose the correct answer:
a. Pressure does not appear in the momentum equation
b. Pressure does not appear in the continuity equation
c. Only velocity needs to be solved
d. None of the above [4 Marks]

Answer 17

. b. Pressure does not appear in the continuity equation ******

Question 18

MCQ2: When should a coupled density based solver be used instead of a segregated pressure based solver? Choose the correct answer:
a. If the fluid is compressible and there is weak coupling between the Navier-Stokes equations and the energy equation
b. If pressure does not need to be solved
c. If density is not important for the solution
d. If the fluid is compressible and there is strong coupling between the Navier-Stokes equations and the energy equation [4 Marks]

Answer 18

d. If the fluid is compressible and there is strong coupling between the Navier-Stokes equations and the energy equation. *********

Question 19

MCQ3: What is the most likely reason for the residual monitors from a SIMPLE solution to appear as a series of saw tooth graphs where the residuals repeatedly increase sharply before reducing more smoothly? Choose the correct answer:
a. The solution relies on a steady time formulation and a maximum number of iterations per time step set at 10.
b. The solution relies on a transient time formulation with a continuity residual threshold of 10−1
c. The solution relies on a transient time formulation with residual thresholds all below 10−4 and a maximum number of iteration per time step set at 10.
d. The energy and momentum equations are strongly coupled. [4 Marks]

Answer 19

c. The solution relies on a transient time formulation with residual thresholds all below 10−4 and a maximum number of iterations per time step set at 10.

Question 20

MCQ4: Which of the following terms is neglected in the derivation of the pressure correction equation from the momentum equation in the SIMPLE method? Choose the correct answer:
a. ∑ (anbunb’)nb
b. aeu’e’
c. (pE’ - pP’)δy
d. (pE’)δy [4 Marks]

Answer 20

a. ∑ (anbunb’)nb

Question 21

MCQ5: What does the Rφi defined below represent? Choose the correct answer:
Rφi = ∑ |aPi φPi - ∑ anbi φnbi nb - bi / aPi|
a. The result of the conservation equation for the transported quantity φ evaluated at time index i for a single CV.
b. The global result of the conservation equation for the transported quantity φ evaluated at time index i
c. The residual for the conservation equation for the transported quantity φ evaluated at solution iteration i for a single CV
d. The global residual for the conservation equation for the transported quantity φ evaluated at solution iteration i [3 Marks]

Answer 21

d. The global residual for the conservation equation for the transported quantity φ evaluated at solution iteration i.

Question 22

MCQ6: Which one of the following terms is the velocity correction in the SIMPLE method? Choose the correct answer:
a. u’
b. u
c. u*
d. uΔt [3 Marks]

Answer 22

a. u’.

Question 23

MCQ1 (from Question 3): At a wall boundary in turbulent flow, we know that µt = 0. Why is that the case? Choose the correct answer:
a. Turbulent flow velocity fluctuations vanish at the wall while the mean flow velocity gradients are not zero.
b. Boussinesq’s model is not valid at the wall.
c. Turbulent flow velocity fluctuations vanish at the wall while the shear stress at the wall is infinite.
d. None of the above [3 Marks]

Answer 23

a. Turbulent flow velocity fluctuations vanish at the wall while the mean flow velocity gradients are not zero.

Question 24

MCQ2 (from Question 3): Consider a shear layer for 2D flow so that ∂u/∂y < 0. What can be said of the value of the Reynolds stress u’v’? Choose the correct answer:
a. u’v’ < 0
b. u’v’ = 0
c. u’v’ > 0
d. u’v’ = µt*u [3 Marks]

Answer 24

a. u’v’ < 0.

Question 25

MCQ3 (from Question 3): What is the definition of the friction velocity uτ in turbulent flow? Choose the correct answer: a. τw/ρ b. U∞√(Cf/2) c. ν/√(τw/ρ) d. none of the above [3 Marks]

Answer 25

b. U∞√(Cf/2)

Question 26

MCQ1: Why is a pressure-velocity coupling required when solving the incompressible form of the Navier-Stokes equations with a segregated method? a. Pressure does not appear in the continuity equation. b. Pressure and velocity are not physically coupled in incompressible flow. c. Pressure does not appear in the momentum equation. d. Pressure does not need to be solved. [4 Marks]

Answer 26

a. Pressure does not appear in the continuity equation.

Question 26

MCQ2: When can the segregated incompressible solvers be used? a. If the fluid is compressible. b. If the fluid is incompressible. c. If the fluid is compressible or incompressible and the flow is isothermal. d. If the fluid is compressible or incompressible whether the energy equation is activated or not. [2 Marks]

Answer 26

b. If the fluid is incompressible.

Question 27

MCQ3: The SIMPLE method of solution relies on this equation p_i = p_i−1 + α_p p'. What do the terms α_p and p' represent? a. α_p is the pressure under-relaxation factor and p' is the pressure correction from the current iteration. b. α_p is the diffusivity at the cell center and p' is the pressure correction from the current iteration. c. α_p is the diffusivity at the cell center and p' is the time derivative of pressure. d. None of these. [4 Marks]

Answer 27

a. α_p is the pressure under-relaxation factor and p' is the pressure correction from the current iteration.

Question 28

MCQ4: Which of the following terms is simplified in the derivation of the pressure correction equation from the momentum equation in the SIMPLEC method? a. a_e u_e' b. (p_E' - p_P')δy c. (p_E')δy d. ∑ (a_nb u_nb')_nb [4 Marks]

Answer 28

a. a_e u_e'

Question 29

MCQ7: Why were staggered grids for pressure and momentum first introduced instead of collocated grids in cell center Finite Volume Solvers? a. Collocated grids make it impossible to calculate fluxes through CV face. b. Standard interpolations of cell centered flow variables to CV faces which are needed to calculate fluxes, introduce solution instabilities. c. It simplifies meshing of the computational domain. d. None of the above. [2 Marks]

Answer 29

b. Standard interpolations of cell centered flow variables to CV faces which are needed to calculate fluxes, introduce solution instabilities.

Question 30

MCQ5: The momentum predictor step of the SIMPLE method involves solving for algebraic equations of this type: a_e u_e* = ∑(a_nb u_nb*)_nb - (p_E* - p_P*)δy + b_u. Why is the equation linear while the Navier-Stokes equations are non-linear? a. The non-linear terms are neglected. b. The transient term is neglected. c. The pressure velocity coupling removes the non-linearity. d. The coefficients a_e, a_nb, and b_u are evaluated from the guess/initial values of the unknown variables. [4 Marks]

Answer 30

d. The coefficients a_e, a_nb, and b_u are evaluated from the guess/initial values of the unknown variables.

Question 31

MCQ6: What is the key benefit of the SIMPLEC method compared to the SIMPLE method? a. It is designed for compressible flow. b. It removes all simplifying assumptions of the SIMPLE method and removes the need for under-relaxation. c. It removes one simplifying assumption of the SIMPLE method and relaxes the need for under-relaxation. d. None of the above [2 Marks]

Answer 31

c. It removes one simplifying assumption of the SIMPLE method and relaxes the need for under-relaxation.

Question 32

MCQ2: Which of the following statements on the k-omega SST turbulence model is correct? a. The model over-predicts the spread rate of round jets. b. The model is highly sensitive to free stream values of omega. c. At high Reynolds number, the model is highly sensitive to the wall adjacent cell height when y+ is between 1 and 5 in prediction of wall shear stress. d. Curvature correction is required for flow over flat surfaces. [3 Marks]

Answer 32

a. The model over-predicts the spread rate of round jets

Question 32

MCQ1: Which of the following assumptions made in the derivation of the k-epsilon turbulence model is correct? a. The flow experiences strong swirl and pressure gradients. b. The flow is assumed to be a low Reynolds number flow. c. Near surfaces, the boundary layer is assumed to be in equilibrium which means that the rates of production and dissipation of k are predominant in the transport equation for k. d. Turbulence is assumed to be non isotropic. [3 Marks]

Answer 32

c. Near surfaces, the boundary layer is assumed to be in equilibrium which means that the rates of production and dissipation of k are predominant in the transport equation for k.

Question 33

MCQ3: Consider a two-dimensional boundary layer over a flat surface aligned with the x coordinate of the Cartesian Coordinate system with du/dy > 0. Which of the following statements is correct? a. The Reynolds stresses have the same sign as the mean velocity gradient. b. The Reynolds stresses have the opposite sign to the mean velocity gradient. c. The Reynolds stresses are independent of the mean velocity gradient. d. The Reynolds stresses are inversely proportional to the mean velocity gradient. [3 Marks]

Answer 33

a. The Reynolds stresses have the same sign as the mean velocity gradient.

Question 34

MCQ4: Which of the following equations approximates the integral length scale of turbulence from quantities modeled by the k-epsilon turbulence closure model? a. k epsilon b. k^(3/2)/epsilon c. nu/sqrt(tau_w/rho) d. 1/nu [3 Marks]

Answer 34

b. k^(3/2)/epsilon

Question 35

MCQ1: Consider a conservation problem with the following integral governing equation: 𝜕/𝜕𝑡 ∫ (𝜌 𝜙) 𝐶𝑉 𝑑𝛺 = ∫ [𝛤𝜙 grad (𝜙)] · 𝑛 𝐶𝑆 𝑑𝐴 + ∫ 𝑆 𝐶𝑉 𝑑𝛺. Which statement is correct? a. ∫ ( ) 𝐶𝑆 𝑑𝐴 represents a volume integral and ∫ ( ) 𝐶𝑉 𝑑𝛺 represents a surface integral. b. There is no convective transport and there is a uniform and constant source term. c. There is no diffusive transport and there is uniform and constant source term. d. There is no diffusive transport. [4 Marks]

Answer 35

b. There is no convective transport and there is a uniform and constant source term.

Question 36

MCQ2: When should a coupled density based solver be used? a. If pressure does not need to be solved. b. If density is not important for the solution. c. If the fluid is compressible and there is strong coupling between the Navier-Stokes equations and the energy equation. d. If the fluid is incompressible and there is strong coupling between the Navier-Stokes equations and the energy equation. [4 Marks]

Answer 36

c. If the fluid is compressible and there is strong coupling between the Navier-Stokes equations and the energy equation.

Question 37

MCQ3: Iterative segregated solution methods can test convergence against a criterion defined by 𝑅𝜙 = ∑ |∑ 𝑎𝑛𝑏𝜙𝑛𝑏 + 𝑏 − 𝑎𝑃𝜙𝑃𝑛𝑏 |/𝑐𝑒𝑙𝑙𝑠 𝑃 ∑ |𝑎𝑃𝜙𝑃|𝑐𝑒𝑙𝑙𝑠 𝑃. What does the 𝑅𝜙 represent? a. 𝑅𝜙 is the scaled residual which measures the sum, over all cells in the domain, of the absolute imbalance of the governing equation for the transport of 𝜙. In this definition, it is scaled relative to the sum of the contribution from the cell centre values to the governing equation. b. 𝑅𝜙 is the scaled residual which measures the sum, over all cells in the domain, of the absolute imbalance of the governing equation for the transport of 𝜙. In this definition, it is scaled relative to the sum of the contribution from the cell neighbouring values to the governing equation c. 𝑅𝜙 is the absolute residual which measures the sum, over all cells in the domain, of the absolute imbalance of the governing equation for the transport of 𝜙. d. None of these. [4 Marks]

Answer 37

a. 𝑅𝜙 is the scaled residual which measures the sum, over all cells in the domain, of the absolute imbalance of the governing equation for the transport of 𝜙.In this definition, it is scaled relative to the sum of the contribution from the cell centre values to the governing equation.

Question 38

MCQ4: Which of the following assumptions is made in the derivation of the pressure correction equation from the momentum equation in the SIMPLE method? a. There is no diffusive transport. b. There is no convective transport. c. The momentum source terms are unchanged when evaluated from the initial guess and from the corrected value of the transported quantity at each iteration. d. Velocity does not need to be corrected after the solution of the pressure correction equation. [4 Marks]

Answer 38

c. The momentum source terms are unchanged when evaluated from the initial guess and from the corrected value of the transported quantity at each iteration.

Question 39

MCQ5: The discrete equations in the SIMPLE method are derived by linearizing the governing equations. What does this mean? a. The non-linear terms are neglected. b. The pressure velocity coupling removes the non-linearity. c. The transient term is neglected. d. Non-linear terms are approximated as linear terms using initial guess values. [4 Marks]

Answer 39

d. Non-linear terms are approximated as linear terms using initial guess values.

Question 40

MCQ6: Which of the following statements is correct? a. The source term of the pressure correction equation becomes zero when the discrete velocity solution satisfies the discrete continuity equation. b. The source term of the pressure correction equation is zero at the first iteration only. c. The discrete pressure correction equation has no source term. d. None of these. [2 Marks]

Answer 40

a. The source term of the pressure correction equation becomes zero when the discrete velocity solution satisfies the discrete continuity equation.

Question 41

MCQ1 Which of the following statements on the 𝑘-𝜀 turbulence model is correct? [2 marks] a. The flow is assumed to be a low Reynolds flow over flat surfaces. b. The transport equation for 𝑘 is modelled on the transport equation for 𝜀. c. The rate of production and dissipation are predominant in the transport equation for 𝑘. d. None of these.

Answer 41

c. The rate of production and dissipation are predominant in the transport equation for 𝑘.

Question 42

MCQ2 (Q 3): Which of the following statements on the k-omega SST turbulence model is correct? a. The model over-predicts the spread rate of round jets. b. The model is highly sensitive to free stream values of omega. c. At high Reynolds number, the model is highly sensitive to the wall adjacent cell height when y+ is between 1 and 5 in prediction of wall shear stress. d. Curvature correction is required for flow over flat surfaces. [3 Marks]

Answer 42

a. The model over-predicts the spread rate of round jets.

Question 43

Consider the two dimensional boundary layer sketched below where 𝝏𝒖 𝝏𝒚 > 𝟎. Decide which of the four following coloured arrows represent the turbulent velocity fluctuation 𝒗′𝟑 < 𝟎 . Note that the notations used are the same as in the lecture notes. [2 marks] a. Orange arrow b. Green arrow c. Blue arrow d. Red arrow