Computational Fluid Dynamics I Exam questions PDF

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COMPUTATIONAL FLUID DYNAMICS 1 (14MMC802) Summer 2015

2 Hours Answer ONLY THREE questions out of five. All questions carry equal marks. Any approved University calculator is permitted.

Candidates may use the course text and one folder with study notes during the examination.

1. Fully-developed flow in an annular gap between two co-axial circular cylinders satisfies the following axial momentum equation in cylindrical coordinates: µ ∂  ∂VZ  ∂p r = r ∂r  ∂r  ∂z where Vz = axial fluid velocity, µ = dynamic viscosity of liquid, ∂ p/∂z = pressure gradient, r = radius. Consider the flow between two co-axial cylinders with inner radius RI = 9 mm and outer radius RO = 18 mm (see Figure Q1, Page 5). The outer cylinder is stationary, so VZ,O = 0 and the inner cylinder velocity moves along its axis with a velocity VZ,I = 1 m/s. The fluid -1 -1 has dynamic viscosity µ = 1.5 kgm s . (a) Using the finite volume method, develop discretization equations for the three control volumes of the radial grid between r = RI and r = RO sketched in Figure Q1 (Page 5). Assume constant pressure throughout the flow domain. (Hint: multiply both sides of the axial momentum equation by r/µ prior to carrying out the control volume integration). [10 marks] (b) Estimate the fluid velocity at the three grid nodes and compute the error in the numerical solution by comparing it with the analytical solution given by: ln (r RO ) [6 marks] V Z = V Z ,I ln (RI RO ) (c) Verify if the viscous shear forces on the inner and outer wall balance in the numerical solution. Comment on your result. If appropriate, make reference to the comparison in Part (b). [4 marks] /continued… Page 1 of 9

2. In many practically important flows the effects of viscous stresses are concentrated in thin boundary layers with τxy >> τxx. Such flows can be described by a simplified form of the Navier-Stokes equations. Consider an unsteady, two-dimensional, laminar boundary layer of a Newtonian fluid without heat transfer in a large expanse of fluid, which is flowing uniformly in the x-direction. The flow interacts with a thin plate with a finite length L = 1 m along the positive x-axis. (a) Taking as a starting point the unsteady continuity and x-momentum equations, show that these equations can be simplified to the following form for the viscous flow inside the boundary layer along the flat plate if the fluid viscosity and density are constant: ∂U ∂V + = 0 ∂x ∂y

∂U ∂U ∂U µ ∂2 U 1 ∂P +U +V =− + ρ ∂x ρ ∂y 2 ∂t ∂x ∂y (b) Classify the unsteady boundary layer equations.

[6 marks] [2 marks]

(c) Make a sketch illustrating the main features of a subsonic flow at high Reynolds number past a flat plate set at an incidence angle of 5 degrees with respect to the xaxis. The flow is steady and incoming flow far upstream from the plate follows the xdirection. Clearly indicate the edge of the boundary layer and the main directions of information flow inside and outside the boundary layer. [4 marks] (d) Make a clear sketch of the computational domain that would be used in a model of the flat plate at incidence (as introduced in Part (c)) developed with a commercial CFD code such as Star-CCM+, which solves the Navier-Stokes equations. State the physical conditions at the edge of the domain. [3 marks] (e) State the boundary conditions.

[2 marks]

(f) Make a separate sketch of the computational mesh. Highlight the main mesh design

features by means of clear labels.

[3 marks]

/continued…

Page 2 of 9

3. Shown in Figure Q3.1 (Page 5) is a cylindrical rod with uniform cross-sectional area A. The base is at a temperature of 200 °C ( TA ) and one half of the cylindrical surface is insulated. The other half is exposed to an ambient temperature of 20 °C (T∞ ). Onedimensional heat transfer in this situation is governed by: d  dT   kA  − hP (T − T∞ ) = 0 dx  dx  Where h is the convective heat transfer coefficient, P the perimeter, k the thermal conductivity of the material and T∞ the ambient temperature. The length of the cylinder is 1.2 m, h = 100 W/m2 °C, k= 50 W/m °C, diameter of the cylinder is 0.1 m. The control volume method is to be used with 6 control volumes as shown in Figure Q3.2 (Page 5) to obtain a set of discretised equations that could be solved to obtain the temperature distribution. (a) Write down the discretised form of the above equation for a general node P and provide expressions for its coefficients. [4 marks] (b) Show how you incorporate boundary conditions by modifying the general discretised equation at points 1, 2, 3 and 6. [10 marks] (c) Calculate values of the coefficients at each node and tabulate values of the coefficients for all nodes. Complete Table Q3 (Page 6) and attach the table to your answer book. [4 marks] (d) Write a set of discretised equations for all nodes.

[2 marks]

4. A property φ generated by a source distribution S is transported by means of convection and diffusion through the one-dimensional domain sketched in Figure Q4 (Page 7). The governing equation is: d  dφ  d (ρ uφ ) = Γ  + S dx  dx  dx A computational grid as shown in Figure Q4 (Page 7) with five equally spaced control volumes is used to formulate discretised equations. The boundary conditions are at x=0 φ = 1 and at x = L, φ = 0 . The source distribution for this case is shown in Figure Q4 3

(Page 7). The data are L = 1.0 m, u = 2.5 m/s ρ = 1.0 kg/m and Γ = 0.1 kg/m/s. (a) Use the hybrid differencing scheme to calculate coefficients of the discretised equations at each node and complete Table Q4 (Page 8). Attach the table to your answer book. [15 marks] (b) Formulate a set of equations that could be solved to obtain the distribution of φ . [2 marks] (c) Obtain a solution to the set of equations by simple elimination.

[3 marks] /continued…

Page 3 of 9

5. The staggered grid shown in Figure Q5 (Page 9) may be used to calculate onedimensional flow through the converging duct shown. The cross-sectional areas (in m2) at different sections of the nozzle are indicated in Figure Q5 (Page 9). The velocity is calculated at locations A, B, C, D and E while the pressure p is calculated for node locations 1, 2, 3, 4 and 5. In the solution algorithm a guessed pressure field p * is used to solve the relevant momentum equation to give an intermediate velocity field u * . The correct velocity is obtained from: ui = ui* + di ( pI′−1 − pI′ ) where locations I-1 and I lie on either side of the u-velocity node and p′ stands for pressure correction. At an intermediate stage of the calculation uB∗ = 2.0 m/s, uC∗ = 2.0m/s , uD∗ = 2.0 m/s and uE∗ = 2.0 m/s. Here d = 1 and ρ = 1 kg/m3 everywhere and the boundary conditions are u A = 2.0 m/s and p5′ = 0.

(a) Using the methodology of the SIMPLE algorithm, formulate pressure correction equations for locations 1, 2, 3 and 4. [12 marks] (b) Use the TDMA algorithm to solve the set of equations to obtain p1′, p2′ , p3′ and p′4 . [6 marks] (c) Using your answers, calculate the corrected velocities at B, C, D and E, and show that they satisfy continuity. [2 marks]

TDMA method For a set of equations of the form: − β jφ j−1 + D jφ j −α jφ j+1 = C j where β j , α j and Dj are coefficients, j = 2,3,4 ....(n-1) are points along a line.

φ j can be obtained from the recurrence formulae: φj = Aj φj +1 + C j′ Aj = C′j =

αj (Dj − β j Aj −1 )

(β C ′ + C ) ; (D − β A ) j

j

j −1

j

j

A1 = 0 and C1′′ = φ1

j −1

H K Versteeg W Malalasekera

Page 4 of 9

Outer cylinder; RO = 18 mm; VZ,O = 0

Fluid

1 2 3 Fluid Inner cylinder; RI = 9 mm; V Z,I = 1 m/s Figure Q1

Figure Q3.1

Figure Q3.2

Page 5 of 9

Control volumes

Attach This to the Answer Book. ID Number of the candidate: ___________________________

TABLE Q3

Node

aW

aE

Su

1 2 3 4 5 6

Page 6 of 9

Sp

aP

Figure Q4

Page 7 of 9

Attach This to the Answer Book. ID Number of the candidate: ___________________________

TABLE Q4

Node

aW

aE

Su

1 2 3 4 5

Page 8 of 9

Sp

aP

Figure Q5

Page 9 of 9...