1. The document is an exam for a Computational Fluid Dynamics course covering topics like FDM, FVM, FEM, direct/iterative solving methods, turbulence models, and commercial CFD software. 2. It contains 6 questions - the first being a short answer question on various CFD topics, and questions 2-6 involving derivations, analyses, and programming related to computational fluid flow and heat transfer problems. 3. Students are instructed to answer question 1 and any 4 of the remaining 5 questions, showing work and using diagrams where needed. Problems involve finite volume discretization, 1D and 2D conduction/convection analyses, and implementing the SIMPLE algorithm.

1. VEERMATA JIJABAI TECHNOLOGICAL INSTITUTE
Matunga, Mumbai – 400 019
[Autonomous]
End Semester Examination
Sem & Programme VIII Final Year B. Tech (Mechanical) Duration 03 Hours
Course code & Course ME0453 Computational Fluid Dynamics Max.Marks 100
Date of Exam 3-5-2011
Instructions: 1. Question 1 is compulsory. Solve any 4 out of the remaining.
2. Figures to the right indicate full marks.
3. Assume suitable data if necessary.
4. Illustrate your answers with neat sketches wherever necessary.
1. Answer in brief. 2
0
a) Compare FDM, FVM and FEM.
b) Compare direct methods with iterative methods for solving large number of
equations.
c) What is the difference between SIMPLE, SIMPLER and SIMPLEC?
d) What are advantages and disadvantages of k-ε turbulence model?
e) Explain the relevance of CFD theory in successful use of commercial software.
2. a) Derive energy equation and express it in conservation form. 1
5
b) Explain the finite difference method for any governing equation with suitable 5
boundary conditions.
3. The governing differential equation for a thin fin with uniform cross-sectional
area, where the heat loss to the surrounding is purely by convection, is given by
d 2T hP
− ( T − T∞ ) = 0
d 2 x kA
a) Following the standard procedure of the Finite Volume discretization process,
derive the discretization equation (DE) in usual form 6
a P TP = ∑ a nb Tnb + b
b) Use the following data: A = 2 x 10-5 m2, P = 0.015 m, k = 20 W/m-k, 9
h = 10 W/m2-K, T∞ = 250C, The temperature at the left end (”base”) is
Tb = 1000C, while that at the right end (“tip”) is TL = 400C. The length of the
fin is L = 20 mm. Use ∆x = 4 mm. Obtain the complete system of algebraic
equations and solve for the temperature distribution within the fin.
c) The analytical expression for the temperature distribution is given by 2
 T − T∞ 
  sinh mx + sinh m( L − x )
T − T∞  Tb − T∞ 
 
=
Tb − T∞ sinh mL
hP
where m = . Compare your numerical solution with the analytical solution
kA
at any one internal point in the fin.
d) If right end (“tip”) is insulated or open to atmosphere, how the algebraic
equation will change for the boundary point? 3
4. a) A one-dimensional slab of 1 m width and a constant thermal diffusivity of 1 m2/hr 1
is initially at a uniform temperature of 1000C. The surface temperatures of the left 0
(x = 0) and right (x = L) faces are suddenly increased and maintained at 300 0C.
There are no sources. Determine the temperature distribution within the wall at

2. 0.1 hr and 0.2 hr. Use a grid size of 0.25 m and time step of 0.1 hr. Solve the
problem with the fully implicit method.
b) Write a general program to get temperature distribution for two dimensional 1
steady state conduction. The west boundary is at constant temperature, east 0
boundary receives steady heat flux, North boundary is exposed to atmosphere
and South boundary is insulated. The program should take user input for
dimensions, material properties, no. of nodes in 2 directions and boundary
conditions.
5. Consider Finite volume discretization of 1-D steady convection-diffusion equation
for a variable φ , with no sources and a constant diffusion coefficient Γ . The
diffusion term is evaluated using the usual central differencing scheme. For the
convection term, a Second Order Upwind (SOU) scheme is used. In particular,
the CV face value is evaluated by extrapolating values at two upstream points
with respect to that face.
a) For a uniform grid of size ∆x , considering flow in both the directions derive
the generalized DE in the form 8
a P TP = ∑ a nbTnb
and provide expressions for the coefficients a.
b) Consider that left boundary condition is specified in terms of a known φ L , 4
obtain the appropriate DE for the CV adjacent to boundary using mirror node
concept for the case of flow going from left to right.
c) Write a general program to implement the SOU scheme using terms obtained 8
in a) and b).
6. a) Explain SIMPLE algorithm with a flowchart. 1
0
b) 1
0
1 B 2 C 3
A one-dimensional flow through a porous material is governed by
c | u | u + dp / dx = 0 , where c is a constant. The continuity equation is d (uA) / dx = 0
, where A is the effective area for the flow. Use SIMPLE procedure for the grid
shown in fig. to calculate P2 , u B and u C from the following data:
x 2 − x1 = x3 − x 2 = 2
c B = 0.25, cC = 0.2, AB = 5, AC = 4, P1 = 200, P3 = 38
As an initial guess, set u B = u C = 15 and P2 = 120
(Calculate pressure correction based on imbalance of mass flow rate and correct
velocity based on new values of pressure. Perform few iterations. Mass source
will reduce in each iteration. Take values of pressure and velocity to at least 3
decimal places.)