2marks:

1. Define control volume?It is defined as a closed volume drawn within a finite region of the flow. This volume is defines as

a control volume, V. 2. Distinguish between conservation and non-conservation forms of fluid flow.

Conservation forms Non-conservation formsThe equations obtained from finite control volume fixed in space, in either integral or partial differential form, are called the conservative form.

The equations obtained from finite control volume moving with the fluid, in either integral or partial differential form, are called the conservative form.

3. Define local derivative?∂/∂t is called the local derivative, which is physically the time rate of change at a fixed point.

4. What are the important applications of CFD in engineering? Automobile Engineering Aerospace and Aeronautical engineering Industrial engineering Biomedical Civil Engineering Marine Engineering

5. Define Courant number?

C is called the Courant number. This equation says that Δt≤ Δx/c for the numerical solution to be stable. The above equation is also called the Courant–Friedrichs–Lewy condition, generally written as the CFL condition. It is an important stability criterion for hyperbolic equations.

6. What are the fundamental governing equations of fluid dynamics?Continuity equationMomentum equationEnergy equation

7. What are the types of panel method?Source panel methodVortex panel methodDoublet panel method

8. Define control surface?It is defined as a closed surface which bounds the volume. The control volume may be fixed in

space with the fluid moving through it; it is define as a control surface, S.

9. Write the Classification of Partial Differential Equations.1. Hyperbolic Partial Differential Equations2. Parabolic Partial Differential Equations3. Elliptic Partial Differential Equations

10. Define grid point?

Analytical solutions of partial differential equations involve closed-form expressions which give the variation of the dependent variables continuously throughout the domain. In contrast, numerical solutions can give answers at only discrete points in the domain, called grid points.

11. Write down the significance of Taylor series expansion. Easy to solve higher order equations Can solve any type of equations.

12. Define round-off error?The numerical error introduced after a repetitive number of calculation in which the computer is

constantly rounding the number to some significant figure.

16 marks:13. Write down elliptic, parabolic and hyperbolic partial differential equations as

applicable to CFD. (16)HYPERBOLIC EQUATION:

PARABOLIC EQUATION:

ELLIPTICAL EQUATION:

14. Explain the grid generation technique based on PDE and summarize the advantages of the elliptic grid generation method. (16)Grid can be defined as the region or cell which is formed with the help of intersection of lines and those lines are called grid lines. Elements and nodes.

Hyperbolic Partial Differential Equations Parabolic Partial Differential Equations Elliptic Partial Differential Equations

Elliptic partial differential equations

determines the potential   at each point in space   of a region, provided that boundary conditions are specified on the closed boundary of the region. The differential operator   is called elliptic. To understand this terminology, consider two dimension problem. The locus of   values in  -space of a given eigenvalue

Various types of boundary conditions include: Dirichlet boundary conditions: the value of   is specified on the closed

boundary. In electrostatics this might correspond to specifying the potential on a conducting surface.

Neumann boundary conditions: the normal component   is specified on the boundary. In electrostatics, this corresponds to specifing the normal component of the electric field   on a conductor.

Periodic boundary conditions: are often used to model finite regions of large systems.

Parabolic partial differential equationsGiven a source   as a function of space and time, and a diffusion coefficient  , the diffusion equation

determines the concentration   in a closed region of space, provided thatinitial conditions are specified at some time  , andboundary conditions are specified at the boundaries of the closed region for all times  .The terminology parabolic can be understood by considering one spatial dimension, a constant  , and

for which the differential operator on the left hand side has the eigenvalue

which represents a parabola in   space.Hyperbolic partial differential equationsThe wave equation

is classified as hyperbolic} because the eigenvalues of the differential operator

define hyperboloidal surfaces in   space.Unique solutions of the wave equation are obtained by specifying

initial conditions on the solution and} its first derivative with respect to   at some time  , and

boundary conditions at the boundaries of a closed region for all times  .

15. Discuss the vortex panel method applied to lifting flows over a flat plate.

16. Derive the continuity equation for inviscid flow in partial differential non-conservation form.

17. Derive the energy equation for a viscous flow in partial differential non-conservation form.

. Net heat transfer is given as,

(OR)18. Obtain the 2D steady compressible continuity equation in transformed coordinates

for the transformationξ=x , η= ln ( y+1) .

19. State and explain the difference between explicit and implicit methods with suitable examples.

1.Explicit approach.(a) Advantage. Relatively simple to set up and program.(b) Disadvantage. In terms of our above example, for a given Δx, Δt must be less than some limit imposed by stability constraints. In many cases, Δt must be very small to maintain stability; this can result in long computer running times to make calculations over a given interval of t.2. Implicit approach.

(a) Advantage. Stability can be maintained over much larger values of Δt, hence using considerably fewer time steps to make calculations over a given interval of t. This results in less computer time.(b) Disadvantage. More complicated to set up and program.(c) Disadvantage. Since massive matrix manipulations are usually required at each time step, the computer time per time step is much larger than in the explicit approach.(d) Disadvantage. Since large Δt can be taken, the truncation error is larger, and the use of implicit methods to follow the exact transients (time variations of the independent variable) may not be as accurate as an explicit approach. However, for a time-dependent solution in which the steady state is the desired result, this relative time-wise inaccuracy is not important

19. Discuss the source panel method for the flow past an oscillating cylinder.

20. Write down the procedure for the calculation of pressure coefficient distribution around a circular cylinder using the source panel technique.

21. Explain boundary layer equation and its methodology?

22. Derive the momentum equation in partial differential non-conservation form.

23. Explain the stability properties of an implicit and explicit method?

Consider

, the solution can be written

as,