Concept of Computational Fluid Dynamics Material

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file:///D|/chitra/nptel_phase2/mechanical/cfd/lecture13/13_1.htm[6/20/2012 4:39:56 PM]
Module 1: Introduction to Finite Difference Method and Fundamentals of CFD
Lecture 13:
The Lecture deals with:
Some more Suggestions for Improvement of Discretization Schemes
Some Non-Trivial Problems with Discretized Equations

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Lecture 13:
Third-order Upwind Differencing
Another widely suggested improvement is known as third-order upwind differencing (see
Kawamura et al. 1986).
The following example illustrates the essence of this discretization scheme.
(13.1)
Higher order upwind is an emerging area of research in Computational Fluid Dynamics.
However, so far no unique suggestion has been evolved as an optimal method for a wide
variety of problems. Interested readers are referred to Vanka (1987), Fletcher (1988) and
Rai and Moin (1991) for more stimulating information on related topics.
One of the most widely used higher order schemes is known as QUICK (Leonard, 1979).
The QUICK scheme may be written in a compact manner in the following way
(13.2)
The fifth-order upwind scheme (Rai and Moin, 1991) uses seven points stencil along with
sixth-order dissipation. The scheme is expressed as
(13.3)

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Lecture 13:
Some Non-Trivial Problems with Discretized Equation
The discussion in this section is based upon some ideas indicated by Hirt (1968) which are
applied to model Burger's equation as
(13.4)
From this, the modified equation becomes
(13.5)
We define
Courant number
It is interesting to note that the values and C=1 (which are extreme conditions of
Von Neumann stability analysis) unfortunately eliminates viscous diffusion completely in
Eq. (13.5) and produce a solution from Eq. (13.4) directly as which is
unacceptable. From Eq. (13.5) it is clear that in order to obtain a solution for convection
diffusion equation, we should have
For meaningful physical result in the case of inviscid flow we require
Combining these two criteria, for a meaningful solution
(13.6)
Here we define the mesh Reynolds-number or cell-peclet number as

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So, we get
or
(13.7)
Figure 13.1: Limiting Line ( )
The plot of C vs is shown in Fig. 13.1 to describe the significance of Eq. (13.7).
From the CFL condition, we know that the stability requirement is Under such a
restriction, below the calculation is always stable. The interesting information is
that it is possible to cross the cell Reynolds number of 2 if C is made less than unity.

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Lecture 13:
Thomas algorithm
In Crank Nicolson solution procedure, we get a system of algebraic equations which
assumes the form of a tridiagonal matrix problem. Here we shall discuss a very well known
solution procedure known as Thomas algorithm (1949) which utilizes efficiently the
advantage of the tridiagonal form. A tridiagonal system is:
The Thomas Algorithm is a modified Gaussian matrix-solver applied to a tridingonal system.
The idea is to transform the coefficient matrix into a upper triangular form. The
intermediate steps that solve for x1, x2, ...xN .
Change di and ci arrays as
i = 2,3,....N
and
Similarly
i = 2,3,....N
and
At this stage the matrix in upper triangular form. The solution is them obtained by back
substitution as
and

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Lecture 13:
Problems
(1) Consider the nonlinear equation
(13.8)
where µ is a constant and u the x component of velocity. The normal direction is y.
(a) Is this equation in conservative from? If not, suggest a conservative from of the equation.
(b) Consider a domain in to x ( x = 0 to x = L) and y (y = 0 to y = H) and assume that all the
value of the dependent variable are known at x = 0 (along y = 0 to y = H at every y interval).
Develop an implicit expression for determining u at all the points along (y=0 to y=H) at the
next (x+Δx)
(2) Establish the truncation error of the following finite-difference approximation to
at the point for a uniform mesh
What is the order of the truncation error ?
If you want to apply a second-order-accurate boundary condition for at the
boundary (refer to Fig. 13.2), can you make use of the above mentioned expression? If
yes, what should be the expression for at the boundary?
(3) The lax-Wendroff finite difference scheme (Lax and Wendroff, 1960) can be derived
from a Taylor series expansion in the following manner:
Using the wave equations
the Taylor series expression may be written as

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Prove that the CFL condition is the stability requirement for the above discretization scheme.
Figure 13.2: Grid points at a boundary
(4) A three-level explicit discretization of
can be written as
Expand each term as a Taylor series to determine the truncation error of the complete
equation for arbitary values of d. Suggest the general technique where for a functional
relationship between d and the scheme will be fourth-order accurate in
.
(5) Consider the equation

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where T is the dependent variable which is convected and diffused. The independent
variable, x and y, are in space while t is the time (evolution) coordinate. The coefficient
u,v and a can be treated as constant. Employing forward difference for the first-order
derivative and central-second difference for the second derivatives, obtain the finite-
difference equation. What is the physical significance of the difference between the above
equation and the equation actually being solved? Suggest any method to overcome this
difference.
(6) Write down the expression for the Finite Difference Quotient for the convective term of
the Burger's Equation given by
(13.9)
Use upwind differencing on a week conservative from of the equation. The upwind
differencing is known to retain the transportive property. Show that the formulation preserves
the conservative property of the continuum as well [you are allowed to exclude the diffusive
term from the analysis].

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Lecture 13:
Bibliography
1. Anderson, D.A., Taannehill, J.C, and Pletcher, R.H., Computational Fluid Mechanics
and Heat Transfer, Hemisphere Publishing Corporation, New York, USA, 1984.
2. Burgers, J.M., A Mathematical Model Illustrating the Theory of Turbulence, Adv. Appl.
Mech., Vol. 1, pp. 171-199, 1948.
3. Dufort, E.C. and Frankel, S.P., Stability Conditions in the Numerical Treatment of
Parabolic Differential Equations, Mathematical Tables and Others Aids to
Computation, Vol 7, pp. 135-152, 1953.
4. Fletcher, C.a.j., Computational Techniques for Fluid Dynamics, Vol. 1 (Fundamentals
and General Techniques), Springer Verlag, 1988.
5. Gentry, Ra., Martin, R.E. and Daly, B.J., An Eulerian Differencing Method for
Unsteady Compressible Flow Problems, J. Comput. Phys., Vol.1, pp. 87-118,1966.
6. Hirt, C.W., Heuristic Stability Theory of Finite Difference Equation, J. Comput. Phys.,
Vol. 2, pp. 339-335, 1968.
7. Kawamura, T., Takami, H. and Kuwahara, K., Computation of High Reynolds Number
Flow around a Circular Cylinder with Surface Roughness, Fluid Dynamics Research,
Vol. 1. pp. 145-162, 1986.
8. Khosla, P.K. and Rubin, S.G., A Diagonally Dominant Second Order Accurate Lmplicit
Scheme, Computer and Fluids Vol. 2, pp. 2.7-209, 1974.
9. Lax, P.D. and Wendroff, B. Systems of Conservation Laws, Pure Appl. Math, Vol. 13,
pp. 217-237, 1960.
10. Leonard, B.P., A Stable and Accurate Convective Modelling Procedure based on
Quadratic Upstream Interpolation, Comp. Method Appl. Mech. Engr., Vol. 19, pp. 59-
98, 1979.
11. Rai, M.M. and Moin, P., Direct Simulations of urbulent Flow Using Finite Difference
Schemes, J. Comput. Phys., Vol. 96, pp. 15-53, 1991.
12. Raithby, G.D. and Torrance , K.E., Upstream-weighted Differencing Scheme and
Their Applications to Elliptic Problems Involving Fluid Flow, Computers and Fluids,
Vol. 2, pp. 191-206, 1974.
13. Roache, P.J., Computational Fluid Dynamics, Hermosa, Albuquerque , New Mexico ,
1972 (revised printing 1985).
14. Runchal, A.k. and Wolfshtein, M., Numerical Integration Procedure for the Steady
State Navier-Strokes Equations, J. Mech Engg. Sci., Vol. 11, pp. 445-452, 1969.
15. Thomas, L.H., Elliptic Problems in Linear Difference Equations Over a Network,
Waston Sci. Comput. Lab. Rept., Columbia University , New York , 1949.
16. Vanka, S.P., Second-Order Upwind Differencing in a Recirculating Flow, AIAA J ., Vol
25, pp. 1441, 1987.
Congratulations, you have finished Lecture 13. To view the next lecture select it from the
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Concept of Computational Fluid Dynamics Material

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