Module 1: Introduction to Finite Difference Method and Fundamentals of CFD Lecture 13: The Lecture deals with:

Transcription

1 The Lecture deals with: Some more Suggestions for Improvement of Discretization Schemes Some Non-Trivial Problems with Discretized Equations file:///d /chitra/nptel_phase2/mechanical/cfd/lecture13/13_1.htm[6/20/2012 4:39:56 PM]

2 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) file:///d /chitra/nptel_phase2/mechanical/cfd/lecture13/13_2.htm[6/20/2012 4:39:56 PM]

3 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 file:///d /chitra/nptel_phase2/mechanical/cfd/lecture13/13_3.htm[6/20/2012 4:39:57 PM]

4 So, we get or (13.7) Figure 13.1: Limiting Line ( ) The plot of C vs is shown in Fig 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. file:///d /chitra/nptel_phase2/mechanical/cfd/lecture13/13_3.htm[6/20/2012 4:39:57 PM]

5 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 x 1, x 2,...x N. Change d i and c i 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 file:///d /chitra/nptel_phase2/mechanical/cfd/lecture13/13_4.htm[6/20/2012 4:39:57 PM]

6 file:///d /chitra/nptel_phase2/mechanical/cfd/lecture13/13_4.htm[6/20/2012 4:39:57 PM]

7 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 file:///d /chitra/nptel_phase2/mechanical/cfd/lecture13/13_5.htm[6/20/2012 4:39:57 PM]

8 Prove that the CFL condition is the stability requirement for the above discretization scheme. (4) A three-level explicit discretization of Figure 13.2: Grid points at a boundary 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 file:///d /chitra/nptel_phase2/mechanical/cfd/lecture13/13_5.htm[6/20/2012 4:39:57 PM]

9 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 finitedifference 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]. file:///d /chitra/nptel_phase2/mechanical/cfd/lecture13/13_5.htm[6/20/2012 4:39:57 PM]

10 Bibliography 1. Anderson, D.A., Taannehill, J.C, and Pletcher, R.H., Computational Fluid Mechanics and Heat Transfer, Hemisphere Publishing Corporation, New York, USA, Burgers, J.M., A Mathematical Model Illustrating the Theory of Turbulence, Adv. Appl. Mech., Vol. 1, pp , 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 , Fletcher, C.a.j., Computational Techniques for Fluid Dynamics, Vol. 1 (Fundamentals and General Techniques), Springer Verlag, Gentry, Ra., Martin, R.E. and Daly, B.J., An Eulerian Differencing Method for Unsteady Compressible Flow Problems, J. Comput. Phys., Vol.1, pp , Hirt, C.W., Heuristic Stability Theory of Finite Difference Equation, J. Comput. Phys., Vol. 2, pp , 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 , Khosla, P.K. and Rubin, S.G., A Diagonally Dominant Second Order Accurate Lmplicit Scheme, Computer and Fluids Vol. 2, pp , Lax, P.D. and Wendroff, B. Systems of Conservation Laws, Pure Appl. Math, Vol. 13, pp , Leonard, B.P., A Stable and Accurate Convective Modelling Procedure based on Quadratic Upstream Interpolation, Comp. Method Appl. Mech. Engr., Vol. 19, pp , Rai, M.M. and Moin, P., Direct Simulations of urbulent Flow Using Finite Difference Schemes, J. Comput. Phys., Vol. 96, pp , 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 , 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 , Thomas, L.H., Elliptic Problems in Linear Difference Equations Over a Network, Waston Sci. Comput. Lab. Rept., Columbia University, New York, Vanka, S.P., Second-Order Upwind Differencing in a Recirculating Flow, AIAA J., Vol 25, pp. 1441, Congratulations, you have finished Lecture 13. To view the next lecture select it from the left hand side menu of the page or click the next button. file:///d /chitra/nptel_phase2/mechanical/cfd/lecture13/13_6.htm[6/20/2012 4:39:57 PM]