The Lecture deals with:

Transcription

1 Lecture 11 The Lecture deals with: The upwind scheme Transportive Property Upwind Differencing and Artificial Viscosity file:///d /chitra/nptel_phase2/mechanical/cfd/lecture11/11_1.htm[6/20/2012 4:36:58 PM]

2 Lecture 11: The Upwind Scheme Once again, we shall start with the inviscid Burger's equation. (9.12) Regarding discretization, we can think about the following formulations (11.1) (11.2) If Von Neumann's stability analysis is applied to these schemes, we find that both are unconditionally unstable. A well known remedy for the difficulties encountered in such formulations is the upwind scheme which is described by Gentry, Martin and Daly (1966) and Runchal and Wolfshtein (1969). Eq. (11.1) can be made stable by substituting the forward space difference by a backward space difference scheme, provided that the carrier velocity u is positive. If u is negative, a forward difference scheme must be used to assure stability. For full Burger's equation. (9.11), the formulation of the diffusion term remains unchanged and only the convective term (in conservative form) is calculated in the following way (Figure 11.1): viscous term, for (11.3) viscous term, for (11.4) Figure 11.1: The Upwind Scheme It is also well known that upwind method of discretization is very much necessary in convection (advection) dominated flows in order to obtain numerically stable results. As such, upwind bias retains transportative property of flow equation. Let us have a closer look at the transportative property and related upwind bias. file:///d /chitra/nptel_phase2/mechanical/cfd/lecture11/11_2.htm[6/20/2012 4:36:58 PM]

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4 Lecture 11: Transportive Property A finite-difference formulation of a flow equation possesses the transportive property if the effect of a perturbation is convected (advected) only in the diprection of the velocity. Consider the model Burger's equation in conservation form (11.5) Let us examine a method which is central in space. Using FTCS we get (11.6) Consider a perturbation in. A perturbation will spread in all directions due to diffusion. We are taking an inviscid model equation and we want the perturbation to be carried along only in the direction of the velocity. So, for (perturbation at m th space location), all other. Therefore, at a point (m+1) downstream of the perturbation which is acceptable. However, at the point of perturbation ( i=m) which is not very reasonable. But at the upstream station ( i = m-1 ) we observe which indicated that the transportive property is violated. On the contrary, let us see what happens when an upwind scheme is used. We know that for u>0 (11.7) Then for at the downstream location (m+1) file:///d /chitra/nptel_phase2/mechanical/cfd/lecture11/11_3.htm[6/20/2012 4:36:59 PM]

5 which follows the rational for the transport property. At point m of the disturbance which means that the perturbation is being transported out of the affected region. Finally, at ( m-1) station, we observe that This signifies that no perturbation effect is carried upstream. In other words, the upwind method maintains unidirectional flow of information. In conclusion, it can be said that while space centred difference are more accurate than upwind differences, as indicated by the Taylor series expansion, the whole system is not more accurate if the criteria for accuracy includes the tranportive property as well. file:///d /chitra/nptel_phase2/mechanical/cfd/lecture11/11_3.htm[6/20/2012 4:36:59 PM]

6 Lecture 11: Upwind Differencing and Artificial Viscosity Consider the model Burger's equation. (9.11) and focus the attention on the inertia terms As seen, the simple upwind scheme gives for u > 0 for u < 0 From Taylor series expansion, we can write (11.8) (11.9) Substituting Eqns. (11.8) and (11.9) into (11.3) gives (dropping the subscript i and superscript n) [Diffusive terms] or which may be rewritten as file:///d /chitra/nptel_phase2/mechanical/cfd/lecture11/11_4.htm[6/20/2012 4:36:59 PM]

7 higher order terms (11.10) where C (Courant number) In deriving Eq. (11.10), was taken as. However, the nonphysical coefficient leads to diffusion like term which is dependent on the discretization procedure. This is known as the numerical or artificial viscosity. Let us look at the expression somewhat more critically.., for u > 0 (11.11) On one hand we have considered that u > 0 and on the other CFL condition demands that C < 1 (so that the algorithm can work). As a consequence, work). is always a positive non-zero quantity ( so that the algorithm can If, instead of analyzing the transient equation, we put in Eq. (11.3) and expand it in Taylor series, we obtain (11.12) file:///d /chitra/nptel_phase2/mechanical/cfd/lecture11/11_4.htm[6/20/2012 4:36:59 PM]

8 Lecture 11: Let us now consider a two-dimensional convective-diffusive equation with viscous diffusion in both directions (Eq. (8.13) but with. For upwind differencing gives (11.13) The Taylor series procedure as was done for Eq. (11.10) will produce (11.14) where with As such for and CFL condition is This indicate that for a stable calculation, artificial viscosity will necessarily be present. However, for a steady-state analysis, we get (11.15) We have observed that some amount of upwind effect is indeed necessary to maintain transportive property of flow equations while the computations based on upwind differencing often suffer from false diffusion (inaccuracy!). One of the plausible improvements is the usage of higher-order upwind method of differencing. In the next lecture we'll discuss this aspect of improving accuracy. Congratulations, you have finished Lecture 11. 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/lecture11/11_5.htm[6/20/2012 4:36:59 PM]