I have a simple problem which I have not understood.

It is well known that the Peclet number ( Peclet = u dx / \nu) must be smaller than 2.0 to maintain numerical stability in a linear convective-diffusive problem, where the convective velocity remains constant. However for a nonlinear problem such as boundary layer flows, as the flow goes away from the wall, the local velocity is increased. To satisfy the condition of Peclet number < 2, does it mean that the spatial size dx should be decreased? Certaninly, it does not. But why?

Remember that at each point there are three velocity components, so, there are three such numbers. In a more general convection-diffusion equation, they are related to the coefficient of the first-order differential terms. The coefficient does not have to be the velocity components. But there are always three first-order terms. The numbers associated with each direction ( coordinates) are not the same. For 1-D, there is one cell Reynolds number, for 2-D, there are two cell Reynolds number, and for 3-D, there are three cell Reynolds number,etc... A good exercise is : try to solve a standard 2-D cavity flow problem. " try it and you will like it !"

Let's discuss the most simple case, 1-D convection-diffusion equation of temperature,

u ( dT/dx ) - a ( d^2 T/dx^2 ) = 0,

where u has boundary-layer-like property, i.e., u(0) = 0 and u(1) = 1. By using 2nd-order central differential scheme, we get the Peclet number = u dx/a. To satisfy the condition Pe < 2, dx should be decreased as u is increased. How to explain?

In fact for a more general problem of convection-diffusion , your pde has the following form:

d/dx(ro*u*T)=d/dx(nu*dT/dx), u and T are your flowfield variables.

then you can integrate this pde on a control volume .

Thev problem you'll have is the correct estimation of the variables values at the interfaces of your control volume.

to do this you can use different discretisation schemes:

1- central difference scheme: limitation Pe<2 2-exponential scheme 3-upwind scheme: not appropriate for low Pe, for high values of Pe ,the diffusion is over estimated. 4-hybrid scheme: maximum error for Pe=2

and finally the power law scheme, which is recommended ( no dependance to Pe)

for more details , check the book of Patankar " "numerical heat transfer and fluid flow"

good luck karim

Patankar's blending scheme is very effective to maintain numerical stability, so it has been widely used in industrial applications. But it often results in losing accuracy of computation when Peclet number > 2. In this case, the accuracy is blended between first and second order.

Maybe the answer of my question is that, as the distance is increased away from the wall, the diffusion terms become smaller and smaller, thus the constraint of Peclet number becomes less important. Consquently a large spatial size is allowed.

But, this answer is incomplete in mathematics.

(1). specify the complete u-profile from x=0 to x=1,(2). specify T at x=0, and T at x=1,(3). write the finite-difference equation or the finite-volume equation from your 1-D convection-diffusion equation,using central difference for convection term and diffusion term,(4).write a Fortran code,(5). run the program, solve the 1-D problem,(6). plot the temperature distribution, examine the temperature profile, (7). change the mesh size based on the condition derived when necessary. (8). tell us your result about the temperature profile.

Hi, This is a very good suggestion from John C. and along the way, it may be useful to look at Example section 3.11 in Ferziger and Peric' "Computational Methods for Fluid Dynamics" which solves the 1-D Advection-Diffusion equation for a constant advection velocity(U) with UDS and CDS constant and variable grid spacing and compare to the exact solution......................................Duan e

Dear Zhong Lei,

As I know, Fromm and Roache derived incorrectly this stability criteria (Pe<2) from the first Von Neumann stability condition (see Charls Hirsch "Numerical Computaions of Internal and External flows". The correct criteria of Hirt is: Pe<2/CFL, where CFL (Couran-Friedrich-Lewy) number is a product of eigen value and time step. For example, for the explicit method, CFL should be smaller than 1, that means, if CFL is sufficient small, Pe can be very large. But the Peclet number can give the impression about the accuracy.

X. Ye

The CFL number is proportional to time step, the sum of sound and gas velocity but oppositional to the grid size. So, if you reduce the grid size, the stability will be reduced.

X. Ye