Burger equation
 

Dear All,

I am trying to solve 1-D Burgers equation. EQ: DELU/DELt + u delu/delx = (1/Re)del^2u/delx^2 Is there any physical significance for Re (Reynolds number). As you all know this problem is a good precursor to solving N-S equations.

with best regards, hary

Re: BurgerS equation
 

The Burgers equation is a good way to validate a numerical scheme. Analytical solutions for both 1d and 2d cases are obtained using a Cole-Hopf transformation, though the solutions obtained are usually for infinite domains.

As for 1d case, the most common used analytical solution represent a shock wave. The higher Re number is, the steeper the wave. For that reason, it becomes more and more difficult to capture the propagation of the shock wave, as viscosity decreases. Almost all numerical approaches exhibit oscillations due to false diffusion, and/or shift due to dispersion error, while re-solving a shock.

You may want to try a 2d case as well. Refer to the paper (C. A. J. Fletcher, `Generating Exact Solutions of the Two-dimensional Burgers' Equations', International Journal for Numerical Methods in Fluids, 3(3), 213-216, 1983) for an exact solution.

As for "Is there any physical significance for Re...", I'm affraid I didn't get your question. If it is not what I wrote above, please re-formulate your question.

Re: BurgerS equation
 

Sir,

I am grateful for the insights provided. In a simple pipe flow problem, Re = u*d/nu ; d: pipe diameter. In Burgers equation, is there any correspondance for 'd'?. I will look into the paper of Prof.Fletcher. Thankyou once again, hary

Re: BurgerS equation
 

Hary, It's always preferable to deal with the non-dimensionalized equations and variables. When you consider the Navier-Stokes equations and non-dimensionalize them, you usually end up with 1/Re (Reynolds number) as a factor for the viscous terms. But not always. For same cases, it may be the Prandtl number, or even something else. It strongly depends on how you actually choose reference velocity, length, and etc. As you may see, the choice is related to the problem under consideration, i.e. the definition of it. Coming back to your case, you need to specify what kind of problem you are trying to solve, i.e. set the boundary and initial conditions (solving transient problem, right?). After that, you may get your set of governing dimensionless equations easily. Just pick the reference velocity that does not change in space and time, pick reference length to scale co-ordinates, and that's it. Automatically, you'll get your "Re" number. If you struggle to specify the reference quantities, simply provide us with the defintion of your current problem, and let us see what we could do for you. enjoy colorized fluid dynamics...