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Resolution with a non constant v velocity

Now, the v velocity is not a constant and the 2-D Burgers equation possesses two nolinear terms. We propose here to calculate this speed with the continuity equation defined by 

 
\begin{displaymath}\frac{\partial u} {\partial x} + \frac{\partial v}{\partial y} = 0\end{displaymath} (2.4)
 So the new problem to solve has no physical representation. In fact, if we consider a physical problem describes by the 2-D Burgers equation issue to the simplification of the Navier-Stokes equations, it signified that the v velocity does not depends of x, y and z variables. The introduction of the continuity equation imposed the dependence of these variables for v velocity. Physically, it is impossible to consider the continuity equation and the Burgers' equation together.
However, it may be interresting to take a non constant v velocity in order to see a not uniform transversal convection.

Equation 2.4 gives 


 
\begin{displaymath}v = - \int \frac{\partial u}{\partial x} dy\end{displaymath} (2.5)
 with v=v(x,y,t)

We can see that the v velocity is all more so great since the u velocity gradients are important. This is true where the shock is because it is defined by its discontinuity, so its velocity gradients. With this v velocity we can expect to see the highest transversal convection where the shock is.

Numerically, we approximate the term $\frac{\partial u} {\partial x}$ by a central difference defined by

\begin{displaymath}\left( \frac{\partial u} {\partial x} \right)_{i,j} \simeq \frac{u_{i+1,j}-u_{i-1,j}}{2 \Delta x} \end{displaymath}

And v can be approximated by a trapezoidal method defined by
 
 

\begin{displaymath}v = - \int \frac{\partial u}{\partial x} dy \simeq - \frac{\D...... \left( \frac{\partial u} {\partial x} \right)_{i,j+1} \right] \end{displaymath}

To solve the problem, we must initialized the calculation (see appendix Appendix B) by an arbitrary v velocity : we choose$v^{(0)}=0$.


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Next:The numerical problem Up:The physical problem Previous:Resolution with a constant   Contents
 
Alban Depoutre
2000-11-21