Hi everybody

I found in a paper a test case called the Taylor problem. It's used to check the accuracy of a numerical scheme for the two dimensional incompressible viscous flows. In a square [2Pi][2Pi], the initial velocity field is given by :

u(x,y,0) = -cos(x)sin(x)

v(x,y,0) = sin(x)cos(x)

And the analytical solution is :

u(x,y,t) = -cos(x)sin(y)exp(-2t/Re)

v(x,y,t) = sin(x)cos(y)exp(-2t/Re)

P(x,y,t) = -0.25[cos(2x)+cos(2y)]exp(-4t/Re)

Unfortunaly, the authors didn't give any reference to this problem, so I tried to find by myself on the net but no way.

So please, if anyone have any informations about it, i'll be very grateful.

Sorry, i've made a mistake int the initial values:

u(x,y,0) = -cos(x)sin(y)

v(x,y,0) = sin(x)cos(y)

The original work by Taylor: \bibitem{tay23} G.~I. Taylor. Stability of a viscous liquid contained between two rotating cylinders. {\em Phil. Trans. Roy. Soc. Lond A}, 223:289--343, 1923.

You have to have the boundary conditions time dependent. I had a (0,1)(0,1) square

horizontal velocity: v_x=-\cos{(2\pi x)}\sin{(2\pi y)}e^{-\frac{8\pi^2}{Re}t}

vertical velocity v_y=+\cos{(2\pi y)}\sin{(2\pi x)}e^{-\frac{8\pi^2}{Re}t}

vorticity: \omega=4\pi\cos{(2\pi x)}\cos{(2\pi y)}e^{-\frac{8\pi^2}{Re}t}

stream function

\psi=\frac{\omega}{8\pi^2}

with best regards, Jure. iepoi.uni-mb.si/ravnik/research.html