Taylor-Green vortex

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In fluid dynamics, the Taylor-Green vortex is a 2-dimensional, unsteady flow of a decaying vortex, which has the exact closed form solution of incompressible Navier-Stokes equations in Cartesian coordinates. It is named after the British physicists and mathematicians Geoffrey Ingram Taylor and George Green.

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[edit] Incompressible Navier-Stokes equations

The incompressible Navier-Stokes equation in the absence of body force is given by

\frac{\partial u}{\partial x}+ \frac{\partial v}{\partial y} = 0
\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \nu \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right)
\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} = -\frac{1}{\rho} \frac{\partial p}{\partial y} + \nu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} \right)

The first of the above equation represents the continuity equation and the other two represent the momentum equations.

[edit] Taylor-Green vortex solution

In the domain 0 \le x,y \le \pi , the solution is given by

u = \sin x \cos y F(t) \qquad \qquad v = -\cos x \sin y F(t)

where F(t) = e − 2νt, ν being the kinematic viscosity of the fluid. The pressure field p can be obtained by substituting the velocity solution in the momentum equations and is given by

p = \frac{\rho}{2} \left( \cos 2x + \sin 2y \right) F^2(t)

The Taylor-Green vortex solution may be used for testing and validation of temporal accuracy of Navier-Stokes algorithms.[1] [2]

[edit] References

  1. ^ Chorin, A. J., Numerical solution of the Navier-Stokes equations, Math. Comp., 22, 745-762 (1968).
  2. ^ Kim, J. and Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59, 308-323 (1985).

[edit] See also