Taylor–Green vortex

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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Incompressible Navier-Stokes equations

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

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

Taylor-Green vortex solution

In the domain 0 \le x,y \le 2\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^{-2\nu t}, \nu 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}{4} \left( \cos 2x %2B \cos 2y \right) F^2(t)

The stream function of the Taylor–Green vortex solution, i.e. which satisfies \mathbf{v} = \nabla \times \boldsymbol{\psi} for flow velocity \mathbf{v}, is

\psi = \sin x \sin y F(t)\, \hat{\mathbf{z}}.

Similarly, the vorticity, which satisfies \mathbf{\omega} = \nabla \times \mathbf{v} , is given by

\mathbf{\omega} = 2\sin(x)\sin(y)F(t)\hat{\mathbf{z}}.

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

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).

See also