Taylor–Green vortex

  1. Fluid dynamics
  2. Computational fluid dynamics


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.

Incompressible Navier-Stokes equations

The incompressible Navier–Stokes equations in the absence of body force are 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.

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

\[ p = \frac{\rho}{4} \left( \cos 2x + \cos 2y \right) F^2(t) \]

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


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