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

Geoffrey Ingram Taylor

Sir Geoffrey Ingram Taylor OM was a British physicist, mathematician and expert on fluid dynamics and wave theory. His biographer and one-time student, George Batchelor, described him as "one of the most notable scientists of this century".-Biography:Taylor was born in St. John's Wood, London...

 and George Green

George Green

George Green was a British mathematical physicist who wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism...

.

Incompressible Navier-Stokes equations

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

Body force

A body force is a force that acts throughout the volume of a body, in contrast to contact forces.Gravity and electromagnetic forces are examples of body forces. Centrifugal and Coriolis forces can also be viewed as body forces.This can be put into contrast to the classical definition of surface...

 is given by


The first of the above equation represents the continuity equation

Continuity equation

A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...

 and the other two represent the momentum equations.

Taylor-Green vortex solution

In the domain , the solution is given by


where , being the kinematic viscosity of the fluid. The pressure field can be obtained by substituting the velocity solution in the momentum equations and is given by


The stream function

Stream function

The stream function is defined for two-dimensional flows of various kinds. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. Streamlines are perpendicular to equipotential lines...

 of the Taylor–Green vortex solution, i.e. which satisfies for flow velocity , is

Similarly, the vorticity, which satisfies , is given by

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