In fluid dynamics, the Taylor–Green vortex is an unsteady flow of a decaying vortex, which has an exact closed form solution of the incompressible Navier–Stokes equations in Cartesian coordinates. It is named after the British physicist and mathematician Geoffrey Ingram Taylor and his collaborator A. E. Green.
Original work In the original work of Taylor and Green, a particular flow is analyzed in three spatial dimensions, with the three velocity components v = ( u , v , w ) {\displaystyle \mathbf {v} =(u,v,w)} at time t = 0 {\displaystyle t=0} specified by
u = A cos a x sin b y sin c z , {\displaystyle u=A\cos ax\sin by\sin cz,}
v = B sin a x cos b y sin c z , {\displaystyle v=B\sin ax\cos by\sin cz,}
w = C sin a x sin b y cos c z . {\displaystyle w=C\sin ax\sin by\cos cz.}
The continuity equation ∇ ⋅ v = 0 {\displaystyle \nabla \cdot \mathbf {v} =0} determines that A a + B b + C c = 0 {\displaystyle Aa+Bb+Cc=0} . The small time behavior of the flow is then found through simplification of the incompressible Navier–Stokes equations using the initial flow to give a step-by-step solution as time progresses. An exact solution in two spatial dimensions is known, and is presented below.
Incompressible Navier–Stokes equations The incompressible Navier–Stokes equations in the absence of body force, and in two spatial dimensions, are given by
∂ u ∂ x + ∂ v ∂ y = 0 , {\displaystyle {\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}=0,}
∂ u ∂ t + u ∂ u ∂ x + v ∂ u ∂ y = − 1 ρ ∂ p ∂ x + ν ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 ) , {\displaystyle {\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),}
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