How it Works
The Taylor-Green vortex starts from a periodic sinusoidal initial condition in a box [0, 2π]². The velocity field u(x,y,0) = U·sin(kx)cos(ky) and v(x,y,0) = -U·cos(kx)sin(ky) satisfies incompressibility exactly. We solve the vorticity-streamfunction formulation of the 2D Navier-Stokes equations on a grid using finite differences.
The vorticity ω evolves as Dω/Dt = ν∇²ω. Energy initially concentrated at wavenumber k cascades to higher wavenumbers as vortex structures stretch and fold, while viscosity dissipates energy at the smallest scales. Enstrophy (integral of ω²) first grows then decays.
v(x,y,0) = −U cos(kx) sin(ky)
ω = ∂v/∂x − ∂u/∂y
∂ω/∂t + u·∇ω = ν∇²ω
Frequently Asked Questions
What is the Taylor-Green vortex?
The Taylor-Green vortex is an exact solution to the incompressible Navier-Stokes equations at t=0, defined by a sinusoidal velocity field. It is a canonical test case for turbulence models and numerical methods.
What is an energy cascade in turbulence?
Energy cascade is the process by which kinetic energy is transferred from large eddies to progressively smaller ones until viscous dissipation converts it to heat. Kolmogorov's theory describes the cascade spectrum E(k) ~ k-5/3.
What is enstrophy?
Enstrophy is the volume integral of vorticity squared: E_n = 0.5·∫ω²dV. In 2D turbulence it is conserved; in 3D it grows during the energy cascade and peaks before viscous dissipation dominates.
What are the Navier-Stokes equations?
The Navier-Stokes equations describe the motion of viscous fluids: ρ(Du/Dt) = −∇p + μ∇²u + f. They encode conservation of momentum and, with the continuity equation, conservation of mass.
Why is the Taylor-Green vortex used as a benchmark?
Its analytic initial conditions, periodic boundary conditions, and known energy decay rates make it an ideal test for DNS codes, LES models, and spectral methods without the complexity of boundary-layer flows.
What is Reynolds number and how does it affect the simulation?
Reynolds number Re = UL/ν measures the ratio of inertial to viscous forces. At higher Re the energy cascade extends to smaller scales, enstrophy peaks later, and the flow becomes more chaotic before decaying.
What is Direct Numerical Simulation (DNS)?
DNS resolves all scales of turbulence from the energy-containing eddies down to the Kolmogorov length scale without turbulence modelling. For the Taylor-Green vortex at Re=1600 it requires roughly 512³ grid points.
What is the Kolmogorov length scale?
The Kolmogorov scale η = (ν³/ε)1/4 is the smallest length scale where turbulent energy is dissipated by viscosity. Below this scale the flow is smooth and laminar.
How does vortex stretching affect turbulence?
Vortex stretching occurs when a vortex filament is aligned with a strain-rate eigenvector. Stretching intensifies the vorticity and transfers energy to smaller scales, a key mechanism absent in 2D turbulence.
What happens to kinetic energy over time in the Taylor-Green vortex?
Kinetic energy decays monotonically from its initial value. Enstrophy first rises as vortex structures form and stretch, then decays as viscosity dissipates the small-scale fluctuations.