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Kelvin-Helmholtz instability : Benchmark computations

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Introduction

 

     

This benchmark simulates the Kelvin-Helmholtz instability problem. We provide reference computations for this problem. On this page we describe the problem set up and give on overview on the quantities of interest. For simulation results and details that allow for a comparison of computational results we refer to our repository with our reference results

Geometry and flow setup

The computation domain is a square Ω=[0,1]2.
The initial condition for the velocity u is given by 

u0(x,y)=(utanh(2y1δ0)0)+cn(yψ(x,y)xψ(x,y))

with the stream function
ψ(x,y)=uexp((y0.5)2δ02)[cos(8πx)+cos(20πx)].Here, δ0=1/28 denotes the initial vorticity thickness, u=1 is a reference velocity and cn=103 is a scaling/noise factor.

For the solution to the problem we consider the incompressible Navier-Stokes equations:

ut+(u)uνΔu+p=0 div(u)=0

At x=0 and x=1, periodic boundary conditions are prescribed and at y=0 and y=1, free-slip boundary conditions. 

 

The Reynolds number Re of the Kelvin--Helmholtz instability flow is usually calculated on the bases of the characteristic length scale δ0 and the characteristic velocity scale u, i.e., Re=δ0u/ν=1/(28ν)
We consider different values for the viscosity ν, such that we consider Re{100,1000,10000}
For the simulations and their evaluation, the time unit t¯=δ0/u is introduced. The time interval of interest is [0,400t¯]. The flow is only driven by the inital data. There are no additional forces acting.

Quantities of interest

In the following we list the quantities of interested that have been collected for comparison:

1. Vorticity field at selected points in time

Vorticity, ω:=curl(u)=xu2yu1 as a field on Ω. Time points of interest are t/t¯{5,10,17,34,36,180,220,257,400}. The pictures on the top of this page correspond to t/t¯{10,34,36}.

2. Kinetic energy over time:

K(t,u)=u(t)L2(Ω)2,t/t¯{0,1,..,400}

3. Kinetic energy spectra at selected points in time

E(κ,t)=01|01u1(t,x,y)eiκxdx|2dy Here, u1 is the velocity in x-direction. Time points of interest are t/t¯{5,10,17,34,36,180,220,257,400}.

4. Vorticity thickness over time

δ(t)=2u/supy(0,1)|ω(t,y)|,t/t¯{0,1,..,400} where the integral mean vorticity in x-direction, ω is defined as ω:=01ω(t,x,y)dx where ω=curl(u)=xu2yu1 is the vorticity of the flow.

5. Enstrophy over time

E(t,u):=12ω(t)L2(Ω)2=12curl(u)L2(Ω)2,t/t¯{0,1,..,400}

6. Palinstrophy over time

P(t,u):=12ω(t)L2(Ω)2=12curl(u)L2(Ω)2,t/t¯{0,1,..,400}

For further details we refer to the paper [1] and to our repository with our reference results

References