Co-rotating vortices



In [1]:

    
using ViscousFlow



In [2]:

    
using Plots
pyplot()
clibrary(:colorbrewer)
default(grid = false)

Set up a grid and a Reynolds number



In [3]:

    
xlim = (-2,2); ylim = (-2,2)
Δx = 0.02
Re = 200









    Out[3]:





200



In [4]:

    
Δt = min(0.5*Δx,0.5*Δx^2*Re)









    Out[4]:





0.01



In [5]:

    
sys = NavierStokes(Re,Δx,xlim,ylim,Δt)









    Out[5]:





Navier-Stokes system on a grid of size 202 x 202



In [6]:

    
w₀ = Nodes(Dual,size(sys));



In [7]:

    
xg,yg = coordinates(w₀,dx=Δx,I0=Systems.origin(sys))









    Out[7]:





(-2.0100000000000002:0.02:2.0100000000000002, -2.0100000000000002:0.02:2.0100000000000002)

Make a Gaussian patch function



In [8]:

    
using LinearAlgebra
gaussian(x,x0,σ) = exp(-LinearAlgebra.norm(x.-x0)^2/σ^2)/(π*σ^2)









    Out[8]:





gaussian (generic function with 1 method)

Set up the integrator

First set up the integrating factor plan



In [9]:

    
plan_intfact(t,w) = Systems.plan_intfact(t,w,sys)









    Out[9]:





plan_intfact (generic function with 1 method)

The right-hand side function



In [10]:

    
r₁(w,t) = Systems.r₁(w,t,sys)









    Out[10]:





r₁ (generic function with 1 method)

And finally, the integrator



In [11]:

    
ifrk = IFRK(w₀,sys.Δt,plan_intfact,r₁,rk=TimeMarching.RK31)









    Out[11]:





Order-3 IF-RK integator with
   State of type Nodes{Dual,202,202}
   Time step size 0.01

Now solve the problem



In [12]:

    
t = 0.0
x01 = (-0.5,0); x02 = (0.5,0); σ = 0.2; Γ = 1
w₀ .= [Γ*gaussian((x,y),x01,σ) + Γ*gaussian((x,y),x02,σ) for x in xg, y in yg]*Δx
w = deepcopy(w₀);



In [16]:

    
tf = 5
T = 0:Δt:tf;



In [17]:

    
@time for ti in T
    global t, w = ifrk(t,w)
end









    



 36.625689 seconds (43.59 k allocations: 4.258 GiB, 1.84% gc time)



In [18]:

    
plot(xg,yg,w)









    Out[18]:



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