[selig5576]

Hi All,

I am writing a CFD code to simulate 2D turbulence via a Fourier spectral method, however am running into issues in terms of the initial condition. I am currently trying to implement McWilliams 2000 initial condition, however have 2 questions regarding setting up the initial condition.

According to McWilliams, the initial condition is a "zero mean, Gaussian vorticity field with random Fourier Phases." The initial energy spectrum is defined as


My interpretation of his initial condition is


The "theta" represents a random Gaussian distribution. In terms of the code, I have implemented it as follows

Lap_hat = sqrt(kx**2 + ky**2)
Energy0 = C0*Lap_hat**6/(1+Lap_hat/60)**18
Vort0 = Energy0*exp(2*1j*np.random.normal(0,1)) + 1j*Energy0*exp(2*1j*np.random.normal(0,1))

Vort_hat = rfft2(Vort0)
Vort_Temp = irfft(Vort0)

Vort = Vort_Temp

Vort represents is an array np.array(nk,nl). In this case I'm running a simulation of 256 x 256. With that said, with my implementation I do not get accurate results. In stead of np.random.normal I have also used np.random.uniform(0,1). I was curious if someone knew of an example code or could point out my mistake. I can also provide my main body of code.


Paper of interesting:
"Revisting Freely Decaiying two-dimensiona turbulence at millennial resolution," J. C. McWilliams, 2000, Physics of Fluids

[FMDenaro]

Originally Posted by selig5576

Hi All,

I am writing a CFD code to simulate 2D turbulence via a Fourier spectral method, however am running into issues in terms of the initial condition. I am currently trying to implement McWilliams 2000 initial condition, however have 2 questions regarding setting up the initial condition.

According to McWilliams, the initial condition is a "zero mean, Gaussian vorticity field with random Fourier Phases." The initial energy spectrum is defined as


My interpretation of his initial condition is


The "theta" represents a random Gaussian distribution. In terms of the code, I have implemented it as follows

Lap_hat = sqrt(kx**2 + ky**2)
Energy0 = C0*Lap_hat**6/(1+Lap_hat/60)**18
Vort0 = Energy0*exp(2*1j*np.random.normal(0,1)) + 1j*Energy0*exp(2*1j*np.random.normal(0,1))

Vort_hat = rfft2(Vort0)
Vort_Temp = irfft(Vort0)

Vort = Vort_Temp

Vort represents is an array np.array(nk,nl). In this case I'm running a simulation of 256 x 256. With that said, with my implementation I do not get accurate results. In stead of np.random.normal I have also used np.random.uniform(0,1). I was curious if someone knew of an example code or could point out my mistake. I can also provide my main body of code.


Paper of interesting:
"Revisting Freely Decaiying two-dimensiona turbulence at millennial resolution," J. C. McWilliams, 2000, Physics of Fluids

Have you considered that E(k) is a kinetic energy and you are setting the intial vorticity field? Be careful of the dimensions.
I suggest to reconstruct first a velocity field by means of the energy spectrum, if it works fine you should get the same energy spectra. Then you can compute the vorticity field.