# Generating initial velocity field

I am trying to generate initial velocity field which satisfies incompressible flows condition. To formulate this I am using Rogallo's procedure formulation which is given by $$\tilde{u}(k) = \frac{\alpha k k_2 + \beta k_1 k_3}{k \sqrt {k_1^2 + k_2^2}} \hat{k_1} + \frac{\beta k_2 k_3 - \alpha k_1 k}{k \sqrt {k_1^2 + k_2^2}}\hat{k_2} - \frac{\beta \sqrt {k_1^2 + k_2^2}}{k}\hat{k_3}$$ where $\alpha$ and $\beta$ are given by $$\alpha = \sqrt{\frac{E_0(k)}{4\pi k^2}} e^{i \theta_1} \cos \phi,\hspace{0.5cm} \beta = \sqrt{\frac{E_0(k)}{4\pi k^2}} e^{i \theta_1} \sin \phi$$ in which $E_0(k)$ is the assumed energy spectrum in my case its $$E_0(k) = A k^4 e^{-0.14 k^2}, \hspace{1cm} k \in [k_a,k_b]$$ My doubt is what are the terms $k , k_1,k_2,k_3$ in this formulation? Secondly, after substituting these terms and obtaining the velocities in fourier space what is the procedure to enforce conjugate symmetry so that real velocities are obtained in the physical space for simulation?

The method produces a velocity field in Fourier space, where $k_1,k_2,k_3$ are the components of the wave vector $k \in {\mathbb R}^3$. In order to get a velocity field in physical space, you need to do an inverse Fourier transform. This typically done via the inverse Fast Fourier Transform (iFFT).
• The question to answer is whether $\text{Re}(iFFT(\tilde u(k)))$ has the same power spectrum as $iFFT(\tilde u(k))$. I don't know the answer to this question, though. – Wolfgang Bangerth Aug 29 '16 at 16:29