Milestone 2: Taylor-Green vortex

In the second milestone, you will implement the spectral solution of the Navier-Stokes equation and test it on a Taylor-Green vortex. The goal is to…

…learn how work with explicit time stepping schemes and to…
…understand that viscosity describes the decay of the velocity field

Navier-Stokes equation

You will now implement the first solver for the incompressible Navier-Stokes equations. We use the rotational form of the equations, which is given by

\[\frac{\partial\vec{u}}{\partial t}=2\vec{u}\times \vec{\omega}+\nu \nabla^2 \vec{u}-\nabla P\]

where $\nu$ is the (kinematic) viscosity and $P(x,y,z)$ the pressure field. The vorticity $\vec{\omega}=\frac{1}{2}\nabla\times\vec{u}$ was already introduced in the previous milestone. This equation is solved subject to the incompressibility condition,

\[\nabla\cdot\vec{u}=0.\]

We can use this incompressibility condition to compute the pressure. Taking the divergence of the Navier-Stokes equations and using incompressibility, we obtain the auxiliary equation

\[\nabla^2 P=2\nabla\cdot\left(\vec{u}\times \vec{\omega}\right).\]

This equation is known as the pressure Poisson equation. You have seen how to solve equations of this type in tutorial 3 and tutorial 4.

Task 1: Spectral Navier-Stokes equations

Rewrite the Navier-Stokes equations and the pressure Poisson equation in terms of the Fourier-transformed velocity field $\tilde{\vec{u}}$. Since $\vec{\omega}$ depends on $\vec{u}$ (see milestone 1), the term $\vec{u}\times\vec{\omega}$ is nonlinear in $\vec{u}$. This term is best computed in real (and not Fourier) space and it makes sense to leave the Fourier transform of this term, $\widetilde{\vec{u}\times\vec{\omega}}$, as is in the equations. Because this implies that part of the calculation is carried out in Fourier space and part in real space, the resulting numerical scheme is often denoted as pseudo spectral. Once you have written everything in the Fourier representation, you can use the pressure Poisson equation to eliminate the pressure in the Navier-Stokes equations.

Hint: The final equation can be found for example in Mortensen & Langtangen, Comput. Phys. Comm. (2016). The first report of a pseudo-spectral Navier-Stokes solver in the literature is from Orszag & Patterson, see Statistical Models and Turbulence (1972) and Physical Review Letters (1972). Make sure to include the full derivation of the this equation into your report.

Time stepping

You are now almost ready to solve the Navier-Stokes equations. You are still missing a time stepping algorithm that propagates the velocity field $\vec{u}$ (or rather $\tilde{\vec{u}}$) forward in time. We suggest to use a simple explicit, fourth-order Runge-Kutta integrator for this. The following code is an implementation of this specific Runge-Kutta method:

def rk4(f, t: float, y: np.ndarray, dt: float) -> np.ndarray:
    """
    Implements the fourth-order Runge-Kutta method for numerical integration
    of multidimensional fields.

    Parameters
    ----------
    f : function
        The function to be integrated. It should take two arguments: time t
        and field y.
    t : float
        The current time.
    y : array_like
        The current value of the field.
    dt : float
        The time step for the integration.

    Returns
    -------
    dy : np.ndarray
        The increment of the field required to obtain the value at t + dt.
    """
    k1 = f(t, y)
    k2 = f(t + dt / 2, y + dt / 2 * k1)
    k3 = f(t + dt / 2, y + dt / 2 * k2)
    k4 = f(t + dt, y + dt * k3)
    return dt / 6 * (k1 + 2 * k2 + 2 * k3 + k4)

Task 2: Fourth-order Runge-Kutta

Use the code snippet above and implement the main loop that steps the simulation forward in time.

Taylor-Green vortex

The Taylor-Green vortex is a simple sinosoidal flow field. It is given by the velocity field

\[\begin{aligned}u_x(x,y,z)&=A\cos ax\sin by\sin cz\\u_y(x,y,z)&=B\sin ax\cos by\sin cz\\u_z(x,y,z)&=C\sin ax\sin by\cos cz\end{aligned}.\]

The two-dimensional Taylor-Green vortex

\[\begin{aligned}u_x(x,y,z)&=A\cos ax\sin by\\u_y(x,y,z)&=B\sin ax\cos by\\u_z(x,y,z)&=0\end{aligned}\]

has an analytical solution that can be used to test whether your code works.

Task 3: Analytical solution

Show that the Taylor-Green vortex represents incompressible flow. The incompressibility condition will link the prefactors $A$, $B$ and $C$ (in the case of the three-dimensional vortex). Then derive the analytical solution for the time evolution of the two-dimensional Taylor-Green vortex. Assume that the prefactor is time-dependent and seek a solution for that time dependence.

Task 4: Numerical solution

Run a numerical calculation of the two-dimensional Taylor-Green vortex and check whether the results agree with your analytical solution.

Hint: You can run the two-dimensional vortex on a system consisting only of a few grid points in one direction to save computational time.

Task summary

We ask you to provide the following results in your final report:

Derivation of the spectral representation of the incompressible Navier-Stokes equations
Derivation of the analytical solution of the two-dimensional Taylor-Green vortex
Plot illustrating a numerical test of your code, carried out by comparing analytical and numerical solutions of the Taylor-Green vortex