Classical Turbulence & KS

For the past 5 years I have been involves in an ERCOFTAC special interest group, SIG42. The groups focus is synthetic models of turbulence, and in particular the KS flow.

Here the flow is obtained through the summation of a predefined set of unsteady Fourier modes. More formally

(1)
\begin{align} \mathbf{v}_n({\mathbf{s}},t)= \sum_{m=1}^M\left(\mathbf{A}_m \times \mathbf{k}_m \cos\phi_m + {\bf B}_m \times \mathbf{k}_m \sin\phi_m \right), \end{align}

with $\phi_m=\mathbf{k}_m \cdot \mathbf{s} + \omega_m t$, where $\mathbf{k}_m$ and $\omega_m=\sqrt{k_m^3 E(k_m)}$ are wavevectors and frequencies. It is typical, through an appropriate choice of $\mathbf{A}_m$ and $\mathbf{B}_m$, to impose an energy spectrum of which reduces to the Kolmogorov form $E(k_m)\propto k_m^{-5/3}$ for $1\ll k\ll k_M$, with $k=1$ at the integral scale and $k_M$ at the cut-off scale. The effective Reynolds number

(2)
\begin{align} \mathrm{Re}=(k_M/k_1)^{4/3}, \end{align}

is defined by the condition that the dissipation time equals the eddy turnover time at $k = k_M$.

KS is perhaps most commonly used in the numerical investigation of the statistical properties of both fluid and inertial particles advected by a turbulent flow. I have focused on the use of KS to model the turbulent normal fluid in studies of quantum turbulence at finite temperatures.
The effect of the normal fluid on the motion of the quantised vortices, characteristic of quantum turbulence, can be considered by balancing the magnus and drag forces acting on the vortex. It is then typical to seek a numerical solution (using the vortex filament method) to the resulting equation of motion,

(3)
\begin{align} \frac{d\mathbf{s}}{dt}=\mathbf{v}_s+\alpha\mathbf{s}' \times (\mathbf{v}_n-\mathbf{v}_s) -\alpha'\mathbf{s}' \times \left[ \mathbf{s}' \times (\mathbf{v}_n-\mathbf{v}_s)\right], \end{align}

the so-called Schwarz equation. Here $\alpha$ and $\alpha'$ are temperature dependent friction coefficients, $\mathbf{v}_n$ is the normal fluid velocity (i.e. the KS flow) and $\mathbf{v}_s$ is the superfluid velocity, obtained through the Biot-Savart integral,

(4)
\begin{align} \mathbf{v}_s(\mathbf{s})=\frac{1}{4\pi}\nabla \times \int_V \frac{\boldsymbol{\omega}(\mathbf{r})}{|\mathbf{s}-\mathbf{r}|} \; \mathbf{dr}. \end{align}

Which we can rewrite as a line integral,

(5)
\begin{align} \mathbf{v}_s=-\frac{\kappa}{4 \pi} \oint_{\cal L} \frac{(\mathbf{s}-\mathbf{r}) } {\vert \mathbf{s} - \mathbf{r} \vert^3} \times {\bf dr}, \end{align}

note we must de-singularise the above equation before use.

Please consult my publications page for details of publications. At present I have been looking at structures, bundles of quantised vortices, that are present in systems of quantised vortices that exhibit a Kolmogorov spectrum at large scales. This can then be contrasted with systems of quantised vortices with no large scale flow, counterflow turbulence is an example. Do they also exhibit structures? If so are they different?

As an example let us look at a model of quantum turbulence at 2.16K, where the normal fluid is a prescribed KS flow ($\mathrm{Re} \sim 200$). After reaching a steady state, where energy injected through the normal fluid is balance by dissipation due to reconnections, we take a snapshot of the quantised vortices:

We may think we can see bundles with our eyes, are they real or just an optical illusion? To find out we perform a Gaussian blur, i.e. we convolve the system with a Gaussian kernel. The smoothing length is the taken as the intervortex spacing, $\ell$, the only appropriate length scale.
Below we present a volume rendering of this smoothed 'vorticity' field, the structures/bundles are apparent. Paraview is my weapon of choice for this form of visualisation.

As always please email with an comments and questions on the material above.