If you are a masters student in KTH or SU and want to do your masters project with me
you are welcome to contact me. Below I sketch a few possible projects. The purpose
is to give you an idea of the kind of problem we might attack. They are not set it stone.
Singularities of hyper-viscous Burgers equation
In simulations of Navier--Stokes equation with hyperviscosity
it is observed that the intermediate range between
dissipation range and the inertial range has a hump. This
is called ``bottle-neck''. The explanation of this phenomenon
might be along the following line.
The inertial range has a power-law scaling. For pure viscosity
the dissipation range is an exponential fall-off of the
form $e^{-k \delta}$ where $\delta$ is the distance to the
nearest singularity in the complex space. Here the intermediate
range is a matching between the exponential and the decreasing
power-law function and the matching happens smoothly.
In presence of hyper-viscosity the nature of the complex space
singularity will be changed. Hence the nature of the dissipation
range will be changed and might have the form
$k^{\beta}e^{-k \delta}$ where $\beta$ is a positive power.
Then the matching problem becomes more difficult and we
might expect a hump. This idea cannot be easily put to test
in the Navier--Stokes case, but can be put to test in the
case of Burgers equation in the following way.
- First we have to check whether the Burgers equation shows
the ``bottle-neck'' effect or not. This can be easily confirmed
by careful numerical spectral simulation of the Burgers equation
with and without hyperviscosity.
- Next for non-hyperviscous Burgers equation we proceed
in the following way. In the frame of a shock, throw-away the
time-derivative. Then you have the equation
\begin{equation}
\partial_x(u^2) = \nu \partial_{xx} u
\end{equation}
which can be integrated once to obtain,
\begin{equation}
u^2 - a^2 = \nu \partial_x u
\end{equation}
which can again be integrated to yield
\begin{equation}
u(x) = a \tanh \left(\frac{x}{4\nu}\right)
\end{equation}
The Fourier transform of this function should give the
correct form of the tail of the spectrum in Fourier space.
- To obtain the correct form of the spectrum in the presence
of hyperviscosity is more complicated. We need to study the
ordinary differential equation
\begin{equation}
\partial_x(u^2) = \nu \partial_{xx} u + \nu_h \partial_{xxxx} u
\end{equation}
on which one integration gives
\begin{equation}
u^2 - a^2 = \nu \partial_x u + \nu_h \partial_{xxx} u
\end{equation}
It is the third-derivative term which creates the problem.
We might study the fixed points of this differential equation
to obtain some insight into the problem.
A way to tackle this equation in a perturbative way
is the following. Rescale $x$ to transform $x \to x/\nu$,
and $\epsilon = \nu_h/\nu^3$. Then we have
\begin{equation}
u^2 - a^2 = \partial_x u + \epsilon \partial^3_x u
\label{u}
\end{equation}
Then start perturbation is $\epsilon$ by
\begin{equation}
u = u_0 + \epsilon u_1 + \epsilon^2 u_2 + \ldots
\end{equation}
Substituting this in Eq.~\ref{u} we get the following
equation in different orders of $\epsilon$.
\begin{eqnarray}
u_0^2 - a^2 &=& \partial_x u_0 \hspace{1cm} {\rm (zeroth order)} \\
2u_0u_1 &=& \partial_x u_1 + \partial_x^3 u_0 \hspace{1cm} {\rm (first order)} \\
2u_0u_2 + u_1^2 &=& \partial_x u_2 + \partial_x^3 u_1
{\rm (second order) }
\end{eqnarray}
The zeroth order equation has the solution
\begin{equation}
u_0 = a \tanh\left(\frac{x}{4}\right)
\end{equation}
We next need to solve the first order equation.
- Otherwise we might use dominant ballance and Painleve
analysis to study the nature of the complex space singularities
of the hyperviscous equation. The cube-derivative would probably
give a cubic pole in the complex plane which will contribute
to $k^2exp(-\delta k)$ to the tail of the spectrum. This would
illustrate the positive power of $k$ prefactor to the
exponential and then we might try to match this with a power-law
in the inertial range and gain insight into the problem.
Both the methods are somewhat hazy after the initial confidence
but can perhaps be carried to conclusion.