% $Id: notes.tex,v 1.11 2021/04/27 05:33:32 brandenb Exp $
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\title{Drag reduction by a turbulent dynamo?}
\author{}
\date{\today,~ $ $Revision: 1.11 $ $}
\begin{document}
\maketitle

\section{}

In pipe flows, drag reduction, for example by the addition of polymers,
allows the mass flow to be increased while keeping the pressure drop
unchanged.
To address this problem numerically, we consider a turbulent flow between
non-slip boundaries.
We begin by considering a situation where the flow is periodic in one of
the two cross-stream directions ($x$) and also periodic in the streamwise
direction ($y$).
Boundary conditions are therefore applied only in the $z$ direction
at $z=\pm L/2$, which are chosen to be at $z=\pm\pi$.
Turbulence is produced by forcing the flow in the volume through a
forcing function that consists of random plane waves that can be helical
or nonhelical.
A magnetic field can emerge solely from dynamo action if the magnetic
Reynolds number is large enough.
We apply insulating boundary conditions, i.e., the magnetic field on
the boundary vanishes, $\nn\times\BB=0$, and there for no electric field
into the boundary, $\nn\cdot\EE=0$.

\section{The model}

We solve the forced isothermal MHD equations for the velocity $\UU$,
the logarithmic density $\ln\rho$, and the magnetic vector
potential $\AAA$,
\EQ
{\DD\UU\over\DD t}=-\cs^2\nab\ln\rho+\ff+\FF+{1\over\rho}
\left(\JJ\times\BB+\nab\cdot2\nu\rho\SSSS\right),
\EN
\EQ
{\DD\ln\rho\over\DD t}=-\nab\cdot\UU,
\EN
\EQ
{\partial\AAA\over\partial t}=\UU\times\BB+\eta\nabla^2\AAA,
\EN
where $\cs=\const$ is the sound speed,
$\ff$ and $\FF$ are forcing functions,
$\BB=\nab\times\AAA$ is the magnetic field,
$\JJ=\nab\times\BB/\mu_0$ is the current density,
$\mu_0$ is the vacuum permeability,
$\nu$ is the viscosity,
${\sf S}_{ij}=(\partial_i U_j+\partial_j U_i)/2+\delta_{ij}\nab\cdot\UU$
are the components of the rate of strain tensor $\SSSS$, 
and $\eta$ is the magnetic diffusivity.
Turbulence is driven by stochastic random waves $\ff$ that
have a different direction at each time step and a mean wavenumber $\kf$,
and the mean flow is driven by the function $\FF=\yyy F$.

In the absence of turbulence, the flow profile is parabolic.
\begin{equation}
U=A\,(\pi-z)(\pi+z)
\end{equation}
with $\max U=A\pi^2$ and $U''=2A$.
Use $F=\nu U''=2\nu A$.
Want $\max U\approx0.5$, so choose $A=0.05$,
so, for $\nu=0.01$, we have $F=2\times0.01\times0.05=10^{-3}$.
The mean work done against the laminar viscous force
is given by $W_\nu=(\pi F)^2/3\nu$.

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{pcomp_prof}
\end{center}\caption[]{
Blue shows the profile for the non-magnetic run, and
red is the a magnetic run with $\Pm=1$ and no helicity.
The green line is also a magnetic run, but with $\Pm=2$.
The red dashed line is a magnetic run scaled up
by a factor 1.082, to show that the profile has
also a different shape.
Likewise, the dashed green line is scaled up by a factor 1.936.
\url{pcomp_prof}
}\label{pcomp_prof}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{peps_512b_nomag_cont_nof3}
\end{center}\caption[]{
Evolution of $\epsK$ for the non-magnetic run (black)
and the magnetic run (blue), showing also $\epsM$ (red)
and $\epsT=\epsK+\epsM$ for this run.
Run~C with $\Pm=2$ (dotted lines) has been restarted from Run~B at $t=6000$.
\url{peps_512b_nomag_cont_nof3_nohel}
}\label{peps_512b_nomag_cont_nof3}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{ppxyaver2_512b_nomag_cont_nof3}
\end{center}\caption[]{
Mean flow profiles for nonmagnetic (upper line) and dynamo-generated
magnetic turbulence (lower line).
}\label{ppxyaver2_512b_nomag_cont_nof3}\end{figure}

\begin{figure}[h!]\begin{center}
\includegraphics[width=\columnwidth]{ppxyaver3_512b_nomag_cont_nof3}
\end{center}\caption[]{
Profile of rms velocity.
}\label{ppxyaver3_512b_nomag_cont_nof3}\end{figure}

\begin{table*}[htb]\caption{
}\vspace{12pt}\centerline{\begin{tabular}{ccccccccl}
Run & $F$ & $\nu$ & $\Pm$ & $\Rey$ & $W_F/W_\nu$ & $\epsT/W_\nu$ & $\epsM/W_\nu$ & comment \\
\hline
A & $5\times10^{-4}$ & $2\times10^{-4}$ & 1 & 317 & 0.075 & $0.325$ &  ---  & non-magnetic \\%512b_nomag
B & $5\times10^{-4}$ & $2\times10^{-4}$ & 1 & 291 & 0.069 & $0.311$ & 0.134 & small-scale dynamo \\%512b_nohel
%C & $5\times10^{-4}$ & $2\times10^{-4}$ & 2 & 171 & 0.038 & $0.270$ & 0.133 & larger $B$ field \\%512b_nohel_Pm
\label{Ttimescale}\end{tabular}}\end{table*}

\Tab{Ttimescale} gives a summary of runs.
The fluid Reynolds number is in the range 200--300;
the smaller values are a result of the suppression of turbulence by the dynamo.
The turbulent suppression of the mean flow can be seen by the smallness of the
values of $W_F/W_\nu$.
In the absence of turbulence they would be unity, but at the Reynolds numbers
considered here, turbulence suppresses the flow speed to between 3\% and 8\%
of the laminar value.

%r e f
\begin{thebibliography}{}

\bibitem[Ogilvie, \& Proctor(2003)]{OP03}
Ogilvie, G. I., \& Proctor, M. R. E.\yjfm{2003}{476}{389}
{409}{On the relation between viscoelastic and magnetohydrodynamic flows and their instabilities}

\bibitem[Ogilvie, \& Potter(2008)]{OP08}
Ogilvie, G. I., \& Potter, A. T.\yprlN{2008}{100}{074503}
{Magnetorotational-type instability in Couette-Taylor flow of a viscoelastic polymer liquid}

%Biskamp, D., \& M\"uller, W.-C.\yprl{1999}{83}{2195}

\end{thebibliography}

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