Incompressible viscous hydrodynamics solver
===========================================

pyro's incompressible viscous solver solves:

.. math::

   \frac{\partial U}{\partial t} + U \cdot \nabla U + \nabla p &= \nu \nabla^2 U \\
   \nabla \cdot U &= 0

This solver is based on pyro's incompressible solver, but modifies the
velocity update step to take viscosity into account.

The main parameters that affect this solver are:

.. include:: incompressible_viscous_defaults.inc

Examples
--------

shear
^^^^^

The same shear problem as in incompressible solver, here with viscosity
added.

.. prompt:: bash

   pyro_sim.py incompressible_viscous shear inputs.shear

.. image:: shear_viscous.png
   :align: center

Compare this with the inviscid result. Notice how the velocities have 
diffused in all directions.

cavity
^^^^^^

The lid-driven cavity is a well-known benchmark problem for hydro codes
(see e.g. :cite:t:`ghia1982`, :cite:t:`Kuhlmann2019`). In a unit square box 
with initially static fluid, motion is initiated by a "lid" at the top 
boundary, moving to the right with unit velocity. The basic command is:

.. prompt:: bash

   pyro_sim.py incompressible_viscous cavity inputs.cavity

It is interesting to observe what happens when varying the viscosity, or, 
equivalently the Reynolds number (in this case :math:`\rm{Re}=1/\nu` since 
the characteristic length and velocity scales are 1 by default).

|pic1| |pic2| |pic3|

.. |pic1| image:: cavity_Re100.png
   :width: 32%

.. |pic2| image:: cavity_Re400.png
   :width: 32%

.. |pic3| image:: cavity_Re1000.png
   :width: 32%

These plots were made by allowing the code to run for longer and approach a
steady-state with the option ``driver.max_steps=1000``, then running 
(e.g. for the Re=100 case):

.. prompt:: bash

   python incompressible_viscous/problems/plot_cavity.py cavity_n64_Re100_0406.h5 -Re 100 -o cavity_Re100.png

convergence
^^^^^^^^^^^

This is the same test as in the incompressible solver. With viscosity,
an exponential term is added to the solution. Limiting can again be 
disabled by adding ``incompressible.limiter=0`` to the run command.
The basic set of tests shown below are run as:

.. prompt:: bash

   pyro_sim.py incompressible_viscous converge inputs.converge.32 vis.dovis=0
   pyro_sim.py incompressible_viscous converge inputs.converge.64 vis.dovis=0
   pyro_sim.py incompressible_viscous converge inputs.converge.128 vis.dovis=0

The error is measured by comparing with the analytic solution using
the routine ``incomp_viscous_converge_error.py`` in ``analysis/``. To 
generate the plot below, run

.. prompt:: bash

   python incompressible_viscous/tests/convergence_errors.py convergence_errors.txt

or ``convergence_errors_no_limiter.txt`` after running with that option. Then:

.. prompt:: bash

   python incompressible_viscous/tests/convergence_plot.py

.. image:: incomp_viscous_converge.png
   :align: center

The solver is converging but below second-order, unlike the inviscid case. Limiting
does not seem to make a difference here.

Exercises
---------

Explorations
^^^^^^^^^^^^

* In the lid-driven cavity problem, when does the solution reach a steady-state?

* :cite:`ghia1982` give benchmark velocities at different Reynolds number for the
  lid-driven cavity problem (see their Table I). Do we agree with their results?