more flow solver details and numerical considerations

docs/source/user_guide/fluid_solver.rst

@@ -28,24 +28,28 @@ The nondimensional version is
 
    - \frac{1}{\mathrm{Re}}\nabla^2 \mathbf u + \mathbf u \cdot \nabla \mathbf u + \nabla p = 0,
 
-where all quantities have been nondimensionalized, and :math:`\mathrm{Re}` is the Reynolds number. In both cases, we also have the incompressibility condition
+where all quantities have been nondimensionalized, and :math:`\mathrm{Re}=
+\frac{\bar {\mathbf u} L}{\nu}` is the Reynolds number, where :math:`L` is the
+characteristic length.
+
+In both cases, we also have the incompressibility condition
 
 .. math::
 
    \nabla \cdot \mathbf{u} = 0.
 
 Physical parameters are passed as a
 :py:class:`dict` in the ``fluid_params`` argument.
+Specifically, for the dimensional version, the user must pass the following keys:
 
-For the dimensional version, the user must pass the following keys:
+* ``"density"``: :math:`\rho`
 
-* ``"density"``
+* ``"dynamic viscosity"`` or ``"kinematic viscosity"``: :math:`\mu` and :math:`\nu`, respectively
 
-* ``"dynamic viscosity"`` or ``"kinematic viscosity"``
 
 To use the nondimensional version, the user must pass the following key:
 
-* ``"Reynolds number"``
+* ``"Reynolds number"``: :math:`\mathrm{Re}`
 
 Navier-Stokes-Brinkman Solver
 -----------------------------
@@ -61,33 +65,35 @@ The dimensional version of the momentum equation is
    -\nabla\cdot\left(\nu_\mathrm{eff} \nabla\mathbf u \right)+ \mathbf u \cdot \nabla \mathbf u + \frac{1}{\rho} \nabla p + \nu K^{-1} \mathbf u = 0,
 
 where :math:`\nu_\mathrm{eff}` is the effective viscosity in the porous medium,
-and :math:`K`, its permeability. The inverse permeability can be provided
-directly in cases where it is zero in some regions, i.e. liquid-only regions.
-It is common to take :math:`\nu_\mathrm{eff}=\nu` for simplicity (the default
-here).
+and :math:`K`, its permeability. It is also common to use the effective dynamic
+viscosity :math:`\mu_\mathrm{eff} = \nu_\mathrm{eff} \rho`
+
+The inverse permeability :math:`K^{-1}` can be provided directly in cases where
+it is zero in some regions, i.e. liquid-only regions. It is common to take
+:math:`\nu_\mathrm{eff}=\nu` for simplicity (the default here).
 
 The nondimensional implementation currently assume :math:`\nu_\mathrm{eff}=\nu`
-and :math:`K>0`, such that
+and :math:`K^{-1}>0`, such that
 
 
 .. math::
 
    - \frac{1}{\mathrm{Re}}\nabla^2 \mathbf u + \mathbf u \cdot \nabla \mathbf u + \nabla p + \frac{1}{\mathrm{Re}\mathrm{Da}} \mathbf u = 0,
 
-where all quantities have been nondimensionalized and :math:`\mathrm{Da}` is the Darcy number.
+where all quantities have been nondimensionalized and
+:math:`\mathrm{Da}=\frac{K}{L^2}` is the Darcy number.
 
 In addition to the parameters provided for Navier-Stokes, the physical
 parameters below must be provided.
-
 For the dimensional version, the user must also pass the following keys:
 
-* ``"permeability"`` or ``"inverse permeability"``
+* ``"permeability"`` or ``"inverse permeability"``: :math:`K` and :math:`K^{-1}`, respectively
 
-* Optional: ``"effective kinematic viscosity"`` or ``"effective dynamic viscosity"``
+* Optional: ``"effective kinematic viscosity"`` or ``"effective dynamic viscosity"``: :math:`\nu_\mathrm{eff}` and :math:`\mu_\mathrm{eff}`, respectively
 
 To use the nondimensional version, the user must also pass the following key:
 
-* ``"Darcy number"``
+* ``"Darcy number"``: :math:`\mathrm{Da}`
 
 Boundary conditions
 -------------------
@@ -110,3 +116,27 @@ keys.
 
 * ``"outlet pressure"``: Sets outlet pressure, :math:`p = p_\mathrm{out}`,
   which is passed in ``fluid_params`` as ``"outlet pressure"``.
+
+Numerical considerations
+------------------------
+
+The discretization is done using Taylor-Hood elements (piecewise linear
+pressure, piecewise quadratic velocity), which satisfy the inf-sup condition.
+
+To achieve convergence other than at low a Reynolds number, it may be required
+to do continuation on the parameter so that Newton's method has better initial
+guesses. This can be done for example, by passing the Reynolds number as a
+:class:`firedrake.constant.Constant`, and assigning a larger value after each
+solve.
+
+The highest Reynolds number that can be provided is dictated by the validity of
+the steady-sate assumption. Indeed, as the Reynolds number increases, steady
+flow becomes unstable. The critical Reynolds number at which transient effects
+occur is problem dependent, typically somewhere within
+:math:`10^2<\mathrm{Re}<10^4`.
+
+For the Darcy number, using a very small value (:math:`\mathrm{Da}\sim
+10^{-6}`) can be used to simulate a nearly impermeable wall, which is commonly
+done in topology optimization. For smaller values, convergence issues can
+arise. Reversly, as :math:`\mathrm{Da}\to\infty`, we simply recover
+Navier-Stokes.