Fluid Variables

Variable	Definition
\(\rho\)	Fluid density
\(U\)	Fluid velocity
\(p\)	Pressure
\(\eta\)	Viscosity (incflo.mu for a Newtonian fluid)
\(\tau\)	Viscous stress tensor, \(\eta (\nabla U + (\nabla U)^T)\)
\(\mu_s\)	scalar diffusivity
\(\mu_T\)	thermal conductivity
\(c_p\)	Specific heat (incflo.cp)
\({\bf g}\)	Gravitational acceleration
\({\bf H}_U\)	\(= (H_x , H_y , H_z )\), External Forces
\(H_s\)	External sources
\(H_T\)	External heat sources

Fluid Equations

Conservation of fluid mass:

\[\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho U) = 0\]

Conservation of fluid momentum:

\[\frac{ \partial (\rho U)}{\partial t} + \nabla \cdot (\rho U U) + \nabla p = \nabla \cdot \tau + {\bf H}_U\]

By default, incflo solves the equivalent convective form, \(\rho ( \partial U / \partial t + U \cdot \nabla U ) + \nabla p = \nabla \cdot \tau + {\bf H}_U\); incflo.advect_momentum = true selects the conservative form above. With incflo.use_tensor_solve = false, the velocity components are diffused separately, with \(\nabla \cdot (\eta \nabla U)\) in place of \(\nabla \cdot \tau\) (the two are equal for a constant viscosity).

Velocity constraint:

\[\nabla \cdot U = 0\]

or, with incflo.use_divu_source = 1, the constraint

\[\nabla \cdot U = S\]

for a given source \(S\), as in low Mach number models. Mass is still conserved, so the density changes along particle paths as \(D\rho/Dt = -\rho S\). With incflo.constant_density = 1 the density is not evolved, so mass is not conserved: the constraint adds mass at the rate \(\rho_0 S\) per unit volume. That is not a low Mach number model, and incflo warns about it, but it has exact solutions that are useful for testing. \(S\) comes from incflo::compute_divu_source, which by default uses an analytic \(S\) defined for the chosen incflo.probtype. It is virtual, so a class derived from incflo can compute \(S\) from the state instead. In a closed domain (only walls and periodic boundaries) the projections can only enforce \(\nabla \cdot U = S - \bar{S}\), where \(\bar{S}\) is the mean of \(S\); incflo warns when \(\bar{S} \ne 0\). That is a uniform change of the thermodynamic pressure, which incflo does not evolve.

Tracer(s): for conservative,

\[\frac{\partial \rho s}{\partial t} + \nabla \cdot (\rho U s) = \nabla \cdot \mu_s \nabla s + \rho H_s\]

or, for non-conservative (see incflo.trac_is_conservative),

\[\frac{\partial s}{\partial t} + U \cdot \nabla s = \nabla \cdot \mu_s \nabla s + H_s\]

Optional temperature:

\[\rho c_p \left( \frac{\partial T}{\partial t} + U \cdot \nabla T \right) = \nabla \cdot \mu_T \nabla T + H_T\]

By default, \(H_s = 0\), \(H_T = 0\), and \({\bf H}_U = {\bf 0}\). If gravity is set (incflo.gravity), then \({\bf H}_U = \rho {\bf g}\), except for the Boussinesq problems (incflo.probtype = 11, 111, 112, 113 and 500), which use \({\bf H}_U = \rho s_0 {\bf g}\), where \(s_0\) is the first tracer.

Background pressure

Except for the Boussinesq problems, incflo writes the pressure as \(p = p_0 + p'\), where the background pressure \(p_0\) has a constant gradient, and solves for \(p'\):

• in a direction with gravity, \(\nabla p_0 = \rho_0 {\bf g}\), the hydrostatic pressure gradient for the density \(\rho_0\) (incflo.ro_0);

• in the direction in which incflo.delp is non-zero, \(\nabla p_0 = - \mathrm{delp} / L\), where \(L\) is the length of the domain in that direction. This pressure drop drives the flow, e.g. in a periodic channel;

• in a direction with a pressure inflow at one end and a pressure outflow at the other, \(\nabla p_0 = (p_{hi} - p_{lo}) / L\), where \(p_{lo}\) and \(p_{hi}\) are the boundary pressures.

Only one direction can have a delp or a pressure inflow/outflow pair, and gravity cannot be in that direction. The pressure written to plotfiles (p) is \(p'\), while the pressure gradient (gpx, gpy and gpz) includes \(\nabla p_0\).