Fluid Variables

The fluid variables are defined as follows.

Variable	Definition
\(\rho\)	Fluid density
\(U\)	Fluid velocity
\(\tau\)	Viscous stress tensor
\({\bf H}_U\)	\(= (H_x , H_y , H_z )\), External Forces
\(H_s\)	External sources
\(\phi\)	Level set field
\(\alpha_p\)	Particle volume fraction
\(\mathbf{F}_p\)	Eulerian force

Compared with IAMR, IAMReX adds a level set field \(\phi\) for the two-phase flow, and the particle volume fraction \(\alpha_p\), and the Eulerian force \(\mathbf{F}_p\) for the particle-fluid interaction.

Fluid Equations

Conservation of fluid mass:

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

Conservation of fluid momentum:

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

Velocity constraint:

\[\nabla \cdot U = S\]

where \(S\) is zero by default, to model incompressible flow. The \(S \ne 0\) case is discussed below.

Tracer(s):

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

for conservatively advected scalars and

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

for passively advected scalars. In general, one could advect an arbitrary number of scalars.

Level set equation:

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

IAMReX has the ability to incorporate general, user-defined external forcing and source terms. The default behaviour is that \(H_s=0\), and \({\bf H}_U\) represents gravitational forces, with \({\bf H}_U= (0 , 0 , -\rho g )\) in 3d and \({\bf H}_U= (0 , -\rho g )\) in 2d, where \(g\) is the magnitude of the gravitational acceleration. However, since by default, \(g=0\), \({\bf H}_U = 0\) unless ns.gravity is set.

By default, IAMReX solves the momentum equation in convective form. The inputs parameter ns.do_mom_diff = 1 is used to switch to conservation form. Tracers are passively advected by default. The inputs parameter ns.do_cons_trac = 1 switches the first tracer to conservative. A second tracer can be included with ns.do_trac2 = 1, and it can be conservatively advected with ns.do_cons_trac2 = 1.

IAMReX also has the option to solve for temperature, along with a modified divergence constraint on the velocity field:

\[ \begin{align}\begin{aligned}\rho c_p \left( \frac{\partial T}{\partial t} + U \cdot \nabla T \right) = \nabla \cdot \lambda \nabla T + H_T\\\nabla \cdot U = \frac{1}{\rho c_p T} \nabla \cdot \lambda \nabla T\end{aligned}\end{align} \]

Here, the divergence constraint captures compressibily effects due to thermal diffusion. To enable the temperature solve, use ns.do_temp = 1 and set ns.temp_cond_coef to represent \(\lambda / c_p\), which is taken to be constant. More sophiticated treatments are possible; if interested, please open an issue on github: https://github.com/ruohai0925/IAMReX/issues

Time Step - Godunov

In IAMReX, the canonical projection is applied to solve the fluid equations. Note that we only use the time-centered Godunov advection [11, 12], and there no longer needs the predictor and corrector steps.

• Define the time-centered face-centered (staggered) MAC velocity which is used for advection: \(U^{MAC,n+1/2}\)

• Define the new-time density, \(\rho^{n+1} = \rho^n - \Delta t (\rho^{n+1/2,pred} U^{MAC,n+1/2})\) by setting

• Define an approximation to the new-time state, \((\rho U)^{\ast}\) by setting

  \[\begin{split}(\rho^{n+1} U^{\ast}) &= (\rho^n U^n) - \Delta t \nabla \cdot (\rho U^{MAC} U) + \Delta t \nabla {p}^{n-1/2} \\ &+ \frac{\Delta t}{2} (\nabla \cdot \tau^n + \nabla \cdot \tau^\ast) + \Delta t \rho g\end{split}\]

  (for implicit diffusion, which is the current default)

• Project \(U^{\ast}\) by solving

\[\nabla \cdot \frac{1}{\rho} \nabla \phi = \nabla \cdot \left( \frac{1}{\Delta t} U^{\ast}+ \frac{1}{\rho} \nabla {p}^{n-1/2} \right)\]

then defining

\[U^{n+1} = U^{\ast} - \frac{\Delta t}{\rho} \nabla \phi\]

and

\[{p}^{n+1/2} = \phi\]