 version 0.5.0
Brief description of models and equations

This section describes the models and equations solved by Notus.

Fluid-flows models that can be solved are:

• incompressible,
• monophasic or multiphasic (one-fluid model),
• isothermal or not,
• laminar or turbulent,
• composed of scalar species that optionally interact with it (thermosolutal flows).

The International System of Units is used.

# The Navier-Stokes equations

The Navier-Stokes equations take the following form:

$\rho \Big( \frac {\partial \mathbf{u}} {\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \Big) = - \nabla p + \nabla \cdot \Big( \mu \left( {\nabla \mathbf{u} + \nabla^\top \mathbf{u}} \right) \Big) + \mathbf{f} \\ \nabla \cdot \mathbf{u} = 0$

where $$\rho$$ is the density, $$\mathbf{u}$$ is the velocity, $$p$$ is the pressure, $$\mu$$ is the dynamic viscosity, and $$\mathbf{f}$$ is a source term.

Optionally, the following terms can be added:

• The gravity term

$\rho \mathbf{g}$

where $$\mathbf{g}$$ is the gravity field. The gravity field is assumed to be uniform. In this case, the Boussinesq approximation is used, i.e. $$\rho$$ is taken as constant except in the gravity term.

• The surface tension term

Following the Continuum Surface Force model , this term is written as:

$\sigma \kappa \delta_{\Gamma} \mathbf{n} = \sigma \kappa \nabla c$

where $$\sigma$$ is the surface tension between the two phases, $$\kappa$$ the curvature of the interface $$\Gamma$$, $$\mathbf{n}$$ the normal to the interface, $$\delta_{\Gamma}$$ the Dirac function that localizes the force on the interface.

• The eddy-viscosity term

For turbulent flows, the dynamic viscosity $$\mu$$ is replaced by $$(\mu + \mu_t)$$ where $$\mu_t$$ is the turbulent viscosity computed by a suited model.

## Boundary conditions

A set of boundary conditions are available. If $$\mathbf{n}$$ is the normal to a boundary and $$\mathbf{\tau}$$ its tangent, they can be written as:

• Wall or adherence of the flow: $$\mathbf{u} = 0$$
• Inlet: $$\mathbf{u} = \mathbf{u_0}$$
• Neumann: $$\nabla \mathbf{u} \cdot \mathbf{n} = 0$$
• Slip: $$\mathbf{u} \cdot \mathbf{n} = 0$$ and $$( \nabla \mathbf{u} \cdot \mathbf{n} ) \times \mathbf{n} = 0$$
• Moving: $$\mathbf{u} \times \mathbf{n} = \mathbf{u_0} \times \mathbf{n}$$ and $$\mathbf{u} \cdot \mathbf{n} = 0$$
• Periodic

# The energy equation

When activated, the energy equation takes the following form:

$\rho C_p \left( \frac {\partial T} {\partial t} + \mathbf{u} \cdot \nabla T \right) = \nabla \cdot \left( \lambda \nabla T \right)$

where $$T$$ is the temperature, $$C_p$$ is the heat capacity at constant pressure, $$\lambda$$ is the thermal diffusion coefficient.

## Boundary condition

Available boundary conditions are:

• Dirichlet: $$T = T_0$$
• Neumann: $$\frac {\partial T} {\partial n} = f$$
• Robin
• Periodic
Note
Turbulence is not (yet) handled in this equation.

# The species transport equation

When activated, the species transport equation takes the following form:

$\omega_i \left( \frac {\partial S_j} {\partial t} + \mathbf{u} \cdot \nabla S_j \right) = \nabla \cdot \left( D_j \nabla S_j \right)$

where $$S_j$$ is one of the species, and $$D_j$$ is the corresponding diffusion coefficient. Both passive and active species are modeled this way.

Note
Turbulence is not (yet) handled in this equation.

## Boundary conditions

Available boundary conditions are:

• Dirichlet: $$S_i = S_{i,0}$$
• Neumann: $$\frac {\partial S_i} {\partial n} = g$$
• Robin
• Periodic
Note
Turbulence is not (yet) handled in this equation.

When solving multiphase flow of immiscible fluids, the advection of the $$i$$-th phase is done by:

$\frac {\partial C_i} {\partial t} + \mathbf{u} \cdot \nabla C_i = 0$

where $$C_i$$ is the $$i$$-th fluid volume fraction.

Depending on the model, there may be a non-advected phase filling the remaining volume fraction $$(1 - \sum_{i=1}^{n} C_i)$$.

# Turbulence models

Large Eddy Simulation approach is proposed to model the subgrid energy dissipation (to be validated). The only model available so far is the mixed scale one which is a combination of the Smagorinsky and Turbulent Kinetic Energy ones.

Reynolds Average approach is also under development ( $$k-\omega~SST$$ and $$v^2-f$$)

# Physical properties laws

All physical properties are considered as constant per phase, except density. Density may vary with temperature and species concentration:

$\rho = \rho_0 \Big( 1 - \beta_T ( T - T_{\text{ref}} ) + \sum_{j=1}^{m} \beta_{S_j} ( S_j - S_{j,\text{ref}} ) \Big)$

where $$T_{\text{ref}}, S_{1,\text{ref}}, \ldots, S_{m,\text{ref}}$$ are the reference temperature and references concentrations, respectively.

In multiphase flows, physical properties ( $$\rho_0, \mu, C_p, \lambda, \text{etc.})$$ depend on volume fractions $$C_i$$ according to the following law:

$\phi = \phi_0 (1 - \sum_{i=1}^{n} C_i) + \sum_{i=1}^{n} \phi_i C_i$

where $$\phi = \rho, \mu, C_p, \lambda, etc.$$, the index 0 represent the non-advected phase.

# Bibliography

 J. U. Brackbill, D. B. Korma, and C. Zemach, A Continuum Method for Modeling Surface Tension, Journal of Computational Physics, 100, 335-354 (1992).