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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 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 [1], 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.

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*

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.

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.

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.

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 \( (1 - \sum_{i=1}^{n} C_i) \).

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 \(v^2-f\))

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.

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