Links equations & Data set

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The code Sunfluidh solves the Navier-Stokes equations by means of an incremental projection method. In the prediction step, the Navier-Stokes equations are solved in order to estimate the velocity field without ensuring the mass conservation ( for incompressible flows or for low Mach number flows). In the projection step, The mass conservation is ensured by solving a Poisson's equation. The result corresponds to the time increment of the “dynamical” pressure (the part of the pressure associated to the dynamics) and its gradient is the velocity correction that ensures the mass conservation. For more details on the projection methods, see the document here.

The user can read in this page : the different sets of governing equations that depend on the flow is either incompressible or dilatable (low Mach number hypothesis). the different formulations of the Poisson's equation in respect with the problem treated. the links between the equations, physical quantities and the data set

The governing equations

Incompressible flow formulation

Simulation of flows at moderate velocity ( ) where the hypothesis is valid.

Isothermal flows

Flows with heat transfer under the Boussinesq's hypothesis

Flows with free interface (in progress)

The equation set considered is

is uniform (except in the case of flows with free interface). Flows with free interface are solved with a level set method (in progress) is ensured by solving a Poisson's equation (see projection methods). When is associated to the gravity/buoyancy force, it depends on the temperature variation, the thermal expansion coefficient and the gravity constant (see the Namelist "Gravity"), except for free interface flows : it also depends on the density variation. The dynamic viscosity and the thermal conductivity can be constant or depend on the temperature by means of the Sutherland's law (See the Namelist "Fluid_Properties").

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Low Mach number formulation

Simulation of flows at moderate velocity ( ) with a noticeable variation of density (the hypothesis is no longer valid).

Homogeneous gas with high temperature variation

Multi-component gas with large density variation

Multi-component gas with temperature variation

Reactive flows

We remind the user that the low Mach hypothesis is based on the hypothesis of the scale splitting between the thermodynamics and dynamics phenomena. As a consequence the pressure is read as , where is the thermodynamical pressure and the dynamical pressure. is supposed to be uniform over the domain and is defined by the equation of state. is solved from the Poisson's equation (see the projection methods).

The global governing equations are :

is uniform except in the case of incompressible two-phase flows. the mass conservation is ensured by solving a Poisson's equation (see projection methods). When is associated to the gravity/buoyancy force, it depends on the density variation and the gravity constant (see the Namelist "Gravity"). The dynamic viscosity, the thermal conductivity and diffusion coefficients of species can be constant, only depend on the temperature (Sutherland's law) or depend on the temeperature and species concentrations by means of relations coming from the kinetic theory of gases (See the Namelist "Fluid_Properties").

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The different formulations of the Poisson's equation

We consider

the time increment of pressure

the fluid density

the estimated velocity field solved in the prediction step

the time step

a time-coefficient of time scheme used for solving the Navier-Stokes equations.

The usual form for the incompressible flows is :

The form for low Mach number flows, when the density variation is moderate :

The form for low Mach number flows or incompressible two-phase flows, when the density variation can be large :

where is estimated from the differential equation of state.

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Link between the data set & the variables in equations

The knowledge of the data setup is recommanded in order to better understand the links. Click here to go to the data-set page

List of variables	Definition	namelists where the physical quantity is defined
the fluid density	"Fluid_Properties", "Numerical_Methods"
the velocity field	"Velocity_Initialization", "Velocity_Wall_Boundary_Condition_Setup", "Inlet_Boundary_Condition", "Numerical_Methods"
the dynamic viscosity of the fluid	"Fluid_Properties"
the field of pressure variation related to the mass conservation (solved by a Poisson's equation – Projection method)	"Numerical_Methods"
the force leading the fluid motion (gravity/buoyancy effect, external force …)	"Gravity", "External_Force"
the temperature field	"Fluid_Properties", "Heat_Wall_Boundary_Condition_Setup", "Inlet_Boundary_Condition", "Numerical_Methods"
the uniform thermodynamic pressure (low Mach number Hypothesis). Either or	"Fluid_Properties" (variable “Constant_Mass_Flow)
the mass heat capacity of the fluid	"Fluid_Properties"
the thermal conductivity	deduced from other data with the relation
the Prandtl number	"Fluid_Properties"
the chemical source term in the enthalpy equation	Namelist "Chemical_Reactions_Features"
the radiative source term in the enthalpy equation	in progress
mass fraction field of the i-species	"Fluid_Properties", Namelist "Species_Properties", Namelist "Species_Initialization", Namelist "Species_Wall_Boundary_Condition_Setup", "Inlet_Boundary_Condition", "Numerical_Methods"
the diffusion coefficient of the species in the gas mixture	"Fluid_Properties"
chemical reaction rate	Namelist "Chemical_Reactions_Features"
the perfect gas constant	parameter of the code
the molecular mass (it is uniform if the fluid is homogeneous or depends on the species mass fractions)	"Fluid_Properties"

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