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The code Sunfluidh solves the Navier-Stokes equations by means of an incremental projection method.
$$ P_{dyn}^{n+1} = P_{dyn}^{n} + \phi$$ $$ \vec{V}^{n+1}= \vec{V}^* - \frac{\Delta t}{\rho} \nabla \phi$$
For more details on the projection methods, see the document on the numerical methods used in Sunfluidh (click here).
Simulation of flows at moderate velocity ($ V < 0.1 \cdot Ma$ ) where the hypothesis $\frac {d\rho}{dt}=0$ is valid.
The equation set considered is \begin{align*} \nabla \cdot \left( \vec{V} \right) & =0 \\ \rho_0 (\frac{\partial \vec{V}}{\partial t} + \left(\vec{V}\cdot \nabla \right)\vec{V} )& = -\nabla P_{dyn} + \nabla \cdot \mu \nabla(\vec{V}+^{t}\vec{V})+\vec{f}_V \\ \rho_0 C_{p0} \left(\frac{\partial T}{\partial t} + \left(\vec{V}\cdot \nabla\right)T \right)&= \nabla \cdot \lambda \nabla T \end{align*}
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Simulation of flows at moderate velocity ($ V < 0.1 \cdot Ma$ ) with a noticeable variation of density (the hypothesis $\frac {d\rho}{dt}=0$ is no longer valid).
The global governing equations are :
\begin{align*} \frac{\partial \rho}{\partial t} + \nabla \cdot \left( \rho\vec{V} \right) & =0 \\ \rho (\frac{\partial \vec{V}}{\partial t} + \left(\vec{V}\cdot \nabla \right)\vec{V} )& = -\nabla P_{dyn} + \nabla \cdot \tau+\vec{f}_V \\ \text{where }\tau_{ij}&=\mu ((\frac{\partial V_i}{\partial x_j}+\frac{\partial V_j}{\partial x_i})-\frac{2}{3}\frac{\partial V_k}{\partial x_k}\delta_{ij}) \\ \rho C_p \left(\frac{\partial T}{\partial t} + \left(\vec{V}\cdot \nabla\right)T \right)&= \nabla \cdot \lambda \nabla T + \frac{dP_{th}}{dt} + S_{c} + S_{r} \\ \rho \left(\frac{\partial Y_i}{\partial t} + \left(\vec{V}\cdot \nabla\right)Y_i \right)&= \nabla \cdot D_{i,m} \nabla Y_i + S_{r} + W_i \\ P_{th}&=\cdot \rho \cdot \frac{R}{M}\cdot T \end{align*}
We consider
The usual form for the incompressible flows is :
$$ \nabla \cdot \nabla \Phi= \frac{ \nabla \cdot \vec{V}^*}{\alpha \Delta t} $$
The form for low Mach number flows, when the density variation is moderate :
$$ \nabla \cdot \nabla \Phi= \frac{ \nabla \cdot \vec{V}^* - \frac{\partial \rho}{\partial t}}{\alpha \Delta t} $$
The form for low Mach number flows or incompressible two-phase flows, when the density variation can be large :
$$ \nabla \frac{1}{\rho}\cdot \nabla \Phi= \frac{ \nabla \cdot \vec{V}^* - \nabla \cdot \vec{V}^{n+1} }{\alpha \Delta t} $$
where $\nabla \cdot \vec{V}^{n+1}$ is estimated from the differential equation of state.
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Information on numerical methods used for solving these equations is available in the pdf document present here.
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 |
---|---|---|
$\rho $ | the fluid density | "Fluid_Properties", "Numerical_Methods" |
$\vec{V}$ | the velocity field | "Velocity_Initialization", "Velocity_Wall_Boundary_Condition_Setup", "Inlet_Boundary_Condition", "Numerical_Methods" |
$\mu$ | the dynamic viscosity of the fluid | "Fluid_Properties" |
$P_{dyn}$ | the field of pressure variation related to the mass conservation (solved by a Poisson's equation – Projection method) | "Numerical_Methods" |
$\vec{f}_V$ | the force leading the fluid motion (gravity/buoyancy effect, external force …) | "Gravity", "External_Force" |
$T$ | the temperature field | "Fluid_Properties", "Heat_Wall_Boundary_Condition_Setup", "Inlet_Boundary_Condition", "Numerical_Methods" |
$P_{th}$ | the uniform thermodynamic pressure (low Mach number Hypothesis). Either $P_{th}=P_0$ or $P_{th}= \rho.\frac{R}{M}.T $ | "Fluid_Properties" (variable “Constant_Mass_Flow) |
$C_p$ | the mass heat capacity of the fluid | "Fluid_Properties" |
$\lambda$ | the thermal conductivity | deduced from other data with the relation $ \lambda = \frac{\mu\cdot C_p}{Pr}$ |
$Pr$ | the Prandtl number | "Fluid_Properties" |
$S_{c}$ | the chemical source term in the enthalpy equation | Namelist "Chemical_Reactions_Features" |
$S_{r}$ | the radiative source term in the enthalpy equation | in progress |
$Y_i$ | 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" |
$D_{i,m}$ | the diffusion coefficient of the species in the gas mixture | "Fluid_Properties" |
$W_i$ | chemical reaction rate | Namelist "Chemical_Reactions_Features" |
$R$ | the perfect gas constant | parameter of the code |
$M$ | the molecular mass (it is uniform if the fluid is homogeneous or depends on the species mass fractions) | "Fluid_Properties" |