Summary of basic models
The Mach number is defined by
(331)\[Ma=\frac{\bigl|\boldsymbol{u}\bigr|}{c_{s}}\]
where \(\boldsymbol{u}\equiv\boldsymbol{u}(\boldsymbol{x},t)\) is the fluid velocity and \(c_{s}\) is the sound speed. At 20°C the sound speed is 344 m/s (1240 km/h) in air and 1500 m/s (5400 km/h) in
water.
In the rest of this section, two popular models of Navier-Stokes equations are detailed for \(Ma\ll1\). The first one is the “low Mach model” and the second one is the incompressible model.
Low Mach Navier-Stokes
(332)\[\partial_{t}\rho+\boldsymbol{\nabla}\cdot(\rho\boldsymbol{u})=0\]
(333)\[\partial_{t}(\rho\boldsymbol{u})+\boldsymbol{\nabla}\cdot(\rho\boldsymbol{u}\boldsymbol{u})=-\boldsymbol{\nabla}p+\boldsymbol{\nabla}\cdot\left[\eta(\boldsymbol{\nabla}\boldsymbol{u}+(\boldsymbol{\nabla}\boldsymbol{u})^{T})+\left(\eta_{B}-\frac{2}{3}\eta\right)(\boldsymbol{\nabla}\cdot\boldsymbol{u})\boldsymbol{I}\right]+\boldsymbol{F}\]
where \(\rho\equiv\rho(\boldsymbol{x},t)\) is the fluid density, \(\boldsymbol{u}\equiv\boldsymbol{u}(\boldsymbol{x},t)\) is the
fluid velocity, \(p\equiv p(\boldsymbol{x},t)\) is the pressure, \(\eta=\rho\nu\) is the dynamic viscosity and \(\nu\) is the
kinematic viscosity, \(\eta_{B}\) is the bulk viscosity and \(\boldsymbol{I}\) is the identity tensor. In Eq.
(333), \(\boldsymbol{F}\) is the force term and for isothermal fluid, the system is closed by the Equation of State
(EoS)
\[p=\rho RT_{0}\]
where \(T_{0}\) is the constant temperature and \(R\) is the specific gas constant.
Incompressible Navier-Stokes
(334)\[\boldsymbol{\nabla}\cdot\boldsymbol{u}=0\]
(335)\[\rho_{0}\left[\partial_{t}\boldsymbol{u}+\boldsymbol{\nabla}\cdot(\boldsymbol{u}\boldsymbol{u})\right]=-\boldsymbol{\nabla}p_{h}+\boldsymbol{\nabla}\cdot\left[\rho_{0}\nu(\boldsymbol{\nabla}\boldsymbol{u}+(\boldsymbol{\nabla}\boldsymbol{u})^{T})\right]+\boldsymbol{F}\]
where \(p_{h}(\boldsymbol{x},t)\) is the hydrodynamic pressure and \(\rho_{0}\) is the constant density.
Advection-Diffusion equations
Levelset or phase-field equations
(336)\[\frac{\partial\phi}{\partial t}+\boldsymbol{\nabla}\cdot(\boldsymbol{u}\phi)=\boldsymbol{\nabla}\cdot\left[M_{\phi}\left(\boldsymbol{\nabla}\phi-\frac{4}{W}\phi(1-\phi)\boldsymbol{n}\right)\right]\]
where the unknown is the phase-field \(\phi\equiv\phi(\boldsymbol{x},t)\) and the inputs are the fluid velocity \(\boldsymbol{u}\) and two parameters: the interface mobility \(M_{\phi}\) and the interface width \(W\). The unit normal vector of interface is defined by
(337)\[\boldsymbol{n}\equiv\boldsymbol{n}(\boldsymbol{x},t)=\frac{\boldsymbol{\nabla}\phi}{\bigl|\boldsymbol{\nabla}\phi\bigr|}\]
Advection-Diffusion Equation (ADE)
(338)\[C_{p}\frac{\partial T}{\partial t}+\boldsymbol{\nabla}\cdot(T\boldsymbol{u})=\boldsymbol{\nabla}\cdot(\kappa\boldsymbol{\nabla}T)+\mathscr{S}_{T}\]
where the unknown is the temperature \(T\equiv T(\boldsymbol{x},t)\) and the inputs are the fluid velocity \(\boldsymbol{u}\) and two parameters: the thermal conductivity \(\kappa\) and the specific heat \(C_{p}\). A source term \(\mathscr{S}_{T}\) can be defined for phase change problems.
Boundary Conditions (BC)
For Navier-Stokes equations
Dirichlet BC
\[\boldsymbol{u}(\boldsymbol{x}_{b},t)=\boldsymbol{U}_{b}(\boldsymbol{x}_{b},t)\]
\[(\boldsymbol{u}-\boldsymbol{U}_{b})\cdot\boldsymbol{n}_{w}=0\]
\[(\boldsymbol{u}-\boldsymbol{U}_{b})\cdot\boldsymbol{t}_{w}=0\qquad\text{(no-slip)}\]
where \(\boldsymbol{n}_{w}\) is normal boundary vector and
\(\boldsymbol{t}_{w}\) tangential boundary vector (\(w\): wall)
Neumann BC
\[\boldsymbol{n}_{w}\cdot\boldsymbol{\sigma}(\boldsymbol{x}_{b},t)=\boldsymbol{T}_{b}(\boldsymbol{x}_{b},t)\]
For ADE
A generic formulation of bounday conditions for ADE writes:
\[a_{1}\left.\frac{\partial f}{\partial n}\right|_{\boldsymbol{x}_{b},t}+a_{2}f(\boldsymbol{x}_{b},t)=a_{3}\]
where \(f\) is the differentiable function (here \(\phi\) or
\(T\)), and \(a_{1}\), \(a_{2}\), \(a_{3}\) three scalar
values.
Name |
\(a_{1}\) |
\(a_{2}\) |
\(a_{3}\) |
|---|
Dirichlet |
0 |
– |
– |
Neumann |
– |
0 |
– |
Robin |
– |
– |
– |
where “–” means non zero value
Section author: Alain Cartalade