Computational Fluid Dynamics

Governing Equations

The mass conservation equation

The mass conservation equation is given by

\[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0 \]

where \(\rho\) is the density of the fluid and \(\vec{u}\) is the velocity vector. For an incompressible fluid, the density is constant, the equation simplifies to

\[ \nabla \cdot \vec{u} = 0 \]

The momentum conservation equation

The momentum conservation equation is given by

\[ \frac{\partial \rho u}{\partial t} + \nabla \cdot (\rho \vec{u} u) = \nabla\cdot(\mu\nabla u)-\frac{\partial p}{\partial x}+S_{Mx} \] \[ \frac{\partial \rho v}{\partial t} + \nabla \cdot (\rho \vec{u} v) = \nabla\cdot(\mu\nabla v)-\frac{\partial p}{\partial y}+S_{My} \] \[ \frac{\partial \rho w}{\partial t} + \nabla \cdot (\rho \vec{u} w) = \nabla\cdot(\mu\nabla w)-\frac{\partial p}{\partial z}+S_{Mz} \]

where \(\rho\) is the density of the fluid, \(\vec{u} = (u, v, w)\) is the velocity vector, \(\mu\) is the dynamic viscosity of the fluid, \(p\) is the pressure of the fluid, and \(S_{Mx}\), \(S_{My}\), and \(S_{Mz}\) are the source terms which describe the effects of body forces such as the centrifugal force.

The energy conservation equation

The energy conservation equation is given by

\[ \frac{\partial \rho i}{\partial t} + \nabla \cdot (\rho \vec{u} i) = \nabla\cdot(k\nabla T)-p\nabla\cdot\vec{u}+S_{\mathrm{Dis}} \]

where \(i\) is the spedific internal energy of the fluid, \(k\) is the thermal conductivity of the fluid, \(T\) is the temperature of the fluid, and \(S_{\mathrm{Dis}}\) is the source term which describes the effects of body forces

Equations of State

\(p = p(\rho, T)\) and \(i = i(\rho, T)\)