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CFD - Governing Equations


This topic introduces you to the 3-dimensional unsteady Naier-Stokes equations for CFD. Prepare for some math!

3-dimensional unsteady Navier-Stokes equations:

Coordinates: (x,y,z)
Velocity components: (u,v,w)
Time: t
Density: \rho
Total Energy: E_T
Pressure: p
Stress: \tau
Heat flux: q
Reynolds number: Re
Prandtl number: Pr


\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x} + \frac{\partial (\rho v)}{\partial y} + \frac{\partial (\rho w)}{\partial z} = 0


\frac{\partial (\rho u)}{\partial t} + \frac{\partial (\rho u^2)}{\partial x} + \frac{\partial (\rho uv)}{\partial y} + \frac{\partial (\rho uw)}{\partial z} = - \frac{\partial \rho}{\partial x} + \frac{1}{Re}\left(\frac{\partial \tau_{xx}}{\partial x}+ \frac{\partial \tau_{xy}}{\partial y} + \frac{\partial \tau_{xz}}{\partial z} \right)


\frac{\partial (\rho v)}{\partial t} + \frac{\partial (\rho uv)}{\partial x} + \frac{\partial (\rho v^2)}{\partial y} + \frac{\partial (\rho uw)}{\partial z} = - \frac{\partial \rho}{\partial y} + \frac{1}{Re}\left(\frac{\partial \tau_{xy}}{\partial x}+ \frac{\partial \tau_{yy}}{\partial y} + \frac{\partial \tau_{yz}}{\partial z} \right)


\frac{\partial (\rho w)}{\partial t} + \frac{\partial (\rho uw)}{\partial x} + \frac{\partial (\rho vw)}{\partial y} + \frac{\partial (\rho w^2)}{\partial z} = - \frac{\partial \rho}{\partial z} + \frac{1}{Re}\left(\frac{\partial \tau_{xz}}{\partial x}+ \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \tau_{zz}}{\partial z} \right)


\frac{\partial(E_T)}{\partial t} + \frac{\partial(uE_T)}{\partial x} + \frac{\partial(vE_T)}{\partial y} + \frac{\partial(wE_T)}{\partial z} = -\frac{\partial(up)}{\partial x} - \frac{\partial(vp)}{\partial y} - \frac{\partial(wp)}{\partial z} - \frac{1}{Re Pr}\left(\frac{\partial q_x}{\partial x} + \frac{\partial q_y}{\partial y} + \frac{\partial q_z}{\partial z} \right) + \frac{1}{Re} \left[\frac{\partial}{\partial x} (u \tau_{xx} + v \tau_{xy} + w \tau_{xz}) +\frac{\partial}{\partial y} (u \tau_{xy} + v \tau_{yy} + w \tau_{yz}) + \frac{\partial}{\partial z} (u \tau_{xz} + v \tau_{yz} + w \tau_{zz}) \right]

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Literature References:

CFD Analysis Process