Franz J. Vesely > CompPhys Tutorial > Selected Applications > Hydrodynamics  

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Part III: Ch. 8

8 Hydrodynamics


How do you describe a flow?

Some basic truths:
General equation of motion for the flow field $\vec{v}(\vec{r},t)$ in a compressible viscous fluid: Navier-Stokes equation

$ \frac{\textstyle \partial }{\textstyle \partial t} \rho \vec{v} + \nabla \cdot \left[ \rho \vec{v} \vec{v} \right] + \nabla p - \mu \nabla \cdot {U} = 0 $    (8.1)

Here, $\mu$ is the viscosity, and

$ {U}\equiv\nabla \vec{v} + (\nabla \vec{v})^{\textstyle T} -\frac{\textstyle 2}{\textstyle 3}(\nabla \cdot \vec{v}) {I} $    (8.2)

defines the Navier-Stokes tensor. (In 2 dimensions, write $1$ in place of $2/3$).
Character of Navier-Stokes PDE:
  • Contains advective (hyperbolic) and diffusive (parabolic) terms
  • Small viscosity: advective terms dominate $\longrightarrow$ hyperbolic
  • High viscosity: diffusive terms important $\longrightarrow$ parabolic
  • Stationary case ($\partial / \partial t = 0$): $\longrightarrow$ elliptic

The NS equation results from the conservation of momentum. In addition, we have conservation of mass,

$ \frac{\textstyle \partial \rho}{\textstyle \partial t}+ \nabla \cdot \rho \vec{v}=0 $    (8.3)

and conservation of energy,

$ \frac{\textstyle \partial e}{\textstyle \partial t} + \nabla \cdot \left[ (e+p) \vec{v} \right]=0 \;\;\;\; \;\;\;\; {\rm with}\;\;\;\; e \equiv \rho \epsilon + \frac{\textstyle \rho v^{2}}{\textstyle 2} \;\;\;\; $    (8.4)

where $e$ is the energy density ($\epsilon$ ... internal energy per unit mass of the fluid).

To close the set of equations some equation of state $p=p(\rho,\epsilon)$ is assumed.

The following approaches will be discussed:

  • Conventional methods of solving the Navier-Stokes PDE
  • Discretized "Lattice gas" or "Lattice Boltzmann" dynamics
  • Direct Simulation Monte Carlo


vesely 2006

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