HOW TO SOLVE THE NAVIER-STOKES EQUATION Benk Janos Department of Informatics, TU München JASS 2007, course 2: Numerical Simulation: From Models to Software Based on: ON PRESSURE BOUNDARY CONDITIONS FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATION Phlilp M. Gresho and Robert L. Sani International Journal For Numerical Methods In Fluids, Vol 7, 1111- 1145(1987)
HOW TO SOLVE THE NAVIER-STOKES EQUATION. Based on: ON PRESSURE BOUNDARY CONDITIONS FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATION Phlilp M. Gresho and Robert L. Sani International Journal For Numerical Methods In Fluids, Vol 7, 1111-1145(1987). Benk Janos Department of Informatics, TU München - PowerPoint PPT Presentation
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HOW TO SOLVE THE NAVIER-STOKES EQUATION
Benk JanosDepartment of Informatics, TU München
JASS 2007, course 2:Numerical Simulation: From Models to Software
Based on:
ON PRESSURE BOUNDARY CONDITIONS FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONPhlilp M. Gresho and Robert L. Sani
International Journal For Numerical Methods In Fluids, Vol 7, 1111-1145(1987)
Content
Short introduction Analysis of the continuum equation Pressure Poisson equation Boundary conditions Discrete approximation to the
continuum equation
Short introduction
- Important field of application of the numerical simulation- The flow is a result of different physical processes - Numerical flow simulation has a various fields of
application, real scenario simulations.
http://www.cfd-online.com/Links/misc.html#picts
Analysis of the continuum equation
gugradPugradut
u
Re
1
0udiv
The second equation is the continuity equation
The momentum equation for incompressible fluids
If it would be compressible fluid
tudiv
(1)
(2)
(3)
Analysis of the continuum equation
t
u
Each part from the (1) equation has a contribution to the momentum
The velocity change describing the acceleration of a infinite mass point. It must be in balance with …
ugradu … the convective term describing the frictionless acceleration induced by the velocity filed.
gradP … the gradient of the pressure. (by definition is an acceleration)
Analysis of the continuum equation
uRe
1 This component reflects the interior drag of the fluid.
Re is the Reynolds number.
The drag appears between two layers of fluid with different velocity.
- The friction force is acting against the velocity gradient
- This laminar flow, which opposes turbulent flow
Analysis of the continuum equation
- At gas flow this term can be neglected, but by fluids not (e.g. honey)
-The internal friction is also called viscosity, which characterize each fluid.
vs - mean fluid velocity, L - characteristic length, μ - (absolute) dynamic fluid viscosity, ν - cinematic fluid viscosity: ν = μ / ρ, ρ - fluid density.