Modeling governing equations Navier Stokes equations 3D in Cartesian

# Modeling governing equations navier stokes equations

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Modeling (governing equations) Navier-Stokes equations (3D in Cartesian coordinates) 2 2 2 2 2 2 ˆ z w y w x w z p z w w y w v x w u t w 2 2 2 2 2 2 ˆ z u y u x u x p z u w y u v x u u t u 2 2 2 2 2 2 ˆ z v y v x v y p z v w y v v x v u t v 0 z w y v x u t RT p L v p p Dt DR Dt R D R 2 2 2 ) ( 2 3 Convection Piezometric pressure gradient Viscous terms Local acceleration Continuity equation Equation of state Rayleigh Equation Mechanical Engineering MVGR Computational Fluid Dynamics 10 Modeling (flow conditions) Based on the physics of the fluids phenomena, CFD can be distinguished into different categories using different criteria Viscous vs. inviscid (Re) External flow or internal flow (wall bounded or not) Turbulent vs. laminar (Re) Incompressible vs. compressible (Ma) Single- vs. multi-phase (Ca) Thermal/density effects (Pr, g , Gr, Ec) Free-surface flow (Fr) and surface tension (We) Chemical reactions and combustion (Pe, Da) etc… Mechanical Engineering MVGR Computational Fluid Dynamics 11 Modeling (initial conditions) Initial conditions (ICS, steady/unsteady flows) ICs should not affect final results and only affect convergence path, i.e. number of iterations (steady) or time steps (unsteady) need to reach converged solutions. More reasonable guess can speed up the convergence For complicated unsteady flow problems, CFD codes are usually run in the steady mode for a few iterations for getting a better initial conditions Mechanical Engineering MVGR Computational Fluid Dynamics 12 Modeling(boundary conditions) Boundary conditions: No-slip or slip-free on walls, periodic, inlet (velocity inlet, mass flow rate, constant pressure, etc.), outlet (constant pressure, velocity convective, numerical beach, zero-gradient), and non-reflecting (for compressible flows, such as acoustics), etc. No-slip walls: u=0,v=0 v=0, dp/dr=0,du/dr=0 Inlet ,u=c,v=0 Outlet, p=c Periodic boundary condition in spanwise direction of an airfoil o r x Axisymmetric Mechanical Engineering MVGR Computational Fluid Dynamics 13 Modeling (selection of models) CFD codes typically designed for solving certain fluid phenomenon by applying different models Viscous vs. inviscid (Re) Turbulent vs. laminar (Re, Turbulent models ) Incompressible vs. compressible (Ma, equation of state ) Single- vs. multi-phase (Ca, cavitation model, two-fluid model ) Thermal/density effects and energy equation (Pr, g , Gr, Ec, conservation of energy ) Free-surface flow (Fr, level-set & surface tracking model )  #### You've reached the end of your free preview.

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