cylindrical system r \u03b8 z and spherical systemr \u03b8 \u03a6 should be appropriately

# Cylindrical system r θ z and spherical systemr θ φ

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cylindrical system (r, θ, z), and spherical system(r, θ, Φ) should be appropriately chosen for a better resolution of the geometry (e.g. cylindrical for circular pipe). 10 Modeling (coordinates) x y z x y z x y z (r, ,z) z r (r, , ) r (x,y,z) Cartesian Cylindrical Spherical General Curvilinear Coordinates General orthogonal Coordinates 11 Modeling (governing equations) Navier-Stokes equations (3D in Cartesian coordinates) 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 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 12 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, , Gr, Ec) Free-surface flow (Fr) and surface tension (We) Chemical reactions and combustion (Pe, Da) etc… 13 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 14 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 15 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, , Gr, Ec, conservation of energy ) Free-surface flow (Fr, level-set & surface tracking model ) and surface tension (We, bubble dynamic model ) Chemical reactions and combustion ( Chemical reaction 16 Modeling (Turbulence and free surface models) Turbulent models : DNS : most accurately solve NS equations, but too expensive for turbulent flows  #### You've reached the end of your free preview.

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