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 •
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- Summer '18