Computational Fluid Dynamics (CFD, CHD)*

Computational Fluid Dynamics (CFD, CHD)*PDE (Shocks 1st); Part I: Basics, Part II: Vorticity FieldsRubin H LandauSally Haerer, Producer-DirectorBased onA Survey of Computational PhysicsbyLandau, Páez, & Bordeianuwith Support from the National Science FoundationCourse: Computational Physics II1/1

Problem: Placement of Boulders for Migrating SalmonWake Block “Force” of eep, wide, fast-flowing streams“Boulder” long rectangular beam, platesObjects not disturb surface/bottom flowProblem: large enough wake for 1m salmon2/1

Theory: HydrodynamicsAssumptions; Continuity m ρ(x, t) ·j 0 tdefj ρ v(x, t)(1): Continuity equation ρ constant1st eqtn hydrodynamicsFriction (viscosity)Incompressible fluidSteady state, v 6 v (t)(1)(2)3/1

Navier–Stokes: 2nd Hydrodynamic EquationHydrodynamic Time DerivativeDv def v (v · )vDt t(1)For quantity within moving fluidRate of change wrt stationary frameVelocity of material in fluid elementChange due to motion explicit t dependenceDv/Dt: 2nd O v nonlinearities Fictitious (inertial) forcesFluid’s rest frame accelerates4/1

Now Really the Navier–Stokes EquationTransport Fluid Momentum Due to Forces & FlowDv1 T , x) ν 2 v P(ρ,Dtρ(1)zzX 2 vxX vx1 P vx ν vj2 t xjρ x xjj xj xν viscosity, P pressureRecall dp/dt Fdef v/ tDv/Dt (v · )vv · v:transport via flowv · v:advection :change P(Vector Form)due to P(x component)ν 2 v:(2)due to viscosityP(ρ, T , x):equation stateAssume P(x)Steady-state t vi 0Incompressible t ρ 05/1

Resulting Hydrodynamic EquationsAssumed: Steady State, Incompressible, P P(x) ·v X vi 0 xi(Continuity)(1)i ν 2 v 1 P (v · )vρ(Navier–Stokes)(2)(1) Continuity equation: Incompressibility, in outStream width beam z dimension z v ' 0 vy vx 0 x y 2 vx vx1 P vx 2 vxν vx vy x 2 y 2 x yρ x ν 2 vy 2 vy x 2 y 2 vx vy vy1 P vy x yρ y(3)(4)(5)6/1

Boundary Conditions for Parallel PlatesPhysics Determines BC Unique SolutionLHConstant stream velocity Upstream unaffectedLow V0 , high viscosity Solve rectangular regionLaminar: smooth, no cross streamlines of motionL, H Rstream uniformdownThin plates laminar flowFar top, bot symmetry7/1

Analytic Solution for Parallel Plates (See Text)Bernoulli Effect: Pressure Drop Through PlatesRiversurfaceRiversurfaceyyLxxbottombottomvx (y ) HL1 P 2(y yH)2ρν x P known constant xV0 1 m/s, ρ 1 kg/m3 , ν 1 m2 /s, H 1 m P 12, xvx (y ) 6y (1 y )(1)(2)(3)(4)8/1

Finite-Difference Navier–Stokes Algorithm SORRectangular grid x ih,y jh3 Simultaneous equations 2 (v y 0)yyxxvi 1,j vi 1,j vi,j 1 vi,j 1 0(1)xxxxxvi 1,j vi 1,j vi,j 1 vi,j 1 4vi,j(2) hh x xh y xxxvi,j 1 vi,j 1 [Pi 1,j Pi 1,j ]vi,j vi 1,j vi 1,j vi,j222Rearrange as algorithm for Successive Over Relaxationxxxxx4vi,j vi 1,j vi 1,j vi,j 1 vi,j 1 h x xxvi,j vi 1,j vi 1,j2 hh y xxv vi,j 1 vi,j 1 [Pi 1,j Pi 1,j ]2 i,j2(3)Accelerate convergence SOR; ω 2 unstable9/1

End Part I: 0 / 1

Part II: Vorticity Form of Navier–Stokes Equation2 HD Equations in Terms of Stream Function u(x) ·v 0 Continuity(1)1 (v · )v P ν 2 vρNavier–Stokes(2)Like EM, simpler via (scalar & vector) potentialsIrrotational Flow: no turbulence, scalar potentialRotational Flow: 2 vector potentials; 1st stream functiondef v u(x) ˆx uy uz y z(3) ˆy ux uz z x · ( u) 0 automatic continuity equation (4)11 / 1

2 HD Equations in Terms of Stream Function (cont)2-D flow: u Constant Contour Lines Streamlinesdef v u(x) ˆx (1) uy uz y z ˆy ux uz z x (2)vz 0 u(x) u ˆz(3) u x(4)vx u, yvy 12 / 1

Introduce Vorticity w(x) ω Vortex: Spinning, Often Turbulent Fluid Flowdef w v(x) wz (1) vy vx x y (2)Measure of v ’s rotationw 0 uniformRH rule fluid elementMoving field linesw 0 irrotationalRelate to stream function:13 / 1

Introduce Vorticity w(x) ω Poisson’s equation 2 φ 4πρw(x,y)012y-16200040x80xy50 0def v(x)w v ( u) ( · u) 2 uw ·u 0yet u u(x, y ) ˆz 2u w (1)(2)(3)(4)Like Poisson with ea w component source14 / 1

Vorticity Form of Navier–Stokes EquationTake Curl of Velocity Formh1 (v · )v ν 2 v Pρ (Navier–Stokes) u) · ]w ν 2 w [( (1)(2)In 2-D only z components: 2u 2u w x 2 y 2 ν 2w 2w 2 x y 2 u w u w y x x y(3)(4)Simultaneous, nonlinear, elliptic PDEs for u & w Poisson’s wave equation friction variable ρ15 / 1

Relaxation Algorithm (SOR) for Vorticity Equationsx ih,y jhCD Laplacians, 1st derivatives 1 (1)ui 1,j ui 1,j ui,j 1 ui,j 1 h2 wi,j41R (wi 1,j wi 1,j wi,j 1 wi,j 1 ) {[ui,j 1 ui,j 1 ]416ui,j wi,jR [wi 1,j wi 1,j ] [ui 1,j ui 1,j ] [wi,j 1 wi,j 1 ]}(2)1V0 h νν(3)(in normal units)R grid Reynolds number (h Rpipe ); measure nonlinearSmall R: smooth flow, friction damps fluctuationsLarge R (' 2000): laminar turbulent flowOnset of turbulence: hard to simulate (need kick)16 / 1

Boundary Conditions for BeamSurface Gvy -du/dx 0vx du/dy V0w 0vy -du/dx 0w u 0yxEw 0du/dx 0u 0CHalfBeamDdw/dx 0Outlet HInlet Fvx du/dy V0u 0w u 0Bcenter lineAWell-defined solution of elliptic PDEs requires u, w BCAssume inlet, outlet, surface far from beamFreeflow: No beamNB w 0 no rotationSymmetry: identical flow above, below centerline, not thru17 / 1

Boundary Conditions for Beam (cont)See Text for More ExplanationsCenterline: streamline, u const 0 (no v No flow in, out beam to it u 0 all beam surfacesSymmetry vorticity w 0 along centerlineInlet: horizontal fluid flow, v vx V0 :Surface: Undisturbed free-flow conditions:Outlet: Matters little; convenient choice: x u x wBeamsides: v u 0; viscous vk 0Yet, over specify BC only no-slip vorticity w:Viscosity vx u y 0(beam top)Smooth flow on beam top vy 0 no x variation: vy vx 2u 0 w 2 x y y(1)18 / 1

Implementation & Assessment:SOR on a GridBasic soltn vorticity form Navier–Stokes: Beam.pyNB relaxation simple, BC 6 simpleSeparate relaxation of stream function & vorticityExplore convergence of up & downstream uDetermine number iterations for 3 place with ω 0, 0.3Change beam’s horizontal position so see wave developMake surface plots of u, w, v with contours; explainIs there a resting place for salmon?19 / 1

Resultsw(x,y)120y-16200040x80xy50 020 / 1

Computational Fluid Dynamics (CFD, CHD)* PDE (Shocks 1st); Part I:Basics, Part II:Vorticity Fields Rubin H Landau Sally Haerer, Producer-Director Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from the National Science Fo