Computational Fluid Dynamics (CFD, CHD)*

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1 1 / 1 Computational Fluid Dnamics (CFD, CHD)* PDE (Shocks 1st); Part I: Basics, Part II: Vorticit Fields Rubin H Landau Sall Haerer, Producer-Director Based on A Surve of Computational Phsics b Landau, Páez, & Bordeianu with Support from the National Science Foundation Course: Computational Phsics II

2 Problem: Placement of Boulders for Migrating Salmon Wake Block Force of River? River surface River surface L L H bottom bottom Deep, wide, fast-flowing streams Boulder = long rectangular beam, plates Objects not disturb surface/bottom flow Problem: large enough wake for 1m salmon 2 / 1

3 3 / 1 Theor: Hdrodnamics Assumptions; Continuit Equation River surface River surface L L H bottom bottom ρ(, t) t + j = 0 (1) j def = ρ v(, t) (2) (1): Continuit equation 1st eqtn hdrodnamics Incompressible fluid ρ = constant Friction (viscosit) Stead state, v v(t)

4 4 / 1 Navier Stokes: 2nd Hdrodnamic Equation Hdrodnamic Time Derivative Dv Dt def = (v )v + v t (1) For quantit within moving fluid Rate of change wrt stationar frame Velocit of material in fluid element Change due to motion + eplicit t dependence Dv/Dt: 2nd O v nonlinearities Fictitious (inertial) forces Fluid s rest frame accelerates

5 5 / 1 Now Reall the Navier Stokes Equation Transport Fluid Momentum Due to Forces & Flow Dv Dt = ν 2 v 1 ρ P(ρ, T, ) (Vector Form) (1) v t + z j= v j v j = ν z j= 2 v 2 j 1 ρ P ( component) (2) ν = viscosit, P = pressure Recall dp/dt = F Dv/Dt def = (v )v + v/ t v v: transport via flow v v: advection P :change due to P ν 2 v: due to viscosit P(ρ, T, ): equation state Assume = P() Stead-state tv i = 0 Incompressible tρ = 0

6 Resulting Hdrodnamic Equations Assumed: Stead State, Incompressible, P = P() v i v i i = 0 (Continuit) (1) (v )v =ν 2 v 1 ρ P (Navier Stokes) (2) (1) Continuit equation: Incompressibilit, in = out Stream width beam z dimension z v 0 v + v = 0 (3) ( ) 2 v ν + 2 v 2 2 ( ) 2 v ν + 2 v 2 2 v = v + v v + 1 P ρ v = v + v v + 1 P ρ (4) (5) 6 / 1

7 7 / 1 Boundar Conditions for Parallel Plates Phsics Determines BC Unique Solution H L Constant stream velocit + Low V 0, high viscosit Laminar: smooth, no cross streamlines of motion Thin plates laminar flow Upstream unaffected Solve rectangular region L, H R stream uniform down Far top, bot smmetr

8 Analtic Solution for Parallel Plates (See Tet) Bernoulli Effect: Pressure Drop Through Plates River surface River surface L L H bottom bottom v () = 1 P 2ρν ( 2 H) (1) P = known constant (2) V 0 = 1 m/s, ρ = 1 kg/m 3, ν = 1 m 2 /s, H = 1 m (3) (4) P = 12, v() = 6(1 ) 8 / 1

9 9 / 1 Finite-Difference Navier Stokes Algorithm + SOR Rectangular grid = ih, = jh 3 Simultaneous equations 2 (v 0) v i+1,j v i 1,j + v i,j+1 v i,j 1 = 0 (1) v i+1,j + v i 1,j + v i,j+1 + v i,j 1 4v i,j (2) = h 2 v i,j [ v i+1,j vi 1,j ] h + 2 v [ i,j v i,j+1 vi,j 1 ] h + 2 [P i+1,j P i 1,j ] Rearrange as algorithm for Successive Over Relaation 4v i,j = vi+1,j + vi 1,j + vi,j+1 + vi,j 1 h 2 v i,j [ v i+1,j vi 1,j ] h 2 v [ i,j v i,j+1 vi,j 1 ] h 2 [P i+1,j P i 1,j ] (3) Accelerate convergence + SOR; ω > 2 unstable

10 10 / 1 End Part I: Basics River surface River surface L L H bottom bottom

11 Part II: Vorticit Form of Navier Stokes Equation 2 HD Equations in Terms of Stream Function u() v = 0 Continuit (1) (v )v = 1 ρ P + ν 2 v Navier Stokes (2) Like EM, simpler via (scalar & vector) potentials Irrotational Flow: no turbulence, scalar potential Rotational Flow: 2 vector potentials; 1st stream function v def = u() (3) ( ) ( ) uz = ˆɛ u u + ˆɛ z z uz (4) ( u) 0 automatic continuit equation 11 / 1

12 2 HD Equations in Terms of Stream Function (cont) 2-D flow: u = Constant Contour Lines = Streamlines v def = u() (1) ( ) ( ) uz = ˆɛ u u + ˆɛ z z uz (2) v z = 0 u() = u ˆɛ z (3) v = u, u v = (4) 12 / 1

13 13 / 1 Introduce Vorticit w() ω Vorte: Spinning, Often Turbulent Fluid Flow w def = v() (1) w z = ( ) v v (2) Measure of v s rotation RH rule fluid element w = 0 irrotational w = 0 uniform Moving field lines Relate to stream function:

14 14 / 1 Introduce Vorticit w() ω Poisson s equation 2 φ = 4πρ w(,) w def = v() (1) w = v = ( u) = ( u) 2 u (2) et u = u(, ) ˆɛ z u = 0 (3) 2 u = w (4) Like Poisson with ea w component = source

15 15 / 1 Vorticit Form of Navier Stokes Equation Take Curl of Velocit Form [ (v )v = ν 2 v 1 ] P ρ (Navier Stokes) (1) ν 2 w = [( u) ]w (2) In 2-D + onl z components: 2 u + 2 u = w (3) 2 2 ( ) 2 w ν + 2 w 2 2 = u w u w Simultaneous, nonlinear, elliptic PDEs for u & w Poisson s + wave equation + friction + variable ρ (4)

16 Relaation Algorithm (SOR) for Vorticit Equations = ih, = jh CD Laplacians, 1st derivatives u i,j = 1 ) (u i+1,j + u i 1,j + u i,j+1 + u i,j 1 + h 2 w i,j 4 (1) w i,j = 1 4 (w i+1,j + w i 1,j + w i,j+1 + w i,j 1 ) R 16 {[u i,j+1 u i,j 1 ] [w i+1,j w i 1,j ] [u i+1,j u i 1,j ] [w i,j+1 w i,j 1 ]} (2) R = 1 ν = V 0h ν (in normal units) (3) R = grid Renolds number (h R pipe ); measure nonlinear Small R: smooth flow, friction damps fluctuations Large R ( 2000): laminar turbulent flow Onset of turbulence: hard to simulate (need kick) 16 / 1

17 H Boundar Conditions for Beam Surface G v =du/d=v 0 w = 0 F Inlet v = -du/d = 0 v =du/d=v 0 w = 0 v = -du/d = 0 u = 0 C Half Beam u = 0 du/d = 0 dw/d = 0 w = u = 0 D B w = u = 0 E A center line Well-defined solution of elliptic PDEs requires u, w BC Assume inlet, outlet, surface far from beam Freeflow: No beam NB w = 0 no rotation Smmetr: identical flow above, below centerline, not thru Outlet 17 / 1

18 Boundar Conditions for Beam (cont) See Tet for More Eplanations Centerline: = streamline, u = const =0 (no v No flow in, out beam to it u = 0 all beam surfaces Smmetr vorticit w = 0 along centerline Inlet: horizontal fluid flow, v = v = V 0 : Surface: Undisturbed free-flow conditions: Outlet: Matters little; convenient choice: u = w Beamsides: v = u = 0; viscous v = 0 Yet, over specif BC onl no-slip vorticit w: Viscosit v = u = 0 (beam top) Smooth flow on beam top v = 0 + no variation: v = 0 w = v = 2 u 2 (1) 18 / 1

19 19 / 1 Implementation & Assessment:SOR on a Grid Basic soltn vorticit form Navier Stokes: Beam.p NB relaation = simple, BC simple Separate relaation of stream function & vorticit Eplore convergence of up & downstream u Determine number iterations for 3 place with ω = 0, 0.3 Change beam s horizontal position so see wave develop Make surface plots of u, w, v with contours; eplain Is there a resting place for salmon?

20 20 / 1 Results w(,)

7. TURBULENCE SPRING 2019

7. TURBULENCE SPRING 2019 7. TRBLENCE SPRING 2019 7.1 What is turbulence? 7.2 Momentum transfer in laminar and turbulent flow 7.3 Turbulence notation 7.4 Effect of turbulence on the mean flow 7.5 Turbulence generation and transport