Fast preconditioned solution of Navier-Stokes equations for compressible flows with physics

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1 Fast recoditioed solutio of Navier-Stoes equatios for comressible flows with hysics Eli Turel & Ore Peles Deartmet of Mathematics, Tel Aviv Uiversity Mathematics, Comutig & Desig Jameso 80 th Birthday Staford Uiversity November 20, 204

2 Lecture outlie RK/Imlicit smoother scheme Turbulet models Low Mach flows Time deedet - Dual time ste Real gas equatio of state Chemical reactios source terms Two-hase flow Results Coclusios

3 Why Accelerate Comlicated multile idustrial alicatios Dual time ste Multile trial rus to imrove accuracy (artial) History JST Jameso multigrid Jameso Imlicit Residual Smoothig Jameso & Caughey Rossow, Swaso & Turel High order accurate schemes (?)

4 Navier-Stoes Equatios N-S equatios i coservative form: S z H H y G G x F F t Q V V V ) ( ) ( ) ( F, G ad H are the iviscid fluxes F v, G v ad H v are the viscous fluxes The Navier-Stoes equatios for the coservative variables: T e w v u Q... 2 i i ;

5 ) ( ) ( ) ( ) ( ) ( ) ( V P E P w V P V v P V u V V F z y x N h V q F N x x x x x N v ) ( - - N S 2 2 u P E for ideal gas 2 2 u T e E N for a real gas j are the diffusio velocities of the secies j P is the ressure ad V is the velocity of the gas j h are the secific ethaly er uit mass of the secies j τ x is the stress tesor ad q is the heat flux

6 RK/Imlicit smoother with source terms The equatios set we solve i coservative form is: Q F t S Q Alyig the Gauss theorem for a arbitrary cotrol volume ad discretizatio i time yields the basic scheme: Q S τ V S F ds S V all faces F ds

7 The Ruge-Kutta time marchig scheme is: Q Q 0 tr The residual of the -th ste is: R S F ds V all faces For acceleratig the calculatios usig large CFL umbers the residual is relaced by a satially smoothed residual R ~

8 Followig Rossow 2006, Swaso et al. 2007, we start with the satially discretized equatio: Q t V all faces F ds S 0 Liearizig F ad S i time we obtai: I t Ads V all faces t S Q Q R F A Q S Q - the flux Jacobia - the source Jacobia

9 A A A where A A A 2 Trasformig the equatios to rimitive variables, the flux Jacobia is writte as: Fially, the imlicit smoothig scheme is give by: faces all NB local faces all ds Q A R Q Q S ds A Vol I ~ ~

10 0 j j j T j j T j x x F x x S u t x x P u t * -ω SST model equatios - ocoservative Salart-Allmaras model equatio - ocoservative ~ ~ ~ ~ ~~ ~ ~ d f C f C C S f C u t t b w w b t b ν κ ν ν ν ν σ ν ν ν

11 Low seed artificial flux: f A u u A u u 2 2 i/2 i i i i where A A c' c' q M c M r 2 r M mi max M, M, r ad i the RK imlicit smoother oerator R R c '

12 Time deedet - Dual time ste For steady state use seudo time aroach Dual time steig used for time deedet flow to use acceleratio methods develoed for steady state flow Physical time derivative is source term i equatios Aroximate hysical time derivative with a bacward differece scheme: Q 3Q 4Q Q t 2t

13 Sice we do ot ow Q + we aroximate by with Q. The residual is ow 3Q 4Q Q R F 2t For dual time stes the smoothig oerator is slightly modified: I t ε Vol R all A faces εt ds all A faces c t α ~ Q τ I t ds NB ~ Q local

14 Real gas equatio of state The equatio of state is: RT P W RT ω, w - Molar cocetratio ad molecular weight of the secies i i The mea molecular weight is: W i i w i The mea iteral eergy er uit mass is: e T N i y e i i T

15 Chemical reactios source terms Arrheius model The source term vector, S, describes the rate of chage of secies : d dt w q i iall reactios i,2,..., N ω, w - Molar cocetratio ad molecular weight of the secies ν i - Stochiometric coefficiets of the secies i the reactio i q i - The rate of rogress variables give by - q f,i ad q r,i are defied by: q i q f, i qr, i q f, i f, i ' sieces i ; q r, i r, i sieces '' i

16 The forward reactio rate f ad the reverse rate r are emirically ow fuctios of the temerature. The forward costat is give by a Arrheius exressio of the tye: AT i e Ei / RT f, i i A i, β i ad E i are the Arrheius costats: A i is the rate costat β i is the temerature exoet E i is the activatio eergy. The source term for the temerature is r, i K P RT atm K i i K S ex R H RT T c v h

17 Two-hase flow Solutio of reactive, turbulet two-hase flow Gaseous mai hase Cotiuous disersed hase icludes mixture of liquid ad solid articles Particles void fractio is egligible 7

18 Couled Navier-Stoes ad disersed hase system I N-S equatios a source term due to the disersal hase is added Q ( F F ) ( G G ) ( H H ) V V V t x y z S - S f cr.. 8

19 The EDP equatio model fluidized equatio for the disersed hase The variables we solve for are, v, T {, v, T, } are the diserse hase desity, velocity ad temerature is the solid fractio 0 T whe the articles are totally solid, whe it s equal to zero, the articles are liquid. 9

20 20 0 v t Coservatio of mass, 0 otherwise melt Q T T T mc v T t i i g u D i i v v f v v t v,,,, z y x i,, ; f T N T T f C m Q, Mometum Temerature where Solid fractio 0 otherwise melt m Q T T ml v t

21 2 The source terms for the N-S EDP equatios are f u D f T N f u D f u D f u D cos f u u u f T T f C w w f v v f u u f S, Re ; Re 000 Re 0.5 Re r N D P F else F where

22 Results Turbulet flow Trasoic flow over RAE 2822 airfoil Turbulet reactive flow rocet motor lume Time deedet - high agle of attac over NACA002 Low Mach Flow Two hase flow iside ballistic evaluatio motor Cojugate heat trasfer flow i covergig-divergig ozzle

23 SST Turbulet flow Trasoic flow over RAE 2822 airfoil M Re ~7x0 6 Comariso betwee exerimetal ad comutatioal ressure coefficiet - C

24 Log (err) Covergece History for Various CFL Numbers Differet CFL # are used for the N-S eqs. ad the turb. eqs. N-S CFL = x0 5 ; Turb. CFL = 2x0 4 N-S CFL = 000; Turb. CFL = 000 N-S CFL = 00; Turb. CFL = 00

25 Turbulet reactive flow rocet motor lume A rocet motor lume exitig from the motor ozzle ito a low Mach umber free stream flow. The flow velocity o the ozzle throat is soic ad the secies mass fractios are defied. The secies used for this roblem are: H, O, OH, H2, O2, CO, CO2, H2O, HCl ad N2. About 50 chemical reactios were cosidered

26 Covergece history fluid CFL =00,000 ad turbulet CFL =200 Mixture desity Turbulet variable ω

27 Turbulet ietic eergy Temerature

28 CO2 mass fractio (A) ad OH radical (B) A B

29 Dual time ste for flow over NACA002 M =0., agle of attac = 30, Re = 3x hysical time stes. Δt hysical = 0.2 sec. CFL = 00, relaxatio factor ε = 0.5

30 Residual reduced 3 orders Residual reduced 4 orders I summary: the smoothed dual time ste code eeds oly about 0% of the CPU time of the origial code.

31 Covergece history for series of CFL umbers Time Deedet Riema roblem 3

32 Efficiecy of MG for dual time stes Deeds o t TOL0 3 TOL0 4 # MG levels Avg # subiteratio CPU Avg # subiteratio CPU o MG mi 8 sec t mi 58 sec mi 5 sec mi 22 sec mi 3 sec mi 2 sec mi 7 sec mi 6 sec 32

33 Low Mach Flows NACA002 M=

34 34

35 Two hase flow iside a Ballistic Evaluatio Motor (BEM)- EDP The roblem of flow iside a solid rocet motor is solved usig: Fiite volume, local time ste, 3 levels sequecig ad multigrid acceleratio The schemes that were used are first order uwid for the EDP system ad AUSM+UP for the gas hase system High temerature, subsoic ijectio Viscous walls Suersoic exit 35

36 Tyical covergece history (Disersed hased desity) 36

37 Global view of cotour ma for diserse hase desity (5 micros) 37

38 Tyical cotour ma for gas hase temerature ad Mach umber T M 38

39 Coclusios RK/imlicit smoothig allows larger time ste The smoother allows faster covergece of comlex ad stiff flow roblems icludig turbulet reactive flow Reactive flow equatios are solved without oerator slittig Two Phase Flow Faster covergece i sub-iteratios of dual time ste Low Mach M=0.00 solved More robust solutios