Verification of a CFD benchmark solution of transient low Mach number flows with Richardson extrapolation procedure 1

Transcription

1 Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure S. Beneboula, S. Gounand, A. Beccanini and E. Suder DEN/DANS/DMS/STMF Commissaria à l Energie Aomique CEA-Saclay ASME Verificaion & Validaion Symposium May -4, Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure

2 Moivaions & Physical problem Moivaion & Physical problem Nuclear reacor safey: conainmen pressuriaion due o loss-of-coolan acciden in nuclear reacor Fluid injecion a low Mach number ino an axisymmeric caviy Mixed convecion flow (ho fluid injeced in cold amosphere) H. Paillère e al., Modelling of naural convecion flows wih large emperaure differences : A Benchmark problem for low Mach number solvers. ESAIM: Mah. Mod. & Num. Anal., 5; 9:67-6 A. Beccanini e al., Numerical simulaions of ransien injecion flow a low Mach number regime. In. J. Numer. Meh. Engng 8; 76: S. Beneboula, Modeling of ideal gas injecion ino a closed domain a low Mach number: Fracional sep and pressure based solvers. Tech. Repor: CEA - SFME/LTMF/RT8-4/A 8. Objecives Modeling flow injecion wih large densiy gradiens caused by hermal effecs Compuing numerical benchmark soluion of he flow wih wo differen CFD codes Esimaing numerical errors and he mehod convergence order 4 Applying Richardson exrapolaion (R.E) o deermine a reference soluion Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure

3 Low Mach number model Compressibiliy effecs are characeried by he Mach number M = v c Compressible flow (M >.): evoluion equaion for ρ, ρ(p, T ) Incompressible flow (M ): v = consrain, ρ consan Low Mach flow M <. Low Mach formulaion (A. Majda and J.Sehian (985)) Asympoic analysis γm Navier-Sokes equaions of order ero Asympoic resuls Pressure decomposiion: hermodynamic P (), dynamic p (x, ) Densiy variaions decoupled from pressure gradiens, ρ(t ), v Advanages Allows large variaions of ρ due o emperaure gradiens Removes acousic waves Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure

4 CFD codes modeling Model : Conservaive unseady axisymmeric N-S equaions S. Beneboula, G. Lauria, Numerical simulaions of anisohermal laminar vorex rings wih large densiy variaions. In. J. Hea & Fluid Flow 9; : ρ + (ρ u) = ρ u + (ρ u u) = p + Re τ + Fr (ρ ρ ) T + γt u T ρ (ρ u) = γ RePr (λ T ) P = ρt Numerical mehod : Finie Difference nd cenered Saggered grid (MAC) Coninuous projecion mehod Time inegraion: Explici nd order A-B/A-M predicor-correcor scheme Model : Non-conservaive unseady axisymmeric N-S equaions A.Beccanini, e.al, Numerical simulaions of ransien injecion flow a low Mach number regime. In. J. Numer. Meh. Engng 8; 76: dp γp d + u = γ γp (λ T ) u RT ( + u u = p ) + τ P ( + g T P ) T P T γ T dp + u T = γ P d + γ T γ P (λ T ) P = rρt Numerical mehod : FE Finie Elemen Q/P Regular mesh Algebraic projecion (Quareroni) Time inegraion: Implici BDF scheme Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure 4

5 Tes case - Ho fluid injecion Physical parameers Compuaional domain R j =.5 m, L = m, L = 7 m. Injeced fluid T j = 6 K, ṁ j =. kg/m /s, V j = ṁj =.7 m/s. ρ j (Re =, Pe = 4 and Fr =.) injecion = 6. s Caviy amosphere T a = K, P = 5 Pa. Boundary condiions Inle: T = T j, ρu = 6ṁj R j ( R j r). Walls: T =, ρ u =. n Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure 5

6 Numerical parameers & Resuls Numerical parameers Mehod N r N Consisency order User ime 5 8.6E-4.5 hour 7 4.5E-4 for ρu r, ρu, T, p 5.45 hours 4 4.5E hours E- FE E-4 for u r, u, T E-4 for p 58 hours Insananeous emperaure fields =.5 s = s = s =.5 s = 6 s Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure 6

7 Verificaion mehod Richardson Exrapolaion (RE): Assumpions f exac regular enough o apply Taylor expansion f h = f exac + C h + O(h + ) h i small enough o ge asympoic convergence C h i O(h+ ) Uniform space and ime refinemen, so for grids h h = h ( fh f h = ln f h f h ) / ln ( ) h h h :, C = f h f h h h, f ex = f h C h R.E advanages Increase of accuracy by eliminaing he runcaion errors Error esimae corresponding o C h fine Mehod verificaion by esimaing he convergence orders Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure 7

8 Resuls & Verificaions: Time dependen quaniies Thermodynamic pressure: P = rρt. Divergence inegral: ( u) dv Ω.6. R.E R.E FE FE -5.P P ex - P fg.8.4 R.E R.E FE FE Errors & convergence order err err FE FE 4 In-U.n In U.n ex -In U.n fg Errors & convergence order err Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure 8

9 Resuls & Verificaions: Time dependen quaniies 8 6 Kineic energy: ρu dv Ω 6 Ensrophy: ( u) Ω R.E R.E FE FE dv E kin 4 -.Ens 8 R.E R.E FE FE Maximum velociy Zoom on he maximum velociy U max R.E R.E FE FE R.E R.E FE FE Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure 9 U max

10 Resuls & Verificaions A = s Mehod P u E kin Ens U max fine grid R.E f /f fg 6.7 E E-.7E E-4.6E (4.) fine grid FE R.E f /f fg.8e-6.e-5.8e-4.5e-4.8e Remarks Relaive errors of he same order of magniude for boh and FE solver soluions solver: convergence order given by R.E close o he consisency order of he scheme () obained for P, u and E kin FE solver: convergence order given by R.E differen from he consisency order of he scheme () Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure

11 Resuls & Verificaions: Local variables ( = 6 s) Temperaure and axial velociy on he axis T FE Nr=8 Nr=44 Nr=9 R.E R.E verificaion FE Err Err FE R.E verificaion FE Err Err FE T ex - T fg.5 U 9.6. U ex - U fg 9.55 Nr= Nr= Nr=4 R.E Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure

12 Resuls & Verificaions: Local variables ( = 6 s) Radial velociy in he crosswise direcion a = Ur -.4 Ur -.84 Nr= -.8 Nr= Nr=4 R.E Nr= Nr= Nr=4 R.E R.E works beer for variable profiles far enough from he inle and he impac area (caviy ceiling) Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure

13 Resuls & Verificaions: Local variables ( = 6 s) Radial velociy in he crosswise direcion a = FE FE. -.8 Ur -.4 Ur -.84 Nr=8 -.8 Nr=44 Nr=9 R.E Nr=4 FE Nr= Fine grid soluions obained wih and FE solvers are in good agreemen even if he R.E doesn work evrywhere in he compuaional domain Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure

14 Conclusions Numerical soluion of low Mach number fluid injecion is compued wih wo CFD codes using: Two differen formulaions of he low Mach number Navier-Sokes equaions Two differen numerical mehods Very good agreemen is obained for local and global quaniies characeriing he flow compued wih and FE solvers Difficulies in applying Richardson exrapolaion on he whole compuing domain since he physical problem doesn saisfy he R.E assumpions: Siffness in ime: impulsive injecion s Siffness in space: large gradiens in paricular a he je impac regions Disconinuiy of he boundary condiions on emperaure a he inle (Dirichle for < r < R j, Neumann for R j < r < L r) Inersecion of he variable profiles resuling from he successively refined meshes 4 solver: in i s validiy domain, R.E gives convergence order abou (formal order) for primiive variables, P and u 5 FE solver: even if R.E resuls are no relevan, he compued soluion is quie accurae Verificaion of a CFD benchmark soluion of ransien low Mach number flows wih Richardson exrapolaion procedure 4