2122 6 15. CFD Error Evaluation and Monotonic Convergence in Numerical Simulation of Flow Toshiyuki HAYASE 1 3 CFD 1 5 CFD 6 98-8577 2-1-1 E-mail: hayase@ifs.tohoku.ac.jp Richardson Extrapolation Grid Convergence IndexGCI 3 2 Richardson Extrapolation Grid Convergence IndexGCI 2 2.1 Validation Verification 2 Validation Verification 4 1Validation 2Verification 1 validation
7 2 verification verification validation verification validation verification 2.2 Grid Convergence IndexGCI Grid Convergence IndexGCI Roache 3 Richardson Extrapolation f h Taylor f = f + g 1 h + g 2 h 2 + g 3 h 3 + 1 f h = 2 g 1 = 2 h 1 h 2 (h 1 < h 2 ) f 1 f 2 f f = f 1 + f 1 f 2 r 2 1 + O ( h 3) r 2 / r = h 2 h1 (> 1) 2 3 2 3 f Richardson Extrapolation Roache p f 1 f 2 3 GCI 1 GCI 2 4 h 2 f 2 GCI 1 = 3 ε r p 1, GCI 2 = 3 ε r p r p 1, ε = f 2 f 1 f 1 5 GCI 7 1 2 QUICK 3 11 SIMPLER 2 QUICK 2 12 2 2 3% 2 κ ε 2 2 13 LES κ ε 14 17 f = f 1 + f 1 f 2 r p 1 + O (hp+1 ) 4
8 3.1 1 X 1 X 2 X 3 U 1 U 2 U 3 div U = 6 U T + ( U grad ) U = grad P + 1 2 U 7 Re Re Re = Ũ m b / ν 8 Blasius Ũ m = 2 P / λl ρ, λ = 2 P / L =.316Re 1/4 9 X 2 1 X 3 O 1 U 3 U 2 1 U 1 L X 1 L P SIMPLER 18 QUICK 2 19 2 2 CRAY C916 1 X 1 1 Present DNSHuser and Biringen Grid system GridA GridB GridC DNSA DNSB Grid points N 1 N 2 N 3 2 1 2 4 2 2 8 4 2 64 81 2 96 1 2.1 (.4.1 (.3 Grid spacing h 1 h 2 h 3.2.1 2.1.5 2.5.25 2.21) 2.17) 2 Mean grid size h mean = 3 h 1 h 2 h 3.126.63.32.11-.35.1-.3 Time step QUICK ht for 5 % error torelance in U m CENTRAL.1.94.5.42.5.44.25.3.25.27.1.17.157.126 Convective term discretization 3rd-order QUICK or 2nd-order central 5th-order upwind Time derivative term discretization 2nd-order 3-time-level implicit 4th-order Runge-Kutta Total residual at convergence.1.1.15 - - CPU time [s] for one time step 2 15 1 - - Periodical length L 4 6.4 Pressure difference P.649.125 Standard Reynolds number Re (Re τ ) 9573 94846
9 L 2 τ = ν / ε ( Ũ m / b ) 1 4 Su 15 6.3Madabhushi 14 6.4Huser 17 L = 4 L GRIDAB C 3 P Re 9 1 1 22 τ =.15 2 AB QUICK CENTRAL 1.8 21 1.7 1.6 1.5 1.4 1.3 1.2 3.2 1.1 Kolmogorov time scale 1...5.1.15.2 aquick 2 1.4 1.3 QUICK 2 CENTRAL AC 1.2 1.1 1. QUICK.9.8.7 QUICK Kolmogorov time scale Choi 2.6..5.1.15.2 2 bcentral U m U m
1 (a) QUICK, (b) QUICK, (c) QUICK,.2 64.2 64.24 (d) CENTRAL, (e) CENTRAL, (f) CENTRAL,.2 64.2 64.24 3 U 1 S 1 (ω n ) = E U 1 (t 2 m) exp ( jω n t m ) m=1 (n = 1,, ), 11 ω n = 2πn / ( ) E[ ] Nd N d N d (α n ) = U c S 1 (ω n ) 12 / α n = ω n Uc 3afQUICK CENTRAL 3 Huser DNS 17 3a L/k k = 1, 2, 3, α B 1 4 4 QUICK CENTRAL 3 2
11 18 18 16 14 Kolmogorov time scale 16 14 Kolmogorov time scale 12 1 12 1 B 8 6 4 2 B 8 6 4 2..5.1.15.2..5.1.15.2 aquick bcentral 4.2.1 * = h mean h mean = 3 h1 h 2 h 3 14 *.2.1 Bulk velocity Bandwidth QUICK CENTRAL.2.2.1.2 h mean 1/5 CENTRAL QUICK 1/2 ht 1 5 ht = φ 2 1 13 φ 2 φ 1 φ 1 φ 2 1 2 φ 13 5 2 5% ht 3.3 6 2 QUICK CENTRAL U τ = P / (4L) y + = U τ Re y Huser DNS / U 1 Uτ = y + 6a QUICK A DNS BC DNS
12 35 35 U 1 /U 3 25 2 15 DNS (B) U 1 /U 3 25 2 15 DNS (B) 1 1 5 5 1 1 y + y + aquick bcentral 6-5/3-5/3.5 5 aquick 7.5 5 bcentral 6b CENTRAL A DNS B DNS C DNS QUICK CEN- TRAL 7 7a QUICK CENTRAL QUICK CENTRAL C α 5/3 8 QUICK CENTRAL 8aQUICK CENTRAL 1 2 8aCENTRAL h mean =.63 QUICK
13 1.7 2 1.5 Kolmogorov length scale QUICK 18 16 14 CENTRAL U m 1. DNS (B) B 12 1 8 6 QUICK CENTRAL 4 2 Kolmogorov length scale.5..5.1.15..5.1.15 h mean a 8 CENTRAL QUICK CENTRAL 2 1 CENTRAL QUICK 8a 2 QUICK Huser DNS 8a 3 ( ν / ε ) 1/4 η = b = ν 3/4 b (.1Ũ 3 m / b ) 1/4 15 DNSB C DNSB CDNS 8bQUICK CENTRAL 2 Scheme QUICK p = 2 CENTRAL p = 2 h mean b GCI Grid Error%GCI 1 %GCI 2 % A 64-138 B 21 34 54 C 9 14 - A 7-15 B 23 38 32 C 16 8 - DNSA CENTRAL h mean α B QUICK 2 GCI 8a GCI 2 QUICK 3 2 p = 2 QUICK GCI GCI
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