Error Evaluation and Monotonic Convergence in Numerical Simulation of Flow

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1 CFD Error Evaluation and Monotonic Convergence in Numerical Simulation of Flow Toshiyuki HAYASE 1 3 CFD 1 5 CFD hayase@ifs.tohoku.ac.jp Richardson Extrapolation Grid Convergence IndexGCI 3 2 Richardson Extrapolation Grid Convergence IndexGCI Validation Verification 2 Validation Verification 4 1Validation 2Verification 1 validation

2 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 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 O ( h 3) r 2 / r = h 2 h1 (> 1) 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 QUICK 3 11 SIMPLER 2 QUICK % 2 κ ε LES κ ε f = f 1 + f 1 f 2 r p 1 + O (hp+1 ) 4

3 X 1 X 2 X 3 U 1 U 2 U 3 div U = 6 U T + ( U grad ) U = grad P 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 CRAY C916 1 X 1 1 Present DNSHuser and Biringen Grid system GridA GridB GridC DNSA DNSB Grid points N 1 N 2 N (.4.1 (.3 Grid spacing h 1 h 2 h ) 2.17) 2 Mean grid size h mean = 3 h 1 h 2 h Time step QUICK ht for 5 % error torelance in U m CENTRAL 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 CPU time [s] for one time step Periodical length L Pressure difference P Standard Reynolds number Re (Re τ )

4 9 L 2 τ = ν / ε ( Ũ m / b ) 1 4 Su Madabhushi Huser 17 L = 4 L GRIDAB C 3 P Re τ =.15 2 AB QUICK CENTRAL Kolmogorov time scale aquick QUICK 2 CENTRAL AC QUICK QUICK Kolmogorov time scale Choi bcentral U m U m

5 1 (a) QUICK, (b) QUICK, (c) QUICK, (d) CENTRAL, (e) CENTRAL, (f) CENTRAL, 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 QUICK CENTRAL 3 2

6 Kolmogorov time scale Kolmogorov time scale B B aquick bcentral * = h mean h mean = 3 h1 h 2 h 3 14 *.2.1 Bulk velocity Bandwidth QUICK CENTRAL h mean 1/5 CENTRAL QUICK 1/2 ht 1 5 ht = φ φ 2 φ 1 φ 1 φ φ % ht QUICK CENTRAL U τ = P / (4L) y + = U τ Re y Huser DNS / U 1 Uτ = y + 6a QUICK A DNS BC DNS

7 U 1 /U DNS (B) U 1 /U DNS (B) y + y + aquick bcentral 6-5/3-5/3.5 5 aquick 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

8 Kolmogorov length scale QUICK CENTRAL U m 1. DNS (B) B QUICK CENTRAL 4 2 Kolmogorov length scale 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 B C A 7-15 B C DNSA CENTRAL h mean α B QUICK 2 GCI 8a GCI 2 QUICK 3 2 p = 2 QUICK GCI GCI

9 14 CENTRAL C GCI B BC CENTRAL GCI GRID A CENTRAL 4 2 QUICK QUICK J. A. C. HumphreyR. Greif 1U. B. Mehta : Some Aspects of Uncertainty in Computational Fluid Dynamics Results, Trans. ASME, J. Fluids Eng A. O. Demuren & R. V. Wilson : Estimating Uncertainty in Computations of Two-Dimensional Separated Flows, Trans. ASME, J. Fluids Eng P. J. Roache : Perspective: A Method for Uniform Reporting of Grid Refinement Studies, Trans. ASME, J. Fluids Eng P. J. Roache : Quantification of Uncertainty in Computational Fluid Dynamics, Ann. Rev. Fluid Mech P. J. Roache : Verification of Codes and Calculations, AIAA Journal C. J. Freitas : Policy Statement on the Control of Numerical Accuracy, Trans. ASME, J. Fluids Eng I. Celik & W. Zhang : Calculation of Numerical Uncertainty Using Richardson Extrapolation: Application to Some Simple Turbulent Flow Calculations, Trans. ASME, J. Fluids Eng I. Celik & O. Karatekin : Numerical Experiments on Application of Richardson Extrapolation with Nonuniform Grids, Trans. ASME, J. Fluids Eng J. Bergstrom & R. Gebart : Estimation of Numerical Accuracy for the Flow Field in a Draft Tube, Int. J. Numerical Method for Heat and Fluid Flow C. Chen, R. Lotz & B. Thompson : Assessment of Numerical Uncertainty Around Shocks and Corners on Blunt Trailing-Edge Supercritical Airfoils, Computers and Fluids T. Hayase : Monotonic Convergence Property of Turbulent Flow Solution With Central Difference and QUICK Schemes, Trans. ASME, J. Fluids Eng H. Schlichting : Boundary Layer Theory, 7th English Ed.McGraw Hill, A. O. Demuren & W. Rodi : Calculation of Tur-

10 15 bulence Driven Secondary Motion in Non Circular Ducts, Trans. ASME, J. Fluids Eng R. Madabhushi & S. Vanka : Large Eddy Simulation of Turbulence-Driven Secondary Flow in a Square Duct, Physics of Fluids A M. Su & R. Friedrich : Investigation of Fully Developed Turbulent Flow in a Straight Duct with Large Eddy Simulation, Trans. ASME, J. Fluids Eng H. K. Myong & T. Kobayashi : Prediction of Three-Dimensional Developing Turbulent Flow in a Square Duct with an Anisotropic Low-Reynolds- Number κ ε Model, Trans. ASME, J. Fluids Eng A. Huser & S. Biringen : Direct Numerical Simulation of Turbulent Flow in a Square Duct, J. Fluid Mechanics S. V. Patankar : Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC/New York, T. Hayase, J. A. C. Humphrey & R. Greif : A Consistently Formulated QUICK Scheme for Fast and Stable Convergence Using Finite Volume Iterative Calculation Procedures, J. Computational Physics C. A. J. Fletcher : Computational Techniques for Fluid Dynamics, 1Springer Verlag, T. Hayase & Y. Suematsu : Direct Numerical Simulation of Turbulent Flow in a Square Pipe, Proceedings of 3rd Triennial International Symposium on Fluid Control, Measurement and Visualization Ed. W. C. Yang and R. L. Woods, H. Choi & P. Moin : Effects of the Computational Time Step on Numerical Solutions of Turbulent Flow, J. Computational Physics

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