TFAWS AEROTHERMAL Paper Session Evaluating the Performance of an Improved Finite Volume Method for Solving the Fluid Dynamic Equations F. Ferguson, J. Mendez, D. Amoo & M. Dhanasar Mechanical Engineering Department, NCAT, Greensboro, NC 27411 Presented By Frederick Ferguson Thermal & Fluids Analysis Workshop TFAWS 2017 TFAWS August 21-25, 2017 NASA Marshall Space Flight Center MSFC · 2017 Huntsville, AL
Presentation Outline ü Introduction ü CFD: Its Importance & Challenges ü The NS System of Equations: Its Closure ü Expectations: Flow Physics Prediction Capabilities ü The NS Non-Dimensionalization Process ü The IDS Concepts: From Grids to Points, Cells & Control Volumes; Spatial & Temporal Cells ü The IDS – Solving the Integral form of the NS Equations ü IDS – An Explicit Numerical Method, its execution ü IDS Applications o The Hypersonic Flat Plate, the Shock Boundary Layer Interaction and the Cross Flow Injection Problems ü Conclusion 2 TFAWS 2017 – August 21-25, 2017
The Importance of CFD Ref. 4 Data TFAWS 2017 – August 21-25, 2017
CFD Challenges Engineering Perspective: Analysis, Not Design Ref. 4 Data Potential Benefit Elimination of Complex Grids Long Cycle Times Due to 1. Grids Generation Req. & 2. Fidelity of Simulation Tools Final Product TFAWS 2017 – August 21-25, 2017
State-of-the-Art in the CFD Industry !Use DNS/LES Data to Develop Turbulence Models for CFD Applications http://www.bcs.org/content/conBlogPost/2035 Computer Speed vs. Time (Year) Industrial Applications vs CFD Models Why Cartesian Grids? Why Algebraic Turbulence Models? -fast grid generation, numerical & - numerical & memory efficiency memory efficiency during parallelization - capture the basic physics of the process TFAWS 2017 – August 21-25, 2017
Introduction Opportunity • Historically only Analytical Fluid Dynamics (AFD) and Experimental Fluid Dynamics (EFD). • CFD is the computer simulation of fluid dynamic systems through the use of engineering models (mathematical formulation) and numerical methods (discretization methods and grid generations) • CFD made possible by the advent of digital computer and advancing with improvements of computer resources (1947, 500 Flops, à 2003 Teraflops, 2003 à 2015 Petaflops à 2020 Exaflops ) Data, Ref. 2 1947 - 500 flops Computer 2003 – 20 Teraflops Computer TFAWS 2017 – August 21-25, 2017
Introduction Research Objective: To Solve the NS Equations - Applicable to a wide Class of Fluid Dynamic Problems - Accurate Flow Physics Capturing Capabilities - Efficient Grid Generation & Solution Process NS Eqs: System of Conservations Laws ¶ òòò òò r + r = dv V d s 0 ¶ t v s ¶ ( ) òòò òò òò òò r + r = - + t ˆ V dv V . d s V Pd s d s ¶ t v s s s ¶ òòò òò òò òò òò r + r = - + t + ! ˆ Edv E V . d s P V . d s . V d s q d s ¶ t v s s s s Plus the Boundary & Initial Conditions TFAWS 2017 – August 21-25, 2017
Viscous Relations æ ö ¶ ¶ u v æ ö t = t = µ ç + ÷ 2 µ ¶ ¶ ¶ u v w ç ÷ t = ç - - ÷ 2 xy yx ¶ ¶ ç ÷ y x è ø xx ¶ ¶ ¶ 3 x y z è ø ¶ ¶ æ ö u w t = t = µ + ç ÷ 2 µ æ ¶ ¶ ¶ ö v u w xz zx ¶ ¶ è z x ø t = ç - - ÷ 2 ç ÷ yy ¶ ¶ ¶ 3 y x z è ø æ ö ¶ ¶ w v ç ÷ t = t = µ + æ ö 2 µ ¶ ¶ ¶ w u v ç ÷ t = ç - - ÷ yz zy 2 ¶ ¶ y z ç ÷ è ø zz ¶ ¶ ¶ 3 z x y è ø ¶ T ¶ T ¶ T = - ! q z k = - ! q y k = - ! q x k ¶ z ¶ y ¶ x Equations and formulas from: J. D. Anderson(1995) : “Computational Fluid Dynamics-The basics with applications”, McGraw-Hill, Inc. TFAWS 2017 – August 21-25, 2017
The Closed System of the Equations Ø Additional relations = r P RT § Equation of state = e C T § Internal energy v 1 . 5 æ ö + T T 110 ç ÷ § Sutherland’s law for viscosity µ = µ ¥ ç ÷ ¥ + T T 110 è ø ¥ § Prandtl number ( ) k = k T Ø The closed system of the equations has only five unknowns ( ) r , u , v , w , T TFAWS 2017 – August 21-25, 2017
Capability: Predicted Flow Physics ! Examples of Problems to be solved Shock/Boundary Layer Interaction Flow Over Blunt Body 10 TFAWS 2017 – August 21-25, 2017
Capability: Predicted Flow Physics ! Examples of Problems to be solved Ref. Y. You, et al: Flow physics of a low momentum jet in supersonic crossflow Computational Flow Physics@Caltech 11 TFAWS 2017 – August 21-25, 2017
IDS: Non-Dimensionalization Process Fluid Properties Geometric Variables µ k z y x µ = = k z = y = x = µ k L L L ¥ ¥ Fluid Parameters/IDS Solution Variables w v r u T = w = = u = u r = r v T u u T ¥ ¥ ¥ ¥ ¥ IDS Code Inputs: r µ g L , V , , T , , k , , R + ¥ ¥ ¥ ¥ ¥ air Grids & I&B Cdts
IDS: Non-Dimensionalization Process Derived No-Dim Variables g P a RT t L u t = = = = P a T = = = = ¥ t t r 2 u g ¥ a RT t u L ¥ ¥ ¥ ¥ ¥ ¥ Properties Computed from Inputs µ g r C R R V L V ¥ = = = = p = ¥ ¥ ¥ C air , C air , Re , Pr ; M ( ) ( ) ¥ ¥ ¥ v p g - g - µ g 1 1 k RT ¥ ¥ ¥ Non-Dim Properties to be Computed µ + ( ) 1 . 0 110 T 1 . 5 µ = = ¥ T µ + T 110 T ¥ ¥ ( ) ( ) ( ) ( ) 3 2 k + k + k + k k T T C T T C T T C T T C = = ¥ ¥ ¥ ¥ k 3 2 1 0 k k ¥ ¥ TFAWS 2017 – August 21-25, 2017 13
IDS: Non-Dimensionalization Process Non-Dim Properties to be Computed 1 e 1 h 1 = = = = = r e T h T P T ( ) ( ) g g - g - 2 2 2 2 V 1 M V 1 M g 2 M ¥ ¥ ¥ ¥ ¥ 2 ( ) V 1 = = + + 2 2 2 = + = + k e u v w E e k e ; H h k e 2 Total Total V 2 ¥ IDS NS Coefficients 1 1 1 = = = NS NS NS C ; C ; C ; ( ) ( ) 1 g g - 2 g 3 g - 2 2 2 1 M M 1 M ¥ ¥ ¥ 2 1 1 1 1 1 1 NS = NS = NS = C C ; C ( ) 4 5 6 g - 2 3 Re Re 1 Re Pr M ¥ ¥ ¥ ¥ ¥ TFAWS 2017 – August 21-25, 2017 14
IDS - Solving the Integral Equations on Cartesian Grids The conservation of mass , momentum, and energy equations ¶ r òòò òò + r = dv V d s 0 ¶ t v s ¶ ( ) òòò òò òò òò r + r = - + t ˆ V dv V . d s V Pd s d s ¶ t v s s ¶ òòò òò òò òò òò r + r = - + t + ˆ ! Edv E V . d s P V . d s . V d s q d s ¶ t v s s s s 15 TFAWS 2017 – August 21-25, 2017
An IDS Cell != Control Volume - It’s A Physics based Approach - It consists of points, cells & surfaces Consider a rectangular Prism … Point Surface Cell TFAWS 2017 – August 21-25, 2017
IDS Control Volume Representation - The Spatial & Temporal Cells - The 2D Control Volume TFAWS 2017 – August 21-25, 2017
An IDS Control Volume Key Features ü Points z ü Cells ü Control Volume i,j,k x y TFAWS 2017 – August 21-25, 2017
The Mass Conservation Equation in an IDS Cell [ ] ( ) ( ) ( ) ( ) r + r + r + r - ì u u u u ü average ¶ r ¶ æ ö ï ï 1 ¢ ¢ ¢ ¢ 1 2 3 4 min us = òòò òò ç ÷ í ý r + r = [ ( ) ( ) ( ) ( ) ] dv V d s 0 r + r + r + r ¶ D ï u u u u ï è t ø 4 x î þ ¶ 1 2 3 4 t cell plus [ ] ( ) ( ) ( ) ( ) v s r + r + r + r - ì ü v v u v ï ï 1 ¢ ¢ ¢ ¢ 1 2 3 4 min us í ý [ ] ( ) ( ) ( ) ( ) r + r + r + r D v v v v 4 y ï ï î þ 1 2 3 4 plus [ ( ) ( ) ( ) ( ) ] r + r + r + r - ì ü w w w w ï ï 1 Flow leaves from the upper side ¢ ¢ ¢ ¢ 1 2 3 4 min us í ý [ ] ( ) ( ) ( ) ( ) r + r + r + r D ï w w w w ï 4 z î þ 4 ’ 3 ’ 1 2 3 4 plus Flow enters from Flow leaves from the left side the right side 1 ’ 2 ’ 4 3 a dy dz 1 2 dx Flow enters from the lower side TFAWS 2017 – August 21-25, 2017 19
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