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Applied Computational Fluid Dynamics FACE8 – Applied CFD 2003 Institute of Energy Technology Aalborg University 1 Course objectives • To give an introduction to the possibilities of Computational Fluid Dynamics • To provide an appreciation for the complexity of fluid mechanics and CFD methods • To introduce important aspects related to the quality and trust in CFD FACE8 – Applied CFD • To give an idea about the potential of CFD in fluid mechanics 2 Course outline ∙ 1 MM ▫ Introduction to applied CFD ▫ Review of the finite volume method ▫ Geometry creation and CAD import ∙ 2 MM FACE8 – Applied CFD ▫ Grid generation ▫ Grid quality ▫ Grid adaption – static and dynamic ▫ Boundary conditions ∙ 3 MM ▫ Solution methods ▫ Solution quality ∙ 4 MM ▫ Advanced physical models ∙ 5 MM ▫ Post-processing ▫ Validation 3 Hands – on (room 5) ∙ Available software (Win2K or Win-XP platforms) ▫ FLUENT currently ver. 6.1 ◦ Gambit ▫ FIDAP FACE8 – Applied CFD ◦ Gambit ▫ FIELDVIEW 7 ▫ CFX4 ◦ CFX-Build ◦ CFX-Analyse ▫ CFX5 ◦ CFX-Visualize ∙ Simple problems ∙ CFD in projects 4 Exercise 1 symmetry =1 FACE8 – Applied CFD Outflow =0 symmetry Outflow =1 =0 Outflow 5 Useful homepages FACE8 – Applied CFD Course homepage www.cfd-online.com www.qnet-cfd.net www.fluent.com 6 What is CFD (the history)? ∙ Colorful fluid dynamics, colors for directors, ... or? FACE8 – Applied CFD LES and DNS simulation Explosions Multiphase flow Single phase Disperse multiphase flow Combustion Multiple frame of reference models GUI Fluid-solid interaction Optimization algorithms Process coupling techniques 1960’s – aircraft industry Unstructured codes Flow around CAD import 2D structured, single aerodynamic profiles 3D structured, single block codes block codes 1-2 eq. turbulence models Multi-block codes Timeline 7 Why use CFD? Pure/classical Experimental fluid dynamics fluid dynamics FACE8 – Applied CFD Limited knowledge Expensive Unsolvable sets of equations Few industrial type ? Pre-prototype testing Timeconsuming Difficult to isolate subproblems problems solvable even Virtual prototyping through approximations Prohibitive to perform Reduced time-to-market full-scale tests Subproblem testing Impossible to measure in Agressive environments Full-scale testing expensive Sensitivity analysis Computational Agressive environments fluid dynamics Complex physics ... 8 CFD application areas Power generation Aerodynamics (cars, aeroplanes etc.) Air conditioning Electronics FACE8 – Applied CFD Safety issues Purpose Reliability Efficiency Forecasts Velocities Leading CFD codes on the market: Turbulence Pressure - CFX (CFX4, CFX5, CFX-Tascflow Properties Temperature - Fluent (Fluent6, Fluent4, FIDAP) Species - STAR-CD Heat fluxes Shear stresses A minimum of time should be spent on pre-processing! - CAD import, automatic grid generation Trajectories 9 CFD – short and sweet Pre-processor FACE8 – Applied CFD CFD Solver Post-processor Wall boundary - ”no-slip” Udløb Neumann condition Dirichlet inlet 0 kst condition n kst p kst Wall boundary - ”no-slip” 10 FLUENT program structure FACE8 – Applied CFD 11 The CFD user must be an expert... Physical models Numerical methods - Turbulence - Truncation errors - Two-phase flow - Stability - Chemistry FACE8 – Applied CFD - Class of PDE - Compressibility - …………….. - Heat transfer - Radiation - …………… Boundary conditions - Experience from similar problems - Measurements - Knowledge from “simple” cases - …………….. ? The Engineer - Physical insight (choice of models) - Experience - Review / evaluation of results - Choice of approach (exp. or num) - …………….. 12 FLUENT recommendations FACE8 – Applied CFD 13 Quality and trust Required level of detail - prediction of small losses (e.g. in pumps) FACE8 – Applied CFD - Overall flow direction (recirculation zones etc.) Even qualitative properties can reveal weak points in a design! Limit Predicted Error bands 14 Quality and trust cont’d Model uncertainties: - incomplete knowledge of underlaying physics FACE8 – Applied CFD Errors (these are the ones we can minimize): - discretization error - convergence error (iteration) - round-off (numerical precision) - user (garbage in - garbage out) - coding 15 FACE8 – Applied CFD 16 Applications Applications cont’d FACE8 – Applied CFD 17 Applications cont’d FACE8 – Applied CFD 18 Applications cont’d FACE8 – Applied CFD 19 Applications cont’d FACE8 – Applied CFD 20 Applications cont’d FACE8 – Applied CFD 21 Applications cont’d FACE8 – Applied CFD 22 Applications cont’d FACE8 – Applied CFD 23 An overview the governing equations Quantity/ Equation property Mass 0 divu FACE8 – Applied CFD t p divgradu S Mx Du x-momentum Dt x p divgradu S My Dv y-momentum Dt y p divgradu S Mz Dw z-momentum Dt z p div u divk grad T Si Di Internal energy Dt Equations of p p , T , i i , T state p RT , i cvT General transport divu div grad S equation t Diffusion coefficient 24 Governing Equations Recall the general transport equation: ( ) FACE8 – Applied CFD div( u ) div( grad ) S t rate of change convection diffusion source term with time Examples: Continuity: =1 =0 S=0 U-momentum: =u = S = -dp/dx 25 Spatial discretization (1) diffusion We are beginning with the diffusion term (not only due to a concentration gradient but in general): ( ) div( u ) div( grad ) S t FACE8 – Applied CFD As an example, consider 1-dimensional conduction of heat: d dT k S 0 dx dx where: k = thermal conductivity T = temperature S = rate of heat generation per unit volume 26 Spatial discretization (2) diffusion Consider the 1-dimensional grid system (y and z = 1): (x) w (x) e FACE8 – Applied CFD w e W P E x We integrate the heat equation over this volume: dT dT e k k w S dx 0 dx e dx w 27 Spatial discretization (3) diffusion dT dT e k k w S dx 0 dx e dx w FACE8 – Applied CFD We need to assume a profile to evaluate the gradient terms: W w P e E W w P e E dT/dx NOT defined for this profile! 28 Spatial discretization (4) diffusion If we use the linear profile assumption: dT dT e k k w S dx 0 dx e dx w FACE8 – Applied CFD W w P e E We arrive at the following discretized equation (with uniform cells): k e (TE TP ) k w (TP TW ) S V 0 (x ) e (x ) w (x) w (x) e w e W P E 29 x Spatial discretization (5) diffusion The discretized equation is written as: aPTP aETE aW TW b anbTnb b FACE8 – Applied CFD where: ke kw aE = aW = aP = aE + aW b = S Dx (dx )e (dx )w In 2 dimensions, this is often illustrated as an amoeba: 30 Spatial discretization (6) diffusion and convection We extend the discretization equation to include convection also: ( ) FACE8 – Applied CFD div( u ) div( grad ) S t We look at the 1-dimensional case for simplicity: d d d ( u ) dx dx dx 31 Spatial discretization (7) diffusion and convection Consider again the 1-dimensional grid system (y and z = 1): FACE8 – Applied CFD (x) w (x) e w e W P E x We integrate the diffusion-convection equation over this volume: dT dT ( u ) e ( u ) w dx e dx w 32 Spatial discretization (8) diffusion and convection If we use the same linear profile assumption for: ( u ) e ( u ) w FACE8 – Applied CFD W w P e E We find that (assuming P to be midways between W and E): e ½( E P ) and w ½( P W ) (x) w (x) e w e W P E x This approximation is termed central differencing - it is termed second-order accurate since error O(x2) in terms of the Taylor series 33 Spatial discretization (9) diffusion and convection Now inserting these discretized diffusion and convection terms yields: 1 1 ( P ) w ( P W ) ( u ) e ( E P ) ( u ) w ( P W ) e E FACE8 – Applied CFD 2 2 (x ) e (x ) w If we introduce: F u D x we can again write: aPP aEE aWW b anbnb b now with Fe F a E De , aW Dw w 2 2 34 Spatial discretization (10) diffusion and convection If we consider the case where: De Dw 1 and Fe Fw 4 FACE8 – Applied CFD We can calculate the value at point P from those at points E and W: (a ) E 200 and W 100 P 50! (b) E 100 and W 200 P 250! This is clearly unrealistic! Conclusion: Central differencing does not always work for the convection term (specifically it can be shown that -2<F/D<2 is required => low velocity + fine grids) We need another approximation! 35 Spatial discretization (11) diffusion and convection The upwind scheme identifies flow direction: FACE8 – Applied CFD W w P e E Flow from right to left (Fe< 0 and Fw< 0) e E and w P Similarly, if flow from left to right (Fe< 0 and Fw< 0) e P and w W 36 Spatial discretization (12) diffusion and convection For the flux across the east face, e, this scheme leads to: Fee P Fe ,0 E Fe ,0 FACE8 – Applied CFD Again we can write: aPP aEE aWW b anbnb b Now, for the upwind scheme the coefficient are: aE De Fe ,0 , aW Dw Fw ,0 The Upwind scheme is only first-order accurate hence the error O(x) 37 Spatial discretization (13) diffusion and convection There is a large number of differencing schemes out there: •the Hybrid scheme combines the Central difference and the Upwind schemes •the Power law scheme approximates the exponential behavior (not exact in 2 or 3D) FACE8 – Applied CFD More advanced schemes extend the amoeba to include more neighbour cells: parabola W w P e E •second-order Upwind •QUICK •and many others 38 Spatial discretization (14) false diffusion Often it is stated that the Central difference scheme is superior to the Upwind scheme because it is second-order accurate whereas the Upwind scheme is only first-order accurate. FACE8 – Applied CFD If we compare the Central difference and Upwind schemes: ux Upwind 2 This added diffusion is considered bad, however it actually corrects the solution at large Peclet number (large cells)! 39 Spatial discretization (15) false diffusion, the proper view False diffusion is a multidimensional phenomenon FACE8 – Applied CFD Consider the following case where = 0: =0 No false diffusion =1 False diffusion False diffusion occurs when the flow is NOT perpendicular to the grid lines! We will investigate this further in the exercise 40 Spatial discretization (16) Quality and trust The convection terms are of primary concern The accuracy stated may be reduced on non-uniform grids (e.g. QUICK scheme) FACE8 – Applied CFD Halving the cell size reduces the error with a factor of: - 8 for third-order schemes - 2 for first-order schemes Higher-order schemes more unstable (perhaps due to physially unsteady flow) Guidelines: - use a least second-order accurate schemes - first-order schemes may be used to start the calculation - use grid refinement test to estimate discretization error - look at distribution of residuals 41 Temporal discretization (1) We have now discretized the spatial terms: ( ) div( u ) div( grad ) S FACE8 – Applied CFD t time convection diffusion dependence For simplicity we consider time-dependent heat conduction: T T c k t x x 42 Temporal discretization (2) We integrate in time (from t to t+t) and space: e t t t t e T T c dt dx k dx dt t t x FACE8 – Applied CFD w t t w (x) w (x) e w e W P E x If we assume the value of T prevails over the entire volume: e t t T c dt dx cx (TP TP0 ) 1 w t t 43 Temporal discretization (3) For the diffusion term: t t k e (TE TP ) k w (TP TW ) cx (TP TP0 ) 1 (x ) (x ) w dt e FACE8 – Applied CFD t We need an assumption for the variation of T in time (between t and t+t): t t 1 TP dt f TP (1 f ) TP0 t t where 0 < f < 1 44 Temporal discretization (4) f =0: Explicit scheme (first-order accurate) f = 0.5: Crank-Nicolcon (second-order accurate) f = 1: Implicit (first-order accurate) FACE8 – Applied CFD Explicit T P0 Crank-Nicolson T P1 Implicit t t+dt 45 Temporal discretization (5) comments For stability, the Explicit scheme requires (on uniform grid): ( x ) 2 t FACE8 – Applied CFD 2 To give realistic results, the Crank-Nicolson scheme requires: ( x ) 2 t The Imlicit scheme is always stable (but still only first-order)! Temporal and spatial discretization are strongly coupled 46 Temporal discretization (6) comments Schemes that are higher-order accurate in time exist, e.g.: FACE8 – Applied CFD T 1 (3 T n 1 4 T n T n 1 ) t 2 t 47 Temporal discretization (7) quality and trust Physically time-dependent flows often fail to converge using steady-state methods The order of the scheme and the time-step size determine the: FACE8 – Applied CFD - phase error - amplitude error Guidelines: - second-order accuracy is recommended - check the influence from temporal discretization on frequency etc. - check the influence from time-step size - ensure the time-step is sufficient to capture feature of interest - begin with small time-steps (e.g. CFL = t v/x < 1) 48 CAD import ∙ Be aware of FACE8 – Applied CFD ▫ Blueprints are up-to-date ▫ last minute design modifications ▫ imported CAD geometry is sufficiently accurate ▫ tolerances e.g. in turbomachinery ▫ changes in geomtry due to mechanical loading ▫ changes in geometry due to wear or fouling ▫ UNITS 49 CAD import ∙ Short Gambit demo FACE8 – Applied CFD 50

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computational fluid dynamics, fluid dynamics, fluid mechanics, vortex dynamics, applied mechanics, numerical analysis, mechanical engineering, fluid flow, turbulent flow, turbulence model, numerical simulation, computer methods in applied mechanics and engineering, large eddy simulation, wright-patterson afb, scientific computing

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posted: | 6/1/2010 |

language: | English |

pages: | 50 |

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