Paola CINNELLA. DynFluid Laboratory, Arts et Métiers ParisTech, Paris, France and Università del Salento, Lecce, Italy.

Transcription

1 Paola CINNELLA DynFluid Laboratory, Arts et Métiers ParisTech, Paris, France and Università del Salento, Lecce, Italy CISM, 23rd January 2014

2 Introduction Recent progress in HiFi-CFD Progress in turbulence modelling Progress in numerical methods Progress in uncertainty quantification and data assimilation Numerical results: some recent contributions High-accurate numerical schemes for scale-resolving simulations Efficient hybrid RANS/LES simulations of separated flows Predictive RANS simulations using Bayesian inference Conclusions and perspectives 2

3 1 Introduction 3

4 ACARE Vision 2020 recommendations for aeronautical engines 50% reduction in CO2 emissions 80% reduction in NOx emissions 50% reduction in perceived aircraft noise Need for innovative concepts and advanced design tools Improvement of Energetic efficiency 20% reduction in greenhouse gases 20% renewable energy technology 10% reduction in energy consumption Need for efficient renewable energy conversion systems Aerodynamic/hydrodynamic design plays a key role Need to take into account at an early stage of design sources of variability (fluctuating operating conditions, uncertain geometry, model deficiencies) European Vision for Aeronautic transport and Energy 4

5 Need for modelling flows of increasing geometrical and physical complexity Use of more realistic physical models (turbulence effects, real-gas effects, multiphase flow phenomena, aeroacoustic, thermal and aeroelastic couplings) Need for more reliable, accurate and efficient numerical tools High-fidelity (error-free, quantified uncertainty) CFD Advanced uncertainty quantification and robust optimization tools Strong interactions among mathematical/physical/computational aspects Gottlich et al. (2004), internal-aerodynamics.html J. Turbom. 126:

6 Today : focus on some aspects of HiFi-CFD methods for industrial applications 6

7 2 Progress in turbulence modelling 7

8 Reynolds Averaged Navier-Stokes equations are the most widely used approach for industrial simulations Robust, cheap, work fine for «simple» flows Separating and reattaching flows dominated by a low-frequency unsteady behavior related to large flow structures Not in the «genes» of RANS!! Use LES/DNS?? 8

9 High computational cost of wall-bounded LES due to the necessity of resolving tiny energetic structures in the near wall layer This layer is often well represented (in average) by RANS simulations! IDEA #1: use RANS as a wall model for LES IDEA #2: more generally, use RANS everywhere as the grid is fine enough to resolve the relevant part of the energy spectrum Based on the formal similarity of RANS and LES equations Hierarchy of turbulence modelling strategies, by 9

10 Two possibilities: «zonal» vs «global» hybrid modelling global methods: continuous treatment of the flow variables at the interface LES content generated progressively through a grey zone zonal methods: discontinuous treatment of the RANS/LES interface construct a transfer operator at the interface Automatic global methods more attractive for industrial applications Extremely strong impact of numerical ingredients (implicit spurious filtering introduced by the numerical scheme) Examples of «zonal» methods. [Spalart Ann Rev 2009} 10

11 High resolution discretization methods Schemes introduce dissipation and dispersion errors Numerical dissipation «drains» energy after a given cutoff frequency Numerical cutoff has to be higher than filter cutoff + Self-adaptive turbulence model Tends to RANS in poorly resolved regions Tends to DNS in fully resolved regions Tends to LES in partially resolved regions Allows backscatter of energy from small to large scales Resolved (blue) vs modelled scales (white) 11

12 Based on the classical k-ε model but extendable to other models Model «sensitized» to grid resolution Backscattering mechanism if a too large portion of energy is modelled compared to local grid resolution (k=modelled energy, k r = resolved energy) If a fine grid simulation is initialized with RANS, the sensor becomes negative and amplifies fluctuations (increases the amount of resolved energy) while lowering the modelled one (negative «production» coefficient) 12

13 3 Progress in numerical methods 13

14 What s high-order? Roughly, order greater or equal 3 Why high order? Increasing the operation count to improve the order is more efficient than increasing the number of grid points Large meshes + massive parallelism memory, storage and post-treatment problems; massively parallel computer not always readily available, high energy consumption High-accurate numerical schemes higher cost per mesh point, robustness, ability to handle complex geometries, parallel performance 3 rd order 5 th order In high-order we trust! Slow down of Moore s law, supercomputers energy consumption issues Taylor-Green Vortex, grid, t=12, Q- criterion = 3: top RBC3, bottom RBC5 14

15 Finite differences ( accurate, cheap, simple; complex geometries, conservation issues) Increase the stencil Standard high-order DF, optimized schemes Use gradient information (Padé) Compact schemes Finite volumes ( conservation, flexibility; cost, accuracy) Use high-order cell-wise reconstructions MUSCL, K-exact methods, radial basis functions + least mean squares, Finite elements ( accuracy, flexibility; cost, memory, shocks) Continuous FE need stabilization for fluid mechanics problems Discontinuous Galerkin, spectral differences, spectral volumes, Your choice depends on what you are looking for!! 15

16 Develop a family of high-order schemes with the following characteristics High resolvability Good shock capturing capabilities Ability to handle complex geometries Robustness Moderate computational cost and memory consumption requirements Design strategy Structured grids low memory, cost Use of compact schemes low error constants, spectral-like accuracy Use of intrinsically dissipative schemes stability and shock capturing without tuning parameters Use of overset grids complex geometries, parallelism 16

17 Initially developed by Lerat and Corre (JCP 2001) Compact stencil First-order compact dissipation in the transient robustness, convergence speed High-order accuracy at steady state Design principles given for the hyperbolic system of conservation laws w + f + g = 0 t x y A= f, B= g Jacobian matrices w w Residual-based scheme expressed only in terms of approximations of the exact residual: r = wt + fx + gy Precisely, it writes like: ( ) ( ) w state vector, f w, g w fluxes in x and y ( r ) = d 0 jk, jk, ( r ) centered approximation of rat point ( jk, ) d 0 jk, jk, residual-based numerical dissipation 17

18 Given a Cartesian grid we introduce the standard difference operators in the j and k directions: The numerical dissipation term is defined as r, r 1 2 Φ, Φ 1 2 d 1 1 jk, = jk, 1 x 2 y 2 δ Φ r + δ Φ r = δ 2 x Φ r + δ y Φ r + O h jk, p ( ) ( ) ( ) ( ) ( ) mid-point residuals, centered approximations of r dissipation matrices depending on the eigensystem of A, B ( xjk,, yjk, ) = ( jδxk, δy), with steps δx, δy O( h) ( ) 1 = ( ) ( ), ( ) 1 = ( ) + ( ) δ µ 1 j+, k j+ 1, k jk, 1 j+, k j+ 1, k jk, 2 2 ( ) 1 = ( ) ( ), ( ) 1 = ( ) + ( ) δ µ 2 jk, + jk, + 1 jk, 1 jk, + jk, + 1 jk, 2 2 = 0 r r r Main and mid-point residuals approximated through high-order Padé formulae 18

19 [Lerat, Grimich, Cinnella JCP 2013; Grimich, Cinnella, Lerat JCP 2013] Genuinely multidimensional Centred, but intrinsically dissipative (no need for artificial dissipation, filters or limiters) High cutoff dissipation Low dispersion Advection along a mesh direction Resolvability Dispersion accuracy limit Dissipation accuracy limit Dispersion Dissipation 19

20 Extension to general grids via a finite volume approach [Rezgui, Cinnella, Lerat C&F 2001; Grimich, Michel, Cinnella, Lerat C&F 2014] Schemes up to 3rd order: weighted formulation taking into account mesh deformations rigorously 2nd-order accurate on highly deformed meshes needs computation of interpolation coefficients of flux densities from cell centers to the nodes (additional memory load) Not straightforward for higher order schemes Overset grid framework [Outtier, Content, Cinnella 2013] computational grids made by several interconnected structured blocks Conformal joins 1 to 1 or point to point communication Non-conformal joins blocks share information on a variety of dimension n-1 (for a n-dimensional problem) Overset joins blocks share information on a n-dimensional variety; multiply defined points exist in the domain 20

21 4 Progress in uncertainty quantification and data assimilation 21

22 Fluid Dynamics equations typically require numerical resolution: they are affected by both errors and uncertainties ERROR = recognizable deficiency in any phase of simulation that is not due to a lack of knowledge. Generally, it can be reduced UNCERTAINTY = potential deficiency in any phase or activity of the modelling and simulation due to a lack of knowledge (Definitions from AIAA Guide G , 1998) Kinds of errors and uncertainties: Numerical approximation errors, solution errors, round-off errors can be improved Model definition uncertainties (geometry, operating conditions) Errors/uncertainties specific to the physical/mathematical model Fluid properties (density, viscosity, compressibility,...) Submodels describing fluid behavior (EOS, turbulence models, viscosity,...) 22

23 Consider transonic flow over a wing section Flow conditions are random variables described by a pdf (not always known) We want the code to return pdf of Quantities of Interest QoI AoA M, Re Mach number isolines 23

24 Geometrical and operating condition uncertainties are essentially irreductible aleatoric uncertainties Physical/mathematical models: error or uncertainty? Modeling errors : conscious use of a possibly unsuitable/partially suitable model for a given problem e.g. use of an inviscid or incompressible flow model, use of turbulence models, use of the ideal polytropic gas model Modelling uncertainties : does a model fit a given problem? How close it is to reality? lack of knowledge that could be improved epistemic uncertainty 24

25 Epistemic uncertainties Choice of the appropriate level : essentially expert judgement For a given level o Up to now Several possible models, which differ by Their mathematical structure The associated closure parameters o Model structure chosen by expert judgement source of uncertainty o Model constants not univocally determined source of uncertainty Literature focuses essentially on the second point How to deal with the first one? 25

26 Montecarlo methods Sample input random variables according to their pdf Solve a deterministic model for each sample Compute solution statistics Unacceptably expensive for CFD applications (deterministic run O(1) to O(10) CPU h) May be performed on a surrogate model (ANN, radial basis, Kriging, Co-Kriging, ). Errors? Polynomial chaos expansions Intrusive approach Non intrusive (collocation) approach Sensitivity methods (Method of Moments) Approach low-order moments of the output pdf by their Taylor-series expansion about the mean value of input pdfs Second-order approximation requires 1st and s2nd-order sensitivity derivatives 26

27 Quantification of the global modelling uncertainty Parameter uncertainty find pdf of model parameters, propagate to the solution Structural uncertainty find probabilities associated to a model (plausibility) For many applications, this is expected to be important (e.g. turbulence models, equations of state, cavitation models, ) Model calibration Map data errors into numerical input errors and correct the input to achieve a better agreement with observed data (posterior pdfs) Mathematical framework: Bayesian framework modelling uncertainties treated in probabilistic terms 27

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29 Model calibration: Ingredients y = (, ) M x θ Explanatory variables x (here assumed as non random) Model random inputs θ described through the prior joint pdf p(θ) Experimental observations z of y characterized by their joint pdf p(z) Mathematical model M maps x into y with some probability p(y θ,m) Bayes theorem p( θ) p( z θ, M) p( θ zm, ) = (1) p( z) where p(θ) is the input prior probability and p(y θ,m) is the likelihood function; p(y) can be treated as a normalization constant Equation (1) is a statistical calibration : it infers the posterior pdf of the parameters that fits the model to the observations y. 29

30 Sometimes several models available to describe the same phenomenon Not always possible to identify the «best model» a priori How to account for this uncertainty in predictions? Bayesian model averaging (BMA) describes the pdf of a QoI as a weighted average of predictions provided by different models Let M i be a model in a (finite and discrete) set M, S k a calibration scenario in a set S and Z the set of all experimental data The BMA prediction of the expectancy of a QoI is [Draper 1998]: i, k k i k k k i M k S expectation of for posterior model prior scenario a given model and data set probability probability ( Z) = ( S, ) (, ) ( ) E E M z p M S z p S ( ) ( i, k k) ( i k k) ( k) Its variance is: var Z = var M S, z p M S, z p S + i M k S in-model, in-scenario variance 2 ( E( Mi, Sk, zk) E( Sk, zk) ) p( Mi Sk, zk) p( Sk) i M k S k S + between-model, in-scenario variance 2 ( E( Sk, zk) E( z k) ) p( Sk ) Between-scenario variance 30

31 5 Numerical results 31

32 On the impact of high-order schemes Resolving fine scales : the viscous Taylor-Green vortex problem Toward geometrical complexity : from an isolated airfoil to a rotor/stator interaction problem High-order hybrid RANS/LES simulation of a backward facing step : when models and numerics interact Predictive RANS simulations using Bayesian modelscenario averaging 32

33 Viscous case, Re=1600 Model of transition to turbulence via vortex stretching mechanism Integral quantities: kinetic energy Vortex stretchin g Transition to turbulence Fully developed turbulence Q=0 RBC5, mesh 33

34 RBC5 scheme, different mesh resolutions mesh, different RBC schemes 34

35 RBC5 scheme, different mesh resolutions RBC3 scheme, mesh mesh, different RBC schemes 5th-order scheme overperforms the third-order one by using 8 times less degrees of freedom 35

36 M=0.85, α=1 (ou alors M=0.8, a=1.25). RBC3 scheme Grid IsoMach lines Wall Mach number Results in good agreement with the literature Sharp and non-oscillatory shock profiles 36

37 carter hub rotor motion Complex unsteady test case High vane exit Mach number: pressure ratio of 5.11 experimentally 3.5M points Chorochronic B.C. (except inlet) URANS (k-l, t= ) Demonstrates the capability of computing 3D complex cases 37

38 hub carter rotor motion 38

39 [Pont, Cinnella, Robinet, Brennet, HRLM 2014] 39

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42 A smart combination of modelling and numerics enables accurate computations of complex flows with an affordable computational cost 42

43 Objective: predict velocity profiles developing in the turbulent boundary layer close to the wall Boundary layer Governing equations: Reynolds-Averaged Navier-Stokes equations supplemented by a turbulence model Algebraic Baldwin-Lomax (1972) model Launder-Jones s (1972) k-e model Menter s (1992) k-w SST model Spalart-Allmaras (1992) one-equation model 43

44 Calibration based on experimental data (velocity measures) for 15 boundary layers subject to both positive and negative pressure gradients One calibration per model and per scenario Numerical solutions obtained through a fast boundary-layer code, more complex flow topologies will require the use of a surrogate model. Use Markov-Chain Monte-Carlo method to draw samples from p(y θ) p(θ) Used these samples of θ to construct approximate pdfs through a kernel-density estimation. 44

45 Posterior distribution of y Posterior distribution of ηy 45

46 Posterior model plausibilities computed for all models in M for each S k using samples from Can be considered as a measure of consistency of calibrated model M i with data z k Large spread in model plausibilities, according to the pressure gradient scenario 46

47 The spread in most-likely closure coefficients due to different pressure gradients is significant, thus there is no such thing as a true value for the closure coefficients. There is no such a thing as a best model no more! How to summarize the effect of both parametric and model-form uncertainty to make predictions of new cases? 47

48 BMA prediction BMA prediction for a validation case (not included in the calibration set) Strong adverse pressure gradient Uniform pmf over the calibration scenarios Good prediction, variance strongly over-estimated Significant contribution of the between-scenario variance 48

49 BMA prediction BMA prediction for a validation case (not included in the calibration set) Strong adverse pressure gradient Non-uniform pmf over the calibration scenarios scenarios with a large betweenmodel, in-scenario variance are penalized through the error measure Prediction closer to the validation data Variance consistent with the experimental uncertainty 49

50 6 Conclusions and Perspectives 50

51 HiFi-CFD requires advanced modelling Hybrid RANS/LES modelling may improve predictions of separated flow IF the numerics is accurate enough Other modelling problems may exhibit a strong dependence on numerical ingredients (dense gas flows, cavitation, ) HiFi-CFD requires high-resolution schemes Compact finite difference schemes + overset grids enable accurate solutions using a reduced number of grid points Other strategies are possible according to the problem you want to study HiFi-CFD requires quantifying uncertainties Bayesian statistical framework seems a promising tool for predictive simulation with quantified modelling uncertainty 51

52 Further research required to: Achieve full industrialization of high-order methods Extend UQ methods to large-scale problems Reduce computational costs using high-performance computation strategies development of new methods cannot be done without taking into account hardware!! Accurately and efficiently predict multidisciplinary problems (aeroacoustics, multiphase flows, real gas flows, fluid/structure interaction, ) Interactions among scientists of different specialties (numerical analysis, statistics, fluid mechanics, informatics, signal processing, ) essential ingredient for further progress in CFD 52

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