Paola CINNELLA DynFluid Laboratory, Arts et Métiers ParisTech, Paris, France

and Università del Salento, Lecce, Italy

paola.cinnella@ensam.eu

CISM, 23rd January 2014

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

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1 – Introduction

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)

4 European Vision for

Aeronautic transport and Energy

Need for modelling flows of increasing geometrical and physical complexity Use of more realistic physical models (turbulence effects, real-gas effects, multi-

phase 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

5 http://www.nrc-cnrc.gc.ca/eng/programs/iar/ Gottlich et al. (2004), internal-aerodynamics.html J. Turbom. 126:297-305.

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

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2 – Progress in turbulence modelling

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??

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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 S Deck

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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)

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Examples of « zonal » methods. [Spalart Ann Rev 2009}

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

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+ 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)

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

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)

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(k=modelled energy, kr= resolved energy)

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3 – Progress in numerical methods

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

◦ Slow down of Moore’s law, supercomputers energy consumption issues…

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In high-order we trust!

3rd order

5th order

Taylor-Green Vortex, 1283 grid, t=12, Q-criterion = 3: top RBC3, bottom RBC5

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, …

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Your choice depends on what you are looking for!!

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

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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

Residual-based scheme expressed only in terms of approximations of the exact residual:

Precisely, it writes like:

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( ) ( )0

state vector, , fluxes in and

, Jacobian matrices

t x yw f g

w f w g w x yf gA Bw w

+ + =

→ →

∂ ∂= = →∂ ∂

t x yr w f g= + +

( )0 ,, j kj kr d=

( ) ( )0 ,

,

centered approximation of at point ,

residual-based numerical dissipationj k

j k

r r j k

d

→

→

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Given a Cartesian grid we introduce the standard difference operators in the j and k directions:

The numerical dissipation term is defined as

mid-point residuals, centered approximations of r

dissipation matrices depending on the eigensystem of A, B

Main and mid-point residuals approximated through high-order Padé formulae

( ) ( ) ( ) ( ) ( ), 1 1 1 2 2 2 1 2, ,

0

1 12 2

pj k x yj k j k

d r r x r y r O hδ δ δ δ

=

= Φ + Φ = Φ + Φ +

( ) ( ) ( ), ,, , , with steps ,j k j kx y j x k y x y O hδ δ δ δ=

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

1 11 1, 1, , , 1, ,2 2

1 12 1, , 1 , , , 1 ,2 2

1, 21, 2

j k j k j k j k j k j k

j k j k j k j k j k j k

δ µ

δ µ

+ + + +

+ + + +

• = • − • • = • + •

• = • − • • = • + •

1 2,r r

0

1

2

rrr

1 2,Φ Φ

Genuinely multidimensional Centred, but intrinsically dissipative (no need for artificial dissipation, filters or

limiters) High cutoff dissipation Low dispersion

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Advection along a mesh direction

Dispersion Dissipation

Resolvability

Dispersion accuracy limit

Dissipation accuracy limit

[Lerat, Grimich, Cinnella JCP 2013; Grimich, Cinnella, Lerat JCP 2013]

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

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4 – Progress in uncertainty quantification and data assimilation

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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-077-1998, 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, ...)

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

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Mach number isolines

AoA

M ∞, Re∞

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

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Choice of the appropriate level : essentially “expert judgement”

For a given level o Several possible models, which differ by

• Their mathematical structure • The associated closure parameters

Up to now

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?

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Epistemic uncertainties

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

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

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Model calibration: ◦ Ingredients 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

◦ 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.

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( ),y M x θ=

( ) ( ) ( )( )

| , | , (1)

p p z Mp z M

p zθ θ

θ =

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 Mi be a model in a (finite and discrete) set M, Sk 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]:

◦ Its variance is:

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( ) ( ) ( ) ( ),posterior model prior scenarioexpectation of forprobability probabilitya given model and data set

| | , | ,Z SM S∈ ∈ ∆

∆ = ∆∑∑ i k k i k k ki k

E E M z p M S z p S

( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )( )

,

in-model, in-scenario variance

2

,

between-model, in-scenario variance

var | var | , | ,

| , | , | ,

| , |

Z S

S S

S

M S

M S

∈ ∈

∈ ∈

∆ = ∆ +

∆ − ∆ +

∆ − ∆

∑∑

∑∑

i k k i k k ki k

i k k k k i k k ki k

k k k

M z p M S z p S

E M z E z p M S z p S

E z E z ( )2

Between-scenario varianceS∈∑ kk

p S

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5 – Numerical results

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 model-

scenario averaging

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Vortex stretching

Transition to turbulence

Fully developed turbulence

Q=0

RBC5, 1283 mesh

Viscous case, Re=1600 Model of transition to turbulence via vortex stretching mechanism Integral quantities: • kinetic energy

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RBC5 scheme, different mesh resolutions

1283 mesh, different RBC schemes

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RBC5 scheme, different mesh resolutions

RBC3 scheme, 1283 mesh

1283 mesh, different RBC schemes

5th-order scheme overperforms the third-order one by using 8 times less degrees of freedom

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

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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=3.5 10-1)

hub

carter rotor

motion

Demonstrates the capability of computing 3D complex cases

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hub

carter

rotor motion

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

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A smart combination of modelling and numerics enables accurate computations of complex flows with

an affordable computational cost

Objective: predict velocity profiles developing in the turbulent boundary layer close to the wall

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

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Boundary layer

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.

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Posterior distribution of y Posterior distribution of ηy

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Posterior model plausibilities computed for all models in M for each Sk using samples from

Can be considered as a measure of consistency of calibrated model Mi with data zk

Large spread in model plausibilities, according to the pressure gradient scenario

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?

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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

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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 between-model, in-scenario variance are penalized through the error measure

Prediction closer to the validation data Variance consistent with the experimental uncertainty

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BMA prediction

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6 – Conclusions and Perspectives

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

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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

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