97 Notes on Numerical Fluid Mechanics and...

97 Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM)

EditorsE.H. Hirschel/München

W. Schröder/AachenK. Fujii/Kanagawa

W. Haase/MünchenB. van Leer/Ann Arbor

M.A. Leschziner/LondonM. Pandolfi/Torino

J. Periaux/ParisA. Rizzi/StockholmB. Roux/Marseille

Yu. Shokin/Novosibirsk

Advances in HybridRANS-LES ModellingPapers Contributed to the 2007 Symposiumon Hybrid RANS-LES Methods, Corfu,Greece, 17–18 June 2007

Shia-Hui PengWerner Haase(Editors)

ABC

Prof. Shia-Hui PengDepartment of Computational PhysicsSwedish Defence Research Agency, FOISE-16490 StockholmSwedenE-mail: [email protected]

Dr. Werner HaaseHöhenkirchener Str. 19 d85662 HohenbrunnGermanyE-mail: [email protected]

ISBN 978-3-540-77813-4 e-ISBN 978-3-540-77815-8

DOI 10.1007/978-3-540-77815-8

Notes on Numerical Fluid Mechanicsand Multidisciplinary Design ISSN 1612-2909

Library of Congress Control Number: 2007943140

c© 2008 Springer-Verlag Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication orparts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, inits current version, and permission for use must always be obtained from Springer. Violations are liable forprosecution under the German Copyright Law.

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective laws andregulations and therefore free for general use.

Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India.

Printed in acid-free paper

5 4 3 2 1 0

springer.com

NNFM Editor Addresses

Prof. Dr. Ernst Heinrich Hirschel(General Editor)Herzog-Heinrich-Weg 6D-85604 ZornedingGermanyE-mail: [email protected]

Prof. Dr. Wolfgang Schröder(Designated General Editor)RWTH AachenLehrstuhl für Strömungslehre undAerodynamisches InstitutWüllnerstr. zw. 5 u. 752062 AachenGermanyE-mail: [email protected]

Prof. Dr. Kozo FujiiSpace Transportation Research DivisionThe Institute of Spaceand Astronautical Science3-1-1, Yoshinodai, SagamiharaKanagawa, 229-8510JapanE-mail: [email protected]

Dr. Werner HaaseHöhenkirchener Str. 19dD-85662 HohenbrunnGermanyE-mail: [email protected]

Prof. Dr. Bram van LeerDepartment of Aerospace EngineeringThe University of MichiganAnn Arbor, MI 48109-2140USAE-mail: [email protected]

Prof. Dr. Michael A. LeschzinerImperial College of ScienceTechnology and MedicineAeronautics DepartmentPrince Consort RoadLondon SW7 2BYU.K.E-mail: [email protected]

Prof. Dr. Maurizio PandolfiPolitecnico di TorinoDipartimento di IngegneriaAeronautica e SpazialeCorso Duca degli Abruzzi, 24I-10129 TorinoItalyE-mail: [email protected]

Prof. Dr. Jacques Periaux38, Boulevard de ReuillyF-75012 ParisFranceE-mail: [email protected]

Prof. Dr. Arthur RizziDepartment of AeronauticsKTH Royal Institute of TechnologyTeknikringen 8S-10044 StockholmSwedenE-mail: [email protected]

Dr. Bernard RouxL3M – IMT La JetéeTechnopole de Chateau-GombertF-13451 Marseille Cedex 20FranceE-mail: [email protected]

Prof. Dr. Yurii I. ShokinSiberian Branch of theRussian Academy of SciencesInstitute of ComputationalTechnologiesAc. Lavrentyeva Ave. 6630090 NovosibirskRussiaE-mail: [email protected]

Preface

Turbulence modelling has long been, and will remain, one of the most important top-ics in turbulence research, challenging scientists and engineers in the academic world and in the industrial society. Over the past decade, Detached Eddy Simulation (DES) and other hybrid RANS-LES methods have received increasing attention from the turbulence-research community, as well as from industrial CFD engineers. Indeed, as an engineering modelling approach, hybrid RANS-LES methods have acquired a remarkable profile in modelling turbulent flows of industrial interest in relation to, for example, transportation, energy production and the environment.

The advantage exploited with hybrid RANS-LES modelling approaches, being po-tentially more computationally efficient than LES and more accurate than (unsteady) RANS, has motivated numerous research and development activities. These activities, together with industrial applications, have been further facilitated over the recent years by the rapid development of modern computing resources. As a European initiative, the EU project DESider (Detached Eddy Simulation for Industrial Aerodynamics, 2004-2007), has been one of the earliest and most systematic international R&D effort with its focus on development, improvement and applications of a variety of existing and new hybrid RANS-LES modelling approaches, as well as on related numerical issues. In association with the DESider project, two subsequent international symposia on hybrid RANS-LES methods have been arranged in Stockholm (Sweden, 2005) and in Corfu (Greece, 2007), respectively.

The present book is a result of the Second Symposium on Hybrid RANS-LES Methods, held in Corfu, Greece, 17-18 June 2007. The symposium has covered a num-ber of relevant topics in the field, including Unsteady RANS and LES, Improved DES Methods, Hybrid RANS-LES Methods, Embedded LES, DES-related Numerical Is-sues, Performance of the New SAS Model, as well as Industrial Applications of DES. 32 papers have been selected to address these topics, which represent a leading part of current studies and achievements on the fundamentals of hybrid RANS-LES methods in general, as well as on their applications to industrial flow problems. Along with 29 papers selected from the symposium presentations (all being reviewed and further revised after the symposium), three invited keynote papers (by B. Geurts, U. Piomelli and P. Sagaut, respectively) plus an invited overview paper by P. Spalart are also presented in this book. The mailing addresses of all authors are listed at the end of the book to allow readers-to-authors communications when needed.

The editors are confident that the present book represents a viable part of the state-of-the-art in the development and application of DES and other hybrid RANS-LES modelling approaches. The contributors include a number of well-known leading academic researchers and industrial experts in the field. It is our wish to offer this book as a useful reference for researchers, university graduate students and industrial engineers in their work on advanced turbulence modelling approaches, as well as on numerical analysis of unsteady industrial flow problems.

Preface

VIII

The success of the symposium rests primarily with the participants, who have fur-ther facilitated the publication of the present book. In particular, the editors are very grateful to P. Spalart for taking the burden of reviewing all the full papers by reading through hundreds of pages and writing comprehensive comments on each paper. We are also grateful to L. Davidson, B. Geurts and D. Laurence for supporting us in that respect.

Furthermore, we would like to express our thankfulness to M. Braza, co-chairwoman of the symposium, who has ardently supported the preparation and arrangement of the symposium, the interaction with the IUTAM symposium, including a joint round-table discussion on “The future of CFD”.

Thanks are also due to all the partners of the DESider project consortium for their valuable support.

Finally, we wish to express our sincere gratefulness to the European Commission for supporting and monitoring the DESider project, in particular to A. Podsadowski being the EC scientific officer for the DESider project and to the financial support of the symposium by ANSYS, Eurocopter, the KATnet-II EU project and last but not least the Numeca company.

October 2007 Shia-Hui Peng Werner Haase

Table of Contents

The 2007 Hybrid RANS-LES Symposium: An Outsider’s ViewP.R. Spalart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Invited Keynote Papers

Reliability of LES in Complex ApplicationsBernard J. Geurts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Turbulent Eddies in the RANS/LES Transition RegionUgo Piomelli, Senthilkumaran Radhakrishnan, Giuseppe De Prisco . . . . . . 21

Uncertainty Modeling, Error Charts and Improvement ofSubgrid ModelsP. Sagaut, J. Meyers, D. Lucor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Hybrid RANS-LES Modelling and RelatedNumerical Issues

Hybrid LES-RANS Method Based on an Explicit AlgebraicReynolds Stress ModelMichael Breuer, Benoıt Jaffrezic, Antonio Delgado . . . . . . . . . . . . . . . . . . . . 45

Hybrid LES-RANS: Inlet Boundary Conditions for Flowswith RecirculationLars Davidson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

On the Use of Stimulated Detached Eddy Simulation (SDES)for Spatially Developing Boundary LayersS. Deck, P.E. Weiss, M. Pamies, E. Garnier . . . . . . . . . . . . . . . . . . . . . . . . . 67

Synthetic Inflow Boundary Conditions for Wall BoundedFlowsN. Jarrin, J.-C. Uribe, R. Prosser, D. Laurence . . . . . . . . . . . . . . . . . . . . . . 77

X-LES Simulations Using a High-Order Finite-Volume SchemeJohan C. Kok, Bambang I. Soemarwoto, Harmen van der Ven . . . . . . . . . . 87

One-Equation RG Hybrid RANS/LES ModellingC. De Langhe, J. Bigda, K. Lodefier, E. Dick . . . . . . . . . . . . . . . . . . . . . . . . . 97

X Table of Contents

Scrutinizing Velocity and Pressure Coupling Conditions forLES with Downstream RANS CalculationsDominic von Terzi, Wolfgang Rodi, Jochen Frohlich . . . . . . . . . . . . . . . . . . . 107

Comparison of Different Modelling Approaches

Computation of the Helicopter Fuselage Wake with the SST,SAS, DES and XLES ModelsF. Le Chuiton, A. D’Alascio, G. Barakos, R. Steijl, D. Schwamborn,H. Ludeke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Numerical Simulation of the Flow in the Wake of AhmedBody Using Detached Eddy Simulation and URANS ModelingG. Martinat, R. Bourguet, Y. Hoarau, F. Dehaeze, B. Jorez,M. Braza . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

DES and Hybrid RANS-LES Modelling of Unsteady PressureOscillations and Flow Features in a Rectangular CavityShia-Hui Peng, Stefan Leicher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Comparative Assessment of Hybrid LES/RANS Models inTurbulent Flows Separating from Smooth SurfacesS. Saric, B. Kniesner, A. Mehdizadeh, S. Jakirlic, K. Hanjalic,C. Tropea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

Numerical Investigation of a Laboratory Combustor ApplyingHybrid RANS-LES MethodsA. Widenhorn, B. Noll, M. Stohr, M. Aigner . . . . . . . . . . . . . . . . . . . . . . . . . 152

Applications of DES and DES Variants

DES of a Cavity with SpoilerRichard Ashworth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

DES Analysis of Confined Turbulent Swirling Flows in theSub-critical RegimeA.C. Benim, M.P. Escudier, A. Nahavandi, K. Nickson, K.J. Syed . . . . . . 172

Zonal-Detached Eddy Simulation of Transonic Buffet on aCivil Aircraft Type ConfigurationV. Brunet, S. Deck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

Detached Eddy Simulation of Separated Flow on a High-LiftDevice and Noise PropagationC. Lubon, M. Kessler, S. Wagner, E. Kramer . . . . . . . . . . . . . . . . . . . . . . . . 192

Table of Contents XI

Unsteady CFD Analysis of a Delta Wing Fighter Configurationby Delayed Detached Eddy SimulationHeinrich Ludeke, Stefan Leicher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Comparison of DES and LES on the Transitional Flow ofTurbine BladesF. Magagnato, B. Pritz, M. Gabi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Demonstration of Improved DES Methods for Generic andIndustrial ApplicationsC. Mockett, B. Greschner, T. Knacke, R. Perrin, J. Yan, F. Thiele . . . . . 222

Towards a Successful Implementation of DES Strategies inIndustrial RANS SolversL. Temmerman, Ch. Hirsch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

Delayed Detached-Eddy Simulation of Supersonic Inlet BuzzS. Trapier, S. Deck, P. Duveau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

DES Applied to an Isolated Synthetic Jet FlowH. Xia, N. Qin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

Advanced URANS Modelling and Applications

Development and Application of SST-SAS Turbulence Modelin the DESIDER ProjectY. Egorov, F. Menter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Numerical Simulation of the Dynamic Stall of a NACA 0012Airfoil Using DES and Advanced OES/URANS ModellingG. Martinat, Y. Hoarau, M. Braza, J. Vos, G. Harran . . . . . . . . . . . . . . . . . 271

Turbulence Modelling of Strongly Detached Unsteady Flows:The Circular CylinderA. Revell, T. Craft, D. Laurence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

SGS Modelling and LES Applications

A New Variant of Subgrid Dissipation for LES Method andSimulation of Laminar-Turbulent Transition in Subsonic GasFlowsT.G. Elizarova, P.N. Nikolskii, J.C. Lengrand . . . . . . . . . . . . . . . . . . . . . . . . 289

Formulation of Subgrid Stochastic Acceleration Model(SSAM) for LES of a High Reynolds Number FlowV. Sabel’nikov, A. Chtab, M. Gorokhovski . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

XII Table of Contents

Flow Around a Surface-Mounted Finite Cylinder: AChallenging Case for LESS. Krajnovic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Compressibility Effects on Turbulent Separated Flow in aStreamwise-Periodic Hill Channel – Part 2J. Ziefle, L. Kleiser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

Addresses of Corresponding Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

S.-H. Peng and W. Haase (Eds.): Adv. in Hybrid RANS-LES Modelling, NNFM 97, pp. 1–9, 2008. springerlink.com © Springer-Verlag Berlin Heidelberg 2008

The 2007 Hybrid RANS-LES Symposium: An Outsider’s View

P.R. Spalart

Boeing Commercial Airplanes, USA

Abstract

The contributions, lessons, and trends found in the DESider Symposium Proceedings are discussed. Three broad classifications can be made among the papers:

• The first is between studies which widen the code- and user-base of DES and its range of applications to establish the current level of quality, and those which bring out neighbouring hybrid concepts such as SAS for discussions and improvements.

• The second is between “natural” uses of DES with boundary layers fully covered by RANS, and uses with RANS only as a wall model in LES.

• The third is between single-model methods such as DES, and zonal methods even if they are closely related.

A few studies are directed at non-hybrid issues, such as the fundamental accuracy of LES or numerics. Issues of concern in the simulation of complex turbulent flows are addressed, notably related to treating laminar boundary layers, to grid density in DES versus LES, and to verifying the nature of a simulation (SRANS, URANS, LES) using visualisations. An emerging theme is the generation of LES content inside an attached boundary layer. This is intentional, in the absence of a rapid separated-layer instability, and aimed at Wall-Modelled LES ahead of separation.

A safe conclusion is that the hybrid-method applications here are certainly pioneering, but just as certainly preliminary, and would justify much higher computing power than has been applied in many cases.

1 Generalities

Detached-Eddy Simulation has received the most appreciation and nurture in the European Community, even though the Good Fairies standing over its cradle had come from the United States and Russia, in 1997. This may be appropriate, since DES made its first baby steps in Corsica in 1999. The first independent implementations came from France. The DESider EU research program, so well conceived and directed, is a prime example of this nurture. While some DESider members are missing here, this program has much overlap with the present symposium, which is also characterized by Chinese and Australian participation, but a lone US contributor.

This author has had access to the contents of the proceedings and email exchanges with some of the authors, but did not attend the symposium, and was not involved in DESider other than at the Stockholm meeting. In writing this, he naturally drew

2 P.R. Spalart

predominantly on his favourite papers and the ones closest to the theme of the symposium, and on the authors which were responsive to his informal comments.

The list of authors is dominated by academics, which provide about 75%, while government labs provide about 20%. Industry and CFD-vendor presence is small, and the automobile industry is absent, a curious finding since hybrid methods are more directly useful for such flows than for aerospace flows. Many of these firms rely on Lattice Boltzmann methods, one of which is definitely approaching DES in its treatment of turbulent separated flows. In any case, immersed-boundary methods with locally uniform cubic grids are a temporary “easy solution to a difficult problem.” The future belongs to unstructured adaptive grids.

Although the mode of operation of DESider is characterised by substantial sharing of tools and by extensive comparisons of results between groups, these highly instructive comparisons are not shown here and will be saved for a limited distribution.

The first impression is that hybrid methods are alive and well, in both the major types of use, “Natural” DES and “Wall-Modelled” LES. A strong RANS influence in WMLES is sensible because grid cells contain numerous eddies, thus making the nature of the filtering evolve towards Reynolds-Averaging. However, many methods are capable of WMLES, and not of NDES. Conversely, it appears all NDES formulations are capable of WMLES. It’s only that their results are not perfect; this is the well-known Log-Layer Mismatch first shown by Nikitin et al. in 2000. There is a whole body of work on WMLES, which shows no interest in NDES. In fact, Deardorff and Schumann were practicing WMLES in the 1970’s. This body of work is relevant to internal flows.

The second impression is that the various RANS-LES hybrid concepts are mathematically different, but remain physically quite similar. The most meaningful difference is between concepts which inject the grid spacing into the model and those which do not, as discussed below. The zonal issue is similarly important. It remains a central goal to refine an accurate non-zonal method. Consider the complexity of the external and internal fluid mechanics of an automobile.

The third impression is that newer and unfortunately somewhat more complex versions of DES now have a following, and could potentially become the primary version. This is good news in terms of performance, but a protracted “arms race” would not be in the interest of users.

2 The Wider Exercise of DES

This volume definitely contains new applications of DES, cavity flows probably being the most successful area in the past few years. DES is available in an increasing number of government and vendor CFD codes; this was, of course, the hope of its creators. Since it is simple to code, it is likely that actual errors are rare, with the possible exception of the definition of the cell size especially in unstructured grids (there is also a lingering attachment to the definition with the cube root of the cell volume).

What will be much more difficult to establish is good practices. It is mixed news that DES is starting to be used as a “black box,” possibly only “one click away” from RANS in a User Interface. A similar situation has existed between different RANS

The 2007 Hybrid RANS-LES Symposium: An Outsider’s View 3

models, the UI sometimes hiding subtleties, for instance a larger number of boundary conditions.

However, LES function has more of an on/off character. DNS and LES research writers have always felt the need to illustrate the LES Content, eddies, and spectra of their simulated turbulence. It was essential that the simulation was three-dimensional and unsteady, with a wide range of scales, and also a time sample sufficient for smooth profiles and balance in the budgets. In some cases here, the writers demonstrate no such interest, and simply present the final solution, averaged in an unspecified manner.

Sometimes, solutions are labelled RANS without mention of conclusive convergence to steady state governed by a RANS model, or labelled URANS without conclusive unsteadiness, even one with a very simple spectrum. This is the case in the work of Le Chuiton, D’Alascio, Barakos, Steijl, Schwamborn and Lüdecke. It may stand at the very first application of DES to helicopter bodies and such applications have a very bright future, but the nature itself of each simulation in turbulence terms is essentially left to the imagination.

Ideally, every simulation would be unquestionably identified as steady, or periodic, or blessed with broadband resolved turbulence. Admittedly, this demand could be a case of a group of scientists attempting to impose their intuition and graphic or even aesthetic “vision” of turbulence on the community (or, worse, arranging job security). Still, the decades of experience in DNS and LES must be a source of guidance, and a key contribution from the research to the engineering community.

This is not to imply that RANS publications are immune to quality problems; the essential requirements of a sufficiently small first y+ and grid-stretching ratio over the entire turbulence region are too easily violated locally in complex flows, particularly when the boundary-layer thickness cannot be predicted at the grid-design stage. Schedule pressure and page limits hamper careful verification of these aspects.

The other pitfall in pure RANS is the use of a model in a type of flow it has not competency in, for instance one dominated by streamline curvature. This is still less egregious than believing that a CFD solution took advantage of the power of LES, when it in fact had no LES content anywhere.

A double-edged feature of LES and DES in external flows is that they necessarily involve the convection of turbulence into coarser “departure” grid regions (Spalart, 2001), in which case it is fully accepted for the SGS viscosity and/or numerical dissipation (in Implicit LES) to suppress the smaller eddies. It becomes very difficult to judge whether such a grid is a “quality” one or not. Only grid and time-step refinement by a uniform ratio would begin to prove that these solutions are numerically well understood. Most often, the authors argue that such unquestionable refinement would be too onerous so that they, the referees and the readers are left with significant uncertainty. This is even not to mention additional issues of numerical quality such as the domain size in periodic simulations, particularly of ‘2D’ bluff-body flows, or the time sample for flows with strong modulations (Travin et al., 2000). Once the experimental uncertainties are added, a devil’s advocate could find fault and request rework in any study except for the simplest of flows. Not that the flow past a smooth cylinder is simple at all, physically.

The contribution of Mockett, Greschner, Knake, Perrin, Yan and Thiele in itself covers much ground and reflects much competence. It would have justified two if not

4 P.R. Spalart

three papers. Different solvers are used including an unstructured one; LES content is everywhere in evidence; modern and more complex versions such as Delayed DES (DDES, of Spalart et al., 2006) and Improved DDES (IDDES, of Travin et al., 2006) are assessed with positive results. The author’s concerns over the complexity of IDDES may be unjustified. Low cell Reynolds numbers are accommodated. Aero-acoustic applications are beginning. Attached boundary layers are not avoided. This is all very positive and instructive.

The paper of Brunet and Deck certainly displays LES content, and is quite successful in an industrially crucial area, namely wing buffet, and in a type of flow which in the past has revealed weaknesses of DES: shallow separation. The superiority over RANS appears undisputable. There is a slight concern over possible Grid-Induced Separation, and the zonal character of the simulations makes them somewhat less user-friendly than simple DES. It remains impressive and encouraging work.

The work of Lüdecke and Leicher is similar: DES is applied to a full aircraft configuration; it is reminiscent of that of Forsythe and co-workers on the F-15 fighter. The unstructured grid has over 107 nodes, and LES content is plentiful. Both DES97 and DDES are applied, with moderate differences in the results. Reynolds stresses are rather good. Some of the velocity spectra are in poor agreement with experiment. This is disappointing but showing such results, especially for a relatively new quantity, is a valuable service to the scientific community.

Trapier, Deck, and Duveau are equally ambitious and thorough in verifying the LES nature of their simulations over a significant region, using a very large grid. They simulate supersonic inlet buzz, and have close agreement with the experiment on the frequency of the shock motion. URANS is far from failing on this flow, but DDES results for amplitude are somewhat better at low frequencies, and much better at high frequencies. It is unusual for URANS to give a solution that is not periodic in time; the wide range of scales of the geometry probably contributes to this.

As mentioned, cavity flows have been good to DES, probably because of the favourable ratio of scales between cavity depth and boundary-layer thickness. This is evident in Ashworth’s paper. Similarly Peng and Leicher obtain pressure-rms levels about only 2dB from the experiment. In truth, the dominant cavity dynamics are inviscid and the only role of the turbulence model is to bring a boundary layer with the right thickness. If that were set by inflow conditions, ILES would capture most of the physics. However, DES will be capable of predicting the boundary-layer thickness in full-geometry cases, whether military or commercial in the air or ground vehicles, and not only on wind-tunnel “cavity only” models.

Šarić, Kniesner, Mehdizadeh, Jakirlić, Hanjalić and Tropea apply several hybrid methods to some delicate problems of separation from smooth surfaces, some with suction. They display very good LES content (even though the incoming boundary layer had none, and the separation was from a smooth surface) and revealing results, even with suction. They find themselves constrained to a large lateral grid spacing in the DES, in order to avoid Grid-Induced Separation, and DES gives poor results with suction. It is possible DDES would be helpful here. Their Instability-Sensitized k-ε model borrows from SAS, but is starting to evolve independently, and performed well. It does not use the grid spacing which is, as mentioned, very intriguing for those versed in LES and DES.


Widenhorn, Noll and Aigner studied internal flows, which is not too common for DES, but justified when thin boundary layers are present. Although a vigorous unsteadiness may have been expected, it does not appear to be found by CFD; it is one of the many promising studies which would benefit from grid refinement.

The same is likely for that of Benim, Escudier, Nahavandi, Nickson and Syed of a confined swirling flow. Grid clustering in the shear layers appears absent. Yet, the agreement with experiment is quite good almost everywhere, and begins to suggest an advantage for LES and DES over RANS using the SSG Reynolds-Stress Model. This might be expected in such a complex situation.

Lübon, Kessler, Wagner and Kramer apply DES to an airfoil with a small Trailing-Edge Device, similar to a Gurney Flap. The study has wide unexplained inconsistencies between the lift and pressure results, but continues with convincing LES content and encouraging agreement with PIV in the near wake.

Not all the simulations in this volume are immune to criticism, and there is no sense in suppressing such concerns when they are motivated. Some practices simply must disappear over time, when comparing LES and DES. The first is to provide DES a coarser grid than LES, in the separated region. This is unfair, inasmuch as DES as a SGS model away from walls has no reason to surpass any other SGS model. The tolerance of coarser grids by DES resides only in attached boundary layers. Therefore, if a region of massive separation justifies a grid of, say, 323 in LES, it also justifies 323 in DES.

The same applies to the time step, naturally, but now and then authors increase the DES step over the one for LES with much abandon. Another risk factor derives from using different codes for the two approaches. There are often historical reasons for it, but this practice comes close to making comparisons meaningless, especially knowing how grid convergence is still essentially impossible to achieve for such three-dimensional, unsteady equations, and how numerical dissipation is often as powerful as the SGS model.

Conversely, the practice of adjusting the “constant” CDES to a new code is not truly objectionable. In fact, the approach of “Implicit LES” can be very successful away from walls, just like its precursor, MILES. ILES can be viewed as using a very low value of CDES, but is often obtained with a zonal approach instead.

Another danger resides in simulating flows at Reynolds numbers so low that DNS is possible or nearly so, at least in the most influential region. If the cell Reynolds number is low over the entire key region, the model has so little leverage that nothing can be learned about it except maybe the fortuitous effect of its low-Reynolds-number term, which was designed for the standard viscous sub-layer in any case. It must be remembered also that high cell Reynolds numbers often actually simplify the task of designing an SGS model, and certainly simplify the task of initially assessing its performance.

3 Neighbouring Concepts

The eXtra-Large Eddy Simulation (XLES by Kok, Soewarmoto, and van der Ven) and Organized-Eddy Simulation (OES by Martinat, Hoarau, Braza, Vos, and Harran) are quite experimental and have a narrow user base at this time, and it is possible they

6 P.R. Spalart

will evolve significantly yet. Their physical motivation is not fundamentally different from that of DES, but the formulations are, and the intent when creating new methods seems to be to cause a deliberate “mutation” and avoid being locked in an approach such as DES which might, after all, eventually be viewed as pioneering but too simple or limiting. Thorough students of hybrid approaches will also explore the LNS concept, which is not represented here.

In the entire field of turbulence, the hope of a decisive improvement based on an increased reliance on first principles and mathematics over intuition unfortunately remains slim, but nobody wishes to give up the intellectual pleasure of trying.

4 Scale-Adaptive Simulation

A surprise in this volume is the evolution of SAS presented by Egorov and Menter. The essential feature of SAS, relative to DES, has been not to involve the grid spacing. It is a near-philosophical distinction, and much of the thinking has centred on how a model that is formally a RANS model can allow the generation of LES content. The key is the von-Karman length scale, which involves higher derivatives of the velocity field than common models do.

The new version of SAS now involves the grid size, creating the appearance of an evolutionary convergence with DES. However one should not read too much into it, inasmuch as in DES, the grid size Δ sets an upper bound for a length scale, which enters the turbulence model. This scale is essentially the mixing length. In contrast, in “SAS07,” Δ limits a length scale from below. If so, the concepts appear completely opposite, but closer scrutiny leads to the following speculation.

The scale Δ limits from below a length scale lvk, which limits the model’s length scale from above. If, as there is some evidence of, lvk has a disruptive tendency to tend to zero (precisely the tendency which motivated the limiter), then everything considered, Δ will limit the model’s scale from above. This issue will elicit further discussions and tests. This author has also speculated that the original SAS effectively introduces the grid spacing by low-pass filtering the higher derivatives; this would not be wrong in itself, but would make the approach more code-dependent than most.

The favourable situation here is that DES and SAS continue independent lines of thinking by groups which are both experienced, very aware of practical requirements, and often openly communicating.

5 Separate Issues in Complex Flows

An area for definite improvements in fidelity is the treatment of boundary layers in Natural DES and in LES. The subcritical flow past a cylinder, say with a smooth surface and a Reynolds number of 140,000, has been known since Prandtl to hinge on laminar separation. The critical and higher regimes with some measure of transition in the attached boundary layer, or more likely mingled with separation, are very delicate. The subcritical regime is not, and there is no doubt that in careful CFD, all eddy viscosity should be disabled up to separation. Ignoring this fact leads to “Turbulent Separation” DES, in the terminology of Travin et al., 2000, or to LES with the SGS


model active in the boundary layer. Examples before DESider exist, of course. However this practice amounts to using a SGS model, for instance Smagorinsky’s, as a RANS model in, effectively, a hybrid method. This would not be justified even for turbulent boundary layers. In particular, the eddy viscosity is grid-dependent. The dependence of results on the flow Reynolds number is also likely to be highly spurious.

6 Wall Modelling

An objective declared by this author years ago in the field of WMLES is to allow grid spacings which are unlimited in wall units, at least in the wall-parallel directions. Some authors are still reticent to subject their new Wall Modelling proposals to the complete test, sometimes curiously leading to LES exercises that are limited to Reynolds numbers lower than those already reached by DNS; for instance, Reτ = 2000 has been conclusively reached by DNS in a channel, and the Ekman layer is being simulated in the same region.

Very significant improvements in WMLES are demonstrated by Piomelli, Radhakrishnan, and De Prisco with several types of forcing, and both for RANS regions “under” and “upstream of” the LES region. The improvements extend into a separated shear layer, possibly because the forcing removed excessive two-dimensionality, which is known to raise the Reynolds shear stress in shear layers. The use of a “controller” tasked with leading the resolved Reynolds stress to the needed level, using information from the RANS model, is creative.

Deck, Weiss, Pamies and Garnier focus on the flow of a boundary layer from RANS into LES, and compare different approaches aimed at the immediate generation of high-quality eddies, ideally reflected in a seamless skin-friction distribution. All approaches are zonal, by necessity, and the term Stimulated DES is introduced. IDDES is included. The recycling method first proposed by Lund et al. in 1998 has the shortest adjustment, as seen in other studies, but will also be difficult to extend to complex geometries.

7 Pure LES Studies

Two studies, by Geurts and by Sagaut, Meyers and Lucor address homogeneous turbulence. They reflect a struggle, which is hardly new, to make quantitative comparisons meaningful in chaotic flows. They are limited to decaying unbounded turbulence, using a reference DNS solution and obtaining initial fields for LES via systematic filtering. The principal result is to challenge the universality of the Smagorinsky and similar constants.

Krajnović is treating a cylinder finite length, standing on a plate like a tall building; this has the substantial advantage of removing the dilemma of choosing a spanwise period in simulations. He is not applying a hybrid method; such a method is not needed with a diameter Reynolds number of 2×104. However, as already mentioned, the Smagorinsky model should then be disabled in the boundary layers.

8 P.R. Spalart

The simulation has good LES content and detailed visualisations, and also some degree of grid refinement to 14 106 points, with a dramatic effect on the pressure near the cylinder tip. This region appears very sensitive, if judged also by the disagreement between two experiments.

A very positive development is that full 3D LES and shock-capturing are compatible, with extreme care in the algorithm, as displayed by Ziefle and Kleiser. This had been demonstrated by Shur et al. (2006, 2007) in jets, but is seen here in an independent code and with walls. They simulated supersonic hill flows, with the approximate-deconvolution SGS model, and good results.

8 Outlook

When it comes to application-ready hybrid methods, the highest priority might be to aggressively increase the computing expense. Were cost no object, every paper would show a grid and time step such as the present ones, which the authors view as publishable, and results with twice as fine a resolution in every direction, costing 16 times as much. If these do not agree well, a run costing 256 times as much would readily be started. Here, none of the studies went this far, and most had no grid study at all. In general, the grid description includes valid practices for the first y+, and probably for the wall-normal stretching, but little mention of the other directions and time.

Many contributions have the rushed feeling of a conference paper, although they are being finalized many weeks after the meeting. Again, revealing flow visualisations of the LES content would be most welcome in all papers, and would certainly expose some of the solutions as having very little LES content. This would send the authors back to their grid design, or in some cases their numerical dissipation.

In terms of the next challenges to the methods themselves, the log-layer mismatch in Wall-Modelled LES has been resilient, especially if a method that is general and independent of flow direction is demanded. Yet, the progress has been significant, notably for Piomelli and Strelets (not shown). On the other hand, the variety in the proposed solutions is striking.

Another area, which will be very active, is the generation of LES content to obtain a deliberate and as short as possible switch from RANS mode to LES mode in an attached boundary layer. It has been called Large-Eddy Stimulation by Batten. It is preferable to avoid the word “transition,” as tempting as it is. Again the variety of approaches, from random numbers to multiple Fourier modes to random vortices and to recycling, is very wide (not all these are represented in this book). Some are just too complex, or too dependent on databases, which are limited in domain size and/or Reynolds number. Generally, “synthetic turbulence” fails to develop rapidly; humans seem to lack the imagination of the Navier-Stokes equations. This LES-content generation capability could be a key to success in flows such as that in the LESfoil workshop, but in that flow, even the most capable methods exhaust the best current computers. The Ahmed body is similar: valuable and delicate CFD exercises, with faint questions over the true accuracy level of the experiment, and the possibility of extreme sensitivity near some values of the angle of incidence or inclination.


A likely controversy in the coming years is between the convenience of non-zonal methods such as basic DES, and the higher power of zonal methods. The latter are especially tempting in relatively simple “academic” geometries, and there is increasing evidence that, barring massive separation, the generation of LES content is quite difficult without allowing any zonal features in the algorithm. Measures such as very sudden and/or irregular refinement of the grid, which may collaborate well with basic DES, may or may not be called zonal. Applications with LES or DES in a sub-domain surrounded by a RANS calculation are also becoming active.

A general conclusion is that progress is certain in this field, although fairly chaotic, and that the societal value of accurate CFD in complex turbulent flows is less in doubt than ever. This justifies the vision, the expense and the intellectual power and diversity applied by DESider and similar programs. Imagine an idea which ultimately reduces by 1% the fuel burn of every vehicle on Earth.

References

Lund, T.S., Wu, X., Squires, K.D.: Generation of turbulent inflow data for spatially-developing boundary layer simulations. Journal of Computational Physics 140, 233–258 (1998)

Nikitin, et al.: An approach to wall modeling in large-eddy simulations. Physics of Fluids 12(7), 1629–1632 (2000)

Shur, M.L., et al.: Further Steps in LES-Based Noise Prediction for Complex Jets. In: AIAA Paper-2006-0485, 44th Aerospace Sciences Meeting and Exhibit, Reno, NV (2006)

Shur, M.L., et al.: Analysis of jet-noise-reduction concepts by large-eddy simulation. International Journal of Aeroacoustics 6(3), 1–44 (2007)

Spalart, P.R.: Young person’s guide to Detached-Eddy Simulation (2001), http:// techreports.larc.nasa.gov/ltrs/PDF/2001/cr/NASA-2001-cr211032.PDF

Spalart, P.R., et al.: A new version of Detached-Eddy Simulation, resistant to ambiguous grid densities. Theoretical and Computational Fluid Dynamics 20(3), 181–195 (2006)

Travin, A., Shur, M., Splalart, P.R.: Detached-eddy simulations past a circular cylinder. International Journal of Flow Turbulence and Combustion 63, 293–313 (2000)

Travin, A.K., et al.: Improvement of delayed detached-eddy simulation for LES with wall modelling. In: Wesseling, P., Onate, E., Periaux, J. (eds.) CD-ROM Proc. of European Conference on Computational Fluid Dynamics, ECCOMAS CFD 2006, The Netherlands (2006)

S.-H. Peng and W. Haase (Eds.): Adv. in Hybrid RANS-LES Modelling, NNFM 97, pp. 10–20, 2008. springerlink.com © Springer-Verlag Berlin Heidelberg 2008

Reliability of LES in Complex Applications

Bernard J. Geurts

Multiscale Modeling and Simulation, Faculty EEMCS, University of Twente, The Netherlands

and Anisotropic Turbulence, Fluid Dynamics Laboratory,

Department of Applied Physics, Eindhoven University of Technology, The Netherlands

Abstract

The accuracy of large-eddy simulations is limited, among others, by the quality of the subgrid parameterisation and the numerical contamination of the smaller retained flow-structures. We review the effects of discretisation and modelling errors from two different perspectives. First, we review a database-approach to assess the total simulation error and its numerical and modelling contributions. The interaction between the different sources of error in the kinetic energy is shown to lead to their partial cancellation. An ‘optimal refinement strategy’ for given subgrid model, given discretisation method and given flow conditions is identified, leading to minimal total simulation error at given computational cost. We provide full detail for homogeneous decaying turbulence in a `Smagorinsky fluid'. The optimal refinement strategy is compared with the error-reduction that arises from grid-refinement of the dynamic eddy-viscosity model. Dynamic modelling yields significant error reduction upon grid refinement. However, at coarse resolutions high error-levels remain. To address this deficiency in eddy-viscosity modelling, we then consider a new successive inverse polynomial interpolation procedure with which the optimal Smagorinsky constant may be efficiently approximated at any given resolution. The computational overhead of this optimisation procedure is well justified in view of the achieved reduction of the error-level relative to the ‘no-model’ and dynamic model predictions.

1 Introduction

Direct numerical simulation and large-eddy simulation are two important strategies for the numerical investigation of turbulent flows. Within the constraints of present-day computing infrastructure, the direct simulation approach is adopted for full resolution of flow-problems of modest complexity, e.g., to under-pin theoretical and modelling studies. Instead, the focus in large-eddy simulation is on a computationally more accessible coarsened flow description, which is obtained by low-pass spatial filtering (Geurts, 2003; Sagaut, 2001). This allows an external control over the required spatial resolution. However, low-pass filtering gives rise to the well-known closure problem for the turbulent stress, which represents the dynamic effects of the filtered-out small-scale turbulence on the retained flow-structures. Viewed entirely from the PDE-level corresponding to the spatially filtered Navier-Stokes equations,

Reliability of LES in Complex Applications 11

the remaining task is to close the system of equations by modelling these small-scale dynamic effects in terms of the resolved flow. Considerable effort has been put into construction, testing and tuning of such so-called subgrid models over the past years (see, e.g., Geurts, 2003; Meneveau, Katz, 2000; Lesieur, Metais, 1996; Sagaut, 2001). In such testing procedures one frequently compares predictions from large-eddy simulations to filtered data from direct numerical simulation and/or experimental data.

The above sketch of direct and large-eddy simulation is incomplete in at least one important respect, as it does not contain the unavoidable subsequent discretisation step. In fact, since the numerical representation in large-eddy simulations is typically associated with only marginal subgrid resolution, a significant alteration of the resolved scales’ dynamics may be introduced in the computational model (Rogallo, Moin, 1984; Salvetti, Beux, 1998; Geurts, Froehlich, 2002; Meyers, Geurts, Baelmans, 2003).

A principal difficulty in comparing two turbulent solutions lies in their sensitivity to slight alterations in the initial conditions. Therefore, we consider DNS and LES that start from the same (filtered) initial condition, and restrict to decaying turbulence in this paper. By adhering to rather modest Reynolds numbers a comparison of the deterministic evolution of model turbulent flows is feasible. In these situations both DNS and LES are available. A judicious combination of these results allows arriving at an approximate decomposition of the total effect of the errors in terms of a discretization component and a modelling component. These effects may be of opposite signs (Vreman, Geurts, Kuerten, 1996) and counteract. This leads to significant resolution dependence of optimal model parameters that yield lowest total error at given resolution, e.g., measured as L2 norm. In particular, strong dependencies are observed in case the grid spacing is near the ‘large-scale border’ of an inertial range.

It is the purpose of this paper to review and quantify the numerical error dynamics explicitly. Next to the filter-width Δ, the specification of the numerical method implies the introduction of a second length-scale h, which characterises the (local) computational grid spacing. Correspondingly, the discretisation step induces a second element of possible flow filtering (Geurts, Van der Bos, 2005). The difficulty hence resides in assessing the modelling and discretisation errors and their dynamic interaction in order to arrive at a specification of simulation and modelling parameters which are optimal, i.e., yield minimal total simulation error in the quantities of interest, at given computational cost. In this paper we review consequences of the interactions that occur between discretisation and subgrid modelling errors. Moreover, we propose a simple optimisation procedure to approximate optimal simulation parameters.

The relative importance of the turbulent stresses compared to the numerically induced contributions depends strongly on the sub-filter resolution r=Δ/h (Ghosal, 1996). If r is sufficiently large, the grid-independent large-eddy solution, consistent with the assumed value of Δ and the adopted subgrid model, may be accurately approximated. However, large-eddy simulation of applications with a realistic complexity is typically hampered by only marginal resolution corresponding to r=1-2. In that case the numerically induced effects can be comparable to or even larger than the turbulent stresses for typical discretisation methods such as central or upwind

12 B.J. Geurts

finite difference or finite volume methods. Thereby, the computational large-eddy closure contains an important contribution, which is sensitive to the adopted spatial discretisation.

Central to a framework for assessing the error-behaviour in an actual large-eddy implementation is the evaluation of the total simulation error and its decomposition into numerical and modelling components. By appropriately comparing large-eddy and direct simulation predictions, the total simulation-error can be quantified (Geurts, Froehlich, 2002). Recently, a database of both direct and large-eddy simulations of decaying homogeneous, isotropic turbulence was analysed at two different Reynolds numbers (Meyers, Geurts, Baelmans, 2003). In particular, for the Smagorinsky eddy-viscosity model combined with second order finite volume discretisation, the dependence of modelling - and numerical errors on simulation parameters was discussed. The interaction between these two basic sources of error was shown to lead to their partial cancellation for several flow properties such as kinetic energy and velocity skewness.

In this paper we review the interacting error-dynamics in terms of a so-called error-landscape, which provides a concise visualisation of the induced errors. The ‘optimal refinement strategy’ that can be extracted from such a landscape, yields minimal total simulation error at given computational effort. Compared to the optimal refinement strategy, the error induced by the dynamic eddy-viscosity model (Germano et al., 1991) at different resolutions, is about a factor two larger. However, the rate by which the error reduces with increased resolution was shown to be quite strong, particularly at high resolutions (Meyers, Geurts, Baelmans, 2005). To compensate for the remaining high error-levels at coarse resolutions, a new successive inverse polynomial interpolation (SIPI) procedure was proposed (Geurts, 2006). This allows approximating efficiently the optimal Smagorinsky constant at given resolution. As point of reference, the procedure starts from predictions using either no subgrid model at all or the dynamic eddy-viscosity model at a given spatial resolution. The proposed iterative procedure rapidly converges toward the optimal model-parameter. The computational overhead of this procedure is well justified by the increased accuracy.

The organisation of this paper is as follows. In section 2 we introduce a database analysis of large-eddy simulation of homogeneous decaying turbulence in a ‘Smagorinsky fluid’. An exhaustive database-approach is rather expensive, but allows identifying the optimal refinement strategy. To achieve more practical error-reduction approaches, the degree of optimality of the dynamic procedure is discussed. Further improvements in accuracy may be achieved by directly approximating the optimal model parameters at given resolution. The SIPI iteration is discussed in section 4. Concluding remarks are collected in section 5.

2 Database Approach to Interacting Errors

In this section we review the database approach for LES errors that gives an overview of the total simulation errors. This methodology was recently adopted to assess simulation errors in homogeneous turbulence (Meyers, Geurts, Baelmans, 2003). We illustrate the effect of partial error-cancellation. Throughout, we adopt the classical filtering approach to large-eddy simulation, and consider subgrid closure based on the


classical Smagorinsky model (Smagorinsky, 1963), or on the dynamic eddy-viscosity model (Germano et al., 1991). Excellent text-books exist that review the filtering approach (Pope, 2000; Sagaut, 2001; Geurts, 2003).

We focus on the error-landscape associated with the Smagorinsky model and identify the so-called optimal refinement strategy. At given spatial resolution, the optimal refinement strategy corresponds to the value of the Smagorinsky parameter at which the total error is minimal. This refinement strategy also allows interpreting the degree of optimality of the popular dynamic eddy-viscosity model (Germano et al., 1991). In fact, this subgrid model will be shown to display too high levels of eddy-viscosity and yields significant errors at coarse resolutions. However, at higher resolutions, the dynamic modelling provides a strong reduction of the error-levels.

The database approach provides an ‘experimental’ quantification of the errors that arise in large-eddy simulation. In particular, it allows a detailed decomposition of the total error in a numerical - and a subgrid modelling contribution. This clarifies which effects form the dominant limitations for the overall accuracy of large-eddy simulations, and under what computational settings and flow-conditions this identification applies. An extensive large-eddy simulation database of homogeneous decaying turbulence was generated at a variety of resolutions and filter-widths using grids with N by N by N cells where N is up to 128. Initial conditions at Taylor Reynolds numbers 50 and 100 were adopted. For the direct simulation N=384 or N=512 was used and the level of convergence for various flow quantities was established.

In terms of an explicit filter it is straightforward to define the different error contributions in a large-eddy simulation. We restrict ourselves to the resolved kinetic energy E(t), which is a volume average of the velocity product (u,u). To quantify the error-behaviour, we introduce some basic notation. The total error in E resulting from a ‘Smagorinsky-fluid’ is concisely expressed as function of the so-called ‘Smagorinsky length-scale’, i.e., the product of the filter-width Δ and the Smagorinsky parameter. The Smagorinsky subgrid resolution ξ is the ratio between this Smagorinsky length and the grid-spacing h=1/N. Finally, the relative error δ is defined in terms of the difference between the filtered DNS result and the actual LES prediction. Note that this is a convenient choice, adhered to in this paper – in case DNS data are not available, alternative error-measures can be considered, e.g., based on experimental data, external constraints or concepts of ‘expected’ smoothness of the numerical solution, e.g., giving a high error-weight to artificial grid-oscillations.

In the sequel we express the total error behaviour in terms of the Smagorinsky subgrid resolution ξ. This error can be further decomposed into a contribution due to the discretisation and one due to the subgrid modelling. To quantify these measures for the simulation error a direct numerical simulation and several large-eddy simulations at different Smagorinsky lengths and spatial resolutions are required. These data form the heart of the database approach. A central role is played by the grid-independent prediction of the decay of the resolved kinetic energy E obtained with the Smagorinsky model at fixed Smagorinsky length and ‘infinite’ spatial resolution. This grid-independent large-eddy prediction provides a central point of reference in the error decomposition, next to the reference direct numerical simulation data. It is well approximated at resolutions where Δ/h=4-6 (Geurts, Froehlich, 2002). This grid-independent large-eddy simulation should not be confused with an actual, more practical LES in which the ratio between Smagorinsky length and grid spacing

14 B.J. Geurts

is much smaller. Grid-independent LES certainly is not optimal in the sense of computational effort – its accuracy reflects the accuracy of the subgrid model and allows separate reasoning about which procedure for subgrid closure is physically consistent.

The total relative error δ is time-dependent. In order to arrive at a single number to characterise the error in any specific large-eddy simulation we introduce as measure the time-average square norm of the difference between the filtered DNS result and the LES prediction (Geurts, Froehlich, 2002). The error-behaviour can conveniently be described in terms of the ‘error-landscape’. For the Smagorinsky fluid the error-landscape can be expressed using the subgrid resolution of the Smagorinsky length ξ and the spatial resolution N as basic parameters. Each combination of ξ and N corresponds to a single large-eddy simulation. The error landscape is formed by contours of the error-measure on the (ξ, N) plane, and can be constructed from systematic variation of the simulation parameters. It provides the ‘optimal refinement strategy’, which identifies the optimal values of simulation parameters that leads to the smallest total error at a particular resolution.

The direct numerical simulation data and the grid-independent large-eddy predictions allow full separation of the modelling and discretisation contributions to the total error. In figure 1 a decomposition is collected in which we consider simulations at constant Smagorinsky length hξ and various resolutions. In this case we observe that a higher spatial resolution may yield a larger total error. In fact, the discretisation error effect decreases with increasing resolution and the total error approaches the modelling error. However, on the coarser grids, this modelling error is itself larger than the total error. On coarse grids the comparably large discretisation error effects partially cancel the modelling error effects.

Fig. 1. Error decomposition at Smagorinsky length 0.00625 for resolution N=32 (no symbol) and N=64 (o) displaying total error with solid curves, modelling error with dashed curves and discretisation error with dash-dotted curves


Fig. 2. Error-landscape based on the time-integrated L2 norm of the kinetic energy differences at initial Taylor Reynolds number 100. The label on the contours refers to the relative error in %. The thick dashed line corresponds to the optimal refinement strategy.

Collecting the total errors as arise at various parameters settings in the (ξ, N) plane, produces an overview as shown in the error-landscape in figure 2. From these contour plots the ‘optimal refinement strategy’ can be determined straightforwardly. At fixed, coarse resolutions, we observe a fairly sharp increase in the total simulation error in case the Smagorinsky parameter is below the optimal trajectory, while a slightly more gradual increase is observed in case hξ is larger than optimal.

The degree of optimality of the famous dynamic model (Germano et al., 1991) may readily be interpreted in relation to the error-landscape of the Smagorinsky fluid. In fact, the dynamic procedure gives rise to a self-induced length-scale, which can directly be translated into a ‘dynamic refinement trajectory’ in the error-landscape. Of particular importance is how this ‘dynamic trajectory’ relates to the ‘optimal refinement strategy’.

In figure 3 we collected the dynamic refinement trajectories (Fig. 3a) for two different Taylor Reynolds numbers. The general trend in the dynamic length-scale is found to be similar to the optimal refinement strategy, although the dynamic refinement path is at somewhat too large values of the eddy-viscosity. In Fig. 3b we observe that an increase in the resolution, with a test-filter width equal to Δ=2h, yields a strong error-reduction.

The errors along the dynamic refinement trajectories are about twice as large as the errors along the optimal refinement trajectory. An effective error-reduction arises from grid refinement. Particularly at higher resolutions a high rate of error-reduction with increased resolution is observed (further details may be found in (Meyers, Geurts, Baelmans, 2005). Conversely, at coarse resolutions the errors are still quite considerable and further improvement would be desired. This is the subject of the next section.

16 B.J. Geurts

Fig. 3. In (a) we show the optimal refinement strategies at Taylor Reynolds number 50 (dashed) and 100 (solid) compared to the dynamic refinement trajectories (triangles and squares respectively). In (b) the induced relative errors are shown as function of resolution.

3 Iterative Approximation of the Optimal Smagorinsky Constant

We propose an iterative procedure to improve predictions based on eddy-viscosity modelling at coarse subgrid resolutions. This iterative method is based on determining the minimum of the total simulation error. It allows efficient approximation of the optimal Smagorinsky constant at a particular resolution and leads to a significant error-reduction compared to the dynamic eddy-viscosity predictions.

To approximate the lowest total simulation error for the Smagorinsky fluid at given spatial resolution N we may adapt the Smagorinsky constant iteratively. The first task is to obtain an interval [a, c], which contains the optimal value for the resolution of the Smagorinsky length ξ. Subsequently, this interval will iteratively be reduced in size until an acceptable approximation of the minimal simulation error is achieved. As points of reference we start from the ‘no-model’ simulation which corresponds to ξ=0=a. This is the first simulation that is required in our iterative approach and it characterises the effects of discretization-error only. A second point of reference is obtained at fairly large ξ. A ‘practical upper-bound’ for ξ may be obtained by taking ξ (=c) to be the resolution of dynamic model length-scale. In the previous section we illustrated that this provides a reliable upper bound for the optimum. The evaluation of this error requires a second large-eddy simulation, now based on the dynamic eddy-viscosity model. Note that the dynamic eddy-viscosity model is used only to provide this upper-bound estimate; all other simulations employ the Smagorinsky model. The optimum is now bracketed by [a, c].

In order to start the process of successive approximation of the optimal value for ξ at the given resolution N, we evaluate the total error at an interior point of [a, c]. For this we select the mid-point ξ=b=(a+c)/2=c/2. The simulation at ξ=b is the third large-eddy simulation in our iterative approach, which finalizes the initial set-up. Further


improvements in ξ may be obtained iteratively. In view of the high computational effort that is required to evaluate the ‘error-function’, only minimisation algorithms that do not rely on the explicit use of derivatives will be considered (see Brent, 1973).

Fig. 4. Successive inverse parabolic interpolation (SIPI) is illustrated, approximating the optimal resolution of the Smagorinsky length-scale ξ. The initial triplet (a, b, c) defines an interpolating polynomial (dashed), whose minimum yields a next approximation d at which a new large-eddy simulation should be performed.

Locally around its minimum we assume that a parabola may approximate the error-function. This motivates the use of successive inverse parabolic interpolation (SIPI) to obtain a next estimate for ξ. Referring to figure 4 we start by constructing an interpolating parabola through the original bracketing triplet. The location of the minimum of this parabola is at d. A new simulation is performed corresponding to ξ=d and the total error can be obtained at this new parameter value. From this information a new bracketing triplet may be identified which, in turn, defines a new interpolating polynomial, and the process may be continued. Successive inverse parabolic interpolation and evaluation of the error-function leads to a sequence of bracketing triplets, which quite rapidly converges to the optimum (Geurts, 2006). If the error-function has a continuous second derivative, which is positive at the minimum, then convergence is super-linear (Brent, 1973). The application of this method to the Smagorinsky fluid is illustrated in figure 5. SIPI yields a rapid convergence. After about 4-6 large-eddy simulations the optimum is well approximated and a relative error of about 1-2 % remains.

The computational overhead of the additional large-eddy simulations required for the iterations is well justified. This may be quantified by incorporating the scaling of CPU time with resolution, i.e., proportional to the fourth power of N (Geurts, Froehlich, 2002). In fact, the completion of 4-6 simulations at resolution N is comparable in cost to just one simulation at resolution 3N/2. However, the reduction

18 B.J. Geurts

in the total simulation error arising from SIPI is typically much more pronounced than the error-reduction that may be achieved by increasing the resolution by 50 % in each direction. For example, comparison of the optimal results at N=32 with results obtained with the dynamic model at N=48 shows that the error is reduced by a factor of about 4-5 as a result of applying SIPI.

Fig. 5. Application of SIPI to the total simulation error at Taylor-Reynolds number 100 and N=32 (solid) and N=48 (dashed). The initial triplet is indicated with open circles, the first iterand with *, the second with diamond and the third with square.

The optimisation procedure described here provides a very simple illustration of the consequences of interacting errors. The error-reduction relies specifically on the structure of the error-landscape. Further developments are required to render this approach suitable for more complex applications and to allow for ‘context-specific’ extensions to cases in which DNS data are not available. Here, we compared LES to available DNS data - this constitutes a rather academic setting. Moreover, we only considered deviations with respect to one flow property. In general, one encounters situations in which the accurate prediction of a collection of ‘monitoring quantities’ is desired. This requires simultaneous accuracy for each of these quantities and implies an appropriate weighing of individual errors as part of the total error-measure (Meyers, Sagaut, Geurts, 2006).

4 Concluding Remarks

In this paper we considered the modifications of the large-eddy closure problem arising from the spatial discretisation at coarse subgrid resolutions. Moreover, a new


iterative method for approximating the optimal Smagorinsky constant at low resolutions was proposed and applied to homogeneous, decaying turbulence.

In case the sub-filter resolution r=Δ/h is low, the particular discretisation scheme that is adopted in the computational model, was shown to have a large dynamic effect. The difference between the actual computational stress-tensor and the turbulent stress-tensor may be expressed most directly in terms of the modified equation (Geurts, Van der Bos, 2005). Fourier analysis showed that for values as low as r=1-2 the induced numerical contribution is comparable to or even larger than the term that requires closure in large-eddy simulation. This was observed earlier in a posteriori analysis of turbulent mixing (Vreman, Geurts, Kuerten, 1996). In case r > 4 it appears that the adverse dynamic consequences of the discretization errors can largely be neglected.

The basic modelling and discretisation errors were found to counteract, e.g., in predictions of the resolved kinetic energy. This leads to an intriguing paradox related to possible strategies that should be followed to further improve large-eddy predictions compared to some reference simulation. While it is tempting to think that a higher resolution, a better numerical method or a more precise subgrid model would always lead to improved accuracy of the predictions, the counteracting property of the errors can completely distort this impression. Rather, the total error arises from a balance between modelling – and discretisation errors and it is not an easy matter to predict a priori whether these errors will or will not counteract and what the magnitude of the individual error contributions is.

The optimal working conditions for large-eddy simulations may be inferred from an error-landscape. The use of optimal refinement strategies as a point of reference for the evaluation of the dynamic procedure was reviewed. The dynamic procedure was found to provide a build-in ‘dynamic trajectory’, which follows the main Reynolds-number and resolution trends seen in the ‘optimal refinement strategy’ relatively well. However, an over-prediction of the optimal resolution of the Smagorinsky length-scale is obtained, which was found to lead to errors about twice as high as the optimal errors.

To improve upon these shortcomings of the popular eddy-viscosity model, a new iterative optimisation procedure for the Smagorinsky constant was proposed. This procedure is based on successive inverse polynomial interpolation (SIPI) and yields strongly improved accuracy compared to predictions based on the dynamic eddy-viscosity model. Initially, this method requires a bracketing interval for which simulations without subgrid model and with the dynamic eddy-viscosity model were adopted. The computational overhead associated with this iterative procedure is well justified in view of the increased accuracy compared to the dynamic eddy-viscosity model. Without the optimisation, such error-level would require much higher resolutions and computational costs. Further analysis of this new iterative procedure is subject of ongoing research.

Acknowledgement

The author gratefully acknowledges stimulating discussions with Johan Meyers (KU Leuven) and Jochen Froehlich (Karlsruhe). Simulations were performed at the Netherlands supercomputing center, SARA, and made possible through grant SG-213 of the Dutch National Computing Foundation (NCF).

20 B.J. Geurts

References

Brent, R.: Algorithms for Minimization without Derivatives. Prentice-Hall, Englewood Cliffs (1973)

Germano, M., et al.: A dynamic subgrid-scale model. Phys. Fluids 3, 1760 (1991) Geurts, B.J.: Elements of direct and large eddy simulation. Edwards Publishing, Inc. (2003) Geurts, B.J., van der Bos, F.: Numerically induced high-pass dynamics in large-eddy

simulation. Phys. of Fluids 17, 125103 (2005) Geurts, B.J., Froehlich, J.: A framework for predicting accuracy limitations in large eddy

simulation. Phys. of Fluids 14, L41(2002) Geurts, B.J., Meyers, J.: Successive inverse polynomial interpolation to optimize

Smagorinsky’s model for large-eddy simulation of homogeneous turbulence. Physics of Fluids 18, 118102 (2006)

Ghosal, S.: An analysis of numerical errors in large-eddy simulations of turbulence. J. Comp. Phys. 125, 187 (1996)

Lesieur, M., Metais, O.: New Trends in Large-Eddy Simulations of Turbulence. Ann. Rev. Fluid Mech. 28, 45 (1996)

Meneveau, C., Katz, J.: Scale-invariance and turbulence models for large eddy simulation. Ann. Rev. Fluid Mech. 32, 1 (2000)

Meyers, J., Geurts, B.J., Baelmans, M.: Database analysis of errors in large eddy simulation. Phys. of Fluids 15, 2740 (2003)

Meyers, J., Geurts, B.J., Baelmans, M.: Optimality of the dynamic procedure for large-eddy simulation. Phys. Fluids 17, 045108 (2005)

Meyers, J., Geurts, B.J., Sagaut, P.: Optimal model parameters for multi-objective large-eddy simulatons. Phys. of Fluids 18, 095103 (2006)

Pope, S.B.: Turbulent flows. Cambridge University Press, Cambridge (2000) Rogallo, R.S., Moin, P.: Numerical simulation of turbulent flows. Ann.Rev. Fl. Mech. 16, 99

(1984) Sagaut, P.: Large eddy simulation for incompressible flows; an introduction. In: Scientific

Computation, Springer, Heidelberg (2001) Salvetti, M.V., Beux, F.: The effect of the numerical scheme on the subgrid scale term in large-

eddy simulation. Phys. of Fluids 10, 3020 (1998) Smagorinsky, J.: General circulation experiments with the primitive equations. Mon. Weather

Rev. 91, 99 (1963) Vreman, A.W., Geurts, B.J., Kuerten, J.G.M.: Comparison of numerical schemes in Large Eddy

Simulation of the temporal mixing layer. Int. J. Num. Meth. in Fluids 22, 299 (1996)

97 Notes on Numerical Fluid Mechanics and...

Documents

Transcript of 97 Notes on Numerical Fluid Mechanics and...