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On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body

Published online by Cambridge University Press:  23 April 2014

Jan Östh ,
Bernd R. Noack ,
Siniša Krajnović ,
Diogo Barros  and
Jacques Borée
Jan Östh*
Affiliation:
Division of Fluid Dynamics, Department of Applied Mechanics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
Bernd R. Noack
Affiliation:
Institut PPRIME, CNRS – Université de Poitiers – ENSMA, UPR 3346, Départment Fluides, Thermique, Combustion, CEAT, 43 rue de l’Aérodrome, F-86036 Poitiers CEDEX, France
Siniša Krajnović
Affiliation:
Division of Fluid Dynamics, Department of Applied Mechanics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
Diogo Barros
Affiliation:
Institut PPRIME, CNRS – Université de Poitiers – ENSMA, UPR 3346, Départment Fluides, Thermique, Combustion, CEAT, 43 rue de l’Aérodrome, F-86036 Poitiers CEDEX, France PSA Peugeot Citroën, Centre Technique de Velizy, 78943 Vélizy-Villacoublay CEDEX, France
Jacques Borée
Affiliation:
Institut PPRIME, CNRS – Université de Poitiers – ENSMA, UPR 3346, Départment Fluides, Thermique, Combustion, CEAT, 43 rue de l’Aérodrome, F-86036 Poitiers CEDEX, France
*
†Email address for correspondence: ojan@chalmers.se
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Abstract

We investigate a hierarchy of eddy-viscosity terms in proper orthogonal decomposition (POD) Galerkin models to account for a large fraction of unresolved fluctuation energy. These Galerkin methods are applied to large eddy simulation (LES) data for a flow around a vehicle-like bluff body called an Ahmed body. This flow has three challenges for any reduced-order model: a high Reynolds number, coherent structures with broadband frequency dynamics, and meta-stable asymmetric base flow states. The Galerkin models are found to be most accurate with modal eddy viscosities as proposed by Rempfer & Fasel (J. Fluid Mech., vol. 260, 1994a, pp. 351–375; J. Fluid Mech. vol. 275, 1994b, pp. 257–283). Robustness of the model solution with respect to initial conditions, eddy-viscosity values and model order is achieved only for state-dependent eddy viscosities as proposed by Noack, Morzyński & Tadmor (Reduced-Order Modelling for Flow Control, CISM Courses and Lectures, vol. 528, 2011). Only the POD system with state-dependent modal eddy viscosities can address all challenges of the flow characteristics. All parameters are analytically derived from the Navier–Stokes-based balance equations with the available data. We arrive at simple general guidelines for robust and accurate POD models which can be expected to hold for a large class of turbulent flows.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Turbulent Flows: Turbulence simulation Wakes/Jets: Wakes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 747 , 25 May 2014 , pp. 518 - 544
Copyright
© 2014 Cambridge University Press 

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References

Ahmed, S., Ramm, G. & Faltin, G.1984 Some salient features of the time-averaged ground vehicle wake. SAE Tech. Rep. 840300.Google Scholar
Aider, J.-L., Beaudoin, J.-F. & Wesfreid, J. E. 2010 Drag and lift reduction of a 3D bluff-body using active vortex generators. Exp. Fluids 48, 771–789.CrossRefGoogle Scholar
Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115–173.CrossRefGoogle Scholar
AVL Fire 2013 CFD Solver. FIRE CFD Solver Users Guide, v2013.Google Scholar
Balajewicz, M. J., Dowell, E. H. & Noack, B. R. 2013 Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation. J. Fluid Mech. 729, 285–308.Google Scholar
Beaudoin, J.-F. & Aider, J.-L. 2008 Drag and lift reduction of a 3D bluff body using flaps. Exp. Fluids 44, 491–501.Google Scholar
Boberg, L. & Brosa, U. 1988 Onset of turbulence in a pipe. Z. Naturforsch. 43a, 697–726.Google Scholar
Brunn, A., Nitsche, W., Henning, L. & King, R.2008 Application of slope-seeking to a generic car model for active drag control. AIAA Paper 2008-6734.CrossRefGoogle Scholar
Busse, F. H. 1991 Numerical analysis of secondary and tertiary states of fluid flow and their stability properties. Appl. Sci. Res. 48, 341–351.Google Scholar
Cazemier, W., Verstappen, R. W. C. P. & Veldman, A. E. P. 1998 Proper orthogonal decomposition and low-dimensional models for driven cavity flows. Phys. Fluids 10, 1685–1699.Google Scholar
Cordier, L., Noack, B. R., Tissot, G., Lehnasch, G., Delville, J., Balajewicz, M., Daviller, G. & Niven, R. K. 2013 Identification strategies for model-based control. Exp. Fluids 54 (8), 1–21.Google Scholar
Davidson, L. 2010 How to estimate the resolution of an LES of recirculating flow. In Quality and Reliability of Large-Eddy Simulations II (ed. Salvetti, M. V., Geurts, B., Meyers, J. & Sagaut, P.), Ercoftac Series, vol. 16, pp. 269–286. Springer.Google Scholar
Deane, A. E., Kevrekidis, I. G., Karniadakis, G. E. & Orszag, S. A. 1991 Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders. Phys. Fluids A 3, 2337–2354.Google Scholar
Duell, E. G. & George, A. R.1999 Experimental study of a ground vehicle body unsteady near wake. SAE Paper 1999-01-0812.Google Scholar
Galletti, G., Bruneau, C. H., Zannetti, L. & Iollo, A. 2004 Low-order modelling of laminar flow regimes past a confined square cylinder. J. Fluid Mech. 503, 161–170.Google Scholar
Grandemange, M., Gohlke, M. & Cadot, O. 2013 Turbulent wake past a three-dimensional blunt body. Part 1. Global modes and bi-stability. J. Fluid Mech. 722, 51–84.Google Scholar
Hemida, H.2008 Numerical simulations of flows around trains and buses in cross winds. PhD thesis, Chalmers University of Technology, Gothenburg, Sweden, ISBN/ISSN 978-91-7385-187-9.Google Scholar
Hemida, H., Gil, N. & Baker, C. 2010 LES of the slipstream of a rotating train. Trans. ASME J. Fluids Engng 132 (5), 051103–051103-9.CrossRefGoogle Scholar
Hemida, H. & Krajnović, S. 2008 LES study of the influence of a train-nose shape on the flow structures under cross-wind conditions. Trans. ASME J. Fluids Engng 130 (9), 091101–091101-12.Google Scholar
Hemida, H. & Krajnović, S. 2010 LES study of the influence of the nose shape and yaw angles on flow structures around trains. J. Wind Engng Ind. Aerodyn. 98, 34–46.Google Scholar
Holmes, P., Lumley, J. L., Berkooz, G. & Rowley, C. W. 2012 Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn Cambridge University Press.Google Scholar
Joseph, D. D. 1976 Stability of Fluid Motions I & II. Springer-Verlag.Google Scholar
Kolmogorov, A. N. 1941a Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 16–18 (translated and reprinted 1991 in Proc. R. Soc. Lond. A 434, 15–17).Google Scholar
Kolmogorov, A. N. 1941b The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Dokl. Akad. Nauk SSSR 30, 9–13 (translated and reprinted 1991 in Proc. R. Soc. Lond. A 434, 9–13).Google Scholar
Kraichnan, R. H. & Chen, S. 1989 Is there a statistical mechanics of turbulence? Physica D 37, 160–172.Google Scholar
Krajnović, S.2002 Large-Eddy Simulation for computing the flow around vehicles. PhD thesis, Chalmers University of Technology, Gothenburg, Sweden, ISBN/ISSN 91-7291-188-3.Google Scholar
Krajnović, S. 2009 LES of flows around ground vehicles and other bluff bodies. Phil. Trans. R. Soc. A 367 (1899), 2917–2930.Google Scholar
Krajnović, S. 2011 Flow around a tall finite cylinder explored by large eddy simulation. J. Fluid Mech. 676, 294–317.Google Scholar
Krajnović, S.2013 LES investigation of passive flow control around an Ahmed body. In ASME 2013 International Mechanical Engineering Congress & Exposition, San Diego, CA, USA, vol. 1, pp. 305–315, Paper IMECE2013-62373.Google Scholar
Krajnović, S. & Davidson, L. 2003 Numerical study of the flow around the bus-shaped body. Trans. ASME J. Fluids Engng 125, 500–509.CrossRefGoogle Scholar
Krajnović, S. & Davidson, L. 2005a Flow around a simplified car. Part 1. Large Eddy Simulation. Trans. ASME J. Fluids Engng 127, 907–918.CrossRefGoogle Scholar
Krajnović, S. & Davidson, L. 2005b Flow around a simplified car. Part 2. Understanding the flow. Trans. ASME J. Fluids Engng 127, 919–928.Google Scholar
Krajnović, S. & Fernandes, J. 2011 Numerical simulation of the flow around a simplified vehicle model with active flow control. Intl J. Heat Fluid Flow 32 (5), 192–200.Google Scholar
Lahaye, A., Leroy, A. & Kourta, A. 2014 Aerodynamic characterisation of a square back bluff body flow. Intl J. Aerodyn. 4 (1–2), 43–60.Google Scholar
Lienhart, H. & Becker, S.2003 Flow and turbulent structure in the wake of a simplified car model. SAE Paper No. 2003-01-0656.Google Scholar
Lumley, J. L. 1970 Stochastic Tools in Turbulence. Academic Press.Google Scholar
Ma, X. & Karniadakis, G. E. 2002 A low-dimensional model for simulating three-dimensional cylinder flow. J. Fluid Mech. 458, 181–190.CrossRefGoogle Scholar
Noack, B. R., Afanasiev, K., Morzyński, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.Google Scholar
Noack, B. R. & Eckelmann, H. 1994 A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330.Google Scholar
Noack, B. R., Morzyński, M. & Tadmor, G. 2011 Reduced-Order Modelling for Flow Control. (CISM Courses and Lectures) , vol. 528. Springer.CrossRefGoogle Scholar
Noack, B. R., Papas, P. & Monkewitz, P. A. 2005 The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339–365.Google Scholar
Noack, B. R., Schlegel, M., Ahlborn, B., Mutschke, G., Morzyński, M., Comte, P. & Tadmor, G. 2008 A finite-time thermodynamics of unsteady fluid flows. J. Non-Equilib. Thermodyn. 33, 103–148.Google Scholar
Östh, J. & Krajnović, S. 2012 The flow around a simplified tractor-trailer model studied by Large Eddy Simulation. J. Wind Engng Ind. Aerodyn. 102, 36–47.Google Scholar
Östh, J. & Krajnović, S. 2014 A study of the aerodynamics of a generic container freight wagon using Large-Eddy Simulation. J. Fluid. Struct. 44, 31–51.Google Scholar
Östh, J., Krajnović, S., Barros, D., Cordier, L., Noack, B. R., Borée, J. & Ruiz, T.2013 Active flow control for drag reduction of vehicles using Large Eddy Simulation, experimental investigations and reduced order modelling. Proceedings of the 8th International Symposium on Turbulent and Shear Flow Phenomena (TSFP-8), Poitiers, France.Google Scholar
Pastoor, M., Henning, L., Noack, B. R., King, R. & Tadmor, G. 2008 Feedback shear layer control for bluff body drag reduction. J. Fluid Mech. 608, 161–196.Google Scholar
Podvin, B. 2009 A proper-orthogonal-decomposition-based model for the wall layer of a turbulent channel flow. Phys. Fluids 21, 015111 1 $\ldots$ 18.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. 1st edn Cambridge University Press.Google Scholar
Rajaee, M., Karlsson, S. K. F. & Sirovich, L. 1994 Low-dimensional description of free-shear-flow coherent structures and their dynamical behaviour. J. Fluid Mech. 258, 1–29.Google Scholar
Rempfer, D. & Fasel, F. H. 1994a Evolution of three-dimensional coherent structures in a flat-plate boundary-layer. J. Fluid Mech. 260, 351–375.Google Scholar
Rempfer, D. & Fasel, F. H. 1994b Dynamics of three-dimensional coherent structures in a flat-plate boundary-layer. J. Fluid Mech. 275, 257–283.Google Scholar
Schlegel, M. & Noack, B. R. 2013 On long-term boundedness of Galerkin models. J. Fluid Mech. (submitted).Google Scholar
Schlegel, M., Noack, B. R., Comte, P., Kolomenskiy, D., Schneider, K., Farge, M., Scouten, J., Luchtenburg, D. M. & Tadmor, G. 2009 Reduced-order modelling of turbulent jets for noise control. In Numerical Simulation of Turbulent Flows and Noise Generation: Results of the DFG/CNRS Research Groups FOR 507 and FOR 508, Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), vol. 1, pp. 3–27. Springer.Google Scholar
Sirovich, L. 1987a Turbulence and the dynamics of coherent structures, part I: coherent structures. Q. Appl. Math. 45, 561–571.Google Scholar
Sirovich, L. 1987b Turbulence and the dynamics of coherent structures, part II: symmetries and transformations. Q. Appl. Math. 45, 573–582.CrossRefGoogle Scholar
Smagorinsky, J. 1963 General circulation experiments with the primitive equations. Mon. Weath. Rev. 91 (3), 99–165.2.3.CO;2>CrossRefGoogle Scholar
Spohn, A. & Gillieron, P.2002 Flow separations generated by a simplified geometry of an automotive vehicle. IUTAM Symposium: Unsteady Separated Flows, Toulouse, France.Google Scholar
Ukeiley, L., Cordier, L., Manceau, R., Delville, J., Bonnet, J. P. & Glauser, M. 2001 Examination of large-scale structures in a turbulent plane mixing layer. Part 2. Dynamical systems model. J. Fluid Mech. 441, 61–108.Google Scholar
Verzicco, R., Fatica, M., Laccarino, G. & Moin, P. 2002 Large Eddy Simulation of a road vehicle with drag-reduction devices. AIAA J. 40, 2447–2455.Google Scholar
Wang, Z., Akhtar, I., Borggaard, J. & Iliescu, T. 2011 Two-level discretizations of nonlinear closure models for proper orthogonal decomposition. J. Comput. Phys. 230, 126–146.Google Scholar
Wang, Z., Akhtar, I., Borggaard, J. & Iliescu, T. 2012 Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison. Comput. Meth. Appl. Mech. Engng 237–240, 10–26.Google Scholar
Wassen, E. & Thiele, F.2009 Road vehicle drag reduction by combined steady blowing and suction. AIAA Paper 2009-4174.Google Scholar