Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T16:15:35.010Z Has data issue: false hasContentIssue false
  • English
  • Français

Effective Boundary Conditions: A General Strategy and Application to Compressible Flows Over Rough Boundaries

Part of: Homogenization, determination of effective properties General aerodynamics and subsonic flows Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  07 February 2017

Giulia Deolmi ,
Wolfgang Dahmen  and
Siegfried Müller
Giulia Deolmi*
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany
Wolfgang Dahmen*
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany
Siegfried Müller*
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany
*
*Corresponding author.Email addresses:[email protected] (G. Deolmi), [email protected] (W. Dahmen), [email protected] (S. Müller)
*Corresponding author.Email addresses:[email protected] (G. Deolmi), [email protected] (W. Dahmen), [email protected] (S. Müller)
*Corresponding author.Email addresses:[email protected] (G. Deolmi), [email protected] (W. Dahmen), [email protected] (S. Müller)
Get access

Abstract

Determining the drag of a flowover a rough surface is a guiding example for the need to take geometric micro-scale effects into account when computing a macroscale quantity. A well-known strategy to avoid a prohibitively expensive numerical resolution of micro-scale structures is to capture the micro-scale effects through some effective boundary conditions posed for a problem on a (virtually) smooth domain. The central objective of this paper is to develop a numerical scheme for accurately capturing the micro-scale effects at essentially the cost of twice solving a problem on a (piecewise) smooth domain at affordable resolution. Here and throughout the paper “smooth” means the absence of any micro-scale roughness. Our derivation is based on a “conceptual recipe” formulated first in a simplified setting of boundary value problems under the assumption of sufficient local regularity to permit asymptotic expansions in terms of the micro-scale parameter.

The proposed multiscale model relies then on an upscaling strategy similar in spirit to previous works by Achdou et al. [1], Jäger and Mikelic [29, 31], Friedmann et al. [24, 25], for incompressible fluids. Extensions to compressible fluids, although with several noteworthy distinctions regarding e.g. the “micro-scale size” relative to boundary layer thickness or the systematic treatment of different boundary conditions, are discussed in Deolmi et al. [16,17]. For proof of concept the general strategy is applied to the compressible Navier-Stokes equations to investigate steady, laminar, subsonic flow over a flat plate with partially embedded isotropic and anisotropic periodic roughness imposing adiabatic and isothermal wall conditions, respectively. The results are compared with high resolution direct simulations on a fully resolved rough domain.

Keywords

Homogenizationupscaling strategyeffective boundary conditionsNavier wall lawcompressible flow

MSC classification

Secondary: 74Q15: Effective constitutive equations 76G25: General aerodynamics and subsonic flows 35Q30: Navier-Stokes equations
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 2 , February 2017 , pp. 358 - 400
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Achdou, Y., Pironneau, O., Valentin, F., Effective boundary conditions for laminar flows over periodic rough boundaries, J. Comp. Phys., 147, 187–218, 1998.CrossRefGoogle Scholar
[2] Achdou, Y., Pironneau, O., Valentin, F., New wall laws for unsteady incompressible Navier-Stokes equations, ECCOMAS 2000, 2000.CrossRefGoogle Scholar
[3] Achdou, Y., Le Tallec, P., Valentin, F., Pironneau, O., Constructing wall laws with domain decomposition or asymptotic expansion techniques, Comput. Methods Appl. Mech. Engrg., 151, 215–232, 1998.CrossRefGoogle Scholar
[4] Achdou, Y., Pironneau, O., Valentin, F., Comparison of wall laws for unsteady incompressible Navier-Stokes equations, First MIT Conference on Computational Fluid and Solid Mechanics, Boston, USA, June 12-15, 2001.Google Scholar
[5] Anderson, J.D., Hypersonic and High Temperature Gas Dynamics, McGraw-Hill Series in Aeronautical and Aerospace Engineering, 1989.Google Scholar
[6] Bangerth, W., Hartmann, R., Kanschat, G., deal.II — A general-purpose object-oriented finite element library, ACM Trans. Math. Softw., 33(4), 24/1–24/27, 2007.CrossRefGoogle Scholar
[7] Barrenechea, G.R., Le Tallec, P., Valentin, F., New wall laws for the unsteady incompressible Navier-Stokes equations on rough domains, Mathematical Modeling and Numerical Analysis, 36, 177–203, 2002.CrossRefGoogle Scholar
[8] Basson, A., Gerard-Varet, D., Wall laws for fluid flows at a boundary with random roughness, Commun. Pure Appl. Math., 61, 941–987, 2008.CrossRefGoogle Scholar
[9] Bramkamp, F., Lamby, Ph., Müller, S., An adaptive multiscale finite volume solver for unsteady and steady state flow computations, J. Comp. Phys., 197(2), 460–490, 2004.CrossRefGoogle Scholar
[10] Bensoussan, A., Lions, J.L., Papanicolaou, G., Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978.Google Scholar
[11] Brix, K., Mogosan, S., Müller, S., Schieffer, G., Parallelization of multiscale-based grid adaptation using space-filling curves, in Multiresolution and Adaptive Methods for Convection-Dominated Problems, Coquel, F., Maday, Y., Müller, S., Postel, M., Tran, Q.H. (eds.), ESAIM: Proceedings, 29, 108–129, 2009.Google Scholar
[12] Bresch, D., Milisic, V., High order multi-scale wall-laws, Part I: The periodic case, Quarterly of Applied Mathematics, 68, 229–253, 2010.CrossRefGoogle Scholar
[13] Bresch, D., Milisic, V., Towards implicit multi-scale wall-laws, Comptes Rendus Mathematique, 346, 833–838, 2008.CrossRefGoogle Scholar
[14] Casado-Diaz, J., Fernandez-Cara, E., Simon, J., Why viscous fluids adhere to rugose walls: A mathematical explanation, J. Differential Equations, 189, 526–537, 2003.CrossRefGoogle Scholar
[15] Dahmen, W., Plesken, Ch., G. Welper Double greedy algorithms: Reduced basis methods for transport dominated problems, ESAIM:Mathematical Modelling and Numerical Analysis, 48(3), 623–663, 2014.Google Scholar
[16] Deolmi, G., Dahmen, W., Müller, S., Effective boundary conditions for compressible flows over rough surface, Math. Models Methods Appl. Sci., 25, 1257–1297, 2015.CrossRefGoogle Scholar
[17] Deolmi, G., Dahmen, W., Müller, S., Effective boundary conditions for compressible flows over rough surface: Isothermal case, Proceedings of the XXIV Congress on Differential Equations and Applications - XIV Congress on Applied Mathematics, ISBN 978-84-9828-527-7, 2015.Google Scholar
[18] Du, Y., Karniadakis, G.E., Suppressing wall turbulence by means of a transverse traveling wave, Science 288, 1230–1234, 2000.CrossRefGoogle ScholarPubMed
[19] Du, Y., Karniadakis, G.E., Drag reduction in wall bounded turbulence via a transverse travelling wave, J. Fluid. Mech. 457, 1–34, 2002.CrossRefGoogle Scholar
[20] Efendiev, Y., Hou, T.Y., Multiscale Finite Element Methods: Theory and Applications, Springer, 2009.Google Scholar
[21] W. E, , Engquist, B., The heterogeneous multiscale methods, Commun. Math. Sci. 1(1), 87–132, 2003.Google Scholar
[22] W. E, , Engquist, B., The heterogeneous multi-scale method for homogenization problems, in Multiscale Methods in Science and Engineering, 89–110, Lecture Notes Comput. Sci. Eng., Vol. 44, Springer, Berlin, 2005.Google Scholar
[23] Friedmann, E., Riblets in the viscous sublayer, Inaugural Dissertation, University of Heidelberg, 2005.Google Scholar
[24] Friedmann, E., The optimal shape of riblets in the viscous sublayer, J. Math. Fluid Mech., 12, 243–265, 2010.CrossRefGoogle Scholar
[25] Friedmann, E., Richter, T., Optimal microstructures drag reducing mechanism of riblets, J. Math. Fluid Mech., 13, 429–447, 2011.CrossRefGoogle Scholar
[26] Iwamoto, K., Fukagata, K., Kasagi, N., Suzuki, Y., Friction drag reduction achievable by near-wall turbulence manipulation at high Reynolds numbers, Physics of Fluids 17, 011702, doi:10.1063/1.1827276, 2005.CrossRefGoogle Scholar
[27] Jäger, W., Mikelic, A., On the boundary conditions at the contact interface between a porous medium and a free fluid, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 23, 403–465, 1996.Google Scholar
[28] Jäger, W., Mikelic, A., On the interface boundary conditions of Beavers, Joseph and Saffman, SIAM J. Appl. Math., 60, 1111–1127, 2000.Google Scholar
[29] Jäger, W., Mikelic, A., On the roughness-induced effective boundary conditions for an incompressible viscous flow, J. Differential Equations, 170, 96–122, 2001.CrossRefGoogle Scholar
[30] Jäger, W., Mikelic, A., Neuss, N., Asymptotic analysis of the laminar viscous flow over a porous bed, SIAM J. Sci. Comput., 22, 2006–2028, 2001.CrossRefGoogle Scholar
[31] Jäger, W., Mikelic, A., Couette flows over a rough boundary and drag reduction, Commun. Math. Phys., 232, 429–455, 2003.CrossRefGoogle Scholar
[32] Jäger, W., Mikelic, A., On the boundary conditions at the contact interface between a porous medium and a free fluid, Ann. Scuola Norm. Super. Pisa Cl. Fis. Mat. (IV), 23, 403–465, 1996.Google Scholar
[33] Jikov, V., Kozlov, S., Oleinik, O., Homogenization of Differential Operators and Integral Functionals, Springer, Berlin, 1995.Google Scholar
[34] Jung, W.J., Mangiavacchi, N., Akhhavan, R., Suppression of turbulence in wall-bounded flows by high frequency spanwise oscillations, Physics of Fluids A 4, 1605–1607, 1992.CrossRefGoogle Scholar
[35] Kang, S., Choi, H., Active wall motions for skin-friction drag reduction, Physics of Fluids 12, 3301–3304, 2000.CrossRefGoogle Scholar
[36] Kenig, C., Prange, C., Uniform Lipschitz estimates in bumpy half-spaces, Archive for Rational Mechanics and Analysis, November 2014.CrossRefGoogle Scholar
[37] Kevorkian, J., Cole, J.D., Perturbation Methods in Applied Mathematics, Applied Mathematical Sciences, Vol. 34. New York-Heidelberg-Berlin: Springer-Verlag, 1981.CrossRefGoogle Scholar
[38] Kweon, J.R., Kellogg, R.B., Regularity of solutions to the Navier-Stokes system for compressible flows on a polygon, SIAM J. Math. Anal. 35(6), 1451–1485, 2004.CrossRefGoogle Scholar
[39] Lamby, Ph., Parametric multi-block grid generation and application to adaptive flow simulations, Diss. RWTH Aachen, 2007.Google Scholar
[40] Madureira, A., Valentin, F., Asymptotics of the Poisson problem in domains with curved rough boundaries, SIAM J. Math. Anal., 38, 1450–1473, 2006/07.CrossRefGoogle Scholar
[41] Mikelic, A., Rough boundaries and wall laws, in Qualitative properties of solutions to partial differential equations, Lecture Notes of Necas Center for Mathematical Modeling, Feireisl, E., Kaplicky, P. and Malek, J. (eds), 5, 103–134, 2009.Google Scholar
[42] Mohammadi, B., Pironneau, O., Valentin, F., Rough boundaries and wall laws, Int. J. Numer. Meth. Fluids, 27, 169–177, 1998.3.0.CO;2-4>CrossRefGoogle Scholar
[43] Müller, S., Adaptive Multiscale Schemes for Conservation Laws, Lecture Notes Comput. Sci. Eng., Vol. 27, first edition, Springer Verlag, 2003.CrossRefGoogle Scholar
[44] Palmer, G., Technical Java, Prentice Hall, 2003.Google Scholar
[45] Sanchez-Palencia, E., Non-Homogeneous Media and Vibration Theory, Springer, 1980.Google Scholar
[46] Schlichting, H., Gersten, K., Boundary Layers Theory, Springer, 8th edition, 2000.CrossRefGoogle Scholar
[47] Schultz, M.P., Flack, K.A., The rough-wall turbulent boundary layer from the hydraulically smooth to the fully rough regime, J. Fluid Mech., 580, 381–405, 2007.CrossRefGoogle Scholar
[48] Tartar, L., The General Theory of Homogenization. A Personalized Introduction, Lecture Notes of the Unione Matematica Italiana, 7. Springer-Verlag, Berlin; UMI, Bologna, 2009.Google Scholar
[49] Veran, S., Aspa, Y., Quintard, M., Effective boundary conditions for rough reactive walls in laminar boundary layers, Int. J. Heat Mass Transfer, 52, 3712–3725, 2009.CrossRefGoogle Scholar
[50] Walsh, M.J., Riblets, in Viscous Drag Reduction in Boundary Layers, Bushnell, D.M., Hefner, J.N. (eds.), AIAA, New York, NY, 203–261, 1990.Google Scholar
[51] Choi, K.-S., European drag-reduction research-recent developments and current status, Fluid Dyn. Res., 26, 325–335, 2000.CrossRefGoogle Scholar
[52] Bushnell, D.M., Aircraft drag reduction — A review, Proc. Inst. Mech. Eng., 217, 1–18, 2003.CrossRefGoogle Scholar
[53] Bruse, M., Bechert, D.W., der Hoeven, J.G.T.V., Hage, W., Hoppe, G., Experiments with conventional and with novel adjustable drag-reducing surfaces, in Near-Wall Turbulent Flows, So, R.M., Speziale, C.G., Launder, B.E. (eds.), Elsevier, Amsterdam, The Netherlands, 719–738, 1993.Google Scholar
[54] Bechert, D.W., Bruse, M., Hage, W., Meyer, R., Biological surfaces and their technological application — Laboratory and flight experiments on drag reduction and separation control, AIAA paper 97-1960, 1997.CrossRefGoogle Scholar
[55] Itoh, M., Tamano, S., Igushi, R., Yokota, K., Akino, N., Hino, R., Kubo, S., Turbulent drag reduction by the seal fur surface, Phys. Fluids, 18, 065102, 2006.Google Scholar
[56] Lee, S.-J., Jang, Y.-G., Control of flow around a NACA 0012 airfoil with a micro-riblet film, J. Fluids Struct., 20, 659–672, 2005.CrossRefGoogle Scholar
[57] Viswanath, P.R., Aircraft viscous drag reduction using riblets, Prog. Aerosp. Sci., 38, 571–600, 2002.CrossRefGoogle Scholar
[58] Szodruch, J., Viscous drag reduction on transport aircraft, AIAA paper 91-0685, 1991.CrossRefGoogle Scholar
[59] Reif, W.-E., Dinkelacker, A., Hydrodynamics of the squamation in fast swimming sharks, Neues Jahrbuch für Geologie und Paläontologie Abhandlungen, 164, 184–187, 1982.CrossRefGoogle Scholar