Top

Acta Mechanica Sinica

Published in:

22-02-2019 | Research Paper

Numerical solutions of 2-D steady compressible natural convection using high-order flux reconstruction

Authors: Yaojie Yu, Feng Liu, Tingbo Zhou, Chao Gao, Ya Liu

Published in: Acta Mechanica Sinica | Issue 2/2019

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

Low Mach number flows are common and typical in industrial applications. When simulating these flows, performance of traditional compressible flow solvers can deteriorate in terms of both efficiency and accuracy. In this paper, a new high-order numerical method for two-dimensional (2-D) state low Mach number flows is proposed by combining flux reconstruction (FR) and preconditioning. Firstly, a Couette flow problem is used to assess the efficiency and accuracy of preconditioned FR. It is found that the FR scheme with preconditioning is much more efficient than the original FR scheme. Meanwhile, this improvement still preserves the numerical accuracy. Using this new method and without the Boussinesq assumption, classic natural convection is directly simulated for cases of small and large temperature differences. For the small temperature difference, a p and h refinement study is conducted to verify the grid convergence and accuracy. Then, the influence of the Rayleigh number (Ra) is analyzed. By comparing with the reference results, the numerical results of preconditioned FR is very close to that calculated by incompressible solvers. Furthermore, a large temperature difference test case is calculated and analyzed, indicating this method is not limited by the Boussinesq assumption and is also applicable to heat convection with large temperature differences.

previous article Direct numerical simulation of a turbulent boundary layer over an anisotropic compliant wall

next article Physical mechanism for origin of streamwise vortices in mode A of a square-section cylinder

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Chorin, A.J.: A numerical method for solving incompressible viscous flow problems. J. Comput. Phys. 2, 12–26 (1967). https://doi.org/10.1016/0021-9991(67)90037-X CrossRefMATH

Turkel, E.: Preconditioning techniques in computational fluid dynamics. Annu. Rev. Fluid Mech. 31, 385–416 (1999). https://doi.org/10.1146/annurev.fluid.31.1.385 MathSciNetCrossRef

Ke, G., Aulisa, E., Bornia, G., et al.: Block triangular preconditioners for linearization schemes of the Rayleigh-Bénard convection problem. Numer. Linear Algebra Appl. 24, e2096 (2017). https://doi.org/10.1002/nla.2096

Bhowmik, S.K.: A multigrid preconditioned numerical scheme for a reaction-diffusion system. Appl. Math. Comput. 254, 266–276 (2015). https://doi.org/10.1016/j.amc.2014.12.062 MathSciNetMATH

Ke, G., Aulisa, E.: New preconditioning techniques for the steady and unsteady buoyancy driven flow problems. J. Comput. Phys. 371, 244–260 (2018). https://doi.org/10.1016/j.jcp.2018.05.037 MathSciNetCrossRef

Weiss, J.M., Smith, W.A.: Preconditioning applied to variable and constant density flows. AIAA J. 33, 2050–2057 (1995). https://doi.org/10.2514/3.12946 CrossRefMATH

Choi, Y.H., Merkle, C.L.: The application of preconditioning in viscous flows. J. Comput. Phys. 105, 207–223 (1997). https://doi.org/10.1006/jcph.1993.1069 MathSciNetCrossRefMATH

Campobasso, M.S., Drofelnik, J.: Compressible Navier–Stokes analysis of an oscillating wing in a power-extraction regime using efficient low-speed preconditioning. Comput. Fluids. 67, 26–40 (2012). https://doi.org/10.1016/j.compfluid.2012.07.002 MathSciNetCrossRefMATH

Huynh, H.T.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA Paper, 2007–4079 (2007). https://doi.org/10.2514/6.2007-4079

10.

Huynh, H.T.: A reconstruction approach to high-order schemes including Discontinuous Galerkin for diffusion. AIAA Paper, 2009–403 (2009). https://doi.org/10.2514/6.2009-403

11.

Wang, Z.J., Gao, H.: A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids. J. Comput. Phys. 228, 8161–8186 (2009). https://doi.org/10.1016/j.jcp.2009.07.036 MathSciNetCrossRefMATH

12.

Jameson, A.: A proof of the stability of the spectral difference method for all orders of accuracy. J. Sci. Comput. 45, 348–358 (2010). https://doi.org/10.1007/s10915-009-9339-4 MathSciNetCrossRefMATH

13.

Vincent, P.E., Castonguay, P., Jameson, A.: A new class of high-order energy stable flux reconstruction schemes. J. Sci. Comput. 47, 50–72 (2011). https://doi.org/10.1007/s10915-010-9420-z MathSciNetCrossRefMATH

14.

Castonguay, P., Williams, D.M., Vincent, D.M., et al.: Energy stable flux reconstruction schemes for advection-diffusion problems. Comput. Methods Appl. Mech. Eng. 267, 400–417 (2013). https://doi.org/10.1016/j.cma.2013.08.012

15.

Williams, D.M., Castonguay, P., Vincent, P.E.: Energy stable flux reconstruction schemes for advection-diffusion problems on triangles. J. Comput. Phys. 250, 53–76 (2013). https://doi.org/10.1016/j.jcp.2013.05.007 MathSciNetCrossRefMATH

16.

Huynh, H.T., Wang, Z.J., Vincent, P.E., et al.: High-order methods for computational fluid dynamics: a brief review of compact differential formulations on unstructured grids. Comput. Fluids 98, 209–220 (2014). https://doi.org/10.1016/j.compfluid.2013.12.007

17.

Wang, Z.J.: High-order methods for the Euler and Navier–Stokes equations on unstructured grids. Prog. Aerosp. Sci. 43, 1–41 (2007). https://doi.org/10.1016/j.paerosci.2007.05.001 CrossRef

18.

Lu, Y.L., Wan, H., Zhuang, L.X.: Three dimensional finite element analysis for Rayleight–Bénard convection in rectangular enclosures. Acta Mech. Sin. 2, 114–125 (2002). https://doi.org/10.1007/BF02487466

19.

He, X.Z., Tong, P.: Space-time correlations in turbulent Rayleigh–Bénard convection. Acta Mech. Sin. 4, 457–467 (2014). https://doi.org/10.1007/s10409-014-0068-z CrossRefMATH

20.

Kizildag, D., Rodrguez, I., Oliva, A., et al.: Limits of the Oberbeck–Boussinesq approximation in a tall differentially heated cavity filled with water. Int. J. Heat Mass Transf. 68, 489–499 (2014). https://doi.org/10.1016/j.ijheatmasstransfer.2013.09.046 CrossRef

21.

Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981). https://doi.org/10.1016/0021-9991(81)90128-5 MathSciNetCrossRefMATH

22.

Liou, M.S., Steffen, C.: A new flux splitting scheme. J. Comput. Phys. 107, 23–39 (1993). https://doi.org/10.1006/jcph.1993.1122 MathSciNetCrossRefMATH

23.

Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4, 25–34 (1994). https://doi.org/10.1007/BF01414629 CrossRefMATH

24.

Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998). https://doi.org/10.1137/S0036142997316712 MathSciNetCrossRefMATH

25.

Bassi, F., Rebay, S.: High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. 138, 251–285 (1997). https://doi.org/10.1006/jcph.1997.5454 MathSciNetCrossRefMATH

26.

Peraire, J., Persson, P.O.: The compact discontinuous Galerkin method for elliptic problems. SIAM J. Sci. Comput. 30, 1806–1824 (2009). https://doi.org/10.1137/070685518 MathSciNetCrossRef

27.

Bassi, F., Rebay, S.: A high-order discontinuous Galerkin method forcompressible turbulent flows. In: Cockburn, G.K.B., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation, and Application. Lecture Notes in Computational Science and Engineering, pp. 77–88. Springer, Berlin (2000)CrossRef

28.

Davis, G.D.V., Jones, I.P.: Natural convection in a square cavity: a comparison exercise. Int. J. Numer. Methods Fluids 3, 227–248 (1983). https://doi.org/10.1002/fld.1650030304 CrossRefMATH

29.

Dennis, S.C.R., Hudson, J.D.: Compact finite-difference approximations to operators of Navier–Stokes type. J. Comput. Phys. 85, 390–416 (1989). https://doi.org/10.1016/0021-9991(89)90156-3 MathSciNetCrossRefMATH

30.

Becker, R., Braack, M.: Solution of a stationary benchmark problem for natural convection with high temperature difference. Int. J. Therm. Sci. 41, 428–439 (2002). https://doi.org/10.1016/S1290-0729(02)01335-2 CrossRef

31.

Vierendeels, J., Riemslagh, K., Dick, E.: A multigrid semi-implicit line-method for viscous incompressible and low-Mach number flows on high aspect ratio grids. J. Comput. Phys. 154, 310–341 (1999). https://doi.org/10.1006/jcph.1999.6315 CrossRefMATH

32.

Le Quéré, P., Masson, R., Perrot, P.: A Chebyshev collocation algorithm for 2D non-Boussinesq convection. J. Comput. Phys. 103, 320–335 (1992). https://doi.org/10.1016/0021-9991(92)90404-M CrossRefMATH

Title: Numerical solutions of 2-D steady compressible natural convection using high-order flux reconstruction
Authors: Yaojie Yu
Feng Liu
Tingbo Zhou
Chao Gao
Ya Liu
Publication date: 22-02-2019
Publisher: The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences
Published in: Acta Mechanica Sinica / Issue 2/2019
Print ISSN: 0567-7718
Electronic ISSN: 1614-3116
DOI: https://doi.org/10.1007/s10409-018-00835-w

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Other articles of this Issue 2/2019

Tissue cell differentiation and multicellular evolution via cytoskeletal stiffening in mechanically stressed microenvironments

Mechanical microenvironment as a key cellular regulator in the liver

Fabrication of three-dimensional islet models by the geometry-controlled hanging-drop method

Imaging oxygen microenvironment in hydrogel microwell array

Harnessing structural instability for cell durotaxis

Effects of hypoxia on the biological behavior of MSCs seeded in demineralized bone scaffolds with different stiffness

Premium Partners