High-Resolution Finite Volume Methods on Unstructured Grids for Turbulence and Aeroacoustics
Tóm tắt
In this paper we focus on the application of a higher-order finite volume method for the resolution of Computational Aeroacoustics problems. In particular, we present the application of a finite volume method based in Moving Least Squares approximations in the context of a hybrid approach for low Mach number flows. In this case, the acoustic and aerodynamic fields can be computed separately. We focus on two kinds of computations: turbulent flow and aeroacoustics in complex geometries. Both fields require very accurate methods to capture the fine features of the flow, small scales in the case of turbulent flows and very low-amplitude acoustic waves in the case of aeroacoustics. On the other hand, the use of unstructured grids is interesting for real engineering applications, but unfortunately, the accuracy and efficiency of the numerical methods developed for unstructured grids is far to reach the performance of those methods developed for structured grids. In this context, we propose the FV-MLS method as a tool for accurate CAA computations on unstructured grids.
Tài liệu tham khảo
Godunov SK (1959) A finite difference method for the computation of discontinuous solutions of the equations of fluid dynamics. Mat Sb 47(89):271–306
Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103:16–42
Rizzetta DP, Visbal MR, Blaisdell GA (1999) Application of a high-order compact difference scheme to large-eddy and direct numerical simulation. AIAA paper 99-3714
Visbal MR, Gaitonde DV (1999) High-order-accurate methods for complex unsteady subsonic flows. AIAA J 37(10):1231–1239
Visbal MR, Rizzeta DP (2002) Large-eddy simulation on curvilinear grids using compact differencing and filtering schemes. J Fluids Eng 124:836–847
Canuto C, Hussaini MY, Quarteroni A, Zang TA (2007) Spectral methods. Evolution to complex geometries and applications to fluid dynamics. Springer, New York
Karniadakis GE, Sherwin SJ (2005) Spectral/hp element methods for computational fluid dynamics, 2nd edn. Oxford University Press, New York
Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, New York
Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195
Piegl L, Tiller W (1997) The NURBS book, 2nd edn. Springer, New York
Lipton S, Evans JA, Bazilevs Y, Elguedj T, Hughes TJR (2010) Robustness of isogeometric structural discretizations under severe mesh distortion. Comput Methods Appl Mech Eng 199:357–373
Cottrell JA, Reali A, Bazilevs Y, Hughes TJR (2006) Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Eng 195:5257–5296
Van Leer B (1979) Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method. J Comput Phys 32:101–136
Van Leer B (1982) Flux vector splitting for the Euler equations. Lecture notes in physics, vol 170. Springer, Berlin
Roe PL (1981) Approximate Riemann solvers, parameter vectores and difference schemes. J Comput Phys 43:357–372
Venkatakrishnan V (1995) Convergence to steady state solutions of the Euler equations on unstructured grids with limiters. J Comput Phys 118:120–130
Barth TJ (1995) Aspects of unstructured grids and finite-volume solvers for the Euler and Navier-Stokes equations. VKI lecture series 1994–95
Barth TJ, Frederickson PO (1990) Higher-order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA paper 90-0013
Barth TJ, Jespersen DC (1989) The design and application of upwind schemes on unstructured meshes. AIAA paper 89-0366
Colella P, Woodward P (1984) The piecewise parabolic method (PPM) for gas-dynamical simulations. J Comput Phys 54:174–201
Jameson A, Baker TJ (1983) Solution of the Euler equations for complex configurations. AIAA paper 83-1929
Frink NT (1992) Upwind scheme for solving the Euler equations on unstructured tetrahedral meshes. AIAA J 30:70
Ollivier-Gooch CF, Van Altena M (2002) A high-order accurate unstructured mesh finite volume scheme for the advection-diffusion equation. J Comput Phys 181:729–752
Bassi F, Rebay S (1997) A higher-order accurate discontinuous finite element solution of the 2D Euler equations. J Comput Phys 138:251–285
Bassi F, Rebay S (1997) A higher-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J Comput Phys 131:267–279
Bassi F, Rebay S (1997) High-order accurate discontinuous finite element solution of the 2D Euler equations. J Comput Phys 138:251–285
Cockburn B, Shu C-W (1998) The local discontinuous Galerkin finite element method for convection-diffusion systems. SIAM J Numer Anal 35:2440–2463
Cockburn B, Shu C-W (1989) TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math Comput 52:411–435
Cockburn B, Lin SY, Shu C-W (1989) TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one dimensional systems. J Comput Phys 84:90–113
Cockburn B, Hou S, Shu C-W (1990) TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math Comput 54:545–581
Cockburn B, Shu C-W (1998) The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J Numer Anal 35:2440–2463
Cockburn B, Kanschat G, Perugia I, Schötzau D (2001) Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids. SIAM J Numer Anal 39(1):264–285
Cockburn B, Shu C-W (2001) Runge-Kutta discontinuous Galerkin methods for convection dominated problems. J Sci Comput 16:173–261
Crivellini A, Bassi F (2003) A three-dimensional parallel discontinuous Galerkin solver for acoustic propagation studies. International Journal of aeroacoustics 2:157–174
Dolejší V (2004) On the discontinuous Galerkin method for the numerical solution of the Navier-Stokes equations. Int J Numer Methods Fluids 45:1083–1106
Zhang M, Shu C-W (2003) An analysis of three different formulations of the discontinuous Galerkin method for diffusion equations. Math Models Methods Appl Sci 13(3):395–413
Peraire J, Persson P-O (2008) The compact discontinuous Galerkin (CDG) method for elliptic problems. SIAM J Sci Comput 30(4):1806–1824
Persson P-O, Peraire J (2008) Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the Navier-Stokes equations. SIAM J Sci Comput 30(6):2709–2733
Harten A, Osher S (1987) Uniformly high order accurate non-oscillatory schemes I. SIAM J Numer Anal 24:279–309
Harten A, Engquist B, Osher S, Chakravarthy S (1987) Uniformly high order essentially non-oscillatory schemes III. J Comput Phys 71:231–303
Hu CQ, Shu CW (1999) Weighted essentially non-oscillatory schemes on triangular meshes. J Comput Phys 150:97–127
Shu CW, Osher S (1988) Efficient implementation of essentially non-oscillatory shock-capturing schemes. J Comput Phys 77:439–471
Shu CW, Osher S (1989) Efficient implementation of essentially non-oscillatory shock-capturing schemes II. J Comput Phys 83:32–78
Shu CW (1997) Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. ICASE Report 97-65
Shu CW, Osher S (1989) Efficient implementation of essentially non-oscillatory shock-capturing schemes II. J Comput Phys 83:32–78
Abgrall R (1994) On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J Comput Phys 114:45–58
Borges R, Carmona M, Costa B, Don WS (2008) An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J Comput Phys 227(6):3101–3211
Capdeville G (2008) A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes. J Comput Phys 227:2977–3014
Dumbser M, Käser M (2007) Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. J Comput Phys 221:693–723
Dumbser M, Käser M, Titarev VA, Toro EF (2007) Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 226:204–243
Henrick AK, Aslam TD, Powers JM (2005) Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J Comput Phys 207:542–567
Sonar T (1997) On the construction of essentially non-oscillatory finite volume approximations to hyperbolic conservation laws on general triangulations: polynomial recovery, accuracy and stencil selection. Comput Methods Appl Mech Eng 140:157–181
Zhang YT, Shu CW (2009) Third order WENO scheme on three dimensional tetrahedral meshes. Commun Comput Phys 5:836–848
Wang ZJ (2002) Spectral (finite) volume method for conservation laws on unstructured grids. Basic formulation. J Comput Phys 178:210–251
Wang ZJ, Liu Y (2002) Spectral (finite) volume method for conservation laws on unstructured grids II: extension to two-dimensional scalar equation. J Comput Phys 179:665–697
Wang ZJ, Liu Y (2004) Spectral (finite) volume method for conservation laws on unstructured grids III: one-dimensional systems and partition optimization. J Sci Comput 20:137–157
Wang ZJ, Liu Y (2004) Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional systems. J Comput Phys 194:716–741
Liu Y, Vinokurb M, Wang ZJ (2006) Spectral (finite) volume method for conservation laws on unstructured grids V: extension to three-dimensional systems. J Comput Phys 212(2):454–472
Wang ZJ, Liu Y (2006) Spectral (finite) volume method for conservation laws on unstructured grids VI: extension to viscous flow. J Comput Phys 215:41–58
Kannan R, Wang ZJ (2009) A study of viscous flux formulations for a p-multigrid spectral volume Navier Stokes solver. J Sci Comput 41(2):165–199
Roe PL (1982) Fluctuations and signals—a framework for numerical evolution problems. In: Morton KW, Baines MJ (eds) Numerical methods for fluid dynamics. Academic Press, San Diego, pp 219–257
Roe PL (1987) Linear advection schemes on triangular meshes. Cranfield Institute of Technology, Report 8720
Roe PL (1994–1995) Multidimensional upwinding. Motivation and concepts. VKI lecture series
Deconinck H, Paillère H, Struijs R, Roe PL (1993) Multidimensional upwind schemes based on fluctuation-splitting for systems of conservation laws. Comput Mech 11:323–340
Paillère H, Boxho J, Degrez G, Deconinck H (1996) Multidimensional upwind residual distribution schemes for the convection-diffusion equation. Int J Numer Methods Fluids 23:923–936
Issman E, Degrez G, Deconinck H (1996) Implicit upwind residual-distribution Euler and Navier-Stokes solver on unstructured meshes. AIAA J 34(10):2021–2028
Hubbard ME, Roe PL (2000) Compact high-resolution algorithms for time-dependent advection on unstructured grids. Int J Numer Methods Fluids 33(5):711–736
Deconinck H, Sermeus K, Abgrall R (2000) Status of multidimensional upwind residual distribution schemes and applications in aeronautics. AIAA paper 2000–2328
Abgrall R (2001) Toward the ultimate conservative scheme: following the quest. J Comput Phys 167:277–315
Abgrall R, Roe PL (2003) Construction of very high order fluctuation schemes. J Sci Comput 19:3–36
Abgrall R, Mezine M (2003) Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems. J Comput Phys 188:16–55
Abgrall R, Mezine M (2004) Construction of second order accurate monotone and stable residual distribution schemes for steady problems. J Comput Phys 195:474–507
Abgrall R, Barth TJ (2002) Weighted residual distribution schemes for conservation laws via adaptive quadrature. SIAM J Sci Comput 24:732–769
Abgrall R (2006) Essentially non-oscillatory residual distribution schemes for hyperbolic problems. J Comput Phys 214:773–808
Abgrall R, Roe PL (2003) High order fluctuation schemes on triangular meshes. J Sci Comput 19:3–36
Abgrall R, Adrianov N, Mezine M (2005) Towards very high-order accurate schemes for unsteady convection problems on unstructured meshes. Int J Numer Methods Fluids 47:679–691
Ricchiuto M, Csìk Á, Deconinck H (2005) Residual distribution for general time dependent conservation laws. J Comput Phys 209:249–289 2005
De Palma P, Pascazio G, Rubino DT, Napolitano M (2006) Multidimensional upwind residual distribution schemes for the convection-diffusion equation. J Comput Phys 218:159–199
Abgrall R, Marpeau F (2007) Residual distribution schemes on quadrilateral meshes. J Sci Comput 30:131–175
Cueto-Felgueroso L, Colominas I, Fe J, Navarrina F, Casteleiro M (2006) High order finite volume schemes on unstructured grids using moving least squares reconstruction. Application to shallow waters dynamics. Int J Numer Methods Eng 65:295–331
Cueto-Felgueroso L, Colominas I, Nogueira X, Navarrina F, Casteleiro M (2007) Finite volume solvers and moving least-squares approximations for the compressible Navier-Stokes equations on unstructured grids. Comput Methods Appl Mech Eng 196:4712–4736
Lancaster P, Salkauskas K (1981) Surfaces generated by moving least squares methods. Math Comput 37(155):141–158
Liu WK, Li S, Belytschko T (1997) Moving least square reproducing kernel method part I: methodology and convergence. Comput Methods Appl Mech Eng 143:113–154
Liu WK, Hao W, Chen Y, Jun S, Gosz J (1997) Multiresolution reproducing kernel particle methods. Comput Mech 20:295–309
Nogueira X, Cueto-Felgueroso L, Colominas I, Gómez H, Navarrina F, Casteleiro M (2009) On the accuracy of finite volume and discontinuous Galerkin discretizations for compressible flow on unstructured grids. Int J Numer Methods Eng 78:1553–1584
Cueto-Felgueroso L, Colominas I (2008) High-order finite volume methods and multiresolution reproducing kernels. Arch Comput Methods Eng 15(2):185–228
Nogueira X, Cueto-Felgueroso L, Colominas I, Navarrina F, Casteleiro M (2010) A new shock-capturing technique based on moving least squares for higher-order numerical schemes on unstructured grids. Comput Methods Appl Mech Eng 199(37–40):2544–2558
George WK (2005) Lectures in turbulence for the 21st century. Department of Thermo and Fluid Engineering, Chalmers University of Technology, Göteborg, Sweden
Sagaut P (2005) Large eddy simulation for incompressible flows. An introduction, 3rd edn. Springer, Berlin
Garnier E, Adams N, Sagaut P (2009) Large-eddy simulation for compressible flows. Scientific computation series. Springer, Berlin
Pope SB (2000) Turbulent flows. Cambridge University Press, Cambridge
Wilcox DC (1994) Turbulence modelling for CFD. DCW industries
Richardson LF (1922) Weather prediction by numerical process. Cambridge University Press, Cambridge (republished by Dover in 1965)
Kolmogorov AN (1941) The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl Akad Nauk SSSR 30:301–305 (in Russian), translated to English in Kolmogorov AN (1991) in The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Proceedings of the Royal Society of London, Series A: mathematical and physical sciences 434(1890):9–13
Davidson PA (2004) Turbulence. An introduction for scientist and engineers. Oxford University Press, London
Orszag SA, Patterson GS (1972) Numerical simulation of turbulence. In: Lecture notes in physics, vol 12. Springer, London, pp 127–147
Kolmogorov AN (1942) Equations of turbulent motion of an incompressible flow. Izv Akad Nauk Uzbekskoi SSR Ser Fiziko-Mat Nauk 6:56–58 (in Russian), A translation is found in Spalding, DB, Kolmogorov’s two-equation model of turbulence, Proc Math Physical Sci, 434(1890):211–216. Turbulence and stochastic process: Kolmogorov’s ideas 50 years on 1991
Launder BE, Sharma BI (1974) Application of the energy dissipation model of turbulence to the calculation of flow near a spinning disc. Lett Heat Mass Transf 1:131–138
Spalding DB (1991) Kolmogorov’s two-equation model of turbulence. Proc Math Phys Sci 434(1890):211–216. Turbulence and stochastic process: Kolmogorov’s ideas 50 years
Menter FR (1993) Zonal two equation k−ω turbulence models for aerodynamic flows. AIAA paper 93-2906
Lien FS, Durbin PA (1996) Non-linear \(k-\epsilon-\overline{\nu^{2}}\) modeling with application to high-lift. CTR summer proceedings
Barenblatt GI (1996) Scaling, self-similarity, and intermediate asymptotics. Cambridge University Press, Cambridge
Yakhot V, Orszag SA (1986) Renormalization group analysis of turbulence. J Sci Comput 1(1):3–51
Liou WW (1991) Modeling of compressible turbulent shear flows. NASA technical report 19920014097
Lele SK (1994) Compressibility effect on turbulence. Annu Rev Fluid Mech 26:211–254
Lele SK (1993) Notes on the effect of compressibility on turbulence. Center for turbulence research manuscripts, 145, Standford University
Batchelor GK (1953) The theory of homogeneous turbulence. Cambridge University Press, Cambridge
Kovasznay LSG (1953) Turbulence in supersonic flow. J Aeronaut Sci 20(10):657–682. Reprinted in the AIAA J Spec Suppl: Centennial of powered flight: a retrospective of aerospace research, GM Faeth, Library of flight series, vol 41, 2003
Pope SB (1975) A more general effective-viscosity hypothesis. J Fluid Mech 72:331–340
Yoshizawa A (1984) Statistical analysis of the derivation of the Reynolds stress from its eddy-viscosity representation. Phys Fluids 27:1377–1387
Rubinstein R, Barton JM (1990) Nonlinear Reynolds stress models and the renormalization group. Phys Fluids A 2:1472–1476
Deardorff JW (1970) A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers. J Fluid Mech 41(2):453–480
Ferziger JH (1976) Large eddy numerical simulations of turbulent flows. AIAA paper 76-347. San Diego, CA
Wagner GJ, Liu WK (2000) Turbulence simulation and multiple scale subgrid models. Comput Mech 25:117–136
Haselbacher A, Vasilyev OV (2003) Commutative discrete filtering on unstructured grids based on least-squares techniques. J Comput Phys 187:197–211
Marsden AL, Vasilyev OV, Moin P (2002) Construction of commutative filters for LES on unstructured meshes. J Comput Phys 175:584–602
Stolz S (2005) High-pass filtered eddy-viscosity models for large-eddy simulations of compressible wall-bounded flows. J Fluids Eng 127:666–673
Vreman AW (2003) The filtering analog of the variational multiscale method in large-eddy simulation. Phys Fluids 15(8):L61–L64
Vreman B, Geurts B, Kuerten H (1995) A priori tests of large Eddy simulation of the compressible plane mixing layer. J Eng Math 29(4):199–327
Smagorinsky J (1963) General circulation experiments with the primitive equations. Mon Weather Rev 91(3):99–164
Germano M, Piomelli U, Moin P, Cabot WH (1991) A dynamic subgrid-scale eddy viscosity model. Phys Fluids A 3(7):1760–1765
Moin P, Squires K, Cabot WH, Lee S (1991) A dynamic subgrid-scale model for compressible turbulence and scalar transport. Phys Fluids A 3(11):2746–2757
Stolz S, Adams NA, Kleiser L (1999) Analysis of sub-grid scales and sub-grid scale modeling for shock-boundary-layer interaction. In: Banerjee S, Eaton J (eds) Turbulence and Shear Flow I. Begell House, New York, pp 881–886
Hughes TJR, Mazzei L, Jansen KE (2000) Large-eddy simulation and the variational multiscale method. Comput Vis Sci 3:47–59
Hughes TJR, Mazzei L, Oberai AA, Wray AA (2001) The multiscale formulation of large-eddy simulation: decay of homogeneous isotropic turbulence. Phys Fluids 13:505–512
Hughes TJR, Oberai AA, Mazzei L (2001) Large-eddy simulation of turbulent channel flows by the variational multiscale method. Phys Fluids 13:1784–1799
Hughes TJR, Sangalli G (2007) Variational multiscale analysis: the fine-scale Green’s function, projection, optimization, localization and stabilized methods. SIAM J Numer Anal 45:539–557
Collis SS (2001) Monitoring unresolved scales in multiscale turbulence modeling. Phys Fluids 13(6):1800–1806
Bazilevs Y, Calo VM, Cottrell JA, Hughes TJR, Reali A, Scovazzi G (2007) Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng 197:173–201
Voke PR (1990) Multiple mesh simulation of turbulent flow. Technical report QMW EP-1082. Queen Mary and Westfield College, University of London, UK
Terracol M, Sagaut P, Basdevan C (2001) A multilevel algorithm for large-eddy simulation of turbulent compressible flows. J Comput Phys 167(2):439–474
Sagaut P, Labourasse E, Quémére P, Terracol M (2000) Multiscale approaches for unsteady simulation of turbulent flows. Int J Nonlinear Sci Numer Simul 1(4):285–298
Stolz S, Adams NA (1999) An approximate deconvolution procedure for large-eddy simulation. Phys Fluids 11:1699–1701
Stolz S, Schlatter P, Meyer D, Kleiser L (2003) High-pass filtered eddy-viscosity models for LES. In: Friedrich VR, Geurts BJ, Métais O (eds) Direct and large-eddy simulation. Kluwer Academic, Dordrecht, pp 81–88
Mathew J (2003) An explicit filtering method for LES of compressible flows. Phys Fluids 15:2279–2289
Beaudan P, Moin P (1994) Numerical experiments on the flow past a circular cylinder at sub-critical Reynolds numbers. Dept. of Mechanical Engineering, Rept. TF-62. Stanford University, Stanford, CA
Mittal R, Moin P (1997) Suitability of upwind-biased finite difference schemes for large-eddy simulation of turbulent flows. AIAA J 35(8):1415–1417
Nogueira X, Cueto-Felgueroso L, Colominas Gómez H (2010) Implicit large eddy simulation of non-wall-bounded turbulent flows based on the multiscale properties of a high-order finite volume method. Comput Methods Appl Mech Eng 199:315–624
Boris JP, Grinstein FF, Oran ES, Kolbe RJ (1992) New insights into large eddy simulation. Fluid Dyn Res 10:199–228
Oran ES, Boris JP (1993) Computing turbulent shear flows—a convenient conspiracy. Comput Phys 7:523–533
Porter DH, Pouquet A, Woodward PR (1994) Kolmogorv-like spectra in decaying three-dimensional supersonic flows. Phys Fluids 6:2133–2142
Margolin LG, Smolarkiewicz PK, Sorbjan Z (1999) Large eddy simulations of convective boundary layers using nonoscillatory differencing. Physica D 133:390–397
Grinstein FF, Fureby C (2002) Recent progress on MILES for high Reynolds number flows. J Fluids Eng 124:848–861
Margolin LG, Rider WJ (2002) A rationale for implicit turbulence modelling. Int J Numer Methods Fluids 39:821–841
Porter DH, Pouquet A, Woodward PR (1992) A numerical study of supersonic turbulence. Theor Comput Fluid Dyn 4:13–49
Johnsen E et al. (2010) Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves. J Comput Phys 229(4):1213–1237
Maaloum A, Kouidri S, Rey R (2004) Aeroacoustic performances evaluation of axial fans based on the unsteady pressure field on the blades surface. Appl Acoust 65:367–384
Moon Young J, Cho Y, Nam H-S (2003) Computation of unsteady viscous flow and aeroacoustic noise of cross flow fans. Comput Fluids 32:995–1015
Khelladi S, Kouidri S, Bakir F, Rey R (2008) Predicting tonal noise from a high speed vaned centrifugal fan. J Sound Vib 313(1–2):113–133
Farassat F, Myers MK (1988) Extension of Kirchhoff’s formula to radiation from moving surfaces. J Sound Vib 123:451–560
Colonius T, Lele SK (2004) Computational aeroacoustics: progress on nonlinear problems of sound generation. Prog Aerosp Sci 40:345–416
Bogey C, Bailly C, Juvé D (2002) Computation of flow noise using source terms in linearized Euler’s equations. AIAA J 40(2):235–243
Williams JE, Hawkings DL (1969) Sound generation by turbulence and surfaces in arbitrary motion. Phil Trans R Soc Lond A 264(1151):321–342, doi:10.1098/rsta.1969.0031
Moon Young J, Seo Jung H (2006) Linearized perturbed compressible equations for low Mach number aeroacoustics. J Comput Phys 218(2):702–719
Lynam EJ, Webb HA (1919) The emission of sound by airscrews. R. & M., No 624
Bryan GH (1920) The acoustics of moving sources with application to airscrews. R. & M., No 684, British A.R.C.
Gutin L (1936) On the sound field of a rotating propeller NACA TM1195 (Traduction de Über das Schallfeld einer rotierenden Luftschraube. Phys Z Sowjetunion 9(1):57–71
Lighthill MJ (1952) On sound generated aerodynamically, I. General theory. Proc R Soc Lond Ser A 211:564–587
Lighthill MJ (1954) On sound generated aerodynamically, II. Turbulence as a source of sound. Proc R Soc A 222(1148):1–32, doi:10.1098/rspa.1954.0049
Curle N (1955) The influence of solid boundaries upon aerodynamic sound. Proc R Soc Lond Ser A 231:505–514
Harten A, Lax P, Van Leer B (1983) On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev 25:35–61
Jahawar P, Kamath H (2000) A high-resolution procedure for Euler and Navier-Stokes computations on unstructured grids. J Comput Phys 164:165–203
Barth TJ (1993) Recent developments in high order k-exact reconstruction on unstructured meshes. AIAA paper 93-0068
Ollivier-Gooch CF, Nejat A, Michalak K (2007) On obtaining high-order finite volume solutions to the Euler equations on unstructured meshes. In: 18th AIAA computational fluid dynamics conference. AIAA, Washington
Dumbser M, Balsara DW, Toro EF, Munz CD (2008) A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes. J Comput Phys 227:8209–8253
Dumbser M (2010) Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations. Comput Fluids 39(1):60–76
Nogueira X, Cueto-Felgueroso L, Colominas, Khelladi S. (2010) On the simulation of wave propagation with a higher-order finite volume scheme based on reproducing kernel methods. Comput Methods Appl Mech Eng 199:1471–1490
Khelladi S, Martin S, Nogueira X, Bakir F (2010) Higher-order preserving methods for unsteady finite volume solvers based on reproducing kernels: application to aeroacoustic problems. In: 16th AIAA/CEAS, Aeroacoustics conference, Stockholm, Sweden, AIAA paper 2010-3817
Venkatakrishnan V, Mavriplis D (1996) Implicit method for the computation of unsteady flows on unstructured grids. J Comput Phys 127:380–397
Bailly C, Bogey C (2006) An overview of numerical methods for acoustic wave propagation. In: Wesseling P, Oñate E, Périaux J (eds) European conference on computational fluid dynamics, ECCOMAS CFD
Hardin JC, Ristorcelli JR, Tam CKW (1995) ICASE/LaRC workshop on benchmark problems in computational aeroacoustics. NASA conference publication, vol 3300
Viswanathan K, Sankar LN (1995) A comparative study of upwind and MacCormack schemes for CAA benchmark problems. In: ICASE/LaRC workshop on benchmark problems in computational aeroacoustics, pp 185–195
Tam CKW, Shen H (1993) Direct computation of nonlinear acoustic pulses using high order finite difference schemes. AIAA paper 93-4325
Ducros F, Ferrand V, Nicoud F, Weber C, Darracq D, Gacherieu C, Poinsot T (1999) Large-eddy simulation of the shock/turbulence interaction. J Comput Phys 152:517–549
Pirozzoli S (2002) Conservative hybrid compact-WENO schemes for shock-turbulence interaction. J Comput Phys 178:81–117
Harten A (1978) The artificial compression method for computation of shocks and contact discontinuities. III. Self adjusting hybrid schemes. Math Comput 32:363–389
Adams NA, Shariff K (1996) A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems. J Comput Phys 127:27–51
Sjögreen B, Yee HC (2004) Multiresolution wavelet based adaptive numerical dissipation control for high order methods. J Sci Comput 20:211–255
Spyropoulos ET, Blaisdell GA (1996) Evaluation of the dynamic model for simulations of compressible decaying isotropic turbulence. AIAA J 34(5):990–998
Sarkar S, Erlebacher G, Hussaini MY, Kreiss HO (1991) The analysis and modelling of dilatational terms in compressible turbulence. J Fluid Mech 227:473–493
Bataille F (1994) Etude d’une turbulence faiblement compressible dans le cadre d’une modelisation Quasi-Normale avec Amortissement Tourbillonaire. Thèse Ecole Central de Lyon
Hussaini MY (1998) On large-eddy simulation of compressible flows. AIAA 29th fluid dynamics conference, Albuquerque, New Mexico, Paper AIAA 98-2802
Colonius T, Lele SK, Moin P (1993) Boundary conditions for direct computation of aerodynamic sound generation. AIAA J 31(9):1574–1582
Hu FQ (1996) On absorbing boundary conditions for linearized Euler equations by a perfectly matched layer. J Comput Phys 129:201–219
Tam CKW, Webb JC (1993) Dispersion-relation-preserving finite difference schemes for computational aeroacoustics. J Comput Phys 107:262–281
Bernacki M, Lanteri S, Piperno S (2006) Time-domain parallel simulation of heterogeneous wave propagation on unstructured grids using explicit, non-diffusive, discontinuous Galerkin methods. J Comput Acoust 14(1):57–82
Nogueira X, Cueto-Felgueroso L, Colominas I, Khelladi S, Navarrina F, Casteleiro M (2010) Resolution of computational aeroacoustics problem on unstructured grids with high-order finite volume scheme. J Comput Appl Math 234(7):2089–2097
Bailly C, Juvé D (2000) Numerical solution of acoustic propagation problems using linearized Euler equations. AIAA J 38(1):22–29
Khelladi S, Kouidri S, Bakir F, Rey R (2005) Flow study in the impeller-diffuser interface of a vaned centrifugal fan. ASME J Fluids Eng 127:495–502