## A variational multiscale higher-order finite element formulation for turbomachinery flow computations.(English)Zbl 1093.76032

Summary: The variational multiscale approach for finite elements addresses the inclusion of the effect of fine scales of the solution in the coarse problem. In this framework advective-diffusive-reactive equations modelling turbomachinery flows feature both advection and reaction induced instabilities to be tackled. To this end, this work deals with a new method called V-SGS (Variable-SubGrid Scale), designed for quadratic elements, with a variable ‘intrinsic time’ parameter. Two-dimensional tests have been considered to compare V-SGS against SUPG and other stabilization devices, including the flow around a NACA 4412 airfoil to assess its reliability in the handling of advanced turbulence closures on cruder meshes.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76U05 General theory of rotating fluids 76F60 $$k$$-$$\varepsilon$$ modeling in turbulence

### Keywords:

turbulence closure model; advection; reaction; NACA 4412 airfoil
Full Text:

### References:

 [1] Borello, D.; Corsini, A.; Rispoli, F., A finite element overlapping scheme for turbomachinery flows on parallel platforms, Comput. & fluids, 32, 1017-1047, (2003) · Zbl 1137.76410 [2] Brezzi, F.; Franca, L.P.; Hughes, T.J.R.; Russo, A., $$b = \int g$$, Comput. methods appl. mech. engrg., 145, 329-339, (1997) · Zbl 0904.76041 [3] Brezzi, F.; Marini, D.; Russo, A., Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion problems, Comput. methods appl. mech. engrg., 166, 51-63, (1998) · Zbl 0932.65113 [4] Codina, R., A stabilized finite element method for generalized stationary incompressible flows, Comput. methods appl. mech. engrg., 190, 2681-2706, (2001) · Zbl 0996.76045 [5] Codina, R.; Oñate, E.; Cervera, M., The intrinsic time for the streamline upwind/Petrov-Galerkin formulation using quadratic elements, Comput. methods appl. mech. engrg., 94, 239-262, (1992) · Zbl 0748.76082 [6] Coles, D.; Wadcock, A.J., Flying hot-wire study of flow past a NACA 4412 airfoil at maximum lift, Aiaa j., 17, 321-329, (1979) [7] A. Corsini, F. Rispoli, A. Santoriello, A new stabilized finite element method for advection-diffusion-reaction equations using quadratic elements, in: T. Lajos, J. Vad (Ed.), CMFF’03 Conference Proceedings, vol.2, Department of Fluid Mechanics, Budapest University of Technology and Economics, 2003, pp. 791-799. · Zbl 1112.76385 [8] Durbin, P.A., Separated flow computations with the κ-ε-v2 model, Aiaa j., 33, 659-664, (1995) [9] Franca, L.P.; Farhat, C., Bubble functions prompt unusual stabilized finite element methods, Comput. methods appl. mech. engrg., 123, 299-308, (1995) · Zbl 1067.76567 [10] Franca, L.P.; Valentin, F., On an improved unusual stabilized finite element method for the advective-reactive-diffusive equation, Comput. methods appl. mech. engrg., 190, 1785-1800, (2001) · Zbl 0976.76038 [11] Guilmineau, E.; Piquet, J.; Queutey, P., Two-dimensional turbulent viscous flow simulation past airfoils at fixed incidence, Comput. & fluids, 26, 135-162, (1997) · Zbl 0893.76055 [12] Hanjalic, K.; Hadzic, I.; Jakirlic, S., Modeling turbulent wall flows subjected to strong pressure variations, J. fluids engrg., 121, 57-64, (1999) [13] Harari, I.; Hughes, T.J.R., Stabilized finite element methods for steady advection-diffusion with production, Comput. methods appl. mech. engrg., 115, 165-191, (1994) [14] Hauke, G., A simple subgrid scale stabilized method for the advection-diffusion-reaction equation, Comput. methods appl. mech. engrg., 191, 2925-2947, (2001) · Zbl 1005.76057 [15] Hauke, G.; Garcia-Olivares, A., Variational subgrid scale formulations for the advection-diffusion-reaction equation, Comput. methods appl. mech. engrg., 190, 6847-6865, (2001) · Zbl 0996.76074 [16] Hughes, T.J.R., Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origin of stabilized methods, Comput. methods appl. mech. engrg., 127, 387-401, (1995) · Zbl 0866.76044 [17] Hughes, T.J.R.; Brooks, A.N., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041 [18] Hughes, T.J.R.; Brooks, A.N., A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions: application to the streamline-upwind procedure, Finite elem. fluids, 4, 47-65, (1982) [19] Hughes, T.J.R.; Feijòo, G.R.; Mazzei, L.; Quincy, J.-B., The variational multiscale method—a paradigm for computational mechanics, Comput. methods appl. mech. engrg., 166, 3-24, (1998) · Zbl 1017.65525 [20] Idelsohn, S.; Nigro, N.; Storti, M.; Buscaglia, G., A Petrov-Galerkin formulation for advection-reaction-diffusion problems, Comput. methods appl. mech. engrg., 136, 27-46, (1996) · Zbl 0896.76042 [21] Kythe, P.K.; Puri, P.; Schaferkotter, M.R., Partial differential equations and boundary value problems with Mathematica, (2003), CRC press · Zbl 1037.35002 [22] Leonard, B.P., A survey of finite differences of opinion on numerical muddling of the incomprehensible defective confusion equation, () · Zbl 0435.76003 [23] F.S. Lien, G. Kalitzin, P.A. Durbin, RANS modeling for compressible and transitional flows, in: Proceedings of the Summer Program 1998, Center for Turbulence Research-Stanford University, 1998, pp. 267-286. [24] Manceau, R.; Hanjalic, K., Elliptic blending model: A near-wall Reynolds-stress turbulence closure, Phys. fluids, 14-3, 1-11, (2001) · Zbl 1184.76343 [25] Masud, A.; Khurram, R.A., A multiscale/stabilized finite element method for the advection-diffusion equation, Comput. methods appl. mech. engrg., 193, 1997-2018, (2004) · Zbl 1067.76570 [26] Oberai, A.A.; Pinski, P.M., A multiscale finite element method for the helmoltz equation, Comput. methods appl. mech. engrg., 154, 281-297, (1998) · Zbl 0937.65119 [27] Roach, G.F., Green’s functions—introductory theory with applications, (1970), VNR London · Zbl 0186.47104 [28] Saad, Y., A flexible inner-outer preconditioned gmres algorithm, SIAM J. sci. statistical comput., 14, 461-469, (1993) · Zbl 0780.65022 [29] Sani, R.L.; Gresho, P.M.; Lee, R.L.; Griffiths, D.F., The cause and cure (?) of spurious pressures generated by certain FEM solutions of the incompressible Navier-Stokes equations: part 1, Int. J. num. meth. fluids, 1, 17-43, (1981) · Zbl 0461.76021 [30] Tezduyar, T.E.; Mittal, S.; Ray, S.E.; Shih, R., Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements, Comput. methods appl. mech. engrg., 95, 221-242, (1992) · Zbl 0756.76048 [31] Tezduyar, T.E.; Park, Y.J., Discontinuity capturing finite element formulations for nonlinear convection-diffusion-reaction problems, Comput. methods appl. mech. engrg., 59, 307-325, (1986) · Zbl 0593.76096
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.