A massively parallel hybrid scheme for direct numerical simulation of turbulent viscoelastic channel flow. (English) Zbl 1452.76075
Summary: This paper describes in detail a numerical scheme designed for direct numerical simulation (DNS) of turbulent drag reduction. The hybrid spatial scheme includes Fourier spectral accuracy in two directions and sixth-order compact finite differences for first and second-order wall-normal derivatives, while time marching can be up to fourth-order accurate. High-resolution and high-drag reduction viscoelastic DNS are made possible through domain decomposition with a two-dimensional MPI Cartesian grid alternatively splitting two directions of space (‘pencil’ decomposition). The resulting algorithm has been shown to scale properly up to 16384 cores on the Blue Gene/P at IDRIS-CNRS, France.
Drag reduction is modeled for the three-dimensional wall-bounded channel flow of a FENE-P dilute polymer solution which mimics injection of heavy-weight flexible polymers in a Newtonian solvent. We present results for four high-drag reduction viscoelastic flows with friction Reynolds numbers \(Re_{\tau 0} = 180, 395, 590\) and 1000, all of them sharing the same friction Weissenberg number \(We_{\tau 0} = 115\) and the same rheological parameters. A primary analysis of the DNS database indicates that turbulence modification by the presence of polymers is Reynolds-number dependent. This translates into a smaller percent drag reduction with increasing Reynolds number, from 64% at \(Re_{\tau 0} = 180\) down to 59% at \(Re_{\tau 0} = 1000\), and a steeper mean current at small Reynolds number. The Reynolds number dependence is also visible in second-order statistics and in the vortex structures visualized with iso-surfaces of the \(Q\)-criterion.
Drag reduction is modeled for the three-dimensional wall-bounded channel flow of a FENE-P dilute polymer solution which mimics injection of heavy-weight flexible polymers in a Newtonian solvent. We present results for four high-drag reduction viscoelastic flows with friction Reynolds numbers \(Re_{\tau 0} = 180, 395, 590\) and 1000, all of them sharing the same friction Weissenberg number \(We_{\tau 0} = 115\) and the same rheological parameters. A primary analysis of the DNS database indicates that turbulence modification by the presence of polymers is Reynolds-number dependent. This translates into a smaller percent drag reduction with increasing Reynolds number, from 64% at \(Re_{\tau 0} = 180\) down to 59% at \(Re_{\tau 0} = 1000\), and a steeper mean current at small Reynolds number. The Reynolds number dependence is also visible in second-order statistics and in the vortex structures visualized with iso-surfaces of the \(Q\)-criterion.
MSC:
| 76F65 | Direct numerical and large eddy simulation of turbulence |
| 76A10 | Viscoelastic fluids |
| 65Y05 | Parallel numerical computation |
| 82D60 | Statistical mechanics of polymers |
References:
| [1] | Toms, B. A., Some observations on the flow of linear polymer solutions through straight tubes at large Reynolds numbers, (Proc. of the 1st international congress of rheology, vol. 2 (1949), North Holland: North Holland Amsterdam), 135-141 |
| [2] | Sureshkumar, R.; Beris, A. N.; Handler, R. A., Direct numerical simulation of the turbulent channel flow of a polymer solution, Phys Fluids, 9, 743-755 (1997) |
| [3] | Dimitropoulos, C. D.; Sureshkumar, R.; Beris, A. N., Direct numerical simulation of viscoelastic turbulent channel flow exhibiting drag reduction: effect of the variation of rheological parameters, J Non-Newtonian Fluid Mech, 79, 433-468 (1998) · Zbl 0960.76057 |
| [4] | Dimitropoulos CD, Sureshkumar R, Beris AN, Handler RA. Budgets of Reynolds stress, kinetic energy and streamwise enstrophy in viscoelastic turbulent channel flow. Phys Fluids 2001;13(4):1016-24.; Dimitropoulos CD, Sureshkumar R, Beris AN, Handler RA. Budgets of Reynolds stress, kinetic energy and streamwise enstrophy in viscoelastic turbulent channel flow. Phys Fluids 2001;13(4):1016-24. · Zbl 1184.76137 |
| [5] | Dimitropoulos, C. D.; Dubief, Y.; Shaqfeh, E. S.G.; Moin, P.; Lele, S. K., Direct numerical simulation of polymer-induced drag reduction in turbulent boundary layer flow, Phys Fluids, 17 (2005), 011705-1-011705-4 · Zbl 1187.76127 |
| [6] | Dimitropoulos, C. D.; Dubief, Y.; Shaqfeh, E. S.G.; Moin, P., Direct numerical simulation of polymer-induced drag reduction in turbulent boundary layer flow of inhomogeneous polymer solutions, J Fluid Mech, 566, 153-162 (2006) · Zbl 1145.76027 |
| [7] | De Angelis, E.; Casciola, C. M.; L’vov, V. S.; Piva, R.; Procaccia, I., Drag reduction by polymers in turbulent channel flows: energy redistribution between invariant empirical modes, Phys Rev Let, 67 (2003), 056312-1-056312-11 |
| [8] | Min, T.; Yoo, J. Y.; Choi, H., Maximum drag reduction in a turbulent channel flow by polymer additives, J Fluid Mech, 492, 91-100 (2003) · Zbl 1063.76579 |
| [9] | Dubief, Y.; White, C. M.; Terrapon, V. E.; Shaqfeh, E. S.G.; Moin, P.; Lele, S. K., On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows, J Fluid Mech, 514, 271-280 (2004) · Zbl 1067.76052 |
| [10] | Housiadas, K. D.; Beris, A. N., Polymer-induced drag reduction: effects of the variations in elasticity and inertia in turbulent viscoelastic channel flow, Phys Fluids, 15, 2369-2384 (2003) · Zbl 1186.76235 |
| [11] | Housiadas, K. D.; Beris, A. N., An efficient fully implicit spectral scheme for DNS of turbulent viscoelastic channel flow, J Non-Newtonian Fluid Mech, 122, 243-262 (2004) · Zbl 1143.76330 |
| [12] | Housiadas, K. D.; Wang, L.; Beris, A. N., A new method preserving the positive definiteness of a second order tensor variable in flow simulations with application to viscoelastic turbulence, Comput Fluids, 39, 2, 225-241 (2010) · Zbl 1242.76220 |
| [13] | Pinho, F. T.; Sadanandan, B.; Sureshkumar, R., One equation model for turbulent channel flow with second order viscoelastic corrections, Flow Turbul Combust, 81, 3, 337-367 (2008) · Zbl 1257.76032 |
| [14] | Thais, L.; Tejada-Martı´nez, A. E.; Gatski, T. B.; Mompean, M., Temporal large eddy simulations of viscoelastic drag reduction flows, Phys Fluids, 22, 1, 013103.1-013103.13 (2010) · Zbl 1183.76516 |
| [15] | Jiménez, J.; Moin, P., The minimal flow unit in near-wall turbulence, J Fluid Mech, 225, 213-240 (1991) · Zbl 0721.76040 |
| [16] | Monin, A. S.; Yaglom, A. M., Statistical fluid mechanics: mechanics of turbulence (1975), The M.I.T. press: The M.I.T. press Cambridge (MA) |
| [17] | Housiadas, K. D.; Beris, A. N.; Handler, R. A., Viscoelastic effects on higher order statistics and on coherent structures in turbulent channel flow, Phys Fluids, 17, 035106-035125 (2005) · Zbl 1187.76219 |
| [18] | Sureshkumar, R.; Beris, A. N., Effect of artificial stress diffusivity on the stability of numerical calculations and the dynamics of time-dependent viscoelastic flows, J Non-Newtonian Fluid Mech, 60, 53-80 (1995) |
| [19] | Armfield, S. W.; Street, R. L., Fractional step methods for the Navier-Stokes equations on non-staggered grids, ANZIAM J, 42, E, C134-C156 (2000) · Zbl 1008.76058 |
| [20] | Guermond, J. L.; Minev, P.; Jie, S., An overview of projection methods for incompressible flows, Comput Methods Appl Mech Eng, 195, 44-47, 6011-6045 (2006) · Zbl 1122.76072 |
| [21] | Kundu, P. K.; Cohen, I. M., Fluid mechanics (2004), Elsevier Academic Press: Elsevier Academic Press San Diego (CA) |
| [22] | Oden, J. T.; Carey, G. F., Finite elements: mathematical aspects, vol. IV (1984), Prentice-Hall: Prentice-Hall Englewood Cliffs (NJ) · Zbl 0558.73064 |
| [23] | Tejada-Martı´nez, A. E.; Grosch, C. E., Langmuir turbulence in shallow water: part II. large eddy simulations, J Fluid Mech, 576, 63-108 (2007) · Zbl 1178.76207 |
| [24] | Tejada-Martı´nez, A. E.; Grosch, C. E.; Gatski, T. B., Temporal large-eddy simulation of unstratified and stratrified turbulent channel flows, Int J Heat Fluid Flow, 28, 1244-1261 (2007) |
| [25] | Tejada-Martı´nez, A. E.; Grosch, C. E.; Gargett, A. E.; Polton, J. A.; Smith, J. A.; MacKinnon, J. A., A hybrid spectral/finite-difference large-eddy simulator of turbulent processes in the upper ocean, Ocean Modell, 30, 115-142 (2009) |
| [26] | Polton, J. A.; Smith, J. A.; MacKinnon, J. A.; Tejada-Martfnez, A. E., Rapid generation of high frequency internal waves beneath a wind and wave forced oceanic surface mixed layer, Geophys Res Lett, 35, 13602 (2008) |
| [27] | Lamballais, E.; Metais, O.; Lesieur, M., Spectral-dynamic model for large-eddy simulations of turbulent rotating channel flow, Theor Comput Fluid Dyn, 121, 149-177 (1998) · Zbl 0941.76044 |
| [28] | Shih, T. M.; Tan, C. H.; Hwang, B. C., Effects of grid staggering on numerical schemes, Int J Numer Methods Fluids, 9, 193-212 (1989) · Zbl 0661.76024 |
| [29] | Garg, R. P.; Ferziger, J. H.; Monismith, S. G., Hybrid spectral finite difference simulations of stratified turbulent flows on distributed memory architectures, Int J Numer Meth Fluids, 24, 1129-1158 (1997) · Zbl 0886.76058 |
| [30] | Hirsh, R. S., Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J Comput Phys, 19, 90-109 (1975) · Zbl 0326.76024 |
| [31] | Adam, Y., Highly accurate compact implicit methods and boundary conditions, J Comput Phys, 24, 10-22 (1977) · Zbl 0357.65074 |
| [32] | Slinn, D. N.; Riley, J. J., A model for the simulation of turbulent boundary layers in an incompressible stratified flow, J Comput Phys, 144, 2, 550-602 (1998) · Zbl 0936.76027 |
| [33] | Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J Fluid Mech, 177, 133-166 (1987) · Zbl 0616.76071 |
| [34] | Lele, S. K., Compact finite difference schemes with spectral-like resolution, J Comput Phys, 103, 1, 16-42 (1992) · Zbl 0759.65006 |
| [35] | Carpenter, M. H.; Gottlieb, D.; Abarbanel, S., The stability of numerical boundary treatments for compact high-order finite-difference schemes, J Comput Phys, 108, 2, 272-295 (1993) · Zbl 0791.76052 |
| [36] | Canuto, C.; Hussaini, M. Y.; Quarteroni, A.; Zang, T. A., Spectral methods in fluid dynamics (1988), Springer: Springer New York · Zbl 0658.76001 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
