# zbMATH — the first resource for mathematics

One equation model for turbulent channel flow with second order viscoelastic corrections. (English) Zbl 1257.76032
Summary: A modified second order viscoelastic constitutive equation is used to derive a $$k-l$$ type turbulence closure to qualitatively assess the effects of elastic stresses on fully-developed channel flow. Specifically, the second order correction to the Newtonian constitutive equation gives rise to a new term in the momentum equation involving the time-averaged elastic shear stress and in the turbulent kinetic energy transport equation quantifying the interaction between the fluctuating elastic stress and rate of strain tensors, denoted by $$P_w$$, for which a closure is developed and tested. This closure is based on arguments of isotropic turbulence and equilibrium in boundary layer flows and a priori $$P_w$$ could be either positive or negative. When $$P_w$$ is positive, it acts to reduce the production of turbulent kinetic energy and the turbulence model predictions qualitatively agree with direct numerical simulation (DNS) results obtained for more realistic viscoelastic fluid models with memory which exhibit drag reduction. In contrast, $$P_w < 0$$ leads to a drag increase and numerical breakdown of the model occurs at very low values of the Deborah number, which signifies the ratio of elastic to viscous stresses. Limitations of the turbulence model primarily stem from the inadequacy of the $$k-l$$ formulation rather than from the closure for $$P_w$$. An alternative closure for $$P_w$$, mimicking the viscoelastic stress work predicted by DNS using the finitely extensible nonlinear elastic-peterlin fluid model, which is mostly characterized by $$P_w>0$$ but has also a small region of negative $$P_w$$ in the buffer layer, was also successfully tested. This second model for $$P_w$$ leads to predictions of drag reduction, in spite of the enhancement of turbulence production very close to the wall, but the equilibrium conditions in the inertial sub-layer were not strictly maintained.

##### MSC:
 76F60 $$k$$-$$\varepsilon$$ modeling in turbulence 76A10 Viscoelastic fluids
Full Text:
##### References:
 [1] Warholic, M.D., Heist, D.K., Katcher, M., Hanratty, T.J.: A study with particle-image velocimetry of the influence of drag-reducing polymers on the structure of turbulence. Exp. Fluids 31, 474–483 (2001) · doi:10.1007/s003480100288 [2] Warholic, M.D., Massah, H., Hanratty, T.J.: Influence of drag reducing polymers on turbulence: effects of Reynolds number, concentration and mixing. Exp. Fluids 27, 461–472 (1999) · doi:10.1007/s003480050371 [3] Ptasinski, P.K.M., Nieuwstadt, F.T., Brule, B.H.A.A.V.D., Hulsen, M.A.: Experiments in turbulent pipe flow with polymer additives at maximum drag reduction. Flow Turbul. Combust 66, 159–182 (2001) · Zbl 1094.76506 · doi:10.1023/A:1017985826227 [4] Escudier, M.P., Presti, F., Smith, S.: Drag reduction in the turbulent pipe flow of polymers. J. Non-Newton. Fluid Mech 81, 197–213 (1999) · Zbl 0948.76523 · doi:10.1016/S0377-0257(98)00098-6 [5] Ptasinski, P.K., Boersma, B.J., Nieuwstadt, F.T.M., Hulsen, M.A., den Brule, B.H.A.A.V., Hunt, J.C.R.: Turbulent channel flow near maximum drag reduction: simulations, experiments and mechanisms. J. Fluid Mech 490, 251–291 (2003) · Zbl 1063.76580 · doi:10.1017/S0022112003005305 [6] 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-Newton. Fluid Mech 79, 433–468 (1998) · Zbl 0960.76057 · doi:10.1016/S0377-0257(98)00115-3 [7] Dimitropoulos, C.D., Sureshkumar, R., Beris, A.N., Handler, R.A.: Budgets of Reynolds stress, kinetic energy and streamwise enstrophy in viscoelastic turbulent channel flow. Phys. Fluids 13, 1016–1027 (2001) · Zbl 1184.76137 · doi:10.1063/1.1345882 [8] 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, 11705 (2005) · Zbl 1187.76127 · doi:10.1063/1.1829751 [9] Housiadas, K.D., Beris, A.N.: An efficient fully implicit spectral scheme for DNS of turbulent viscoelastic channel flow. J. Non-Newton. Fluid Mech 122, 243–262 (2004) · Zbl 1143.76330 · doi:10.1016/j.jnnfm.2004.07.001 [10] Angelis, E.D., Casciola, C.M., Piva, R.: DNS of wall turbulence: dilute polymers and self-sustaining mechanisms. Comput. Fluids 31, 495–507 (2002) · Zbl 1075.76556 · doi:10.1016/S0045-7930(01)00069-X [11] Kim, K., Li, C.-F., Sureshkumar, R., Balachandar, S., Adrian, R.: Effects of polymer stresses on eddy structures in drag-reduced turbulent channel flow. J. Fluid Mech 584, 281–299 (2007) · Zbl 1175.76069 · doi:10.1017/S0022112007006611 [12] Li, C.-F., Gupta, V.K., Sureshkumar, R., Khomami, B.: Turbulent channel flow of dilute polymeric solutions: drag reduction scaling and an eddy viscosity model. J. Non-Newton. Fluid Mech 139, 177–189 (2006) · Zbl 1195.76034 · doi:10.1016/j.jnnfm.2006.04.012 [13] Shaqfeh, E.S., Iaccarini, G., Shi, M.: A RANS model for turbulent drag reduction by polymer injection and comparison to DNS. Paper FM2, The Society of Rheology 78th Annual Meeting, October 8–12, 2006, Portland, Maine, USA (2006) [14] Cruz, D.O.A., Pinho, F.T., Resende, P.R.: Modeling the new stress for improved drag reduction predictions of viscoelastic pipe flow. J. Non-Newton. Fluid Mech 121, 127–141 (2004) · Zbl 1115.76303 · doi:10.1016/j.jnnfm.2004.05.004 [15] Resende, P.R., Escudier, M.P., Presti, F., Pinho, F.T., Cruz, D.O.A.: Numerical predictions and measurements of Reynolds normal stresses in turbulent pipe flow of polymers. Int. J. Heat Fluid Flow 27, 204–219 (2006) · doi:10.1016/j.ijheatfluidflow.2005.08.002 [16] Resende, P.R., Pinho, F.T., Cruz, D.O.A.: Performance of the k–e and Reynolds stress models in turbulent flows with viscoelastic fluids. Proceedings of the 11th Brazilian Congress of Thermal Sciences and Engineering–ENCIT 2006, Curitiba, Brazil, Dec 5–8 2006, paper CIT06-0805.pdf (2006) [17] Elata, C., Poreh, M.: Momentum transfer in turbulent shear flow of an elastico-viscous fluid. Rheol. Acta 5, (2), 148–151 (1966) · doi:10.1007/BF01968496 [18] Roy, A., Morozov, A., van Saarloos, W., Larson, R.G.: Mechanism of polymer drag reduction using a low-dimensional model. Phys. Rev. Lett 97, 234501 (2006) · doi:10.1103/PhysRevLett.97.234501 [19] Tanner, R.I.: Engineering Rheology. Clarendon, Oxford (1985) · Zbl 1171.76300 [20] Bird, R.B., Armstrong, R.C., Hassager, O.: Dynamics of Polymeric Liquids. Volume 1: Fluid Mechanics. Wiley, New York (1987) [21] Prandtl, L.: Über ein neues Formelsystem für die ausgebildete Turbulenz, pp. 6–19. Mathematik–Physik Klasse, Nachrichten Akademisches Wissenschaft, Göttingen (1945) · Zbl 0061.45401 [22] Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge, UK (2000) · Zbl 0966.76002 [23] Mathieu, J., Scott, J.: An Introduction to Turbulent Flow, pp. 267–268. Cambridge University Press, Cambridge, UK (2000) · Zbl 0955.76001 [24] Hoyt, J.W.: The effect of additives on fluid friction. J. Basic Eng 94, 258–285 (1972) · doi:10.1115/1.3425401 [25] Patel, V.C., Rodi, W., Scheuerer, G.: Turbulence models for near-wall and low Reynolds number flows: a review. AIAA J 23, 1308–1319 (1984) · doi:10.2514/3.9086 [26] Van Driest, R.E.: On turbulent flow near a wall. J. Aeronaut. Sci 23, 1007–1011 (1956) · Zbl 0073.20802 · doi:10.2514/8.3713 [27] Younis, B.A.: A computer program for two-dimensional turbulent boundary layer flows. Internal report, Department of Civil Engineering, City University, London, UK (1987) [28] Patankar, S.V., Spalding, D.B.: A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. Int. J. Heat Mass Transfer 15, 1787–1806 (1972) · Zbl 0246.76080 · doi:10.1016/0017-9310(72)90054-3 [29] Roache, P.J.: Quantification of uncertainty in computational fluid mechanics. Annu. Rev. Fluid Mech 29, 123–160 (1997) · doi:10.1146/annurev.fluid.29.1.123 [30] Virk, P.S., Mickley, H.S., Smith, K.A.: The ultimate asymptote and mean flow structure in Toms Phenomena. J. Appl. Mech 92, 488–493 (1970) · doi:10.1115/1.3408532 [31] Mansour, N.N., Kim, J., Moin, P.: Reynolds stress and dissipation rate budgets in a turbulent channel flow. J. Fluid Mech 194, 15–44 (1988) · doi:10.1017/S0022112088002885
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.