Elasto-inertial wall mode instabilities in viscoelastic plane Poiseuille flow.

*(English)*Zbl 1430.76037Summary: A linear stability analysis of plane Poiseuille flow of an upper-convected Maxwell (UCM) fluid, bounded between rigid plates separated by a distance \(2L\), has been carried out to investigate the interplay of elasticity and inertia on flow stability. The stability is governed by the following dimensionless groups: the Reynolds number \(Re=\rho U_{\max}L/\eta\) and the elasticity number \(E\equiv W/Re=\lambda\eta /(\rho L^2)\), where \(W=\lambda U_{\max}/L\) is the Weissenberg number. Here, \( \rho\) is the fluid density, \( \eta\) is the fluid viscosity, \( \lambda\) is the micro-structural relaxation time and \(U_{\max}\) is the maximum base-flow velocity. The stability is analysed for two-dimensional perturbations using both pseudo-spectral and shooting methods. We also analyse the linear stability of plane Couette flow which, along with the results for plane Poiseuille flow, yields insight into the structure of the complete elasto-inertial eigenspectrum. While the general features of the spectrum for both flows remain similar, plane Couette flow is found to be stable over the range of parameters examined \((Re\leqslant 10^4,E\leqslant 0.01)\). On the other hand, plane Poiseuille flow appears to be susceptible to an infinite hierarchy of elasto-inertial instabilities. Over the range of parameters examined, there are up to seven distinct neutral stability curves in the \(Re-k\) plane (here \(k\) is the perturbation wavenumber in the flow direction). Based on the symmetry of the eigenfunctions for the streamwise velocity about the centreline, four of these instabilities are antisymmetric, while the other three are symmetric. The neutral stability curve corresponding to the first antisymmetric mode is shown to be a continuation (to finite \(E)\) of the Tollmien-Schlichting (TS) instability already present for Newtonian channel flow. \(As E\) is increased beyond \(0.0016\), a new elastic mode appears at \(Re\sim 10^4\), which coalesces with the continuation of the TS mode for a range of \(Re\), thereby yielding a single unstable mode in this range. This trend persists until \(E\sim 0.0021\), beyond which this neutral curve splits into two separate ones in the \(Re-k\) plane. The new elastic mode which arises out of this splitting has been found to be the most unstable, with the lowest critical Reynolds number \(Re_c\approx 1210.9\) for \(E=0.0066\). The neutral curves for both the continuation of the original TS mode, and the new elastic antisymmetric mode, form closed loops upon further increase in \(E\), which eventually vanish at sufficiently high \(E\). For \(E\ll 1\), the critical Reynolds number and wavenumber scale as \(Re_c\sim E^{-1}\) and \(k_c\sim E^{-1/2}\) for the first two of the symmetric modal families, and as \(Re_c\sim E^{-5/4}\) for first two of the antisymmetric modal families; \(k_c\sim E^{-1/4}\) for the third antisymmetric family. The critical wave speed for all of these unstable eigenmodes scales as \(c_{r,c}\sim E^{1/2}\) for \(E\ll 1\), implying that the modes belong to a class of “wall modes” in viscoelastic flows with disturbances being confined in a thin region near the wall. The present study shows that, surprisingly, even in plane shear flows, elasticity acting along with inertia can drive novel instabilities absent in the Newtonian limit.

##### MSC:

76A10 | Viscoelastic fluids |

76E05 | Parallel shear flows in hydrodynamic stability |

76F06 | Transition to turbulence |

##### Keywords:

viscoelasticity
PDF
BibTeX
XML
Cite

\textit{I. Chaudhary} et al., J. Fluid Mech. 881, 119--163 (2019; Zbl 1430.76037)

Full Text:
DOI

##### References:

[1] | Avila, K., Moxey, D., Lozar, A. D., Barkley, D. & Hof, B.2011The onset of turbulence in pipe flow. Science333, 192-196. · Zbl 1411.76035 |

[2] | Bender, C. M. & Orszag, S. A.2013Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory. Springer Science & Business Media. |

[3] | Bertola, V., Meulenbroek, B., Wagner, C., Storm, C., Morozov, A., van Saarloos, W. & Bonn, D.2003Experimental evidence for an intrinsic route to polymer melt fracture phenomena: a nonlinear instability of viscoelastic Poiseuille flow. Phys. Rev. Lett.90, 114502. |

[4] | Bistagnino, A., Boffetta, G., Celani, A., Mazzino, A., Puliafito, A. & Vergassola, M.2007Nonlinear dynamics of the viscoelastic Kolmogorov flow. J. Fluid Mech.590, 61-80. · Zbl 1141.76335 |

[5] | Budanur, N. B., Short, K. Y., Farazmand, M., Willis, A. P. & Cvitanović, P.2017Relative periodic orbits form the backbone of turbulent pipe flow. J. Fluid Mech.833, 274-301. · Zbl 1419.76261 |

[6] | Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A.1988Spectral Methods in Fluid Dynamics. Springer. |

[7] | Carlson, D. R., Windall, S. E. & Peters, M. F.1982A flow-visualization study of transition in plane Poiseuille flow. J. Fluid Mech.121, 487-505. |

[8] | Castillo, H. A. & Wilson, H. J.2017Towards a mechanism for instability in channel flow of highly shear-thinning viscoelastic fluids. J. Non-Newtonian Fluid Mech.247, 15-21. |

[9] | Chandra, B., Shankar, V. & Das, D.2018Onset of transition in the flow of polymer solutions through microtubes. J. Fluid Mech.844, 1052-1083. |

[10] | Chokshi, P. & Kumaran, V.2009Stability of the plane shear flow of dilute polymeric solutions. Phys. Fluids21, 014109. |

[11] | Choueiri, G. H., Lopez, J. M. & Hof, B.2018Exceeding the asymptotic limit of polymer drag reduction. Phys. Rev. Lett.120, 124501. |

[12] | Cromer, M., Fredrickson, G. H. & Leal, L. G.2014A study of shear banding in polymer solutions. Phys. Fluids26, 063101. |

[13] | Cromer, M., Villet, M. C., Fredrickson, G. H. & Leal, L. G.2013Shear banding in polymer solutions. Phys. Fluids25, 051703. · Zbl 1321.76007 |

[14] | Doering, C. R., Eckhardt, B. & Schumacher, J.2006Failure of energy stability in Oldroyd-b fluids at arbitrarily low Reynolds numbers. J. Non-Newtonian Fluid Mech.135 (2), 92-96. · Zbl 1195.76174 |

[15] | Drazin, P. G.2002Introduction to Hydrodynamic Stability. Cambridge University Press. |

[16] | Drazin, P. G. & Reid, W. H.2004Hydrodynamic Stability, 2nd edn. Cambridge University Press. |

[17] | Dubief, Y., Terrapon, V. E. & Soria, J.2013On the mechanism of elasto-inertial turbulence. Phys. Fluids25, 110817. |

[18] | Dubief, Y., Terrapon, V. E., White, C. M., Shaqfeh, E. S. G., Moin, P. & Lele, S. K.2005New answers on the interaction between polymers and vortices in turbulent flows. Flow Turbul. Combust.74, 311-329. · Zbl 1200.76106 |

[19] | Dubief, Y., White, C. M., Terrapon, V. E., Shaqfeh, E. S. G., Moin, P. & Lele, S. K.2004On the coherent drag-reducing and turbulence-enhancing behaviour of polymers in wall flows. J. Fluid Mech.514, 271-280. · Zbl 1067.76052 |

[20] | Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J.2007Turbulence transition in pipe flow. Annu. Rev. Fluid Mech.39, 447-468. · Zbl 1296.76062 |

[21] | Garg, P., Chaudhary, I., Khalid, M., Shankar, V. & Subramanian, G.2018Viscoelastic pipe flow is linearly unstable. Phys. Rev. Lett.121, 024502. |

[22] | Gorodtsov, V. A. & Leonov, A. I.1967On a linear instability of a plane parallel Couette flow of viscoelastic fluid. Z. Angew. Math. Mech. J. Appl. Math. Mech.31, 310-319. · Zbl 0166.45101 |

[23] | Graham, M. D.1998Effect of axial flow on viscoelastic Taylor-Couette instability. J. Fluid Mech.360, 341-374. · Zbl 0947.76023 |

[24] | Graham, M. D.2014Drag reduction and the dynamics of turbulence in simple and complex fluids. Phys. Fluids26, 625-656. |

[25] | Grillet, A. M., Bogaerds, A. C. B., Peters, G. W. M. & Baaijens, F. P. T.2002Stability analysis of constitutive equations for polymer melts in viscometric flows. J. Non-Newtonian Fluid Mech.103, 221-250. · Zbl 1058.76528 |

[26] | Ho, T. C. & Denn, M. M.1977Stability of plane Poiseuille flow of a highly elastic liquid. J. Non-Newtonian Fluid Mech.3, 179-195. · Zbl 0414.76008 |

[27] | Hoda, N., Jovanovic, M. R. & Kumar, S.2008Energy amplification in channel flows of viscoelastic fluids. J. Fluid Mech.601, 407-424. · Zbl 1151.76372 |

[28] | Hoda, N., Jovanovic, M. R. & Kumar, S.2009Frequency responses of streamwise-constant perturbations in channel flows of Oldroyd-B fluids. J. Fluid Mech.625, 411-434. · Zbl 1171.76364 |

[29] | Hof, B., de Lozar, A., Avila, M., Tu, X. & Schneider, T. M.2010Eliminating turbulence in spatially intermittent flows. Science327, 1491-1494. |

[30] | Hof, B., de Lozar, A., Kuik, D. J. & Westerweel, J.2008Repeller or attractor? Selecting the dynamical model for the onset of turbulence in pipe flow. Phys. Rev. Lett.101, 214501. |

[31] | Jeong, J., Hussain, F., Schoppa, W. & Kim, J.1997Coherent structures near the wall in a turbulent channel flow. J. Fluid Mech.332, 185-214. · Zbl 0892.76036 |

[32] | Joo, Y. L. & Shaqfeh, E. S. G.1992The effects of inertia on the viscoelastic Dean and Taylor-Couette flow instabilities with application to coating flows. Phys. Fluids A4, 2415-2431. · Zbl 0762.76025 |

[33] | Jovanovic, M. R. & Kumar, S.2010Transient growth without inertia. Phys. Fluids22, 023101. · Zbl 1183.76263 |

[34] | Jovanovic, M. R. & Kumar, S.2011Nonmodal amplification of stochastic disturbances in strongly elastic channel flows. J. Non-Newtonian Fluid Mech.166, 755-778. · Zbl 1282.76052 |

[35] | Kumar, A. S. & Shankar, V.2005Instability of high-frequency modes in viscoelastic plane Couette flow past a deformable wall at low and finite Reynolds number. J. Non-Newtonian Fluid Mech.125, 121-141. · Zbl 1187.76656 |

[36] | Larson, R. G.1988Constitutive Equations for Polymer Melts and Solutions. Butterworths. |

[37] | Lee, K. C. & Finlayson, B. A.1986Stability of plane Poiseuille and Couette flow of a Maxwell fluid. J. Non-Newtonian Fluid Mech.21, 65-78. · Zbl 0587.76059 |

[38] | Lin, C. C.1945On the stability of two-dimensional parallel flows. I. General theory. Q. Appl. Maths3, 117-142. |

[39] | Liu, R. & Liu, Q.2010Non-modal instabilities of two-dimensional disturbances in plane Couette flow of a power-law fluid. J. Non-Newtonian Fluid Mech.165, 1228-1240. · Zbl 1274.76206 |

[40] | Mack, L. M.1976A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech.73, 497-520. · Zbl 0339.76030 |

[41] | Metzner, A. B. & Park, M. G.1964Turbulent flow characteristics of viscoelastic fluids. J. Fluid Mech.20, 291-303. |

[42] | Meulenbroek, B., Storm, C., Bertola, V., Wagner, C., Bonn, D. & van Saarloos, W.2003Intrinsic route to melt fracture in polymer extrusion: a weakly nonlinear subcritical instability of viscoelastic Poiseuille flow. Phys. Rev. Lett.90, 024502. |

[43] | Meulenbroek, B., Storm, C., Morozov, A. N. & van Saarloos, W.2004Weakly nonlinear subcritical instability of viscoelastic Poiseuille flow. J. Non-Newtonian Fluid Mech.116, 235-268. · Zbl 1106.76367 |

[44] | Morozov, A. N. & van Saarloos, W.2005Subcritical finite-amplitude solutions for plane Couette flow of viscoelastic fluids. Phys. Rev. Lett.95, 024501. |

[45] | Morozov, A. N. & van Saarloos, W.2007An introductory essay on subcritical instabilities and the transition to turbulence in viscoelastic parallel shear flows. Phys. Rep.447, 112-143. |

[46] | Muller, S. J.2008Elastically-influenced instabilities in Taylor-Couette and other flows with curved streamlines: a review. Korea-Australi Rheology J.20, 117-125. |

[47] | Mullin, T.2011Experimental studies of transition to turbulence in a pipe. Annu. Rev. Fluid Mech.43, 1-24. · Zbl 1210.76005 |

[48] | Nishioka, M. & Asai, M.1985Some observations of the subcritical transition in plane Poiseuille flow. J. Fluid Mech.150, 441-450. |

[49] | Orszag, S. A. & Kells, L. C.1980Transition to turbulence in plane Poiseuille and plane Couette flow. J. Fluid Mech.96, 159-205. · Zbl 0418.76036 |

[50] | Page, J. & Zaki, T. A.2014Streak evolution in viscoelastic Couette flow. J. Fluid Mech.742, 520-551. |

[51] | Pan, L., Morozov, A., Wagner, C. & Arratia, P. E.2013Nonlinear elastic instability in channel flows at low Reynolds numbers. Phys. Rev. Lett.110, 174502. |

[52] | Patel, V. C. & Head, M. R.1969Some observations on skin friction and velocity profiles in fully developed pipe and channel flows. J. Fluid Mech.38, 181-201. |

[53] | Poole, R. J.2012The Deborah and Weissenberg numbers. Rheol. Bull.53, 32-39. |

[54] | Poole, R. J.2016Elastic instabilities in parallel shear flows of a viscoelastic shear-thinning liquid. Phys. Rev. Fluids1, 041301. |

[55] | Porteous, K. C. & Denn, M. M.1972Linear stability of plane Poiseuille flow of viscoelastic liquids. Trans. Soc. Rheol.16, 295-308. · Zbl 0362.76079 |

[56] | Renardy, M.1992A rigorous stability proof for plane Couette flow of an upper convected Maxwell fluid at zero Reynolds number. Eur. J. Mech. (B/Fluids)11 (4), 511-516. · Zbl 0850.76196 |

[57] | Renardy, M. & Renardy, Y.1986Linear stability of plane Couette flow of an upper convected Maxwell fluid. J. Non-Newtonian Fluid Mech.22, 23-33. · Zbl 0608.76006 |

[58] | Robinson, S. K.1991Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech.23, 601-639. |

[59] | Roy, A., Morozov, A., van Saarloos, W. & Larson, R. G.2006Mechanism of polymer drag reduction using a low-dimensional model. Phys. Rev. Lett.97, 234501. |

[60] | Sadanandan, B. & Sureshkumar, R.2002Viscoelastic effects on the stability of wall-bounded shear flows. Phys. Fluids14, 41-48. |

[61] | Samanta, D., Dubief, Y., Holzner, M., Schäfer, C., Morozov, A. N., Wagner, C. & Hof, B.2013Elasto-inertial turbulence. Proc. Natl Acad. Sci. USA110, 10557-10562. |

[62] | Schmid, P. J. & Henningson, D. S.2001Stability and Transition in Shear Flows. Springer. |

[63] | Seyer, F. A. & Metzner, A. B.1969Turbulence phenomena in drag reducing systems. AIChE J.15, 426-434. |

[64] | Shaqfeh, E. S. G.1996Purely elastic instabilities in viscometric flows. Annu. Rev. Fluid Mech.28, 129-185. |

[65] | Shekar, A., McMullen, R. M., Wang, S. N., McKeon, B. J. & Graham, M. D.2019Critical-layer structures and mechanisms in elastoinertial turbulence. Phys. Rev. Lett.122, 124503. |

[66] | Sid, S., Terrapon, V. E. & Dubief, Y.2018Two-dimensional dynamics of elasto-inertial turbulence and its role in polymer drag reduction. Phys. Rev. Fluids3, 011301(R). |

[67] | Srinivas, S. S. & Kumaran, V.2017Effect of viscoelasticity on the soft-wall transition and turbulence in a microchannel. J. Fluid Mech.812, 1076-1118. · Zbl 1383.76193 |

[68] | Stone, P. A. & Graham, M. D.2003Polymer dynamics in a model of the turbulent buffer layer. Phys. Fluids15, 1247-1256. |

[69] | Stone, P. A., Roy, A., Larson, R. G., Waleffe, F. & Graham, M. D.2004Polymer drag reduction in exact coherent structures of plane shear flow. Phys. Fluids16, 3470-3482. |

[70] | Stone, P. A., Waleffe, F. & Graham, M. D.2002Toward a structural understanding of turbulent drag reduction: nonlinear coherent states in viscoelastic shear flows. Phys. Rev. Lett.89, 208301. |

[71] | Sureshkumar, R.2000Numerical observations on the continuous spectrum of the linearized viscoelastic operator in shear dominated complex flows. J. Non-Newtonian Fluid Mech.94, 205-211. · Zbl 1004.76032 |

[72] | Sureshkumar, R. & Beris, A. N.1995aEffect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of time-dependent viscoelastic flows. J. Non-Newtonian Fluid Mech.60, 53-80. |

[73] | Sureshkumar, R. & Beris, A. N.1995bLinear stability analysis of viscoelastic Poiseuille flow using an Arnoldi-based orthogonalization algorithm. J. Non-Newtonian Fluid Mech.56, 151-182. |

[74] | Sureshkumar, R., Beris, A. N. & Handler, R. A.1997Direct numerical simulation of the turbulent channel flow of a polymer solution. Phys. Fluids9, 743-755. |

[75] | Virk, P. S.1975Drag reduction fundamentals. AIChE J.21, 625-656. |

[76] | Waleffe, F.2001Exact coherent structures in channel flow. J. Fluid Mech.435, 93-102. · Zbl 0987.76034 |

[77] | Wen, C., Poole, R. J., Willis, A. P. & Dennis, D. J. C.2017Experimental evidence of symmetry-breaking supercritical transition in pipe flow of shear-thinning fluids. Phys. Rev. Fluids2, 031901. |

[78] | White, C. M. & Mungal, M. G.2008Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech.40, 235-256. · Zbl 1229.76043 |

[79] | Wilson, H. J. & Loridan, V.2015Linear instability of a highly shear-thinning fluid in channel flow. J. Non-Newtonian Fluid Mech.223, 200-208. |

[80] | Wilson, H. J. & Rallison, J. M.1999Instability of channel flow of a shear-thinning whitemetzner fluid. J. Non-Newtonian Fluid Mech.87, 75-96. |

[81] | Wilson, H. J., Renardy, M. & Renardy, Y.1999Structure of the spectrum in zero Reynolds number shear flow of the UCM and Oldroyd-B liquids. J. Non-Newtonian Fluid Mech.80, 251-268. · Zbl 0956.76025 |

[82] | Zhang, M., Lashgari, I., Zaki, T. A. & Brandt, L.2013Linear stability analysis of channel flow of viscoelastic Oldroyd-B and FENE-P fluids. J. Fluid Mech.737, 249-279. · Zbl 1294.76119 |

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.