×

Entrainment and growth of vortical disturbances in the channel-entrance region. (English) Zbl 1475.76036

Summary: The entrainment of free-stream unsteady three-dimensional vortical disturbances in the entry region of a channel is studied via matched asymptotic expansions and by numerical means. The interest is in flows at Reynolds numbers where experimental studies have documented the occurrence of intense transient growth, despite the flow being stable according to classical stability analysis. The analytical description of the vortical perturbations at the channel mouth reveals how the oncoming disturbances penetrate into the wall-attached shear layers and amplify downstream. The effects of the channel confinement, the streamwise pressure gradient and the viscous/inviscid interplay between the oncoming disturbances and the boundary-layer perturbations are discussed. The composite perturbation velocity profiles are employed as initial conditions for the unsteady boundary-region perturbation equations. At a short distance from the channel mouth, the disturbance flow is mostly confined within the shear layers and assumes the form of streamwise-elongated streaks, while farther downstream the viscous disturbances permeate the whole channel although the base flow is still mostly inviscid in the core. Symmetrical disturbances exhibit a more significant growth than anti-symmetrical disturbances, the latter maintaining a nearly constant amplitude for several channel heights downstream before growing transiently, a unique feature not reported in open boundary layers. The disturbances are more intense as the frequency decreases or the bulk Reynolds number increases. We compute the spanwise wavelengths that cause the most intense downstream growth and the threshold wall-normal wavelengths below which the perturbations are damped through viscous dissipation.

MSC:

76E05 Parallel shear flows in hydrodynamic stability
76E15 Absolute and convective instability and stability in hydrodynamic stability
76D10 Boundary-layer theory, separation and reattachment, higher-order effects
76D17 Viscous vortex flows
76M45 Asymptotic methods, singular perturbations applied to problems in fluid mechanics
76F06 Transition to turbulence
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alizard, F., Cadiou, A., Le Penven, L., Di Pierro, B. & Buffat, M.2018Space-time dynamics of optimal wavepackets for streaks in a channel entrance flow. J. Fluid Mech.844, 669-706. · Zbl 1461.76201
[2] Andersson, P., Berggren, M. & Henningson, D.S.1999Optimal disturbances and bypass transition in boundary layers. Phys. Fluids11 (1), 134-150. · Zbl 1147.76308
[3] Asai, M. & Floryan, J.M.2004Certain aspects of channel entrance flow. Phys. Fluids16 (4), 1160-1163. · Zbl 1186.76035
[4] Beavers, G.S., Sparrow, E.M. & Magnuson, R.A.1970Experiments on hydrodynamically developing flow in rectangular ducts of arbitrary aspect ratio. Intl J. Heat Transfer13, 689-702.
[5] Bodoia, J.R. & Osterle, J.F.1962Finite difference analysis of plane Poiseuille and Couette flow developments. Appl. Sci. Res.10 (1), 265-276. · Zbl 0100.22701
[6] Borodulin, V.I., Ivanov, A.V., Kachanov, Y.S. & Roschektayev, A.P.2021aDistributed vortex receptivity of a swept-wing boundary layer. Part 1. Efficient excitation of CF modes. J. Fluid Mech.908, A14. · Zbl 1461.76202
[7] Borodulin, V.I., Ivanov, A.V., Kachanov, Y.S. & Roschektayev, A.P.2021bDistributed vortex receptivity of a swept-wing boundary layer. Part 2. Receptivity characteristics. J. Fluid Mech.908, A15. · Zbl 1461.76203
[8] Brandt, L., Schlatter, P. & Henningson, D.S.2004Transition in boundary layers subject to free-stream turbulence. J. Fluid Mech.517, 167-198. · Zbl 1131.76326
[9] Buffat, M., Le Penven, L., Cadiou, A. & Montagnier, J.2014DNS of bypass transition in entrance channel flow induced by boundary layer interaction. Eur. J. Mech. - B/Fluids43, 1-13. · Zbl 1297.76082
[10] Carlson, D.R., Widnall, S.E. & Peeters, M.F.1982A flow-visualization study of transition in plane Poiseuille flow. J. Fluid Mech.121, 487-505.
[11] Cebeci, T.2002Convective Heat Transfer. Springer-Verlag. · Zbl 1057.76001
[12] Chen, T.S. & Sparrow, E.M.1967Stability of the developing laminar flow in a parallel-plate channel. J. Fluid Mech.30 (2), 209-224. · Zbl 0153.29802
[13] Collins, M. & Schowalter, W.R.1962Laminar flow in the inlet region of a straight channel. Phys. Fluids5, 1122-1124. · Zbl 0109.18704
[14] Davies, S.J. & White, C.M.1928An experimental study of the flow of water in pipes of rectangular section. Proc. R. Soc. London119 (781), 92-107. · JFM 54.0924.01
[15] Dietz, A.J.1999Local boundary-layer receptivity to a convected free-stream disturbance. J. Fluid Mech.378, 291-317.
[16] Drazin, P.G. & Reid, W.H.2004Hydrodynamic Stability. Cambridge Mathematical Library. · Zbl 1055.76001
[17] Dryden, H.L.1936 Air flow in the boundary layer near a plate, vol. 562. NACA Rep.
[18] Dryden, H.L.1955 Transition from laminar to turbulent flow at subsonic and supersonic speeds. In Conference on High-Speed Aeronautics, vol. 41. Polytechnic of Brooklyn.
[19] Duck, P.W.2005Transient growth in developing plane and Hagen Poiseuille flow. Proc. R. Soc. Lond. Ser. A461, 1311-1333. · Zbl 1145.76358
[20] Garg, V.K. & Gupta, S.C.1981aNonparallel effects on the stability of developing flow in a channel. Phys. Fluids24 (9), 1752-1754.
[21] Garg, V.K. & Gupta, S.C.1981bStability of the nonparallel developing flow in a channel. Comput. Meth. Appl. Mech. Engng29, 259-269. · Zbl 0482.76047
[22] Goldstein, M.E.1978Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech.89, 433-468. · Zbl 0401.76018
[23] Goldstein, M.E.1983The evolution of Tollmien-Schlichting waves near a leading edge. J. Fluid Mech.127, 59-81. · Zbl 0524.76046
[24] Goldstein, M.E.1985Scattering of acoustic waves into Tollmien-Schlichting waves by small streamwise variations in surface geometry. J. Fluid Mech.154, 509-529. · Zbl 0576.76064
[25] Grosch, C.E. & Salwen, H.1978The continuous spectrum of the Orr-Sommerfeld equation. Part 1. The spectrum and the eigenfunctions. J. Fluid Mech.87, 33-54. · Zbl 0383.76031
[26] Gupta, S.C. & Garg, V.K.1981aLinear spatial stability of developing flow in a parallel plate channel. J. Appl. Mech.48, 192-194.
[27] Gupta, S.C. & Garg, V.K.1981bStability of developing flow in a two-dimensional channel - symmetric vs. antisymmetric disturbances. Comput. Meth. Appl. Mech. Engng27, 363-368.
[28] Gustavsson, L.H.1991Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech.224, 241-260. · Zbl 0717.76044
[29] Hahneman, E., Freeman, J.C. & Finston, M.1948Stability of boundary layers and of flow in entrance section of a channel. J. Aerosp. Sci.15 (8), 493-496.
[30] Hunt, J.C.R.1973A theory of turbulent flow round two-dimensional bluff bodies. J. Fluid Mech.61 (04), 625-706. · Zbl 0282.76048
[31] Jacobs, R.G. & Durbin, P.A.2001Simulation of bypass transition. J. Fluid Mech.428, 185-212. · Zbl 0983.76027
[32] Kao, T.W. & Park, C.1970Experimental investigations of the stability of channel flows. Part 1. flow of a single liquid in a rectangular channel. J. Fluid Mech.43 (1), 145-164.
[33] Kemp, N.1951 The laminar three-dimensional boundary layer and a study of the flow past a side edge. MSc Thesis, Cornell University.
[34] Kendall, J.M.1991 Studies on laminar boundary layer receptivity to free-stream turbulence near a leading edge. In Boundary Layer Stability and Transition to Turbulence (ed. D.C. Reda, H.L. Reed & R. Kobayashi), vol. 114, pp. 23-30. ASME FED.
[35] Kim, J., Moin, P. & Moser, R.1987Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech.177, 133-166. · Zbl 0616.76071
[36] Klebanoff, P.S.1971Effect of free-stream turbulence on a laminar boundary layer. Bull. Am. Phys. Soc.16, 1323.
[37] Leib, S.J., Wundrow, D.W. & Goldstein, M.E.1999Effect of free-stream turbulence and other vortical disturbances on a laminar boundary layer. J. Fluid Mech.380, 169-203. · Zbl 0951.76032
[38] Luchini, P.1996Reynolds-number-independent instability of the boundary layer over a flat surface. J. Fluid Mech.327, 101-115. · Zbl 0883.76034
[39] Luchini, P.2000Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech.404, 289-309. · Zbl 0959.76022
[40] Marensi, E., Ricco, P. & Wu, X.2017Nonlinear unsteady streaks engendered by the interaction of free-stream vorticity with a compressible boundary layer. J. Fluid Mech.817, 80-121. · Zbl 1383.76403
[41] Matsubara, M. & Alfredsson, P.H.2001Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech.430, 149-168. · Zbl 0963.76509
[42] Morkovin, M.V.1984 Bypass transition to turbulence and research desiderata. NASA CP-2386 Transition in Turbines, pp. 161-204.
[43] Nishioka, M. & Asai, M.1985Some observations of the subcritical transition in plane Poiseuille flow. J. Fluid Mech.150, 441-450.
[44] Nishioka, M., Iida, S. & Ichikawa, Y.1975An experimental investigation of the stability of plane Poiseuille flow. J. Fluid Mech.72, 731-751.
[45] Orszag, S.A.1971Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech.50, 689-703. · Zbl 0237.76027
[46] 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 (1), 181-201.
[47] Reed, H.L., Reshotko, E. & Saric, W.S.2015 Receptivity: the inspiration of Mark Morkovin. In 45th AIAA Fluid Dynamics Conference, p. 2471.
[48] Reynolds, O.1883An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. R. Soc.35 (224-226), 84-99.
[49] Ricco, P.2009The pre-transitional Klebanoff modes and other boundary layer disturbances induced by small-wavelength free-stream vorticity. J. Fluid Mech.638, 267-303. · Zbl 1183.76751
[50] Ricco, P., Luo, J. & Wu, X.2011Evolution and instability of unsteady nonlinear streaks generated by free-stream vortical disturbances. J. Fluid Mech.677, 1-38. · Zbl 1241.76236
[51] Ricco, P., Walsh, E.J., Brighenti, F. & Mceligot, D.M.2016Growth of boundary-layer streaks due to free-stream turbulence. Intl J. Heat Fluid Flow61, 272-283.
[52] Ruban, A.I.1984On Tollmien-Schlichting wave generation by sound. Izv. Akad. Nauk SSSR Mekh. Zhid. Gaza5, 44. · Zbl 0566.76069
[53] Ruban, A.I.1985On the generation of Tollmien-Schlichting waves by sound. Fluid Dyn.25 (2), 213-221.
[54] Rubin, S.G., Khosla, P.K. & Saari, S.1977Laminar flow in rectangular channels. Comput. Fluids5, 151-173. · Zbl 0368.76038
[55] Schlichting, H.1934Laminar channel entrance flow. Z. Angew. Math. Mech.14, 368-373. · JFM 60.0731.01
[56] Schmid, P.J. & Henningson, D.S.2001Stability and Transition in Shear Flows. Applied Mathematical Sciences, vol. 142. Springer. · Zbl 0966.76003
[57] Smith, F.T. & Bodonyi, R.J.1980On the stability of the developing flow in a channel or circular pipe. Q. J. Mech. Appl. Maths33 (3), 293-320. · Zbl 0522.76049
[58] Sparrow, E.M., Hixon, C.W. & Shavit, G.1967Experiments on laminar flow development in rectangular ducts. J. Basic Engng89, 116-124.
[59] Sparrow, E.M., Lin, S.H. & Lundgren, T.S.1964Flow development in the hydrodynamic entrance region of tubes and ducts. Phys. Fluids7 (3), 338-347. · Zbl 0118.20603
[60] Taylor, G.I.1939 Some recent developments in the study of turbulence. In Proceedings of the Fifth International Congress for Applied Mechanics, pp. 294-310. · JFM 65.0989.01
[61] Van Dyke, M.1969Entry flow in a channel. J. Fluid Mech.44, 813-823. · Zbl 0205.57001
[62] Westin, K.J.A., Boiko, A.V., Klingmann, B.G.B., Kozlov, V.V. & Alfredsson, P.H.1994Experiments in a boundary layer subjected to free stream turbulence. Part 1. Boundary layer structure and receptivity. J. Fluid Mech.281, 193-218.
[63] Wilson, S.D.R.1970Entry flow in a channel. Part 2. J. Fluid Mech.46, 787-799. · Zbl 0222.76027
[64] Wu, X.2001Receptivity of boundary layers with distributed roughness to vortical and acoustic disturbances: A second-order asymptotic theory and comparison with experiments. J. Fluid Mech.431, 91-133. · Zbl 1008.76017
[65] Wu, X., Moin, P., Adrian, R.J. & Baltzer, J.R.2015Osborne Reynolds pipe flow: direct simulation from laminar through gradual transition to fully developed turbulence. PNAS112 (26), 7920-7924.
[66] Wundrow, D.W. & Goldstein, M.E.2001Effect on a laminar boundary layer of small-amplitude streamwise vorticity in the upstream flow. J. Fluid Mech.426, 229-262. · Zbl 1010.76029
[67] Xu, D., Liu, J. & Wu, X.2020Görtler vortices and streaks in boundary layer subject to pressure gradient: excitation by free stream vortical disturbances, nonlinear evolution and secondary instability. J. Fluid Mech.900, A15. · Zbl 1460.76227
[68] Zanoun, E.-S., Kito, M. & Egbers, C.2009A study on flow transition and development in circular and rectangular ducts. J. Fluids Engng131, 061204.
[69] Zhang, Y., Zaki, T., Sherwin, S. & Wu, X.2011 Nonlinear response of a laminar boundary layer to isotropic and spanwise localized free-stream turbulence. In 6th AIAA Theoretical Fluid Mechanics Conference, vol. 3292.
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.