Prandtl System of Equations with Self-Induced Pressure for the Case of Non-Newtonian Fluid: Dynamics of Boundary Layer Separation

R. K. Gaydukov

Russian journal of nonlinear dynamics, Tome 20 (2024) no. 1, pp. 113-125.

Résumé

The problem of flow of a non-Newtonian viscous fluid with power-law rheological properties along a semi-infinite plate with a small localized irregularity on the surface is considered for large Reynolds numbers. The asymptotic solution with double-deck structure of the boundary layer is constructed. The numerical simulation of the flow in the region near the surface was performed for different fluid indices. The results of investigations of the flow properties depending on the fluid index are presented. Namely, the boundary layer separation is investigated for different fluid indices, and the dynamics of vortex formation in this region is shown.

Détail

Keywords: double-deck structure, boundary layer separation, power-law fluid, localized perturbations, asymptotics, numerical simulation

Bibliographie

[1] Schlichting, H. and Gersten, K., Boundary-Layer Theory, with contributions from Egon Krause and Herbert Oertel Jr., translated from the German by Katherine Mayes, 9th ed., Springer, Berlin, 2017, xxviii, 805 pp. | MR | Zbl

[2] Neiland, V. Ya., “Theory of Laminar Boundary Layer Separation in Supersonic Flow”, Fluid Dyn., 4:4 (1969), 33–35 | DOI

[3] Stewartson, K. and Williams, P. G., “Self-Induced Separation”, Proc. Roy. Soc. London Ser. A, 312:1509 (1969), 181–206 | DOI | Zbl

[4] Smith, F. T., “Laminar Flow over a Small Hump on a Flat Plate”, J. Fluid Mech., 57:4 (1973), 803–824 | DOI | Zbl

[5] Mauss, J., “Asymptotic Modelling for Separating Boundary Layers”, Asymptotic Modelling in Fluid Mechanics: Proc. of the Symp. in Honor of J.-P. Guiraud (Paris, Apr 1994), Lecture Notes in Phys., 442, eds. P.-A. Bois, E. Dériat, R. Gatignol, A. Rigolot, Springer, Berlin, 1995, 239–254 | DOI | MR | Zbl

[6] Danilov, V. G. and Makarova, M. V., “Asymptotic and Numerical Analysis of the Flow around a Plate with Small Periodic Irregularities”, Russian J. Math. Phys., 2:1 (1994), 49–56 | MR | Zbl

[7] Danilov, V. G. and Gaydukov, R. K., “Asymptotic Multiscale Solutions to Navier – Stokes Equations with Fast Oscillating Perturbations in Boundary Layers”, Russian J. Math. Phys., 29:4 (2022), 431–455 | DOI | MR | Zbl

[8] Danilov, V. G. and Gaydukov, R. K., “Double-Deck Structure of the Boundary Layer in Problems of Flow around Localized Perturbations on a Plate”, Math. Notes, 98:4 (2015), 561–571 | DOI | Zbl

[9] Mat. Zametki, 112:4 (2022), 521–533 (Russian) | DOI | MR | Zbl

[10] Gaydukov, R. K., “A Numerical Algorithm for Solving the Prandtl Equations with Induced Pressure in the Periodic Case”, Numer. Analys. Appl., 15:2 (2022), 79–89 | DOI | MR

[11] Gaydukov, R. K., “Double-Deck Structure in the Fluid Flow Problem over Plate with Small Irregularities of Time-Dependent Shape”, Eur. J. Mech. B Fluids, 89 (2021), 401–410 | DOI | MR | Zbl

[12] Yapalparvi, R., “Double-Deck Structure Revisited”, Eur. J. Mech. B Fluids, 31 (2012), 53–70 | DOI | MR | Zbl

[13] Gaydukov, R. K., “Double-Deck Structure in the Problem of a Compressible Flow along a Plate with Small Localized Irregularities on the Surface”, Eur. J. Mech. B Fluids, 71 (2018), 59–65 | DOI | MR | Zbl

[14] Cousteix, J. and Mauss, J., Asymptotic Analysis and Boundary Layers, Springer, Berlin, 2007, xviii, 432 pp. | MR | Zbl

[15] Gaydukov, R. K. and Fonareva, A. V., “Nonstationary Double-Deck Structure of Boundary Layers in Compressible Flow Problem inside a Channel with Small Irregularities on the Walls”, Russian J. Math. Phys., 28:2 (2021), 224–243 | DOI | MR | Zbl

[16] Smith, F. T., “Flow through Constricted or Dilated Pipes and Channels: Pt. 1”, Q. J. Mech. Appl. Math., 29:3 (1976), 343–364 | DOI | Zbl

[17] Smith, F. T., “Flow through Constricted or Dilated Pipes and Channels: Pt. 2”, Q. J. Mech. Appl. Math., 29:3 (1976), 365–376 | DOI | Zbl

[18] Chicchiero, C., Segalini, A., and Camarri, S., “Triple-Deck Analysis of the Steady Flow over a Rotating Disk with Surface Roughness”, Phys. Rev. Fluids, 6:1 (2021), Art. 014103, 25 pp. | DOI

[19] Gaydukov, R. K. and Fonareva, A. V., “Double-Deck Structure in the Fluid Flow Induced by a Uniformly Rotating Disk with Small Symmetric Irregularities on Its Surface”, Eur. J. Mech. B Fluids, 94 (2022), 50–59 | DOI | MR | Zbl

[20] Prikl. Mat. Mekh., 41:6 (1977), 1007–1023 (Russian) | DOI | MR

[21] Meyer, R. E., “A View of the Triple Deck”, SIAM J. Appl. Math., 43:4 (1983), 639–663 | DOI | MR | Zbl

[22] Nayfeh, A. H., “Triple-Deck Structure”, Comput. Fluids, 20:3 (1991), 269–292 | DOI | MR | Zbl

[23] Sychev, V. V., Ruban, A. I., Sychev, Vic. V., and Korolev, G. L., Asymptotic Theory of Separated Flows, translated from the 1987 Russian original by E. V. Maroko and revised by the authors, Cambridge Univ. Press, Cambridge, 1998, x, 334 pp. | MR | Zbl

[24] Mengaldo, G., Kravtsova, M., Ruban, A. I., and Sherwin, S. J., “Triple-Deck and Direct Numerical Simulation Analyses of High-Speed Subsonic Flows past a Roughness Element”, J. Fluid Mech., 774 (2015), 311–323 | DOI | MR

[25] Aljohani, A. F. and Gajjar, J. S. B., “Subsonic Flow past Localised Heating Elements in Boundary Layers”, J. Fluid Mech., 821 (2017), R2, 10 pp. | DOI | MR | Zbl

[26] Aljohani, A. F. and Gajjar, J. S. B., “Transonic Flow over Localised Heating Elements in Boundary Layers”, J. Fluid Mech., 844 (2018), 746–765 | DOI | MR | Zbl

[27] Stewartson, K., “Multistructured Boundary Layers on at Plates and Related Bodies”, Adv. Appl. Mech., 14 (1974), 145–239 | DOI

[28] Astarita, G. and Marrucci, G., Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, London, 1974, 289 pp.

[29] Böhme, G., Non-Newtonian Fluid Mechanics, North-Holland Ser. Appl. Math. Mec., 31, North-Holland, Amsterdam, 1987, xii, 351 pp. | MR | Zbl

[30] Acrivos, A., Shah, M. J., and Petersen, E. E., “Momentum and Heat Transfer in Laminar Boundary-Layer Flows of Non-Newtonian Fluids past External Surfaces”, AIChE J., 6:2 (1960), 312–317 | DOI

[31] Bognár, G., “Similarity Solution of Boundary Layer Flows for Non-Newtonian Fluids”, Int. J. Nonlin. Sci. Numer., 10 (2009), 1555–1566 | DOI

[32] Benlahsen, M., Guedda, M., and Kersner, R., “The Generalized Blasius Equation Revisited”, Math. Comput. Modelling, 47:9–10 (2008), 1063–1076 | DOI | MR | Zbl

[33] Zhizhin, G. V. and Ufimtsev, A. A., “Flow in the Plane Laminar Boundary Layer of Dilatant Liquids”, Fluid Dyn., 12 (1977), 780–784 | DOI

[34] Uspekhi Mat. Nauk, 12:5(77) (1957), 3–122 (Russian) | MR | Zbl | Zbl

[35] Cebeci, T., Mosinskis, G. J., and Smith, A. M. O., “Calculation of Separation Points in Incompressible Turbulent Flows”, J. Aircraft, 9:9 (1972), 618–624 | DOI