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Receptivity to Thermal Noise of the Boundary Layer over a Swept Wing

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The commonly accepted description of spatial transition to turbulence in a convectively unstable boundary layer involves as one of its stages the permanent excitation of instabilities by external noise (receptivity). The idea that thermal fluctuations of microscopic origin provide a sufficient amount of permanent noise to play a role in this process has made only scant and isolated appearances in the literature. Yet, the contribution of thermal noise to receptivity has its amplitude determined from physical first principles; it provides the lower bound beyond which disturbances cannot be reduced (and the upper bound beyond which transition cannot be delayed) lest thermodynamics is violated. As it is also not significantly harder to compute than a typical N-factor plot, thermal-noise receptivity can be adopted as a standard against which to compare other forms of receptivity through a noise figure analogous to the one that is of commonplace use in electronics. Elaborating on previous computations of the effect of thermal noise upon the two-dimensional boundary layer past a flat plate, it is shown by a few real-world examples that the receptivity to thermal noise of the crossflow boundary layer over a swept wing is even larger, and by a substantial amount.


  • [1] Smith A. M. O. and Gamberoni N., “Transition, Pressure Gradient, and Stability Theory,” Proceedings of 9th International Congress of Applied Mechanics, Vol. 4, Free Univ. of Brussels, Brussels, Belgium, Sept. 1956, pp. 234–244; also Douglas Aircraft Co. Rept.  ES26388, El Segundo, CA, Aug. 1956. Google Scholar

  • [2] van Ingen J. L., “A Suggested Semi-Empirical Method for the Calculation of the Boundary-Layer Transition Region,” Aeronautical Engineering Dept., Delft Rept.  VTH-74, Delft, The Netherlands, 1956. Google Scholar

  • [3] White F., Viscous Fluid Flow, McGraw–Hill, New York, 1991. Google Scholar

  • [4] Saric W. S., “Boundary-Layer Receptivity to Freestream Disturbances,” Annual Review of Fluid Mechanics, Vol. 34, Jan. 2002, No. 1, pp. 291–319. doi: ARVFA3 0066-4189 CrossrefGoogle Scholar

  • [5] Betchov R., “Thermal Agitation and Turbulence,” Rarefied Gas Dynamics, edited by Talbot L., Academic Press, New York, 1960, pp. 308–321. Google Scholar

  • [6] Zavol’skii N. A. and Reutov V. P., “Thermal Excitation of Waves in a Boundary Layer,” Fluid Dynamics, Vol. 18, No. 5, 1984, pp. 701–705. doi: FLDYAH 0015-4628 CrossrefGoogle Scholar

  • [7] Luchini P., “The Role of Microscopic Fluctuations in Shear-Flow Transition,” April 2008 (unpublished). Google Scholar

  • [8] Luchini P., “The Role of Molecular Agitation in Boundary-Layer Transition,” Transition Study Group Open Forum of the 5th AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 2008. Google Scholar

  • [9] Luchini P., “Receptivity to Molecular Agitation in Boundary-Layer Transition,” 61st Annual Meeting of the APS Division of Fluid Dynamics, San Antonio, TX, Nov. 2008; also Bulletin of the American Physical Society, Vol. 53, No. 15, 2008, pp. 179–180. Google Scholar

  • [10] Luchini P., “A Thermodynamic Lower Bound on the Level of Transition-Triggering Disturbances,” Seventh IUTAM Symposium on Laminar-Turbulent Transition, Vol. 18, Springer, New York, 2009, pp. 11–18. Google Scholar

  • [11] Fedorov A. V. and Averkin S. N., “Receptivity of Compressible Boundary Layer to Kinetic Fluctuations,” Seventh IUTAM Symposium on Laminar-Turbulent Transition, Springer, New York, 2009, pp. 485–488. Google Scholar

  • [12] Fedorov A., “Prediction and Control of Laminar-Turbulent Transition in High-Speed Boundary-Layer Flows,” Procedia IUTAM, Vol. 14, IUTAM-ABCM Symposium on Laminar-Turbulent Transition, edited by Medeiros M. A. F. and Meneghini J. R., Elsevier, New York, 2015, pp. 3–14. CrossrefGoogle Scholar

  • [13] Landau L. D. and Lifschitz E. M., “The Fluctuation–Dissipation Theorem,” Course of Theoretical Physics, 3rd ed., Vol. 5, Statistical Physics, Pergamon, Oxford, England, U.K., 1980, Chap. 124. Google Scholar

  • [14] de Groot S. R. and Mazur P., “The Fluctuation Dissipation Theorem,” Non-Equilibrium Thermodynamics, Dover, New York, 1984, Chap. VIII. Google Scholar

  • [15] Landau L. D. and Lifschitz E. M., “Hydrodynamic Fluctuations,” Soviet Physics, Journal of Experimental and Theoretical Physics, Vol. 5, pp. 512–513, Zh. Eksp. Teor. Fiz. 32 618 [Sov. Phys. JETP 5 512], 1957; also in Course of Theoretical Physics, Vol. 6 Fluid Mechanics, Pergamon Press, 1959, Chap. XVII; also Course of Theoretical Physics, Vol. 9 Statistical Physics Part 2, Pergamon Press, 1980, Sec. 88. Google Scholar

  • [16] Luchini P. and Bottaro A., “Adjoint Equations in Stability Analysis,” Annual Review of Fluid Mechanics, Vol. 46, No. 1, 2014, pp. 493–517. doi: ARVFA3 0066-4189 CrossrefGoogle Scholar

  • [17] De Matteis P., Donelli R. S. and Luchini P., “Application of the Ray-Tracing Theory to the Stability Analysis of Three-Dimensional Incompressible Boundary Layers,” Proceedings of XIII Congresso Nazionale AIDAA, Vol. 1, Università di Roma La Sapienza, Rome, Italy, Sept. 1995, pp. 1–0. Google Scholar

  • [18] Zuccher S., “Receptivity and Control of Flow Instabilities in a Boundary Layer,” Ph.D. Dissertation, Politecnico di Milano, Milan, 2001. Google Scholar

  • [19] Zuccher S. and Luchini P., “Boundary-Layer Receptivity to External Disturbances in the Multiple-Scale Approximation,” Meccanica, Vol. 49, No. 2, 2014, pp. 441–467. doi: MECCB9 0025-6455 CrossrefGoogle Scholar

  • [20] Gaster M., “On the Effects of Boundary-Layer Growth on Flow Stability,” Journal of Fluid Mechanics, Vol. 66, No. 3, 1974, pp. 465–480. doi: JFLSA7 0022-1120 CrossrefGoogle Scholar

  • [21] Saric W. S. and Nayfeh A. H., “Non-Parallel Stability of Boundary-Layer Flows,” Physics of Fluids, Vol. 18, No. 8, 1975, pp. 945–950. doi: CrossrefGoogle Scholar

  • [22] Reibert M., Saric W. S., Carrillo R. B. and Chapman K. L., “Experiments in Nonlinear Saturation of Stationary Crossflow Vortices in a Swept-Wing Boundary-Layer,” AIAA Paper  1996-0184, 1996. LinkGoogle Scholar

  • [23] Hunt L. E. and Saric W. S., “Boundary-Layer Receptivity of Three-Dimensional Roughness Arrays on a Swept-Wing,” AIAA Paper  2011-3881, 2011. LinkGoogle Scholar

  • [24] Bonfigli G., Kloker M. and Wagner S., “3-D-Boundary-Layer Transition Induced by Superposed Steady and Traveling Crossflow Vortices,” High Performance Computing in Science and Engineering ‘02, edited by Krause E. and Jäger W., Springer, New York, 2002, pp. 255–271. Google Scholar