Skip to main content
SpringerLink
Log in

HTR approach to the asymptotic solutions of supersonic boundary layer problem: the case of slow acoustic waves interacting with streamwise isolated wall roughness

  • Original Research
  • Published:
Mathematical Sciences Aims and scope Submit manuscript

Abstract

We address supersonic boundary layer flow to slow acoustic waves using homotopy renormalization method. The model involves an ordinary equation system, which allows us to investigate the problem analytically and find asymptotic solutions in explicit form. At first, we rewrite the original problem in the form of a system of inhomogeneous variable coefficients homotopy equations and then handle them by the traditional perturbation theory, using renormalization group methods to eliminate the secular terms. We prove analytically and numerically the high accuracy of our solutions, and study in some details the effects of Mach number and wall temperature. Finally, we discuss how the explicit expression of boundary thickness makes our solutions suitable for practical applications. To the best of our knowledge, this is the first time that the HTR method is applied to supersonic boundary flow, and the analytic solutions are obtained. These solutions are easy to apply, and they could also help provide deeper insight into the supersonic boundary layer model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Kuiken, H.K.: A backward free-convective boundary layer. Q. J. Mech. Appl. Math. 34(3), 397–413 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  2. Kechil, S.A., Hashim, I.: Non-perturbative solution of free-convective boundary-layer equation by Adomian decomposition method. Phys. Lett. A 363(1), 110–114 (2007)

    Article  MATH  Google Scholar 

  3. Ibrahim, W., Makinde, O.D.: Magnetohydrodynamic stagnation point flow of a power-law nanofluid towards a convectively heated stretching sheet with slip. Proc. Instit. Mech. Eng., Part E: J. Process Mech. Eng. 230(5), 345–354 (2016)

    Article  Google Scholar 

  4. Rajagopal, K.R., Na, T.Y., Gupta, A.S.: Flow of a viscoelastic fluid over a stretching sheet. Rheologica Acta 23(2), 213–215 (1984)

    Article  Google Scholar 

  5. Seth, G.S., Bhattacharyya, A., Kumar, R., et al.: Entropy generation in hydromagnetic nanofluid flow over a non-linear stretching sheet with Naviers velocity slip and convective heat transfer. Phys. Fluids 30(12), 122003 (2012)

    Article  Google Scholar 

  6. Kai, Y., Zheng, B.L., Zhang, K.: Exact and asymptotic solutions to magnetohydrodynamic flow over a nonlinear stretching sheet with a power-law velocity by the homotopy renormalization method. Phys. Fluids 31(5), 063606 (2019)

    Article  Google Scholar 

  7. Makinde, O.D., Khan, Z.H., Ahmad, R., et al.: Numerical study of unsteady hydromagnetic radiating fluid flow past a slippery stretching sheet embedded in a porous medium. Phys. Fluids 30(8), 083601 (2018)

    Article  Google Scholar 

  8. Chiam, T.C.: Hydromagnetic flow over a surface stretching with a power-law velocity. Int. J. Eng. Sci. 33(3), 429–435 (1995)

    Article  MATH  Google Scholar 

  9. Afzal, N., Varshney, I.S.: The cooling of a low-heat-resistance stretching sheet moving through a fluid. Wärme- und Stoffübertragung 217(17), 217–219 (1983)

    Article  Google Scholar 

  10. Sakiadis, B.C.: Boundary-layer behavior on continuous solid surfaces: I. boundary-layer equations for two-dimensional and axisymmetric flow. AIChE J. 7(1), 26–28 (1961)

    Article  Google Scholar 

  11. Duck, P.W., Lasseigne, D.G., Hussaini, M.Y.: On the interaction between the shock wave attached to a wedge and freestream disturbances. Theor. Comput. Fluid Dyn. 7(2), 119–139 (1995)

    Article  MATH  Google Scholar 

  12. Duck, P.W., Lasseigne, D.G., Hussaini, M.Y.: The effect of three-dimensional freestream disturbances on the supersonic flow past a wedge. Phys. Fluids 9(2), 456–467 (1997)

    Article  Google Scholar 

  13. RUBAN, A.I.: On Tollmien-Schlichting wave generation by sound. In: Laminar-Turbulent Transition, Springer, Berlin, Heidelberg (1985)

  14. GOLDSTEIN, M.E.: Scattering of acoustic waves into Tollmien-Schlichting waves by small streamwise variations in surface geometry. J. Fluid Mech. 154(1), 509–529 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, Y., Dong, M., Wu, X.: Generation of first Mack modes in supersonic boundary layers by slow acoustic waves interacting with streamwise isolated wall roughness. J. Fluid Mech. 888(10), 1–42 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. Schwinger, J.: On quantum-electrodynamics and the magnetic moment of the electron. Phys. Rev. 73(4), 416–417 (1948)

    Article  MATH  Google Scholar 

  17. Tomonaga, S.I., Oppenheimer, J.R.: On infinite field reactions in quantum field theory. Phys. Rev. 74(2), 224–225 (1948)

    Article  MATH  Google Scholar 

  18. Feynman, R.: Relativistic cut-off for quantum electrodynamics. Phys. Rev. 74(10), 1430–1438 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  19. Goldenfeld, N., Martin, O., Oono, Y.: Intermediate asymptotics and renormalization group theory. J. Sci. Comput. 4(4), 355–372 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Goldenfeld, N., Martin, O., Oono, Y., et al.: Anomalous dimensions and the renormalization group in a nonlinear diffusion process. Phys. Rev. Lett. 64(12), 1361 (1990)

    Article  Google Scholar 

  21. Goldenfeld, N., Martin, O., Oono, Y.: Asymptotics of partial differential equations and the renormalization group. asymptotics beyond all orders. Springer, New York (1991)

    Google Scholar 

  22. Kunihiro, T.: A geometrical formulation of the renormalization group method for global analysis. Progr. Theor. Phys. 94(4), 503–514 (1995)

    Article  MathSciNet  Google Scholar 

  23. Kunihiro, T.: A geometrical formulation of the renormalization group method for global analysis II: partial differential equations. Japan J. Indus. Appl. Math. 14(1), 51–69 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kunihiro, T.: The renormalization-group method applied to asymptotic analysis of vector fields. Progr. Theor. Phys. 97(2), 179–200 (1997)

    Article  MathSciNet  Google Scholar 

  25. Liu, C.S.: The renormalization method based on the Taylor expansion and applications for asymptotic analysis. Nonlinear Dyn. 88(2), 1099–1124 (2017)

    Article  MATH  Google Scholar 

  26. Kai, Y.: Exact solutions and asymptotic solutions of one-dimensional domain walls in nonlinearly coupled system. Nonlinear Dyn. 92(4), 1665–1677 (2018)

    Article  MATH  Google Scholar 

  27. Kai, Y., Zheng, B.L., Zhang, K., et al.: Exact and asymptotic solutions to magnetohydrodynamic flow over a nonlinear stretching sheet with a power-law velocity by the homotopy renormalization method. Phys. Fluids 31(6), 063606 (2019)

    Article  Google Scholar 

  28. Liu, C.S.: The renormalization method from continuous to discrete dynamical systems: asymptotic solutions, reductions and invariant manifolds. Nonlinear Dyn. 94(2), 873–888 (2018)

    Article  Google Scholar 

  29. Morales-Delgado, V.F., Gómez-Aguilar, J.F., Yépez-Martínez, H., et al.: Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular. Adv. Diff. Eq. 2006(1), 164–180 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gómez-Aguilar, J.F., Yépez-Martínez, H., et al.: Series solution for the time-fractional coupled mKdV equation using the homotopy analysis method. Math. Probl. Eng. 2006(11), 1–21 (2016)

    Article  MathSciNet  Google Scholar 

  31. Gómez-Aguilar, J.F., Yépez-Martínez, H., et al.: Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel. Adv. Diff. Eq. 2007(1), 1–18 (2017)

    MathSciNet  MATH  Google Scholar 

  32. Yépez-Martínez, H., Gómez-Aguilar, J.F.: A new modified definition of Caputo-Fabrizio fractional-order derivative and their applications to the multi step homotopy analysis method (MHAM). J. Comput. Appl. Math. 346, 247–260 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  33. Saad, K.M., Al-Shareef, E.H.F.: On exact solutions for time-fractional Korteweg-de Vries and Korteweg-de Vries-Burgers equations using homotopy analysis transform method. Chin. J. Phys. 63, 149–162 (2020)

    Article  MathSciNet  Google Scholar 

  34. Qin, H., Dong, M.: Boundary-layer disturbances subjected to free-stream turbulence and simulation on bypass transition. Appl. Math. Mech. 37(8), 967–986 (2016)

    Article  MathSciNet  Google Scholar 

  35. Fazio, R.: A Non-Iterative Transformation Method for an Extended Blasius Problem. Math. Method Appl. Sci. 44(2), 1996–2001 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  36. Gajjarand, J., Turkyilmazo, M.: On the absolute instability of the triple-deck flow over humps and near wedged trailing edges. Philos. Trans. R. Soc., Part A: Math. Phys. Eng. Sci. 358(1777), 3113–3128 (2000)

    Article  MathSciNet  Google Scholar 

  37. Turkyilmazoglu, M.: Analytic approximate solutions of parameterized unperturbed and singularly perturbed boundary value problems. Appl. Math. Modell. 35(8), 3879–3886 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Turkyilmazoglu, M.: Is homotopy perturbation method the traditional Taylor series expansion. Hacettepe J. Math. Stat. 44(3), 651–657 (2005)

    MathSciNet  MATH  Google Scholar 

  39. Turkyilmazoglu, M., Uygun, N.: Compressible modes of the rotating-disk boundary-layer flow leading to absolute instability. Stud. Appl. Math. 115(1), 1–20 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Turkyilmazoglu, M.: On the purely analytic computation of laminar boundary layer flow over a rotating cone. Int. J. Eng. Sci. 47(9), 875–882 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. De, T.N., Ruban, A.I.: A numerical evaluation of the asymptotic theory of receptivity for subsonic compressible boundary layers. J. Fluid Mech. 771(21), 520–546 (2015)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 62072296.

Author information

Authors and Affiliations

  1. Center of Intelligent Computing and Applied Statistics, School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai, 201620, China

    Yue Kai & Zhixiang Yin

  2. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, China

    Kai Zhang

Authors
  1. Yue Kai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kai Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhixiang Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhixiang Yin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kai, Y., Zhang, K. & Yin, Z. HTR approach to the asymptotic solutions of supersonic boundary layer problem: the case of slow acoustic waves interacting with streamwise isolated wall roughness. Math Sci 17, 21–30 (2023). https://doi.org/10.1007/s40096-021-00436-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40096-021-00436-y

Keywords

Search

Navigation