×

zbMATH — the first resource for mathematics

Analysis and application of ellipticity of stability equations on fluid mechanics. (English) Zbl 1145.76381
Summary: By using characteristic analysis of the linear and nonlinear parabolic stability equations (PSE), PSE of primitive disturbance variables are proved to be parabolic in total. By using sub-characteristic analysis of PSE, the linear PSE are proved to be elliptical and hyperbolic-parabolic for velocity \(U\) in subsonic and supersonic, respectively; the nonlinear PSE are proved to be elliptical and hyperbolic-parabolic for velocity \(U+u\) in subsonic and supersonic, respectively. Methods are gained that remove the remaining ellipticity from the PSE by characteristic and sub-characteristic theories, the results for the linear PSE are consistent with the known results, and the influence of the Mach number is also given out. At the same time, the methods of removing the remained ellipticity are further obtained from the nonlinear PSE.
MSC:
76E30 Nonlinear effects in hydrodynamic stability
35Q35 PDEs in connection with fluid mechanics
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Herbert Th, Bertolotti F P. Stability analysis of non-parallel boundary layers[J].Bull American Phys Soc, 1987,32(8):2097. · Zbl 0748.76050
[2] GAO Zhi. Grade structure theory for the basic equations of fluid mechanics (BEFM) and the simplied Navier-Stokes equations (SNSE) [J].Acta Mechanica Sinica, 1988,20(2), 107–116. (in Chinese)
[3] Herbert Th. Nonlinear stability of parallel flows by high-order amplitude expansions[J].AIAA J, 1980,18(3):243–248. · Zbl 0427.76046 · doi:10.2514/3.50755
[4] Haj-Hariri H. Characteristics analysis of the parabolic stability equations [J].Stud Appl Math, 1994,92(1):41–53. · Zbl 0802.76025
[5] Chang C L, Malik M R, Erleracher G,et al. Compressible stability of growing boundary layers using parabolic stability equations[Z].AAIA 91–1636, New York: AAIA, 1991.
[6] GAO Zhi. Grade structure of simplified Navier-Stokes equations and its mechanics meaning and application[J].Science in China, Ser A, 1987,17(10):1058–1070. (in Chinese)
[7] GAO Zhi, ZHOU Guang-jiong. Some advances in high Reynolds numbers flow theory, algorithm and application[J].Advances in Mechanics, 2001,31(3):417–436.
[8] GAO Zhi, SHEN Yi-qing. Discrete fluid dynamics and flow numerical simulation [A]. In: F Dubois, WU Hua-mu Eds.New Advances in Computational Fluid Dynamics [C]. Beijing: Higher Education Press, 2001, 204–229.
[9] Schlichting H.Boundary-Layer Theory[M]. 7th ed. New York: McGraw-Hill, 1979. · Zbl 0434.76027
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.