Asymptotic equations of gas dynamics: qualitative analysis, construction of solutions, and applications

P. A. Vel'misov ; J. А. Tamarova ; E. P. Semеnova

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the IV International Scientific Conference "Actual Problems of Applied Mathematics". Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26, 2018. Part I, Tome 165 (2019), pp. 47-62

Résumé

In this paper, we propose asymptotic expansions for the velocity potential and obtain asymptotic equations of gas dynamics for irrotational isentropic flows of an ideal gas: an equation of linear theory, a nonlinear equation for supersonic flows, and a nonlinear transonic equation. We construct some exact particular solutions for the asymptotic nonlinear transonic equation, which takes into account transverse perturbations. Based on a linear asymptotic equation, we examine the dynamic stability of an elastic deformable element of a channel at a subsonic flow rate of a gas or liquid. The study of stability is carried out in a statement corresponding to small perturbations of a homogeneous flow and small deformations of an elastic element, and is based on the construction of a positive definite functional. Sufficient stability conditions are obtained.

Détail

Keywords: aerodynamics, partial differential equation, asymptotic expansion, transonic gas flow, channel, de Laval nozzle, aerohydroelasticity, dynamic stability.

Bibliographie

[1] Ankilov A. V., Velmisov P. A., Ustoichivost vyazkouprugikh elementov stenok protochnykh kanalov, UlGTU, Ulyanovsk, 2000

[2] Ankilov A. V., Velmisov P. A., Dinamika i ustoichivost uprugikh plastin pri aerogidrodinamicheskom vozdeistvii, UlGTU, Ulyanovsk, 2009

[3] Ankilov A. V., Velmisov P. A., Matematicheskoe modelirovanie v zadachakh dinamicheskoi ustoichivosti deformiruemykh elementov konstruktsii pri aerogidrodinamicheskom vozdeistvii, UlGTU, Ulyanovsk, 2013

[4] Ankilov A. V., Velmisov P. A., “Ustoichivost reshenii nekotorykh klassov integro-differentsialnykh uravnenii v chastnykh proizvodnykh”, Vestn. Samar. gos. un-ta. Ser. estestvennonauch., 8:1 (67) (2008), 331–344

[5] Ankilov A. V., Velmisov P. A., “Matematicheskoe modelirovanie dinamiki i ustoichivosti uprugikh elementov kryla”, Vestn. Saratov. gos. tekhn. un-ta., 1:37 (2009), 7–16

[6] Ankilov A. V., Velmisov P. A., “Ob ustoichivosti reshenii uravnenii vzaimodeistviya uprugikh stenok kanalov s protekayuschei zhidkostyu”, Vestn. Samar. gos. tekhn. un-ta. Ser. fiz.-mat. nauki., 1:22 (2011), 179–185 | DOI

[7] Ankilov A. V., Velmisov P. A., “Issledovanie dinamiki i ustoichivosti uprugogo elementa konstruktsii pri sverkhzvukovom obtekanii”, Vestn. Saratov. gos. tekhn. un-ta., 1:3 (57) (2011), 59–67

[8] Ankilov A. V., Velmisov P. A., Funktsionaly Lyapunova v nekotorykh zadachakh dinamicheskoi ustoichivosti aerouprugikh konstruktsii, UlGTU, Ulyanovsk, 2015

[9] Ankilov A. V., Velmisov P. A., Kazakova Yu. A., “Ustoichivost reshenii odnoi nelineinoi nachalno-kraevoi zadachi aerouprugosti”, Vestn. Samar. gos. tekhn. un-ta. Ser. fiz.-mat. nauki., 2:31 (2013), 120–126 | DOI

[10] Ankilov A. V., Velmisov P. A., Semenova E. P., “Issledovanie dinamicheskoi ustoichivosti uprugikh elementov stenok kanala”, Vestn. Saratov. gos. tekhn. un-ta., 1:2 (38) (2009), 7–17 | MR

[11] Ankilov A. V., Velmisov P. A., Semenova E. P., “O resheniyakh integro-differentsialnykh uravnenii v zadache dinamiki odnoi aerouprugoi sistemy tipa «tandem»”, Vestn. Samar. gos. tekhn. un-ta. Ser. fiz.-mat. nauki., 2011, no. 2 (23), 266–271 | DOI

[12] Ankilov A. V., Velmisov P. A., Tamarova Yu. A., “Matematicheskaya model vibratsionnogo ustroistva”, Avtomat. protsessov upravl., 2014, no. 3 (37), 58–67

[13] Ankilov A. V., Velmisov P. A., Tamarova Yu. A., “Dinamicheskaya ustoichivost uprugogo elementa protochnogo kanala”, Izv. vuzov. Povolzh. region. Fiz.-mat. nauki., 2014, no. 3 (31), 40–55

[14] Barmetov Yu. P., Dobodeich I. A., “K raschetu nestatsionarnykh techenii szhimaemoi zhidkosti v truboprovode”, Izv. vuzov. Aviats. tekhn., 2006, no. 1, 18–21

[15] Velmisov P. A., “K voprosu o neustanovivshemsya dvizhenii gaza v soplakh Lavalya”, Izv. vuzov. Mat., 1976, no. 12, 3–10

[16] Velmisov P. A., “O edinstvennosti resheniya pryamoi zadachi sopla Lavalya”, Izv. vuzov. Mat., 1979, no. 1, 15–17 | Zbl

[17] Velmisov P. A., “Asimptoticheskoe issledovanie nelineinykh effektov v zadache o nestatsionarnom sverkhzvukovom obtekanii profilya”, Prikl. mat. mekh., 43 (1979), 30–37 | Zbl

[18] Velmisov P. A., Kazakova Yu. A., “O parametricheskikh resheniyakh differentsialnykh uravnenii s chastnymi proizvodnymi; prilozheniya v transzvukovoi gazovoi dinamike”, Zh. Srednevolzh. mat. ob-va., 2010, no. 4, 30–36

[19] Velmisov P. A., Molgachev A. A., Matematicheskoe modelirovanie v zadachakh dinamicheskoi ustoichivosti vyazkouprugikh elementov protochnykh kanalov, UlGTU, Ulyanovsk, 2012

[20] Velmisov P. A., Manzhosov V. K., Matematicheskoe modelirovanie v zadachakh dinamiki vibroudarnykh i aerouprugikh sistem, UlGTU, Ulyanovsk, 2014

[21] Velmisov P. A., Matsenko P. K., “O nekotorykh zadachakh vneshnego prostranstvennogo obtekaniya tel okolozvukovym potokom gaza”, Izv. vuzov. Mat., 1986, no. 9, 10–16 | MR | Zbl

[22] Velmisov P. A., Gorbokonenko V. D, Reshetnikov Yu. A., “Matematicheskoe modelirovanie mekhanicheskoi sistemy «truboprovod—datchik davleniya»”, Datchiki i sistemy., 2003, no. 6 (49), 12–15

[23] Velmisov P. A., Reshetnikov Yu. A., Ustoichivost vyazkouprugikh plastin pri aerogidrodinamicheskom vozdeistvii, Izd-vo Saratov. un-ta, Saratov, 1994

[24] Velmisov P. A., Tamarova Yu. A., “Asimptoticheskie uravneniya nelineinykh transzvukovykh techenii gaza i ikh resheniya”, Zh. Srednevolzh. mat. ob-va., 15:4 (2013), 70–76 | Zbl

[25] Velmisov P. A., Tamarova Yu. A., “Asimptoticheskie modelnye uravneniya transzvukovykh techenii gaza”, Vestn. UlGTU., 2013, no. 3 (63), 53–58

[26] Velmisov P. A., Tamarova Yu. A., “Matematicheskoe modelirovanie transzvukovykh techenii”, Avtomat. protsessov upravl., 2014, no. 1 (35), 47–54 | MR

[27] Velmisov P. A., Falkovich S. V., “Neustanovivshiesya techeniya gaza v soplakh Lavalya s mestnymi sverkhzvukovymi zonami”, Prikl. mat. mekh., 39:2 (1975), 271–279 | Zbl

[28] Velmisov P. A., Falkovich S. V., “K teorii okolozvukovykh techenii vyazkogo gaza”, Izv. vuzov. Mat., 1974, no. 5, 52–61 | Zbl

[29] Velmisov P. A., Reshetnikov Yu. A., Kolmanovskii E. E., “Ustoichivost uravnenii vzaimodeistviya vyazkouprugikh plastin s zhidkostyu”, Differ. uravn., 30:11 (1994), 1966–1981 | MR | Zbl

[30] Guderlei K. G., Teoriya okolozvukovykh techenii, IL, M., 1960

[31] Ershov B. A., Kuteeva G. A., “Kolebaniya idealnoi zhidkosti v pryamougolnom sosude s uprugoi vstavkoi na stenke. Uchet vnutrennego treniya v materiale vstavki”, Vestn. Sankt-Peterb. un-ta., 1:2 (2005), 86–94

[32] Zaslavskii B. I., Klepikova N. A., “Ob odnom klasse tochnykh chastnykh reshenii uravnenii okolozvukovykh techenii gaza”, Prikl. mat. tekhn. fiz., 1965, no. 6, 65–68

[33] Zvyagin A. V., “Dvizhenie vyazkoi zhidkosti v kanale s uprugimi granitsami”, Vestn. Mosk. un-ta. Ser. 1. Mat. mekh., 2005, no. 1, 50–54 | Zbl

[34] Kazakova Yu. A., “O nekotorykh klassakh reshenii uravnenii aerogidromekhaniki”, Vestn. Samar. gos. tekhn. un-ta. Ser. fiz.-mat. nauki., 2011, no. 2 (23), 289–294 | DOI

[35] Kollatts L., Zadachi na sobstvennye znacheniya, Nauka, M., 1968

[36] Lifshits Yu. B., Ryzhov O. S., “Ob asimptoticheskom tipe ploskoparallelnogo techeniya v okrestnosti sopla Lavalya”, Dokl. AN SSSR., 154:2 (1964), 290–293 | Zbl

[37] Lifshits Yu. B., Ryzhov O. S., “O prichinakh obrazovaniya udarnykh voln v soplakh Lavalya”, Dokl. AN SSSR., 154:5 (1964), 1052–1055 | Zbl

[38] Lifshits Yu. B., Ryzhov O. S., “O nekotorykh tochnykh resheniyakh uravnenii transzvukovykh techenii gaza”, Zh. vychisl. mat. mat. fiz., 4:5 (1964), 954–958 | Zbl

[39] Mogilevich L. I., Popova A. A., “Dinamika vzaimodeistviya uprugoi tsilindricheskoi obolochki s laminarnym potokom zhidkosti vnutri nee primenitelno k truboprovodnomu transportu”, Nauka i tekhn. transport., 2007, no. 2, 69–72

[40] Mokeev V. V., “Konechno-elementnoe reshenie zadachi gidrouprugosti dlya vyazkouprugoi zhidkosti”, Probl. mashinostr. nadezhn. mashin., 2005, no. 2, 9–86

[41] Ovsyannikov L. V., “Issledovanie gazovykh techenii s pryamoi dozvukovoi liniei”, Tr. Leningrad. Voenno-Vozdushn. inzh. akad. im. A. F. Mozhaiskogo., 1950, no. 33, 3–24

[42] Ryzhov O. S., “Nekotorye vyrozhdennye okolozvukovye techeniya”, Prikl. mat. mekh., 22:2 (1958), 260–264 | Zbl

[43] Ryzhov O. S., Issledovanie transzvukovykh techenii v soplakh Lavalya, VTs AN SSSR, M., 1965

[44] Ryzhov O. S., “O rabote sopel Lavalya v neraschetnykh rezhimakh”, Zh. vychisl. mat. mat. fiz., 7:4 (1967), 859–866 | Zbl

[45] Ryzhov O. S., Shefter G. M., “O vliyanii vyazkosti i teploprovodnosti na strukturu szhimaemykh techenii”, Prikl. mat. mekh., 28:6 (1964), 996–1007

[46] Sokolov V. G., Bereznev A. V., “Uravneniya dvizheniya krivolineinogo uchastka truboprovoda s potokom zhidkosti”, Izv. vuzov. Neft i gaz., 2004, no. 6, 76–80

[47] Falkovich S.V., “K teorii sopla Lavalya”, Prikl. mat. mekh., 1946, no. 10, 503–512 | Zbl

[48] Frankl F. I., Izbrannye trudy po gazovoi dinamike, Nauka, M., 1973

[49] Shifrin E. G., Potentsialnye i vikhrevye transzvukovye techeniya idealnogo gaza, Fizmatlit, M., 2001

[50] Adamson T. C., “Unsteady transonic flows in two-dimensional channels”, J. Fluid Mech., 52:3 (1972), 437–449 | DOI | Zbl

[51] Vel'misov P. A., Tamarova Ju. A., “Asymptotic research of transonic gas flows”, AIP Conf. Proc., 1910 (2017), 040002 | DOI

[52] Vel'misov P. A., Todorov M. D., Kazakova Ju. A., “Some classes of the solutions of aerohydromechanic equations”, AIP Conf. Proc., 1067 (2009), 427–441 | MR

[53] Karman T., “The similarity law of transonic flow”, J. Math. Phys., 26:3 (1947), 182–190 | DOI | MR | Zbl

[54] Lin C. C., Reissner E., Tsien H. S., “On two-dimensional non-steady motion of a slender body in a compressible fluid”, J. Math. Phys., 27:3 (1948), 220–231 | DOI | MR | Zbl

[55] Paidoussis M. P., “The canonical problem of the fluid-conveying pipe and radiation of the knowledge gained to other dynamics problems across Applied Mechanics”, J. Sound Vibr., 310:3 (2008), 462–492 | DOI | MR

[56] Sichel M., “Structure of weak non-Hygoniot shocks”, Phys. Fluids., 6:5 (1963), 653–662 | DOI | MR | Zbl

[57] Sichel M., “The effect of longitudinal viscosity on the flow at a nozzle throat”, J. Fluid. Mech., 25:4 (1966), 769–786 | DOI | Zbl

[58] Tomotika S., Tamada K., “Studies on two-dimensional transonic flows of compressible fluids”, Quart. Appl. Math., 7:4 (1949), 381–397 | DOI | MR

[59] Tomotika S., Hasimoto Z., “On the transonic flow of a compressible fluid through an axially symmetrical nozzle”, J. Math. Phys., 29:2 (1950), 105–117 | DOI | MR | Zbl