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On non-stationary viscous incompressible flow through a cascade of profiles. (English) Zbl 1124.35054
The incompressible Newtonian flows in blade machines (turbines) is studied, related with some recent results of the authors. The flow domain is given by “cells” periodically spaced in one direction. Particular Dirichlet conditions are used at the end of “cells”. A weak formulation is obtained, using Sobolev and Sobolev-Slobodetskii spaces. An existence theorem of approximate solutions is given, using a time semi-discretization. A priori estimations and a limit process are used to obtain the existence of the solution.
However, some used inequalities are not so clear. For example, consider Lemma 1 (page 1914), in the particular case of a constant \( u \in H^1(D) \), where \(D\) is a domain in \(\mathbb{R}^N\). We obtain the following result: there exists a constant \(C\) such that (length of boundary of \(D)\) is less than \hbox{\(C\cdot\)(area of \(D)\).} It is clear that \(C\) depends on \(D\). Moreover, for thin domains (with very long boundary and small area), the constant \(C\) must be very large. Then all subsequent estimates give us a very slow convergence rate.

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI
[1] Feistauer, Applications of Mathematics 29 pp 423– (1984)
[2] Mathematical Methods in Fluid Dynamics. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 67. Longman Scientific & Technical: Harlow, 1993.
[3] Czibere, Acta Technica 4 pp 215– (1963)
[4] Hoffmeister, Maschinenbautechnik 10 pp 416– (1961)
[5] Kazimierski, Archiwum Budowy Maszyn (1966)
[6] Mani, Transactions of the ASME, Journal of Engineering for Power, Series A 90 pp 165– (1968)
[7] Rauchman, Izv. AN U.S.S.R., MZG 1 pp 83– (1971)
[8] Vallander, Dan U.S.S.R. 84 pp 673– (1972)
[9] , . A note on the influence of axial velocity ratio on cascade performance. Proceedings of the Symposium, ’Theoretical Prediction of Two- and Three-Dimensional Flows in Turbomachinery’, NASA-SP-304, Pennsylvania State University, Session I, Part I, 1974.
[10] Martensen, Zeitschrift fur Angewandte Mathematik und Mechanik 37 pp 302– (1957)
[11] Martensen, Archive for Rational Mechanics and Analysis 3 pp 253– (1959)
[12] Martensen, Mitt. Max-Planck-Inst. Strömungsforsch. Aerodynam 23 (1959)
[13] Berechnung der incompressiblen Potentialströmung für Einzel- und Gitterprofile nach Einer Variante des Martensens-Verfahrens. Bericht 63 RO 2 der Aerodynamischen, Versuchsanstalt Göttingen, 1963.
[14] Polášek, Applications of Mathematics 17 pp 295– (1972)
[15] Deconinck, Transactions of the ASME, Journal of Engineering for Power 103 pp 665– (1981)
[16] Dolejší, Zeitschrift fur Angewandte Mathematik und Mechanik 76 pp 301– (1996)
[17] Dolejší, Applications of Mathematics 47 pp 301– (2002)
[18] Ecer, AIAA Journal 19 pp 1174– (1981)
[19] Feistauer, Applications of Mathematics 26 pp 345– (1981)
[20] Feistauer, Zeitschrift fur Angewandte Mathematik und Mechanik 65 pp 191– (1985)
[21] Feistauer, Applications of Mathematics 34 pp 318– (1989)
[22] Feistauer, Zeitschrift fur Angewandte Mathematik und Mechanik 68 pp 381– (1988)
[23] Feistauer, Applications of Mathematics 31 pp 309– (1986)
[24] Feistauer, Numerische Mathematik 50 pp 655– (1987)
[25] Feistauer, Journal of Computational and Applied Mathematics 44 pp 131– (1992)
[26] Fürst, Mathematica Bohemica 2 pp 379– (2001)
[27] Fořt, Task Quarterly 6 pp 127– (2002)
[28] . Numerical Solution of Transonic Flows Through 2D and 3D Turbine Cascades. Computing and Visualization in Science, vol. 4(3). Springer: Berlin, 2002. · Zbl 1009.76061
[29] . On the numerical solution of the compressible subsonic gas flow. Report No. 1/84, Mathematics, University of Oulu.
[30] Numerical Computation of Internal and External Flows. Wiley: Chichester, New York, Brisbane, Toronto, Singapore, 1989.
[31] . A finite element method for flow calculations in turbomachines. Report V.U.B. STR. 5, Vrije Universiteit Brussel, July 1974.
[32] Hirsch Ch, Transactions of the ASME, Journal of Fluids Engineering 98 pp 403– (1976)
[33] The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach: New York, London, Paris, 1969 (newer Russian edition: Nauka, Moscow, 1970).
[34] . Finite Element Approximation of the Navier–Stokes Equations. Lecture Notes in Mathematics, vol. 749. Springer: Berlin, Heidelberg, New York, 1979. · Zbl 0413.65081
[35] Navier–Stokes Equations. North-Holland: Amsterdam, New York, Oxford, 1977.
[36] . Experimental analysis data on the transonic flow past a plane turbine. ASME Paper 90-GT-313, New York, 1990.
[37] , . Mathematical and Computational Methods for Compressible Flow. Clarendon Press: Oxford, 2003. · Zbl 1028.76001
[38] Bruneau, Mathematical Modelling and Numerical Analysis 30 pp 815– (1996)
[39] Les Méthodes Directes en Theorie des Équations Elliptiques. Academia: Prague, Masson: Paris, 1967.
[40] , . Function Spaces. Academia: Prague, 1977.
[41] . On Some Aspects of Analysis of Incompressible Flow through Cascades of Profiles. Operator Theory Advances and Applications, vol. 147. Birkhäuser: Basel/Switzerland, 2004; 257–276. · Zbl 1054.35051
[42] Method of Rothe in Evolution Equations. Leipzig: Teubner, 1985. · Zbl 0582.65084
[43] The Method of Discretization in Time and Partial Differential Equations. Reidel: Dodrecht, 1982.
[44] . Regularity of viscous Navier–Stokes flows in nonsmooth domains. In Boundary Value Problems and Integral Equations in Nonsmooth Domains, et al. (ed.), Proceedings of the Conference, Held at the CIRM, Luminy, France, May 3–7, 1993. Lecture Notes Pure Applied Mathematics. vol. 167. Marcel Dekker: New York, NY, 1995; 185–201 [ISBN 0-8247-9320-X/pbk].
[45] Regularitätsuntersuchungen und FEM–Fehlerabschätzungen für algemeine Randwertprobleme der Navier–Stokes–Gleichungen. Ph.D. Dissertation, Mathematisch-Naturwissenschaftliche Fakultät, Universität Rostock, 1997.
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