Handling of temperature dependence of viscosity in problems of incompressible medium flow around a cylinder
Tóm tắt
The viscous incompressible medium (water, air) flow past a circular cylinder is considered with regard for the temperature T dependent viscosity v. The influence of different boundary conditions for temperature on flow structure, the drag coefficient and its components due to the pressure and viscosity is investigated in the problem of the flow past a cylinder at rest for the (diameter-based) Reynolds number ReD = 40. A relation between the viscosity gradient along a normal to the body surface and the integral vorticity flux from the body surface into the boundary layer is discussed. Unlike the constant viscosity case the vorticity flux may be different from zero, which must lead because of the integral conservation law for the vorticity to an alteration of the far-field boundary conditions for the velocity. In the same connection, the problem is analysed on the heat spot entry into the computational region under consideration for the flow past a circular cylinder. The examples of the symmetrization of separated flow past a cylinder performing rotation oscillations in a uniform free stream (the Taneda problem) are considered. A comparison with flow computations for low Mach numbers M « 1 for the flow of a medium past a cylinder at rest is carried out. At the computation of the equation for heat transfer under the assumption of incompressibility of such media as air, it is proposed to retain the pressure derivative, which is typical of gases. In this case, a better agreement with the computations of compressible flows (for M « 1) is achieved, for example, at the determination of the sizes of a symmetric zone of flow separation past a circular cylinder. An unsteady flow in the neighborhood of the point of joining the zero streamline bounding a closed region of separated flow (the cavity) in a wake of the cylinder at rest is obtained by a numerical simulation at the Reynolds number equal to 40.
Tài liệu tham khảo
N.A. Slezkin, Viscous Fluid Dynamics, Gostekhteoretizdat, Moscow, 1985.
Yu.D. Chashechkin, Yu.S. Ilyinykh, Yu.V. Kistovich, and V.V. Milyutin, Short Internal Waves in a Continuously Stratified Fluid. A Textbook and Methodical Instructions for Laboratory Works, Prepr. of the Inst. of Problems in Mechanics RAS, No. 633, Inst. of Problems in Mechanics RAS, Moscow, 1998.
C. Torres, H. Hanazaki, J. Ochoa, J. Castillo, and M. van Woert, Flow past a sphere moving vertically in a stratified diffusive fluid, J. Fluid Mech, 2000, Vol. 417, P. 211–236.
P. Roache, Computational Fluid Dynamics, Hermosa, Albuquerque, N.M., 1976.
P.M. Gresho and R.I. Lee, Don’t suppress the wiggles — they’re telling you something, Comp. and Fluids, 1981, Vol. 9, P. 223–253.
G.I. Marchuk, Methods of Numerical Mathematics, Nauka, Moscow, 1977.
A. Arakava, Computational design of longterm numerical integration of the equations of fluid motion. I. Two-dimensional incompressible flow, J. Comput. Phys., 1966, Vol. 1, No. 1, P. 119–143.
A.A. Samarsky and Yu.P. Popov, Difference Schemes of Gas Dynamics, Editorial URSS, 2004.
Yu.I. Shokin and N.N. Yanenko, Differential Approximation Method. Application to Gas Dynamics, Nauka, Novosibirsk, 1985.
N.J. Chung and J.I. Sohn, Interactions of Coupled Acoustic and Vortical Instability, AIAA J., 1986, Vol. 24, No. 10, P. 1582–1596.
Wang Meng, K. Sanjiva, K. Lele, and M. Parviz, Computation of quadropole noise using acoustic analogy, AIAA J., 1996, Vol. 34, No. 11, P. 2247–2254.
M.N. Zakharenkov, Thermal processes in viscous incompressible flow around a circular cylinder performing rotational oscillations, in: The Fourth Int. Conf. on New Energy Systems and Conversions (NESC’99), Osaka, 1999.
M. Kawaguti, Numerical solution of the Navier — Stokes equations for the flow around a circular cylinder at Reynolds number 40, J. Phys. Soc. Japan, 1953, Vol. 8, No. 6, P. 747–757.
S. Taneda, Experimental investigation of the wakes behind cylinders and plates at low Reynolds numbers, J. Phys. Soc. Japan, 1956, Vol. 11./
C.J. Apelt, The steady flow of a viscous fluid past a circular cylinder at Reynolds number 40 and 44, Aero. Res. Council., London, R&M, 1958, No. 3175, P. 1–28.
D.J. Tritton, Experiments on the flow past a circular cylinder at low Reynolds numbers, J. Fluid Mech., 1959, Vol. 6, No. 4, P. 547–567.
A.S. Grove, F.H. Shair, E.E. Peterson, and A. Acrivos, An experimental investigation of the steady separated flow past a circular cylinder, Ibid., 1964, Vol. 19, Pt 1, P. 60–80.
M. Kawaguti and P. Jain, Numerical study of a viscous fluid flow past a circular cylinder, J. Phys. Soc. Japan, 1966, Vol. 21.
H.B. Keller and H. Takami, Numerical studies of steady viscous flow about cylinders, in: Numerical solutions of nonlinear differential equations, D. Greenspan (Ed.), Willey, New York, 1966.
A. Acrivos, L.G. Leal, D.D. Snowden, and P. Pan, Further experiments on steady separated flows past bluff objects, J. Fluid Mech., 1968, Vol. 34, Pt 1, P. 25–48.
J.S. Son and T.J. Hanratty, Numerical solutions for the flow around a cylinder at Reynolds numbers of 40, 200 and 500, Ibid., 1969, Vol. 35, P. 369–386.
P.C. Jain and K.S. Rao, Numerical solution of unsteady viscous incompressible fluid flow past a circular cylinder, Phys. Fluids Suppl., 1969, Vol. 12, Pt II, Paper No. 57.
A.E. Hamielec and J.D. Roal, Numerical study of viscous flow around circular cylinder, Phys. Fluids, 1969, Vol. 12, No. 1, P. 11–17.
S.C.R. Dennis and G.Z. Chang, Numerical solution for steady flow past a circular cylinder at Reynolds numbers up to 100, J. Fluid Mech., 1970, Vol. 42.
W.M. Collins and S.C.R. Dennis, Flow past an impulsively started circular cylinder, Ibid., 1973, Vol. 60.
F. Nieuwstadt and H.B. Keller, Viscous flow past circular cylinder, Comp. and Fluids, 1973, Vol. 1, P. 59–71.
Ta Phuok Loc. Etude numerique de l’ecoulement d’un fluide visqueux incompressible autour d’un cylin-dre-fixe ou en rotation. Effect Magnus, J. Mec., 1975, Vol. 14, P. 109–134.
M.M. Zdrawkovich, Flow around circular cylinder. Vol. Fundamentals, Oxford University Press, Oxford, 1977.
M. Coutanceau and R. Bourd, Experimental determination of the main features of the viscous flow in the wake of a circular cylinder in uniform translation. Pt 2. Unsteady flow, J. Fluid Mech., 1977, Vol. 79, No. 2, P. 257–272.
B. Fornberg, A numerical study of steady viscous flow past a circular cylinder, Ibid., 1980, Vol. 98, P. 819.
J.R. Chaplin, History forces and the unsteady wake of a cylinder, Ibid., 1999, Vol. 393, No. 25, P. 99–121.
H. Schlichting, Boundary Layer Theory, 6th Ed., McGraw Hill, New York, 1968.
A.A. Samarsky, Theory of Difference Schemes, Nauka, Moscow, 1977.
M.N. Zakharenkov, Unsteady detached separation from a circular cylinder performing rotational oscillations in a uniform viscous incompressible flow, Int. J. Numer. Meth. in Fluids, 1997, Vol. 25, P. 125–142.
M.N. Zakharenkov, Pressure uniqueness at the solution of the Navier — Stokes equations in the variables of stream function and vorticity, Matem. modelirovanie, 1998, Vol. 10, No. 1, P. 3–10.
D. Potter, Computational Physics, Wiley, London, 1973.
M.N. Zakharenkov, Approximation of the boundary condition for vorticity on the solid body surface at the solution of the Navier — Stokes equations in the variables of stream function and vorticity, Numer. Methods of Continuum Mechanics, IT AM SB USSR, Novosibirsk, 1980, Vol. 11, No. 7, P. 56–74.
E.L. Tarunin, Analysis of approximation formulas for vorticity on a solid boundary, Uchen. Zap. Perm Pedagogic Inst. Hydrodynamics, 1976, Vol. 152, Iss. 9, P. 167–178.
N. Kasagi, Application of direct numerical simulation in fundamental research of turbulent transport phenomena, in: Proc. Ninth Int. Symp. on Transport Phenomena (ISTP-9) in Thermal-Fluids Engng., Vol. 1, Singapore, 1996, P. 32–45.
B. Fornberg, Steady viscous flow past a circular cylinder up to Reynolds number 600, J. Comput. Phys., 1985, Vol. 61, No. 2, P. 297–320.
A.V. Volkov and S.V. Lyapunov, Private communication, Zhukovsky GUP TsAGI, Zhukovsky, Moscow Region, 2001.
A. Thom and K. Apelt, Numerical Computations of Fields in Technology and Physics, Energiya, Moscow, Leningrad, 1969.
M.N. Zakharenkov, Realization of dynamic boundary conditions for Navier — Stokes equations in the variables of stream function and vorticity. Realization of necessary differential constraints on a solid surface in the fourth-order difference schemes at the solution of the N-S equations in f-Q variables, in: Abstr. XII All-Russ. Conf. “Theoretical Fundamentals of the Construction of Numerical Algorithms for Solving the Mathematical Physics Problems”, Novorossiysk, 1998.
M.N. Zakharenkov, Implementation of perfect and partial slip boundary conditions in solving the Navier — Stokes equations in terms of stream function and vorticity, Comp. Math. Math. Phys. 2001, Vol. 41, No. 5, P. 751–761
M.N. Zakharenkov, On the question of satisfaction of dynamic boundary conditions in the problem of viscous incompressible fluid flow around a body, in: Abstr. XII Int. Conf. on Comput. Mechanics and Advanced Applied Program Systems (VMSPPS’2003), Vol. 1, 30 June–5 July 2003, Vladimir, 2003, P. 288–290.
M.J. Lighthill, On displacement thickness, J. Fluid Mech., 1959, P. 383–392.
M.N. Zakharenkov, The heat processes in attached vortical layer around a circular cylinder performing rotational oscillations in the uniform viscous incompressible flow, in: Abstr. Int. Conf. “Flows and Structures in Fluid”, St. Petersburg, 1999, P. 146–148.
M.N. Zakharenkow, Unsteady incompressible viscous flow past an airfoil, Arch. Mech., 1990, Vol. 42, Nos. 4–5, P. 609–615.
J.C. Wu, Theory for aerodynamic force and moment in viscous flows, AIAA J., 1981, Vol. 19, No. 4, P. 432–441.
I.V. Zelenov and V.Ya. Shkadov, Viscous fluid flow around an airfoil, Fluid Dyn., 1986, Vol. 21, No. 4, P. 29–37.
C.R. Anderson, Domain decomposition techniques and the solution of Poisson’s equation in infinite domains, in: Domain Decomposition Methods, T.F. Chan, R. Glowinski, J. Periaux, and O.B. Widlund (Eds.), SIAM, Philadelphia, 1989, P. 129–139.
H. Schlichting, Grenzschicht — Theorie, 5-te Auflage. Verlag G. Braun, Karlsruhe, 1965.
H. Schlichting, Boundary Layer Theory, 7th ed., McGraw-Hill, New York, 1979.
V.V. Sychev, Asymptotic theory of separated flows, Fluid Dyn., 1982, Vol. 17, No. 2, P. 20–30.
S. Tomotika and T. Aoi, Steady motion of sphere and cylinder in viscous incompressible fluid at low Re numbers, in: Mekhanika. Collection of Shortened Translations and Abstracts of Foreign Periodical Literature, Iss. 2 (XII), Izdatel’stvo inostr. lit., Moscow, 1952, P. 46–63.
I.V. Sturova, Problems of radiation and diffraction for a circular cylinder in stratified fluid, Fluid Dyn., 1999, Vol. 34, No. 4, P. 81–94.
J.B. Keller and D. Jivoli, Exact non-reflecting boundary conditions, J. Comput. Phys., 1989, Vol. 82, P. 172–192.
G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, London, 1967.