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Specified discharge velocity models for numerical simulations of laminar vortex rings. (English) Zbl 1234.76015

Summary: We numerically and theoretically investigate the flow generated at the exit section of a piston/cylinder arrangement that is generally used in experiments to produce vortex rings. Accurate models for the velocity profile in this section (also called specified discharge velocity, SDV models) are necessary in (i) numerical simulations of laminar vortex rings that do not compute the flow inside the cylinder and (ii) in slug-models that provide a formula for the total circulation of the flow. Based on the theoretical and numerical analysis of the flow evolution in the entrance region of a pipe, we derive two new and easy to implement SDV models. A first model takes into account the unsteady evolution of the centerline velocity, while the second model also includes the time variation of the characteristics of the boundary layer at the exit plane of the vortex generator. The models are tested in axisymmetric direct numerical simulations of vortex rings. As distinguished from classical SDV model, the new models allow to accurately reproduce the characteristics of the flow. In particular, the time evolution of the total circulation is in good agreement with experimental results and previous numerical simulations including the vortex generator. The second model also provides a more realistic time evolution of the vortex ring circulation. Using the classical slug-model and the new correction for the centerline velocity, we finally derive a new and accurate analytical expression for the total circulation of the flow.

MSC:

76D17 Viscous vortex flows
76M20 Finite difference methods applied to problems in fluid mechanics
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[1] Abramowitz M., Stegun I.A.: Handbook of mathematical functions with formulas, graphs and mathematical tables. National Bureau of Standards, Dover (1972) · Zbl 0543.33001
[2] Archer P.J., Thomas T.G., Coleman G.N.: Passive scalar mixing in vortex rings. J. Fluid Mech. 598, 201–226 (2008) · Zbl 1151.76516 · doi:10.1017/S0022112007009883
[3] Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (1988) · Zbl 0958.76001
[4] Danaila I., Hélie J.: Numerical simulation of the postformation evolution of a laminar vortex ring. Phys. Fluids 20, 073602 (2008) · Zbl 1182.76178 · doi:10.1063/1.2949286
[5] Dabiri J.O., Gharib M.: A revised slug model boundary layer correction for starting jet vorticity flux. Theor. Comput. Fluid Dyn. 17, 293–295 (2004) · Zbl 1082.76031 · doi:10.1007/s00162-004-0106-8
[6] Dabiri J.O., Gharib M.: Starting flow through nozzles with temporally variable exit diameter. J. Fluid Mech. 538, 111–136 (2005) · Zbl 1108.76308 · doi:10.1017/S002211200500515X
[7] Fargie D., Martin B.W.: Developing laminar flow in a pipe of circular cross-section. Proc. R. Soc. Lond.. Ser. A, Math. Phys. Sci. 321, 461–476 (1971) · doi:10.1098/rspa.1971.0043
[8] Gharib M., Rambod E., Shariff K.: A universal time scale for vortex ring formation. J. Fluid Mech. 360, 121–140 (1998) · Zbl 0922.76021 · doi:10.1017/S0022112097008410
[9] Heeg R.S., Riley N.: Simulations of the formation of an axisymmetric vortex ring. J. Fluid Mech. 339, 199–211 (1997) · Zbl 0890.76052 · doi:10.1017/S002211209700517X
[10] Hettel M., Wetzel F., Habisreuther P., Bockhorn H.: Numerical verification of the similarity laws for the formation of laminar vortex rings. J. Fluid Mech. 590, 35–60 (2007) · Zbl 1141.76359 · doi:10.1017/S0022112007007677
[11] James S., Madnia C.K.: Direct numerical simulation of a laminar vortex ring. Phys. Fluids 8, 2400–2414 (1996) · Zbl 1027.76525 · doi:10.1063/1.869041
[12] Jiang H., Grosenbaugh M.A.: Numerical simulation of vortex ring formation in the presence of background flow with implications for squid propulsion. Theor. Comput. Fluid Dyn. 20, 103–123 (2006) · Zbl 1168.76316 · doi:10.1007/s00162-006-0010-5
[13] Kim J., Moin P.: Application of a fractional step method to incompressible navier–stokes equations. J. Comput. Phys. 59, 308–323 (1985) · Zbl 0582.76038 · doi:10.1016/0021-9991(85)90148-2
[14] Krueger P.S.: Circulation and trajectories of vortex rings formed from tube and orifice openings. Phys. D 237, 2218–2222 (2008) · Zbl 1143.76427 · doi:10.1016/j.physd.2008.01.004
[15] Krueger P.S., Gharib M.: An over-pressure correction to the slug model for vortex ring calculation. J. Fluid Mech. 545, 427–443 (2005) · Zbl 1085.76507 · doi:10.1017/S0022112005006853
[16] Lim, T.T., Nickels, T.B.: Vortex Rings, vol. Vortices in Fluid Flows. pp. 95. Kluwer, Dordrecht (1995)
[17] Michalke A.: Survey on jet instability theory. Prog. Aerospace Sci. 21, 159–199 (1984) · doi:10.1016/0376-0421(84)90005-8
[18] Moffatt H.K.: Generalised vortex rings with and without swirl. Fluid Dyn. Res. 3, 22–30 (1988) · doi:10.1016/0169-5983(88)90040-8
[19] Mohanty A.K., Asthana S.B.L.: Laminar flow in the entrance region of a smooth pipe. J. Fluid Mech. 90, 433–447 (1978) · doi:10.1017/S0022112079002330
[20] Mohseni K., Gharib M.: A model for universal time scale of vortex ring formation. Phys. Fluids 10, 2436–2438 (1998) · doi:10.1063/1.869785
[21] Mohseni K., Ran H., Colonius T.: Numerical experiments on vortex ring formation. J. Fluid Mech. 430, 267–282 (2001) · Zbl 0989.76012 · doi:10.1017/S0022112000003025
[22] Orlandi P.: Fluid Flow Phenomena: A Numerical Toolkit. Kluwer, Dordrecht (1999) · Zbl 0985.76001
[23] Orlanski I.: A simple boundary condition for unbounded hyperbolic flows. J. Comput. Phys. 21, 251–269 (1976) · Zbl 0403.76040 · doi:10.1016/0021-9991(76)90023-1
[24] Rai M., Moin P.: Direct simulations of turbulent flow using finite-difference schemes. J. Comput. Physics 96, 15–53 (1991) · Zbl 0726.76072 · doi:10.1016/0021-9991(91)90264-L
[25] Rosenfeld M., Rambod E., Gharib M.: Circulation and formation number of a laminar vortex ring. J. Fluid Mech. 376, 297–318 (1998) · Zbl 0935.76040 · doi:10.1017/S0022112098003115
[26] Ruith M.R., Chen P., Meiburg E.: Development of boundary conditions for direct numerical simulations of three-dimensional vortex breakdown phenomena in semi-infinite domains. Comp. Fluids 33, 1225–1250 (2004) · Zbl 1103.76350 · doi:10.1016/j.compfluid.2003.04.001
[27] Saffman P.G.: Vortex Dynamics. Cambridge University Press, Cambridge (1992) · Zbl 0777.76004
[28] Sau R., Mahesh K.: Passive scalar mixing in vortex rings. J. Fluid Mech. 582, 449–461 (2007) · Zbl 1177.76106 · doi:10.1017/S0022112007006349
[29] Schlichting H.: Boundary-Layer Theory. McGraw-Hill, New York (1979) · Zbl 0434.76027
[30] Shadden S.C., Katija K., Rosenfeld M., Marsden J.E., Dabiri J.O.: Transport and stirring induced by vortex formation. J. Fluid Mech. 593, 315–331 (2007) · Zbl 1151.76350 · doi:10.1017/S0022112007008865
[31] Shariff K., Leonard A.: Vortex rings. Ann. Rev. Fluid Mech. 24, 235–279 (1992) · Zbl 0743.76024 · doi:10.1146/annurev.fl.24.010192.001315
[32] Shusser M., Gharib M., Rosenfeld M., Mohseni K.: On the effect of pipe boundary layer growth on the formation of a laminar vortex ring generated by a piston–cylinder arrangement. Theor. Comput. Fluid Dyn. 15, 303–316 (2002) · Zbl 1006.76022 · doi:10.1007/s001620100051
[33] Sullivan I.S., Niemela J.J., Hershberger R.E., Bolster D., Donnelly R.J.: Dynamics of thin vortex rings. J. Fluid Mech. 609, 319–347 (2008) · Zbl 1147.76011 · doi:10.1017/S0022112008002292
[34] Verzicco R., Orlandi P.: A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123, 402–414 (1996) · Zbl 0849.76055 · doi:10.1006/jcph.1996.0033
[35] Zhao W., Steven H.F., Mongeau L.G.: Effects of trailing jet instability on vortex ring formation. Phys. Fluids 12, 589–596 (2000) · Zbl 1149.76598 · doi:10.1063/1.870264
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