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The Evolution of the Viscous Vortex Ring with the Elliptic Core

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The Evolution of the Viscous Vortex Ring with the Elliptic Core The evolution of the viscous vortex ring with the elliptic core Felix Kaplanski1 1Tallinn University of Technology, Estonia Workshop on vortex rings and related problems, Brighton, Friday, 27th November, 2015 Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 1 / 55 Outline Introduction Analytical models for vortex ring-like flows Unconfined vortex ring with a core of circular cross- section (model I) Vortex ring with a core of circular cross- section in a tube (model II) Unconfined vortex ring with a core of elliptical cross- section (model III) Vortex ring with a core of elliptic cross- section in a tube (model IV) Prediction of the integral characteristics for high Reynolds ring on the basis of the models with a core of circular and elliptic cross- section (models I-IV) extension of the Saffman’s formula, Fukumoto&Moffatt 2008 experimental data for range (830 ≤ Re ≤ 1650), Weingand&Gharib 1997 numerical data for Re=1700;3400, Danaila et al. 2015 Prediction of the formation numbers for unconfined and confined vortex rings on the basis of the models with a core of elliptic cross- section (models III and IV) and modified pinch-off criterium, Danaila et al. 2008 Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 2 / 55 Vortex rings Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 3 / 55 Vortex ring generator Tait 1876 ! Gharib et al., Journal of Fluid Mechanics 1998 Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 4 / 55 Optimal Vortex Formation as a Unifying Principle Gharib et al., Journal of Fluid Mechanics 1998 Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 5 / 55 Flow over a cyliner Jeon&Charib, Journal of Fluid Mechanics 2004 Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 6 / 55 Blood flow in the cardiac left ventrile Gharib et al., PANS 2006 Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 7 / 55 Birds and insects flying (flapping flight models ) Rayner, Journal of Fluid Mechanics 1979 Ellington, Philos. Trans. R. Soc. London, Ser.B 1984 Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 8 / 55 Vortex ring- like structures of spray from a low pressure , gazoline fuel injector (PFI) Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 9 / 55 Scheme of a vortex ring Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 10 / 55 Previous work on vortex rings Early work: Helmholtz (1858), Kelvin (1869), Hill (1894), Lamb(1916) Evolution and formation: Fraenkel (1972), Norbury (1973), Saffman (1975, 1978), Maxworthy(1972, 1974, 1977), Moore (1980), Rott&Cantwell (1993), Gharib et. al (1998), Fukumoto&Moffatt(2000,2008) Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 11 / 55 Purpose of our analytical study derive analytical models of vortex ring taking into account the effects of high Reynolds numbers/confinement/swirl/temperature gradient and apply them to the analysis of vortex ring-like structures in natural gas engine-like conditions Kaplanski&Rudi, Physics of Fluids 2005 Fukumoto&Kaplanski, Physics of Fluids 2008 Begg, Kaplanski, Sazhin, Hindle&Heikal, Int. J. Engine Res. 2009 Kaplanski, Sazhin, Fukumoto, Begg&Heikal, Journal of Fluid Mechanics 2009 Kaplanski, Sazhin, Begg, Fukumoto&Heikal, Eur. J. Mech. B-FLUID 2010 Kaplanski, Fukumoto&Rudi, Physics of Fluids 2012 Danaila, Kaplanski&Sazhin, Journal of Fluid Mechanics 2015 Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 12 / 55 Analytical models: Model based on the linear first-order solution of the Navier−Stokes equation for the axisymmetric geometry and arbitrary times (Model I): Γ θ3 σ2 + η2 + θ2 ! = p 0 exp − I (σθ) ; VR 2 2 1 2πR0 Z 1 Γ0R0σ ΨVR = F(µ, η)J1(θµ)J1(σµ)dµ, 4 0 where µ + η µ − η F(µ, η) = exp(ηµ)erfc( p ) + exp(−ηµ)erfc( p ); 2 2 (x − X ) r R R η = c ; σ = ;θ = 0 ; L = p 0 or (L = atb; 1=4 ≤ b ≤ 1=2); L L L 2νt M Z 1 Z 1 Γ = ; = π 2! ; 0 2 M r dxdr πR0 0 −∞ Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 13 / 55 Characteristics of the vortex ring with a circular core θ2 M Γ = Γ − − ; Γ = ; 0 1 exp 0 2 2 πR0 Γ2R θ 1 p 3 3 5 E = 0 0 [ πθ2 F f ; g; f ; 3g; −θ2 ]; 2 12 2 2 2 2 2 p 2 2 Γ0θ π θ θ U = [3 exp(− )I1 4πR0 2 2 θ2 3 3 5 3θ2 3 5 7 + F f ; g; f ; 3g; −θ2 − F f ; g; f2; g; −θ2 ; 12 2 2 2 2 2 5 2 2 2 2 2 Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 14 / 55 Vorticity distribution Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 15 / 55 Previous results Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 16 / 55 Previous results Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 17 / 55 Model I unifies Saffman’s (short time) and Rott&Cantwell’s (large time) descriptions of the vortex ring Mathematica: Γ Series[U; (θ; 1; 2)]= 0 [log[θ] + 3=2 − EulerGamma=2 4πR0 Γ 8R −PolyGamma[0; 3=2]] + ::: = 0 [log(p 0 ) − 0:558]+ ::: 4πR0 4νt 7Γ θ3 I Series[U; (θ; 0; 2)]= p0 + ::: = 0:003704 + ::: 3=2 120 πR0 νt Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 18 / 55 Model I unifies Saffman’s (short time) and Rott&Cantwell’s (large time) descriptions of the vortex ring Γ0 8R0 2 Us = [lg(p ) − 0:558]; νt << R0 Saffman; 1970 4πR0 4νt I 2 Uf = 0:003704 ; νt >> R Rott&Cantwell; 1993 νt3=2 0 Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 19 / 55 Viscous vortex ring in a tube x Rw U β L R0 L r Figure: Schematic of a vortex ring in a tube. Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 20 / 55 A model for a viscous vortex ring in a tube: Governing equations: @! @ 1 @Ψ @ 1 @Ψ @2! @2! 1 @! ! + − ! + ! = ν + + − ; @t @r r @x @x r @r @r 2 @x2 r @r r 2 @2Ψ @2Ψ 1 @Ψ + − = −r!; @r 2 @x2 r @r where x, r are the axes of a cylindrical coordinate system and t is time. We consider the following boundary conditions: symmetry at the axis: ! (0; x) = Ψ (0; x) = 0; for r = 0; and no flow through the tube wall: 1 @Ψ ! ! 0; = 0; for r = R : r @x w Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 21 / 55 Model II: Vortex ring with a core of circular cross- section in a tube Streamfuction for CVF (circular vortex filament) in a tube: Z 1 Γ0R0r Ψ = exp (−xµ)J1(R0µ)J1(rµ)dµ 2 0 Z 1 Γ0R0r K1(µRw ) − I1(R0µ)I1(rµ) cos(xµ)dµ π 0 I1(µRw ) Brasseur, PhD thesis, Report JIAA, TR-26, Stanford University 1979 The basic idea behind our model Z 1 Γ0R0θr µ + xθ ΨVR = [exp(µxθ)erfc p 4 0 2 µ − xθ + exp(−µxθ)erfc p ]J1 (θµ) J1 (rθµ) dµ 2 Z 1 p 2 2 Γ0R0r (for θ x + r ! 1) ! exp (−|xjµ)J1(R0µ)J1(rµ)dµ 2 0 Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 22 / 55 Model II: Vortex ring with a core of circular cross- section in a tube Resulting streamfuction: Z 1 Γ0R0θr µ + xθ ΨVRC = [exp(µxθ)erfc p 4 0 2 µ − xθ + exp(−µxθ)erfc p ]J1 (θµ) J1 (rθµ) dµ 2 Z 1 Γ0R0r K1(µRw ) − I1(R0µ)I1(rµ) cos(xµ)dµ; π 0 I1(µRw ) Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 23 / 55 Vortex ring with a core of circular cross- section in a tube Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 24 / 55 Vortex ring with a core of circular cross- section in a tube 2 1 x 0 -1 -2 0.5 1.0 1.5 2.0 2.5 3.0 r Figure: Isocontours of the normalised streamfunctions Ψc=(Ψc)max for a confined ring for " = 1=3; θ = 3 (solid curves), and ΨVR=(ΨVR)max for an unbounded ring with θ = 3 (dashed curves). Contours are shown for Ψc=(Ψc)max from 0 to 0.9 with an increment of 0.1. The vertical line at r1 = 3 represents the streamfunction at the tube wall Ψc=(Ψc)max = 0. Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 25 / 55 Comparison between the DNS and Model II a) b) 1.8 2 1.8 1.6 1.6 x 1.4x 1.4 1.2 1.2 1 0.8 1 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 r r Figure: Comparison between the DNS data (blue solid curves) and predictions of the vortex ring model (red dashed curves). Contours of normalised vorticity !=!max (a) and corresponding normalised streamfunction = max (b). Values of !=!max and = max from 0.1 to 0.9 with increments of 0.1 are shown.
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