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 ω = √ 0 exp − I (σθ) , VR 2 2 1 2πR0 Z ∞ Γ0R0σ ΨVR = F(µ, η)J1(θµ)J1(σµ)dµ, 4 0 where µ + η µ − η F(µ, η) = exp(ηµ)erfc( √ ) + exp(−ηµ)erfc( √ ), 2 2 (x − X ) r R R η = c , σ = ,θ = 0 , L = √ 0 or (L = atb, 1/4 ≤ b ≤ 1/2), L L L 2νt M Z ∞ Z ∞ Γ = , = π 2ω , 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 √ 3 3 5 E = 0 0 [ πθ2 F { , }, { , 3}, −θ2 ], 2 12 2 2 2 2 2
√ 2 2 Γ0θ π θ θ U = [3 exp(− )I1 4πR0 2 2 θ2 3 3 5 3θ2 3 5 7 + F { , }, { , 3}, −θ2 − F { , }, {2, }, −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, ∞, 2)]= 0 [log[θ] + 3/2 − EulerGamma/2 4πR0 Γ 8R −PolyGamma[0, 3/2]] + ... = 0 [log(√ 0 ) − 0.558]+ ... 4πR0 4νt 7Γ θ3 I Series[U, (θ, 0, 2)]= √0 + ... = 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(√ ) − 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:
∂ω ∂ 1 ∂Ψ ∂ 1 ∂Ψ ∂2ω ∂2ω 1 ∂ω ω + − ω + ω = ν + + − , ∂t ∂r r ∂x ∂x r ∂r ∂r 2 ∂x2 r ∂r r 2
∂2Ψ ∂2Ψ 1 ∂Ψ + − = −rω, ∂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:
ω (0, x) = Ψ (0, x) = 0, for r = 0,
and no flow through the tube wall:
1 ∂Ψ ω → 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 ∞ Γ0R0r Ψ = exp (−xµ)J1(R0µ)J1(rµ)dµ 2 0 Z ∞ 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 ∞ Γ0R0θr µ + xθ ΨVR = [exp(µxθ)erfc √ 4 0 2 µ − xθ + exp(−µxθ)erfc √ ]J1 (θµ) J1 (rθµ) dµ 2 Z ∞ p 2 2 Γ0R0r (for θ x + r → ∞) → exp (−|x|µ)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 ∞ Γ0R0θr µ + xθ ΨVRC = [exp(µxθ)erfc √ 4 0 2 µ − xθ + exp(−µxθ)erfc √ ]J1 (θµ) J1 (rθµ) dµ 2 Z ∞ 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 ε = 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 ω/ωmax (a) and corresponding normalised streamfunction ψ/ψmax (b). Values of ω/ωmax and ψ/ψmax from 0.1 to 0.9 with increments of 0.1 are shown. Re = 1700, Dw /D = 1.75, t = 8.
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 26 / 55 Comparison between the DNS and Model II
a) b) 0.8 0.2
0.7 0.15 E Γ 0.6 0.1
Dw/D = 1.75 DNS Dw/D = 2.00 DNS 0.5 Dw/D = 3.00 DNS 0.05 Dw/D = 1.75 model Dw/D = 2.00 model D /D = 3.00 model 0.4 w 0 8 10 12 14 16 18 20 8 10 12 14 16 18 20 τ τ
Figure: Time evolution of the circulation Γ (a) and energy E (b) of the vortex ring obtained by DNS and the model for three confinement parameters Dw /D and Re = 1700.
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 27 / 55 Comparison between the DNS and Model II
D /D = 1.75 DNS 0.4 w Dw/D = 2.00 DNS Dw/D = 3.00 DNS Dw/D = 1.75 model 0.3 Dw/D = 2.00 model
Dw/D = 3.00 model
0.2 U
0.1
0 8 10 12 14 16 18 20 τ
Figure: Time evolution of the translational velocity U of the vortex ring obtained by DNS and the model for three confinement parameters Dw /D and Re = 1700.
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 28 / 55 Comparison between the DNS and Model II
a) b)
3.4 3.8 3.6
3.2 3.4 3.2
x 3 x 3 2.8 2.8 2.6 2.4 2.6 2.2 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 ω/ωmax (a) and corresponding normalised streamfunction ψ/ψmax (b). Values of ω/ωmax and ψ/ψmax from 0.1 to 0.9 with increments of 0.1 are shown. Re = 3400, Dw /D = 1.75, t = 8.
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 29 / 55 Comparison between the DNS and Model II
a) b) D /D = 2.00 DNS 0.8 w 0.2 Dw/D = 3.00 DNS Dw/D = 4.00 DNS
Dw/D = 2.00 model D /D = 3.00 model 0.7 w 0.15 Dw/D = 4.00 model Γ E 0.6 0.1
0.5 0.05
0.4 0 15 20 25 30 35τ40 45 50 55 60 15 20 25 30 35τ40 45 50 55 60
Figure: Time evolution of the circulation Γ (a) and energy E (b) of the vortex ring obtained by DNS and the model for three confinement parameters Dw /D and Re = 3400.
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 30 / 55 Motivation to improve the previous models
Normalized vorticity distribution in the radial and axial directions near the orifice for Re=933 (experiment by Cater,Soria and Lim, 1998)
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 31 / 55 Motivation to improve the previous models
Normalized vorticity distribution in the radial and axial directions far from the orifice for Re=2000 (experiment by Cater,Soria and Lim, 2003)
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 32 / 55 Model III: taking into account the directional asymmetry in the cross section of the vorticity
Γ θ3 (σ2 + (η/β)2 + θ2) ω = 0√e exp − e I (σθ ) , VRE 2 2 1 e R0 β 2π Z ∞ Γ0R0θeσ 2 2 µβ − η/β ΨVRE = exp((β − 1)µ /2)[exp(−ηµ)erfc( √ ) 4 0 2 µβ + η/β + exp(ηµ)erfc( √ )]J1 (µθe) J1 (σµ) dµ, 2
where θe = (R0/Le), with Le the new viscous length scale:
R0 L θe = = λθ =⇒ Le = , Le λ and parameters β > 0 and λ > 0 measure elongation and compression along axes x and r, respectively.
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 33 / 55 Characteristics of a vortex ring with a core of the elliptical cross- section
θ2 Γ = Γ 1 − exp − e , e 0 2 2 Z ∞ Γ0R0πθe 2 2 2 Ee = exp((β − 1)µ ) erfc(βµ) J1(θeµ) dµ, 2 0 Z ∞ Γ0θe 2 √ Ue = exp(−µ )[6 πβµ 4πR0 0 2 2 2 2 2 +π exp(β µ )(1 − 6β µ )erfc(βµ)]J1(θeµ)dµ,
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 34 / 55 Contours of the normalised vorticity and streamfunction for different β
1.5 * * ΩVRE ΩVRE max 10
* * 1.0 H L YVRE YVRE max
5 H L 0.5 1 1
0.0 x 0 x
-0.5 -5
-1.0
-10
-1.5 0.0 0.5 1.0 1.5 2.0 0 5 10 15
r1 r1
Figure: Model of a vortex ring with elliptical core for λ = 1 and θ = 3. ∗ ∗ Normalised vorticity contours VRE/( VRE)max = 0.05. (left part). Isocontours of ∗ ∗ the stream function ΨVRE/(ΨVRE)max = 0.3. (right part). β = 1.5 (dashed curve), β = 1 (dot-dashed curve) and for β = 0.5 (solid curve).
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 35 / 55 Time evolution of the translational velocity and energy for different β
E* * e Ue 0.06 0.05
0.04 0.025 0.02
2 2 νt R0 νt R0 0.05 0.15 0.25 0.35 0.45 0.05 0.15 0.25 0.35 0.45
Figure: Model of a vortex ring with elliptical core for λ = 1 and θ = 3. Time ∗ evolution of the kinetic energy Ee . (left part) Time evolution of the translation ∗ velocity Ue .(right part). β = 1.5 (dashed curve), β = 1 (thick solid curve) and for β = 0.5 (thin solid curve).
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 36 / 55 Modeling of an ideal vortex, which starts from thin-cored structure (Saffman, 1970) and finished as a thick vortex ring (Rott&Cantwell, 1988)
Time dependence of the elipticity parameters β and δ:
β = 1 + δ, λ = 1 + γ,
δ = δ0θ0/θ, γ = γ0θ0/θ (θ > θ0)(short time),
δ = δ0θ/θ0, γ = γ0θ/θ0 (θ ≤ θ0)(large time), where 0 ≤ δ < 1 and 0 ≤ γ < 1
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 37 / 55 Finding of the parameters β and δ from the numerical data
1/2 3/2 Ed = E/(M Γ ) = 0.276, 1/3 2/3 Γd = Γ/(M U ) = 2.128.
θ 1
Ed=0.276
4.2
4.15 Γd = 2.128
4.1
4.05
β 1.36 1.39 1.42 1.45
Intersect of the curves described by the equations for the normalized
energyKaplanski and (TUT, circulation. Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 38 / 55 The translation velocity for short times U 0.3
0.25
0.2
0.15 2 νt R0 0 0.01 0.02 0.03 0.04 0.05
Figure: The dashed line is the large-Reynolds- number asymptotic by Fukumoto&Moffatt(Physica D, 2008) and the thin solid line is the present result with correction (β = 1 + 0.4θ0/θ; λ = 1 + 0.16θ0/θ, θ0 = 3.56). The thick solid line corresponds to Model I.
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 39 / 55 The translation velocity at large times U 0.2 HbL
0.1
logHt*L -2.5 -2 -1.5
Figure: The temporal evolution of the translation velocity at the postformation phase. The dashed line draw predicted by the formula (Saffman,Stud. Appl. Math. 1970) (corresponds to the experimental data by Wengand&Gharib, 0 Exp. in Fluids,1997) with k = 14.4 and k = 7.8, and the thin solid line is the present result with correction (β = 1 + 0.4θ/θ0, λ = 1 + 0.16θ/θ0, θ0 = 3.56). The thick solid line corresponds to Model I. Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 40 / 55 Modeling of a real vortex, which starts from flattened elliptical structure (Cater, Soria and Lim, 1998) and finished as an elongated vortex ring (Weingand&Gharib, 1997)
Time dependence of the elipticity parameters β and δ:
β = 1 + δ, λ = 1 + γ,
δ < 0, γ = γ0θ0/θ (θ > θ0)(short time, very short interval),
δ ≥ 0, γ = γ0θ/θ0 (θ ≤ θ0)(large time, long interval), where |δ| < 1 and 0 ≤ γ < 1
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 41 / 55 The improved asymptotic for large time
Improved Rott&Cantwell(1988) asymptotic velocity for large t :
Γ θ3 7 I U = 0 √ − 0 = (0.0037038 − 0.0011338 ) ef 0 3/2 R0 4 π 30 14 ρ(νt) I M I = U ( = 0.4)= 0.00325027 , Γ = , M = . ef 0 3/2 0 2 ρ ρ(νt) πR0 This asymptotic decay is in agreement with experimental data by Weigand&Gharib, (Exp. in Fluids,1997).
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 42 / 55 Vortex ring with a core of elliptic cross- section in a tube, Model IV
Rewriting the streamfunction of the vortex ring with the elliptical shape of the core in regular coordinates we can obtain the following expression
Z ∞ 2 2 Γ0R0θr β − 1)µ xθ µβ + xθ/(R0β) ΨVRE = exp( [exp(µ )erfc √ 4 0 2 R0 2 xθ µβ − xθ/(R0β) rθ + exp(−µ )erfc √ ]J1 (θµ) J1 µ dµ, R0 2 R0 √ which at the large distances for z = θ x2 − r 2 → ∞ tends to
r Z ∞ R2 Ψ ≈ Γ ((β2 − ) 0 µ2) (−| |µ) ( µ) ( µ) µ. VRE 0R0 exp 1 2 exp x J1 R0 J1 r d 2 0 2θ
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 43 / 55 Finding of the streamfunction induced by the presence of the tube for the elliptic cored vortex ring
The idea behind the Brasseour’s approach was to find such streamfunction (Green function to Laplace’s equation with Neumann boundary condition and treated as induced by the presence of the tube) which being combined with the circular vortex filament ( CVF) would satisfy the corresponding boundary condition of no flow on the wall. The idea behind our model is to find the superposition of the solutions, that correspond to the all terms of the expansion R2 ((β2 − ) 0 µ2) exp 1 2θ2 in 0:
r Z ∞ R2µ2 R2µ2(θ2 + R2µ2)2 Ψ ≈ Γ ( + 0 0 + 0 0 0 + ...) VRE 0R0 1 2 4 2 0 θ 2θ
× exp(−|x|µ)J1(R0µ)J1(rµ)dµ.
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 44 / 55 Model IV: Vortex ring with a core of elliptic cross- section in a tube
The streamfunction corresponding to a confined elliptical vortex ring
Z ∞ 2 2 Γ0R0θr1 β − 1)µ µβ + x1(θ/β) ΨVREC = exp( [exp(µx1θ)erfc √ 4 0 2 2 µβ − x1(θ/β) + exp(−µx1θ)erfc √ ]J1 (θµ) J1 (r1θµ) dµ 2 Γ R r Z ∞ 1 1 1 − 0 0 1 ( − µ2 + (− µ2 + µ4)2 + ...) 1 2 0 2 4 0 π 0 θ 2θ 2θ
K1(µ/ε) × I1(µ)I1(r1µ) cos(x1µ)dµ, β = 1 + 0, ε = R0/Rw I1(µ/ε)
This streamfunction is identical with the Brasseour’s result for 0 = 0.
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 45 / 55 Streamlines for confined vortex rings with the elliptical ring’s cores for different values of 0
The streamfunction Ψ with the YvrecHYvrecLmax VREC
0.5 vorticity ωVR (Model I) may serve as an approximation of the solution of the problem of viscous vortex
0.0 x ring in a tube.
-0.5
0.0 0.2 0.4 0.6 0.8 1.0 r
Figure: Isocontours of the normalised streamfunctions (ΨVREC /ΨVREC max ) for a confined vortex ring ( = 0.5)with elliptical core predicted by Model IV for two values of ε0: ε0 = 0.5 (red solid curves); ε0 = −0.5 (blue dashed curves).
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 46 / 55 Criterion for the pinch-off (kinematic approach by Shusser&Gharib 2000, modified by Danaila et al. 2009)
2 2 W U ≤ (D /R0 )Wf , Wf (t) = √ √ = W Λs(t), 1 − 8 t/ πRe + 8t/Re The normalized energy E(t) Ed (t) = , pI(t)Γ(t)3 Relations from the slug-flow approximation model 3 Z t 3 πD W Wf 2 πD W I(t) = ( ) ds = αs(t), 4 0 W 4 Z t WD Wf 2 WD Γ(t) = ( ) ds = αs(t), 2 0 W 2 3 2 Z t 3 2 πD W Wf 3 πD W E(t) = ( ) ds = bs(t). 8 0 W 8
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 47 / 55 Criterion for the pinch-off
By introducing s πI(t) B(t) = W Γ(t)3 s πΓ(t) b(t) = R , 0 2I(t) we can to express the pinch-off criterium in the other form 3/2 −3/2 2 bs(t) αs(t) 2b(t) B(t) Ed (t) ≥ √ , Λs(t) π
After fitting the DNS vortex with the results of the model at t = τf , we can obtain the time evolution of the integral characteristics of the elliptical-cored vortex ring and predict the formation number L/D Z t W L = D ( f )ds. 0 W L Experiments show3 .5 ≤ D ≤ 4.5 Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 48 / 55 Formation number for varying of β
L D L D
Rw R0=2 3.8ò 3.8 Rw R0=4 ò 3.7ò 3.7 ò ò ò ò ò ò 3.6 ò 3.6 ò Rw Rò0=3 ò 3.5 ò 3.5 ò ò ò ò ò 3.4 3.4 ò ò ò ò ε0 ε0 -0.3 -0.2 -0.1 ò0 -0.3 -0.2 -0.1 ò0 Figure: Formation number L/D of a vortex ring for different confines (a)Re = 1700, Rw /R0 = 1.75; 2; 3 (solid, dotted and dot-dashed curves, (b)Re = 3400, Rw /R0 = 2; 3; 4 (solid, dotted and dot-dashed curves, with elliptical core for 0 = −0.1; −0.2, −0.3. The dashed curve shows the influence of the Reynolds number Re, calculated by make the use of Ionut Danaila modification (I. Danaila et al., 2009) of the Shrusser&Gharib criterium (2000)
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 49 / 55 Fitting procedure for the vortex rings, predicted by DNS and the model at the given instant θ0 En
ME1 0.31 ME2 MC 0.29 N Gn 1.95 1.98 2.01 2.04 2.07
Figure: Comparison of the dimensionless energy (Ed ) and circulation (Γd ). Points correspond to the numerical data ( "N"), ring model with circular core ("MC") and ring model with elliptic core ("ME1", γ = 0.2, β = 1.4), ("ME2", γ = 0.16, β = 1.4, black)
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 50 / 55 Example of the application of the obtained ellipticity parameters β and γ for Re = 1700
U 0.4
0.3 æ æ æ æ 0.2 æ æ æ
0.1
τ 8 10 12 14 16 18 20
Figure: Time evolution of the translational velocity U of the vortex ring. Comparison between DNS data and the model predictions for the confinement parameters Dw/D = 3. Results correspond to the DNS data ( solid curve ), confined vortex ring model with circular core (dotted curve) and confined vortex ring model with the elliptic core (dashed curve, γ0 = 0.2, 0 = 0.4)
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 51 / 55 Calculations for the kinetic energy using ellipticity parameters for different confinement parameters
Re=1700 Re=3400 E E 0.3 0.3 ---- DwD=3, Model IV ---- DwD=4, Model IV ---- DwD=2, Model III ---- DwD=3, Model IV 0.2 ---- DwD=1.75, Model III 0.2 ---- DwD=2, Model III æ æ æ æò æ ò æ ò æ ò æ æ æ òæ æ 0.1à ò 0.1à ò òæ æ æ à ò ò à ò ò æ à ò ò à à ò à à à à à à à à à τ τ 8 10 12 14 16 18 20 15 30 45 60
Figure: Time evolution of the kinetic energy E of the vortex ring for Re = 1700 and Re = 3400. Comparison between the DNS data and prediction of the analytical models.
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 52 / 55 Calculations for the translational velocity using ellipticity parameters for different confinement parameters
Re=1700 Re=3400 U U 0.4 0.4 ---- DwD=3, Model IV --- DwD=4,Model IV ---- DwD=1.75, Model III --- DwD=3,Model IV 0.3 ---- DwD=2, Model III 0.3 --- DwD=2,Model III æ ---- Model II æ --- Model II æ æ ò æ æ ò æ æ æ æ æ ò ò æ 0.2ò ò 0.2à ò æò æ ò ò à à ò æ æ à à à à ò ò ò à à ò ò à à à à à 0.1 0.1 à à
τ τ 8 10 12 14 16 18 20 15 30 45 60
Figure: Time evolution of the translational velocity U of the vortex ring for Re = 1700 and Re = 3400. Comparison between the DNS data and prediction of different models.
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 53 / 55 Concluding Remarks
Models for the unconfined and confined vortex rings are presented. The first type of these models describes rings with a core of the circular cross-section and the second one describes elliptical -cored rings. Comparisons show that the predictions of the models which take into account the ellipticity of the ring’s core enable to improve agreement with DNS and experimental data. This improving is most important for the predictions of the formation number both for unconfined and confined vortex rings.
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 54 / 55 Acknowledgements
The authors are grateful to EPSRC (grants EP/K 005758/1 and EP/M002608/1) for the financial support of this project.
Kaplanski (TUT, Estonia) Workshop on vortex rings and related problems WABI, Brighton, 2015 55 / 55