aa r X i v : . [ phy s i c s . f l u - dyn ] D ec Generalization of the JTZ model to open planewakes
Zuo-Bing Wu ∗ State Key Laboratory of Nonlinear Mechanics,Institute of Mechanics, Chinese Academy of Sciences,Beijing 100190, ChinaFebruary 20, 2018 Author to whom correspondence should be addressed. Tel: 86-10-82543955; fax: 86-10-82543977. Email: [email protected](Z.-B. Wu). bstract The JTZ model [C. Jung, T. T´el and E. Ziemniak, Chaos , (1993)555], as a theoretical model of a plane wake behind a circular cylinderin a narrow channel at a moderate Reynolds number, has previouslybeen employed to analyze phenomena of chaotic scattering. It is ex-tended here to describe an open plane wake without the confined nar-row channel by incorporating a double row of shedding vortices intothe intermediate and far wake. The extended JTZ model is found inqualitative agreement with both direct numerical simulations and ex-perimental results in describing streamlines and vorticity contours. Tofurther validate its applications to particle transport processes, the in-teraction between small spherical particles and vortices in an extendedJTZ model flow is studied. It is shown that the particle size has signif-icant influences on the features of particle trajectories, which have twocharacteristic patterns: one is rotating around the vortex centers andthe other accumulating in the exterior of vortices. Numerical resultsbased on the extended JTZ model are found in qualitative agreementwith experimental ones in the normal range of particle sizes. PACS numer(s) : 05.45.-a, 47.32.-y
Keywords : Plane wake, von K´arm´an Vortex street, JTZ model, Particledynamics 2 he plane wake behind a circular cylinder is one of the most fun-damental phenomena in fluid mechanics, involving periodic vortexshedding from the cylinder, which is known as von K´arm´an vortexstreet. Recently, there emerge vast interests in the chaotic ad-vection of particles in the wake flow, in particular in its diversifiedapplications to such processes as chemical and biological ones. TheJTZ model was introduced to describe the plane wake in a narrowchannel at a moderate Reynolds number and used to investigatechaotic advection of particles near the circular cylinder. In thispaper we extend the JTZ model to describe an open plane wakewithout the confined narrow channel by adding a double row ofshedding vortices in the intermediate and far wake, where the es-sential change is that large damping in the original JTZ model isnow replaced by little damping in the extended one. The results ofextended JTZ model agree qualitatively with both direct numeri-cal simulations and experimental investigations on streamlines andvorticity contours. By using the extended JTZ model, particletransport processes are also simulated on different particle sizes.It is shown that the qualitative features of particle trajectories inthe experimental investigation can be predicted by the numeri-cal simulation, justifying the application of extended JTZ modelto quick and convenient estimations of the qualitative feature ofparticle transports in open plane wake flows. Introduction
The motion of particles in a non-uniform flow has received great attentiondue to its potential applications to both natural and engineering systems[1],as well as its dominant role played in the transport processes of particulateand multi-phase systems[2], such as those found in environmental engineer-ing, geophysical sciences, microfluids and combustion. In the transport pro-cesses, particles are not just passively carried by the background flow, butalso have dynamics of their own. The motion of particles exhibits abundantcharacteristics even if the background flows are very simple. Particles tendto concentrate asymptotically along periodic, quasi-periodic or chaotic tra-jectories for such flows as steady and unsteady cellular flows[3,4], periodicStuart vortex flows[5], the von K´arm´an vortex street flows[6,7] and planewake flows confined in a narrow channel (the JTZ model)[8,9].The classical von K´arm´an vortex street is a well known pattern[10,11],and the viscous plane wake behind a circular cylinder, as an example of com-plex flow containing vortices and shear layers, has been extensively studied.For exhibiting the phenomenon of chaotic scattering, a theoretical model(the JTZ model) was proposed in [12] to describe plane wakes in narrowchannels at a moderate Reynolds number (Re=250). It fits well the wakeflow field given by direct numerical calculations and its application is muchmore convenient in getting the background wake flow field than solving theNavier-Stokes equations. The chaotic advection of particles near a circularcylinder has been investigated by using the JTZ model[13-17], however, rele-vant experiments of particle dispersion were conducted in open plane wakesconsisting of near, intermediate and far regions[18,19]. And thus, we’ll ex-tend in this paper the JTZ model to an open plane wake without the confinednarrow channel. 4
Direct numerical simulation of a plane wake
The incompressible plane wake is governed by the two-dimensional Navier-Stokes equations, as the following non-dimensionalized ones ∇ • u = 0 , ∂ u ∂t = −∇ p + N ( u ) + Re L ( u ) , (1)where u is the fluid velocity and p is the fluid pressure divided by the fluiddensity. The uniform flow velocity U ∞ and the diameter of circular cylinder D are taken as the characteristic variables of the system, and Re(= U ∞ D/ν f , ν f is the fluids kinematic viscosity) is Reynolds number of the plane wake.In Eq. (1), the linear diffusion and nonlinear advection terms are describedby L ( u ) = ∇ u ,N ( u ) = − [ u • ∇ u + ∇ • uu ] . (2)To solve the Navier-Stokes equations, we incorporate a high-order splittingalgorithm into the spectral element method, please refer to [20, 21] for detail.Integrating the Navier-Stokes equations under the boundary conditions givenin [21], we obtain velocity field u and vorticity field ω z at Re=250. Thedrag coefficient C d and Strouhal number Sr are determined as 1.5 and 0.21,respectively, very close to the calculation results of [22, 23].The streamlines and vorticity contours are displayed in Fig. 1, where awave structure and the von K´arm´an vortex street are shown in the down-stream wake with x restricted to the range of x <
20 to have a clear andregular vortex structure. Since the wake is a periodic process with a periodof T c = 1 /Sr = 4 .
8, only figures at time phases 0 and T c / T c are plotted in Fig.1(a)(b) and Fig.1(c)(d), respectively. The correspondingfigures at time phases T c / T c / x axis, respectively.5t is observed that only two vortices can be discerned in the streamline plotof near wake. The regular von K´arm´an vortex street appears in the inter-mediate and far wake as shown in the vorticity contour plots, where thesemimajor axes of the elliptical vortices are in the vertical direction. Closeto the cylinder, the distance between the vortices is about one quarter ofthose further away. Due to viscous decay, the vorticity decreases graduallyin the wake downstream. The maximum absolute value of vorticity is verylarge in the boundary layer, and drops sharply in the near wake, and then,the maximum decreases slowly in the intermediate and far wake, having twoconstants in the range of 2 < x <
10 and 10 < x <
20, respectively. Themaximum in the near wake is about 1.7 times that in the intermediate wakeof 2 < x <
10, which is about 1.5 times that in the far wake of 10 < x < x <
20, there occur two transition zones with varying vorticityand two constant vorticity zones. The first transition zone is the near wakecontaining two vortices, and adjacent to which is the first constant vorticityzone, i.e. the intermediate wake with four vortices. And further downstreamthere occurs the second transition zone containing two vortices, and finallycomes the second constant vorticity zone-the far wake with two vortices. Anexperimental investigation on open plane wakes at Re=250 was given in [24]with time phase around T c /
2, the streamline of which are in close agreementwith the above-given simulation results.6
An extended JTZ model
The classical von K´arm´an vortex street is the most simple model for a planewake behind a circular cylinder, however, its computational zone doesn’tcontain the circular cylinder. The JTZ model for plane wakes does cover thecircular cylinder, and thus the flow field satisfies the no-slip boundary condi-tions at the cylinder surface. In what follows, we give a brief presentation ofthe JTZ model, where a plane wake behind a circular cylinder is consideredwithin a narrow channel at a moderate Reynolds number (Re=250) withno-slip boundary conditions on the cylinder surface. By taking the cylinderradius R and the vortex shedding period T c as the characteristic length andtime of the flow, the dimensionless model stream function[12] is written asΨ( x, y, t ) = f ( x, y ) g ( x, y, t ) , (3)where the first factor f ( x, y ) = 1 − exp[ − a ( q x + y − ] (4)satisfies the no-slip boundary conditions at the cylinder surface. The secondfactor in Eq. (3) is g ( x, y, t ) = − wh ( t ) g ( x, y, t ) + wh ( t ) g ( x, y, t ) + u ys ( x, y ) . (5)The first two terms model the periodic detachment of the vortices, in which w represents average strength of the time-dependent vortices and h ( t ) = | sin πt | ,h ( t ) = h ( t − ) . (6)The factors g ( x, y, t ) = exp − β [( x − x ( t )) + α ( y − y ) ] ,g ( x, y, t ) = exp − β [( x − x ( t )) + α ( y + y ) ] (7)7re the Gaussian forms with dimensionless vortex size β / with its centerlocated at [ x ( t ) , y ] and [ x ( t ) , − y ] in the wake. The vortices move down-stream at a constant velocity x ( t ) = 1 + L mod( t, ,x ( t ) = x ( t − ) . (8) y is the distance of the vortex centers from the x -axis, L is the dimensionlessdistance a vortex traverses during its lifetime and u is the dimensionlessbackground velocity. The last term in Eq. (5) arises from the backgroundflow, and the screening factor s ( x, y ) = 1 − exp[ − ( x − /α − y ] (9)ensures that the effect of the background flow velocity u is reduced in thewake. In the numerical simulation, parameters are chosen as a = 1, α = 2, β = 0 . L = 2, y = 0 . u = 14 and w = 24[15]. The streamlines andvorticity contours given by the JTZ model at time phases 0 and 1 / D .Both the streamlines and vorticity plots show that the JTZ model givesonly two vortices near the circular cylinder, which emerge and diminish ina period. So the JTZ model fits well the flow structures of plane wakesin a narrow channel. However, for open plane wakes without the confinednarrow channel, the emerged vortices won’t die out so quickly, and thus itcan represent the flow structures in the near wake, rather than the structuresof staggered double row vortices in the intermediate and far wake. To extendthe JTZ model to the open plane wake, we will preserve the JTZ model forthe near wake and add a double row of staggered vortices for the intermediateand far wake. 8n the extended stream function Ψ( x, y, t ) = f ( x, y ) g ( x, y, t ), the bound-ary function f ( x, y ) is kept, but g ( x, y ) is replaced by g ( x, y, t ) = γ n X i =1 ( − i h i ( t ) g i ( x, y, t ) + u ys ( x, y ) , (10)where γ = Γ T c /R ,u = U ∞ T c /R ,g i ( x, y, t ) = exp − β { [ x − x i ( t )] + α i [ y − ( − i − y ] } , i = 1 , , · · · , n. (11)Here, n − U ∞ are respectively the magnitude ofvortex and free-stream velocity, and γ and u are their non-dimensionalizedquantities based on R and T c . The reference velocity is R /T c instead of U ∞ ,and the dimensionless inflow velocity is ( u ,
0) instead of (1 , w of the original JTZ model is replaced by γ . The strength of vortices isdefined as time-dependent in view of the viscous decay mentioned in section2, where n (= 10) vortices are observed in the range − < x < h = | sin πt | ,h = 1 − . t, ,h ≤ i ≤ = 0 . ,h = 0 . − . t, ,h = 0 . − . t, ,h ≤ i ≤ = 0 . , if mod( t, < . h = 1 − . t + 0 . , ,h = | sin π ( t + 0 . | = | cos πt | ,h ≤ i ≤ = 0 . ,h = 0 . − . t + 0 . , ,h = 0 . − . t + 0 . , ,h ≤ i ≤ = 0 . , if mod( t, ≥ . . (12)In the first half of the shedding period, the time dependent strength of ashedding vortex starts from 0 and reaches 1. In the second half, it decreasesfrom 1 to 0.6. In the later periods, it is changing or keeps unchanged depend-ing on its downstream distance. The vortex centers are located at positions[ x i ( t ) , ± y ], and move downstream with two different velocities L / L in the first shedding period and the later periods, respectively. x ( t ) = 1 + L mod( t, ,x ( t ) = 1 + L mod( t + 0 . , ,x i ( t ) = L + ( i − L + 2 L mod( t, , if i = 2 k − k ≥ , L + ( i − L + 2 L mod( t + 0 . , , if i = 2 k ( k ≥ , (13)where L / L are the streamwise distance between two neighboringvortices within and outside the near wake region, respectively. Moreover,10 ( x, y ) in the last term of Eq. (10) is written as follows s ( x, y ) = 1 − exp[ − ( x − /α − y ] . (14)In order to clarify essential features of the extended JTZ model, we displayin Table 1 the time-variation of the strength and positions of four vorticesnear the cylinder in a period. In general, the up/down vortices in the doublerow of staggered vortices are denoted by odd/even. Each pair of vorticesexists in a periodic streamwise interval [ x i (=1 , ,... ) ( t ) , x i +2 ( t )), which is definedat time phase 0. For example, at time phase 0 / , vortex i = 1 and vortex i = 2 live in the first periodic streamwise interval between 1 and 1 + L / i = 3 and vortex i = 4 live in the second one between 1 + L / L /
2, and so on. After one half of the period, i.e., at time phase /
0, anew down/up vortex is shed from the circular cylinder and denoted by 2/1.At the same time, the down/up vortices in the double row of vortices moveaway from the original periodic streamwise positions and are re-located at thenext periodic streamwise positions. Therefore, the notation i for the originaldown/up row of vortices at time phase 0 / will be rewritten as i + 2. Forexample, at t = 1 /
2, vortex i = 2 and vortex i = 4 are located at x = 1+ L / L /
2, respectively. i = 4 and i = 6 will be used to denote the vortex i = 2 at x = 1 + L / i = 4 at x = 1 + 5 L /
2, respectively. Atthe same time, a new vortex is shed at (1 , − y ). It will be denoted by i = 2.At t = 1, vortex i = 1 and vortex i = 3 are located at x = 1 + L / L /
2, respectively. i = 3 and i = 5 will be used to denote the vortices i = 1 at x = 1 + L / i = 3 at x = 1 + 5 L /
2, respectively. At the sametime, a new vortex is shed at (1 , y ). It will be denoted by i = 1. The vortexshedding process is periodic with period t = 1.Parameters n = 10, a = 1, α = 2, β = 0 .
35 and y = 0 . α i is defined as follows11 i = − . t, , if i = 1;2 − . t + 0 . , , if i = 2;0 . , if i ≥ , (15)which describes the continuous change of the horizontal axes of ellipticalvortices from the semimajor in the near wake to the semiminor in the in-termediate and far wake. From the definition of Sr (= 2 R /U ∞ T c ), the non-dimensional velocity u is written as u = 2 /Sr . For Re = 250, Sr equals 0.21as given in [22]. Based on the velocity and vorticity fields given by the directnumerical simulation described in section 2, we can determine approximately γ = 2 . u and L = 4 . / For the motion of a small spherical particle in fluid flow, it is well-knownthat the flow field around the particle can be approximated by a Stokes flow12rovided the particle diameter is small enough compared to the characteristiclength of the fluid motion. Therefore, the particle-particle interactions as wellas the effects of particles on the flow are negligible. Under these assumptions,the momentum equation of the motion of the small spherical particle[25] iswritten as follows π d p ( ρ p + 0 . ρ f ) d V dt = π d p ( ρ p − ρ f ) g + π d p ρ f D u Dt +3 πd p ν f ρ f ( u − V ) + ( πν f ) / d p ρ f R t √ t − τ ( d u dτ − d V dτ ) dτ, (16)where V is velocity of the particle, and ρ is density, g is gravitational acceler-ation, and subscripts f and p refer to the fluid and particle, respectively. Thederivatives D/Dt and d/dt are used to denote the time derivative followinga fluid element and the moving sphere, respectively. Introducing the dimen-sionless quantities δ = ρ p /ρ f , ǫ = 1 / (0 . δ ), x ∗ = x /R , t ∗ = t/ ( R /U ∞ ), u ∗ = u /U ∞ , V ∗ = V /U ∞ and g ∗ = g /g , based on R and U ∞ , we cannon-dimensionlize Eq.(16) as follows d V dt = (1 − . ǫ ) F r g + ǫ D u Dt + St ( u − V )+3 q ǫ πSt R t √ t − τ ( d u dτ − d V dτ ) dτ, (17)where F r = U ∞ / √ gR and St = U ∞ T /R ( T is the particle viscous relax-ation time, d p / ǫν f ) are the Froude number and the Stokes number, respec-tively. The density and size of particles in Eq.(16) are taken from [18] as ρ p = 2 . × kg/m and d p = 0(10 − ) m , respectively. Since air is chosen asthe fluid medium in the flow, the fluid properties in Eq.(16) are described by ρ f = 1 . kg/m and ν f = 1 . × − m /s [26]. To emphasize the effects ofviscous force or St in Eq. (17) on particle transport process, the backgroundflow field parameters are chosen as U ∞ = 0 . m/s and R = 1 . × − m .Since δ is fixed and is of the order of 0(10 ), parameter ǫ appears to be of13he order of 0(10 − ). The particle diameters d p is very small [ d p = 0(10 − ) m ],so the parameters 1 /St and 1 /F r appear to be of the order of 0(10 ) and0(10 ), respectively. In this case, we can neglect the following smaller orderterms: the gravity term, the stress tensor term, the Basset history term andreduce the equation of motion (17) to d V dt = 1 St ( u − V ) . (18)In section 3, the physical quantities are non-dimensionalized using T c and R , which are different from the above ones. The function u ( x, y, t ) in Eq.(18) is now replaced by u ( x, y, t ) = ˆ u ( x, y, ˆ t ) /u , ˆ t = Sr t, (19)where ˆ t and ˆ u are the time and velocity field for the extended JTZ model,respectively. In the definition of Sr , the characteristic length is 2 R , ratherthan R , which is the characteristic length of the model. Since ˆ t × T c = t × R /U ∞ , so ˆ t = t × R / ( U ∞ T c ) = t × Sr/
2. In what follows, we willanalyze essential features of particle trajectories in the extended JTZ modelflow.Equation (19) is integrated with time step ∆ t = 0 . − .
01 by using afourth-order Runge-Kutta algorithm. Particles are released along a semicircle(of radius 1.2) with time interval 0 .
05. In each time interval, the releasedparticles total to 100. The initial velocities of particles are taken as the localflow ones. We choose ˆ t = 0 as the time for the initial release of particles, andgive a snapshot of particle distribution in the wake at ˆ t = 3.The particle diameter d p is taken in the range of 8 × − − × − m,which corresponds to St = 0 . − .
27. In the numerical simulation of d p = 8 × − m, 1 × − m, 2 × − m, 3 × − m, 5 × − m and 8 × − m,14he particles rotate around neighboring vortices as well as move downstreamunder the effects of viscous force ( u − V ) /St . In interacting with vortices,they exhibit some characteristic distributions. According to the rotation di-rections of vortices, the particle clusters form foliations near the vortices, andshow two typical patterns depending on the particle sizes. Firstly, when theparticle diameter is small enough (in the range of 8 × − − × − m), thefoliations rotate around the vortex centers with an example shown in Fig.4(a) for d p = 1 × − m ( St = 0 . × − − × − m, the foliations are accumulated in theexterior of vortices with an example shown in Fig. 4(b) for d p = 5 × − m( St = 0 . d p increases, the larger the d p , the smaller the centripetal force provided by the viscous force. Hence,the foliations are moved away from the vortex centers and accumulate inthe exterior of vortices. In the experiments of [18,19], the particle dispersionin a plane wake is investigated in detail. As the particle size increases, theorganized pattern of particle trajectories changes from filling in the vortexstructures to accumulating along the exterior of vortices. In the direct nu-merical simulation of [27], the particle distributions for different particle sizesare similar to the above-mentioned experimental observations, and these be-haviors agree qualitatively with our numerical results based on the extendedJTZ model. 15 Conclusion
In summary, we generalize and extende the JTZ model, which is a theoret-ical model for a plane wake behind a circular cylinder in a narrow channelat a moderate Reynolds number, to describe an open plane wake withoutthe confined narrow channel by incorporating a global distribution of shed-ding vortices. The extended JTZ model is found in qualitative agreementwith both direct numerical simulations and experimental studies in respectof streamlines and vorticity contours. The interaction of small spherical par-ticles with vortices in the extended JTZ model flow is investigated to furthervalidate the moderling in studying particle transport processes. It is foundthat the particle size has significant influences on the features of particle tra-jectories, and the particles exhibit two typical patterns as the particle sizeincreases. When the particle size is small enough, particle clusters wouldrotate around the vortex centers, whereas they would accumulate in the ex-terior of vortices for larger particle size. The numerical results are found inqualitatively agreement with the experimental ones in the normal range ofparticle sizes.
Acknowledgments
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8, (1997) 109. 19able I. Time-variation of the strength and positions of four vortices nearthe cylinder in a period.t 0 1/4 1/2 3/4 h √ h √ h h x L /8 1+ L / L /8 x L /4 1+3 L /8 1 1+ L /8 x L /2 1+ L L /2 1+2 L x L /2 1+2 L L /2 1+ L IGURE CAPTION
Fig. 1. Streamlines and vorticity contours for the plane wake flow at Re =250 obtained from direct numerical simulation. The values of vorticity ω D,U ∞ ,which are non-dimensionalized by D and U ∞ , are shown in the vorticitycontour plots. The time phases are chosen as 0 in (a) − < x < < x <
20 and T c / − < x <
10; (d)10 < x <
20 in a vortexshedding period T c .Fig. 2. Streamlines and vorticity contours for the JTZ model flow at timephases (a) 0 and (b) 1 / − < x < < x <
20 and 1 / − 10; (d)10 < x < 20 in a period. The index i of vortices in the extendedJTZ model (10) is marked at the upper and bottom zones of the vortex streetcentral region. The non-dimensional values ω D,U ∞ in the vorticity contourplots are calculated from the modelling of vorticity ω R ,T c scaled by R and T c by referring to the relation ω D,U ∞ = Srω R ,T c = 0 . ω R ,T c .Fig. 4. Instantaneous particle distribution and vorticity contour in theextended JTZ model flow at time phase 0 in a period with different particlediameters (a) d p = 1 × − m and (b) d p = 5 × − m. The release of particlesstarts at ˆ t = 0 and is completed at ˆ tt