PHYSICS OF FLUIDS

VOLUME 13, NUMBER 4

APRIL 2001

A numerical study of dynamics of a temporally evolving swirling jet Guo-Hui Hua) Department of Thermal Science and Energy Engineering, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China

De-Jun Sun and Xie-Yuan Yin Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China

共Received 8 February 2000; accepted 5 January 2001兲 Direct numerical simulation 共DNS兲 of a swirling jet near the outlet of a nozzle with axisymmetric and non-axisymmetric disturbances is performed to investigate the dynamic characteristics of the flow. The early 共linear兲 stage of the jet evolution agrees well with the predictions of linear stability theory. In the nonlinear stage, the axisymmetric DNS shows that the interaction between the primary vortex ring and the streamwise columnar vortex creates a secondary vortex structure with opposite azimuthal vorticity near the columnar vortex. Then a vortex pair consisting of the primary and secondary vortices forms and travels radially away from the symmetry axis, causing a rapid increase of the thickness of mixing layer. The non-axisymmetric DNS shows that the streamwise vortex layer developed in the early stage of evolution due to azimuthal instability breakdowns into small eddies under the joint stretch of the axial and azimuthal shear. The results reveal several mechanisms of mixing enhancement by swirl, i.e., the radial motion of vortex ring pairs, the rapid growth of streamwise vorticity, and the creation of three-dimensional small eddies. They are all favorable for fluid entrainment in swirling jets. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1350876兴

I. INTRODUCTION

swirl on the Kelvin–Helmholtz instability was analyzed by Loiseleux et al.,7 who applied a normal-mode method to a basic flow with a top-hat axial velocity profile and a solidbody core rotation in the form of a Rankine vortex. An inviscid spatial stability analysis was performed on the same model by Wu et al.,8 and a comparison was made between theoretical results and experimental data for an axisymmetric mode therein. Recently, Lu and Lele9 extended the study of this field to a viscous compressible flow. Martin and Meiburg10 used a different simplified model to study the linear potential stability of a swirling jet. The model is an axial centerline vortex surrounded by a nominally axisymmetric vortex sheet containing both streamwise and circumferential vorticity. An obvious feature of this model is that it allows the competition of the centrifugal and Kelvin–Helmholtz instability waves. In their succeeding works,2,3 Martin and Meiburg employed a vortex filament technique to conduct a three-dimensional numerical simulation of the nonlinear evolution of swirling jets, and studied the growth of azimuthal and helical disturbances of their model. Another study related to the present work is the axisymmetric numerical study on two-phase swirling jets performed by Park et al.,11 in which the motion of large-scale structures and formation of a recirculation bubble were investigated. Despite the increased understanding gained from these studies, some important issues still need be further clarified. These include, among others, the effect of swirl on the onset of azimuthal instability, and the interaction between vortex rings and streamwise vortices. It should be pointed out that these streamwise vortices are not only the columnar vortex at

Swirling jets are commonly encountered in numerous practical applications, e.g., in supersonic combustors and cyclone separators. Earlier investigations on swirling jets mainly focused on measuring their mean velocity profiles or turbulent transport properties, etc. It was then realized that a full understanding of the evolution mechanism of swirling jets is necessary for both engineering and theoretical interests, and hence many studies have recently been carried out on the instability and dynamics of the three-dimensional vortical structures of the flows.1–3 As is well known, the instability properties or the evolution of coherent structures in a swirling jet are very sensitive to the velocity profiles of the basic flow. For a fully developed swirling jet, Batchelor’s trailing vortex model is often regarded as an appropriate approximation. Its temporal and absolute/convective instability have been extensively investigated 共see Refs. 4–6 and references therein兲. For a swirling jet near the nozzle exit, Panda and McLaughlin1 conducted an experimental study on the instability and vortical structures of the flow. They found that the overall growth rate of the instability waves was considerably smaller compared with swirl-free jets, and the vortex pairing was suppressed when the swirl was added. An important contribution of their work is the finding of the crucial role of the dynamics of organized structures in the flow. Then, in the early 共linear兲 developing stage of swirling jets, the effect of a兲

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Phys. Fluids, Vol. 13, No. 4, April 2001

the centerline, but also those created by azimuthal instability. These issues are the main objective of the present study. We conducted a direct numerical simulation 共DNS兲 of a temporally evolving swirling jet near a nozzle exit under axisymmetric and azimuthal disturbances, which enable an assessment of the roles of swirl in the nonlinear development of the Kelvin–Helmholtz and azimuthal instability waves as well as the vortical structures. The present work is also a contribution toward the understanding of the mechanisms responsible to swirlenhanced mixing. It is well known that in a supersonic combustor the high compressibility greatly reduces the growth rate of fuel–air mixing layers. While so far no satisfactory explanation has been given on the reduction of mixing rate in compressible flows, a seemly reasonable consideration is that the compressibility will mainly decrease the mixing caused by spanwise or azimuthal vorticity, but has little influence on the mixing caused by streamwise vortical structures.12 Many experiments have confirmed that swirl is an effective way to enhance the mixing rate in jet flows.1,13–15 However, the key mechanisms for the enhancement are still not fully explored. Naughton et al.13 reviewed several proposed explanations on this issue. One possible mechanism supported by several researchers1,14 is that the centrifugal instability causes the mixing enhancement. However, although experimental and numerical studies have confirmed that this instability does play an active role in the mixing, it may not be the necessary condition since Naughton et al.13 obtained experimentally a mixing enhancement with a centrifugally stable basic flow. A novel explanation proposed by Naughton et al. is that the addition of a tangential velocity to the flow creates an additional component of shear, thus inducing a remarkable component of streamwise vorticity not affected by the compressibility. Now, our numerical study of the nonlinear evolution of the swirling jet provides a clarification of this important issue. This article is organized as follows. In the next section we outline the numerical implementation, including a brief description of numerical algorithm, and initial flow conditions. In Secs. III and IV, numerical results of the vortex dynamics of axisymmetric and non-axisymmetric flows are presented respectively, and a further discussion on the mechanisms of swirl-enhanced mixing is made. The final section summarizes our most important findings. II. NUMERICAL IMPLEMENTATION A. 3-D finite difference scheme

Direct numerical simulation 共DNS兲 is an effective tool for understanding the instability characteristics and vortical structures in circular jet flows with or without swirl.2,3,16,17 In our study the three-dimensional 共3-D兲 unsteady incompressible Navier–Stokes equations in cylindrical coordinates ( ␪ ,r,z) are solved by using a finite difference scheme developed by Verzicco and Orlandi.18 The singularity at the axis r⫽0 is removed by introducing new variables q ␪ ⫽ v ␪ , q r ⫽r v r , q z ⫽ v z , and these unknowns are located on a staggered grid with the pressure at the center of the cell. The fractional-step method and approximation factorization tech-

Hu, Sun, and Yin

nique are adapted to solve the equations for primitive variables. The detailed numerical procedure is given in Ref. 18, and here we only make a brief outline. The computational domain is denoted as R⫽ 兵 ( ␪ ,r,z) 兩 0 ⭐ ␪ ⭐2 ␲ ,0⭐r⭐R max ,0⭐z⭐L max其, where the axial length L max is determined by the length of disturbance waves. R max must be large enough to ensure that the streamwise columnar vortex 共see later in this work兲 decays sufficiently at r ⫽R max , which is set to be 25 in this article. The new variables (q ␪ ,q r ,q z ) defined earlier are substituted into the Navier–Stokes equations in cylindrical coordinates to obtain the governing equations for these new variables. To reduce the computational amount, a periodic condition is imposed in the z direction when we study a temporally evolving flow. A free-slip condition is imposed at r⫽R max . The spatial difference of viscous and advective terms are both second-order centered schemes. The fractional-step method developed by Kim and Moin19 is applied to solve the governing equations. A scalar ⌽ related to the pressure is introduced to project the nonsolenoidal intermediate velocity qˆ i to a velocity field q i with zero divergence. When the intermediate velocity vector qˆ i is computed by using the approximation factorization technique, the viscous and advective terms are discretized temporally by Crank–Nicholson and low-storage third-order Runge–Kutta schemes, respectively. The discrete Fourier transformation is used to solve the scalar ⌽ in ␪ and z directions, which forced us to use uniform grids in these directions. B. Initial conditions and run parameters

In the present article we study the dynamic characteristics of a swirling jet near the nozzle exit. In choosing a velocity profile as the initial condition for numerical simulations, it is naturally expected that it can reflect the basic physical features of the flow comparable with existing experiments. Unfortunately, existing measured velocity profiles are quite dispersed due to researchers’ different experimental devices for swirling jets. For example, the azimuthal velocity is centrifugally unstable in Mehta et al.’s experiments,14 while Naughton et al. observed a centrifugally stable azimuthal velocity profile.13 In this article, we set the azimuthal component of the velocity vector to be a centrifugally stable one, so the effect of swirl can be examined carefully without considering the influence of centrifugal instability. The role of centrifugal instability had also been studied and was reported elsewhere.20 The basic flow model of the swirling jet chosen in this study has a Lamb–Oseen vortex velocity profile in the azimuthal direction and a tanh-velocity profile in axial direction. Thus, in cylindrical coordinates ( ␪ ,r,z), it reads U ␪ 共 r 兲 ⫽q 共 1⫺e ⫺r

2/␳

兲 / 共 rV 0m 兲 ,

U r 共 r 兲 ⫽0,

共1兲

U z 共 r 兲 ⫽0.5兵 1⫹tanh关 0.5R 0 /⍜ 0 共 1⫺r/R 0 兲兴 其 . The above expressions and basic equations are nondimensionalized by the axial velocity excess 共or deficit兲 U * at centerline and a length scale L * ⫽1/3R 0* , then R 0 ⫽3 is the

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Phys. Fluids, Vol. 13, No. 4, April 2001

A numerical study of dynamics

953

FIG. 1. Dependence of computational results on grid number for axisymmetric flows: Time evolution of E KHv r 共a兲 and the maximum azimuthal vorticity 共b兲 for different grids with Re⫽500, q⫽0.5, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽1.

nondimensional jet exit radius. We define a Reynolds number Re based on U * and L * . In the equation for axial velocity, ⍜ 0 is the momentum thickness of the mixing layer. The parameter R 0 /⍜ 0 determines the strength of Kelvin– Helmholtz waves. In 共1兲, V 0m ⫽0.6382/冑␳ is a normalized parameter in azimuthal velocity so that the swirl ratio q characterizes the ratio of the maximum azimuthal velocity and the maximum difference of the streamwise velocity U z . ␳ measures the size of the Lamb–Oseen vortex core and hence determines the distribution of streamwise vorticity. The initial 3-D flow field in DNS contains both axial and azimuthal disturbances to trigger the Kelvin–Helmholtz and azimuthal instabilities. An axial disturbance is introduced by setting the initial azimuthal vorticity following Ref. 18: ⍀ ␪ 共 r,z, ␪ 兲 ⫽⍀ ␪ 0 共 r 兲关 1⫹ ⑀ 1 sin共 k 1 z 兲 ⫹ ⑀ 2 sin共 k 2 z 兲兴

共2兲

and an azimuthal disturbance is imposed by perturbing the shear layer radius to R( ␪ )⫽R 0 ⫹ ⑀ ␪ cos(n␪), in which n is the wave number of azimuthal disturbance. Other components of the vorticity field are determined from its solenoidal feature. In 共2兲 ⍀ ␪ 0 (r) is the basic azimuthal vorticity derived from 共1兲, and k 1 and k 2 are the axial wave numbers of disturbance modes with k 1 ⫽2k 2 . These excited waves are determined by linear stability theory. Here the unstable wave number k 1 ⫽2 for a thinner shear layer R 0 /⍜ 0 ⫽33.9 with L max⫽2␲, and k 1 ⫽1 for a thicker one R 0 /⍜ 0 ⫽11.3 with L max⫽4␲. In the present work we set ⑀ 1 ⫽0.03 and ⑀ 2 ⫽0.006 as the amplitudes of the axial primary mode and its subharmonic wave, respectively. The amplitude ⑀ ␪ of the azimuthal disturbance wave is 0.05 in the fully 3-D computations so that before the growth of the subharmonic wave is saturated the non-axisymmetric effect can almost be ignored. It should be pointed out that the ratio of the amplitudes of axial and azimuthal disturbance waves is a key parameter in the evolution of the flow, because it is known to have a significant effect on the competition between different instability waves, such as Kelvin–Holmholtz and azimuthal instability waves.3,21,22 The mesh dependence of the numerical results has been checked by repeating one typical axisymmetric simulation 共Re⫽500, q⫽0.5, R 0 /⍜ 0 ⫽33.9, ␳ ⫽1兲 with different grid numbers for axisymmetric disturbances. In Fig. 1 the history of perturbed energy E KHv r (k,t) 共for definition see later in this work兲 and the maximum azimuthal vorticity are shown

for different meshes. Only minor differences can be found in these figures for different grid numbers. A linear stability analysis was also performed to compare the obtained growth rate with the DNS results measured in early stage. The linearized Navier–Stokes equations in the cylindrical coordinates ( ␪ ,r,z) was solved by a Chebyshev spectral collocation method developed by Khorrami23 共see also Ref. 6兲 for the normal mode disturbances in the form of

兵 u r ,u ␪ ,u z ,p 其 ⫽ 兵 iF 共 r 兲 ,G 共 r 兲 ,H 共 r 兲 , P 共 r 兲 其 e i(kz⫹n ␪ ⫺ ␻ t) , where F, G, H, P are eigenfunctions of velocity vectors and pressure, k, n are axial and azimuthal wave numbers, respectively, and ␻ ⫽ ␻ r ⫹i ␻ i is frequency. The predicted growth rate ␻ i of Re⫽500, q⫽0.5 and ␳ ⫽1 for axisymmetric mode n⫽0 was used to compare with the growth rates measured from the linear stage of numerical results in Table I. It was found that although a finer grid in radial direction would be helpful to get a more accurate linear growth rate, it has minor effects on the vortical evolution of the flow. Thus the axisymmetric simulations in this article have been conducted using a grid of 256⫻64 for the jet with R 0 /⍜ 0 ⫽33.9 and 256⫻128 for the one with R 0 /⍜ 0 ⫽11.3 in the radial and axial directions, respectively. For the full three-dimensional computations, the grid number in the azimuthal direction equals to 64. To check the azimuthal resolution, we run the code using 64 azimuthal grid points for a swirl-free jet. Comparison to the results of Ref. 16 was made and it looks satisfactory.

TABLE I. Comparison of linear growth rate of the primary wave measured from axisymmetric DNS with different meshes to that obtained from stability theory for axisymmetric mode 共Re⫽500, q⫽0.5, and ␳ ⫽1兲. (r,z) meshes

R 0 /⍜ 0 ⫽33.9

256⫻64 256⫻128 512⫻64 512⫻128 1024⫻64

0.377 0.379 0.400 0.401 0.406

predicted value

0.435

R 0 /⍜ 0 ⫽11.3 0.129 0.132 0.142

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C. Quantities for flow analysis

We define several quantities for the convenience of examining the temporal evolution of the flow field. First, a useful parameter is the root mean square 共RMS兲 of the instantaneous velocity or vorticity components, which reads RMS共 ␾ 兲 ⫽



冕冕冕␾

1

2 ␲ R max L max

2

共 ␪ ,r,z,t 兲 r dr d ␪ dz,

共3兲

where ␾ can be some component of the velocity or vorticity. It measures the contribution of this component to the energy or enstrophy in a unit volume. ˆ be the Fourier coefficients of ␾: Then, let ␾

␾ˆ 共 n,r,z,t 兲 ⫽

1 2␲



2␲

0

␾ 共 ␪ ,r,z,t 兲 e ⫺in ␪ d ␪ ,

共4兲

the energy of the azimuthal disturbance mode with wavenumber n, is defined as 共 E ap␾ 兲共 n,t 兲 ⫽

冕冕 ␾

兩 ˆ 共 n,r,z,t 兲 兩 2 r dr dz

共5兲

when we study the azimuthal instability of the flow. Here the subscript ap denotes azimuthal perturbation. The growth rate of disturbances can then be easily obtained by differentiating log(Eap␾ ) with respect to t. The energy (E KH␾ )(k,t) of axial disturbances with wave number k and its growth rate is also defined in a similar way when we consider the growth of Kelvin–Helmholtz waves and their subharmonics, where the suffix KH stands for Kelvin–Helmholtz. Finally, mean momentum thickness ⍜(t) is broadly used to measure the mixing in circular jet flows, which is given by ⍜共 t 兲⫽

1 2 ␲ L max

冕冕 冕 ⬁

0

2␲

0

L max

0

v z 共 ␪ ,r,z,t 兲

⫻„1⫺ v z 共 ␪ ,r,z,t 兲 … dr d ␪ dz.

共6兲

III. AXISYMMETRIC RESULTS

Since the spanwise vortical structures have been found to significantly affect the fully 3-D flow fields,21 we consider the axisymmetric case as the first step of understanding the three-dimensional flow characteristics before addressing the fully 3-D evolution of swirling jets. The basic velocity profile 共1兲 has three parameters: swirl ratio q, momentum thickness ratio R 0 /⍜ 0 , and vortex core size ␳. If the effect of viscosity is included, we have a fourparameter space. Since our main interest is the effect of the swirl, to simplify the discussion we will focus on the dynamics of the flows for Re⫽500, ␳ ⫽1, and R 0 /⍜ 0 ⫽33.9 with a grid of 256⫻64 in (r,z) directions first. The influences of the parameters will be documented at the end of this section. Figure 2 shows the variation of the perturbed energy E KH v r , the RMS of radial velocity, the momentum thickness ⍜, and the maximum and minimum azimuthal vorticity with time for the above fixed Re, R 0 /⍜ 0 , and ␳ with q ⫽0,0.5,2. Since four nearly linear segments are observed in the RMS( v r )⬃t curve for the swirling jet 关Fig. 2共b兲兴, we suggest that the dynamical process of the axisymmetric flow

can be roughly divided into four stages: 共1兲 the development of Kelvin–Helmholtz instability and rolling-up of mixing layer; 共2兲 the growth of subharmonic mode and vortex pairing; 共3兲 the formation of a secondary vortex ring with negative vorticity; and 共4兲 the outward radial motion of the vortex ring pair with opposite vorticity. A. Rolling-up of mixing layer and vortex pairing

The azimuthal vorticity contours for q⫽0 and q⫽0.5 are drawn in Fig. 3 to show the role of swirl in the early stage of evolution (t⭐23). Figure 2共a兲 has shown that the primary harmonic disturbance wave with wave number k 1 ⫽2 grows linearly and reaches saturation at about t⫽14. This implies the rolling-up of mixing layer and the formation of vortex rings in the flow field 共Fig. 3兲. Then the two vortex rings induce each other and move closely, and the mean momentum thickness ⍜ and the RMS of radial velocity vary slowly during t⫽14– 18. Initially the energy of the subharmonic wave k 2 ⫽1 is about an order of magnitude smaller than the primary one. In the duration of vortex rolling-up, it increases continuously and exceeds the energy of disturbance k 1 ⫽2 at t⫽18, and finally reaches a saturation point at t ⫽26. In the meantime the momentum thickness and RMS( v r ) rise again 共Fig. 2兲, which corresponds to the occurrence of vortex pairing in the flows. These results show that the swirl has little influence on the evolution of the flow in the early stage. Figure 2共a兲 indicates that the linear growth rate of the axisymmetric mode k 1 ⫽2 is slightly smaller than the swirl-free case. This phenomenon is consistent with the prediction of temporal linear stability theory using a Chebyshev spectral collocation method, and is qualitatively agree with the inviscid stability analysis by Loiseleux et al.7 The dispersion relations, ␻ i (k), with different swirl ratio are illustrated in Fig. 4 for the axisymmetric mode. It shows that the decrease can hardly be distinguished for the primary harmonic mode, but more evident for its subharmonic one, especially for larger swirl q ⫽2 关Fig. 2共a兲兴. Therefore, in the axisymmetric flows swirl slightly suppresses the rolling-up and vortex pairing. The contours of the azimuthal vorticity show that the flow field with q⫽0.5 is nearly indifferent from the swirlfree jet in the initial stage 共Fig. 3, t⫽10兲. Furthermore, the vortical structures of the two flows are quite similar during the process of the rolling-up and vortex pairing (t⫽20,23). The swirling jet differs from the jet without swirl in that the negative azimuthal vorticity appears in the streamwise vortex core under the stretch of the stress field produced by the vortex rings. B. Vortex dynamics of the appearance of negative azimuthal vorticity

Since the creation of negative azimuthal vorticity is of vital importance to the nonlinear evolution of the flow field, we now address its mechanism using a vortex dynamics argument. The production of negative azimuthal vorticity is also considered to be necessary for the flow to decelerate on the axis in Brown and Lopez’s study,24 and this might be related to vortex breakdown in swirling jets. Consider a sim-

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Phys. Fluids, Vol. 13, No. 4, April 2001

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FIG. 2. Variation of energy of axial disturbances 共a兲, RMS of radial velocity 共b兲, momentum thickness 共c兲, and the extrema of azimuthal vorticity 共d兲 with time for Re⫽500, ␳ ⫽1, and R 0 /⍜ 0 ⫽33.9.

d␻r ⳵ur ⳵ur ␻r ⫽␻r ⫹␻z ⫹ ␯ ⵜ 2␻ r⫺ 2 , dt ⳵r ⳵z r

共7兲

vorticity at later time, while in region B there appears a positive azimuthal vorticity. Hereafter the relatively weak vortex structures with negative vorticity will be strengthened under the continuous interaction between the primary vortex ring and the columnar vortex, and this causes the formation of a secondary vortex ring with opposite vorticity.

␻␪ d ␻ ␪ ␻ ␪u r 2 ␻ ru ␪ ⫽ ⫺ ⫹ ␯ ⵜ 2␻ ␪⫺ 2 , dt r r r

共8兲

C. Formation of secondary vortex ring and radial fluid motion

plified model consisting of a Lamb–Oseen vortex at the centerline and an array of wrapped thin vortex rings, as sketched in Fig. 5. The incompressible vorticity transport equations in cylindrical coordinate are

冉 冉

⳵uz ⳵uz d␻z ⫽␻r ⫹␻z ⫹ ␯ ⵜ 2␻ z . dt ⳵r ⳵z

冊 冊

共9兲

For a swirl-free jet with u ␪ ⫽0 and ␻ z ⫽0, the second terms on the right-hand side of 共7兲 and 共8兲 are zero. When a swirl is added, both terms cause a variation of the radial and azimuthal components of vorticity. Initially we have u ␪ ⬎0, ␻ z ⬎0, and ␻ ␪ ⫽ ␻ r ⫽0 in the columnar vortex core. It is easily seen that in region A below the vortex ring, there is ⳵ u r / ⳵ z⬎0 due to the induction of vortex ring; while in region B, neighboring A, there is ⳵ u r / ⳵ z⬍0. Thus, by 共7兲 we have d ␻ r /dt⬃ ␻ z ⳵ u r / ⳵ z⬎0 in region A, and hence ␻ r ⬎0 as time advances. In contrast, the second term on the right of 共8兲 is negative and hence d ␻ ␪ /dt⬍0. Therefore, in region A with ␻ ␪ ⫽0 at t⫽0 there will appear a negative azimuthal

During t⫽26– 40, the RMS of radial velocity remains almost constant for the swirling jet 兵see a ‘‘plateau’’ in the RMS( v r )⬃t curve for q⫽0.5 关Fig. 2共b兲兴其. This corresponds to the stage of the formation of a secondary vortex ring with negative azimuthal vorticity. As shown in Fig. 2共d兲, ( ␻ ␪ ) min is very small before t⫽26, whereas its absolute value increases exponentially in the third stage, and eventually exceeds the positive vorticity. At t⫽35, ( ␻ ␪ ) max⫽1.59 while ( ␻ ␪ ) min⫽⫺2.27 for q⫽0.5. Thereafter (t⬎42) it is observed that the momentum thickness ⍜, RMS( v r ), and E KHv r all increase rapidly 关note that the vertical ordinate is logarithmic in Fig. 2共a兲兴. This phenomenon is due to the strong interaction between the vortex ring and columnar vortex. The contours of azimuthal vorticity ␻ ␪ are shown in Fig. 6 for q⫽0.5 and t⬎23. The flow structures at t⫽35 关Fig.

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FIG. 3. Early evolution (t⭐23) of contours of azimuthal vorticity for Re⫽500, ␳ ⫽1, and R 0 /⍜ 0 ⫽33.9 with q⫽0 and 0.5, in which the computational grid is 256⫻64 in (r,z) directions. The maximum, minimum, and increment of the contours are 共a兲 1.72, 0.11, 0.11; 共b兲 1.65, 0.11, 0.11; 共c兲 1.64, 0.10, 0.11; 共d兲 1.714, 0.087, 0.116; 共e兲 1.628, ⫺0.083, 0.122; and 共f兲 1.591, ⫺0.216, 0.129.

6共a兲兴 indicate that the negative azimuthal vorticity in the vortex core region is stretched under the action of vortex ring formed in the vortex pairing process, and erupts deeply into the mixing layer. At t⬇45, the flow field is dominated by the

motion of a pair of vortex rings, i.e., the primary and secondary vortex rings with opposite azimuthal vorticity 关Fig. 6共b兲兴. As is well known, the two rings induce each other just like a pair of point vortices rotating in the opposite direction,

FIG. 4. The dispersion relation ␻ i (k) obtained by a Chebyshev spectral collocation method for axisymmetric mode with different q (Re⫽500, ␳ ⫽1). 共a兲 R 0 /⍜ 0 ⫽33.9; 共b兲 R 0 /⍜ 0 ⫽11.3.

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Phys. Fluids, Vol. 13, No. 4, April 2001

FIG. 5. A sketch of the formation of negative azimuthal vorticity.

and produce a positive radial velocity in the (r, ␪ ) cross plane. It is this outward radial motion of the pair of rings that leads to the third increase of the RMS of radial velocity RMS( v r ) and the momentum thickness ⍜ 共see Fig. 2兲. Figure 6共b兲 shows the pair of vortex rings moves radially outward, carrying a body of fluid from the jet core region which builds up a new shear layer 关Figs. 6共c兲 and 6共d兲兴. This shear layer also rolls up and eventually merges with the primary vortex ring. The vortical structures of flow field at t

A numerical study of dynamics

957

⫽95 mainly consist of two vortex rings with opposite vorticity, as shown in Fig. 6共f兲. This complex phenomenon has also been confirmed by the simulation for a larger swirl ratio q⫽2 共not shown here兲. Figure 7 is the variation of the radial location of the maximum azimuthal vorticity in the primary vortex ring for q⫽0.5 and q⫽2. It is interesting that the radial motion of the flow for q⫽2 is weaker than that for q⫽0.5. This is also in agreement with the evolution of RMS of radial velocity RMS( v r ) in Fig. 2共b兲, in which the curve for q⫽0.5 rises higher than that for q⫽2. The flow induced by an array of periodic vortex rings wrapped around a columnar vortex core has been studied by Marshall.25 He investigated numerically the responses of the columnar vortex to the ring-induced velocity field for the axisymmetric flow. It was found that the ring-induced velocity is initially observed to cause the formation of a wave in the columnar vortex which travels with the rings, and a sharp cusp is observed to form at the wave crest. Vorticity in the columnar vortex is ejected out from this cusp, thus creating a negative azimuthal vorticity there. This result is well consis-

FIG. 6. Contours of azimuthal vorticity for t⭓35 in the flow of q⫽0.5, Re⫽500, ␳ ⫽1, and R 0 /⍜ 0 ⫽33.9. The minimum, maximum, and increment of the contours are 共a兲 ⫺2.029, 1.349, 0.241; 共b兲 ⫺3.635, 2.315, 0.425; 共c兲 ⫺3.574, 1.554, 0.366; 共d兲 ⫺2.587, 1.088, 0.262; 共e兲 ⫺1.832, 1.072, 0.207; and 共f兲 ⫺1.155, 0.875, 0.145.

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FIG. 7. Variation of the radial location of the maximum of azimuthal vorticity of the primary vortex ring for Re⫽500, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽1.

tent with our calculations. The outward radial motion was also noted by Park et al.11 in their axisymmetric computations, and was thought to be beneficial to fluid entrainment or mixing. Martin and Meiburg3 performed a numerical simulation for their swirling jet model, in which the secondary counter-rotating vortex rings were also observed. However, since the initial azimuthal velocity profile in their study was centrifugally unstable, the formation of the counter-rotating vortex rings was thought to be the result of the interaction between the Kelvin–Helmholtz instability and centrifugal instability.

Hu, Sun, and Yin

subharmonic mode exceeds the primary one, and thus implies the vortex pairing in the flow field. The agreement between the numerical simulation and the linear theory is illustrated in Table I for the linear growth rate. As expected, the development of mixing layer for R 0 /⍜ 0 ⫽11.3 is much slower than the case for R 0 /⍜ 0 ⫽33.9. The four stages of evolution are evidently seen by the RMS( v r ) in Fig. 8共b兲, and are also shown in Fig. 9 with the contours of azimuthal vorticity. Figures 9共a兲 and 9共b兲 plot the rolling-up and pairing of vortex rings. The formation of the counter-rotating vortex ring is shown in Fig. 9共c兲, and Fig. 9共d兲 draws the outward movement of the pair of vortex rings. These results show that the nonlinear evolution of vortical structures for the thicker shear layer is similar to that for the thinner one. A difference between the two cases is that the interaction between the vortex ring and columnar vortex for the thicker jet R 0 /⍜ 0 ⫽11.3 is more dramatic than that for R 0 /⍜ 0 ⫽33.9. The dependence of flow evolution on the Reynolds number is reported in Fig. 10 for E KHv r and the extremum of ␻ ␪ with a grid of 256⫻64 in (r,z) directions. As expected, less viscosity leads to larger linear growth rate in the early stage. In Fig. 10共b兲 ␻ ␪ varies slowly for Re⫽200, which implies the viscosity partially suppresses the interaction of vortical structures. To study the role of parameter ␳ on the dynamics, the flow with a larger Lamb–Oseen vortex core was also computed for Re⫽500, q⫽0.5, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽2.25 共figures not shown here兲. The numerical results are similar to the case with ␳ ⫽1 and no essential difference has been found. IV. NON-AXISYMMETRIC RESULTS

D. Influences of parameters on flow evolution

The effects of swirl ratio q on the dynamics of flows have been documented in the previous sections; here we aim to study the influences of other parameters on the evolution for axisymmetric flows. The time history of the perturbed energy E KHv r and the RMS of the radial velocity are displayed in Fig. 8 for the flow with a thicker shear layer 关Re ⫽500, q⫽0.5, R 0 /⍜ 0 ⫽11.3, ␳ ⫽1.0, and the computational grid is 256⫻64 in (r,z) directions兴. It was found that the primary perturbed mode grows linearly initially, then it reaches saturation at about t⬇40. At t⬇50 the energy of the

Previous investigations 共e.g., Ref. 16兲 have confirmed that the mechanism of the onset of secondary instability 共azimuthal instability兲 in a circular jet without swirl is analogous to that of the streamwise vortices in a planar shear layer which have been extensively studied for many years.21,26 However, what modification should be made for a swirling jet is so far still an open question. This also implies that a theory for determining the most unstable azimuthal mode in a fully 3-D jet is still lacking. In the present study, following Ref. 27, the perturbed azimuthal wave number is chosen to be n⫽3 for both swirl and swirl-free jets.

FIG. 8. Variation of energy of axial disturbances 共a兲 and RMS of radial velocity 共b兲 with time for Re⫽500, ␳ ⫽1, and R 0 /⍜ 0 ⫽11.3.

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FIG. 9. Contours of azimuthal vorticity for the flow of q⫽0.5, Re⫽500, ␳ ⫽1, and R 0 /⍜ 0 ⫽11.3 with a grid of 256⫻128 in (r,z) directions. The minimum, maximum, and increment of the contours are 共a兲 ⫺1.343, 0.577, 0.137; 共b兲 ⫺1,944, 1.643, 0.256; 共c兲 ⫺2.961, 1.081, 0.289; and 共d兲 ⫺1.343, 0.577, 0.137.

Before examining the non-axisymmetric results, following Refs. 16 and 21 we define four special planes in the flow fields. Two of them are denoted by MP and CP, representing the (r, ␪ ) planes crossing the middle of the braid region and the core of vortex rings, respectively. The other two are the (r,z) planes, i.e., RP, the plane traversing the rib vortices, and BP, the plane between the rib vortices. A grid of 64 ⫻256⫻64 in ( ␪ ,r,z) directions is adopted to all computations in this section. A. Early evolution

For a fully 3-D swirling jet, it is necessary to distinguish two kinds of streamwise vortices in the flow, i.e., the columnar vortex at the centerline and the streamwise vorticity cre-

ated by the azimuthal instability. Although the Lamb–Oseen vortex always decays due to the viscous dissipation, as we have shown in the preceding section, its interaction with the vortex rings plays a key role in the evolution of vortical structures. In contrast, the latter grows continuously during evolution because of development of azimuthal instability. It should be noted that a basic difference of a threedimensional swirling jet from a swirl-free one is that the swirl, i.e., azimuthal velocity, creates an additional tangential stress field in the flow. So, the 3-D flow are under the action of joint stretches in both axial and azimuthal directions. We will show that the joint stretches are of great importance in relevant dynamics, which promotes the streamwise vortical layer to breakdown into small eddies.

FIG. 10. Variation of energy of axial disturbances 共a兲 and extrema of azimuthal vorticity 共b兲 with time for different Reynolds numbers with q ⫽0.5, ␳ ⫽1, R 0 /⍜ 0 ⫽33.9, and a grid of 256⫻64 in (r,z) directions.

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FIG. 11. Evolution of energy of axial and azimuthal disturbances for Re ⫽500, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽2.25.

First we report the numerical results of the flow with q ⫽0.5, Re⫽500, ␳ ⫽2.25, and R 0 /⍜ 0 ⫽33.9. Figures 11共a兲 and 11共b兲 display the variation of the energy of axial and azimuthal disturbance waves versus time for q⫽0 and q ⫽0.5. The E KHv r (k) curve 关Fig. 11共a兲兴 shows that in the linear stage the axial disturbance grows in a way quite similar to the axisymmetric case, indicating that the dynamics of the flow is dominated by the development of the primary harmonic and its subharmonic modes. The primary harmonic mode k 1 ⫽2 reaches saturation at t⫽14 and its subharmonic one at t⫽32. In this period the growth rates of the disturbances are slightly less than the swirl-free ones, as has been predicted by the linear stability analysis. From Fig. 11共b兲 it can be found the azimuthal disturbance waves develop almost linearly in the early time, with a growth rate slightly larger than the swirl-free case. Before t⫽20 it only has little influence on the dynamics of the flow since the order of magnitude of the azimuthal disturbance energy is only about 10⫺4 . This energy is too small to dominate the 3-D evolution of the flow before the vortex merging has accomplished. B. Development of 3-D vortical structures

The contours of the streamwise vorticity at the CP plane are shown in Fig. 12 for t⫽10 and t⫽15. A streamwise

vortex layer created by the azimuthal instability is evidently seen, which has already been observed in swirl-free jets or a 3-D planar mixing layers. However, compared with swirlfree jets the vortical structures are distorted and dislocated due to the effect of azimuthal velocity. The contours of the streamwise vorticity in CP and MP for both swirl and swirlfree jets are illustrated in Fig. 13 at t⫽20 to show the evolution of the streamwise vortical layer. Further development of the flow under the effect of the joint stretch results in very complicated structures of streamwise vortices. Figures 14 and 15 describe the azimuthal and axial vorticity in the RP and CP planes at t⫽35. Comparing the vortical structures to those in axisymmetric flow, one may find that the vortex ring pair consisting of the primary and secondary vortex structures, although more tanglesome, still dominates the evolution of azimuthal vorticity in the (r, ␪ ) plane. The rib vortices, commonly observed in 3-D jets without swirl, can hardly be found now since they are destroyed by the tangential velocity. The strength of the streamwise vorticity induced by the azimuthal instability can be comparable with that of the Lamb–Oseen vortex column in this stage. This is because the former grows all the way to the late stage while the latter decays continuously. At this moment (t⫽35) the streamwise vortical structures are formed as a series of heli-

FIG. 12. Early evolution of streamwise vorticity in the CP plane for q ⫽0.5, Re⫽500, R 0 /⍜ 0 ⫽33.9, ␳ ⫽2.25, and a grid of 64⫻256⫻64 in ( ␪ ,r,z) directions. The minimum, maximum, and increment of the contours are 共a兲 ⫺0.017, 0.928, 0.068 and 共b兲 ⫺0.160, 0.923, 0.077.

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FIG. 13. Contours of streamwise vorticity in CP and MP planes for q⫽0 and q⫽0.5 共Re⫽500, R 0 /⍜ 0 ⫽33.9 and ␳ ⫽2.25兲 at t⫽20. The minimum, maximum, and increment of the contours are 共a兲 ⫺0.282, 0.282, 0.063; 共b兲 ⫺0.282, 0.282, 0.063; 共c兲 ⫺0.140, 0.822, 0.069; and 共d兲 ⫺0.019, 0.884, 0.065.

cal arm-shaped vortices as well as the columnar vortex at the centerline 共see Fig. 15兲. These results reveal that the streamwise vortex layer evolved from the early stage due to the secondary instability will rapidly breakdown into small eddies under the axial and azimuthal stretches. Actually, the joint stretch can breakdown large-scale organized structures even for very small swirl ratio. Figure 16 plots the evolution of the streamwise vorticity in CP and MP planes for q⫽0.1, Re⫽500, ␳ ⫽1, and R 0 /⍜ 0 ⫽33.9. It can been observed that at t⫽10 the streamwise vorticity in the ring region and in the braids created by the secondary insta-

bility is distributed in the same way but out of phase by ␲ 关Figs. 16共a兲 and 16共b兲兴. Hence, at the same azimuthal location the sign of ␻ z is opposite at rings and braids. The early evolution of the streamwise vortical layer is quite similar to that observed by Brancher et al.27 for a swirl-free jet, since the effect of swirl is too weak 关Figs. 16共c兲 and 16共d兲兴. As time advances, the streamwise vortex layer does not collapse as predicted by the widely accepted theory suggested by Lin and Corcos26 for a planar mixing layer 共which has also been adopted in 3-D round jet successfully兲, but distorts and breakdowns under the action of the additional tangential

FIG. 14. Contours of azimuthal 共a兲 and streamwise vorticity 共b兲 in the RP plane for q⫽0.5, Re⫽500, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽2.25 at t⫽35. The minimum, maximum, and increment of the contours are 共a兲 ⫺1.246, 1.606, 0.204 and 共b兲 ⫺0.347, 3.141, 0.249.

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FIG. 15. Contours of streamwise vorticity in the CP plane for q⫽0.5, Re ⫽500, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽2.25 at t⫽35.

stress field 关Figs. 16共e兲 and 16共f兲兴. The broken streamwise vortex layer is illustrated in Figs. 16共g兲 and 16共h兲 at t⫽40. The onset of the small eddies have been observed by many experiments.1,13,14 In these studies a swirl was thought to be helpful to the transition to turbulence and the increase of the Reynolds stress level. Thus, continuing the computation to this stage may cause the disturbance energy to pile up at high wave numbers, for which a turbulent model or a subgrid scale model should be included in numerical simulations. C. Influences of parameters on flow evolution

Figure 17 displays the evolution of energy for azimuthal disturbances E apv r for different Reynolds number. The figure illustrates viscosity evidently decreases the growth of nonaxisymmetric waves at the linear stage. In Fig. 17共b兲 ␻ ␪ varies slowly for lower Reynolds number Re⫽200, and the streamwise vorticity in CP and MP planes at t⫽40 共Fig. 18兲

FIG. 16. Contours of streamwise vorticity in the CP and MP planes for q ⫽0.1, Re⫽500, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽1, in which the ( ␪ ,r,z) mesh is 64 ⫻256⫻64. The minimum, maximum, and increment of the contours are 共a兲,共b兲 ⫺0.044, 0.256, 0.033; 共c兲,共d兲 ⫺0.220, 0.254, 0.053; 共e兲,共f兲 ⫺2.009, 1.919, 0.281; and 共g兲,共h兲 ⫺2.727, 2.357, 0.363.

FIG. 17. Variation of E apv r 共a兲 and the extrema of azimuthal vorticity 共b兲 for different Reynolds number 共q⫽0.5, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽2.25兲.

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FIG. 18. Contours of streamwise vorticity at CP 共a兲 and MP 共b兲 plane and t⫽40 for Re⫽200, q⫽0.5, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽2.25, in which the ( ␪ ,r,z) mesh is 64⫻256⫻64. The minimum, maximum, and increment of the contours are ⫺0.350, 1.106, 0.104.

shows that the arm-like structures are more regular than the case for Re⫽500. Thus we suggest that the increase of azimuthal instability will be largely suppressed by viscosity. The dependence of swirl ratio q on the flow characteristics is studied in Fig. 19 for Re⫽500, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽2.25. The growth rate of an azimuthal perturbed wave of q⫽0.5 is evidently larger than the case of q⫽0.1. In Fig. 19共b兲, the variation of RMS( v r ) shows the development of the mixing layer is slightly slower for larger swirl ratio in the earlier stage, which has been documented by linear stability. When the flow field is dominated by three-dimensional perturbations after vortex pairing (t⬇27), the RMS( v r ) for q ⫽0.5 increases rapidly and exceeds the curve for q⫽0.1, which implies a larger swirl ratio is helpful to mixing enhancement in full 3-D flows. D. Discussion on mechanisms of swirl-enhanced mixing

The variation of momentum thickness ⍜ for q⫽0.5, Re⫽500, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽2.25 关Fig. 20共a兲兴 shows that swirl is very helpful to the growth of mixing layer and entrainment in the flow. The early stage of the evolution of momentum thickness ⍜ is nearly indifferent from that of swirl-free jets. Once the three dimensionality dominates the development of the flow, the momentum thickness for q ⫽0.5 exceeds that for q⫽0 rapidly.

Although many studies have confirmed that a swirl can effectively raise the rate of fluid entrainment, as we mentioned, a satisfactory physical explanation for the mixing enhancement is not yet available since, e.g., the role of centrifugal instability has not been fully clarified. To understand this issue, a linear stability analysis had been conducted for the basic flow with a centrifugally unstable azimuthal velocity.20 Anyway, since Naughton et al.13 obtained swirlenhanced mixing with a centrifugally stable profile, which is also verified by the present numerical study, we conclude that centrifugal instability might not be the necessary condition for the enhancement of mixing in a 3-D swirling jet. In this study we focus on the centrifugal stable case. For axisymmetric flows, as documented in Sec. III, the interaction between the vortex rings and the columnar vortex, which results in the emergence of counter-rotating vortex ring and the outward radial motion in the flow field, is beneficial for the mixing enhancement. For full threedimensional flows, the entrainment induced by the motion becomes weaker than the axisymmetric case. This is because the joint stretches have partially destroyed the large-scale vortical structures. However, the phenomenon can still be observed in the flow when we examine the mean profile of azimuthal vorticity at the CP plane for the non-axisymmetric flow 关Fig. 20共b兲兴. From that it is seen that the vorticity profile for the swirling jet is wider than that for q⫽0, and the loca-

FIG. 19. Variation of E apv r 共a兲 and the root mean square of v r 共b兲 are plotted to show the influence of q on the flow evolution for Re⫽500, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽2.25.

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FIG. 20. The role of swirl on the mixing in the flow 共Re⫽500, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽2.25兲: 共a兲 momentum thickness and 共b兲 distribution of mean azimuthal vorticity in CP plane at t ⫽40 for the 3-D flow with q⫽0.5.

tion of the maximum of azimuthal vorticity shifts away from the centerline. This result is in agreement with the measurement of Naughton et al.13 As mentioned before, another mechanism for the mixing enhancement in swirling jets proposed by Naughton et al. is that the additional tangential velocity in the flow creates a novel ingredient of shear. The total shear in the mixing layer is therefore increased, causing an additional turbulence production. More importantly, any mixing produced by the axial vorticity is not expected to be affected by compressibility. The maximum and minimum of axial vorticity for Re ⫽500, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽2.25 are plotted in Fig. 21 to show ␻ z becomes much larger in the swirling jet due to azimuthal instability. At t⫽40, ␻ z ⫽5.18 for q⫽0.5 compared with ␻ z ⫽0.85 for the swirl-free jet. Another effect of the addition of swirl is that the joint shears in both axial and azimuthal directions accelerate the transition to small-scale vortical structures in flows, thereby leading to a further enhancement of fluid mixing. According to our axisymmetric and non-axisymmetric results, combined with the view of Naughton and co-

workers, it seems the following issues are both responsible for mixing enhancement in the swirling jet, they are the outward radial motion of vortex ring pair, the rapid production of the small eddies, and growth of streamwise vorticity. However, whether or not these proposed mechanisms are all effective in practical applications needs be further tested under more realistic and complicated conditions, such as the inclusion of compressibility, chemical reaction, and turbulence. Some of these issues will be considered in our future studies. V. SUMMARY

A direct numerical simulation has been conducted for the effects of swirl on the dynamics of axisymmetric and three-dimensional round jets. Results show that the evolution of flows in the early stage compares well with the linear stability theory. In nonlinear stage, in addition to the routinely observed rolling-up of shear layer and pairing of vortex rings, a novel phenomenon beneficial to the fluid entrainment is the outward movement of vortex pairs induced by the interaction of the primary ring and streamwise columnar vortex, thus leading to a rapid increase of momentum thickness. When the responses of azimuthal disturbances are considered, it is found that the shear of the tangential velocity has a significant influence on the dynamics of flows. The streamwise vortical layer developed in the early stage due to azimuthal instability will rapidly breakdown into small eddies under the joint stretch of the axial and azimuthal shear. The numerical results are also used to analyze and evaluate the existing theories on mechanism of swirl-enhanced mixing. ACKNOWLEDGMENTS

FIG. 21. Variation of the extrema of streamwise vorticity ␻ z for the axisymmetric and full 3-D flows with and without swirl 共Re⫽500, R 0 /⍜ 0 ⫽33.9, and ␳ ⫽2.25兲.

This work was supported in part by National Natural Science Foundation of China and U.S.–China International Cooperative Research Program sponsored by National Science Foundation of the United States 共under Grant No. INT9511552兲. The authors wish to express their gratitude to Professor G. F. Carnevale of the University of California and Professor P. Orlandi of the University of Rome for their generosity in

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Phys. Fluids, Vol. 13, No. 4, April 2001

providing computational codes. Useful discussion with Professor J.-Z. Wu of the University of Tennessee Space Institute is also acknowledged. 1

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14

R. D. Mehta, D. H. Wood, and P. D. Clausen, ‘‘Some effects of swirl on turbulent mixing layer development,’’ Phys. Fluids A 3, 2716 共1991兲. 15 S. H. Park and H. D. Shin, ‘‘Measurement of entrainment characteristics of swirling jets,’’ Int. J. Heat Mass Transf. 36, 4009 共1993兲. 16 R. Verzicco and P. Orlandi, ‘‘Direct simulation of the transitional regime of a circular jet,’’ Phys. Fluids 6, 751 共1994兲. 17 I. Danaila, J. Dus˘ek, and F. Anselmet, ‘‘Coherent structures in a round, spatial evolving, unforced, homogeneous jet at low Reynolds number,’’ Phys. Fluids 9, 3323 共1997兲. 18 R. Verzicco and P. Orlandi, ‘‘A finite difference scheme for the three dimensional incompressible flows in cylindrical coordinates,’’ J. Comput. Phys. 123, 402 共1996兲. 19 J. Kim and P. Moin, ‘‘Application of a fractional-step method to incompressible Navier–Stokes equations,’’ J. Comput. Phys. 59, 308 共1985兲. 20 G.-H. Hu, ‘‘The stability analyses and nonlinear evolution of swirling flows,’’ Ph.D. dissertation, University of Science and Technology of China, 1999. 21 M. M. Rogers and R. D. Moser, ‘‘The three dimensional evolution of a plane mixing layer: The Kelvin–Helmholtz rollup,’’ J. Fluid Mech. 243, 183 共1992兲. 22 R. W. Metcalfe, S. A. Orszag, M. E. Brachet, S. Menon, and J. J. Riley, ‘‘Secondary instability of a temporally growing mixing layer,’’ J. Fluid Mech. 184, 207 共1987兲. 23 M. R. Khorammi, M. Malik, and R. Ash, ‘‘Application of spectral collocation techniques to the stability of swirling flows,’’ J. Comput. Phys. 81, 206 共1989兲. 24 G. L. Brown and J. M. Lopez, ‘‘Axisymmetric vortex breakdown part 2. Physical mechanics,’’ J. Fluid Mech. 221, 553 共1990兲. 25 J. S. Marshall, ‘‘The flow induced by periodic vortex rings wrapped around a columnar vortex core,’’ J. Fluid Mech. 345, 1 共1997兲. 26 S. J. Lin and G. M. Corcos, ‘‘The mixing layer: Deterministic models of a turbulent flow. Part 3. The effect of plane strain on the dynamics of streamwise vortices,’’ J. Fluid Mech. 141, 139 共1984兲. 27 P. Brancher, J. M. Chomaz, and P. Huerre, ‘‘Direct numerical simulations of round jet: Vortex induction and side jet,’’ Phys. Fluids 6, 1768 共1994兲.

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The non-axisymmetric DNS shows that the streamwise vortex layer developed in the early stage of evolution due to ... and absolute/convective instability have been extensively in- vestigated see Refs. 4–6 and references ... compared with swirl-free jets, and the vortex pairing was suppressed when the swirl was added.

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Jan 1, 2001 - nature materials are often found confined within host mate- rials. Molecules and ions ... guest molecules insignificant. Table I lists the potential ...

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11–13 Е while the window diameter varies between 4.5 to ... ability per window or bottleneck of the unit cell divided by the probability to be within that cell.

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bath at T0 is represented by the thermal conductance λr . λr may be con- trolled by .... for better thermal contact) obtained through the above auto- mated process in ... 21 National Instruments Corporation, 11500 N. Mopac Expwy., Austin, TX.

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Electron paramagnetic resonance and magnetization in Co doped ... of electron paramagnetic resonance EPR and dc magnetic susceptibility in .... Project No.

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In this scenario, revisited early studies of the 1950s4 on mixed-valence manganese perovskites led to the ... VOLUME 89, NUMBER 3. 1 FEBRUARY 2001. 1746.

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of the dielectric constant at temperatures ranging from 4.2 to 80 K and in the audio frequency range ... Low frequency dielectric measurements are very impor- .... function generator (Stanford Research Systems Model DS360), (b) isolation.

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degree of micellar organization is characterized by. P(nmax)P(nmin), where nmin and ... are attractive. We also associate an energy b. 0 to each bent bond in the ...

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given degree of confidence that the pulse train has ended. As ... a 2.5-ms separation when Fr 100 Hz.) This condition con- ...... ''Dual temporal pitch percepts.

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Understanding diffusion processes in porous media is a subject of fundamental as ... importance.1–3 Examples of such processes in nature are ion diffusion in ...

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all the binary combinations of these five common carbonates: EC, PC, DMC, EMC, DEC. ... binary carbonate systems: propylene carbonate (PC)-dimethyl car-.

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Institute of Physics. S0021-8979(00)09811-X. I. INTRODUCTION ... ing polarization fatigue, becomes critical as the storage ca- pacitor in NV-FRAM decreases in ...

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change from single domain to multidomain magnetic behav- ior as grain size increases.16,17 On the other hand, TC remains constant for all the samples see Fig ...

Using JCP format - Henry Fu
The soft-mode frequencies reveal strong bend-stretch coupling in the complex. Excellent .... potential.11 The analysis of the observed transitions is readily.

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Department of Physics and Department of Materials Science and Engineering,. North Carolina State University, Raleigh, North Carolina 27695-8202. (Received 5 .... stepped through computer control, and the lock-in outputs are recorded by ...

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E-mail: [email protected]. Abstract. .... the photoexcited carriers in the bulk of the material after they ... in heat generation, followed by bulk and surface recombi-.

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May 1, 2000 - host atoms by the spin-orbit interaction, which allows the impurity spin to get ... of the conduction electrons on the nonmagnetic host atoms.

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1A–1C. A small vessel 250 mL, fitted with a one- way valve and nitrogen gas inlet and liquid nitrogen outlets, is located towards the bottom of the supply Dewar.

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Run III attempts to simulate the β- to α-phase transformation in the NPT ensemble while run IV is an NPT ensemble simulation of liquid toluene. Run IIIa is an.

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a)Electronic mail: [email protected]. REVIEW .... length providing a 1 mm cylindrical shell space for condens- ... This shell space is vacuum sealed at both.

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format amplop.pdf
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Aug 15, 2013 - He earned a Master in Business. Administration degree from Boston College and a Master of Professional. Studies degree from Salem State University. Mr. Wentworth has been an intermediate classroom teacher in both Topsfield and Lawrence