International Journal of Fluid Dynamics (2000), Vol. 4, Article 1

Three-Dimensional Vortices of a Spatially Developing Plane Jet

SING HON LO

School of Mechanical, Aeronautical and Production Engineering,
Kingston University, Roehampton Vale, Friars Avenue, London SW15 3DW, U.K.

PETER RYAL VOKE

School of Mechanical and Materials Engineering,
The University of Surrey, Guildford GU2 5XH, U.K.

NICOLE JACQUELINE ROCKLIFF

School of Mechanical and Materials Engineering,
The University of Surrey, Guildford GU2 5XH, U.K.

(First Received 5th May 1999; and in revised form 28th February 2000. Published 1st March 2000)


Abstract

The coherent eddy structures of a plane jet have been investigated by using large-eddy simulation of the incompressible Navier-Stokes equations. The simulation data is presented in an animated form in three plan views that can be studied to follow the evolution of the structures. Without forced periodic excitation, the instability of the jet and transition to turbulence are characterised by the rolling up of anti-symmetric spanwise rollers. The results clearly show the formation of horseshoe-like streamwise vortex pairs and longitudinal ribs through three-dimensional instabilities of the spanwise rolls. The streamwise vortex pairs align at an angle of approximately 135 degrees (or 45 degrees) to the mean flow. The two vortex cores are connected by a thin vorticity layer and the wavelength of the spanwise perturbation is 0.51 of the distance between successive rollers. Conceptual models for the vortical structures are proposed based on the animated flow visualisation.


Keywords

  • Fluid Dynamics
  • Turbulent Plane Jet
  • Coherent Structure
  • Vortices
  • Numerical Simulation

  • Copyright Notice

    This material is the work of the authors listed here. It is original and has not been published previously unless acknowledged. It may be freely copied and distributed provided that the names of the authors, the institution and the journal remain attached.


    Abstract

    1. Introduction

    2. Methods

    3. Results and Discussion

    4. Conclusion

    Acknowledgments

    References


    1. Introduction

    In recent years, large-scale structures in turbulent jets have received a great deal of attention owing to scientific and engineering interest. Coherent structures have a direct influence on entrainment, mixing, turbulence production and noise generation. The understanding of these structures is essential for the refinement of existing theory and models, improvement of engineering systems that use jet concepts, and the development of new methods for controlling turbulence. A general review on the topic of turbulent jets and plumes was given by List (1982). There have been numerous experimental studies on free turbulent jets. While most of the research has concentrated on axisymmetric jets and the effects of harmonic forcing, the coherent structures of the plane jet under natural conditions are also of great interest and have received less attention.

    Evidence of coherent structures in jets can be dated back to as early as the 1930s; Brown (1935) showed orderly vortices being produced on alternate sides of jets under acoustic forcing. One of the most well-known phenomena in a plane jet is the flag-like flapping motion noted by Oler and Goldschmidt (1982) (see also Thomas and Goldschmidt (1986)) which indicates the presence of large-scale structures. Mumford (1982) applied pattern recognition analysis (PRA) to study the fully-developed far field of a plane jet, several types of roller-like eddies being extracted from the experimental data. We have previously (Lo, 1994) performed large-eddy simulation of a turbulent plane jet and applied pattern recognition analysis to the simulation database. This work confirmed the existence of double roller structures in the turbulent region of the jet as proposed by Mumford (1982).

    Antonia et al. (1983) applied conditional averaging to study the structures in the nearly self-preserving region of a heated turbulent jet. They suggested that the double-roller eddies could align with the braids connecting two adjacent spanwise rolls. The presence of primary rollers and longitudinal vortices in the plane mixing layers and wakes has been well documented in experimental studies (Breidenthal, 1981; Bernal and Roshko, 1986; Lasheras and Choi, 1988; Cimbala et al., 1988; Hayakawa and Hussain, 1989) and numerical simulations (Metcalfe et al., 1987; Comte et al., 1992; Rogers and Moser, 1992; Moser and Rogers, 1993). Although we may expect the evolution of structures in the plane jet to be similar to those found in plane mixing layers and wakes, the detailed organisation of coherent structures is not fully clear in the jet. Sakakibara et al. (2000) investigated the near-field vortical structures in a plane jet and noted that the spanwise rollers develop a spanwise perturbation and evolve as an isolated pair of vortex filaments with preferred inclination. They not only observed the 'successive ribs' that connect successive rollers on the same side of the symmetry plane, but also 'cross ribs' which extend from a roller to an opposite roller across the symmetry plane.

    There have been a few numerical studies of axisymmetric jets based on discrete vortex methods (Action, 1980; Oler and Goldschmidt, 1982). Grant (1974) investigated the instability of axisymmetric jets by integrating the time-dependent vorticity transport equation using finite-difference equations. The generation of noise by coherent structures has been the subject of interest in recent years. A two-dimensional simulation of a compressible plane jet at subsonic and supersonic Mach numbers has been performed by Bastin et al. (1997) using semi-deterministic modelling (SDM). Mitchell et al. (1999) have applied direct-numerical simulation (DNS) to compute the sound radiated by subsonic and supersonic axisymmetric jets.

    We have shown (Lo, 1994) that the proper specification of boundary conditions is crucial for the realistic modelling of the entraining potential flow from the surrounding fluid in the large-eddy simulation of the plane jet. The problems of recirculation regions together with the distortion of the outer flow streamlines in the computations were encountered by both Grant (1974) in the axisymmetric jet and Lo and Voke (1996) in the plane jet, despite the fact that different approaches to the treatment of boundary conditions were used in these two studies.

    The present work forms part of a research project on the eduction of coherent structures from large-eddy simulations of the flat-plate boundary layer, plane wake and plane jet. The results of the conditional sampling analyses of the turbulent boundary layer and wake are reported elsewhere (Lo and Voke, 1996). The objective of the part of the study reported here is to elucidate the formation mechanisms of coherent structures and their evolution in a transitional plane jet.


    2 Methods

    The numerical approximations involve the discretization of the Navier-Stokes equations using a finite-volume method of second order (Voke and Potamitis, 1994). The pressure solver consists of a hybrid Fourier-multigrid algorithm (Voke and Zang, 1996). The physical problem comprises the incompressible flow of a plane jet discharging into stationary surrounding fluid. The computational domain is discretised in space by a Cartesian grid Nx x Ny x Nz, where x, y and z are the streamwise, cross-stream and spanwise coordinates respectively.

    The boundary conditions of the inflow profile and the y-walls are based on the analytical solution of the reduced Reynolds-averaged Navier-Stokes equations in two dimensions, following Schlichting (1968). The two-dimensional velocity profiles can be approximated by:

    equation 1

    equation 2

    equation 3

    where s = 7.67 is an empirical constant.

    This inflow profile is assumed to be fully turbulent. This is not strictly true in the simulation as transition takes place at a small distance away from the inflow plane. This only affects the values of growth and entrainment rate of the jet; the dynamics of turbulence is still fully captured in the simulation.

    At the inflow plane, random background noise was imposed to excite the instability of the flow. The Reynolds number based on the half-width of the inflow profile is 265. The simulation was also repeated for a much higher Reynolds number of 10,000 to clarify Reynolds-number effects by comparing the two simulations. It was found that the simulations were very similar but that the large-scale vortical structures that are the focus of interest in this paper were smeared and embedded in finer scale turbulence which masked the coherent motions at the higher Reynolds number. Since the flow structures are far less orderly and visible and the jet boundaries are very irregular, all the visualisations presented here are from the lower Reynolds number simulation.

    The outflow boundary was treated by the application of the convection equation decribed in Voke and Potamitis (1994). The physical width of the computational box was made as wide as possible to accommodate the spreading of the jet and to avoid the formation of unrealistic recirculating flows as discussed by Lo (1994). The boundary conditions are summarised as follows:

    Inflow plane:
    u-profile from analytical solution,
    v-profile from analytical solution,
    dw/dx = 0.
    Y-boundaries
    u-profile from analytical solution,
    du/dy = 0 and dw/dy = 0.
    Z-boundaries periodic.
    Outflow plane: convective:
    du/dt + Uc dui/dx = 0 where Uc is the convection velocity.

    Several subgrid-scale models, the Smagorinsky and 2nd-order velocity structure-function models and a low-Reynolds number model suitable for transitional flows, have been implemented and tested for the jet simulation. For instance for the Smagorinsky model (for which the results are given) the eddy viscosity is

    equation 4

    equation 5

    where nt and sij denote turbulent eddy viscosity and strain rate tensor respectively. Even for this model, which is known to be rather strong, the eddy viscosity is found to be very much lower than the molecular viscosity in all regions except near the outflow boundaries. No significant differences are found for simulations using other models, or none, except in the outflow region where the absence of any model leads to rougher contours where the structures break down.

    Cs is the model constant. Values of 0.1 and 0.2 have been used in the plane jet simulations, but again it was found that the results were not sensitive to the value of Cs . The SGS stress tensor is given by:

    equation 6

    Specifications of the simulation

    The parameters for the simulation are listed below:

    Viscosity n 1.54 x 10-5 m2/s
    Inflow centreline velocity Umax 6.1235 m/s
    Inflow velocity profile half-width lhalf 0.6664 mm
    Inflow Reynolds number Re 265
    Streamwise (x) domain length Lx 0.08 m (120lhalf)
    Cross-stream (y) domain length Ly 0.18 m (270lhalf)
    Spanwise (z) domain length Lz 0.02 m (30lhalf)
    Resolution Nx, Ny, Nz 120, 160, 32
    Inflow random disturbance amplitude ex, ey 2% Umax
    Time step Dt 7.5 x 10-6 s


    3 Results and Discussion

    Laminar jet tests

    The critical Reynolds number of a plane jet depends on the shape of the exit jet profile, disturbance amplitude and dimensionality. (The instability of axisymmetric jets was discussed by Sato (1960), Batchelor and Gill (1962) and Grant (1974)). Andrade (1939) found the Reynolds number of a plane jet for transition to turbulence to be 33. Two-dimensional laminar jet simulations are very cheap computationally and so we performed a series of laminar jet simulations in order to check the suitability of the boundary conditions, to find the optimum mesh configuration and computational domain size.

    A laminar jet of Reynolds number 30 (Umax = 0.30537 m/s, lhalf = 0.0015129 m) was simulated with a resolution of 128 x 128 grid points. The boundary conditions were the analytical profiles of a two-dimensional laminar jet given by Schlichting (1968). The initial velocity fields were set to zero everywhere except the boundaries. The simulation converged to the steady solution after a large number of time steps (150,000 steps; Dt = 0.000007 s). The simulation results agreed with analytical solutions very closely, with hardly any distortion of the streamlines of the potential flow or the velocity profiles at the outflow plane, and the convective outflow boundary condition worked perfectly. It was found that recirculation or instability of the computation would occur if the lateral width of the computational domain was not great enough.

    We also carried out a further series of two-dimensional simulations of the instability of a plane jet without any external forcing at a Reynolds number of 400. The results show very strong patterns of von-Kármán-like vortices dominated by the anti-symmetric mode. The instability mechanism possibly involves nonlinear interaction of the two vortex sheets in the jet. When the flow became turbulent in the downstream region, excess amplication of the vortical structures occurred owing to the fact that a 2-D turbulent jet flow is not physically realistic; vortex stretching and energy cascade are not possible in two dimensions. However the 2-D simulations were useful tests before running the more expensive 3-D simulations.

    Transitional-turbulent jet simulation

    The full three-dimensional jet simulation was run on a Cray YMP-8 for 375,000 time steps, requiring 645 cpu hours. At a Reynolds number of 265, maximum values for the eddy viscosity were found to be up to 3 times the molecular viscosity, and occurred close to the outflow boundary. Values of eddy viscosity were negligible in the regions of primary interest. The second-order time-averaged statistics of the transitional-turbulent jet simulation agree well with the experimental results of fully-developed turbulent jets at much higher Reynolds numbers. The non-dimensionalised Reynolds shear stress and turbulence intensities are in general higher than the experimental measurements, perhaps due to Reynolds number effects together with the influences of the inlet and entrainment boundary conditions (Lo and Voke, 1996).

    There is strong evidence of sinuous flapping motion in the laminar-transition regime as shown in Figure 1 which shows the contours of velocity and pressure patterns at streamwise position x/l = 45 (x = 0.03m). The horizontal axis is the local convection velocity Uc multipled by time. The frequency f of the flapping motions is estimated to be 126 Hz in this streamwise position. The Strouhal number is

    equation 7

    Figure 1

    Sinuous flapping motions in the laminar-transition region of the jet at x / l = 45; (x, y)-plane (side view).
    (a) contours of the streamwise velocity; (b) contours of the pressure fluctuations.

    Such wave-like motions are associated with the alternating positive and negative pressure patterns shown in Figure 1. The negative pressure patterns are signatures of low-pressure vortex cores. The flapping mode of the near jet is linked to the formation of two-dimensional spanwise rolls in an anti-symmetrical arrangement, although instantaneous symmetrical structures were occasionally observed in the flow field.

    Clear patterns of vortex shedding are observed in the transition region of the jet. In an attempt to investigate the effects of the random disturbances, the simulations were run for a long time with and without disturbances. Comparison of the instantaneous flow fields from the two simulations show that transition takes place earlier and the vortical structures are stronger with white noise imposed at the inflow plane at every time step. The effect of the random disturbances is to excite the natural instability modes of the jet earlier, implying that the occurrence of coherent structures depends on the character of the inlet disturbances.

    The spanwise rolls occur alternately about the jet centreplane in the same manner as those in a mixing layer. The initially straight spanwise rolls evolve and develop a spanwise perturbation. The transition process is followed by further contortion of the rolls and development of horseshoe vortices. The wavelength of the spanwise perturbation is approximately in the range between half to one spanwise width ( 30 lhalf )of the periodic domain.

    Figure 2 and Figure 3 show the cross-sections of the typical horseshoe-shaped vortex pairs frequently observed in the transition region of the simulation. The direction of the mean flow is pointing into the plane of the paper. The horseshoe structures are formed because the rollers undergo a spanwise perturbation, which is a common feature of plane shear flows. Pierrehumbert and Widnall (1982) suggested that practically any spanwise wavelength can be excited. The same phenomenon can also be observed in the plane jet (Sakakibara et al., 2000) and mixing layers (Breidenthal, 1981; Jimenez, 1983; Rogers and Moser, 1992). Such deformation of the spanwise rolls was described as outward kinking and inward kinking motion in the turbulent plane wake by Hayakawa and Hussain (1989).

    Figure 2

    Cross-stream slices of a cap-type horseshoe vortex structure. The jet is flowing into the page. Instantaneous velocity vectors in (y, z)-planes at streamwise positions
    x / l = (a) 63.7, (b) 65.7, (c) 67.5, (d) 69.4, (e) 71.3 and (f) 73.

    Figure 3

    Cross-stream slices of a cup-type horseshoe vortex structure. The jet is flowing into the page. Instantaneous velocity vectors in (y, z)-planes at streamwise positions
    x / l = (a) 53.5, (b) 54.4, (c) 55.3, (d) 56.3, (e) 57.2 and (f) 58.1.

    As shown in the succesive slices, two vortex cores are connected by a sinusoidal-like layer characterised by sharp velocity jump across it. The cross-sections of the structure resemble the shape of mushrooms. Similar features have also be seen in flow visualisations in plane wakes and mixing layers (Soria and Wu, 1992; Bernal and Roshko, 1986; Metcalfe et al., 1987; Lasheras and Choi, 1988; Comte et al., 1992).

    The heads of the horseshoe structure are clearly seen in the figures. They are confined to either side of the jet centre plane, having inclination between 136 and 141 degrees to the direction of the mean flow. These structures are strain dominated. No symmetric structure extending across the jet centreplane has been found. The horseshoe structures appear to have two distinctive configurations: cap and cup-types, as shown in Figure 4. The cap-type vortices (Figure 2) have opposite direction of circulation in the (y, z)-plane and 90 degrees difference in orientation as compared with their cup-type counterparts (Figure 3). In both configurations, the direction of spanwise vorticity for the head of horseshoe has the same sense of rotation as the mean velocity gradient dU/dy. For the cap-type vortices, the average inclination is very close to the direction of principal axis of the mean strain-rate tensor (135 degrees), suggesting that these structures are strain dominated; on the other hand, the inclination of cup-type structure is orthogonal to the principal rate-of-strain direction. Both type of structures are shown in the lower half of jet and therefore they are not the result of symmetry; both types also occur in the upper half of the jet. It also appears that the horseshoe vortices are usually staggered underneath the distorted spanwise roll. The spanwise roll displays an undulation along the span perturbed by the velocity field of the streamwise vortices.

    Figure 4

    Conceptual models of the horseshoe vortex structures:
    (a) cap-type horseshoe vortex, (b) cup-type horseshoe vortex.

    Despite the fact that the existence of roller eddies in the fully-developed region of a plane jet was suggested by Mumford (1982) and Antonia et al. (1983) based on pattern recognition analysis and conditional statistics, very little knowledge has been gained about the actual topology of the instantaneous structures. The present simulation provides the first direct evidence of the existence of orderly horseshoe vortices in the planar jet under unforced conditions. In some respects, the transitional horseshoe structures are reminiscent of the inclined double-roller eddy structures (Mumford, 1982; Lo, 1993) found in the self-preserving far-field of a turbulent plane jet. However, the laminar-transition horseshoe structures are far more orderly than the turbulent double-roller eddies. Here the inclination angle is roughly the same as the correlation measurements by Mumford (1982). Thomas and Goldschmidt (1986) also found the inclination to be 130 degrees based on time-delay correlation measurements. The main differences between the observed horseshoe vortices and the double-roller eddies proposed by Mumford (1982) are the absence of the heads and lateral inclination in the latter case. The current results are in good agreement with the topology and generation mechanism of horseshoe vortices proposed by Lo (1993) on the basis of the pattern recognition analysis of structures in a fully-developed plane jet, though the similarity in dynamics between the relatively young horseshoe vortices in the transition region and the much weaker double-roller eddies is not clear.

    Conceptual models of the horseshoe structures are presented in Figure 4. The thin connecting layer observed in Figure 2 and Figure 3 may be interpreted as an undulating vortex sheet. It is wrapped around the horseshoe vortex with the direction of vorticity roughly parallel to the cores of roller vorticity. The origin of this curved vortex sheet can be explained by the fact that vorticity from the jet comes from the boundary layer on the nozzle walls (from the inflexion in the inflow velocity profile in the case of the simulation). Once the plane vortex sheet is free of the nozzle, it rolls up and redistributes itself into concentrated cores of vorticity. In doing so, it has to satisfy the solenoidality condition that vortex lines can end only on themselves or a surface. The cores use up most of the available vorticity, but they remain connected by an ever-thinning vortex sheet. Successive rollers in the jets and mixing layers are connected by a stretched vorticity layer. We can think of the arrangement as a 'chinese scroll' with the initially flat vortex sheet connecting the bottom of a spanwise roller to the top of a successive roller as shown in Figure 5a. In Figure 5b, one of the rollers is distorted and lifted up to form a horseshoe structure. The vortex layer still connects each roller, since vorticity cannot be annihilated unless it is combined with vorticity of opposite sign. The resulting layer exhibits sinusoidal-like distortions connecting to each leg of the horseshoe vortex and also to the adjacent rollers. The existence of similar vorticity layer wrapping around vortex loops was visible in the flow visualisation of a cylinder wake by Cimbala et al. (1988).

    Figure 5

    (a) Vortex sheet rolling up into concentrated cores. (b) Distortion of rollers to form a horseshoe vortex.

    Flow animations

    The enstrophy (the magnitude of the vorticity vector) is computed everywhere in the flow field:

    equation 8

    In order to visualise the evolution of the structures, three animation films of the jet simulation in three orthogonal planes have been produced using MATLAB and Silicon Graphics EXPLORER moviemaker. Two-dimensional slices of the enstrophy w in the (x, y), (y, z), and (x, z)-planes were dumped at the rate of one every 30 time steps (Dt = 0.000007) over 12,000 time steps. The total number of frames in each plane is 400. The non-dimensionalised x, y and z-positions (based on the half-width lhalf of the inflow velocity profile) of the frames for Animations 1, 2 and 3 are listed as follows:

    Animation Plane xmin xmax ymin ymax zmin zmax
    1 (x, y) 27 102 -18.5 +16.6 15 15
    2 (x, z) 27 102 -1.4 -1.4 0.0 30
    3 (y, z) 66.6 66.6 -18.5 +16.6 0.0 0.0

    Animation 1. (2 MB MPEG) The magnitude of the vorticity vector field in a (x,y) plane (side view). For key to contour levels, see Figure 8. (Also available, the full size animation 1, 5 MB MPEG)

    Animation 2. (2 MB MPEG) The magnitude of the vorticity vector field in a (y,z) plane at x/l= 66.6 (front view; the jet is flowing into the screen). For key to contour levels, see Figure 8. (Also available, the full size animation 2 3.6 MB)

    Animation  3. (2 MB MPEG) The magnitude of the vorticity vector field in a (x,z) plane at y/l= -1.42 (top view). For key to contour levels, see Figure 8. (Also available, the full size animation 3, 8.3 MB MPEG)

    The anti-symmetrical spanwise rolls clearly dominate the flow organisation in the transition region as shown in Animation 1 and Figure 6. One of the most interesting features of the instantaneous pictures is the formation of rib vortices (Pierrehumbert and Widnall, 1982; Bernal and Roshko, 1986; Hayakawa and Hussain, 1989; Rogers and Mosers, 1992) formed on the braids between succesive rolls. In the laminar flow region, the spanwise rolls are initially straight and show increasing wavy undulation as they move downstream (see Animation 3). The iso-enstrophy surfaces of a horseshoe vortex and a distorted spanwise roll are shown in Figure 7. Note that the absence of the vorticity layer in the figure is due to the contour threshold for vorticity magnitude. A thin filament of the rib-like structure is adjacent to the horseshoe vortex. The horseshoe vortices often lift up and interact with the adjacent rollers, as observed in Animation 2 and 3 at frame T=0.0 sec. (Figure 8 and Figure 9). The streamwise vortices often occur in pairs (see Animation 2 at T=0.001197, 0.003, 0.0346, 0.03864 and 0.0649 sec, for example) which suggests they are either the legs of horseshoe vortices or longitudinal ribs connecting successive rolls.

    Figure 6

    Contours of the magnitude of vorticity vector field in a (x, y)-plane at T = 0.00756 sec.
    Locations marked "R" and "r" are the spanwise rollers and longitudinal ribs respectively.
    The key to the contour levels is given with Figure 8.

    Figure 7

    Iso-entrophy surfaces of a horseshoe vortex and a distorted spanwise roll (at a level of 15% of the peak vorticity vector magnitude).

    Figure 8

    Cross-stream slice of a horseshoe vortex pair in the (y, z)-plane
    at x / l = 66.6 and T = 0.0 sec (front view).
    Contour levels given as percentage of peak vorticity magnitude.

    Figure 9

    Cross-sectional slice of a horseshoe vortex pair and spanwise rollers in a (x, z)-plane
    at y / l = -1.42 and T = 0.0 sec (top view). For key to contour levels, see Figure 8.

    In Animation 1, the rib-like structures align at an angle of 140 to 160 degrees with the direction of the mean flow, being strongly stretched by the induced strain field of the successive spanwise rolls. The increasing elongation and meandering of these ribs with time can be clearly seen in Animation 1. Further downstream, the convection and entanglement of the streamwise vortices are followed by the formation of finer scale turbulence. From the side view of the jet, the vortex structures clearly display some similarity to the roll/rib arrangement found in wakes or mixing layers (Bernal and Roshko, 1986; Metcalfe et al., 1987); Lasheras and Choi, 1988). However their orientation and the sign of vorticity are opposite to their counterparts found in plane wakes, owing to the opposite direction of the mean streamwise velocity gradient in the jet.

    By following the evolution of individual roller cores, the convection velocity is estimated to be in the range 55 to 75% of the inflow velocity at the centreplane. The trajectories of the concentrated vortex cores indicate that the rollers do not always move away from the centreplane as they convect downstream. Rollers can move across the centreplane interacting with ribs and rollers from the opposite side of the jet. Well-developed rollers tend to be highly deformed and broken up to smaller structures (randomization). Pairing can be observed at time intervals between T=0.01848 to 0.02831 in Animation 1, involving co-rotation of pairs of adjacent rollers from the same side of the jet. Amalgamation of neighbouring rollers are also observed in the further downstream region. Rollers are known to be unstable to subharmonic disturbances leading to pairing (Pierrehumbert and Widnall, 1982). Such a phenomenon has been also observed in the plane jet (Comte et al., 1992), mixing layers (Moser and Rogers, 1993; Metcalfe et al., 1987), and axisymmetric jet (Hussain and Zaman, 1981; Mitchell et al., 1999).

    From the animated flow visualisation, the distinction between horseshoe vortices and highly distorted rollers is not obvious; the streamwise vortices (ribs) could either be interpreted as the legs of the horseshoe vortex pairs or highly contorted rolls. However, the streamwise vortices tend to be elongated and thinner as compared with the primary roller structures as shown in the animations. The simulation reproduces the jet flow under natural condition with random disturbance only. The observed vortex structures are therefore less periodic than vortex shedding in the wake under forced excitation. The roller/rib arrangement can be represented by an idealised topological model as shown in Figure 10. The rib structures tend to be located at the saddle point between adjacent rollers. This model is similar to the counterpart structures in other plane shear flows (e.g. Bernal and Roshko, 1986; Hayakawa and Hussain, 1989). 'Cross ribs' (Jimenez, 1983) are not found in the current study, probably due to the dominant asymmetric mode of the jet.

    Figure 10

    Topological arrangement of the spanwise rolls and ribs.

    The evolution of the vortical structures can be summarised as follows. Initially the two 2-D vortex sheets of opposite sign roll up into spanwise rollers. The three-dimensional instabilities of the rollers lead to the subsequent formation of horseshoe-like vortices and ribs. The ribs are then subjected to the strain field induced by the adjacent rollers. The ribs are rapidly stretched and elongated with increasing downstream distance. The inclination of the ribs remains nearly constant at this stage. The size of the roller cores increases rapidly in the initial roll-up phase but it does not scale with the corresponding growth of the jet width. The coherency of the structures disintegrates when the rolls and ribs start to interact in a complex manner. The evolution of the vortical structures involves stretching, pairing, merging and tearing, and the break-up stage is characterised by formation of smaller random structures.

    Although the flow organisation is dominated by the anti-symmetric mode, symmetric rollers are found occasionally in the instantaneous flow field. This suggests that the flow may contain a mixture of a strongly amplified anti-symmetric mode and a weaker symmetric mode. On the other hand, only a symmetric mode was found in the excited plane jet by Sakakibara et al. (2000). Sato (1960) found that the relative intensity of the two modes was highly dependent on the shape of the inlet velocity profile. Flat top-hat profiles led to the dominance of symmetric mode, while parabolic profiles resulting in the dominance of anti-symmetrical mode. The present result is consistent with the laboratory flow visualisation of a plane jet by Makita and Hayakawa (1992), who also observed that the unexcited natural jet has a mixture of these two modes and hence the coherent structures are far more complicated than those found in the excited jets.

    From the animations in the (x,z) and (y,z)-planes, we see that streamwise vortices occur mostly in counter-rotating pairs, although single vortices or a group of vortices do exist in the flow. For example, a group of two closely clustered vortex pairs or a quartet can be seen in Animations 2 and 3 at time T =0.05670 sec (Figure 11 and Figure 12). Linear stability analysis of the translative instability of Stuart vortices (Pierrehumbert and Widnall, 1982) shows that the most-amplified wavelength is about 2/3 of the streamwise spacing between rollers in a plane mixing layer. Bernal and Roshko (1986) obtained the value of 0.67 in the experimental study of a plane mixing layer. Here the distance ratio between successive rollers and streamwise vortices is approximately equal to 0.53, as compared to the value of 0.71 in a plane impinging jet (Sakakibara et al., 2000). This falls within the range of 0.5 to 0.8 observed in plane mixing layers. The wavelength ratio does not double when the rollers pair, as observed in the experimental study by Sakakibara et al. (2000).

    Figure 11

    Cross-sectional slice of a group of closely clustered vortex pairs in the (x, z)-plane
    at y / l = -1.42 and T = 0.05754 sec (top view). For key to contour levels, see Figure 8.

    Figure 12

    Cross-stream slice of a group of closely clustered vortex pairs in the (y, z)-plane
    at x / l = 66.6 and T = 0.05754 sec (front view). For key to contour levels, see Figure 8.


    Conclusion

    A simulation of a spatially-developing plane jet has been performed at low Reynolds number. The computer animations make possible the tracking of the evolving vortical structures, also helping to understand the complex three-dimensional topology. The simulation clearly captures features of the streamwise rib vortices and spanwise rolls under unforced conditions. In the laminar-transition regime, flapping modes of the jet are observed and the flow is dominated by spanwise rolls which are initially straight and two-dimensional. The flow consists of a mixture of a dominant anti-symmetric mode and weak symmetric modes. The formation of three-dimensional structures is characterised by the development of wavy undulation of the spanwise rolls. They are similar to the vortical structures seen in plane wakes and mixing layers, but the orientation is opposite owing to the opposite mean shear.

    The most interesting finding is the observation of horseshoe vortices in the laminar-turbulent transition region. The legs of the horseshoe vortex appear to be connected by a thin vorticity layer and mostly confined to either side of the jet centreplane. Two basic configurations of vortex pairs have been identified; namely tex2html_wrap_inline317-type and tex2html_wrap_inline319-type. Their mean inclination angles are about 135 degrees and 45 degrees respectively. It appears that the spanwise rolls and streamwise vortices interact with each other, leading to the emergence of complex vortical arrangements and subsequent transition to turbulence. Pairing of adjacent rollers of the same vorticity sign has been observed. The spacing of the ribs is approximately 0.53 of the spacing between successive rolls, which compares well with linear stability theory and other experimental results.

    Further investigations are desirable to repeat the simulation using dynamic subgrid models (Germano et al., 1991; Vreman et al., 1997) since other studies have shown the Smagorinsky model is too dissipative and can delay the onset of transition. Although this is less important in free shear flows than wall-bounded flows, it would be interesting to carry out a critical assessment of the influence of subgrid models on coherent structures. In addition, the spanwise width (and the number of grid points) of the periodic domain could be doubled in order to investigate its effect on the global instability of the spanwise perturbation. Nevertheless, the current large-eddy simulation captures important transitional flow structures such as rib vortices, spanwise rolls and horseshoe vortices in the unforced natural plane jet.


    Acknowledgements

    The authors wish to acknowledge the support of the Defence Evaluation and Research Agency (DERA) Farnborough and the U.K. Engineering Science and Research Council who co-funded this project. We also wish to thank Dr Ian Cowan for his help in making the video animations. We are grateful to Professor W.R.C. Phillips for reading the draft and helpful comments.


    References

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