Bubble and conical forms of vortex breakdown in swirling jets

Abstract

The present study focusses on vortex breakdown (VB), which occurs in axially convected swirling flows and is characterized by the development of an internal stagnation point and regions of reverse flow. VB has been observed or utilized in a variety of situations including flow over delta wings, tornadoes, re whirls, turbomachinery, fuel injectors and combustors. Flows where VB is observed include swirling round jets with non-swirling and negligible co flow. The experimental study of Billant et al., (1998) has shown VB characteristics unique to this family of flows. In addition to the commonly observed bubble form of breakdown (BVB) the experiments revealed a new form, referred to as conical form of breakdown (CVB). Features of these forms (especially the latter) remain mostly unexplored. This serves as motivation for the present numerical investigation of VB in swirling jets. In this study, a survey of different VB states (laminar and turbulent) observed with varying flow parameters has been carried out using numerical simulations, along with investigations on hysteresis effects and bistability phenomenon. Previous theoretical models for VB have been assessed using these results. Helical instabilities that were observed to arise in the simulations have been examined using the tools of stability analysis. Three-dimensional numerical simulations were carried out using the open-source incompressible

flow solver incompact3d in a Cartesian coordinate framework to study a swirling jet entering an adequately large domain to prevent con nement effects. The swirling jet was modelled using the axisymmetric and steady `Maxworthy' in flow pro le (Ruith et al., 2004), with the main control parameter as the swirl number, S, representing the relative rate of rotation in comparison to the centreline axial velocity of the jet. The Reynolds number (Re) was chosen based on the jet radius and centreline axial velocity.

The long-time flow states achieved with varying S and initial conditions are reported for Re = 200 and 1000. For the latter case, where the flow transitioned to turbulence, large eddy simulations were carried out using the explicit fi ltering approach. Both local and global stability analyses have been performed to examine instabilities. Selective frequency damping and axisymmetric simulations using ANSYS Fluent have been used to compute base flows. For Re = 200, BVB and CVB were observed and could be clearly distinguished by the distinct spatial structure of the recirculation zone. These different VB forms could be further classi ed based on unique characteristics into different types. For BVB, the following types were identi ed{ steady one-celled BVB, one-celled BVB with spiral tail, pulsating BVB, two-celled BVB with spiral tail and asymmetric BVB. The categorization as one-celled and two-celled was based on the number of toroidal structures that could be identi fied within the bubble, while the pulsating BVB was observed to be an intermediate state between the two. The term `spiral tail' is used to denote the presence of a helical mode that developed in the wake of the bubble which had no signifi cant effect on the axisymmetric upstream portions of the bubble. In contrast, for the asymmetric BVB, a helical mode was present, which caused asymmetric motion of the entire bubble. Two types of CVB were identifi ed{ regular and wide-open{ the latter with an approximately radial expansion of the flow downstream to the stagnation point. Comparisons with different experimental results showed strong similarities in features for most flow states. Hysteresis studies established the coexistence of different types of BVB with the regular CVB in overlapping ranges of swirl numbers, confo rming that these are bistable forms. Remarkable differences in length scales involved for the two forms could be observed when comparing time-averaged flow structure. A hysteresis plot is provided based on the maximum radius achieved by a streamline starting from the in flow plane at an arbitrary radius. It was additionally seen that the two-celled BVB with spiral tail and asymmetric BVB coexist (along with regular CVB) over a small range of swirls. For Re = 1000, a transition to turbulence was observed, leading to some interesting differences in the flow states observed. At low swirls, the stagnation point occurred only intermittently in time, in contrast to the steady VB observed for equivalent S for Re = 200. For swirls above this, a two-celled BVB with a turbulent wake was observed for a large swirl range. The stagnation point at the bubble's nose was lost at swirls just above those for which BVB was observed. For this range, a spiral coherent structure was seen to arise intermittently in the bubble's wake, accompanied by streamwise oscillatory motions in the flow. At higher swirls, this motion was subdued and the spiral downstream of the bubble was lost, while the stagnation point at the bubble's nose eventually reappeared. The streamwise oscillations of the bubble being a common feature to all long-time flow states observed, these states are collectively referred to as oscillating BVB. For even higher swirls, turbulent regular and wide-open types of CVB were observed. Bistability of oscillating BVB and regular CVB, and additionally, between regular and wide-open CVB were established using hysteresis studies. The stark differences in length scales associated with the bistable bubble and conical forms were reduced due to turbulence, indicating that CVB might be misidenti ed as BVB at high Re. Indeed, it is speculated that some of the VB states reported in experimental studies of Liang and Maxworthy, (2005) are likely the CVB. The bistable regular and wide-open types of CVB were found to have considerable differences in the length scales of respective recirculation zones. The two VB states for which helical modes were observed, the BVB with spiral tail and asymmetric BVB, were examined using stability analysis. A closely related flow state to BVB with spiral tail is the spiral vortex breakdown (SVB). In previous studies, SVB was identi ed to arise due to the instability of a nonlinear steep global mode. Assuming weakly non-parallel flow, a local spatio-temporal analysis coupled with a WKBJ framework was used to show that the global mode associated with BVB with spiral tail differed from that of the SVB, with the linear frequency selection criterion (Chomaz et al., 1991) better predicting the global frequency. Using tools of global stability analysis, the asymmetric type of BVB was shown to arise due to a different unstable mode that has strong energy content in the bubble region. It was observed in the simulations that the base state was strongly modifi ed by the instability. The stability analysis using the mean flow made

better predictions as compared to that based on the base flow for this type of BVB. Two theories developed by Benjamin, (1962) and Brown and Lopez, (1990) towards explaining VB were assessed. It was observed that the prediction from the former theory, that the flow becomes subcritical downstream of VB, was confi rmed based on simulation results. However, other aspects of the theory, which have not been scrutinized in previous studies displayed a trend opposite to that observed in the present simulations as well as other available experimental results. That is, the conjugate states predicted by the theory, modelling the flow downstream of VB, were found to better resemble the primary state upstream of VB with increasing swirl. This contradicts the observation that the bubble increases in size with swirl. Reasons for why the predictions display this trend, based on analogies to gasdynamic shockwaves, are provided. The theory of Brown and Lopez, (1990) was found to give qualitatively similar predictions for BVB, but deviated strongly from the numerical results for CVB. A major highlight of this study is the signifi cance of the initial conditions in determining the VB form or type achieved, with results showing the coexistence of three distinct pairs of long-time flow states in overlapping parametric ranges for swirling jets. Though emphasized by Billant et al., (1998), many later studies on swirling jets have generally neglected this aspect. Similarly, as this study shows, the CVB is a distinct form of VB that has not been clearly identifi ed in previous studies on swirling jets. These results might aid in better understanding features of the elementary flow confi guration of swirling jets and allow for more informed developments of design and control strategies in practical applications.