Swirling Flows Characteristics in a Cylinder Under Effect of Buoyancy

Swirling flows have been investigated extensively aiming at providing further insight into the conditions that cause abrupt flow stagnation and associated breakdown. This latter complex feature may be evidenced, for instance, in the benchmark model flow driven by one end disk of an upright cylinder of given aspect ratio [1, 2]. Several prior studies provided means of controlling breakdown onset as it can be harmful / beneficial, depending upon the application considered. Hereafter, are presented selected investigations based on kinematic, geometric and thermal means of vortex flows control. Of note, Mullin et al. [3] demonstrated the effects of including a tapered center body and showed it to alter significantly breakdown onset and location. In particular, they argued that the inhibition / suppression of the vortex structure depended on the axial pressure gradient sign, induced by the conical axial rod, whose circular base is sealed to either the rotating or the stationary disk of the enclosure. Manungua et al. [4], through numerical simulations and dye visualizations, attributed the effective vortex control to the role of vorticity strength, modified upstream of the vortex by the differential rotation of a partial rotating disk flash mounted into the top lid. More recently, breakdown topology in an enclosure shrouded by a polygonal cross section sidewall was explored numerically and experimentally by Naumov et al. [5]. Despite the drastic change in the lateral geometry, the findings revealed that the core flow which displays breakdown remains axisymmetric and the 3D non axisymmetric behavior was remarked in the bulk flow adjacent to the sidewall. The topology and parameter dependence of breakdown was addressed by Jones et al. [6]. Their analysis led to the identification of the adequate parameters which allowed consistent comparison between bubble breakdown observed in a torsionally driven cylinder flow and the spiral type vortex occurring in flows through openended pipes. More recently, Shtern [7] highlighted the role of the swirl decay as a physical mechanism to control breakdown development. Thermal effects on confined vortex flows were addressed numerically by, for instance, Lugt et al. [8]. Their findings evidenced the sensitivity of breakdown to weak buoyancy caused by an unstable thermal stratification resulting from a slightly heated bottom disk. Under stable thermal stratification, significant effects on the swirling flow topology were reported by Omi et al. [9] who extended the investigation to account for co-/counter-rotating end disks. Compressibility effects were demonstrated by Herrada et al. [10, 11], in the case of a non-Boussinesq swirling flow driven by the bottom disk rotation of a cylinder with and without a central rod. A combination of buoyancy and rod differential rotation provided efficient means of vortex flows control: a forced axial temperature gradient was shown to enhance or eliminate the vortex features; depending on the sign of the temperature gradient. Flow stagnation and associated breakdown which occur in a swirling jet under effect of thermal buoyancy was addressed by Dina et al. [12] who performed PIV measurements and concluded that a temperature gradient between the jet core and its surrounding fluid causes the suppression/enhancement of the vortex pattern; depending on the sign of the gradient. Effects of a density variation on a confined isothermal swirling flow which exhibits on-axis vortex breakdown, driven by the top disk rotation of a cylinder, were recently addressed experimentally by Ismadi et al. [13] using axial injection of dye, denser or lighter than the ambient liquid. Flow visualizations indicated that heavy dye enhanced the formation of breakdown while very light dye injection (density variations of order 0.01%) caused the occurrence of offaxis ring type vortices and modified the threshold of breakdown onset. Concerns over the symmetry/asymmetry of onaxis breakdown in a cylinder were noted by several investigators: dye visualizations often provide images of vortices with asymmetric folds while 3D numerical counterparts, under well controlled conditions, predict axisymmetric patterns [6, 14, 15]. This discrepancy is often attributed to the inevitable imperfections of the set up and operating conditions such as a 0.01% misalignment of the cavity axis [14] or a slight tilt (≈2°) of one end disk [15]. Motivated by the above considerations, the current work explored numerically the sensitivity of a helical flow and associated vortex characteristics to a differential heating in a disk-cylinder system. Buoyancy was first implemented by a warm/cool axial fluid injection then by means of a differentially heated/cooled tiny rod.


Introduction
Swirling flows have been investigated extensively aiming at providing further insight into the conditions that cause abrupt flow stagnation and associated breakdown. This latter complex feature may be evidenced, for instance, in the benchmark model flow driven by one end disk of an upright cylinder of given aspect ratio [1,2]. Several prior studies provided means of controlling breakdown onset as it can be harmful / beneficial, depending upon the application considered. Hereafter, are presented selected investigations based on kinematic, geometric and thermal means of vortex flows control.
Of note, Mullin et al. [3] demonstrated the effects of including a tapered center body and showed it to alter significantly breakdown onset and location. In particular, they argued that the inhibition / suppression of the vortex structure depended on the axial pressure gradient sign, induced by the conical axial rod, whose circular base is sealed to either the rotating or the stationary disk of the enclosure. Manungua et al. [4], through numerical simulations and dye visualizations, attributed the effective vortex control to the role of vorticity strength, modified upstream of the vortex by the differential rotation of a partial rotating disk flash mounted into the top lid. More recently, breakdown topology in an enclosure shrouded by a polygonal cross section sidewall was explored numerically and experimentally by Naumov et al. [5]. Despite the drastic change in the lateral geometry, the findings revealed that the core flow which displays breakdown remains axisymmetric and the 3D non axisymmetric behavior was remarked in the bulk flow adjacent to the sidewall. The topology and parameter dependence of breakdown was addressed by Jones et al. [6]. Their analysis led to the identification of the adequate parameters which allowed consistent comparison between bubble breakdown observed in a torsionally driven cylinder flow and the spiral type vortex occurring in flows through openended pipes. More recently, Shtern [7] highlighted the role of the swirl decay as a physical mechanism to control breakdown development.
Thermal effects on confined vortex flows were addressed numerically by, for instance, Lugt et al. [8]. Their findings evidenced the sensitivity of breakdown to weak buoyancy caused by an unstable thermal stratification resulting from a slightly heated bottom disk. Under stable thermal stratification, significant effects on the swirling flow topology were reported by Omi et al. [9] who extended the investigation to account for co-/counter-rotating end disks. Compressibility effects were demonstrated by Herrada et al. [10,11], in the case of a non-Boussinesq swirling flow driven by the bottom disk rotation of a cylinder with and without a central rod. A combination of buoyancy and rod differential rotation provided efficient means of vortex flows control: a forced axial temperature gradient was shown to enhance or eliminate the vortex features; depending on the sign of the temperature gradient. Flow stagnation and associated breakdown which occur in a swirling jet under effect of thermal buoyancy was addressed by Dina et al. [12] who performed PIV measurements and concluded that a temperature gradient between the jet core and its surrounding fluid causes the suppression/enhancement of the vortex pattern; depending on the sign of the gradient.
Effects of a density variation on a confined isothermal swirling flow which exhibits on-axis vortex breakdown, driven by the top disk rotation of a cylinder, were recently addressed experimentally by Ismadi et al. [13] using axial injection of dye, denser or lighter than the ambient liquid. Flow visualizations indicated that heavy dye enhanced the formation of breakdown while very light dye injection (density variations of order 0.01%) caused the occurrence of offaxis ring type vortices and modified the threshold of breakdown onset.
Concerns over the symmetry/asymmetry of onaxis breakdown in a cylinder were noted by several investigators: dye visualizations often provide images of vortices with asymmetric folds while 3D numerical counterparts, under well controlled conditions, predict axisymmetric patterns [6,14,15]. This discrepancy is often attributed to the inevitable imperfections of the set up and operating conditions such as a 0.01% misalignment of the cavity axis [14] or a slight tilt (≈2°) of one end disk [15].
Motivated by the above considerations, the current work explored numerically the sensitivity of a helical flow and associated vortex characteristics to a differential heating in a disk-cylinder system. Buoyancy was first implemented by a warm/cool axial fluid injection then by means of a differentially heated/cooled tiny rod.

Governing parameters
We consider a vertical cylinder of height H and radius R, filled with an incompressible viscous Newtonian fluid, whose top disk rotates with a uniform angular velocity Ω as sketched in Fig.1 The flows under consideration are assumed to be laminar and axisymmetric which reduces the numerical domain to a meridian plane as illustrated in Fig. 1 which includes a typical on-axis bubble type breakdown and selected 202 vortex characteristics. The main parameters, required to govern the dynamics of the flows under consideration are: the rotational Reynolds, the cavity aspect ratio and the Richardson number which appropriately accounts for thermal buoyancy strength, defined respectively as: where: ΔT=Tj-Ta is the temperature gradient between the jet and the ambient fluid; υ and β denote the kinematic viscosity and thermal expansion coefficients respectively.

Initial and boundary conditions
The well documented isothermal flow driven by the top disk rotation, in the absence of injection/rod, which displays a steady bubble type breakdown for a given couple of parameters (Re, Λh), is defined as a basic state.

Method of solution
The set of time dependent axisymmetric Navier-Stokes equations together with the mass conservation and energy equation [14], subjected to the above boundary conditions, were integrated under the Boussinesq approximation using a finite volume approach, implemented in the CFD package ANSYS Fluent 15. The laminar solver is adopted and the volume of fluid method is used when injection is considered. This code was recently employed successfully to accurately model confined as well as jet like swirling flows, which display breakdown, analogous to the configuration under consideration [16][17][18].
We used a segregated solver in which a second order upwind scheme was employed to discretize the convective terms, a first order implicit scheme was adopted for time marching while central differencing is utilized to approximate the diffusion terms. Pressure-velocity coupling was achieved by using the SIMPLEC algorithm and double precision was considered in all calculations.
The Convergence criteria were based on the asymptotic level reached by the scaled residuals of each equation. Overall, these were found to lie between 10 -10 for the continuity and momentum equations and10 -15 for the swirl and energy equations.
The computational domain is the meridian plane  . An orthogonal grid, especially clustered in the viscous core region of the helical flow was utilized in order to adequately capture the requisite details of the vortex characteristics. The appropriate number of cells and associated time steps were determined on the basis of a grid sensitivity analysis. For the selected configuration with 1 98 h .   which is frequently referred to in this work, the grid sensitivity was evaluated by considering two sets of non-uniform grids, clustered along the cavity axis where breakdown is expected to occur. In particular, we used 45000 and 80000 cells along with a uniform grid of 200 400  elements in the (r, z) plane for comparison. Time marching was carried out using different time steps over the range 4 10 0 005 t.

  
. It was found that, a non-uniform grid of 80000 cells associated to a time increment 0 001 t.

 
were sufficient to predict the main vortex characteristics with a maximum error 1%  compared to the solution predicted with the smallest time step 4 10 t    . In addition, to speed up the convergence process and for computational efficiency, simulations were generally initialized with a solution from a lower Re.

Validation
The numerical accuracy was first assessed by simulating the helical flow in the absence of injection, investigated experimentally by Escudier [1] and Iwatsu et al. [19] and supported quantitatively by numerous numerical predictions [2,19].Validation was then extended to include the configuration under heavy dye injection [13].

Prior configurations without injection
It is well established that for the selected parameters (Re, Ri, Λh)=(1850, 0, 2), the bulk helical flow driven by the uniform rotation of one end disk of a vertical cylinder displays a primary large meridian circulation and two distinct on-axis bubble type vortices characterized by an upstream stagnation point located on the cavity axis at an axial distance Zs away from the non-rotating disk. The current calculations were found to reproduce qualitatively the flow topologies visualized in experiments [1,19] and very accurately the numerical predictions [2,19]. In table1 are reported three selected vortex flows characteristics (Fig.1); namely, the upstream (first) stagnation point location (Zs), the radial extent of the vortex pattern (emax) scaled with the cylinder radius, as well as the vortex strengths ψmin and ψmax (meridian volume flow rates respectively within and outside the vortex structure). It is remarked that the maximum error estimate is within 0.5 % on the stagnation point's location (sensitive criterion for onset /suppression of breakdown) and approximately 1 % for the above volume flow rates which is considered as negligibly small. Table 1 Values of the vortex characteristics: Zs, emax, ψmin and ψmax obtained in this work and in [1,19]

Prior experiments under heavy dye injection
Further comparison was carried out by considering the simulation of the above confined swirling flow subjected to heavy dye injection, applied at the bottom disk center, explored experimentally by Ismadi et al. [13]. To the best of our knowledge, there has been no prior numerical study of this model helical flow.
To allow direct comparison, we adopt the experimental conditions: Λh=1.98, Uj=1.7×10 -3 m/s while varying the density ratio Δρ/ρa = (ρj -ρa) /ρa (ρa and ρj are densities of the jet and ambient fluid respectively) over the range 0 0 02 The validation focused on the behavior of axisymmetric and non-oscillatory flow structures which limits the comparison to 2600 Re  [20]. Fig.2 illustrates the effect of injecting a fluid slightly heavier than the ambient one, i.e. ρa > ρj. Good agreement is observed between the present numerical predictions and experiments [12] as both reveal breakdown enhancement (decrease of Re) with increasing ρj almost the same rate as depicted by the trend of the bounding curve which indicates first breakdown onset.

Isothermal vortex flows under injection
In this section is described briefly the incompressible swirling flow driven by the lid uniform rotation of a cylindrical enclosure with and without neutrally buoyant injection. As the study focused on axisymmetric structures which asymptote steady states, the main parameters were varied over the ranges 0 8 2 5 h .
. , We recall that outside the prescribed range of Re, the flow becomes time dependent (bubble oscillations evidenced by 3D simulations along the cavity axis [20]).

Basic flow driven by the top disk rotation
The model swirling flows under consideration, may sufficiently be described by two main parameters; namely, the rotational Reynolds number Re and the cavity aspect ratio Λh. Qualitatively, for given low rates of top disk rotation, a clockwise large meridian circulation takes place which superimposes to the primary rotational motion. In particular, fluid spirals outward (inward) in the vicinity of the top rotating disk (still bottom disk) with the formation of a thin boundary layer on each disk. Along the cylindrical sidewall, flow is directed downward while the bulk flow is driven axially upward by the Ekman suction effect. Depending on the aspect ratio Λh (Λh>1.2), as the disk rotation rate increases the flow stagnates at the cavity axis and an adverse pressure gradient takes place; giving rise to one, two or three reverse flows; commonly referred to as bubble type vortex breakdown. In the current work, the onset and location of a single bubble is evidenced numerically and viewed qualitatively (Fig.3)  circulation. The direction of the meridian motion within and outside the bubble is independent on that of the top disk rotation.

Effect of neutrally buoyant injection
Prior to investigating thermal effects, it appeared necessary to first consider the case under isothermal injection (neutrally buoyant injection Ri=0) in order to assess the sensitivity of the vortex pattern to the inlet geometry and associated injection conditions. Fig.3 shows a close up of the meridian vortex pattern, before and after neutral injection applied to the above configuration(Λh, Re)= (1.98,1700); assuming experimental conditions [13,15]: uniform injection velocity Uj=1.7×10 -3 m/s at the orifice (radius ratio

Effect of thermally buoyant injection
In this section we focus on the case when the vortex flows are subjected to buoyancy, induced by uniform injection at temperature Tj slightly different from the ambient fluid temperature Ta. The inlet conditions are the same as those implemented in the case of a neutrally buoyant injection (Ri =0) described above. Buoyancy strength is controlled by varying the Richardson number Ri over a range corresponding to small temperature gradients.

Warm injection
When the jet temperature is slightly higher than that of the ambient fluid (Ri > 0), calculations revealed significant changes of the flow pattern due to buoyancy. In particular, as depicted in Fig.4b for the selected parameters (Re, Λh, Ri)=(1700, 1.98, 0.023) including the neutrally buoyant case Ri =0 (Fig.4a), the on-axis stagnation points are suppressed while the bubble evolved into a ring type vortex, which remained robust, characterized by a saddle point at its periphery alongside with a stagnation point located at its center. These findings are qualitatively very consistent with the experimental results performed under light dye injection [13]. However, by varying Ri over the range 0 0 092 Ri .


, for fixed aspect ratio and Re detailed time wise evolution of the flow field revealed the existence of a limited sub-region, involving relatively lower Re values (state diagrams in the next section) where the ring type vortex itself continued to evolve by shrinking until it utterly disappeared; probably as a result of viscous damping. This trend is readily apparent in Fig.4c for (Re, Λh, Ri)=(1700, 1.98, 0.046), showing a bulged region (swelling streamlines) which lacks breakdown. This numerically predicted flow evolution was found to occur for various couples of parameters (Re, Λh, Ri) as discussed in the next section. However, the experimental counterpart [13], performed for a single cavity aspect ratio, reported the bubble transition toward the ring type vortex but did not invoke the particular cases of its suppression; although one would expect, approximately, very similar trends as regards to the very small density variations involved in both studies. Analogous discrepancies between dye experiments on confined vortex breakdown and numerical predictions were reported by Herrada et al. [10,11]. We conjecture that such a discrepancy may mainly be attributed to the inevitable difference between the welldefined boundary conditions implemented in the frame work of axisymmetric simulations and the effective set up and operating conditions in experiments [14,15]. In addition, as a result of local dye diffusion, experimental images [9,13,15,19] show frequently bulged regions of concentrated dye which do not accurately capture the required detailed vortex characteristics (saddle points and associated rings are hardly visible).   and two time stations: the higher time approaches steady state and the lower one was selected as a reference time. We recall that for Re<1400, the basic flow lacks breakdown.
The mapping identifies three distinct flow regimes with bounding curves, brought about by thermal effects. In particular, a region of bubble type (which shrinks with increasing time), another of robust toroidal eddy structure (which expands with time) and finally one which lacks breakdown, involving relatively lower Re values and therefore more viscous damping. For fixed Re over the range 1400 1750 Re  , it is remarked that the bubble type vortex disappears under warm injection and the corresponding bounding curve is almost linear over the sub-range 0<Ri<0.018. Outside this latter interval, i.e. 0.018<Ri<0.092. variations are much less pronounced. We recall that the above numerical predictions showed that the bubble detached from the axis prior to its suppression. However, the remaining range 1700<Re<2600 reveals that bubble bifurcation into a steady eddy structure, which remains observable, with increasing Ri. The bounding curve, in this case, is almost linear over the approximate sub-range 1400<Re<2000. Outside this latter range a decrease of Ri is remarked with increasing Re; indicating higher sensitivity of the bubble to the buoyant injection.

Cool injection
The model flow under cool injection   0 Ri  , is described in this section. Unlike the previous case, buoyancy is shown here to promote flow stagnation conditions as well as vortex strength and size. This is evidenced in Fig. 6, for the selected couple of parameters (Re, Λh, Ri)=(1700, 1.98, Ri<0), which shows qualitatively, a more axially elongated and radially extended flow reversal than that resulting from the neutrally buoyant configuration. Besides, no ring type pattern was observed in this case.

Ri=0
Ri= -0.023 To further highlight the distinct and combined roles of the geometric and dynamic parameters involved, it appeared instructive and convenient to discuss the new (Re, Λh) and (Re, Ri) diagrams.
The effect of the cavity aspect ratio is illustrated in (Fig. 7) [2]. Under a weak cool injection, the breakdown zone is shown to expand and cover wider ranges of (Re, Λh). In particular, the lower bounding curves indicate a significant reduction of the threshold Re (up to 50% reduction) for breakdown onset: much lower top disk rotation rates are sufficient to trigger flow stagnation. Besides, it is well established that configurations without injection display breakdown only if 13 h .   [1,2,19]. In contrast, this limit is shown to reduce considerably when the flow is subjected to cool injection, as depicted in Fig.8   Next, is discussed the sensitivity of breakdown to the combined variation of Re and Ri, for different aspect ratios, focusing on the impact of weak thermal buoyancy. Fig. 8  Breakdown in the disk-cylinder with spinning lid may be associated to the competition between two main local characteristics of the helical jet flow which develops within the viscous core region along the cavity axis [6,21] namely the swirl strength and the meridian flow intensity characterized respectively by the local rotational and axial Accordingly, a consistent local parameter to analyze breakdown development may therefore be constructed as the ratio N r a S Re / Re  , which defines the local swirl number [6]. As a test case, the variation of SN is evaluated approximately in order to predict the sensitivity of the flow to warm injection for the selected configuration Re = 2060, H/R = 2.5 which displays two on-axis breakdown when Ri = 0 [1,17,19]. for breakdown onset. Besides, it can be remarked that the radial velocity magnitude is relatively small and its contribution to the local swirl number can be neglected.

Differentially heated /cooled small rod
In an attempt to confirm buoyancy influence on the stagnation flow conditions, the study was extended to consider a different set up which generates buoyancy by means of a differentially heated /cooled tiny rod sealed to the bottom disc center of the cylinder. Such a configuration and associated conditions are more amenable to numerical modeling and relatively easier to implement in practice.
This configuration introduces a modified Richard-  [13]). In order to obviate the occurrence of undesired additional eddies occurring at the rod tip which may alter breakdown onset conditions, we set the rod aspect ratio to 0 25 hr R h .

 
. Under such geometrical characteristics the presence of the rod did not alter significantly the vortex flows characteristics prior to differential heating and so may be regarded as a non-intrusive means of flow control. Qualitatively, the numerical predictions revealed that small temperature gradients corresponding to the range 0 116 0 116 . Ri , applied between the rod and the ambient fluid, were sufficient to alter significantly the vortex patterns. In particular, when the rod temperature is set slightly higher than that of the ambient fluid (Ri >0) a buoyancy force is generated, inducing an upward convection of swirling co-flow, of week intensity but sufficient to enhance the axial flux and angular momentum transfer, which act to prevent flow stagnation and associated vortex formation. This is depicted qualitatively, for , by the non-uniformly spaced meridian streamlines (Fig. 10, b); the case 0 Ri  (Fig.10, a) corresponds to the isothermal configuration. In contrast, when the rod is differentially cooled, an enhancement of the bubble size and occurrence of a second reverse flow structure were obtained (Fig. 10, c). In addition, to allow comparison of the threshold buoyancy parameter Ri required for breakdown suppression, in both rod and injection models, the axial velocity distribution at the cavity axis is supplied (Fig. 11), before and after thermal control, when R = 1700, Λh = 1.98 . The figure confirms that prior to differential heating both means induce local effects without altering significantly the flow stagnation location. However, for a fixed ambient fluid temperature, to suppress breakdown by means of the rod differential heating, a temperature gradient

Conclusion
Thermal buoyancy, implemented by means of warm/cool uniform injection or by a differentially heated tiny rod, was shown numerically to constitute an effective means of controlling axial flow stagnation and associated breakdown in a disk-cylinder system with a rotating lid. In particular, warm (cool) injection or differentially heated (cooled) rod, applied upstream the vortex, were found to both prevent (favor) breakdown. In addition, under warm injection, buoyancy was found to suppress on-axis stagnation points while the associated bubble bifurcates into an off-axis vortex ring which may remain robust and observable or may continue to evolve until it disappears; depending on buoyancy strength and viscous damping effects. These findings were established for various   0 h ,Re,Ri   parameters. Besides, the differentially heated/cooled tiny rod configuration, considered as a non-intrusive means based on thermal stratification was found to alter significantly the conditions of breakdown onset but, unlike the injection model, did not exhibit any ring type vortex. The current findings, which may be useful for bioreactors, constitute a platform for further investigations which, in perspective, explore oscillatory regimes.