UFR 1-07 Description

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Description

Introduction

Free vertical buoyant plumes and free-jets are related phenomena, both having a core region of higher momentum flow surrounded by shear layers bounding regions of quiescent fluid. However, whereas for jets the driving force for the fluid motion is a pressure drop through an orifice, for plumes the driving force is buoyancy due to gradients in fluid density. Plumes can develop due to density gradients caused by temperature differences, for example in fires, or can be generated by fluids of different density mixing, such as hydrogen releases in air. There are many flows of both engineering and environmental importance that feature buoyant plumes, ranging from flows in cooling towers and heat exchangers to large geothermal events such as volcanic eruptions. There has been considerable attention paid to the mean flow behaviour of plumes in the far field, e.g. Chen & Rodi  [5] or List [6] [7], which are examined in a companion UFR. However, there has been less study of the near-field unsteady dynamics of plumes.

In the present work, only non-reacting plumes are considered. This choice has been made in order to avoid the additional complexities associated with combustion, soot production and radiation in fire plumes. For helium plumes, the difference in density between helium and air is a factor of seven which is similar to that in fire plumes [8]. The principal difference between fire and helium plumes arises from the fact that heat is released locally from the flame in fire plumes whereas in helium plumes the buoyancy is produced only near the source where there are large concentration gradients.

The near-field of buoyant plumes features two key instabilities. The first is the Rayleigh-Taylor instability related to the presence of dense fluid above less-dense fluid. The two layers of different-density fluid are in equilibrium if they remain completely plane-parallel but the slightest disturbance causes the heavier fluid to move downwards under gravity through the lighter fluid. At the interface between the two fluids, irregularities are magnified to form fingers or spikes of dense fluid separated by bubbles of lighter fluid. The size of these irregularities grows exponentially with time and the smaller the density difference, the larger the wavelength of the instability. There has been considerable research into the dynamics of Rayleigh-Taylor instability (e.g. [9][10] [11][12]) as a consequence of its importance in nuclear weapons, atmospheric flows and astrophysics. Figure 2 shows the classic spike and bubble flow structures characteristic of R-T instability produced by two fluids of different density mixing, taken from Cook et al. [13].

Figure 2   Rayleigh-Taylor instability, from Cook et al. [13]. The heavy fluid is in black

The second instability in buoyant plumes is the Kelvin-Helmholtz instability related to the shear-layer interface between the rising plume and the ambient fluid. This forms axisymmetric roll-up vortex sheets on the boundary between the two layers of fluid travelling at different velocities, and is a feature in practically all turbulent shear flows including jets and wakes.

There is some uncertainty over the relative significance of the R-T and K-H instabilities in buoyant plumes. Buckmaster & Peters [14], Ghoniem et al. [15], Coats [16], and Albers & Agrawal [17] have suggested that the K-H instability plays the dominant role in plumes whilst others, including DesJardin et al. [1] , Tieszen et al. [2] and Cetegen & Kasper [18], suggest that the R-T instability is more important. For more details of the instability mechanisms and the transition to turbulence in buoyant flows, see also Gebhart et al. [19].

The Puffing Cycle

Medium to large scale plumes are characterised by the repetitive shedding of coherent vortical structures at a well-defined frequency, a phenomenon known as “puffing”. DesJardin et al. [1] present a detailed analysis of the plume puffing cycle, which they decompose into a number of stages. In the first stage, the less-dense plume fluid is rising close to the plume axis. Near the base of the plume, there is a layer of dense air overlying the less-dense plume fluid. There are two instabilities near the edge of the plume: one related to the misalignment of the vertical pressure-gradient and radial density gradient (the baroclinic torque) and another due to the misalignment of the vertical gravity and the radial density gradient (the gravitational torque). These produce a rotational moment on the fluid, increasing its vorticity and pulling air into the plume. The fluid motion coalesces to produce a large toroidal vortex which is self-propagated vertically upwards. As the vortex shifts vertically, fluid is pumped through to the core of the plume resulting in higher velocities on the plume axis. Radial velocities are induced near the base of the plume and air is drawn in producing an unstable stratification of denser fluid above less-dense fluid, ready for the cycle to begin again.

Using Direct Numerical Simulation (DNS), Jiang & Luo [20] [21] found that the gravitational torque is responsible for much of the initial production of vorticity in plumes. The term is highest towards the edge of the plume where the density gradient vector is pointing radially outwards at right-angles to the gravitational vector. The baroclinic torque was found to dominate the vorticity transport once the puffing structure has been established.

The toroidal vortex structure produced in small puffing plumes of helium in air, with a source diameter of under 10 cm, is relatively coherent. As the size of the plume is increased, the strength of secondary azimuthal instabilities increase which destabilize the toroidal vortex, producing finger-like instabilities. These are shown clearly near the base of the plume in the LES of DesJardin et al. [1] (see Figure 3). The secondary instabilities generate streamwise vorticity that enhances the mixing process. DesJardin et al. suggest that capturing these instabilities may be important in numerical simulations of pool fires where combustion is predominantly mixing-controlled.

Figure 3  An instantaneous snapshot of the puff cycle from DesJardin et al. [1] showing the finger-like azimuthal instabilities near the base of the plume. The isocontour of streamwise vorticity is shown at ±10% of the peak value.

Characteristic Dimensionless Parameters

There are a number of dimensionless parameters which are used to characterise buoyant plumes. For plumes produced by a release of buoyant gas, the inlet Reynolds number, Re, is given by:

 ${\displaystyle Re={\frac {\rho _{0}V_{0}D}{\mu }}}$ ${\displaystyle \left(1\right)}$

where ${\displaystyle \rho _{0}}$ is the plume fluid density, ${\displaystyle V_{0}}$ is the inlet velocity, D is the characteristic inlet length scale or inlet diameter and ${\displaystyle \mu }$ is the dynamic viscosity. The Reynolds number represents the ratio of inertial forces to viscous forces. At high Reynolds numbers, the destabilizing inertial forces dominate the viscous forces and the flow is turbulent. For isothermal pipe flows, this occurs for Re > 3000. Between 2000 < Re < 3000 the flow is transitional, for Re < 2000 the flow is usually laminar.

A useful parameter for describing buoyant flows is the densimetric Froude number, Fr, which represents the ratio of inertial forces to buoyancy forces. It is defined here as:

 ${\displaystyle Fr={\frac {V_{0}}{\sqrt {gD(\rho _{\infty }-\rho _{0})/\rho _{\infty }}}}}$ ${\displaystyle \left(2\right)}$

where g is the gravitational acceleration and ${\displaystyle \rho _{\infty }}$ is the ambient fluid density. The densimetric Froude number varies from near zero for pure plumes to infinity for pure jets. Some texts choose to define Fr using the square of the definition given above (e.g. Chen & Rodi [5]).

The Richardson number, Ri, is simply the inverse of the square of the Froude number:

 ${\displaystyle Ri={\frac {(\rho _{\infty }-\rho _{0})gD}{(\rho _{\infty }V_{0}^{2})}}}$ ${\displaystyle \left(3\right)}$

In some texts, the density difference in the Froude and Richardson numbers is made dimensionless using the plume source density, ${\displaystyle \rho _{0}}$, instead of the ambient density, ${\displaystyle \rho _{\infty }}$.

Subbarao & Cantwell [22] note that the Richardson number can be interpreted as the ratio of two timescales: the time for a fluid element to move one jet diameter due to inertia, ${\displaystyle \tau _{1}=D/V_{0}}$ , and the time for a fluid element to move the same distance under the action of buoyancy, ${\displaystyle \tau _{2}=\left[\rho _{\infty }D/g\left(\rho _{\infty }-\rho _{0}\right)\right]^{1/2}}$ , where:

 ${\displaystyle Ri=\left({\frac {\tau _{1}}{\tau _{2}}}\right)^{2}}$ ${\displaystyle \left(4\right)}$

In addition to Reynolds{}-number effects, the transition from laminar to turbulent flow is affected by the strength of buoyancy. In a buoyant plume that is initially laminar but transitions to turbulent flow at some distance further downstream, the point at which transition occurs moves closer to the source as either the Reynolds number or the Richardson number is increased [22].

Frequency of Pulsatile Plume Motion

The dimensionless Strouhal number, St, is used to describe the oscillation frequency of unsteady plumes. It is defined as follows:

 ${\displaystyle St={\frac {fD}{V_{0}}}}$ ${\displaystyle \left(5\right)}$

where ${\displaystyle f}$ is the frequency of the oscillation.

A number of empirical correlations for the puffing frequency of plumes have been developed based on the Richardson number. Cetegen & Kaspar [18] found that for axisymmetric helium-air plumes with ${\displaystyle Ri<100}$, the Strouhal number was related to the Richardson number by:

 ${\displaystyle St=0.8\left.Ri\right.^{0.38}}$ ${\displaystyle \left(6\right)}$

The graph of St versus Ri taken from their paper showing this relationship is reproduced in Figure 4. Between ${\displaystyle 100 there is a transitional region as the plume becomes more turbulent and mixing is enhanced. For ${\displaystyle Ri>500}$ the Strouhal number was found to scale according to:

 ${\displaystyle St\propto \left.Ri\right.^{0.28}}$ ${\displaystyle \left(7\right)}$

Figure 4   Correlation of puffing frequency in terms of Strouhal number and modified Richardson number, from Cetegan & Kaspar [18].

For planar helium plumes (produced by rectangular nozzles) with Richardson number in the range ${\displaystyle 1, Cetegen et al. [23] found that the Strouhal number varied according to:

 ${\displaystyle St=0.55\left.Ri\right.^{0.45}}$ ${\displaystyle \left(8\right)}$

A similar relationship for planar plumes was obtained in the more recent DNS of planar plumes by Soteriou et al. [24], who obtained the correlation:

 ${\displaystyle St=0.536\left.Ri\right.^{0.457}}$ ${\displaystyle \left(9\right)}$

The difference between the puffing frequency in planar and axisymmetric plumes has been attributed to the difference in mixing rates and the strength of the buoyancy flux in the two cases. If the planar and axisymmetric Strouhal number correlations given by Equations (6) and (8) are extrapolated to higher Richardson numbers, they suggest that planar plumes exhibit higher frequency pulsations for ${\displaystyle Ri>211}$ (where the two correlations cross over).

Figure 5   Puffing frequency of pool fires as a function of the burner diameter D, from Cetegan & Ahmed [25].

For axisymmetric fire plumes, Cetegan & Ahmed [25] found the following relationship between the puffing frequency, ${\displaystyle f}$, and the diameter of the burner or source, ${\displaystyle D}$:

 ${\displaystyle f=1.5\left.D\right.^{-1/2}}$ ${\displaystyle \left(10\right)}$

Their correlation is compared to the experimental data in Figure 5. It is remarkably consistent, considering that the fire plumes used in their study involved solid, liquid and gas fuel sources. The dependence of the puffing frequency on the source diameter is slightly stronger in helium plumes, where ${\displaystyle f\propto D^{-0.62}}$ [18]. For planar helium plumes, Soteriou et al. [24] showed that the frequency varied according to ${\displaystyle f\approx 0.5{\sqrt {g/D}}}$.

Observations from plume experiments  [18][22][26] and CFD simulations [24] have shown that the pulsation frequency in plumes does not strongly depend on the Reynolds number. The relative unimportance of the Reynolds number suggests that the instability mechanism controlling the pulsatile behaviour is essentially inviscid [24]. Once the conditions are met for the plume to become oscillatory, viscosity no longer appears to play a significant role in the puffing frequency. The helium plume experiments and simulations reported by Soteriou et al. [24] showed that the puffing frequency is unaffected by having the nozzle orifice flush to a solid surface or having the pipe from which the buoyant fluid escapes mounted free from the surrounding walls.

Onset of Pulsatile Flow Behaviour

The onset of unsteady flow behaviour in plumes is controlled by the balance of inertial, viscous and buoyancy forces. When viscous forces dominate, the plume remains steady.

Cetegen et al. [23] and Soteriou et al. [24] investigated in depth the transition from steady to unsteady flow behaviour in planar non-reacting plumes using both experiments and direct numerical simulation. Figure 6a shows some of their results, where plumes are characterised as either stable or unstable. The graph axes are the source Reynolds number and the inverse density ratio, ${\displaystyle 1/S=\rho _{0}/\rho _{\infty }}$. Clearly, as either the Reynolds number is increased or the inverse density ratio decreases, the plume becomes less stable.

Experiments with both axisymmetric and planar plumes have found that pulsations are not produced when the density ratio exceeds ${\displaystyle \rho _{0}/\rho _{\infty }\approx 0.6}$ [18][23][27][28]. Simulations by Soteriou et al.[24] showed that pulsations could in fact be produced at density ratios closer to one, but that the Froude and Reynolds numbers at which these pulsations were obtained would not be easily achieved experimentally.

Using their simulations, Soteriou et al. [24] were able to examine separately the effects of the Reynolds number, the density ratio and the Froude number on the onset of transition. They obtained a transition relationship between Reynolds and Richardson numbers of ${\displaystyle Re=183\left.Ri\right.^{-0.627}}$ (see Figure 7). The plume was unsteady for Reynolds or Richardson numbers above the line shown in the graph (i.e. for ${\displaystyle Re>183\left.Ri\right.^{-0.627}}$ or ${\displaystyle {\mathit {Ri}}>\left(Re/183\right)^{0.627}}$ ).

Cetegen et al. [23] showed experimentally that when the nozzle orifice is mounted flush to a wall, the transition from a stable to an oscillatory plume occurs at a lower threshold velocity. The presence of a flat plate surrounding the nozzle prevents any coflow which results in higher induced cross-stream velocities. These cause the plume immediately downstream of the nozzle to contract more and produce a thinner column of buoyant fluid that is more susceptible to perturbations.

In terms of the onset of unsteady flow behaviour, axisymmetric plumes are significantly more stable than planar plumes. This is shown clearly in the results of Cetegen et al.[23] (Figure 6b), where the conditions for stability of axisymmetric plumes are shown in addition to the planar plume behaviour with and without a flat plate.

Figure 6   Stability of buoyant plumes for different density ratios and Reynolds numbers:
(a) experimental and DNS results for planar plumes from Soteriou et al. [24] (top);
(b) experimental results for planar and axisymmetric plumes with and without a flat plate around the nozzle from Cetegan et al. [23] (bottom).

Figure 7   Transition from laminar to turbulent flow in planar plumes as a function of the Reynolds and Richardson numbers, from the DNS of Soteriou et al. [24]. Symbols indicate the results from simulations and the solid line is a fit to the data.

Review of UFR studies and choice of test case

Experiments

Most of the experimental data available on the near-field unsteady behaviour of non-reacting buoyant plumes has originated from the following American groups:

• Cetegen et al.(University of Connecticut) [18][23][24][28][29][30]
• Mell et al. (National Institute for Standards & Technology, NIST) [31][32]
• Subbarao & Cantwell (Stanford University) [22]
• Agrawal et al. (University of Oklahoma/NASA) [33][34]
• Gebhart et al. (Cornell University) [19][35][36]
• O‘Hern et al. (Sandia National Laboratories) [4][37]

Cetegen et al.’s group examined both reacting and non-reacting plumes over a period of nearly a decade. Over that time, a number of significant works were published on axisymmetric helium plumes [18][30], planar helium plumes [23] and the effect of acoustic forcing on helium plumes [28][29]. A website with animations of various plumes is also online[1] Empirical correlations were produced for the puffing frequency of planar and axisymmetric plumes and the causes of transition from steady to oscillatory plume behaviour were investigated (see discussion above). Their work on forced plumes involved using a loudspeaker to impart streamwise velocity fluctuations to the plume fluid. They found that plumes responded readily to the forcing and produced toroidal vortices at the forcing frequency. Interestingly, as the forcing approached the natural frequency of the flow, the large-scale vortices became more unstable and chaotic. This contrasts to other flows, such as jets and mixing layers, where forcing at the natural frequency leads to more spatial and temporal coherence.

In 1994, a series of helium plume experiments were undertaken by Johnson at NIST. Pure helium was released vertically through a 7.29 cm diameter pipe into ambient, quiescent air. The exit velocity was varied to examine different conditions and simultaneous velocity and concentration measurements were made. The data from these experiments has never been published fully in a conference or journal paper, but it has been used in two published computational studies by Mell et al. (also at NIST) [31][32]. The full data is also now available online on Mell's website[2] together with the results from simulations and other data for reacting plumes.

Yep et al.[33] and Pasumarthi & Agrawal [34] performed helium plume experiments in reduced gravity, using a drop tower facility at NASA. They showed that a naturally steady helium plume was up to 70% wider in microgravity than in normal earth gravity [33]. A plume at higher Re and Ri that exhibited puffing behaviour in earth gravity was found to produce steady flow behaviour in microgravity. This was taken as providing direct physical evidence that the oscillatory behaviour of low-density plumes is buoyancy induced.

Subbarao & Cantwell [22] investigated buoyant plumes of helium with a co-flow of air at a fixed velocity ratio of two. They examined the effects of varying the Richardson and Reynolds numbers independently within the range 390 < Re < 772 and 0.58 < Ri < 4.97, and examined the natural frequency of the oscillations and the transition to turbulence. Based on their findings, they proposed a buoyancy Strouhal number of the form:

 ${\displaystyle St={\frac {\left(fD/V_{0}\right)-K_{1}}{{\mathit {Ri}}^{1/2}}}}$ ${\displaystyle \left(11\right)}$

where ${\displaystyle K_{1}}$ is a constant, chosen as 0.445, and the density difference in the Richardson number is made dimensionless using the plume source density. In the range of flows they considered where Ri > 1, the buoyancy Strouhal number was found to be approximately constant at a value of 0.136.

Gebhart et al.’s works [19][35][36] have examined in detail the transition mechanisms and instability of laminar plumes, largely based on theoretical stability analysis and empirical studies. Some very early numerical simulations of plumes were performed in [35] where inviscid solutions of the Orr-Somerfield equations were obtained for symmetric and asymmetric plume disturbances.

O‘Hern et al. [4][37] performed detailed experiments on turbulent helium plumes to help provide data for validation of LES models. Their facility at Sandia National Laboratories involved a main chamber with dimensions 6.1 × 6.1 × 7.3 metres and a 1 metre diameter plume source. The Reynolds number based on the inlet diameter and velocity was 3200, the Richardson number around 76. Measurements taken using Particle Image Velocimetry (PIV) and Planar Laser-Induced Fluorescence (PLIF) produced simultaneous time-resolved velocity and mass fraction data. This was used to calculate density-weighted Favre-averaged and Reynolds-averaged statistics. The detailed measurements were analysed to understand the dynamics of the unsteady plume and the role of the Rayleigh-Taylor instability in producing bubble and spike flow structures. The experiments were subsequently used in computational studies by DesJardin et al. [1], Tieszen et al. [2], Xin [3], Nicolette et al. [38], Chung & Devaud [39] Blanquart & Pitch[40] and Burton [41].

Computational Fluid Dynamics

CFD simulations of the unsteady near-field behaviour of buoyant plumes have mainly used Large-Eddy Simulation (LES) or Direct Numerical Simulation (DNS) rather than traditional Reynolds-Averaged Navier-Stokes (RANS) turbulence models.

There are two notable exceptions. Firstly, the work of Nicolette et al. [38], who performed RANS simulations using a newly-developed buoyancy-modified k – ε model. The cases examined involved large-diameter helium plumes, including those studied experimentally by O‘Hern et al. [4]. The axial velocity was overpredicted in the near-field due to delayed onset of transition to turbulence in the model. Results were also found to be sensitive to the grid resolution, with steady solutions at low resolution and unsteady solutions at high resolutions, using grids with more than 1 million cells. Their modified k – ε model was found to be more numerically stable and gave better predictions over a broader range of grid resolution than the standard k – ε model. The same research group also investigated a Temporally-Filtered Navier Stokes (TFNS) approach for modelling helium plumes [42].

The second notable RANS study is the recent work of Chung & Devaud [39], who used both buoyancy-modifed steady k –ε RANS models and LES to study the large helium plumes examined experimentally by O‘Hern et al. [4]. The RANS simulations were performed using the commercial CFD code, CFX, and the LES simulations using the Fire Dynamics Simulator (FDS) code from NIST[3]. For the RANS simulations, the flow was treated as axisymmetric and details of the experimental geometry, including the location of the co-flow air inlets and the ground plane were included in the model. Both Simple Gradient Diffusion Hypothesis (SGDH) and Generalized Gradient Diffusion Hypothesis (GGDH) models were tested and the sensitivity of the results to the modelling constant Cε3 was assessed[4]. For the LES, a simpler geometry was modelled with only the plume source and ground plane, and the sensitivity of the results to the Smagorinsky constant and the grid size was examined. Four different uniform Cartesian grids were tested for the LES with cell sizes ranging from 1/10 to 1/80 of the plume source diameter, producing grids with between 63,000 and 33 million cells. The RANS results showed very significant sensitivity to the choice of Cε3, with centreline velocities at a distance 0.4 diameters downstream from the source ranging from 0.85 to 5.0 m/s for values of Cε3 from 1.0 to 0.0, respectively, for the SGDH model. The GGDH was found to be even more sensitive to the choice of Cε3. This significant sensitivity to the choice of Cε3 compared to previous studies of the model behaviour in the far-field of buoyant plumes was attributed to the very large density difference in the near-field. Good predictions were obtained using Cε3 = 0.30 for the SGDH model and Cε3 = 0.23 for the GGDH model. The SGDH model gave best agreement with the experiments in terms of the mean concentrations, whilst the GGDH model gave overall slightly better agreement in terms of the streamwise velocity. It was noted by Chung & Devaud [39] that the Cε3 constant may need to be tuned to the particular buoyant plume conditions to obtain the best results. The LES predictions were in good agreement with the experimental measurements both in terms of the puffing frequency and the mean velocity, which was predicted to within the limits of experimental uncertainty up to an axial distance of 0.6 diameters downstream from the plume source. For the mean concentration, the peak centreline values were in good agreement with the measurements at the base of the plume but became overpredicted beyond a distance of 0.2 source diameters, and by 0.6 diameters the peak was more than a factor of two higher than the experimental values. The overprediction of concentration and, to a lesser extent, velocity, on the plume centreline was attributed to under-resolution of buoyancy-induced turbulence, which Chung & Devaud [39] suggested could be improved by using a more sophisticated subgrid-scale model that took into account the effects of backscatter. Best agreement with the experiments was obtained with the finest grid, although results with a grid of 4 million cells (a cell size of 1/40 of the source diameter) were nearly as good, and Chung & Devaud [39] considered them to provide an appropriate balance of accuracy and computational cost. Changing the Smagorinsky constant to values of 0.0, 0.1, 0.2 and 0.3 was found to affect mean velocity and concentration statistics differently at different positions. At an axial distance of 0.4 diameters, a value of Cs = 0.0 provided best agreement with the experiments whilst closer to the source a value of Cs = 0.1 produced better predictions. Overall, it was recommended to use values of Cs between 0.15 and 0.20 with a grid resolution of 4 million cells.

Amongst the earliest DNS studies of plumes are those published in 2000 by Jiang & Luo [20][21]. They examined both plane and axisymmetric non-reacting and reacting plumes with temperature ratios ${\displaystyle \left(T_{0}/T_{\infty }\right)}$ of 2, 3 and 6, and Reynolds number of 1000. The flows were treated as two-dimensional or axisymmetric. This choice was justified on the basis that previous fire-plume studies [43][44] had indicated that buoyancy-induced vortical structures were produced primarily by axisymmetric instability waves and therefore azimuthal wave modes could be ignored. The more recent study of DesJardin et al. [1] has highlighted that azimuthal instabilities are significant near the base of large helium plumes. Two-dimensional/axisymmetric simulations also do not capture the turbulent three-dimensional vortex stretching mechanism.

Jiang & Luo [20][21] used their DNS results to examine the budget of the vorticity transport equation. The production of vorticity near the base of the plume was found to be dominated by the gravitational torque in the initial phase of the vortex formation. Later, when the vortex had become more established and was convecting downstream, the baroclinic torque was found to be the dominant term. The gravitational torque was mainly responsible for the necking phenomenon near the base of the plume whilst the baroclinic torque was more important in forming necking and diverging sections of the vortical structures further downstream.

More recently, Soteriou et al. [24] performed high-resolution two-dimensional simulations of transitional plumes using a Lagrangian Transport Element Method. Simulations were compared to the planar helium plume experiments of Cetegen et al. [23]. The aim of their study was to understand the mechanisms involved in the near-field flow instability. The effects of changing the density ratio, the Reynolds number and Froude number (S, Re and Fr) were explored. The simulations captured the plume pulsation frequency and the correct overall instantaneous flow behaviour. The pulsation frequency was found to be insensitive to the Reynolds number, which confirmed previous observations from plume experiments [18][26]. Whilst experiments had suggested that the pulsation instability does not occur for plumes with density ratios less than ${\displaystyle S\approx 1.7}$ [18][27], the simulations by Soteriou et al. [24] found that pulsations were produced at lower values of S, but that the Froude and Reynolds numbers at which these pulsations were observed could not be easily achieved experimentally. It was also shown that a necessary condition for stable, steady plumes was for the circulation[5] to increase monotonically with height. This leads the flow induced into the plume to be directed inwards towards the plume axis (necking). A non-monotonic increase in the circulation (i.e. a local maximum) leads to vortex formation. Depending upon the relative magnitude of the local convective, buoyant and viscous forces, it was noted that a local circulation maxima could be smoothed out or amplified.

In the mid-1990s, Mell et al. [31][32] studied the behaviour of helium plumes using the FDS code. Axisymmetric simulations were compared to experiments undertaken in-house at NIST for Froude numbers of 0.0015 ≤ Fr ≤ 0.64 and Reynolds numbers based on the exit velocity and nozzle diameter of 22 ≤ Re ≤ 446 (for details, see Mell's website[6]. Results from the simulations were in reasonable agreement with the experiments in terms of flow structures, puffing frequency, mean axial velocity and mean helium concentrations near the nozzle. At distances of more than 3 nozzle diameters downstream from the source, the agreement between simulations and experiments worsened — the axial mean velocity becoming overpredicted by nearly 40%. This was attributed to the increasing importance of three-dimensional turbulent flow structures with downstream distance which were not captured in their axisymmetric simulations. Mell et al. [31][32] also investigated the effect of neglecting the baroclinic torque term on the flow simulations. Neglecting the term was found to cause the plume to pulsate at significantly higher frequencies. More recent simulations by Xin [3], also undertaken using FDS, studied the helium plume experiments of O‘Hern et al. [37] and investigated the influence of the baroclinic torque.

The works of Zhou et al. [45][46] were the first to examine the unsteady motion of plumes using LES all the way from the source to the fully-developed plume region in the far field where the flow exhibits self-similar behaviour. In their simulations, the flow domain extended to a distance of 16 nozzle diameters from the source. Their simulations were compared to the thermal plumes of George et al. [47] and Shabbir & George [48] (R = 1273, Fr = 1.4) in [45] and to those of Cetegen [28] (Re = 730 and 1096, Ri = 0.324 and 0.432) in [46]. In both cases, the simulations used a low-Mach-number approach and a Smagorinsky LES model with constant coefficients (Cs = 0.1 and Prt = 0.3). The same grid of 256 × 128 × 128 ${\displaystyle \approx }$4.2M nodes for the domain of 16 × 8 × 8 diameters was used in both cases. Good agreement was obtained between the LES results and the experiments in terms of the radial profiles of mean velocity and temperature in the self-similar plume region. The decay of mean centreline velocity and temperatures in the simulations followed the -1/3 and -5/3 decay laws characteristic of fully-developed plume behaviour. In the near-field of the plume, the dynamic puffing behaviour was reasonably well-captured when compared to the Cetegen & Kasper [18] correlation (Equation 6). In [46], the LES data was used to present budgets for various terms in the mean axial velocity, temperature, turbulent kinetic energy and temperature-variance equations in the fully-developed plume region. A more recent study by Zhou & Hitt [49] analysed the data obtained in one of their earlier studies using proper orthogonal decomposition.

A more recent study by Pham et al. [50] also simulated the full extent of a plume, from the source to the far field (up to an axial distance of x/D = 80) using DNS and LES. No inlet velocity was prescribed and instead the plume was produced by a circular flat plate heated to 673K, which gave Reynolds and Froude numbers of 7,700 and 1.1, respectively. The DNS grid comprised 660 million nodes, whilst two different LES grids were tested with 1.2 and 2.9 million nodes. The performance of several different subgrid-scale models were assessed including a Smagorinsky model (SM) with coefficients calibrated from the DNS, a dynamic model in which both the Smagorinsky constant and the turbulent Prandtl number were estimated using the dynamic procedure (DM), the Lagrangian dynamic model proposed by Meneveau et al. [51] combined with the dynamic model for the Prandtl number (LDM), and a modified Lagrangian dynamic model which used the Meneveau et al. [51] model for both Smagorinsky constant and Prandtl number (LDMT). The decay of mean velocity and temperature in the DNS were found to follow the -1/3 and -5/3 power law in the fully-developed plume region on the centreline. At an axial distance of 60 source diameters, the power spectrum of temperature fluctuations exhibited a -5/3 Kolmogorov power law decay on the axis, and a more rapid -3 power law decay at a lateral distance of 5 jet diameters, due to enhanced turbulence dissipation driven by buoyancy forces. The DNS solution was filtered using similar filter widths to those used by the LES and used to examine the budgets for the turbulent kinetic energy and heat flux transport equations. The mean values of the Smagorinsky constant and turbulent Prandtl numbers were also extracted along the axis of the plume. Of the four models tested, the LDM and LDMT models were found to produce best agreement with the DNS in the far-field, in terms of both mean and fluctuating quantities. In the near field (x/D < 4), none of the models captured fully the correct behaviour, with all of the models under-predicting the plume width by around 20% and the SM and DM models over-predicting the peak velocity by 15% to 20%. Better predictions of the plume mean velocity and temperature were obtained with the LDM and LDMT models. Turbulence intensities were underpredicted by all models in the near field, by as much a factor of two in some cases for x/D < 4, although good agreement was obtained further downstream for x/D > 4. Overall, it was concluded that the LDMT model provided the best predictions of the purely thermal plume but that particular attention needed to be paid to the grid resolution near the plume source to capture the puffing phenomenon.

Worthy & Rubini [52][53][54] used LES to study the buoyant plumes of Shabbir & George [48] but only extended their flow domain to x/D = 14. They did not present comparisons between the results from their simulations and any experiments or empirical correlations. Instead, they focussed on the differences between various different LES subgrid-scale closure models, including variants of the standard Smagorinsky model, the dynamic Smagorinsky, the structure-function model of Metais & Lesieur [55], the one-equation model of Schumann [56] and mixed models based on the Leonard [57] and Bardina [58] approaches. Different scalar flux models based on the simple gradient diffusion and generalized gradient diffusion hypotheses (SGDH and GGDH) were also tested. They found significant differences between the results obtained using the different models. Purely dissipative SGS models were found to delay the onset of transition compared to mixed models. The grid they used was relatively coarse, composed of 127 × 63 × 63 ${\displaystyle {\approx }}$ 0.5M nodes for the domain size of 14 × 7 × 7 diameters. Compared to the earlier simulations of Zhou et al. [45][46], cells were nearly double the size in each direction. It was also found necessary to use upwind-biased third-order and second-order convection schemes in the momentum and energy equations to obtain a stable solution, whereas Zhou et al. [45][46] were previously able to use central differencing schemes.

DesJardin et al. [1] performed large-eddy simulations of the helium plume experiments of O‘Hern et al. [4] with a fully-compressible code using two different grid resolutions, 512K and 2.5M cells. Results were presented both with without a SGS model. At the base of the plume, the LES was found to overpredict the RMS streamwise velocity and concentration. This was attributed to poor resolution of buoyancy-induced vorticity generation. Tieszen et al. [2] also examined the O‘Hern et al. [4] helium plumes }using an energy-preserving low-Mach-number code, combined with a dynamic Smagorinsky LES model and grids with 250K, 1M and 4M cells. Results were found to improve with grid resolution and it was postulated that this was related to the strong influence on the mean flow behaviour of small-scale Rayleigh-Taylor structures at the base of the plume. The works of DesJardin et al. [1] and Tieszen et al. [2] are discussed in more detail below.

A later study by the same group [40] examined the O‘Hern et al. [4] helium plumes using the Lagrangian dynamic SGS model of Meneveau et al. [51] for turbulent diffusion terms in both the momentum and helium mass-fraction transport equations (modelled in their case as a mixture fraction). The full three-dimensional geometry of the experiments was simulated, including the plume source, ground plane and air co-flow injection flows, using a non-uniform cylindrical mesh with 192 × 187 × 64 ${\displaystyle {\approx }}$2.3M cells. The helium inlet velocity was lowered from the experimental Reynolds-averged value of 0.325 m/s to 0.299 m/s to account for the open area of the honeycomb (92%). The predictions of the mean and RMS velocity, and mean helium mass fraction were in good agreement with the experiments, in most cases within the limits of experimental uncertainty, and better than the earlier simulations of DesJardin et al. [1]. Close to the base of the plume (within 0.1 diameters) the centreline mean helium mass fraction was underpredicted and the RMS mass fraction was overpredicted, by up to a factor of two. These differences did not appear to have a significant effect on the flow downstream and it was noted that results may be improved by modelling more accurately the helium flow through the honeycomb immediately upstream of the plume source. Further downstream from the source, RMS mass fractions tended to be overpredicted and it was noted that an improved SGS model may be needed that takes account of buoyancy-induced turbulence.

A recent study by Burton [41] used a more advanced non-linear LES (nLES) subgrid-scale model to study the O‘Hern et al. [4] helium plumes. Unlike the Smagorinsky class of models, the nLES model does not involve any artificial viscosities or diffusivities and instead models the unknown non-linear term in the filtered Navier-Stokes equations directly [59][60]. A uniform cylindrical grid was used with 128 × 64 × 32 ${\displaystyle {\approx }}$0.3M cells for a flow domain which extended four metres in diameter and ten metres in the axial direction. Using 64 cells across the diameter of the domain, the width of each cell was 1/16 of the plume source diameter, or five times the width of the cells in the finest LES grid used by Chung & Devaud [39]. Despite this relatively coarse grid, the difference in the plume puffing frequency between the model and the experiments was less than 8% and the mean and RMS velocity and concentration profiles were largely within the limits of the experimental uncertainty. Although the results presented by Burton [41] are therefore among the best of the LES model results published to date, radial profiles were not presented at all of the measurement locations, the centreline velocity at a position 0.1 diameters from the source appeared to be overpredicted by around 20% and issues such as grid-dependency were not discussed. Nevertheless, the encouraging results show some promise of what may be achieved with more advanced turbulence closures.

Other related CFD simulations of unsteady plumes include the works of Wen, Kang and colleagues at Kingston University who have studied the transient near-field behaviour of fire plumes [61][62], and Baastians et al. [63] at the J.M. Burgers Centre for Fluid Dynamics in Delft who have performed DNS and LES of plumes in a confined enclosure.

Studies on which this UFR review will be based

This UFR focuses on three separate studies which have each examined the detailed 1-metre-diameter helium plume experiments of O‘Hern et al. [4]:

• DesJardin et al. [1] (2004): simulations using a fully-compressible code with high-order upwind-biased convection schemes and a dynamic Smagorinsky model with grids of 512K and 2.5M cells.
• Tieszen et al. [2] (2004): simulations using an energy-preserving low-Mach-number code, combined with a dynamic Smagorinsky LES model and using grids with 250K, 1M and 4M cells.
• Xin [3] (2005): simulations using the Fire Dynamics Simulator (FDS) code from NIST, a low-Mach-number LES code using a Smagorinsky model with fixed coefficients and a grid of 1.5M cells.

Different numerical methods, grid resolutions and turbulence models are used in these three studies, and they therefore provide useful complementary data on the performance of CFD models which provides a good match for what is required in this UFR. Additional comments on model performance are also provided in the works of Chung & Devaud [39], Blanquart & Pitch [40] and Burton [41], who also simulated the helium plume experiments of O‘Hern et al. [4]. These recent works were published after this UFR was first completed, in 2007, and so are not examined in such detail here.

Footnotes

1. http://www.engr.uconn.edu/~cetegen/cetegen/previous%20research/researchproj.html?plume
2. http://www2.bfrl.nist.gov/userpages/wmell/plumes.html
3. For details of how to download the FDS code and relevant documentation, see http://www.fire.nist.gov/fds
4. For more information on these model details, see the companion UFR on the far-field behaviour of plumes.
5. The circulation, ${\displaystyle {\left.\Gamma \right.}}$, is defined as the integral of vorticity over a surface, ${\displaystyle \Gamma =\int _{S}\omega \cdot {\mathit {dS}}}$
6. http://www2.bfrl.nist.gov/userpages/wmell/plumes.html

Contributed by: Simon Gant — UK Health & Safety Laboratory

© copyright ERCOFTAC 2010