# UFR 1-07 Description

## Contents

# Unsteady Near-Field Plumes

## Free Flows

### Underlying Flow Regime 1-07

# 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:

where is the plume fluid
density, is the inlet velocity, *D*
is the characteristic inlet length scale or inlet diameter and
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:

where *g* is the gravitational acceleration and
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:

In some texts, the density difference in the Froude and Richardson
numbers is made dimensionless using the plume source density, ,
instead of the ambient density, .

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, ,
and the time for a fluid element to move the same distance under the
action of buoyancy, , where:

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:

where 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 ,
the Strouhal number was related to the
Richardson number by:

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

**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 ,
Cetegen *et al.* [23]
found that the Strouhal number varied according to:

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:

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
(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, , and the
diameter of the burner or source, :

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 [18].
For planar helium
plumes, Soteriou *et al.* [24]
showed that the frequency varied
according to .

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,
. 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
[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
(see Figure 7). The plume was unsteady
for Reynolds or Richardson numbers above the line shown in the graph
(i.e. for or
).

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:

*et al.*[24] (top);

*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:

where 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 *C _{ε3}* 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

*C*, 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

_{ε3}*C*from 1.0 to 0.0, respectively, for the SGDH model. The GGDH was found to be even more sensitive to the choice of

_{ε3}*C*. This significant sensitivity to the choice of

_{ε3}*C*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

_{ε3}*C*= 0.30 for the SGDH model and

_{ε3}*C*= 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

_{ε3}*C*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

_{ε3}*C*= 0.0 provided best agreement with the experiments whilst closer to the source a value of

_{s}*C*= 0.1 produced better predictions. Overall, it was recommended to use values of

_{s}*C*between 0.15 and 0.20 with a grid resolution of 4 million cells.

_{s}

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 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
[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 (*C _{s}* = 0.1 and

*Pr*= 0.3). The same grid of 256 × 128 × 128 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.

_{t}

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 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 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 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**[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.**et al.****Tieszen**[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.**et al.****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

- ↑ http://www.engr.uconn.edu/~cetegen/cetegen/previous%20research/researchproj.html?plume
- ↑ http://www2.bfrl.nist.gov/userpages/wmell/plumes.html
- ↑ For details of how to download the FDS code and relevant documentation, see http://www.fire.nist.gov/fds
- ↑ For more information on these model details, see the companion UFR on the far-field behaviour of plumes.
- ↑ The circulation, , is defined as the integral of vorticity over a surface,
- ↑ http://www2.bfrl.nist.gov/userpages/wmell/plumes.html

Contributed by: **Simon Gant** — *UK Health & Safety Laboratory*

© copyright ERCOFTAC 2010