UFR 1-06 Test Case
Axisymmetric buoyant far-field plume in a quiescent unstratified environment
Underlying Flow Regime 1-06
Test Case
Brief Description of the Study Test Case
The experiments used in this UFR are those of George et al. [3] which were conducted in 1974 at the Factory Mutual Research Corporation and were subsequently repeated by Shabbir & George [34] at the University of Buffalo.
- Heated air is discharged through a circular orifice into ambient air that is at rest.
- The plume source temperature is 300°C and the ambient air is 29°C.
- The source has diameter, D = 6.35 cm.
- The hot air is discharged at a velocity of U0 = 67 cm/s with a approximately a top-hat profile.
- Temperature and velocity fluctuations at the inlet are less than 0.1%.
- George et al. [3] present experimentally measured profiles
of both mean and fluctuating components of the temperature and axial velocity in the self-similar region at x/D = 8, 12 and 16 above the source.
Test Case Experiments
The experiments used in this UFR are those of George et al. [3] which were conducted in 1974 at the Factory Mutual Research Corporation and were subsequently repeated by Shabbir & George [34] at the University of Buffalo.
The general arrangement is shown in Figure 4.
Compressed air is passed through a set of heaters and porous mesh screens before exiting through a nozzle into
the enclosure. The nozzle is stated as a 15:1 contraction in [3],
a 12:1 contraction in [sg92]
and appears to be different again in a drawing of the arrangement in [3]
(see Figure 5). It resulted in a velocity
profile through the exit which was uniform to within 2% outside the wall boundary layer. The velocity and
temperature fluctuations at the exit were measured to be very low, less than 0.1%
in [3]
and 0.5% in [34].
The temperature of the source was 300°C and the ambient environment
29°C. Both were controlled to an accuracy of within 1°C. The discharge velocity was 67 cm/s, as
calculated from the measured heat flux. These source conditions corresponded to Reynolds number, Re0 = 870, and densimetric Froude number, Fr0 = 1.23
[#sdfootnote1sym 1].
There was no evidence of laminar flow behaviour at a position two inlet
diameters downstream from the source. The effective origin of the plume, x0, was
found to be at the same location as the exit
(see [3] for details of how this was determined).
The screen enclosure around the plume exit was 2.44 × 2.44 metres in cross-section and 2.44 metres high
(there is, presumably, an error in [3]
which suggests that the enclosure is 2.44 × 2.44 × 2.44 mm).
In the later Shabbir & George experiments, a 2 × 2 × 5 metre enclosure was used. The purpose
of the screens was to minimize the effect of cross-draughts and other disturbances affecting the flow.
Two-wire probes were used by George et al. [3]
to record velocities and temperature.
[#sdfootnote1anc 1]The densimetric Froude number is calculated here from the source and ambient temperatures, the exit velocity and source diameter given by George et al. [gat77], using Equation (1). However, George et al. [gat77] stated that the densimetric Froude number was 1.4. It is unclear how they determined this value. Using the approach taken by Chen & Rodi [cr80] in which the source density instead of the ambient density is used to make the density difference dimensionless, and Froude number is defined using the square of the expression given in Equation (1), this gives a Froude number of 0.80.
George et al. [3] reported that measurement errors, stemming from
directional ambiguity of the hot wire and its thermal inertia, were around 3% for the velocity and lower for
other mean and RMS values. The frequency response of the hot wires was estimated to be around 300 Hz compared
to the frequency of the energy-containing eddies at around 50 Hz and the Kolmogorov microscale at 1 Khz.
It was noted that measurement errors were likely to be higher on the outer edge of the plume where
the velocity fluctuations were higher.
In their review of plume experiments, Chen & Rodi [1]
noted that the data from George et al. differed significantly from earlier measurements by
Rouse et al. [64].
However, they considered it to be more reliable due to its use of more
sophisticated instrumentation.
George [40], describes an experimental program at the University of
Buffalo that was set up following publication of the original
George et al. [3] paper to investigate
possible causes of differences in experimental plume results. Possible sources of errors discussed
included:
- ambient thermal stratification
- the size of the enclosure
- the use of porous screens used to minimise disturbances from the far-field affected the plume source.
- hot wire measurement errors
The most significant concern was ambient thermal stratification.
One of the features of buoyant plumes in neutral environments is that the integral of the
buoyancy across the whole cross-section of the plume, F, should remain constant and equal to the
buoyancy added at the source, F0.
George [40] discussed how thermal stratification involving
small temperature differences of the order of 1°C across a 3 metre vertical span would be sufficient
to cause F to decrease to only 50% of the source value.
This would be likely to cause differences in measured temperature and velocity plume profiles.
In the initial experiments of George et al. [3],
the thermal stratification was not strictly controlled.
However, results from later experiments published in the PhD thesis of
Shabbir [32]
(reproduced in [34] and [40]),
which conserved buoyancy to within 10%, are in good agreement with the earlier results from
George et al. [3].
This suggests that, perhaps fortunately, ambient thermal stratification did not contaminate the
George et al. [3] results significantly.
A summary of the original results from George et al. [3]
and those reproduced later by Shabbir & George [11]
is presented in Table 3.
Also shown are the recommended values from
Chen & Rodi's review [1] and other studies.
The parameters given in Table 3 relate to the following empirical formulae for the mean vertical velocity:
and effective buoyancy acceleration:
where and are Gaussian functions:
The parameters, and are the dimensionless half-widths
of the plume, as defined by the location where the normalized buoyancy or mean velocity falls to half
its centreline value.
The RMS temperature and axial velocity fluctuations normalized by their centreline mean values are
denoted,
and , respectively.
As noted earlier, Dai et al. [10][37]
[38][39][41] disputed the
accuracy of the George et al. [3] experiments and suggested
that they had made measurements too near the source, before the plume had reached a fully-developed state.
Their arguments are disregarded by Shabbir & George [11]
[34].
CFD Methods
Van Maele & Merci: Description of CFD Work
Numerical Methods
Van Maele & Merci [2] used the finite-volume-based commercial CFD code, Fluent, to simulate the plume experiments of George et al. [3]. For the discretization of the convective terms in the momentum, turbulence and energy equations a second-order upwind scheme was used. Diffusion terms were discretized using second-order central differences and the SIMPLE algorithm was used for pressure-velocity coupling. The flow was treated as axisymmetric and elliptic calculations were performed used a Cartesian grid arrangement.
The low-Mach-number form of the Favre-averaged Navier-Stokes equations were used. In this weakly-compressible
approach, the density is treated as only a function of temperature and not pressure. Pressure only affects the
flow field through the pressure-gradient term in the momentum equations. The ideal gas law is used to link the
mean density, , to mean temperature, T as follows:
where p* is taken as constant and equal to the atmospheric pressure. The
low-Mach-number approximation implies that the effect of the mean kinetic energy and the work done by viscous
stresses and pressure are negligible in the energy equation.
Turbulence Modelling
Two turbulence models were used by Van Maele & Merci [2]: the standard k – ε model of Jones & Launder [65] and the realizable k – ε model of Shih et al. [66]. In the former model, the eddy viscosity is given by:
where cμ is a constant equal to 0.09 and the standard k and ε
equations are written:
where cε1 = 1.44,
cε2 = 1.92,
σk = 1.0,
σε = 1.3 and Pk is the production term due to mean shear.
The terms G and SεB are source terms related to the influence of buoyancy on
the k and ε equations. The treatment of these gterms is discussed below.
The Shih et al. [66] model involves two changes to the standard
k – ε model. Firstly, cμ is made a function of strain and
vorticity invariants to ensure that the model always returns positive normal Reynolds stresses and satisfies
the Schwarz inequality for the turbulent shear stresses. The function form of cμ
is given by:
where:
and A0 is a constant equal to 4.04.
Secondly, a different ε-equation is used to resolve the problem of the round-jet/plane-jet
anomaly (see Pope [67]):
where S is the strain-rate invariant as before, cε2 = 1.9,
σk = 1.0 and σε = 1.3.
The Shih et al. [66] model was developed for high
Reynolds number turbulent flows and therefore a zonal or wall-function
approach must be used to bridge the viscous sub-layer near walls.
Compared to the standard k–ε model,
it has been shown to produce improved behaviour in a number of free
shear flows, boundary-layer flows and a backward-facing step flow [66].
One of the major weaknesses of the standard k–ε model
is that it produces too much turbulent
kinetic energy at stagnation points [68].
The Shih et al.
model should in principle suffer less from this weakness since the
functional form of cμ should
reduce the over-production of k. However, its overall
performance in stagnating flows will depend on the type of wall model
used.
Production due to Buoyancy, G
The term G in the k-equation relates to the influence of buoyancy on the turbulent kinetic energy, and is given by:
where gj is the gravitational
acceleration vector. In stably stratified flows, where the temperature
increases with height, G is negative. Conversely, in unstably
stratified flows, where temperature decreases with height, G
is positive and acts to increase k. The unknown
density-velocity correlation, , must be
modelled. The most common approximation of this term is the so-called
Boussinesq Simple Gradient Diffusion Hypothesis (SGDH):
The production due to buoyancy using SGDH is then as follows:
In their paper, Van Maele & Merci [2]
erroneously included
an additional pressure-gradient term in Equation (23) related to the pressure-work rather than
buoyancy (see Wilcox [69]). Since the term is negligible in
incompressible flows, such as the buoyant plumes considered here, it
has therefore been ignored. The ratio of the reference density to the
mean density, , appears in
Equation (23) due to the use of a non-Boussinesq approach and Favre-averaging,
which are discussed later. Van Maele & Merci [2] assumed
that σt was constant and
equal to 0.85.
Instead of writing the buoyancy production in terms of the
density-velocity correlation, , the equation
can be written in terms of the heat flux, :
and the G term is then written:
where t′ is the temperature fluctuation, T is the mean
temperature and β is the volumetric expansion
coefficient, . Other equivalent expressions can also
be formulated using the ideal gas law and the assumption that density
is only a function of temperature, not pressure (the low-Mach-number
approximation). The conversion from mean density to temperature
gradients is then as follows:
The SGDH model predicts zero density-velocity correlation or heat flux
components ( or )
in situations were the density or temperature gradients are zero in
that direction. However, as Ince & Launder [70] noted, in a
simple shear flow in which there are only cross-stream temperature
gradients, the heat flux in the streamwise direction actually exceeds
that in the cross-stream direction. This shortcoming of the SGDH model
was confirmed by the analysis of Shabbir & Taulbee [33], who
showed that the model significantly underestimates the magnitude of the
heat flux in vertical buoyant plumes. The underprediction of
or by the SGDH model
leads to an overly-small production term, G, and hence a
turbulent kinetic energy, k, which is too small, producing too
little mixing in the modelled plume. The study by
Yan & Holmstedt [53]
provides a clear example of how the k – ε
model with SGDH produces buoyant plumes which
are too narrow and with overly high temperatures and velocities in the
core of the flow.
Van Maele & Merci [2]
examined a different model for
G based on the the Generalized-Gradient Diffusion Hypothesis
(GGDH) of Daly & Harlow [52].
This was first used in the
context of practical CFD calculations with the k – ε
model by Ince & Launder [70], and is
written as follows:
which Van Maele & Merci [2] expressed as follows:
again with σt = 0.85. As
previously, Van Maele & Merci [2] included a
pressure-gradient term in the above equation related to pressure-work
but this can effectively be ignored in the present application. The
advantage of the GGDH approach is that transverse density gradients
affect the production term.
In their plume simulations,
Van Maele & Merci [2] used a
slightly modified form of the above relation. They replaced the normal
stress in the streamwise (vertical) direction,
, with the turbulent kinetic energy, k.
They justified this on the basis that the k – ε
model gives poor predictions of normal
stresses in plumes. Experimental measurements indicate that the
streamwise normal stress is approximately twice the magnitude of the
transverse components (i.e. ) whereas the k – ε
model predicts them to be roughly equal. Since
k can be approximated from
and the k – ε model predicts
, they
suggest that it is more appropriate to use k rather than
, to artificially increase the stress to a
more realistic value. This ad-hoc correction may not be appropriate in
more complex flows where the gravitational vector is not aligned to the
Cartesian axes.
A simplification frequently made to the buoyancy treatments described
above is to assume that the mean density is equal to the reference
density, , an approach
known as the Boussinesq approximation. For the SGDH written in
terms of temperature gradients, this gives:
Van Maele & Merci [2] examined the effect of this
simplification on the prediction of the
George et al. [3] buoyant plume experiments.
Buoyancy Source Term in the ε–Equation, SεB
The buoyancy source term, SεB, in the ε–equation is given by:
Unlike other model constants in the k – ε
model, there is still some controversy over the best value or
formula for cε3.
Different approaches have been proposed by different researchers,
partly depending on whether the flows are horizontal or vertical and
whether there is stable or unstable stratification. For a review of the
performance of various models, see Rodi [71],
Markatos et al. [72]
or Worthy et al. [73].
In their paper, Van Maele & Merci [2]
provided a summary of the
values proposed previously in 20 published papers and, based on
analysis of these studies, used a constant value for cε3 of 0.8.
Favre-Averaging
Throughout their paper, Van Maele & Merci [2] refer to Favre-averaged mean velocity, enthalpy and temperature (, and ) instead of the perhaps more familiar Reynolds-averaged values, (U, H and T). The Favre average of a variable , denoted is calculated from:
where overbars represent long time or ensemble averages in the
traditional Reynolds-averaged sense. The turbulent stresses appearing
in the Favre-averaged Navier-Stokes equations are:
This is the same as the usual Reynolds-averaged stress except that the
strain-rate tensor, is now
Favre-averaged. The Favre-averaged strain-rate is calculated from:
where the Favre-averaged mean velocity, , is the
parameter solved for in the momentum equations.
Contributed by: Simon Gant — Lea Associates
© copyright ERCOFTAC 2010