NASA Technical Reports Server (NTRS) 20020043256: Large Eddy Simulation of a Turbulent Jet

4 Pages·0.34 MB·English

by  NASA Technical Reports Server (NTRS)

10-14 December 2001
Large Eddy Simulation of a Turbulent Jet
A. T. Webb _ and N. N. Mansour 2
=School of Civil Engineering
Australian Defense Force Academy, UNSW, ACT, 2600 AUSTRALIA
2Center for Turbulence Research ,
NASA Ames Research Center, CA 94035, USA
Abstract where Uo isthemean verticalvelocityofthefluidleaving
the source (usedtonon-dimensionaiise allvelocities)D,o
Here we presenttheresultsofaLarge Eddy Simulationof
isthediameter oftheorifice(usedtonon-dimensionalise
anon-buoyant jetissuingfrom acircularorificeinawail,
lengths),v isthe kinematic viscosityof the fluid,and
and developing in neutral surroundings. The effectsof
the scalardiffusivity.In the following we use T to
thesubgrid scaleson the largeeddieshave been modeled
represent the dimensionless scalarconcentration (e.g.a
with the dynamic largeeddy simulation model applied
tracerorpollutant).
to the fully3D domain in sphericalcoordinates. The
simulationcaptures the unsteady motions of the large-
Large Eddy Simulation
scaleswithin the jetas well as the laminar motions in
the entrainment region surrounding the jet. The com- To create an LES computational model, the Navier-
puted time-averaged statistics(mean velocity,concentra- Stokes/continuity/tracerequations are filteredwith the
tion,and turbulence parameters) compare wellwith lab- filtercutoffchosen to remove the dissipativescalesbut
oratory data without invoking an empirical entrainment retaintheenergy-containingscales.Inthepresentstudy,
coefficientas employed by lineintegralmodels. The use the subgrid scalesstressesand fluxesare modeled using
ofthe largeeddy simulation technique allowsexamina- the dynamic approach ([6],and [4]),which automatically
tionofunsteady and inhomogeneous featuressuch asthe determines, usingdifferentfilterwidths, the spatialdis-
evolution of eddies and the detailsof the entrainment tributionofthemagnitude ofeddy viscosityas required
process. by thesubgrid-scaieSmagorinsky model. This procedure
obviates theneed forany empiricaldetermination ofthe
Introduction Smagorinsky constant.
Standard engineering designmodels forjetsand plumes A major advantage ofthismethod isthat itismuch more
employ averaging techniques that mask the richdetails likelytobe successfulforinhomogeneous flows,particu-
evident intherealworld. A snapshot ofa plume billow- larlyin the present case where part of the domain is
ingfrom a chimney oreffluentdischargingfrom an ocean laminar. This issuewas explored by Liu etad. [9]who
outfailshows a highlyirregularedge with regionsoflow compared theSmagorinsky and thedynamic models with
concentration near the centre and pockets of high con- particleimage velocimetry laboratory measurements in
centrationsfardownstream from the source. Superim- the farfieldofa turbulent round jet.The Smagorinsky
posed on thisirregularityisastructureevidenced by the model was found tocorrelatepoorly with the realturbu-
boilsor eddieswhich seem torepeat atseemingly regular lentstress,whilethe dynamic model yieldedappropriate
intervals.While experimental techniques are available coefficients.The inadequacy ofthe Smagorinsky model
to explore these unsteady phenomena, sampling islim- was alsodemonstrated by Bastiaans etaL [i]inan LES
ited. For example laser-doppleranemometry isa point model of transientbuoyant plumes in an enclosure. In
method that misses the synoptic, while laser-induced thisstudy, they found they could only get a match of
fluorescenceand particle-imagingvelocimetry only pro- the evolvingplume by tuning the Smagorinsky constant.
vide a sliceof information with sampling limitedto the Such tuning reduces the generalityofa model.
frame rateofthe videocamera and the densityofpixels
inthe charged couple device. An attractivealternative N_mQrical scheme
isto model the fluidmotions at a sufficientlydetailed
We use the LES code of Boersma, an adaptation of
scalesothat the energy-containing eddiesare resolved-
his own jet direct numerical simulation (DNS) code[3].
thismodeling approach iscalledLarge Eddy Simulation
The code was modified by Basu [2] to compute buoy-
(LES). We present here the resultsofsuch a study on
ant plumes. A spherical polar coordinate system (R, e, _b;
a simple momentum jetdischarging intoa semi-infinite
along the radial, lateral and azimuthal directions respec-
quiescentenvironment.
tively) is used here because, for the present flow with
its conical mean growth, a spherical coordinate system
The Model
allows for a well-balanced resolution of the flow field
The equations to be solved are the filtered incompress- without excessive grid points. For presentation purpcees,
ible Navier-Stokes equations together with the continuity however, and for comparison to laboratory data, the re-
equation and a scalar transport equation representing a sults are converted to the axisymmetric cylindrical coor-
passive tracer (or pollutant). dinate system (z axial and r radial). The symbols U and
V are used for velocities in the z and r directions respec-
The non-dimensional governing parameters for this flow
tively. Further details of the computational scheme can
are: be found in [3], [4], and [6].
Reynolds number = Re = UoDo/v ,
Boundary conditions and test parameters
Prandtl number = Pr = v/_ ,
For the chosen staggered grid, only velocities normal to
the boundary need be specified -i.e. six components. The ÷
equation for the tracer (T) is second order in each spatial
÷
direction, requiring six boundary conditions as well.
÷
÷
For the bottom boundary a Dirichlet BC is used with log-
5t ÷
law profile velocity at the orifice, to which white noise of
2% rms is added. At the outflow boundary, or the top
end of the current computational domain, we use the ad-
vective boundary condition and at the lateral boundary
we apply a stress-free condition ([3] and [7]). Periodic
boundary conditions axe used along the azimuthal _ di-
rection boundaries and lateral velocity (u0) is set to mean
value at 8 = 0.
The Keynolds number for the jet source is set at 3500 -
within the range of experimental data for which the jet
-10I 1'0 2_0 3i0 4'0 50 60
is turbulent at or near the nozzle[5].
Z
The computations are carried out in a domain that ex-
tends to 50 diameters downstream of the source. The do-
Figure 1: Growth of jet width in terms of the e-
main encompasses a conical volume of lateral angle lr/12, folding width for z-direction velocity (circles) and tracer
with a virtual origin that is 15 diameters upstream of the (crones).
orifice. For the purpose of LES, this volume is discre-
tised using a grid of size (N, = 128, No = 40, N¢ = 32).
Time stepping is chosen to satisfy CFL conditions [3] The fitted slopes in the central region (beyond the ZFE
which turns out to be typically 0.01 in dimensionless time and short of the apparent influence of the downstream
units (Do/Uo). Spatially the grid is an order of mag- boundary) are 0.105 for U and 0.126 for T (figure 1).
nitude smaller than the expected large scale (typically The matches are remarkably close to the laboratory data
0.4z, although there is an arbitrariness about the defi- and well within the data uncertainty.
nition of the jet edge) and an order of magnitude larger
Other checks carried out [11] show that the centerline ve-
than the Kolmogorov length scale [10] as appropriate for
locity and tracer decay as z-l, and that turbulent fluc-
LES modeling. The temporal sampling is much smaller
tuations and turbulent momentum and scalar transport
than the Kolmogorov time scale.
match laboratory data. In almost every case the match is
The computations are carried out for over 100,000 time- well within data and model uncertainty. We also verified
steps (a non-dimensional time of about t = 300). Aver- that most of the variance can be accounted for by the
aging for statistical quantities was done towards the end resolved portion of the model - as we would hope from
of the simulations when we were convinced that the flow an LES model.
was statistically stationary.
Unsteady features
Model verification
To examine some of the jet features we take sections both
Confidence in the model can only come from comparison through the centerline (meridionai) and across the jet (ra-
with laboratory data. There is a wealth of such data and dial). At the computer, it is instructive to roam through
we use a selection reviewed by List[8]. The data repre- many such sections but even with just two we can see
sent air in air and water in water over a wide range of a great deal. Figure 2a is a meridional section between
Reynolds number (but all in the turbulent regime). De- 6 and 13 diameters from the source while figure 2b is
spite the range of fluids and laboratory conditions there a radial section at about 9 diameters from the source.
is a surprising consistency between the results and hence We overlay the instantaneous tracer concentrations and
we would expect our model to be able to match them velocity vectors.
well.
At the bottom centre of the meridional section (figure
To compare the model results, we have averaged first 2a), between z = 6 and z = 8, we can see what at
moment and second moment statistics over some 10,000 first sight appears to be a core region of almost source
samples. The particular sampling and averaging carried concentration with velocities matching the source veloc-
out yields an uncertainty of about 2% for any given sta- ity. There is actually a peak concentration at z = 6.7,
tistical estimate. r = -0.2 that is at the center of a patch about to break
from the core to form a more or less separate puff. Such
Here we examine two statistics, the mean z-direction ve-
puffs are shed repeatedly, and, in fact, the previously
locity (U) and the tracer concentration (T). The pro-
formed puffcan be seen at z = 11. Given that the source
cedure for reducing the data was to fit Gaussian curves
is uniform and steady with the addition of a small level
(using Matlab's routine ulinfit with emphasis on the in-
of noise it is remarkable how unsteady and irregular are
ner radius) to the r-direction profiles. The fitted e-folding
the features.
radius (re) and the fitted maximum value were used to
scale the data. Of the many checks that we have made Within and about these large scale puffs are eddies of
[11], we here present just two - we recall that the ex- comparable scale. An anti-clockwise eddy with horizon-
perimental jet e-folding width grows linearly as 0.107z tal axis can be seen centered at z = 9, r = -1. At its top
for U and 0.126z for T. (From the laboratory data the this eddy transports jet fluid away from the core while at
uncertainty for the growth rates is 3% for both U and T the bottom, ambient fluid is transported towards the jet
[8]). center. The horizontal extent of the eddy and its three-
13 0.8
12
2
11 15
1
10 0.4 0.5
z y 0
-0.5
-1
-1.5
-22. -1 0 1 2
X
6 3.1
-2 -1 0 1 2
-rto +r
Figure 2: Instantaneous meridional (a) and radial (b) sections of tracer concentrati6n and velocity vectors. The relative
position of each section is indicated by a black line.
dimensional character is evident from the radial section 0.ra
(Figure lb) at x = -1.5, y = -0.8. An example of an
eddy with vertical axis can be seen in the radial section 0OF
centered at x = 1.2, y = 0. As with the other eddy,
this one is clearly advecting away core fluid in the out- gtrm
ward portion and ambient fluid towards the center in the
inward portion. Closer to the core, there appear to be
several smaller scale eddies which further mix the jet fluid
005
with ambient. Image sequences show that any jet fluid 004
that strays too far from the center is readily re-entrained
by the sink flow that exists outside the turbulent jet re- _0_
gion.
Although we have been emphasizing the irregularity of
the flow, there is an underlying apparent dominant size
of the puffs and eddies, and a typical frequency at which 0'¢_I
the puffs are shed. In the following we look to quantifying 0 lO"
10-a 104 10"1
these scales. frequency
Quantifyinl_ scale Figure 3: Spectrum of tracer at z = 30. The dashed line
In an attempt to quantify the time scale we have ex- is the raw spectrum, while the solid is band-averaged.
amined a time series of tracer concentration at a single
point - on the centerline at 30 diameters from the source.
band-average, (2) they represent real higher frequency
This choice is arbitrary as any point should share the
motions - (3) they are spurious peaks associated with
same temporal spectral characteristics. A raw spectrum
the non-sinusoidal shape of the puffs. We will have more
is computed from a sequence of duration 460 time units
to say about this with regard to the spatial spectrum.
at interval of 1.8 time units. Band averaging over 4 ad-
jacent estimates reduces the uncertainty (figure 3). Al- From the mean flow field one could invoke the Taylor
though this still leaves broad confidence intervals, a clear frozen-field assumption to deduce a spatial scale, but an
peak is evident at 0.025 dimensionless frequency units impediment to computing such a scale is the obvious in-
(Uo/Do). There may also be other peaks at about 0.07 homogeneity. However, we can make use of the scaling
and 0.1 for which we offer three possible explanations - determined from the mean and variance to transform an
(1) they are just noise - note that we only use a 4-fold axial sample of the tracer to a homogeneous equivalent.
120 the source, and to be about the same size as the mean
width of the jet. Whether this result is Reynolds number
100 dependent remains to be seen.
Acknowledgments
The first author acknowledges the generosity of the Cen-
ter for Turbulence Research in hosting his visit to par*
ticipate in this study. We are also grateful to Drs. B.
Boersma and A. Basu for providing us with the code used
in this study, and for Dr. Michael P_gers for reading and
,°
helpful comments on the manuscript.
20 References
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rect and Large-Eddy Simulation I, 399-410, Kluwer
Figure 4: Spectrum of centerline scalar concentration in
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