43rd AIAA Aerospace Sciences Meeting and Exhibit
AIAA-2005-0635
Jan 10 - 13 Jan 2005, Reno, Nevada
Numerical Simulation of the Vortical Structures in a Lobed
Jet Mixing Flow
Nathan J. Cooper* and Parviz Merati†
Department of Mechanical and Aeronautical Engineering, Western Michigan University, Kalamazoo, MI 49008
and
Hui Hu‡
Department of Aerospace Engineering, Iowa State University, Ames, IA 50011-2271
Numerical simulations of an incompressible jet mixing flow exhausted from a circular
lobed mixer/nozzle were presented and validated against whole-field quantitative Dual-Plane
Stereoscopic PIV (DP-SPIV) measurement results of the same flow. The numerical
simulations were conducted using a Reynolds Averaged Navier-Stokes approach with a
modest number of unstructured tetrahedral cells and four widely used turbulence models to
predict the vortex structures in a lobed jet mixing flow. The predictability of the turbulence
models to the lobed jet mixing flow were assessed and quantified based on quantitative
comparisons of the numerical predictions with the DP-SPIV measurement results. It is found
that the numerical simulations agree with the experimental measurements reasonably well in
terms of streamwise vorticity and azimuthal (spanwise) vorticity. Although all the four
turbulent models investigated over predict the magnitude of the turbulent kinetic energy
significantly (about 50% to 130% over predicted), the numerical simulations are found to
agree with the experimental results in predicting the locations of the regions with high
turbulent kinetic energy, and the trends associated turbulent kinetic energy production and
decay with downstream distance. It is found that the k-ε Realizable turbulence model
provides the most accurate prediction of the lobed jet mixing flow among the turbulence
models compared.
Nomenclature
D
H
= inner diameter of lobed nozzle, D=40 mm
= lobe height, H=15 mm
k
= turbulent kinetic energy, k =
K(x)
= averaged turbulent kinetic energy, K ( x ) =
ReD
U, V, W
U0
x, y, z
=
=
=
=
=
=
µ
ρ
(
1
ui ⋅ ui
2
2 ⋅U 0
)
∫ ρ ⋅ U ( x, y, z ) ⋅ k ( x, y, z )dydz
∫ ρ ⋅ U ( x, y, z )dydz
Reynolds number from nozzle inner diameter, D
mean velocity components, m/s
mass-flow averaged axial velocity at nozzle exit-plane, U0 = 20 m/s
Cartesian coordinate system
absolute viscosity, 1.789×10-5 kg/m-s
density of air, 1.225 kg/ m3
*
Graduate Research Assistant, Department of Mechanical and Aeronautical Engineering, AIAA Student Member;
Email: nate.cooper@wmich.edu.
†
Professor and Chair, Department of Mechanical and Aeronautical Engineering.
‡
Assistant Professor, Department of Aerospace Engineering, Corresponding Author; Email: huhui@iastate.edu.
1
American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit
AIAA-2005-0635
Jan 10 - 13 Jan 2005, Reno, Nevada
θin
θout
= average inward penetration angle, 14°
= average outward penetration angle, 25°
ωx
= non-dimensional x or streamwise vorticity,
ωy
ωz
ωin-plane
T
ωx =
D
U0
⎛ ∂w ∂v ⎞
⎜⎜
− ⎟⎟
⎝ ∂y ∂z ⎠
D ⎛ ∂u ∂w ⎞
⎜ −
⎟
U 0 ⎝ ∂z ∂x ⎠
D ⎛ ∂v ∂u ⎞
⎜ − ⎟
= non-dimensional z vorticity, ω z =
U 0 ⎜⎝ ∂x ∂y ⎟⎠
= non-dimensional y vorticity,
ωy =
= non-dimensional azimuthal (spanwise) vorticity,
I.
ω in− plane = ω y 2 + ω z 2
Introduction
HERE are numerous aerospace applications where methods to enhance fluid mixing between co-flowing
streams have been identified as critical or enabling technologies. Examples include low emissions combustors,
ejectors for high lift or jet noise reduction, infrared suppressor nozzles, turbofan core-bypass mixers and supersonic
combustion ramjets. It is widely recognized that a powerful mechanism for mixing enhancement is through
introduction of strong streamwise vortices between the co-flowing streams.
Lobed mixers/nozzles, which are essentially splitter plates with corrugated trailing edges, provide an
extraordinary means for efficiently generating strong streamwise vortices and associated cross-stream circulation in
co-flowing streams. Due to their outstanding performance of mixing enhancement, lobed mixers/nozzles have been
given great attention by many researchers in past years. Paynter & Birch1 investigated the benefit of incorporating a
lobed nozzle/mixer into a turbofan exhaust system to decrease jet noise. They found that lobed nozzles/mixers not
only decreased jet noise but also increased thrust output. Further detailed investigations for optimizing the geometry
of lobed nozzles/mixers for bulk mixing enhancement and performance improvement of turbofan exhaust systems
were conducted through the NASA Energy Efficient Engine (E3) program2 initiated in 1978. Additional mixing
benefits were gained by further modifications to the lobed trailing edge, which include notches or scallops in the
lobed trailing edge, cutback lobes, increased penetration and lobe number.
Besides the continuous efforts to optimize the geometry of lobed mixers/nozzles for better mixing enhancement
performance and widen the applications of lobed mixers/nozzles, extensive studies about the mechanism by which
the lobed mixers/nozzles substantially enhance mixing have also been conducted in past years. Based on pressure,
temperature, and velocity measurements of the flow field downstream of a lobed mixer/nozzle, Paterson3, 4 revealed
the existence of large-scale streamwise vortices in lobed mixing flows induced by the special geometry of the lobed
nozzle. Large-scale streamwise vortices were suggested to be responsible for the enhanced mixing. Werle et al.5 and
Eckerle et al.6 found that the streamwise vortices in lobed mixing flows follow a three-step process by which the
streamwise vortices form, intensify, and then break down, and suggested that the high turbulence resulting from the
vortex breakdown improved the overall mixing process. Elliott et al.7 suggested that there are three primary
contributors to the mixing processes in lobed mixing flows. The first is the azimuthal (spanwise) vortices, which
occur in any free shear layers due to the Kelvin-Helmholtz instability. The second is the increased interfacial contact
area due to the convoluted trailing edge of the lobed nozzle/mixer. The last element is the streamwise vortices
produced by the special geometry of the lobed nozzle/mixer. Based on pulsed laser sheet flow visualization with
smoke and three-dimensional velocity measurements with hot film anemometer, McCormick and Bennette8
suggested that it is the interaction of the azimuthal (spanwise) Kelvin-Helmholtz vortices with the streamwise
vortices that produce the high levels of mixing.
Although unsteady vortical and turbulent structures in lobed mixing flows have been suggested to play important
roles for the mixing enhancement, the detailed quantitative experimental data like whole-field velocity and vorticity
distributions to reveal the evolution and interaction characteristics of the unsteady vortical and turbulent structures
have not become available until recent years. By using modern whole-field flow diagnostic techniques like Planar
Laser Induced Fluorescence (PLIF), Particle Image Velocimetry (PIV), Stereoscopic Particle Image Velocimetry
(SPIV) and Dual-plane Stereoscopic PIV techniques, Hu et al. 9-12 conducted a series of studies to address the
evolution and interactive characteristics of various vortical and turbulent structures in lobed jet mixing flows
instantaneously and quantitatively.
2
American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit
AIAA-2005-0635
Jan 10 - 13 Jan 2005, Reno, Nevada
There is a great deal of interest in accurately simulating lobed mixing flows. One obvious reason for doing so is
that designing an optimum mixer/nozzle by experimental evidence alone can be expensive and nontrivial due to the
wide range of mixer parameters involved. By exploiting computational approaches, some of these costs can be
reduced dramatically. Furthermore, numerical simulations can provide detailed information of fluid quantities that
are often difficult or impossible to measure experimentally. Such quantities can be helpful to identify the underlying
physics and further our understanding of lobed mixer flows. Several computational studies of lobed mixing flows
have been conducted by solving the Reynolds-Averaged Navier–Stokes (RANS) equations. The previous studies
include the work of Kotumos and McGuick13, Tsui and Wu14, and Salman et al15.
It is well known that RANS equations introduce additional unknowns in that they require extra equations in
order to solve. These equations incorporate various assumptions in order to estimate the Reynolds stresses and form
a closed set of equations. Therefore, RANS simulations may require thorough validation against experimental data
in order to use the results to predict physically meaningful flows. Salman et al.15 simulated a lobed mixing flow
down stream a two dimensional lobed splitter plate by using a standard k-ε turbulence model to close the RANS
equations. Based on the validation of their simulation results against the experimental measurements of McCormick
and Bennett8, they evaluated the applicability of the standard k-ε turbulence model to predict the planar lobed
mixing flow. The results of Salman et. al.15 revealed that the standard k-ε model generally predicted flow structures
correctly albeit with lower magnitudes of momentum thickness and higher values of streamwise circulation.
In should be noted that many practical applications of lobed mixers are in the form of the circular nozzles instead
of two-dimensional splitter plates. When a two–dimensional lobed splitter plate is used, the vertical components of
the secondary flow velocities are greater than the horizontal components, whereas their magnitudes are comparable
in size for a three-dimensional situation16, as shown in the measurements of Hu et al.12. This effect could result in a
more complex evolution and interaction of vortex structures in lobed jet mixing flows that may be more numerically
challenging.
The main objective of the present study is to assess and quantify the applicability of numerically simulating
circular lobed jet mixing flows with a modest number of unstructured tetrahedral cells and commonly used
turbulence models. In the present paper, we will present our numerical simulation results of streamwise and
azimuthal (spanwise) vortex structures in a circular lobed jet flow by solving a three-dimensional incompressible
form of the Reynolds-Averaged Navier-Stokes (RANS) equations. The Reynolds stresses are modeled using four
widely used two-equation turbulence models: The k-ε standard turbulence model, k-ε Realizable turbulence model,
k-ω standard turbulence model, and k-ω Shear Strain Turbulence (SST) model. Numerical simulation results will be
compared with the Dual-plane Stereoscopic PIV measurement results of Hu et al.12 quantitatively to evaluate the
level of agreement between the predictions and the experimental results of the circular lobed jet mixing flow. The
applicability of the four widely used RANS turbulence models is assessed and quantified based on these
comparisons.
II.
Computational Modeling and Procedure
A. Geometry and flow conditions
Figure 1 shows the geometry of the circular lobed nozzle/mixer used in the present study. The nozzle inner
diameter (D) upstream of the lobed trailing edge is 40 mm. Six lobed structures with straight parallel walls form the
trailing edge of the test nozzle. Lobe width and height are 6 mm and 15 mm respectively. The nozzle has a
contraction ratio of 1.38 corresponding to exit-plane area of 908mm2. Outer and inner penetration angles measure
17° and 22° along the inner surface, and 9° and 28° from the outer surface due to thickness variation between
circular pipe and trailing edge. Reference velocity, U0 taken from Hu et al.12 is 20 m/s at the nozzle exit. The fluid
medium is air, and is passed through a settling chamber with honeycomb mesh structures prior to entering the test
nozzle. The Reynolds number (ReD) of the lobed jet flow is 5.478×104 based on nozzle inner diameter, D.
3
American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit
Jan 10 - 13 Jan 2005, Reno, Nevada
AIAA-2005-0635
Figure 1. Modeled nozzle/mixer geometry (dimension in mm)
B. Computational Modeling
Specific modeling assumptions are made to set
the initial and boundary conditions in the present
numerical simulation. The lobed jet mixing flow
is assumed incompressible based on maximum
experimental velocities recorded of 20 m/s at the
nozzle exit. The incoming velocity in the circular
pipe upstream the lobed trailing edge is calculated
by applying mass conservation of exit velocity
and nozzle contraction ratio. For the present
study, the inlet velocity is set to a constant 14.5
m/s one nozzle diameter upstream of the nozzle
exit plane. The turbulence quantities at the inlet
are estimated to be 3% of the incoming velocities.
The computational domain encompasses one lobe
period (60°) of the nozzle and extends four nozzle
diameters downstream of the exit-plane. Figure 2
shows the nozzle/mixer and the domain modeled.
Constant static pressure is maintained across the
outer domain extents with rotationally periodic
Figure 2. Computational domain and boundary conditions.
boundary condition imposed on each side.
In the present study, the numerical simulations
were conducted by using a commercial CFD solver, Fluent V6.1. The numerical scheme incorporated uses a threedimensional incompressible form of the Reynolds-Averaged Navier-Stokes (RANS) equations similar to Salman et.
al..15 The Reynolds stresses are modeled using four widely used two-equation turbulence models. The k-ε standard
turbulence model17, k-ε Realizable turbulence model, k-ω standard turbulence model, and k-ω Shear Strain
Turbulence (SST) models are chosen for in the present study to form a closed set of the RANS equations. A
4
American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit
AIAA-2005-0635
Jan 10 - 13 Jan 2005, Reno, Nevada
thorough description of the models and constants used is given in reference 18. The four turbulence models were
intentionally chosen to represent those commonly used in industry. The aforementioned models are preferred
because of low computational cost and adequate stability required for industrial applications.
The nozzle and domain are modeled and meshed using commercial software packages, Pro/Engineer and
Gambit. Figure 3 shows the tetrahedral surface meshes on the lobed surface. Unstructured tetrahedral cells are
attractive in circular lobed jet simulations as they can be generated quickly for complicated geometries typical of
lobed nozzle/mixer without creating highly skewed
cells. This is important when the goal of the study is
to determine the best lobe shape for a practical
application. It is noted that the cells are quite coarse
in the boundary layer and cannot physically resolve
the flow near the boundary; thus, it depends upon
standard wall functions incorporated into the
mathematical model to approximate the viscous
sublayer. The meshes were refined until no
significant changes between cases with different
mesh density were observed. The final mesh
number was 434,437 cells in the present study.
Computations were performed on a windows based
single processor Pentium 4 class 3.1Ghz
workstation with 3Gb of physical memory. Model
convergence is obtained within 12-15 hours (5000 –
7000 iterations) after initialization. Convergence is
determined through monitoring flow properties at
the exit plane and at the constant pressure
downstream boundary. Averaged velocity and
turbulent kinetic energy are recorded at these
boundaries and the solution is converged when
Figure 3. The tetrahedral surface mesh on the
values remain constant to at least five significant
lobed surface.
digits.
III.
Results and Discussion
Figure 4 to Fig. 6 compares the results of the present numerical simulations with k-ε Realizable and k-ω SST
turbulence models with the DP-SPIV experimental measurements of Hu et al.12 with respect to streamwise vorticity,
azimuthal (spanwise) vorticity and turbulent kinetic energy distributions. The standard forms of the each turbulence
model is found to show similar behavior, and therefore not presented here. It should be noted that the numerical
simulation results shown in the figures are created by duplicating and periodically rotating the computational
domain (periodic on 60° intervals) through a full 360°.
Figure 7 to 9 give a more detailed quantitative comparison between numerical predictions and the experimental
results. The maximum streamwise vorticity, maximum azimuthal vorticity and averaged turbulent kinetic energy is
plotted versus downstream distance and is used to evaluate the effect of each turbulence model on predicting the
measured quantities.
A. Streamwise vorticity distributions
From the mean streamwise vorticity distributions shown in Fig. 4, it can be seen that the numerical simulations
agree with the DP-SPIV measurement results quite well in revealing the evolution of large-scale streamwise vortex
structures generated by the special geometry of the lobed nozzle. The “signature” of the lobed nozzle is evident by
six pairs of large-scale counter rotating streamwise vortices that correspond to each of the six lobes at the nozzle
trailing edge. The streamwise vortex structures are found to grow in size, and expand radially as they travel
downstream. The strength, contrary to the size, decays rapidly with increasing downstream distance.
In the X/D=0.5 cross plane, both turbulence models predict a more elliptic pattern than the experimental results
which tend to bulge azimuthally near the lobe peaks. Values of maximum streamwise vorticity at the center of each
5
American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit
AIAA-2005-0635
Jan 10 - 13 Jan 2005, Reno, Nevada
streamwise vortex core predicted by numerical simulations are slightly smaller (within 10%) than the experimental
results, i.e. the numerical simulations under predict the mean streamwise vorticity slightly at this cross plane.
The “signature” of the lobed nozzle is still distinguishable in the mean streamwise vorticity distributions of both
numerical simulations and the experimental results in the X/D=1.0 cross plane. Strength of the streamwise vortices
for the numerical simulations is nearly identical to experimental results at this location. The streamwise vortices
predicted expand azimuthally and closely resemble those revealed in Hu et al.’s experimental measurements. The
numerical simulation results agree with the experimental data in predicting the streamwise vortex size as well as the
radially stretching of the streamwise vortices.
Both numerical simulation and measurement results show that the “signature” of the lobed nozzle is less
pronounced in the X/D = 2.0 cross plane compared with upstream sections. The numerical simulation results with
both k-ε Realizable turbulence model and k-ω SST turbulence model indicate similar streamwise vorticity strength
although they are found to be smaller in size in the k-ω SST model prediction. In this cross plane, the numerical
simulation results are found to agree with experimental results quite well in the terms of both vortcity strength and
distribution pattern.
B. Azimuthal (spanwise) vorticity distributions
In order to reveal the azimuthal vortex structures in lobed jet mixing flows more clearly, Hu et al.11 introduced a
quantity ω in− plane , which is defined as ω in − plane =
ω y2 + ωz2
. We adopted the same definition in the present
study in order to make a quantitative comparison of the present numerical simulations with Hu et al.’s experimental
results.
From the azimthal (spanwise) vorticity distribution shown in Fig. 5, it can be seen that the numerical simulations
agree reasonably well with the DP-SPIV measurement results of Hu et al.11 in predicting azimuthal vortex structures
in the lobed jet mixing flow. As it is expected, the azimuthal vortex structures form a vortex ring at the exit of the
lobed nozzle which appear to outline the lobed trailing edge in Fig. 5A. As suggested by McCormick and Bennett8,
the azimuthal vortex ring is pinched off (Fig. 5B) due to the interaction with the streamwise vortex structures in the
lobed mixing flow. This azimuthal ring is found to break down at further downstream locations. The broken
fragments of the azimuthal vortex ring corresponding to the lobe troughs reconnect to form a wreath-like-structure
in the center of the jet flows, while the broken fragments corresponding to the six lobe peaks form six “crescents”
surrounding the wreath-like-structure (Fig. 5C).
Compared with the experimental results, the k-ε Realizable turbulence model is found to over predict (about
10%) the maximum magnitude of the azimuthal vorticity in the X/D=0.5 cross-plane, while the k-ω SST turbulence
model under predicts by about 8% of the value. The locations of the maximum azimuthal (spanwise) vorticity are
found to be at lobe peaks and troughs, which are in good agreement with the experimental results. These results
actually verify the conjectures of Hu et al 8 quantitatively that the interaction of the azimuthal vortex structures with
the streamwise vortex pairs in lobed jet flows will result in uneven stretching of the spanwise vortices in lobe peaks
and troughs.
In the X/D = 1.0 cross plane, the magnitudes and locations of maximum spanwise vorticity are nearly identical
for the numerical predictions and the experimental measurements. Greatest azimuthal vorticity magnitudes remain
located at the lobe peaks and lobe troughs, while the azimuthal vortex ring is found to pinch off at each lobe peak.
The magnitude of the azimuthal vorticity is also shown to dissipate as it travels downstream. The k-ω SST
turbulence model is found to predict a faster dissipation of the azimuthal vorticity compared with k-ε Realizable
turbulence model. Therefore, in the X/D = 2.0 cross plane, the k-ω SST turbulence model predicts a smaller size for
the wreath-like-structure at the center of the jet flow compared with prediction of the k-ε Realizable turbulent model
and the experiment results.
C. Turbulent kinetic energy distributions
Figure 6 shows the results of the numerical simulations compared with Hu et al.’s experimental data for
turbulent kinetic energy distributions in the lobed jet mixing flow. Generally, the numerical simulations agree with
the experimental measurements globally to predict turbulent kinetic energy in terms of the distribution patterns and
their locations. However, the numerical simulations over predict the turbulent kinetic energy significantly in each of
the three investigated cross planes.
The turbulent kinetic energy distributions in the X/D = 0.5 cross plane retain the “signature” of the lobed nozzle
presented in above comparisons. As it is expected, the regions with higher turbulent kinetic energy are found to
concentrate in the mixing layers between the core jet and the ambient flow for both the numerical simulation and
6
American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit
Jan 10 - 13 Jan 2005, Reno, Nevada
AIAA-2005-0635
experiment results. The contour plots indicate distinct differences in magnitude between the two numerical
simulations and experimental measurements with the k-ε Realizable model closer predicting the measured turbulent
kinetic energy over the k-ω SST model.
The numerical simulation results in the X/D = 1.0 cross plane indicate high turbulent kinetic energy distribution
patterns in the same regions as the experimental results. Once again, the numerical simulations over predict the
magnitudes of the turbulent kinetic energy in this cross plane. It is found that the numerical simulation with k-ω
SST turbulence model predicts larger magnitudes of the turbulent kinetic energy compared with the k-ε Realizable
model. In the X/D = 2.0 cross plane, it is found that the regions with higher turbulent kinetic energy levels
concentrate more at the outer regions than at the inner side of the lobed jet flow. Both k-ε Realizable and k-ω SST
models over predict turbulent kinetic energy in the lobe peaks, while the k-ω SST model is also found to over
predict the maximum values at the interface of the core jet region.
D. The Maximum Streamwise Vorticity
The decay profile of the maximum streamwise vorticity in the lobed jet mixing flow versus downstream distance
is plotted in Fig. 7. Compared with the experimental data, the k-ω standard turbulence model is found to provide the
best prediction of the maximum streamwise vorticity near the nozzle exit (X/D = 0.25), which is within 2% of the
experimental value, while the other turbulence models over predict the maximum streamwise vorticity in the range
of 5% to 12%. It is found that the streamwise vorticity is under predicted by all the turbulence models in the regions
of 0.5 < X/D < 2.0 with the lowest amount of error given by k-ε Realizable turbulence model with an averaged value
of 11%. Conversely, the k-ω standard model, which predicts the first mean streamwise vorticity correctly, is found
to provide the worst prediction of the streamwise decay among the four investigated turbulence models, yielding
about 50% under prediction of the streamwise vorticity at the X/D = 2.0. Further downstream (X/D > 2.0), all the
turbulence models continue to under predict the measured values with an increasingly higher error margin. At
x/D=4.0, the predicted values tend to a single value approximately 93% lower than the measured values.
E. The Maximum Azimuthal (spanwise) Vorticity
The decay profiles of the maximum azimuthal vorticity values predicted by the four turbulence models along
with the experimental data are given in Fig. 8. All turbulence models tested over predict the maximum azimuthal
vorticity near the exit of the lobed nozzle. At the downstream region of X/D > 0.50, the k-ω standard turbulence
model, the k-ω SST turbulence model and the standard k-ε turbulent model provide almost the same predictions,
under predicting the maximum azimuthal vorticity by an average of 18%. Interestingly, the k-ε Realizable
turbulence model is found to stand out from other turbulence models by over predicting the maximum azimuthal
vorticity between 10% and 53% in the near field region (X/D < 2.0), but provides good predictions (within 5%) at
farther downstream regions of X/D > 2.0
F. Averaged Turbulent Kinetic Energy (K)
The change of the averaged turbulent kinetic energy in the lobed jet mixing flow with the downstream distance is
shown in Fig. 9. The absolute values of the numerical predictions and the experiment results are plotted in Fig. 9(a).
Fig. 9(b) shows the same data as those used in Fig. 9(b) but normalized by the maximum values for each data set.
From the Fig. 9 (a), it can be seen that all the four turbulence models over predict averaged turbulent kinetic energy
in the lobed jet mixing flow. At the regions near the exit of the lobed nozzle, the k-ω standard model predicts the
greatest magnitude of mass-flow averaged turbulent kinetic energy which deviates from the measurement data by
about 500% at the X/D = 0.25. Interestingly, the k-ω standard turbulence model gives the closest prediction of massflow averaged turbulent kinetic energy further downstream with the predicted values approaching the experimental
data downstream at X/D > 3.0. The predictions of other three models appear overlapped and follow a single profile
downstream of X/D > 2.5. Among the four investigated turbulence models, the k-ε Realizable turbulence model
provides the closest prediction to the experimental data. The k-ε Realizable turbulence model over predicts the
experimental data by approximately 50%.
From Figure 9(b), it can be seen that although all turbulence models over predict the averaged turbulent kinetic
energy in the lobed jet mixing flow significantly, numerical predictions and experimental results follow similar
trends with respect to turbulent kinetic energy production and decay with downstream distance. The averaged
turbulent kinetic energy is found to increase rapidly near the exit of the lobed nozzle, followed by a slower,
moderate growth rate to its maximum value. Once the averaged turbulent kinetic energy reaches its peak, it begins to
decrease slowly with increasing downstream distance. As suggested by Hu et al.12, such trend of the averaged
turbulent kinetic energy production and decay with downstream distance is closely related to the behavior of the
7
American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit
AIAA-2005-0635
Jan 10 - 13 Jan 2005, Reno, Nevada
steamwise vortices and azimuthal vortex structures in lobed mixing flows: The lobe nozzle/mixer generates largescale streamwise vortices at the downstream of the lobed trailing edge. Due to the “stirring effect” of the large-scale
streamwise vortices, the core jet flow and ambient flow mix intensely in the regions near the exit of the lobed
nozzle. The large-scale streamwise vortices generated by the lobed nozzle dissipate very rapidly as the downstream
distance increases (as shown in Fig. 7), i.e, the “stirring effect” of the streamwise vortices weaken. Therefore, the
growth rate of the averaged turbulent kinetic energy slows down, and reaches its peak value. Further downstream at
(X/D>2.0), the streamwise vortices have become so weak (less than 1/10 of those at the nozzle exit) that their
contribution to mixing is minimal. Since most of the turbulent kinetic energy in the lobed jet mixing flow has
dissipated due to intensive mixing upstream, the turbulent kinetic energy is found to decrease at further downstream
locations. Such trends of the averaged turbulent kinetic energy agree with the findings of McCormick and Bennett8
in a two-dimensional planar lobed mixing flow, who suggested that the majority of the enhanced mixing caused by
the lobed mixer occurs within the first six lobed heights.
From the Fig. 9(b), it is evident that the k-ε Realizable turbulence model provides the best prediction among the
four studied turbulence models. Compared with the experimental data, the k-ε Realizable turbulence model not only
predicts the location of the maximum averaged turbulent kinetic energy correctly, but also gives reasonable
estimation of its rapid growth to X/D = 1.5 ~ 2.0 and its slow rate of decay further downstream.
IV.
Conclusion
Numerical simulations of a circular lobed jet mixing flow by solving a three-dimensional incompressible form of
the Reynolds-Averaged Navier-Stokes (RANS) equations with unstructured tetrahedral cells were conducted to
assess and quantify its ability to simulate lobed jet mixing flows. The Reynolds stresses were modeled by using four
widely used turbulence models: k-ε Standard, k-ε Realizable, k-ω Standard, and k-ω Shear Strain Turbulence (SST)
models. The numerical simulation results were validated by using quantitative whole-field DP-SPIV measurement
results of the same circular lobed jet mixing flow to determine the level of agreement between the numerical
predictions and the experiment measurements. The applicability of the four commonly used RANS turbulence
models to lobed jet mixing flows was also evaluated based on the qualitative comparisons of the numerical
predictions with the experimental results.
The most notable result of the present study is that a RANS approach with a commonly used two-equation
turbulence model and a moderate density of unstructured cells can successfully predict many significant features
about the evolution and interaction of the large-scale streamwise vortices and azimuthal (spanwise) vortical
structures in lobed jet mixing flows. The numerical simulation results agree with the experimental data reasonably
well in terms of streamwise vorticity and azimuthal (spanwise) vorticity. Although all the four turbulence models
over predict the turbulent kinetic energy in the lobed jet mixing flow significantly, the numerical simulation
predictions follow a trend similar to the experimental results regarding the production and decay of the averaged
turbulent kinetic energy with downstream distance. The k-ε Realizable turbulence model is found to provide the
most accurate predictions among the four turbulence models investigated in the present study.
References
1
Paynter, G. C. and Birch, S. C., “An Experimental and Numerical Study of the 3-D Mixing Flows of a Turbofan Engine
Exhaust System,” 15th Aerospace Sciences Meeting, AIAA-77-204, Los Angeles, CA., 1977.
2
Rowe, R. K. and Huchar, A. P., “Engineering Efficient Engine (E3) Scaled Mixer Performance Report –Final Report,”
NASA CR-167947, No. 1982.
3
Paterson, R. W., “Turbofan Mixer Nozzle Flow Field Vol. 1- A Benchmark Experimental Study,” NASA CR-3492, 1982.
4
Paterson, R. W., “Turbofan Mixer Nozzle Flow Field – Benchmark Experimental Study,” Journal of Engineering for Gas
Turbines and Power, Vol. 106, July 1984, pp. 692-698.
5
Werle, M. J., Paterson, R. W., and Presz, W. M., “Flow Structure in a Periodic Axial Vortex Array,” 25th Aerospace
Sciences Meeting, AIAA-87-0610, Reno, NV., 1987.
6
Eckerle, W. A., Sheibani, H. and Awad, J., “Experimental Measurement of the Vortex Development Downstream of a
Lobed Forced Mixer,” Journal of Engineering for Gas Turbines and Power, Vol. 114, January 1992, pp. 63-71.
8
American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit
Jan 10 - 13 Jan 2005, Reno, Nevada
AIAA-2005-0635
7
Elliott, J. K., Manning, Y. J., Qiu, Y. J., Greitzer, E. M., Tan, C. S. and Tillman, T. G., “Computational and Experimental
Studies of Flow in Multi – Lobed Forced Mixers,” 28th AIAA/SAE/ASME Joint Propulsion Conference, Nashville, TN., 1992.
8
McCormick, D.C. and Bennett, J. C., “Vortical and Turbulent Structure of a Lobed Mixer Free Shear Layer,” AIAA Journal,
Vol. 32, No. 9, 1994, pp. 1852-1859.
9
Hu H., Kobayashi T., Saga T. and Taniguchi N., “PIV and LIF Measurements on the Lobed Jet Mixing Flows”, Experiments
in Fluids, Vol.29, No.7, 2000, pps141-157.
10
Hu, H., Saga, T., Kobayashi, T. and Taniguchi, N. , “A Study on a Lobed Jet Mixing Flow by Using Stereoscopic PIV
Technique”, Physics of Fluids, Vol.13, No. 11, 2001, pp3425-3441.
11
Hu, H., Saga, T., Kobayashi, T. and Taniguchi, N. , “Mixing Process in a Lobed Jet Flow”, AIAA Journal, Vol. 40, No. 7, ,
2002, pp1339-1345.
12
Hu, H., Saga, T., Kobayashi, T. and Taniguchi, N. “Simultaneous Measurements of All Three Components of Velocity and
Vorticity Vectors in a Lobed Jet Flow by Means of Dual-plane Stereoscopic Particle Image Velocimetry”, Physics of Fluids, Vol.
14, No. 7, 2002, pp2128-2138.
13
Koutmos, P., and McGuirk, J. J., “Turbofan Forced Mixer/Nozzle Temperature and Flow Field Modeling,” International
Journal of Heat andMass Transfer, Vol. 32, No. 6, 1989, pp. 1141–1153.
14
Tsui, Y.-Y., and Wu, P.-W., “Investigation of the Mixing Flow Structure in Multilobe Mixers,” AIAA Journal, Vol. 34, No.
7, 1996, pp. 1386–1391.
15
Salman, H., Page, G. J., and McGuirk, J. J.,“ Prediction of Lobed Mixer Vortical Structures with a k-ε Turbulence Model,”
AIAA Journal, Vol. 41, No. 5, 2003, pp. 878-887.
16
Belovich, V. M., and Samimy, M., “Mixing Processes in a Coaxial Geometry with a Central Lobed Mixer-Nozzle,” 34th
Aerospace Sciences Meeting, AIAA-96-0118, Reno, NV., 1996.
17
Launder, B. E. and Spalding, D. B., “The Numerical Computation of Turbulent Flows,” Computer Methods in Applied
Mechanics and Engineering, Vol. 3, 1974, pp. 269-289.
18
FLUENT, Software and Documentation, Ver. 6.1.18, Fluent, Inc, Centerra Park Lebanon, NH, 2003.
9
American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit
AIAA-2005-0635
Jan 10 - 13 Jan 2005, Reno, Nevada
`
a. The k-ε Realizable turbulence model
b. DP-SPIV measurement
A. X/D=0.50 cross plane
c. The k-ω SST turbulence model
a. The k-ε Realizable turbulence model
b. DP-SPIV measurement
B. X/D=1.00 cross plane
c. The k-ω SST turbulence model
a. The k-ε Realizable turbulence model
b. DP-SPIV measurement
C. X/D=2.00 cross plane
c. The k-ω SST turbulence model
Figure 4. Normalized streamwise vorticity distributions in the lobed jet mixing flow
10
American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit
AIAA-2005-0635
Jan 10 - 13 Jan 2005, Reno, Nevada
a. The k-ε Realizable turbulence model
b. DP-SPIV measurement
A. X/D=2.00 cross plane
c. The k-ω SST turbulence model
a. The k-ε Realizable turbulence model
b. DP-SPIV measurement
B. X/D=2.00 cross plane
c. The k-ω SST turbulence model
a. The k-ε Realizable turbulence model
b. DP-SPIV measurement
C. X/D=2.00 cross plane
c. The k-ω SST turbulence model
Figure 5. Azimutal (spanwise) vorticity distributions in the lobed jet mixing flow
11
American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit
AIAA-2005-0635
Jan 10 - 13 Jan 2005, Reno, Nevada
a. The k-ε Realizable turbulence model
b. DP-SPIV measurement
A. X/D=2.00 cross plane
c. The k-ω SST turbulence model
a. The k-ε Realizable turbulence model
b. DP-SPIV measurement
B. X/D=2.00 cross plane
c. The k-ω SST turbulence model
a. The k-ε Realizable turbulence model
b. DP-SPIV measurement
C. X/D=2.00 cross plane
c. The k-ω SST turbulence model
Figure 6. Turbulent kinetic energy distributions in the lobed jet mixing flow
12
American Institute of Aeronautics and Astronautics
43rd AIAA Aerospace Sciences Meeting and Exhibit
AIAA-2005-0635
Jan 10 - 13 Jan 2005, Reno, Nevada
Figure 7. Decay of the maximum streamwise vorticity
(a). Absolute values
Figure 8. Decay of the maximum azimuthal vorticity
(b). Normalized values
Figure 9. Average turbulent kinetic energy profile
13
American Institute of Aeronautics and Astronautics