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Elmiligui.PDF
Numerical Study of Flow Past a Circular Cylinder
Using RANS, Hybrid RANS/LES
and PANS Formulations
Alaa Elmiligui*
Analytical Services & Materials, Inc.
107 Research Drive, Hampton, VA 23666
Khaled S. Abdol-Hamid†
NASA Langley Research Center
Hampton, VA 23681
Steven J. Massey‡
Eagle Aeronautics, Inc.
13 W. Mercury Blvd., Hampton, 23669
S. Paul Pao§
NASA Langley Research Center
Hampton, VA 23681
Abstract
Two multiscale type turbulence models are implemented in the PAB3D solver. The
models are based on modifying the Reynolds Averaged Navier-Stokes (RANS) equations.
The first scheme is a hybrid RANS/LES model utilizing the two-equation (kε) model with a
RANS/LES transition function dependent on grid spacing and the computed turbulence
length scale. The second scheme is a modified version of the partially averaged NavierStokes (PANS) model, where the unresolved kinetic energy parameter (fk) is allowed to vary
as a function of grid spacing and the turbulence length scale. Solutions from these models
are compared to RANS results and experimental data for a stationary and rotating cylinder.
The parameter fk varies between zero and one and has the characteristic to be equal to one
in the viscous sub layer, and when the RANS turbulent viscosity becomes smaller than the
LES viscosity. The formulation, usage methodology, and validation example are presented to
demonstrate the enhancement of PAB3D's time-accurate and turbulence modeling
capabilities. The models are compared to RANS results and experimental data for turbulent
separated flows (TS) and laminar separated flows (LS) around stationary and rotating
cylinders. For a stationary cylinder, the TS case is accurately simulated using the general
two-equation kε turbulence model (eddy-viscosity model). PAB3D accurately predicts the
drag coefficient (CD), lift coefficient (CL) and the Strouhal number (St). The LS case was a
challenge for the RANS computation with an eddy-viscosity turbulence model. The
*
Senior Research Scientist, AS&M, Inc., 107 Research Drive, Hampton, VA 23666, AIAA Member.
† Aerospace Engineer, Configuration Aerodynamics Branch, M.S. 499, NASA/LaRC, Hampton, VA 23681, Associate Fellow
AIAA
‡
Senior Research Scientist, Eagle Aeronautics, Inc., 13 W. Mercury Blvd., Hampton, VA 23666 AIAA Member
§ Senior Research Scientist, Configuration Aerodynamics Branch, M.S. 499, NASA/LaRC, Hampton, VA 23681, Associate
Fellow AIAA
Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States
under Title 17, U.S. Code. The U.S. Government has a royalty-free license to exercise all rights under the copyright claimed
herein for governmental purposes. All other rights are reserved by the copyright owner.
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American Institute of Aeronautics and Astronautics
RANS/LES and PANS performed well and showed marked improvements over the RANS
solution. The modified PANS model was the most accurate. For the rotating cylinder, the
spin ratio varied from zero to one, and the PANS results were in good agreement with
published experimental data. RANS/LES and PANS capture both temporal and spatial
fluctuations and produce large-scale structures that do not occur in the RANS simulation.
The current results show promise for the capability of PANS in simulating unsteady and
complex flow phenomena.
I. Introduction
The limited capability of the Reynolds Averaged Navier-Stokes (RANS) approach combined with eddy-viscosity
turbulence models to simulate unsteady and complex flows has been well known for some time. The RANS assumes
that most of the energy is modeled through the turbulence transport equations and is resolved in the grid. RANS also
over predicts the eddy viscosity, which results in excessive damping of unsteady motion. Consequently, the eddy
viscosity of the unresolved scales attains unphysicaly large values suppressing most temporal and spatial
fluctuations in the resolved flow field. One of the approaches to overcome this problem is to provide the required
mechanism to resolve the largest scales of motion. Among several methods, the Detached Eddy Simulations (DES)
[1], hybrid Large Eddy Simulation (LES) [2-3], Limited Numerical Scheme (LNS) [4] and Partial Averaged NavierStokes (PANS) [5] are capable of providing the needed mechanism to satisfy this requirement. One of the major
deficiencies associated with the heretofore published use of hybrid schemes is that there is no clear identification of
the different flow regions. These regions need to be clearly defined as RANS regions and hybrid regions in order to
achieve complete simulation, independent of grid resolution. Several researchers observed that in most cases, using
hybrid methods, the use of fine grid might result in incorrect simulations. Abdol-Hamid and Girimaji [6] explored
new approach to improve the accuracy and robustness of creating a simulation of an unsteady flow field based on
the work by [6]. They accomplished that through the development and implementation of a novel two-stage
procedure to efficiently estimate the level of scale resolution possible for a given flow on a given grid for PANS and
other hybrid models.
PAB3D is a structured, multiblock, parallel, implicit, finite-volume solver of the three-dimensional RANS
equations, and advanced turbulence models are available in the code. PAB3D is widely used for internal and
external flow applications by NASA and by the US aerospace industry. Investigations in the area of unsteady flow
control for propulsion applications have led to an increased interest in upgrading PAB3D’s [7-8] time-accurate
capabilities. The current version of the PAB3D code has a second-order time accuracy algorithms scheme by
employing either physical time with sub-iteration or dual time sub-iteration [8].
In an attempt to increase the flow physics fidelity and the accuracy of PAB3D code, hybrid turbulence model
RANS/LES [2-3] has been added. An alternate new feature to PAB3D is the addition of the Partially Averaged
Navier-Stokes (PANS) method, which was suggested by Girimaji et al. [5]. The PANS model was developed to
overcome the grid dependency associated with the customary implementation of the hybrid RANS/LES method. The
addition of improved algorithms for second-order time accuracy, sub-iteration schemes, hybrid turbulence modeling,
and moving boundary conditions provide key modernizing enhancements to the code. The primary objective of this
paper is to assess the performance of these newly implemented approaches in a production CFD code.
The organization of the paper is as follows: The governing equations of the RANS, RANS/LES and PANS
formulation will be presented and discussed in detail. Computational results from RANS, RANS/LES and PANS for
a flow past a stationary and a rotating cylinder will be presented, and compared to experimental data. Flow around a
cylinder is considered as the test case for the hybrid turbulence model [9-13], because it is a basic engineering
problem and is inherently unsteady.
II. Approach
The governing equations of the RANS formulation include the conservation equations for mass, momentum,
energy, and the equation of state. In the present study, the perfect gas law is chosen to represent the air properties,
and the eddy viscosity concept is used to model the Reynolds stresses. The mass, momentum, and energy
conservation equations of the RANS equations can be written in a conservative form as follows:
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American Institute of Aeronautics and Astronautics
∂ρ ∂ρui
+
=0
∂t
∂xi
∂ρui ∂ (ρui u j + pδ ij ) ∂ ( τ ij − ρui u j )
+
=
∂t
∂x j
∂x j
∂ρe0 ∂ (ρe0 ui + pui ) ∂ ( τ ij u j − ρui u j u j ) ∂ ( qi + C P ρ ui θ) ∂
+
=
−
+
∂t
∂x i
∂xi
∂xi
∂xi
(1)

µ t ∂k 
µ l + σ ∂x 
k
i 

To close the RANS equations, the two-equation (kε) turbulence model is used:
c µ k 2 ∂k 
∂ρk ∂ρu j k
∂u
∂ 
+
= − ρ u j ui i +
µl +
 − ρε(1.+ Mτ2 )
∂t
∂x j ∂x j 
σ kε ∂x j 
∂x j
(2)

 ∂ k 2 
cµ k 2 ∂ε 
ε
∂ui ε ∂ 
∂ρε ∂ρu jε
+
= −Cε1ρ u j ui
+
 − f C˜ ρ ε − ν l 
µl +

σεε ∂x j  2 ε 2 k 
∂x j k ∂x j 
∂x j
∂t
 ∂n  
(3)
Cµ = .09, Cε 1 = 1.44, and Cε 2 = Cε 2 = 1.92




−3.41 
k2
f µ = exp
,
R
=
, f 2 = 1.− 0.3exp(−RT2 )
T
2
 RT  
µ tε
1+  
50 

The boundary conditions for ε and k at the wall are:
εwall
 ∂ k 2
= νl

 ∂n 
and kwall = 0. The turbulent stress components are formulated as:
2
3
ρ u j ui = 2 ρυ t S ji − δ ji ρk
1  ∂u
∂u  1 ∂u
S ji =  j + i  − δ ji j
2  ∂x i ∂x j  3 ∂x i
For the purpose of this paper, we will define RANS turbulent viscosity as
υ
RANS
t
= f µ ρCµ
k2
(4)
ε
A. Hybrid RANS/LES
Nichols and Nelson [2] give an example of a hybrid RANS/LES turbulence model. This method was
implemented in conjunction with Menter’s SST two-equation turbulence model and is termed a multiscale (MS)
model. In the present paper, the hybrid model is used with the two-equation model described in equations 2 and 3.
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The turbulent length scale, used in this implementation, is defined as
lt = max(6. υ t / Ω, k 3 / 2 /ε)
(5)
The sub grid turbulent kinetic energy is defined as
k LES = f d k
(6)
The damping function is defined as
f d = {1.+ tanh[2π (Λ − 0.5)]}/2
(7)
where,
Λ=
1
1
=
4 /3
1+ λ4 / 3
 lt 
1+  
∆ 
(8)
λ is the unresolved characteristic ratio, and
∆ = max(∆ x ,∆ y ,∆ z )
(11)
The eddy viscosity is then calculated from:
υ t = f d υ tRANS + (1.− f d )υ tLES
(10)
υ tLES = min(υ tRANS ,0.084∆ k LES )
(12)
Note that this hybrid model allows the transition from RANS to LES as a function of the local grid spacing and the
local turbulent length scale predicted by the RANS model rather than as a function of the grid spacing alone. This
allows the model to detect whether it can resolve the turbulent scales present on the existing grid before its transition
over to the LES mode.
B. PANS Approach
In its original form, PANS [5] replaces the two-equation turbulence model by solving for the unresolved kinetic
energy ku and the dissipation εu. The ku equation is identical to the original k equation. The dissipation equation has
only one major change through:
C˜ε 2 = f k (Cε 2 − Cε 1) + Cε 1
(13)
In the present work, we introduce an attempt to use a variable fk instead of a constant value. We use equation (7)
to compute fk as:
f k = {1.+ tanh[2π (Λ − 0.5)]}/2.
(14)
In this case, turbulent length scale is defined as:
lu = ku3 / 2 /εu , Λ =
1
1+ λ
4 /3
, and
λ=
lu
∆
The function in equation (14) has the characteristic to be equal to 1.0 in the viscous sub layer, as the unresolved
characteristic ratio tends to be of very small value. Also, the value of this function is restricted to 1, in case the
RANS turbulent viscosity becomes smaller than the LES viscosity (12).
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American Institute of Aeronautics and Astronautics
C. Boundary Conditions
In the present study, a characteristic Riemann invariant type boundary condition was used to model the far field
boundaries, while a periodic boundary condition was imposed in the longitudinal direction of the cylinder. A no-slip
velocity boundary condition was used at the cylinder surface so that the fluid would neither slip nor penetrate. For
the case of the stationary cylinder, there is no relative motion between the cylinder surface and the fluid, and the
velocity components were set to zero. For the rotating cylinder, the velocity at the cylinder surface was set to be
−
−
−
−
equal to ω × r where ω is the cylinder angular velocity and r is the position vector connecting the cylinder axis of
rotation to the cylinder surface as shown in Fig. 1a. The interface to the new rotating/spinning boundary condition
resides in the user.cont file. The user has to specify the angular velocity (rad/sec), the axis of rotation, and a point on
−
the axis. Vector r is the vector with the minimum distance between the rotating surfaces to the axis of rotation, and
is defined as:
d=
( x 2 − x1 ) × ( x1 − x0 )
x 2 − x1
where x1 & x2 are points on the axis of rotation while xo is a point on the surface of cylinder.
III. Results and Discussions
Flow past a stationary cylinder as well as a rotating circular cylinder, was computed to verify the time accuracy
of the code and of the relative advantage of the hybrid turbulence models RANS/LES and PANS. The twodimensional grid consisted of 32,256 cells and 24 blocks, and extended 15 diameters into the far field (Fig. 1b). The
three-dimensional grid was the same as the two-dimensional grid with the addition of 40 planes, which covered twocylinder diameter. The same grid was used for all runs, which gave a first grid y+ range of approximately .2 to 2.0.
The diameter of the cylinder D was 40 mm wide at Re = 50,000 and the Mach number for all cylinder cases was set
at a value of M = 0.3. A non-dimensional time step of 0.015 (based on free stream speed and the diameter of the
cylinder) was used for all cases. Based on the Strouhal number range and the time step used for the present cases,
approximately 350 data points in time per cycle of shedding was sampled. Four sub-iterations were used to reduce
the error. Each of the 2D simulations required approximately 3 hours using 24 (2.8 GHz P-4) computers. The 3D
cases each required approximately 48 hours using the same set of 24 computers.
A. Stationary Circular Cylinder in Cross-flow:
The shedding frequencies are determined from the lift coefficient (CL) fluctuation as it varies with time (Fig. 2).
The CL is obtained via the internal force and moment integration algorithm, Post [14]. The strategy for this case was
to first run the simulation at a very coarse grid level (1/4th in each direction) for 10,000 iterations to trigger its
asymmetric vortex shedding instability, and then refine the grid and run the solution for an additional 20,000
iterations. The solutions were averaged over the last 15,000 iterations (approximately 50 shedding cycles). The onset
of asymmetric vortex shedding is seen to occur just after the first 60 time unit, and the switch to fine grid is seen to
coincide with an increase in amplitude. It was observed that approximately 4 sub-iterations per physical time step
produced the optimal convergence per iteration. However, the physics of the specific problem will dictate the sub
iteration number for other cases. In the present results, four sub iterations typically reduced the residual by three
orders of magnitude at that time level, with no improvement using more iteration. The results were compared with
the results using up to 20 sub-iterations, with no substantial difference in the final results.
The Strouhal number (St) was captured with the use of the Power Spectral Density (PSD) of CL as shown in Fig.
3. To verify the capability of PAB3D for simulating unsteady flow problems, the Turbulent Separated (TS) case was
used, which simulates the experimental flow condition in which the boundary layer is tripped well ahead of
separation. Similar to what was found by other researchers [15-17], we achieved this objective numerically by
choosing a free stream turbulence level high enough to cause natural transition (5 times laminar viscosity). We
performed 2D-RANS computations with PAB3D using two-equation kε at the Reynolds number of 140,000. This
flow condition matched the conditions used by Travin et al. [15], Vatsa and Singer [16] and Hansen and Forsythe
[17]. The results from the work by Travin et al. [15] show that there are only small differences observed between the
2D RANS, 3D RANS and 3D-DES results. For this reason, we will not present any 3D results for this case. The
time-averaged surface pressure coefficients (CP) resulting from the 2D computations were compared with the
experimental data of Roshko [10] in Fig. 4. Table 1 compares the present PAB3D results using the 2D kε turbulence
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American Institute of Aeronautics and Astronautics
model with the 3D-DES studies done by Travin et al. [15] and Hansen and Forsythe [16] and the experiment of
Roshko [10]. Based on the results listed in Table 1 and the comparisons of Fig. 4, it was concluded that the current
RANS compared well with Roshko’s experimental data and with other CFD simulations.
Table 1. Re=140,000 TS Time-Averaged Results
Method
CD
Cpb
St
3D-DES Travin et al. [15]
0.57
0.65
0.3
Hansen and Forsythe [17]
0.59
0.72
0.29
PAB3D kε
0.62
0.68
0.27
Experiment
0.62-0.74
0.5-0.9
0.27
The laminar separated (LS) case was chosen to evaluate the implementations of both Hybrid RANS/LES and
PANS formulations into the PAB3D code, and to investigate their capability to simulate such flow. This case is
compared with the experimental data of Nordberg [18] (Re=3000 and 8000) and Cantwell and Coles [19]
(Re=140,000). Fig. 5 shows the PSD variation with St from RANS and PANS results. It is clear that RANS
produced a peak at only one major Strouhal number and no significant spectral energy at higher Strouhal numbers.
This result indicates that the 3D RANS simulation has a two-dimensional flow character. The vorticity plot shown in
Fig. 6 supports this conclusion as it shows the flow with no change in the Y-direction. On the other hand, the PANS
resulted in one major Strouhal number and several minor ones, indicating that more scales were resolved in this
simulation. Similar observations were made for the RANS/LES simulation. Fig. 7 shows the three-dimensional
character of the flow produced as a result of the PANS formulation. Table 2 shows the comparison between the
recent PAB3D simulations, Travin et al. [15] DES solutions, and the experimental data. The PANS formulation in
PAB3D produces the closest results as compared with the experimental data. The 2D PANS and LES results largely
differ from the experimental data, but are similar in character to the 2D-DES simulations of Travin et al. [17].
Table 2. Re=50,000 LS Time-Averaged Results
Method
CD
-Cpb
St
3D-DES Travin et al. [15]
1.27
1.28
0.21
2D-DES Travin et al. [15]
1.77
2.05
0.14/0.20
2D PAB3D RANS/LES
1.69
2.05
0.21
2D PAB3D PANS
1.67
2.1
0.22
3D PAB3D RANS/LES
1.0
0.9
0.22
3D PAB3D PANS
1.1
1.03
0.21
2D/3D PAB3D RANS
1.08
1.03
0.23
0.98-1.25
0.9-1.2
0.18-0.21
Experiment
Figure 8 shows the variations of CL with respect to time for the PANS formulation. The 3D PANS solution
displayed large and random variations, as a result of strong modulation of the vortex shedding. These observations
are similar to the ones reported by Travin et al. [15] and Vatsa and Singer [16], using 3D-DES formulation. The
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American Institute of Aeronautics and Astronautics
time-averaged surface pressure distributions from the 3-D simulations are compared with experimental data of
Nordberg [18] and Cantwell and Coles [19] in Fig. 9. In general, the hybrid RANS/LES and PANS formulations
produced comparable results to the 3D-DES solutions presented in Refs. [15 & 16].
Figure 10 shows the unresolved turbulent kinetic energy resulting from the RANS and PANS formulations. The
levels from the RANS simulation were ten times those produced by the PANS formulation. More energy was
resolved using the PANS formulation as compared with that produced by the RANS simulations. Fig. 11 shows the
time-averaged fk from the PANS simulations. The fk was as low as 0.3 in the region of interest where more energy
was resolved through the use of the PANS formulation. The smaller the fk value is, the more LES-like simulations
are produced. The closer fk is to a value of 1, the more RANS-like the simulation results. This occurs in the viscous
sub-layer and far field regions, as expected, where the grid resolution is enough for the use of RANS.
Rotating Circular Cylinder in Cross-flow:
The computational grid described earlier and shown Fig. 1b was used to compute the flow field around a rotating
cylinder. The flow field has been computed both in 2D and 3D modes using the standard kε turbulence model and
the hybrid turbulence models RANS/LES and PANS. Three main parameters govern the flow around rotating
cylinders: the Reynolds number, the ratio of peripheral to free stream velocity, and the cylinder aspect ratio.
In the present calculation, the rotational speed of the cylinder varied from 0 to 10000 rpm; the free stream Mach
number was 0.3 and the Re=50,000. The same solution strategy used to compute the flow field around the stationary
cylinder was used for the rotating cylinder cases. The variation in the time-averaged CL with respect to the spin ratio
(SR) is shown in Fig. 12a. SR is defined as the ratio between the peripheral velocity (u) of the cylinder to the free
stream velocity (U). The rotation of the cylinder generated lift, and CL increased with increasing SR. Comparison
between the present calculations and the results of Aoki et al. [12] and Tokumaru et al. [13] showed good
agreement. Simulations using 3D Hybrid RANS/PANS yielded results that are closer to the experimental data [12].
Variation of the CD with respect to the SR is shown in Fig. 12b. For the range of speeds computed in the present
study, the coefficient of drag tends to decrease with increasing SR. The St number for the rotating cylinder was
captured using the PSD of CL as explained earlier. The range of St for the rotating cylinder varied between the
values of 0.23 and 0.28. The variation of St with SR is shown in Fig. 12c. Increasing the SR tends to increase the St
number; however, there was no significant change in the St between the 2D and 3D calculations.
Figure 13 shows the time-averaged surface pressure distribution as a result of the 3D PANS simulation for the
SR of 0.0, 0.3, 0.6, and 0.9 respectively. For the stationary cylinder (SR = 0.0), Cp distribution is symmetrical
around the cylinder. As the SR increased, the pressure distribution became asymmetric due to the spinning, and the
fact that the net turning of the flow produced lift. Corresponding pressure contours are shown in Fig. 14. The free
stream flow over the top of the cylinder follows the induced flow from the spinning, while the free stream below the
cylinder is opposed by the induced flow. This result in an accelerated flow on the top half of the cylinder.
IV.Concluding Remarks
A hybrid turbulence model Reynolds Averaged Navier-Stokes/Large Eddy Simulation (RANS/LES) and Partial
Averaged Navier-Stokes (PANS) have been added to PAB3D code. The new capabilities improve the accuracy of
simulating an unsteady flow field. In this paper, a new approach to prescribe the unresolved kinetic energy
parameter (fk) function was proposed. The parameter fk is a function of length scale and grid size, which represents a
characteristic length scale. Some of the drawbacks of such a function are that it varies with time and space and that it
could be affected by grid resolution. The PANS approach is much simpler to implement in most CFD codes than
other approaches such as DES and the RANS/LES. The preliminary results provide a preview of the potential
capability of PANS in simulating unsteady and complex flow phenomena. The PAB3D code utilizing 3D PANS
captured both temporal and spatial fluctuations, and solutions compared well with the experimental data.
The turbulent separated (TS) and laminar separated (LS) flows were simulated for a stationary cylinder. The TS
case was accurately simulated using a general two-equation kε turbulence model (eddy-viscosity model) without any
enhancement. PAB3D accurately predicted the drag coefficient (CD) and the Strouhal number (St). The LS case
posted a great challenge for the eddy-viscosity turbulence models. The RANS/LES and the PANS approaches in the
CFD code provided much better predictions when compared to measured data. We have observed much better
predictions of CD, Strouhal number, and the surface pressure distribution than the results with the kε turbulence
model.
In the case of rotating cylinder, the present calculations indicate that with an increase of the rotational speed, CL
increases while CD decreases. The PANS provided the best comparison with the experimental data. The computed St
number is in the range of 0.22 to 0.28, which is in good agreement with previously published results.
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American Institute of Aeronautics and Astronautics
Future work will involve explorations of new approach to improve the accuracy and robustness of creating a
simulation of an unsteady flow field. This can be accomplishes through the implementation of a novel two-stage
procedure to efficiently estimate the level of scale resolution possible for a given flow on a given grid for PANS and
other hybrid models.
Acknowledgments
The first and third authors would like to acknowledge the support of NASA Langley Research Center for
providing the funding needed to carry out this work. The authors wish to thank Veer Vatsa of NASA Langley
Research Center for providing the grid and many helpful discussions. Furthermore, the authors thankful to Robert
Hansen of the United States Military Academy for providing some of the data presented in this work.
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Hansen, R. P., and Forsythe, J. R., “Large and Detached Eddy Simulations of a Circular Cylinder Using
Unstructured Grids,” AIAA 2003-0775, 2003.
Nordberg, C., “Effects of Reynolds Number and Low Intensity Free-Stream Turbulence on Flow Around a
Circular Cylinder,” Publication 87/2, Department of Applied Thermosciences and Fluid Mechanics,
Chalmers University of Technology, Gothenburg, Sweden, 1987.
Cantwell, B., and Coles, D., “An Experimental Study of Entrainment in the Turbulent near Wake of a
Circular Cylinder,” J. Fluid. Mech., Vol. 136, pp. 321-374.
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Figure 1a. Rotation Boundary Condition
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Figure 1b. Grid Distribution for Circular Cylinder (Ref. 11).
Figure 2. The Lift Coefficient (CL) Fluctuations with Time using 2D PANS Formulation.
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Figure 3. Power Spectral Density vs Strouhal Number using kε RANS
Formulation for Re=140,000.
Figure 4. Comparison between Coefficient of Pressure on the Cylinder Surface
using 2D kε RANS and Experimental Data of Roshko’s (TS Case).
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Figure 5. Power Spectrum Density vs Strouhal Number Results from PANS
and kε RANS Formulations (LS Case).
Figure 6. Three-Dimensional Vorticity Magnitude Results
from RANS Simulation (LS Case)
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Figure 7. Three-Dimensional Vorticity Magnitude Results from PANS
Formulation (LS Case).
Figure 8. Lift Coefficient (CL) Fluctuations with Time Results
from 3D PANS Formulation (LS Case).
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Figure 9a. Coefficient of Pressure on the Cylinder Surface Results from 3D kε RANS,
RANS/LES and PANS Formulation at Y=1.0 (LS Case).
Figure 9b. Coefficient of Pressure on the Cylinder Surface Results
from 3D PANS Formulation Compared with Experimental Data (LS Case).
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Figure 10a. Unresolved Turbulent Kinetic Energy, ku (m2/sec2),
Contours at Y=1.0 Produced from RANS Simulation (LS Case).
Figure 10b. Unresolved Turbulent Kinetic Energy, ku (m2/sec2), Contours at Y=1.0
Produced from PANS Simulation (LS Case).
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Figure 11. Averaged fk Contours at Y=1.0 from PANS Simulation (LS Case).
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Figure 12a. Lift Coefficient for Various Spin Ratios
Figure 12b. Drag Coefficient for Various Spin Ratios
Figure 12c. Computed Strouhal Number
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Figure 13. Effect of Spin Ratio on the Coefficient of Pressure
Figure 14. Averaged Pressure contours for 3D PANS for SR=0.0, 0.3, 0.6 and 0.9
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