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OCR for page 609
High-Incidence and Dynamic Pitch-Up Maneuvering
Characteristics of a Prolate Spheroid - CFD Validation
S. -E. Kim, S. H. Rhee,
(Fluent Inc., Lebanon, NH 03766, U.S.A.)
and D. Cokljat
(Fluent Europe Ltd., Sheffield, U.K.)
ABSTRACT
This paper presents the results of a computational
study of the flow around a 6:1 prolate spheroid at a
range of incidence angles and during a dynamic pitch-
up maneuvering. Several engineering turbulence mod-
els in popular use today are employed for turbulence
closure. Attempts are made to improve the perfor-
mance of second-moment closure models with modi-
fied length-scale equations. The computational results
are compared with the experimental data in terms of
crossbow separation pattern, pressure, skin-friction and
wall-shear angles on the body surface, and lift and pitch-
ing moment characteristics. The prediction accuracy
varies widely depending on the turbulence model em-
ployed. Considering the challenging nature of the flow,
the fidelity of the predictions shown by some turbulence
models such as Wilcox' k-ce model and the Reynolds-
stress transport models with modified length-scale equa-
tions are highly commendable.
INTRODUCTION
Despite its simple geometry, flow around a pro-
late spheroid in maneuvering carries a rich gallery ex-
hibiting a variety of complex three-dimensional tur-
bulent shear flows, featuring stagnation flow, highly
three-dimensional boundary layer under the influence
of strong pressure gradients and streamline curvature,
cross-flow separation, and formation of free-vortex
sheet and ensuing stream-wise vortices. All these fea-
tures of spheroid flows are the archetypes of flows
around airborne and underwater bodies at incidence or
in maneuvering, warranting an in-depth study.
Flow past prolate spheroids has been studied by many
others experimentally and numerically. Among the
most relevant to the present study are the early works
of Meter et al. (1984, 1986) and the more recent works
of the group at Virginia Polytechnic Institute (VPI)
(Chesnakas and Simpson, 1997; Wetzel et al., 1998;
Goody et al., 2000~. The series of experimental studies
conducted by these two groups provide most compre-
1
Fig. 1 Cross-flow separation and streamwise vortices on
a 6: 1 prolate spheroid at a = 30°: The pathlines are
computed using the present CFD solution.
hensive experimental data, revealing important physics
of the flow and offering an invaluable data-set useful
to validate computational fluid dynamics (CFD) codes.
On the numerical side, there have been quite a few
studies using reduced Navier-Stokes equations in ear-
lier days and, more recently, using full Naiver-Stokes
or Reynolds-averaged Navier-Stokes (RANS) equations
(Vatsa et al., 1989; Deng et al., 1990; Kim and Patel,
1991; Gee et al., 1992; Zheng et al., 1997; Rhee and
Hino, 2000~. Yet most of these numerical studies are
not quite up-to-date and comprehensive enough to give
a clear perspective as to what can be achieved by today's
CFD.
The present paper is geared toward contributing to the
ship hydrodynamics community a comprehensive sta-
tus report on the subject flow which will enable one to
gage what today's CFD has to offer in predicting the
turbulent shear flow around a 6:1 prolate spheroid. To
that end, we chose the case measured by the group at
VPI. The present study covers the steady flow for the
entire range of incidence angle experimentally investi-
gated (a = 10° ~ 30°), and the unsteady flow associated
OCR for page 610
with the dynamic pitch-up maneuvering as well. As
with other complex three-dimensional turbulent shear
flows, turbulence modeling plays a significant role, af-
fecting the prediction accuracy, especially in light of
the challenging features of the subject flow noted ear-
lier. In pursuit of the best possible predictions, sev-
eral most popular engineering turbulence models with
good track-records for similar flows were employed, in-
cluding an eddy-viscosity transport model (Spalart and
Allmaras, 1994), two different versions of k-ce models
(Wilcox, 1998; Menter, 1994), and second-moment clo-
sure (SMC) models. In addition to assessing the fidelity
of these models for the present flow, we will also ex-
plore some avenues to improving the performance of
the selected models. All the computations were carried
out using a finite-volume discretization based a RANS
solver (Mathur and Murthy, 1997; Kim et al., 1998~.
The paper is organized as follows. We start by look-
ing at some of the salient features of the flow and pon-
dering upon their implications to turbulence modeling
of the subject flow. The numerical results are utilized
to illustrate some of the significant characteristics of
the flow. This is followed by a brief overview of the
numerical method and turbulence models employed in
this study. Finally the computational results will be pre-
sented along with the discussions.
PREVIEW OF MAIN FLOW FEATURES
How can we characterize the present flow? A vi-
sual impression of the mean flow in question is aptly
portrayed by Figure 1, which was generated using the
numerical solution for the a = 30° case. The figure
serves nicely to highlight the most prominent features of
the flow, i.e. crossbow in the boundary layer, free vortex
sheet, and stream-wise vortices. The whole phenom-
ena depicted here are an embodiment of the so-called
"crossflow" or"open" separation, the significance of
which can be recognized from the fact that the struc-
ture of separation and its change with incidence angle
greatly affect maneuvering characteristics of the body
such as forces and moments acting on it. Another mo-
tivation to characterize the mean flow is that a good
understanding of it, if qualitative, often enables one to
surmise whether or not certain turbulence models will
be adequate for the flow at hand. In this vein, one of the
useful questions to ask for the present flow is: how the
mean flow deforms?
Much insight to the mean flow along this line can be
gained from contours of a normalized invariant of de-
formation tensor (Hunt, 1992) defined as:
SijSij - QijQij
SijSij + QijQii
2.95e~01
1.19~01
-5.76e-0.
-2.34e~1
-4.1 Oe-O]
-5.86e-01
-7.63e-01
1.OOe+O _ ~ _
.24c01~_
Prolate Spheroid at alpha = 20 deg.
Contours of deformation-ratio
Section at x/L = 0.772
Fig. 2 Contours of the deformation invariant (see Equa-
tion (1) for its definition) in the crossflow plane at x/L =
0.772 and a = 20° computed using the RSTM-2 result
where Si; and Qij are strain and rotation tensors
defined by Sij—(aui/axj+aui/Oxi)/2 and Qij—
(OUi/Oxj - OUj/Oxi) /2, respectively.
This invariant, which ranges between -1 and 1, is a
convenient measure of the relative importance of strain,
shear, and rotation. Note that ~ = 1 for pure strain,
Z) = 0 for pure shear, and Z) = - 1 for pure rotation.
Figure 2 shows the contours of this invariant in the
crossflow plane at x/l = 0.772 and the incidence angle
of a = 20°. According to the figure, the subject flow is
largely shear-dominated (~) ~ O) in the leeward bound-
ary layer. However, the flow becomes predominantly
rotational (1) ~—1) near the core of the stream-wise
vortices. Evidently, the mean flow portrayed here is
highly three-dimensional and carries significant extra
rates of strain and/or rotation, providing an acid test for
turbulence models. As regards the effects of rotation
on turbulence at the most fundamental level, say, ho-
mogeneous isotropic turbulence, it is a well-established
fact that rotation inhibits energy transfer from larger to
smaller eddies, decreasing the decay rate of TKE (Wige-
land and Nagib, 1978~. It is also well known that, when
a mean shear exists, rotation can either delay or acceler-
ate the energy transfer depending on the relative orienta-
tions of the mean shear and the rotation, attenuating or
accentuating turbulence accordingly. The well-known
effects of streamline curvature, either convex or con-
cave, which is also relevant to the subject flow, can be
explained in the same way.
How rapid the mean flow is strained, sheared or ro-
tatin~ is also of interest from a turbulence modeling
standpoint. Time-scale of mean flow (1/S or 1/Q)
normalized by turbulence time-scale (k/£), i. e., Sk/£
(S_ ~/~) and Qk/£ (Q_ ~/~), are good
measures for that. The contours of these "relative" strain
(1) and rotation in the crossflow plane at x/L = 0.772 are
2
OCR for page 611
3.24e+01
2.91e+01
2.59e+01
2.27e+01
1.94e+01
1.62e+01
1.30e+01
9.72e+0C
6.48e+0C
3.24e+0C
3.96e-04
6:1 Prolate Spheroid at alpha = 20 deg.
Contours of relativ~strain
AIL = 0.772
Fig. 3 Contours of relative strain (Sk/) at x/L = 0.772
(a = 20°) computed using the RSTM-2 result
1 .23e+01
1.1 1e+01
9.83e+0C
8.60e+0C
~~ .~.
7.37e+0t
6.1 4e+0C
4.91 e+OC
3.68e+OC
2.46e+00
1.23e+00
2.01 e-04
6:1 Prolate Spheroid at alpha = 20 deg.
Contours of relative-rotation
x/L = 0.772
Fig. 4 Contours of relative rotation (Qk/~) at x/L = 0.772
(a = 20°) computed using the RSTM-2 result
shown in Figure 3 and Figure 4. Notwithstanding the
lack of absolute accuracy of these quantities, the level of
these two quantities (Sk/S, Qk/e >> 1) as shown in the
figures suffices to indicate that the subject flow is dis-
torted quite rapidly. This has negative implications for
almost all aspects of turbulence modeling. Both eddy-
viscosity model (EVM) and SMC suffer when the mean
flow under consideration is rapidly sheared, strained, or
rotating (Speziale, 1999~.
All these considerations on the mean flow set the
stage for the following question: how far the turbulence
in a given flow is from equilibrium? It is an important
question to ask, inasmuch as almost all the phenomeno-
logical turbulence models in use today, including most
sophisticated SMC-based models, rely upon an equilib-
rium assumption in one way or another. Equilibrium
refers to a state where production of turbulent kinetic
energy (TKE), which is an event involving larger ed-
dies, is balanced by its dissipation rate occurring at
small scales. In highly complex turbulent flows like the
present one, however, an equilibrium state is less than
likely to be attained. Several mechanisms are respon-
sible for causing nonequilibrium of turbulence. In the
present flow, the salient features of the present flow dis-
cussed so far act to make the turbulence to depart from
equilibrium. It is surmised that rapid strain and rotation,
streamline curvature, and adverse pressure gradient in
the boundary layer play significant roles. Thus, tur-
bulence models that are better capable of representing
these physics and their effects on turbulence are likely
to be more successful for the present flow than others
which do not.
The experimental studies cited in the beginning ex-
tensively discuss and provide valuable insight into the
salient features of the present flow. Among many oth-
ers, we find particularly noteworthy the discussion by
Chesnakas and Simpson (1997) regarding turbulence
anisotropy based on their measurements of the individ-
ual Reynolds stresses and the mean velocity field. By
analysing the stresses and mean rates-of-strain, they at-
tempted to assess the validity of isotropic eddy-viscosity
assumption for the present flow. They found that the
Reynolds stresses are largely aligned with the strain
rates inside the boundary layer at a low incidence angle
(a = 10°~. However, they become grossly misaligned
almost everywhere else, especially along the free vortex
sheet and near the vortices on the leeward side of the
body and at high incidence angle. As they concluded,
this suggests that turbulence models based on isotropic
eddy-viscosity are likely to perform poorly, warranting
use of SMC.
All these observations and thoughts influenced our
overall turbulence modeling strategy adopted in this
study; the model selection and the rationales behind
our attempts to modify some turbulence models.
MATHEMATICAL MODELING
Governing Equations - Ensemble-Averaged
Navier-Stokes (EANS) equations
The present study adopts ensemble-averaged Navier-
Stokes equations as the governing equations, since the
salient features of the flow we are mainly interested in
can be found in averaged or mean flow. It should be em-
phasized that the ensemble averaging retains the local
time-derivative of the averaged velocity and other scalar
fields. Use of unsteady ensemble-averaged Navier-
Stokes equations, therefore, enables one to tackle flows
where there are significant temporal variations of aver-
aged flow fields such as alternate vortex-shedding be-
hind bluff-bodies that can occur in the present flow at
high incidence. In spite of its relevance to the present
flow at high incidence, vortex shedding will not be ad-
dressed in this paper. Yet the unsteady EANS equations
OCR for page 612
are needed to compute the unsteady flow due to the dy-
namic pitch-up motion.
To simulate the dynamic pitch-up motion, we solved
the governing equations in a non-inertial coordinate sys-
tem which rotates with the body at the prescribed angu-
lar velocity. Use of non-inertial frame greatly simplifies
the application of boundary conditions. However, the
body-force terms arising from the use of non-inertial
frame of reference should be accounted for. For the
dynamic pitch-up maneuver considered here, the body-
force terms are due to Coriolis, centripetal and angular
accelerations and can be written as:
fb = - 2Q x V—Q x (Q x r)— i, x r (2)
where r and V are the the displacement and the fluid
velocity vectors in the non-inertial (rotating) coordinate
system, and Q is the angular velocity of the non-inertial
coordinate system.
TURBULENCE MODELS
Among many choices, we screened the turbulence
models based on their relevance and track-records for
aerodynamics applications. So the k-£ models are omit-
ted in the paper. Table 1 summarizes the models chosen
for the paper. The list includes four EVM and three
SMC models. The details of the models can be found
in the cited papers.
Abbreviation Description
SA Eddy-viscosity transport model
(Spalart and Allmaras, 1994)
SST Blended k-m Ik-£ (Menter, 1994)
KO-1 High-Re k-m (Wilcox, 1998)
KO-2 Low-Re k-o (Wilcox, 1998)
RSTM-1
Second-moment closure
(Gibson and Launder, 1978)
Second-moment closure
Shih et al.'s (1995) £ equation
Second-moment closure with
Durbin's (1990) £ equation
RSTM-2
RSTM-3
Table 1 Turbulence models used in this study
Eddy-Viscosity Transport Model of Spalart and
Allmaras
In the Spalart-Allmaras (SA) model, one directly
solves the transport equation for an effective viscos-
ity, v (Spalart and Allmaras, 19941. The SA model
has become rapidly popular especially in the aerospace
community due to its commendable performance for
boundary layer flows subjected to adverse pressure gra-
dient.
The SA model used in this study is identical to the
original model except one thing. In the SA model. the
production of v is computed by:
GV=PCb~SV
where Cal is a model constant, and v the effective
cosity. In the original model, S in Equation (3) is com-
puted from a modulus of rotation rate tensor as:
(3)
vis-
..
S= x/2QijQij _ Q
(4)
For thin boundary layer flows, it would be of no con-
sequence whether the modulus of rotation-rate tensor or
that of strain-rate tensor is used. However, adopting the
vorticity magnitude has a potential to cause problems in
swirling or vortical flows like the present flow where ro-
tation dominates over strain in the vicinity of the cores
of the streamwise vortices, as shown earlier (Figure 2~.
The original S-A model indeed performed very poorly
for the present flow. To avoid this, we adopted what
Dacles-Mariani et al. proposed, i. e.,
3 = Q + Cv min (O. S—Q) (5)
where S(_ >/~35) and Q—(~) are the
moduli of strain-rate and rotation-rate tensors, respec-
tively, with Cv being an adjustable constant of an order
of 1. We simply took the value Cv = 2.0 from the cited
reference.
k-m Models
We adopted here three variants of k-ce models in pop-
ular use today. The first two k-m models, denoted as
KO-1 and KO-2 in Table 1, are based on the recently
revised version (Wilcox, 1998~. The difference between
the original model (Wilcox, 1988) and the revised one
lies in the "shear-correction" and "vortex-stretching"
terms which were added in the new model to improve
the model performance mainly for free shear flows such
as far-wakes, mixing layers, and jets. The new model
also has a re-calibrated low-Reynolds number model de-
signed to account for transitional effects. KO-1 and KO-
2 refer to the revised Wilcox' models without and with
the low-Reynolds number modification, respectively.
Another variant of k-m model adopted in this study is
Menter's k-m model (Menter, 1994) often referred to
as shear-stress transport (SST) k-m model in the litera-
ture. The SST model is essentially a"two-zone" model
that blends a variant of k-m model in the inner layer
and what is tantamount to a traditional k-£ model in the
outer layer. In addition, the SST model clips turbulent
viscosity based on the argument that the structural sim-
ilarity between k and ~uv~ (k/~——al = 0.3) should be
4
OCR for page 613
preserved, which can be used to set an upper limit on
turbulent viscosity as:
vt = min ( k, at k) (6)
It should be noted that, in the SST model, this limit
is applied within the boundary layer only. As for the
present flow, the leeward side of the flow will be mostly
unaffected by the clipping, except the near-wall re-
gion. And the model essentially reduces to a traditional
k-£ model there.
Second-moment closure models
The baseline model of the Reynolds-stress transport
models (RSTM) used in this study is largely based on
the Rotta's model (Rotta, 1951) for the slow redistribu-
tion term, isotropization of production (IP) model of Fu
et al. (1987), and the wall-reflection model of Gibson
and Launder (1978~. The baseline model, which will be
called RSTM-1 hereafter, was implemented in an un-
structured mesh based finite-volume RANS solver, and
has been been validated for a number of complex three-
dimensional internal and external flows (Kim, 2001;
Kim, 2002~. The unique features of the implementa-
tion include: an isotropic turbulent diffusion models
for Reynolds-stress and dissipation equations, a high-
order dissipation term designed to prevent decoupling
of Reynolds stresses, and mean velocity field arising
from co-located, cell-centered finite volume discretiza-
tion scheme. The wall-reflection effects in the pressure-
strain correlation were included with the aid of a wall-
proximity function that allows wall distance and wall
normals to be computed for arbitrary wall configura-
tions.
In the RSTM-1 model, turbulence length-scale is ob-
tained by solving the transport equation for £ given by:
D£ a + _ + c£ —Pii——C£2 —
Dr axj [( ~£) axj] 2 k P k
(7)
where Pij is the production of Reynolds stresses, ~£ =
1.3, Cal = 1.44, C£2 = 1.83.
Despite more-than-modest improvements over linear
k-£ models, the performance of the RSTM-1 model fell
short of our expectation based on the remarkable perfor-
mance of the same model for a similar flow (Kim, 2001~.
Its rather disappointing performance begs a question as
to what the main culprits could be. Among many leads
alluded to earlier, including the shortcomings of the
SMC itself for non-equilibrium flows like the subject
flow, we decided to track down the £-equation. As dis-
cussed earlier, the boundary layer and free vortex sheet
with large shear/strain and strong rotation make con-
ventional length-scale equations, i. e., the £-equation
in Equation (7), less than adequate for the present flow.
To investigate the impact of the length-scale equation on
the prediction, we adopted here two alternative £ model
equations. One of them is the £-equation proposed by
the turbulence modeling group at NASA Glenn (Shih et
al., 1995), which was developed starting from an exact
equation for mean-square vorticity fluctuation (i).
This enstrophy-based £ equation has been used mostly
in the context of two-equation k-£ models, except for the
study by Luo and Lashiminarayana (1997) who adopted
it in conjunction with a RSTM to study duct flows. They
reported that this new £ model equation was capable of
predicting the turbulence enhancement observed in the
boundary layer near concave walls. The new £ model
equation was also claimed to better describe the process
of vortex stretching and spectral energy transfer, and
has actually been found to perform significantly better
than traditional k-£ models for boundary layers involv-
ing rapid strain and severe adverse pressure gradient.
All these benefits seem relevant to the present flow. The
new £-equation reads:
PDt aXi [(lo 6e) axj] P ~ P 2k+
(8)
where 5 - >/~), and
Cat = max [0.43, '~ + 5], 11 = Sk/£
C2 = 1.9, C}k = 1.0, 6£ = 1.2
In the RSTM-2 model, the standard £-equation in the
baseline model was replaced by the new £-equation in
Equation (8~.
The strong non-equilibrium turbulence in the present
flow prompted us to look for models known to better
deal with that. And we employed the modification pro-
posed by Durbin (1990) and Chen and Kim (1987) as
the second alternative. The modified £-equation is given
by:
Ds a + · _ + c£~Pk——C£2 — t9'
Dt axj [( ~£) axe] k P k
where C,£ = 1.3, C£2 = 1.83, and Call is computed from:
Cal = C£i (1 +aPk/£) (10)
where Cal = 1.4 and a = 0.05. As can be noted, the
model parameter, Call in the "production-of-dissipation"
term is a function of Pk/~. This term adds more dissipa-
tion as production of TKE becomes larger than dissipa-
tion rate, suppressing spuriously large TKE frequently
encountered in complex flows. The baseline model with
the standard £-equation replaced by Equation (9) and
Equation (10) is denoted as RSTM-3 in this paper.
s
OCR for page 614
Near-wall treatment
The choice of near-wall treatment depends on the res-
olution of the near-wall mesh in use. For the present
study, we employed both fine and coarse meshes. The
fine meshes are such that the entire near-wall region is
resolved down to wall including viscous sublayer. SA,
SST, KO-1, and KO-2 models, all of which can be nat-
urally integrated to wall, were run on the fine meshes.
When fine meshes are used, the wall boundary con-
ditions for the mean velocity and turbulent quantities
essentially exploit no-slip condition at walls. For ce, we
"fix" the asymptotic value of ce as y ~ O at wall-adjacent
cells, using:
6v
pyp2
where yp is the distance from the wall to the cell center,
andp=0.075.
The coarse meshes were designed to skip the
viscosity-affected region and to place the first grid (cell-
center) points in fully turbulent region. We then can
employ wall functions, namely, the law-of-the-wall and
related hypotheses, to derive the wall boundary condi-
tions for the mean velocity and turbulence quantities
(Kim, 1998; Kim, 2001) such as Reynolds stresses and
co. All RSTM computations were made on the coarse
mesh only. Among the EVMs, SST and KO-1 models
were run on the coarse mesh using the wall functions.
NUMERICAL METHOD
A cell-centered finite-volume method is employed
along with a linear reconstruction scheme that allows
use of computational elements (cells) with arbitrary
polyhedral topology, including quadrilateral, hexahe-
dral, triangular, tetrahedral, pyramidal, prismatic, and
hybrid meshes. The velocity-pressure coupling and
overall solution procedure are based on SIMPLE-type
segregated algorithm adapted to unstructured mesh. The
convection terms are discretized using a third-order
upwind scheme, and the diffusion terms using cen-
tral differencing scheme. The high-order terms are
treated using a deferred correction approach. The dis-
cretized algebraic equations are solved using a point-
wise Gauss-Seidel iterative algorithm. An algebraic
multigrid method is employed to accelerate solution
convergence. The details of the numerical method are
described by Mathur and Murthy (1997), Kim et al.
(1998), and Kim (2001~.
COMPUTATIONAL DETAILS
Body geometry and computational conditions
The experiments conducted at the Virginia Poly-
technic Institute with 1.37m-long model of a 6:1 pro-
late spheroid are simulated in the computations. The
Reynolds number (Re~) based on the freestream veloc-
ity (Uo) and the body-length (L) is 4.2 x 106. In the ex-
periments, the body was supported by a sting mounted
at the rear-end of the body. The body was also mounted
a trip at x/L = 0.2 to help trigger laminar-to-turbulent
transition. The sting-mount were not modeled in the
computations. However, we attempted to mimic the ef-
fects of the trip using a numerical trigger. More on this
will be discussed shortly in the beginning of the "Re-
sults" section.
The coordinate system adopted in this study is such
that the positive x-axis is in the streamwise direction,
`11y y points to the upward vertical direction, and the x—y
plane makes the vertical symmmetry plane. The origin
of the coordinate system is placed at the fore-end of the
spheroid.
Solution domain, mesh, and boundary conditions
The solution domain covers—2.75 3.425
and—2.75 2.75, in the streamwise and
lateral direction, respectively. Single-block hexahedral
meshes are used. Four different sizes of mesh were
used to see the effects of different near-wall modeling
strategies (wall function vis-a-vis near-wall-resolving
approach) and also to ensure mesh-independency of
the numerical solutions. Two of them (540,000 cells,
1,380,200 cells) were designed to penetrate the viscous-
sublayer. Two other meshes (345,000 cells, 1,120,000
cells) were made for the wall function calculations. For
each pair of meshes, mesh-dependency of the numerical
solutions were found to be insignificant. Therefore, we
present here the results for the respective coarser meshes
: 345,000-cell mesh for wall function calculations, and
540,000-cell mesh for the near-wall model calculations.
The y+ values at the wall-adjacent cells of the fine and
the coarse meshes were near y+ = 1.0 and y+ = 30, re-
spectively, over most of the body surface.
The symmetry of the geometry and the flow allowed
us to model only a half of the domain. Thus, the domain
boundary consists of the body surface, upstream/far-
field inlet, vertical plane of symmetry, and exit bound-
ary. The wall boundary conditions were applied as de-
scribed earlier. On the inlet boundary, freestream condi-
tions were specified. On the exit boundary, the solutions
variables were extrapolated.
The steady computations were carried out for three
incidence angles, namely, a = 10°,20O,30°. The nu-
merical solutions were deemed converged when scaled
residuals for all solution variables drop by four orders
of magnitude. The lift and pitching moment were also
monitored to ensure full convergence of the solutions.
6
OCR for page 615
RESULTS
Effects of boundary layer tripping
To mimic the effects of the physical trip mounted at
x/L = 0.2, we employed a numerical trigger by which
the flow in the upstream of the trigger was treated as
being laminar, whereas the turbulence models were en-
forced to become fully effective right at the trigger.
Several runs were made with the k-co models with the
numerical trigger. However. the numerical results did
not show any meaningful differences from the fully-
turbulent results in terms of the global features of the
flow such as the surface quantities and the lift and mo-
ment, except a very small region in the laminar region.
Thus, we made all the subsequent computations assum-
ing that the flow is fully turbulent over the entire body,
which are presented in the paper.
Crossflow separation pattern
Line of crossbow separation is usually defined as a
(hypothetical) line in a given wall-shear vector field
onto which neighboring wall limiting streamlines con-
verge. The separation line can be visualized by sur-
face flow visualizations, as long as convergence of the
wall limiting streamlines is strong enough. The VPI
group looked at several other indicators (Chesnakas and
Simpson, 1997; Wetzel et al., 1998) and studied the
correlations between those and the separation location
determined from the flow visualization. One of the
proposed indicators was the location of minimum wall-
shear (or skin-friction). The locations of skin-friction
minima were found near the separation yet consistently
on the further leeward side of the separation line.
For comparison with the experimental data available
at x/L = 0.729, two circumferential angles were read
off from the numerical results. One is the angle at which
the skin-friction becomes a minimum, which enabled us
to make direct comparison with the measurement. Fig-
ure 5(a) shows the results. It should be mentioned that
all the EVM results shown here were obtained on the
fine mesh, whereas the RSTM results are based on the
coarse mesh. Quite a large scatter is observed among
the results, with SA, KO-2 and RSTM-2 deviating far-
thest from the experimental data. In the experiment, the
location where the circumferential velocity changes its
sign (direction) was found closer to the actual separation
location. Figure 5(b) shows the angles thus-determined
from the numerical results, along with the separation lo-
cation determined from the oil flow visualization. In
comparison to the minimum Cf angles in Figure 5(a),
the angles determined this way shift leeward by a sub-
stantial amount. The overall trends shown by the differ-
ent turbulence models remain largely unchanged from
what were seen earlier. Particularly noteworthy is that
KO-2 and RSTM-2 models better predict the actual sep-
aration locations than the locations of minimum Cf. It
is also seen that the SA and SST models continue to
predict a delayed separation.
The experiments also reveal that, at higher incidence
angle (e.g., a = 20°), another separation line emerges.
The numerical results from some of the turbulence mod-
els also showed a clear evidence for the secondary
separation. Figure 6 depicts the limiting wall stream-
lines numerically visualized using the result of KO-2
model. Note that the body in the figure was rotated
around its axis to provide a better view of the surface
flow pattern. Clearly visible toward the leeward sym-
metry plane is a line starting from a little downstream
of x/L = 0.6, onto which the neighboring streamlines
converge. By x/L = 0.772, the secondary separation
line becomes full-fledged, as Chesnakas and Simpson
(1997) described in their paper. In addition, the diverg-
ing streamlines between the two separation lines imply
that the flow re-attaches to the body surface along a line
(denoted as reattachment line in the figure). The reat-
tachment line is located much closer to the secondary
separation line. All these features predicted by the
KO-2 and RSTM models are in good accordance with
the experimental observations (Chesnakas and Simpson,
1997~.
Surface quantities
The behaviors of the surface quantities like pres-
sure (Cp _ (p—po)/0.5pU02~' skin friction (Cf _
~w/0.5pU02), and wall-shear angle flew—arctan Wu ~ carry
rich information not only on the surface flow itself but
off-the-wall structures like the free vortex layer and
streamwise vortices. At a = 10°, the lowest incidence
angle we computed in this study, the crossbow and
resulting separation are rather weak. And the differ-
ences among the turbulence models were found to be
also small, although the differences become more dis-
cernible as one goes to the rear and leeward side of
the body. Although not shown here, SA model grossly
under-predicted the circumferential variations of all sur-
face quantities, while the KO-2 model tends to overpre-
dict them. Overall, the results of the SST and KO-1
models were about right. The behaviors of the predicted
surface quantities appeared to be largely consistent with
the experimental observations with regard to the incipi-
ent crossbow at x/L = 0.6 and the fully-developed sep-
aration line at x/L = 0.772.
The surface quantities exhibit much richer features at
or = 20°. And the differences among the models also
become far more noticeable. Figure 7, Figure 8, and
Figure 9 depict the circumferential distributions of Cp,
Cf. and Do at a = 20°, respectively. It should be noted
that the results shown in these figures were obtained
7
OCR for page 616
(a) Circumferential angle of minimum Cf
180
leeward
be
150
120
. . Measured (men. Cf)
0 SA
° SST
KO-
~ KO-2
· RSTM-1 (wall fn.)
o RSTM-2 (wall fn.)
· RSTM-3 (wall fn.)
7V5 10 15 20 25 30 35
Angle of attack (deg.)
(b) Circumferential angle of circumferential vet. sign change
180
1 L L. I ~ .
- . ~ · Oil flow
t ~ r ~ SST
. . . · KO-1
. . . '` Key
~ ~ , ~— ,,
r ~ RSTM-1 (wall fn.)
~ To i 0 RSTM-2 (wall fn.)
-:~ i-: :~ :::::::-::::: · RSTM-3 (wall fn.)
. . .
m I I
120 —~ —i— — Y —
90 . ~ t i
5 10
15 20 25
Angle of attack (deg.)
O
-0.1
30 35
Fig. 5 Change in circumferential location of crossflow 0 3
separation with incidence at x/L = 0.729 determined by
(a) minimum Cf (b) circumferential velocity
using the fine (near-wall-resolving) mesh. The exper-
imental data (Chesnakas and Simpson, 1997; Wetzel
et al., 1998) show that the primary vortex is located
near ~ = 158° in the crossbow planes at both x/L = 0.6
and x/L = 0.772. This is manifested by the conspic-
uous dips in the measured Cp data shown in Figure 7.
The predictions widely vary among the models. The
SA model barely captures the fetaure, giving the shal-
lowest dip and yielding the minimum pressure location
more leeward than all the other models and the data.
The KO-2 model seems to closely capture both the lo-
cations and magnitudes of the minimum Cp. KO-2 also
best reproduces Me Cp-plateau seen right after the pn-
mary separation which was found in the experiments to
occur near o = 123° and o = 112° at x/L = 0.6 and
x/L = 0.772, respectively. Furthermore, atx/L = 0.772,
the KO-2 model appears to capture, far closer than other
models, the small kink caused by a secondary vortex
above the surface, which was found to occur at o = 140°
secondary
_ /
y separation line
reattachment line
windward
AL 0.600 x/L = 0.772
primary separation line
Fig. 6 Wall limiting streamlines showing the pattern of
the crossflow separation at a = 20° - based on the KO-2
prediction
(a) X/L = 0.600
-'~'~1I,I,,I,,I,,
O Measured
_ -- SA
SST
--- KO-1
KO-2
\
, , I , . 14/° . I . , I
04
~ 0 30 60 90 120 150 180
1.',
/ ,,
1.~;'
/.',`,o
/,
/.'
(b) x/L = 0.772
0~
-0 1
.
oC-0.2
-0.3
id- ~ ~ ~ P
art''''`"''!
an,
I ~ ~ I i I ~ I
O Measured
SST
--- KO-1
KO-2
W~
_ ~
: ~
-04 ,, 1, , I,,,,, I, , 1 . .
0 30 60
%. ¢r'
~Y
1!
~0_
90 120 150 180
¢(degree)
Fig. 7 Surface pressure (Cp) distributions at a = 20° pre-
dicted with the fine mesh
8
OCR for page 617
(a) x/L = 0.600
0.006
c: 0.004
0.002
OO
0.006
c: 0.004
0.002
OA
it'
30 60 90 120 150 180
(b) x/L = 0.772
n
1
1 . , 1 , , 1 , , I , , 1
30 60 90 120 150 180
¢(degree)
c'
Fig. 8 Skin-friction (Cf) distributions at cc= 20° pre- ~40
dieted with the fine mesh
in the experiment. However, KO-2 model deviates from
the data and other models' results on the windward side
of the primary separation. It is interesting to see large
differences among the three k-m models. Evidently, the
low-Reynolds number modification in the KO-2 model
seems to play a significant role in making a large differ-
ence between the KO-1 and KO-2 model results. The
SST model gives only marginally better results than the
SA model, falling behind the KO-1 and KO-2 models.
As discussed in the foregoing, the circumferential lo-
cations of minimum Cf and the separation locations are
correlated to each other. The experiments indicate that
the minimum Cf locations are located consistently on
the leeward side of the actual separation line. This can
be seen in Figure 8. Note that the primary separation
lines were observed at ~ = 123° and ~ = 112° in the
crossbow planes at x/L = 0.6 and x/L = 0.772, respec-
tively. Again, the predictions widely vary in terms of the
minimum Cf locations, with the maximum difference of
approximately 15° between the SA and the KO-2 mod-
els. The predicted locations of the Cf minima are seen
to shift windward gradually as one goes from SA, SST,
KO-1, KO-2 models. It is also observed that KO-1 and
KO-2 models predict the Cf minima (and separation)
(a) x/L = 0.600
TV
20
O
-20
40 , , 1—~ 1 . . 1 , . 1
~ 0 30 60 90 120 150 180
¢(degree)
0 Measured
- - SA
SST
--- KO-1
— KO-2
, ,—~ 1 , . 1
(b) x/L = 0.772
O Measured
- - SA
SST
--- KO-1
— KO-2
O
. .
a
, , I , , 1 , , 1 , . 1 , , 1
u 30 60 90 120 150 180
¢(degree)
Fig. 9 Wall-shear angle (pw) distributions at a = 20° pre-
dicted with the fine mesh
a little too early in comparison to the data, which is
largely consistent with the results shown in Figure 5.
The distributions of predicted wall-shear angle shown
in Figure 9 give us a measure of the strength (magni-
tude) of the crossbow and the rate of convergence (or
divergence) of the wall limiting streamlines near the
separation or reattachment lines. The separation and
reattachment lines are likely to manifest themselves as
local peaks and zero-crossings in these plots. The com-
parisons among the models are largely in line with what
Cp and Cf distributions show. The crossbow becomes
progressively stronger in the order of SA, SST, KO-1,
and KO-2 models.
Thus far, we have looked at the surface quantities
predicted with the fine mesh without using wall func-
tions. Now we will present the results obtained on the
coarse mesh using the wall function approach. Fig-
ure 10, Figure 11, and Figure 12 depict the distribu-
tions of Cp, Cf. and Do at a = 20°, respectively, pre-
dicted using the coarse (wall function) mesh. The use
of coarse mesh and wall functions precluded the KO-2
model (k-co model with a low-Reynolds number modi-
9
OCR for page 618
(a) AL = 0.600
-0.2
o
-0.1
-0.2
no
O—<- ' 1 ' ' 1 ' ' 1 ' ' I '
. O Measured
- - - SST (wall fn.)
-O. 1 _ ~ KO-1 (wall fn.)
- - RSTM-1 (wall fn.)
RSTM-2 (wall fn.)
RSTM-3 (wall fn.)
\
· ~ 1 ~ '
-0.3 _ \ i) so _ 0.002 _
04 , . 1 ,, 1 , 1,, 1 ,, 1,, O.
. ( ) 30 60 90 120 150 1{ ;0 0
(b) x/L = 0.772
. O Measured
- - - SST (wall fn.)
KO-1 (wall fn.)
- - RSTM-1 (wall fn.)
RSTM-2 (wall fn.)
RSTM-3 (wall fn.)
-04 ,, 1 ,, 1 ,, I,, 1 ,, 1,, 1
0 30 60 90 120 150 180
¢(degree)
Fig. 10 Surface pressure (Cp) distributions at a = 20° pre-
dicted using wall functions
fication) from the comparisons shown here. First of all,
the results from the SST and KO-1 models show only
marginal differences between the coarse (wall function)
mesh results and the corresponding fine mesh results.
Practically, this finding is quite significant, inasmuch
as it gives a credential to the wall function approach
and the results based thereupon. The RSTM-1 model,
the baseline SMC, performs clearly better than the SST
model in terms of capturing the local variations of the
surface quantities. Although not shown here, it should
be mentioned that the RSTM-1 model yields measur-
able improvements over linear k-£ models which failed
to capture the prominent features of the flow. How-
ever, the KO- 1 model seems to outperform the RSTM-1
model. The RSTM-2 and RSTM-3 models are seen
to bring remarkable improvements over the baseline
RSTM model, outperforming the k-co models. It is
quite interesting that seemingly minor changes in the
c-equation lead to such disproportionately large differ-
ences in the results. These improvements in the predic-
tion of the surface quantities are directly carried over to
the prediction of force and moment, as will be shown
later.
(a) x/L = 0.600
u.w-
0.006
0.004
to Into
0.006
0.004
0.002
Or~o
I , , ~ , . ~ ~ ~ 1 ' ' I
)~-'=~ ~
o Measured
- - - SST (wall fn.)
KO-1 (wall fn.)
- - RSTM-1 (wall fn.)
RSTM-2 (wall fn.)
RSTM-3 (wall fn.)
cry
30 60 90 120 150 180
(b) x/L = 0.772
O Measured
- - - SST (wall fn.)
KO-1 (wall fn.)
- - RSTM-1 (wall fn.)
RSTM-2 (wall fn.)
RSTM-3 (wall fn.)
JO 30 60 90 120 150 180
¢(degree)
Fig. 11 Skin-friction (Cf) distributions at a = 20° pre-
dicted using wall functions
Comparisons of the coarse mesh results shown here
and with the fine mesh results presented earlier indicate
that, insofar as high-Reynolds number flows are con-
cerned, CFD predictions based on wall functions are
perhaps better than has been commonly believed. A
similar conclusion was made by Kim (2002) based on
the numerical study of flow around a ship hull which
shares some of the salient features with the present flow.
Turbulence kinetic energy field
The recent experiments by Chesnakas et al. (1998)
and Goody et al.~2000) provide a wealth of turbu-
lence data which include individual Reynolds-stresses,
higher-order correlations, surface pressure fluctuations,
and corresponding spectra. In this paper, we will discuss
only the TKE field in the crossbow plane et x/e = 0.772
for the a = 20° case. The experimental data for the
crossbow plane at x/L = 0.772 (Chesnakas and Simp-
son, 1997) show that the peak value of TKE, 0.022pUo,
occurs in the free-vortex sheet at 0 = 140° and at ap-
proximately 30% of the local radius of the section from
the body surface. The measurement also indicates that
there is a region of relatively low TKE near 0 = 120°
just downstream of the primary separation line occur-
0
OCR for page 619
(a) x/L = 0.600
O Measured
- - - SST (wall fn.)
KO-1 (wall fn.)
- - RSTM-1 (wall fn.)
RSTM-2 (wall fn.)
RSTM-3 (wall fn.)
\
40
20
.
-2()
20
_
~4
5 lo _
-20
-40 , , 1
0 30 60 90 120 150 180 n no
\
(b) x/L = 0.772 ~ it_
a
I ~ ~ O o.os 0.1 0 15
o Measured I °
- - - SST (wall fn.)
_ ~ KO-1 (wall fn.)
- - RSTM-1 (wall fn.)
RSTM-2 (wall fn.)
RSTM-3 (wall fn.)
0.06 _
0.04
x/L=0.772
KO-l
-40` ~ 30 60 90 120
¢(degree)
150 180 0.06
Fig. 12 Wall-shear angle (pw) distributions at a = 20° pre- ° °
dieted using wall functions
nils
ringato=115°.
Figure 13 shows the contours of TKE (k/UO)at the
crossbow plane predicted by all three k-m models with
the fine mesh, whereas Figure 14 depicts the TKE at
the same crossbow plane predicted by the three RSTMs
with the coarse (wall function) mesh. The contours
share some similarities among the results from differ-
ent models, largely reflecting the formation of the free-
vortex layer emanating from the body surface. Yet it can
be seen that the k-m models yield contours that appear to
be more spread out (diffused) than the RSTMs. Perhaps
more important, all the models, except the ASTM-2,
severely underpredict the peak TKE (0.022pUo). The
location and magnitude of the peak TKE predicted by
the RSTM-2 are remarkably close to what the experi-
mental data indicate.
Steady lift and pitching moment predictions
The lift acting on slender bodies like prolate
spheroids is characterized by a nonlinear increase of
lift with incidence angle. The nonlinear lift is often
called "vortex" lift because the augmented lift is due
to the low pressure at the core of the vortices which
x/~=n772
.;2 K0-2-
o
0 0.05 0.1 0.15
n
0.1
>0.08
non
0.04
0.02
I_
0.1 0.15
Fig. 13 Turbulent kinetic energy (k/Uo ) at x/L = 0.772
and a = 20° predicted by k-c,) models
11
OCR for page 620
0.12
0.1
0.08
0.06
0.04
0.02 .
o' 1 0.05
0.14
0.
0.1
0.04
0.02
o
0 o.os
0.14
0.12
0.1
0.06
0.04
0.02
0 0 ,
0 0.05
- x/L=0.772
RSTM-1
l
- x/L=0.772
_ RSTM-2
z
x/L=0.772
RSTM-3
z
0.15
Fig. 14 Turbulent kinetic energy (k/Uo) predictions at
x/L = 0.772 and a = 20° predicted by RSTMs
is impressed upon the near-by body surface. Accurate
prediction of the location and strength of the vortices
is therefore prerequisite to successful prediction of the
force and moment.
The predicted lift (Cr _ FN/O.SPUOL2) and pitching
moment coefficients (CM _ Mz/o.5pUoL3) for the full
range of incidence are plotted in Figure 15 along with
the measured ones. The results shown in Figure 15
were obtained on the fine mesh. Like the results pre-
sented so far, the predictions vary in a wide range. The
differences among the models increase with the inci-
dence. All models underpredict the lift. The SA and the
KO-2 model yield the lowest and the highest lift, respec-
tively. The lift predictions are largely consistent with the
behaviors of the surface quantities discussed before. In-
terestingly, the predicted pitching moment coefficients
shows an opposite trend. The SA model matches closest
with the measurement, while the KO-2 model deviates
most from the data.
Figure 16 depicts the result based on the coarse (wall
function) mesh. It is noteworthy that the lift predictions
by SST and KO-1 models based on wall functions are
slightly higher than the corresponding results based on
the fine mesh. However, the differences are deemed
insignificant, insofar as we are comparing the results
based on two drastically different near-wall modeling
strategies. The RSTM-1 prediction is largely compara-
ble to the SST and KO-1 model results. The RSTM-2
and RSTM-3 results are in commendable agreements
with the data.
Unsteady lift and moment characteristics for
dynamic pitch-up motion
The dynamic pitch-up maneuver has also been simu-
lated by solving unsteady RANS (URANS) equations.
However, the calculation was made using the SST
model only for now. The plot on the right in Figure 17
depicts the time history of the lift as the incidence an-
gle is dynamically increased from 0° to 30° in 11.0
dimensionless time units (t* - tUoo/L). The predicted
unsteady lift was found to lower than the steady lift at
the corresponding incidence angles, which is consistent
with what the experimental data show. One interesting
observation from this preliminary result is that the cal-
culation appears to reproduce the transitory increase in
the lift, which was persistently observed in the experi-
ment (Wetzel and Simpson, 1997) in the early stage of
the pitch-up motion, as reflected by a hump in Figure 17
during t* = 1.0 ~ 4.0. We surmise that this is caused
by a starting vortex generated on the leeward side of
the body by the impulsive start of the pitch-up motion,
which contributes to augmenting the lift until it is swept
away downstream by the flow. The computations with
other turbulence models are reserved for future study.
12
OCR for page 621
(a) Steady lift force
0.o3ot 1 1 1 1 1 1
0.02C
0.02C
V O.01<
0.01C
0.005
0.000` ) 5 10 15 20
Angle of attack (deg.)
· · Measured
_ o SA
: ~ SST
OCR for page 622
nn'Sk
n n2n
- - - - -
o~0.015 _
0.010 -
o.oos t
° °°8
o
o Measured (Wetzel and Simpson)
_ SST (wall fn.) cat
o ~ of
i/
5.0 10.0 15.0
Dimensionless time (t )
Fig.17 Unsteady lift prediction for dynamic pitch-up mo-
tion; Cr vs. dimensionless time, t*
.
. The baseline Reynolds-stress transport model
(RSTM-1), which performed admirably for simi-
lar flows yet with weaker crossbow and rotation,
was less successful for the present flow.
· Minor modifications in the length-scale equation
result in significant improvements of the predic-
tions.
· Resolving the viscous sublayer brings only a
marginal difference. The wall function based re-
sults are largely comparable to the results based on
the fine meshes.
ACKNOWLEDGEMENT
The authors acknowledge use of Fluent Inc.'s soft-
ware and thank the members of the product develop-
ment group at Fluent Inc.
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15
Representative terms from entire chapter:
prolate spheroid
