Proceedings of the 3rd ASME/JSME Joints Fluids Engineering Conference
July 18-23, 1999, San Francisco, California
FEDSM99-6792
SUBGRID-SCALE MODELING IN LARGE-EDDY SIMULATION AND ITS APPLICATION TO FLOW ABOUT YAWED CYLINDER AND CAVITY FLOWS
S. Lee/ Department of Mechanical Engineering, INHA Univ., Incheon City, Korea
A. K. Runchal/ ACRi, 1931 Stradella , Bel Air, CA
J.-O. Han/ Department of Mechanical Engineering, INHA Univ., Incheon City, Korea
ABSTRACT
In this computation, Smagorinsky’s eddy viscosity model combined with a truncated deductive model is used to represent the SGS stress fields in quadrilateral/hexahedral structured/unstructured grid system. The truncated deductive model considers the eddies immediately below the filter size, ∆, which presumably lies within the inertial range, but not those near Kolmogorov scale, η, such that dissipation is not properly accounted for. Therefore it is assumed in this study that the Smagorinsky model describes the dissipation correctly, and the truncated deductive model ensures a smooth cascade process.
To show the applicability of the proposed SGS model to the structured/unstructured grid system, two cases of the flow about a yawed cylinder and a cavity flow were numerically tested and compared with experimental data. The present results of the yawed cylinder were produced at a Reynolds number of 1.3×104 in the sub-critical range. As a second example of validating the code, three configurations of cavity (L/D=1.5, 2.0, 10.0) were used to simulate quasi-periodic flow patterns at a Reynolds number of 1.0×104. The sound generated from turbulent flows around the yawed cylinder and in the cavity was also computed using Lighthill’s acoustic analogy.
The LES results are compared to experimental data in terms of local characteristics such as base pressure coefficients, wall pressure and far-field noise spectra, which were measured in an acoustic wind-tunnel facility.
NOMENCLATURE A surface area C0 sound speed CL lift coefficient CL
2 mean square of lift coefficient CS Smagorinsky constant D domain of the fluid, diameter of cylinder, depth of
cavity f frequency of noise generated G Gaussian filter function
H height of computation domain h enthalpy L length of computation domain, length of cylinder,
cavity length lC correlation length M∞ free stream Mach number m mode number P2 mean square of sound pressure Qk subgrid-scale heat flux Re Reynolds number Rpp cross-correlation S viscous part of the stress tensor
Skl resolved scale strain rate tensor St Strouhal number T absolute temperature (K) stress tensor
U4 free-stream velocity V volume of cell element velocity vector
W width of computation domain β+ fraction of the total surface ∆ width or characteristic length scale of filter δ* displacement thickness φ physical dependent variable γ = αeE/αPE , α is the distance, centroid of correlation
length, η Kolmogorov scale λz wavelength in z direction ν = µ/ρ , kinematic viscosity νT subgrid-scale eddy viscosity ρ density σkl filtered viscous stress tensor τkl subgrid-scale stress tensor ξ spanwise distance Ω integration domain
INTRODUCTION Turbulent flows have been the subject of the experimental
and theoretical studies since the last century. The variety of turbulent flows is enormous and knowledge of them has many important engineering applications. The instantaneous flow variables satisfy the Navier-Stokes equations.
The strategies for dealing with turbulence are many: turbulence models (educated guesses about the needed Reynolds stresses), statistical theory of turbulence (to gain fundamental understanding), direct numerical simulation.
The LES is one of the more promising modes of numerical simulation of turbulence. Most of developments in the LES approach, except recent methodology , e.g. dynamic subgrid-scale modeling, have been reviewed by Rogallo and Moin(1984). One of main differences between the conventional method of RANS(Reynolds-Averaged Navier-Stokes) solver and the LES technique is the averaging process involved. The LES approach does not include the use of ensemble averages as a first step in obtaining equations for mean flows, but uses a space filtering operation applied to the equations of motions.
In this study, the unsteady, viscous flows over a three-dimensional yawed cylinder and in a cavity are computed using ANSWER code which is a general purpose CFD software for simulation of fluid flow, heat and mass transport processes in laminar or turbulent, compressible or incompressible, transient or steady at any speed with subgrid-scale models for LES.
The sound generated by viscous flows over the cylinder or in the cavity has been widely studied but is still difficult to compute at moderate and high Reynolds numbers due to complex vortex dynamics. To predict the sound from unsteady, viscous flow past a circular cylinder assuming a constant correlation length, the RANS equation was solved and combined with the Lighthill acoustic analogy by Cox et al.(1997). The Reynolds-averaged, unsteady, compressible, Navier-Stokes equations were also used by Baysal et al.(1994) to test the effectiveness of devices to suppress the cavity acoustics.
Jordan and Ragab(1998) studied the near wake of a circular cylinder at a Reynolds number of 5600 using the dynamic model of LES in curvilinear coordinates. Recently, the unsteady, turbulent flow-field calculation using LES is employed as input into an acoustic prediction code to predict the near- and far-field noise, e.g. Lee and Meecham(1996). In large-eddy simulations, one calculates directly the large-scale turbulent motions with a relatively coarse time-dependent, three-dimensional computation using the subgrid-scale(SGS) model for the effects of the small scale motions upon the large scale ones. GOVERNING EQUATIONS AND NUMERICAL METHOD
Development of a LES formulation applicable to non-orthogonal grid system begins with the compressible continuity, Navier-Stokes, and energy equation in integral form.
0dSnvdt S
=⋅ρ+Ωρ∂∂
∫∫Ωrr
(1)
∫∫∫∫ ΩΩΩρ+⋅=⋅ρ+Ωρ
∂∂ dbdSndSnvvdvt SS
rrrrrr ı (2)
∫∫∫ ⋅=⋅ρ+Ωρ∂∂
Ω SSdSnTgradkdSnvhdh
trrr
[ ] ∫∫ ΩΩΩρ
∂∂
+Ω+⋅+ dt
dvgrad:Pgradvrr S (3)
where h is the enthalpy per unit mass, T is the absolute temperature(K), S is the viscous part of the stress tensor, ı.
The collocated arrangement is used with Cartesian velocity components of finite volume approach as shown in Fig. 1. In general, the flux is computed from Green’s theorem as
∫ ⋅φ=∂φ∂
SV1
xirrS
ix (4)
For computing fluxes, we also require the gradients at the faces, say, ∠φ!∠x? . If we determine ∠φ/∠x? as a linear inter-
polation of ∠φ!∠x? and ∠φ!∠x? , then it can be seen that the direct effect of nodes, P and E, will be reduced and effects of nodes, ee and w, will enter the formula. This is not desirable either from numerical standpoint or from computational convenience.
e e
P E
There are a number of alternative ways in which ∠φ!∠x? can be determined. The simplest is to define a closed surface extending from node E to node P with other surfaces between these in the y and z direction. The most general approach is to define ∠φ!∠x? on both sides of the e surface as given below and then interpolate.
e
e
Fig. 1. Collocated grid arrangement
[ ] dedeueueseseneneeeEEee
AAAAAAV1
φ−φ+φ−φβ+φ−φ=∂φ∂ +
+
+
x (5)
where β+ is the fraction of the total surface which is contained between e - E connection. We have for simplicity assumed that β+ for all surfaces is constant though in reality it will vary. But for all practical purpose, this choice of constant β+ should suffice since, in any case, ∠φ!∠x? is being obtained from linear interpolation. Further, consistent with the assumption, we shall also assume that:
Now ∠φ!∠x? can be obtained by linear interpolation as: e
+−
∂φ∂
γ−+∂φ∂
γ=∂φ∂
eee)1(
xxx (9)
where γ = αeE / αPE and α is the distance. With the same level of interpolation accuracy, we may write
−+−+
+==γ−=γ eeee
e
e
e VVV,VV
1;VV (10)
Now we may write
eEe
e
Pe
ePE
e
e
e VS
VV
VV)(
VA φ
−+
+∂φ∂
+∂φ∂
+φ−φ=∂φ∂
xxx (11)
where eeP
e
E
e
e
e
e
eeeee
E
eww
P
e AVV
VV
VV
VVA
VVA
VVS φ
−+−+φ−φ=
+−
+
−
−
+−+
φ
EE
ee
E
eeee
e
ee V
VAVVAA
VVA φ
−+−+
−−
+
−
EP
ew
P
ee
e
eee V
VAVVA
VVAA φ
−+−+
++
−
+.
The numerical simulation of both turbulent flows over the yawed cylinder and in the cavity is performed with the ANSWER code (see Runchal et al. (1993)) which accommodate the truncated deductive model proposed by Lee (1992) for LES computation. In ANSWER, the equation of continuity is transformed into an equation for computation of the density. This approach of working directly with the density variable is termed DEFCON for Density Equation Formulation of Continuity equation. The governing Navier-Stokes equations are integrated using Finite Volume Method (FVM) for a collocated, structured/unstructured grid system.
A major advantage of this method is that it intrinsically preserves the mass, material fluxes both on local and global scales. The numerical integration starts with the assumption of an integration profile for the state variable. The CONDIF scheme, which is stable and second order accurate, is employed in this research, see Runchal (1987). In this work, the Alternating Direction Implicit (ADI) solution was used, which solves the set of algebraic equations in three sweeping directions. This temporal discretization of the ADI method is proven to be unconditionally stable in linear sense. But it may not be accurate in time if the time step is large due to the negligence of the third-order term which is essential to the factorization.
SUBGRID-SCALE MODELING
In large-eddy simulation, each flow variable is de-composed into a large-scale component and a subgrid-scale (residual) component. Here ‘grid’ in the keyword ‘subgrid’ refers to a length scales ∆, which is smaller than the correlation
length, L, the size of the large energy-containing eddies. This decomposition is made using flow variables filtering as:
zd)z(F),zx(G)x(FD∫ ∆−= (12)
where G is the filter function, ∆ is the width or characteristic length scale of filter, and D is the domain of the fluid.
For treatment of compressible flows, mass-weighted averaging simplifies the analysis. In this approach mass-averaged variables are defined according to
ρρ
=FF~ (13)
where the decomposition is given by FF~F ′+= . Direct filtering of the continuity equation yields:
0)u~(xt
=ρ∂∂
+∂ρ∂
kk
(14)
Direct filtering of the momentum equation yields:
l
kl
l
kl
klk
lk xxx
P)u~u~(x
)u~(t ∂
τ∂+
∂σ∂
+∂∂
−=ρ∂∂
+ρ∂∂ (15)
where
′′+′+′+−ρ−=τ
~~~~uuuu~uu~u~u~u~u~ lkkllklklkkl .
is called the subgrid-scale stress tensor. If we assume that the kinematic viscosity ν = µ / ρ is constant, the filtered viscous stress tensor can be written as:
∂∂
+∂∂
ρν+δ∂
∂ρν−=σ
k
l
l
kkl
j
jkl x
u~
xu~
xu~
32 (16)
From the equation of state, T~RTRP ρ=ρ= . The filtered momentum equation is now solvable if we provide a model for τkl. Direct filtering of the energy equation yields:
)u~T~(x
RtP)u~T~(
x)T~(
tCP k
kk
kρ
∂∂
+∂∂
=
ρ
∂∂
+ρ∂∂ +
kkkk
k
k
kxx
TKxx
uTRxC
R k
P
~∂∂
−Φ+
∂∂
∂∂
+∂∂
ρ−∂∂ QQ (17)
where
′′+′+′+−ρ−=
~~~~uTu~TuT~u~T~u~T~CP kkkkkkQ . As for
the momentum equation, if we assume that the kinematic viscosity is constant, the filtered viscous dissipation can be written as
~~~xu
xu
xu
xu
xu
xu
32
l
k
k
l
l
k
l
k
j
j
j
j
∂∂
∂∂
ρν+∂∂
∂∂
ρν+∂
∂
∂
∂ρν−=Φ (18)
The filtered energy equation is now closed if we provide models for Qk and for T(∠uk /∠xk). ~
The subgrid-scale stress tensor, τkl, can be decomposed into the subgrid-scale Leonard, cross, and Reynolds stresses based on Favre-filtering. Smagorinsky (1963) was the first to propose a model for the subgrid-scale stresses. His model assumes that they follow a gradient-diffusion process, similar to
molecular motion. It is still the most popular algebraic eddy viscosity model, with τkl given by
S~C,S~2 22sTT ∆ρ=νν=τ klkl (19)
where νT is the subgrid-scale eddy viscosity, CS is the Smagorinsky constant, klS~ is the resolved scale strain-rate
tensor and ? S~ ? = 21
)S~S~2( klkl . The dynamic subgrid-scale turbulence model was
proposed by Germano et al. (1991) to simulate closely the state of the flow by locally calculating the eddy viscosity coefficient through double filtering. The procedures of Germano et al. were used by Jordan et al. (1998) to derive an expression for Smagorinsky’s coefficient in the non-orthogonal curvilinear form. This model exhibits the proper asymptotic behavior near boundaries or in laminar flow without requiring damping or intermittency. However, these models are more demanding from the double filtering at each time step in terms of computation time.
To model the Reynolds subgrid-scale stress tensor, τkl, deductively which is applicable to the compressible flow with unstructured grids, we must consider the quantity ukul - lk u~u~ . ~We have the following relation for Gaussian filtering:
)x(F24!2
124
1)x(F222
22
+
∇∆+∇
∆+= LL (20)
For the filtered products, Gaussian filtering gives
ll x)x(g
x)x(f
12)x(g)x(f)x(g)x(f
2
∂∂
∂∂∆
+=
L+∂∂
∂∂∂
∂
∆+
nmnm xx)x(g
xx)x(f
1221 2222
(21)
Setting f = ρuk and g = ul , we have the terms in Eq.(21) to be replaced as:
( )
∇⋅ρ∇
∆+ρ
ρ= lklkkk uu~
12uu~1uu
2~
( L+
∇∇ρ∇∇
∆+ lk u:u~
1221
22) (22)
The subgrid-scale heat flux can be decomposed into the following components:
(23) )R()C()L( QQQQ kkkk ++=
~where
−ρ= T~u~T~u~CQ P
)L(kkk ,
′+′ρ=
~~Tu~T~uCQ P
)C(kkk ,
and ~
TuCQ P)R( ′′ρ=k , which are called the subgrid-scale
Leonard, cross, and Reynolds heat fluxes, respectively, based on Favre filtering.
To model the subgrid-scale heat flux, Qk deductively, we must quantity Tuk - ku~T~ . Following the procedure used to derive th deductive subgrid-scale stress model, we have
~e
nmnm
k
mm
kkk xx
T~
xxu~
1221
xT~
xu~
12u~T~uT
22222~∂∂
∂∂∂
∂ρ
∆+
∂∂
⋅∂∂
ρ∆
=
−ρ
L+∂∂
∂∂
∂∂ρ∂
∆−
∂∂
∂ρ∂
∂ρ∂
∂∂
ρ
∆+
m
k
nnmnnmm
kxu~
xT~
xx12xT~
xxxu~1
12
22222 (24)
In principle, an exact large-eddy simulation would be found by keeping all terms in the expansion for the deductive model. By keeping all terms, all scales down to the Kolmogorov sclae η would be accounted for. Although viscosity may be very large, it is finite and responsible for the final energy dissipation of the system. But at large Reynolds number, as a practical matter, we can not deal with the dissipation scale directly. So based on the Kolmogorov hypothesis, we make the reasonable assumption of a dynamic similarity, in a statistical sense, between the smallest eddy dictated by viscosity, and the smallest eddy allowed by the filter size ∆ which presumably lies within the inertial subrange. From experiments we know that if the Reynolds number is large enough the inertial subrange begins at about 1/3 the large scale. This is the Smagorinsky model. For the spectrum of eddies below ∆ , this plausibility discussion takes care of the smallest eddies, but not those that are below but still close to ∆. Therefore the use of the Smagorinsky model alone means the loss of the latter, which can conceivably disrupt the cascade process envisioned in the Kolmogorov hypothesis.
Therefore, the truncated deductive model in Eq.(22) and Eq.(24) treats the eddies immediately below ∆, but not those very small eddies which account for the dissipations. Thus it seems fair to conclude that we need the Smagorinsky model to describe the dissipation correctly, and the truncated deductive model to ensure a smooth cascade process. A linear combination of deductive model and eddy viscosity model turned out to correlate better with the exact value than Smagorinsky model alone by Lee and Meecham (1996).
YAWED CYLINDER
The wake flow behind a circular cylinder has been paid attention as a fundamental study about the wake of a bluff body of arbitrary shape. This problem is called Strouhal shedding or K⟨rm⟨n vortices and can find many engineering applications in the flow around rows of tube in a heat exchanger or marine structures. Roshko(1955) reported the inconsistency of Strouhal numbers in the range of Reynolds numbers between 150 and 200. Using flow visualization, Hama(1957) found three dimensionality of wavy structure in the spanwise direction of cylinder at a Reynolds number greater than 150. The cylinder wakes were found to generate counter-rotating Karman vortices leading to three dimensional flows at a certain Reynolds number by Grant(1958).
Bloor(1964) studied the instability of wake flow in a separated shear layer leading to transition of tutbulence at
Reynolds numbers between 200 and 5×104 . While the transition of turbulence occurs in a developed street of vortices where the cylinder wake becomes turbulent at Reynolds number below 400, the Tollimien-Schlichting wave developing in the separated shear layer induces the transition before the formation of vortex above the Reynolds number.
The time-averaged drag from a circular cylinder depends on the Reynolds number because the formation of vortex dependent upon the Reynolds number affects the rate of momentum transport in the wake region. In the subcritical range of Reynolds number between 104 and 3×105, the drag coefficient is constant from the similarity of wake characteristics and the unstable mode leading to transition occurs near the wake region within D/2 apart from the separation. As the Reynolds number increases above 5×105, the irregularity of cylinder wake flow makes the Strouhal numbers to be highly scattered between 0.025 and 0.21. At the supercritical Reynolds number, it is known that relatively dominant Strouhal number of 0.3 can be observed.
If there exists a spanwise velocity component for the yawed cylinder, the distribution of vorticies in the wake is affected and the frequency of vortex generation is thereafter changed. To analyze the flow around yawed cylinder, it is often assumed that only the normal velocity component determines the wake characteristics of cylinder, which is called “independence principle” or “cosine law”. The wake flows of circular cylinder with some yaw angles have been investigated for small Reynolds numbers ranging from ten to a few thousands and their results about the maximum yaw angle allowing “cosine law” fail to agree with each other.
To resolve the spanwise-periodic near-wake structure, we used 30 grid points over the length of 2D based on the formula, λZ/D≅20ReD
-1/2 (Williamson et al.(1995)). The grid number was 141×96×30 (x,y,z directions) with the inflow and outflow boundaries placed at 10D and 15D, respectively (see Fig.2). As an exit boundary condition, the simplified convective one among non-reflecting boundary conditions was employed, which Hayder et al.(1995) proposed.
We investigate the turbulent physics in the near cylinder wake at Re=1.3×104 in the sub-critical range. The base pressure coefficient (Cp)b at the mid-plane in the yawed cylinder with a yaw angle equal to 30o is compared with 2D prediction in Fig. 3. In general, a 2-D computation overpredicts the vortex strength and the shedding frequency. The vortex shedding patterns from 2-D computation results can be seen for one period in Fig. 4 and its shedding Strouhal number is identified as 0.181.
The flow over the cylinder was impulsively started with unit velocity, zero reference pressure, a fixed non-dimensional time step of ∆T=0.005 (T=tU∞/D), and the maximum CFL number of 0.5. The typical instantaneous snapshot of spanwise vorticity contours on the surface of cylinder are shown in Fig.5. Aperiodic, instantaneous vortices are found in the spanwise direction, which may state that the correlation length is at least equal to or greater than 2D for this yawed cylinder flow.
The sound pressure radiated from an obstacle in the flow may be predicted using the formula given as:
)(L2StUCr
cos16C
p C262
L2
2
20
22 γ−
θρ= ∞ l (25)
where ρ is the density of medium, θ is the angle between the normal to the flow and the observation position of distance r,
2LC is the mean square of lift coefficient, L is the length of
cylinder, lC is the correlation length, and γ is its centroid.
∫ ∫ ξξξ=γξξ=L
0
L
0PP
CPPC d)(R2,d)(R2
ll (26)
The spectra of the lift and drag forces obtained by integrating instantaneous pressures over the surface is given in Fig.6. The fundamental frequency at the shedding has most energy for lift and drag forces. By separating the dipole and quadrupole sources, we can see clearly in Fig.7 and Fig.8 that the noise produced at the fundamental frequency is entirely from lift dipole.
CAVITY FLOW
The flow over a cavity introduces pressure oscillations in it. The pressure oscillations inside the cavity have been investigated over 40 years. East (1966) tested the rectangular cavity with high L/D to study the noise generation mechanism. Sarohia and Massier (1977) tried to control the cavity noise using injection of mass at the bottom wall. Recently, Sarno and Franke (1994) performed an experimental study about cavity flow phenomena, noise generation mechanism, and suppression method. Hardin (1995) applied the computational aeroacoustics to simulate two dimensional cavity flow.
The cavity can be viewed as a combination of forward- facing step and backward-facing step. While the flow oscillation occur in the normal direction to the main stream in a deep cavity where the streamwise length,L, is shorter than the depth,D, the oscillation propagate in the streamwise direction in a shallow cavity(L/D ∃ 1). The flow in the shallow cavity may be classified into open cavity flow, transitional cavity flow, and closed cavity flow. The flow oscillation patterns in shallow cavity depend on flow speed(U), cavity length(L), cavity depth(D), width(W) and the inclination angle(θ), among which flow speed and L/D are most dominant factors in cavity noise In the open cavity flow, the stagnant flow in the cavity and upstream flow are separated by a shear layer and circulate when 1 ≤ L/D ≤ 4.
To numerically resolve the spanwise-periodic structure, 30 grid points were placed over the width of 2L. Two open-type cavity models (L/D=1.5,2.0) and closed-type cavity model (L/D=10.0) are simulated with grid numbers of 106×71×30 and 102×71×30 in x,y,z directions, respectively(see Fig.9). The inflow and outflow boundaries were placed at 4/3L and 16L, respectively. The simplified convective boundary conditions as in the yawed cylinder was used as an exit boundary condition.
The flows over cavities were impulsively started with unit velocity, zero reference pressure and a fixed non-dimensional time step of ∆T=5×10-4, 6.67×10-4, 3.33×10-3 with the
maximum CFL number of 0.045 for L/D=l.5, 2.0, 10.0, respectively. The simulated mean static pressure distributions for each cavity are compared with experimental data in Fig. 10, which shows a certain degree of agreement between them only for the cases of L/D=1.5 and 2.0. In general, open cavity flow has a uniform pressure distribution for values of x/L up to approximately 0.6. In a closed cavity flow, the pressure distribution gradually increase from negative values near backward facing step to large positive values near forward facing step. It can be argued that the experimental data measured at L/D=10.0 shows a transitional closed cavity, while the numerical data at the same L/D are close to transitional open cavity because the ratio of W/D also affects these transition, as suggested by Tracy et al.(1997). The mean flow vectors with streamlines are depicted for each cavity in Fig.11.
The resonant frequency for open –type, high Mach number cavity is given by Rossiter(1966) as:
κ+
−γ+
α−=
−∞∞
∞ 1]M)2
1(1[M
mULf
212
m (27)
where fm is the frequency of noise generated, m is the mode number, α is a proportional constant about a delayed time between forward step and vortex motion, κ is the ratio of flow speed and convection speed, M is the Mach number of flow, and γ is specific heat ratio (κ ≅ 0.57, α ≅ 0.25). As L/D increases, cavity flows changes from resonant flow to non-resonant with decrease of resonant amplitudes.
It has been quite difficult to observe the resonant phenomena at low Mach number flows. The open-type cavity of L/D=1.5 was numerically tested to identify the resonant frequency from radiated sound. The power spectra from the drag force acting on the forward and backward faces by instantaneous pressures and far-field sound are shown in Fig.12 and Fig. 13, respectively. The resonant frequency of 132Hz was noticed from both dipole and quadrupole far-field sound spectra for L/D=1.5 and ReL=1.0×104. It is clearly seen that the noise produced at this fundamental frequency is entirely from drag force dipole. We may also notice that quadrupole-type noise sources are active in the resonant open cavity.
CONCLUSIONS
Grid independent results were obtained for turbulent flows behind a yawed cylinder and over cavities of three geometric cases (L/D=1.5, 2.0, 10.0). Aperiodic, instantaneous, large-scale vortices were found in the spanwise direction, stating that the correlation length is at least equal to or greater than 2D for this yawed cylinder flow, which is essential to the prediction of far-field sound level. By separating the dipole and quadrupole sources, the lift dipole is shown to contribute significantly to the noise produced at the fundamental frequency.
The simulation of cavity flow for L/D=1.5 at ReL=1.0×104
showed open-type resonancies from both dipole and quadrupole far-field sound spectra. The noise produced at this frequency was entirely from streamwise, fluctuating forces acting on both
forward and backward faces. The quadrupole-type noise sources were also very active in this resonant open cavity.
ACKNOWLEDGMENTS This work has been partly sponsored by INHA University
Research Program in 1996. The authors would like to express our sincere appreciation to Prof. W.C.Meecham and Dr. D.B.Schein at U.C.L.A. for contributing to the development of compressible deductive model and providing helpful discussions. The first author gratefully acknowledges the support of Prof. J.-S. Choi to provide some experimental data.
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Fig. 2. Boundary conditions and grid system for yawed cylinder(141×96×30)
Fig. 3. Mean pressure coefficient distributions for 2-D and yawed cylinder
Fig. 4. Vortex shedding patterns for one period
(2-D computation, ReD =1.3×104)
Fig. 5. Typical snapshot of spanwise vorticity contours on surface of cylinder (ReD =1.3×104 , yaw angle = 30o)
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