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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
SESSION 8
VISCOUS FLOW: APPLICATIONS 1

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
A Viscous Flow Simulation of Flow About the 1/40-Scale Model of the U.S. Airship Akron at Incidence Angle
C.-I. Yang (David Taylor Model Basin, USA)
ABSTRACT
A three-dimensional incompressible Navier-Stokes code based on an artifical compressibility, implicitupwind-relaxation, flux-splitting algorithm is employed to simulate the flow about a 1/40-scale model of the U.S. airship “Akron” at several incidence angles. The distributions of transverse forces along the hull and the integrated moments about the center of buoyancy are computed and comparisons with the measurements are made.
INTRODUCTION
Purely for mathematical interest, the inviscid flow about a body of revolution has long since been formulated and studied in detail. Practically, because of the predominant viscous effect near the boundary, the related flow pattern is much more complicated, especially if the body is at an incidence with respect to the flow direction. The wake of the body becomes turbulent, and various types of cross flow separation take place. The basic hull form of a modern submersible is typically a body of revolution. While maneuvering at high speed, the hull may be subject to severe hydrodynamic forces. Under certain conditions, the moment of the forces about the center of buoyancy of the body may cause instability. In order to achieve a higher envelope of maneuverability and controllability, the designers of the modern submersible have practical interest in predicting the hydrodynamic response for any given planned movement. Such interest can best be served by parallel efforts in enlarging the data base from controlled laboratory environments and developing accurate computational schemes.
Extensive experiments were carried out by various research parties, some of the representive results were reported in references (1–3). More recently, computational efforts based on newly developed numerical schemes derived from the Reynolds Averaged Navier-Stokes (RANS) formulation offer encouraging predictions (4–8).
This report present a study of the accuracy and feasibility of predicting forces and moment on a body of revolution hull form at incidence with a RANS technique. The data obtained from wind tunnel tests of a 1/40-scale model of the U.S. airship “Akron” are used for the purpose of comparison.
DESCRIPTON OF EXPERIMENT
A series of tests was made on a 1/40-scale model of the U.S. Airship “Akron” at the propeller research wind tunnel, Langley Memorial Aeronautical Laboratory (currently, NASA Langley Research Center) in 1932 (9–11). The purpose of the test was to determine the drag, lift, and pitching moments of the bare hull and the hull equipped with fins.
This particular experiment is attractive to us in some aspects: (1) the hull form is very similar to the modern high performance submersible, (2) the Reynolds number is relatively high due to the large size of the model, and (3) the data are relevent to our study; included are the distributions of the transverse forces along the hull and the moments of the forces about the center of buoyancy.
The model is of hollow wooden construction having 36 sides over the fore part of the hull, fairing into 24 sides near the stern. The length of the hull is 5.98 m.(19.62 ft.), the maximun diameter 1.01 m. (3.32 ft), the fineness ratio 5.9, the

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
volume 3.27m3 (115.61ft3). Four hundred pressure orifices, distributed among 26 stations, were placed along one side of the hull. The orifices were connected inside the hull to two photographic-recording multiple manometers. Each manometer consisted of 200 glass tubes placed about the periphery of a drum, a long incandescent light bulb for making the exposures was placed at the center of the drum.
Tests were conducted at several different wind speeds. The maximun speed was 44.70 m/s (100 miles per hour). The corresponding Reynolds number is about 17 million based on the length of the hull. This value is about 1/34 of the full scale ship at a speed of 37.54 m/s (84 miles per hour). The transition from laminar to turbulent flow occured at a local Reynolds number of 814000 based on the axial distance between the nose and the transition point (10). At a wind speed of 44.70 m/s (100 miles per hour), the transition point is about 0.25 m.(10 inches) from nose.
The maximum departure of the observed wind tunnel velocity from a mean value was about ±0.6 percent. The deflection of the support wire, that is the downstream movement of the model, observed at the maximum velocity of the tunnel with the hull at 0º pitch was approximately 1.5×10−3 m. (0.06 inch). The sources of error and the precision of measurements are discussed in detail in references 9–11.
NUMERICAL APPROXIMATION
The three-dimensional incompressible RANS equations based on primitive variables are formulated in a boundary-fitted curvilinear coordinate system and solved with an artifical compressibility concept (12). The basic operations of converting the set of differential equations to a system of difference equations may be divided into: spatial differencing and time differencing. The procedure can be described as follows.
Spatial Differencing
The three-dimensional differential operator is first split into three independent one-dimensional operators. The spatial differencing of the inviscid flux in each of these one-dimensional operators is then constructed by an upwind flux-differencing scheme based on Roe's approximate Riemann solver approach (13). In each computational cell the differential operator is linearized around an average state such that the flux difference between two adjacent cells satisfies certain conservative properties. As a result, the flux at an interface can be expressed in terms of the direction of the travelling waves. Harten's high-resolution total variation dimishing (TVD) technique (14,15) is then applied to enhance the accuracy of the solution to a higher order in the region where its variation is relatively smooth. The undesirable spurious numerical ocillations associated with high order approximations are suppressed by appling a TVD limiter. The viscous flux is centrally differenced with second-order accuracy. The overall discretization is obtained by summing up all the independent discretizations of the flux derivatives in each dimension.
Time Differencing
Since only the steady-state solutions are of interest, a first-order accurate Euler-implicit time differencing scheme is used. The application of the scheme avoids a overly restrictive time-step size when highly refined grids are used to resolve viscous effects. In addition, a spatially variable time step is used to accelerate convergence.
The governing differential equations are then reduced to a system of difference equations in “delta form”. In each time step, the corrections to the variables, instead of the variables themselves, are solved. The right hand side of the system is defined as residual. It is the explicit part of the system and has four components, one for each variable. As the solutions advance to their steady-state values through time stepping, the corrections and residuals approach zero.
The system is solved iteratively with a hybrid technique which uses approximate factorization in cross planes in combination with a planar Gauss-seidel relaxation in the third direction. The process is highly vectorizable. Presently, the L2 norm of the residual is used as a measurement of convergence of the iteration process.
As a result of upwind-differencing, the coefficient matrix of the system becomes diagonal dominant. In addition, the necessity of adding and tuning of a numerical dissipation term for stability reasons, as in some schemes with central differencing, is alleviated.
BOUNDARY CONDITION
The computational domain defined by a C-O grid extends from two body lengths upstream of the nose to two body lengths downstream of the tail in the longitudinal direction, and two body lengths from the body axis in the radial direction. On the body surface, the no-slip condition

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is imposed and the normal gradient of the pressure is assumed to vanish. Free stream conditions are specified along the outer boundaries except for the outflow boundary, where the values are computed by using extrapolation. Since the flow field is symmetric with respect to the longitudinal plane of symmetry, only the flow field over half of the body is computed. Reflective conditions are then applied on the plane of symmetry. The values of the characteristic variables along the wake line are obtained by first extrapolating from interior points along each radial grid line and then taking the circumferential average. The normal distance between the body surface and the nearest grid line is 1.0×10−5 of the body length; the corresponding y+ is about 4. Computations are first performed on a grid with a 79×81×83 distribution in radial, circumferential and streamwise directions respectively. To determine the effect of gridding on the prediction of lift and cross flow separation, a grid with 79×111×83 distribution is used for a repeat computation. In both cases the circumferential spacing of radial lines is uniformly distributed, The angles between adjacent radial lines are 2.22º and 1.62º respectively.
TURBULENCE MODEL
The algebraic Balwin-Lomax turbulence model was used by Degani and Schiff (16) in computing the turbulent flows around axisymmetric bodies with crossflow separation. In order to predict multiple secondary crossflow separations at high incidence angle, modification was made such that the turbulence length scale of the outer region is determined by the viscous vorticity imbedded in the boundary layer and not the inviscid vorticity shed from the separation line. The modified model has been successfully used in several occasions to compute the turbulent flows over bodies of revolution at an incidence angle (7,16,17). The details of the modification, implementation and the physical justication can be found in Reference 16.
The behavior of the turbulent boundary layer near the stern region of an axisymmetric body has been studied extensively by Huang et al. (19). It was found that as the boundary layer thickens rapidly over the stern region, the turbulence intensity is reduced and becomes more uniformly distributed. The measured mixing length of the thick axisymmetric stern boundary layer was found to be proportional to the square root of the area of the turbulent annulus between the body surface and the edge of the boundary layer. This simple similarity hypothesis for the mixing length improved the prediction of the mean velocity distribution in the entire stern boundary layer.
Based on the above observation and results indicated in Reference 7, it is decided that algebraic Baldwin-Lomax turbulance model with Degani-Schiff's correction and Huang's modification is appropriate for present simuation.
RESULTS
The experimental data reported in References 9, 11 are massive and extensive. Our present interests are limited to the distributions of transverse forces along the bare hull and the pitching moments about the center of buoyancy of the hull at several given incidence angles. The data were presented in terms of the dynamic pressure (denoted by q) of the air stream and were corrected for the difference between the local static pressure in the stream and the reference pressure. The correction consisted simply of substracting from the pressure at any section of the model the static pressure of the air stream, measured in the absence of the model, at the corresponding point along the axis of the model. The correction reduced the pressure at the stagnation point at the nose of the hull, with the model at 0º pitch, to a value equal to the dynamic pressure q. Here, the dynamic pressure q is defined as: , where ρ is the density of the air and V∞ is the air stream velocity. Tests were conducted with the air stream at several different dynamic pressures. The highest value was 1, 225.73 Pa (25.6lb/ft2), the equivalent Reynolds number is about 17 millions based on body length. Based on this condition, the numerical simulations were carried out.
Predictions with the potential-based Munk and Upson equations of the transverse force at 15º of pitch were shown in Reference 11. Both predictions deviated substantially from the measurements near the stern region. The disagreements are not a surprise, since the distribution of force along the hull is strongly influenced by the surface flow separations. An engineering rational flow model based on a discrete vortex cloud method greatly enhanced the prediction (20). The improvement was attributed to a separation line model that defines the body vortex feeding sheets along the body surface. Flow visualizaton indicated that the separation patterns along a smooth surface can be quite complicated. Any further improvement in prediction at current stage may require a turbulent viscous flow approach. The

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present numerical simulation assesses the feasibility and accuracy of RANS's predictions.
The profile and wire-framed perspective view of the hull are shown in Figure 1. Pressure distributions along the hull at 0º of pitch are shown in Figure 2, where the pressure is normalized with the dynamic pressure q and the distance from the nose is normalized with the hull length L. The experimental values were obtained from averaging the circumferential measurements at each of the axial locations where the pressure orifices were placed. The viscous flow solution was obtained from a computation on a 79×81×83 grid. At the mid-section of the hull, there are 28 grid points located inside the boundary layer. The CFL number used in the computation is 10. The L2 norm of the residual and the lift coefficient during the course of the iteration are shown in Figure 3. The lift force is nomalized with q(vol)2/3, where vol is the volume of the hull. The potential flow solution was obtained from a surface panel method VSAERO (21). At zero incidence, experimental data indicate that there exist a small amount of sectional transverse force along the hull at the bow and after portions of the hull. It was assumed that the air flow was not strictly axial or that the model was not exactly symmetrical.
The transverse forces along the hull at several incidence angles are shown in Figure 4. Notice that the integration of the areas underneath the curves gives the total normal forces acting on the hull. The experiment data are obtained from Table V in Reference 11. The computational results are obtained from solutions based on two grids with different densities in circumferential direction. At a given incidence, the difference between the two computational results is insignificant. Noticeable differences between experimental and computational results can be found in the stern region. The predicted and measured total transverse forces on the hull, normalized with q(vol)2/3, are shown in Figure 5 and the values are given in Table 1. The discrepancy is more perceptible at higher incidence. It was reported (11), that at high-speed, high-pitch-angle condition, the model was observed to be quite unsteady.
The axial location of the center of bouyancy of the hull is 2.77 m. (9.10 ft.) from the nose. The pitching moment about the center has two parts: (1) moment (M1) of the transverse force and (2) moment (M2) of the longitudinal force. The experiment value M1 was obtained by taking the moment of the area of the transverse force curves in figure 4 about the center of buoyancy by means of a mechanical integrator. To obtain M2, curves with transverse force at each axial location plotted against the corresponding cross-sectional area, were constructed. M2 values were then obtained by integrating the areas under the curves. The contribution of the longitudinal forces to the total moment is about 4%, and it is opposite in direction to that due to the transverse force. Lift and moment coefficients are shown in Figure 6. The Lift is normalized with q(vol)2/3, and the moment is normalized with q(vol). The values are tabulated in Tables 2 and 3 respectively. The computed values of M1 and M2 are listed in parenthesis.
In general, the measurements and the predictions are in good agreement, Noticeable differences occur only at higher incidence.
Computations have been carried out both on CRAY-YMP and CONVEX-3080 machines. Estimated CPU times are about 40μ sec per grid per iteration on CRAY-YMP and 200μ sec per grid per iteration on CONVEX-3080 in the vector mode.
CONCLUSIONS
Numerical simulations of flow about a 1/40-scale model of U.S. Airship “Akron” were carried out with RANS formulation. The effort is an attempt to predict the hydrodynamic force acting upon a body of revolution type hull form at incidence. Good and encouraging results are obtained. Further enhencement in accuracy and computational efficiency requires a improved turbulence model and a multigrid type approach. Data bases obtained with modern techniques under controlled environments are needed for validation of numerical schemes.
ACKNOWLEDGEMENT
This work was sponsored by Program Element 62323N at David Taylor Model Basin. Computing resources on a CRAY-YMP are provided by the NASA Ames Research Center under the NAS Program. The U.S. Navy Hydrodynamics/Hydroacoustics Technology Center provided computational support on a CONVEX-3080 and various work stations.
REFERENCES
1. Ramaprian, B.R., Patel, V.C., and Choi, D.H., “Mean-flow Measurements in the Three-dimensional Boundary Layer over a

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Body of Revolution at Incidence,” Journal of FluidMechanics, Vol.103, 1981, pp.479–504.
2. Intermann, G.A., “Experimental Investigation of the location and Mechanism of Local Flow Separation on a 3-Caliber Tangent Ogive Cylinder at Moderate Angles of Attack,” M.S. Thesis, Universith of Florida, Gainesville, FL. 1986
3. Kim, S.E., and Patel, V.C., “Separation on a Spheroid at Incidence: Turbulent Flow,” The Second Osaka International Colloquium on Viscous Fluid Dynamics in Ship and Ocean Technology, September 27–30, 1991, Osaka.
4. Vatsa, V.N., Thomas, J.L., and Wedan, B.W., “Navier-Stokes Computations of Prolate at Angle of Attack,” AIAA Journal, Vol. 26, NO.11. 1989, pp.986–993.
5. Degani. D., Schiff, L.B., and Levy, Y., “Numerical Prediction of Subsonic Turbulent Flow over Slender Bodies at High Incidence,” AIAA Journal, Vol.29, No.12, 1991, pp.2054– 2061.
6. Hartwich, P.M., and Hall, R.M., “Navier-Stokes Solution for Vortical Flow over a Tangent-Ogive Cylinder, ” AIAA Journal, Vol.28, No.7, 1990, pp.1171–1179.
7. Sung, C-H., Griffin, M.J., Tsai, J.F., and Huang, T.T., “Incompressible Flow Computation of Force and Moments on Bodies of Revolution at Incidence,” AIAA-93–0787, 31st Aerospace Science Meeting and Exhibit, Jan. 11–14, 1993, Reno, NV.
8. Meir, H.V., and Cebeci, T., “Flow Characteristic of a body of Revolution at Incidence,” 3rd Symposium on Numerical and Physical Aspects of Aerodynamic Flows, Long Beach, California, 1985
9. Freeman, H.B., “Force Measurement on a 1/40-Scale Model of the U.S. Airship “Akron”,” T.R. No.432, NACA, 1932.
10. Freeman, H.B., “Measurements of Flow in the Biundary Layer of a 1/40-Scale Model of the U.S. Airship “Akron”,” T.R. No.430, NACA, 1932.
11. Freeman, H.B., “Pressure-Distribution Measurements on the Hull and Fins of a 1/40-Scale Model of the U.S. Airship “Akron”,” T.R. no.443, NACA, 1933
12. Chorin, A., “A Numerical Method for Solving Incompressible Viscous Flow Problems, ” Journal of Computational Physics, Vol.2, No.1, August, 1967, pp.12–26.
13. Roe, P.L., “Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes, ” Journal of Computational Physics, Vol.43, No.2, 1981, pp.357–372.
14. Harten, A., “High Resolution Scheme for Hyperbolic Conservation Laws,” Journal of Computational Physics, Vol.49, No.3, 1983, pp.357–393.
15. Yee, H.C., Warming, R.F., and Harten, A., “Implicit Total Variation Diminishing (TVD) Schemes for Steady-State Calculations,” Journal of Computational Physics, No.57, 1985, pp.327–360.
16. Degani, D., and Schiff, L.B., “Computation of Turbulent Supersonic Flows around Pointed Bodies Having Crossflow Separation,” Journal of Computational Physics, Vol.66, No.1, 1986, pp.173–196
17. Vatsa, V.N., “viscous Flow Solutions for Slender Bodies of Revolution at Incidence, ” Computers Fluids, Vol.20, No.30, 1991, pp.313,320
18. Gee, K., Cummings, R.M., and Schiff, L.B., “Turbulence Model Effects on Separated Flow about a Prolate Spheroid, ” AIAA Journal, Vol.30, No.3, 1992, pp.655–664
19. Huang, T.T., Santelli, N., and Bolt, G., “Stern Boundary Layer Flow on Axisymmetric Bodies,” 12th Symposium on Naval Hydrodynamics, Washington D.C., June 1978.
20. Mendenhall, M.R., and Perkins, S., “Prediction of the Unsteady Hydrodynamic Characteristics of Submersible Vehicles,” The Proceedings, 4th International onference on Numerical Ship Hydrodynamics, Washington D.C., Sept. 24–27 1985.
21. Maskew, B., “Prediction of Subsonic Aerodynamic Characteristics—A Case for low-order Panel Methods,” AIAA-81–0252, AIAA 19th Aerospace Sciences Meeting, January, 1981.

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Figure 1. Profile and perspective view of bare hull
Figure 2. Pressure distribution along hull at 0º pitch
Figure 3. L2 norm of residual and CL at 15º pitch

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
Figure 4. Transverse force along hull at incidences

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
Figure 6. Lift and moment coefficients
Table 1. Transverse Force Coefficients.
Incidence Angle (degrees)
Experiment
Computation
Grid # 1
Grid # 2
3
0.0127
0.0114
0.0110
6
0.0300
0.0294
0.0297
9
0.0541
0.0536
0.0515
12
0.0845
0.0883
0.0895
15
0.1246
0.1343
0.1330
18
0.1690
0.1939
0.1956
Grid # 1 : 79×81×83
Grid # 2 : 79×111×83
Table 2. Lift Coefficients.
Incidence Angle (degrees)
Experiment
Computation
Grid # 1
Grid # 2
3
0.011
0.011
0.011
6
0.029
0.028
0.027
9
0.054
0.051
0.047
12
0.080
0.083
0.081
15
0.115
0.125
0.121
18
0.155
0.178
0.173
Grid # 1 : 79×81×83
Grid #2 : 79×111×83
Table 3. Pitching Moment Coefficients.
Incidence Angle (degrees)
Experiment
Computation
Grid # 1
Grid #2
3
0.078
0.081
(0.084, −0.003)
0.081
(0.084, −0.003)
6
0.150
0.156
(0.162, −0.006)
0.153
(0.159, −0.006)
9
0.212
0.222
(0.230, −0.008)
0.222
(0.230. −0.008)
12
0.260
0.276
(0.286. −0.010)
0.271
(0.282, −0.010)
15
0.307
0.316
(0.327, −0.011)
0.310
(0.322, −0.012)
18
0.348
0.339
(0.352. −0.013)
0.347
(0.360, −0.013)
Grid # 1 : 79×81×83
Grid # 2 : 79×111×83

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
The Prediction of Nominal Wake Using CFD
A.J.Musker, S.J.Watson, P.W.Bull, and C.Richardsen
(Defence Research Agency, England)
ABSTRACT
A study of the effect of systematically applying different CFD methods and associated parameters is described for the case of the HSVA tanker. Attention is focussed on the propeller plane and the nominal wake in particular. The viscous solutions are compared with an inviscid solution and with experiment data in an attempt to discover how well current codes perform in terms of practical predictions of propeller inflow. It has been found that the flow in the outer region of the propeller disc can be defined with reasonable accuracy. However, the methods fail to describe the flow at the half-radius position.
NOMENCLATURE
Cμ
constant of proportionality for the eddy viscosity
k
turbulence kinetic energy
Lpp
length between forward perpendiculars
r
radial length from propeller axis
RD
radius of grid domain
R
propeller radius
Ut
tangential fluid velocity
Ux
axial fluid velocity at propeller plane
U∞
free-stream fluid velocity
w
Taylor wake fraction
w1
average circumferential wake fraction
w2
volumetric mean wake fraction
x
longitudinal distance from forward perpendicular
turbulence diffusion rate
θ
angle between propeller radius and horizontal radius
INTRODUCTION
In recent years a great deal of effort has been spent on developing numerical techniques to solve the Navier-Stokes equations of fluid motion. For practical reasons, these fundamental equations need to be ‘Reynolds-averaged' and, in so doing, some error is incurred in the modelling. Additional errors are incurred in the choice of the closing turbulence model and also in the various numerical processes invoked to solve the equations. These processes include the discretisation scheme, the grid resolution, cell disposition and quality, choice of solution algorithm and choice of convergence criteria.
This paper describes some recent experiences in predicting the nominal wake of a surface ship using advanced computational fluid dynamics procedures. The paper is the third in a series on CFD validation originating from the CFD Section at the Defence Research Agency, Haslar [1, 2]; these studies concentrate on the issue of numerical verification. A validated CFD capability should enable the designer to make use of the computed velocity field in the propeller plane to aid in the design of a suitable propeller. Not only would this enable more candidate hulls to be assessed and placed in rank order of performance, but it might also permit significant reductions in design costs to be gained.
Whilst the ship hydrodynamics community must continue to support and encourage the development of new methods to aid in ship design, it should, every once in a while, stop to examine the capability that currently exists and then match that capability to the requirements of the naval designer. In this way, any serious shortfall in

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inner and outer boundaries are respectively the solid contours (Cj)j=1,2 and the circles (Σj)j=1,2 of radius Σj centered at the position zi. The radii (∑j) j=1,2 are chosen so that the two outer boundaries (∑j) j=1,2 do not intersect, but they may touch in one point.
The two BVPs are solved successively and separately in the physical plane. The calculation of the boundary conditions for each BVP is described in a coordinate system centered at the position zi. The cartesian and polar coordinates are denoted respectively (x,y) and (,θ) whatever the BVP since those are solved separately and in an exactly similar way.
The interaction between the two bodies appears in the outer boundary conditions. On the boundary (Σj) one has to match the two components of the velocity. This is achieved by using the Green's second identity, which relates the following two functions:
the total stream function ψ,
a test function φ which is harmonic.
The resulting integral equation is:
(19)
where n denotes the outward normal on the contour (∂Ωe) The domain Ωe is limited by the outer boundary (∑j), the boundary of the other cylinder (Ci)i≠j and a circular control surface (∑∞) extending to infinity. The integral over the solid contour (Ci)i≠j vanishes. This follows from the Gauss Divergence Theorem, the fact that this surface is a streamline and that the normal gradient of the stream function (ψ,n) is exactly the tangential velocity. The latter must vanish on the contour due to the no-slip condition. It remains finally the contribution of the surfaces (∑j) and (∑∞) in the integral equation (19). On the surface (∑∞) the vorticity vanishes. Hence, the total stream function ψ is matched on (∑∞) with the stream function of the unperturbed flow plus the stream function of a dipole in order to take into account the presence of the bodies. The behavior of ψ and its radial gradient on the surface (∑∞) is:
(20)
(21)
The test function is otherwise calculated by using the Method of Separation of Variables. The only possible set of solutions is:
(22)
where m is a positive integer number. The function φm is singular at the origin zi, which is outside the domain Ωe.
The stream function is decomposed in a truncated Fourier Series of the polar angle (θ), i.e.
(23)
The number of modes of the Fourier development is and Nθ denotes also the number of nodes in the azimuthal direction of the polar mesh. The expressions of ψ and φ are included in the Green's identity (19):
(24)
Here δ is the Kroenecker symbol and m is an integer number of the interval The equation for the mode 0 reduces to:
(25)
The right hand side of the last equation represents exactly the amount of circulation which lies outside the surface (∑j) and therefore balances the circulation which is inside the polar mesh due to the Kelvin Theorem. In the computations one may increase the stability of the simulations by taking the average of the total circulations lying inside and outside the surface (∑j):
(26)

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The development for the non-zero modes may be pursued by using equation (2). The integral term in the right hand side of equation (24) may be developed as:
(27)
The outer boundary condition is a mixed (or Fourier) condition. This condition is written for a given mode m and depends only on the radius Σj. For body Nºj, the BVP posed in the polar mesh may now be written as:
(28)
where the last condition is defined by equation (24). In order to be consistent, the two first equations are Fourier tranformed. Then, the resulting BVP is classicaly solved for each mode by a Finite Difference Method. The solution ψ gives, after differentiation, the total velocity which is used to convect the vortices lying inside the two polar meshes. The non-homogeneous Dirichlet condition in the BVP defined by equation (28) does not affect the results for the velocity.
NUMERICAL PARAMETERS
Two parameters govern the physics of the problem. These are the Keulegan —Carpenter number and the Reynolds number , where U and T are respectively the amplitude and the period of the oscillating current velocity, L is a characteristic length of the body shape. In the following applications, L denotes the diameter of one cylinder, that is to say L=2a. A third coefficient is usually introduced for the oscillating flows: the Stokes Parameter . In addition, for steady flows the Strouhal number where fo is the frequency of the vortex shedding, will be used.
The force acting on each body is calculated at each time instant of a simulation. The instantaneous force F(t) follows from integration of the Cauchy Stress Tensor whose radial and tangential components correspond respectively to the effect of the pressure and the skin friction.
For steady incident flows, the drag coefficient (CD) is calculated in the direction on the ambient flow, and the lift coefficient (CL) is calculated in the transverse direction.
The force coefficients for oscillating ambient flows are partially defined by the Morison's equation which states that the force F(t) in the direction of the ambient planar flow u(t) can be written:
(29)
Here |.| denotes the absolute value. The velocity oscillates at the circular frequency of oscillation ω and behaves as u(t)= U sin(ωt). The drag coefficient CD characterizes the force in phase with the ambient fluid velocity. The mass coefficient CM is connected with the force in phase with the ambient fluid acceleration. These two main coefficients are calculated from the history of the force by Fourier averaging over one cycle. This is performed according to the following two formulas:
(30)
(31)
RESULTS AND DISCUSSIONS
Computations have been performed for two arrangements of cylinders:
two tandem cylinders in either oscillating or steady ambient flows,
two side-by-side cylinders in a steady ambient flow.
Comparisons have been made with available experimental data. The numerical parameters are given for each arrangement and type of flow.
Two Cylinders in a Tandem Arrangement and in an Oscillating Ambient Flow
The two gaps d/a=4 and d/a=8 are investigated at a Stokes parameter β= 534. Comparisons are made with two sets of experimental data: by (17) at β=534 and by (18) at Re=2.5 · 104 and d/a=3 and 5.
The time simulations are always performed over a number of cycles of oscillations (Nc) as large as possible. Nc decreases mainly with an increasing Keulegan-Carpenter number and to a lesser extent with

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an increasing Reynolds number. The final force coefficient is the mean value over the total number of simulated cycles. Besides, the standard deviations, for each coefficient, are calculated over the total number of cycles of oscillations, in order to quantify how the force coefficients vary from one cycle to another cycle.
The time step has been chosen in order to have 100 time step per period up to KC= 2.5 that is to say . However, for higher KC, the time step is limited to = 0.025.
The simulations for two tandem cylinders have been performed by imposing numerical symmetry about the axis going through the centers of the two cylinders. This is necessary in order to avoid numerical difficulties with a possible orbital current generated during the reversal of the flow. As a matter of fact, a slight asymmetry of the boundary condition on the outer radius (r=r∑j)j=1,2, may induce an orbital current along the body contour. This amplifies very rapidly since the tangential velocity, calculated on the solid contour, provides the strengths of the newly created discrete vortices. Therefore the production of vortices carrying each a very large circulation, affects seriously the stability of the time simulation. The imposed symmetry can be justified up to KC≈4. However beyond that limit the use of the numerical symmetry is more questionnable. As a matter of fact, it has been observed by (19) that the wake behind one circle starts to be asymmetric in the interval .
The figures (3)(a,b) and (4)(a,b) show the results at respectively d/a=4 and d/a=8. The drag coefficients are identical for the two cylinders at small KC numbers. The minimum value of CD appears at KC=2.4 and KC=2 for d/a=4 and d/a=8, respectively. Then the drag coefficient increases with increasing KC-numbers to almost constant values of CD≈1.2 for d/a=4 and CD≈1.3 for d/a=8. The results show a small oscillatory behavior of CD as a function of KC for KC>6. This is most pronounced at d/a=4. The comparison with experimental data can be made in the interval . In that range the present numerical results and the experimental data by (17) do not show the same trend. Beyond KC=6, whatever the spacing, the computed drag coefficient does not increase but remains close to a constant value. However the numerical values match well with the experimental data by (18). These experimental data have been obtained at a high subcritical Reynolds number Re=2.5 · 104 while the highest Reynolds number in the present computations is Re=4272.
Concerning the mass coefficient, an asymptotic limit can be calculated at KC= 0. For the two gaps d/a=4 and d/a= 8 the mass coefficient tends respectively to CM≈1.97 and CM≈2.06. These values can also be compared to theoretical results for a single circular cylinder. For example (20) obtained an asymptotic expansion for small KC-numbers of the mass coefficient as a function of the Stokes parameter (β). According to (20), CM=2.09 for β=534.
For the computed mass coefficients do not follow the decreasing trend shown by the experimental data in (17). But the present numerical data and the experimental data by (18) match well at KC≈8.
Tables (1) to (4) summarize also the performed computations. The results show that the number of simulated cycles decreases as the KC-number increases. The reason is that the maximum number of discrete vortices has been reached (this is fixed to 200.000 in the computer program). In connection with the decreasing number of simulated cycles, one should note the increasing standard deviation with increasing KC-number.
Two Tandem Cylinders in a Steady Incident Flow
For the tandem arrangement in steady incident flow, (21) reported that there exists a critical spacing in the sub critical and laminar flow regimes. This critical spacing may vary in the interval d/a≈[6.8, 7.6]. Below these values, vortex shedding does not occur from the upstream cylinder but only from the downstream cylinder. Beyond this critical value, vortex shedding occurs separately from each cylinder. Results are obtained here for a spacing d/a=6 (that is to say below the critical spacing) at low Reynolds numbers . The numerical parameters of the time simulation are the following:
the time step is and the simulations are performed over 1000 time steps,
the radial discretization in the transformed plane is ,

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in the physical plane the radius of the outer boundary is and the number of radial nodes Nr varies with the Reynolds number from Nr =21 to 60.
The coalescence of vortices takes place in a annulus domain defined by [10., 11.5] and centered at the middle of the line joining the centers of the two cylinders. A grid of 5 radial nodes and Nθ=128 azimuthal nodes covers this annulus. Only that part of the annulus which overlaps the wake is used. The coalescence consists of replacing the vortices enclosed in that region by new vortices at the nodes of the grid. A necessary requirement is that the total circulation is constant before and after the coalescence. In the computations performed in this paper the coalescence occurred at time intervals .
In the range of computed Reynolds numbers and below , the instantaneous force signal, whatever the cylinder, reaches very rapidly a local maximum. Then the drag force of the upstream cylinder stabilizes around a positive constant value while the drag force of the other cylinder decreases rapidly to a negative value. Figure (5) gives an example of the force signal at Re=100. From the time history of the in-line force, the drag coefficient is calculated from the average of the instantaneous force over the interval [20,50].
The present numerical results for CD are compared in figure (6) with:
experimental data by (22) for Re= 3400,
experimental data by (23) for Re ∈
numerical results by (12).
The experimental data (see (24) p17) for a single cylinder are also plotted in order to show the effect of interference. The difference in the drag force between the two cylinders is noticeable. In particular the negative drag force acting on the dowstream cylinder should be noted. This follows from the presence of two large rolled-up vortices between the two cylinders. For higher Reynolds number than Re=1000, instabilities of the large scale vortical structures may occur in the gap. In that case longer time simulations are necessary for good estimations of the average drag coefficients.
Two Side-by-side Cylinders in a Steady Incident Flow
The influence of the gap width on the coefficients of the flow is studied at Re=200. Higher Reynolds number could be handled with the present numerical model, however long simulations are necessary in order to obtain the coefficients of the flow accurately enough. The studied spacings vary in the interval [2.5, 8]. The parameters of the time simulations are the following:
the time step is and the simulations are performed over 1000 time steps,
the radial discretization in the transformed plane varies with the spacing and the number of radial nodes is fixed by the condition ,
in the physical plane the radius of the outer boundary is chosen as large as possible and the number of radial nodes is never less than 10,
the parameters for the coalescence are identical to those used for the tandem cylinders in a steady incident flow.
The force coefficients are calculated in the interval ∈ [20,100] where all transient effects have disappeared. The Strouhal number is obtained by using a spectral analysis of the lift force signal over the same time interval.
The results are shown in figures (7) to (10). Comparisons are made with the experimental data reported by (21) and (25), and the numerical results by (12). The interval of studied spacings covers two typical regimes of vortex shedding for two circular cylinders in a side-by-side arrangement. A classification has been done by (21). In the first regime2 the vortex shedding is characterized by “narrow and wide wakes which are formed behind two identical pipes, respectively, and the gap flow forms a jet biased towards the narrow wake. The flow is bistable, i.e. the biased jet can switch in the opposite direction at irregular time intervals, and the narrow and wide wakes interchange behind the tubes”. In the next regime “both nearwakes are equal in size but the two vortex streets are coupled and mirror along the gap axis. The vortex shedding is synchronized, both in phase
2
the exact limits of the interval of spacing (d/a) for each regime may vary with the Reynolds number

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and frequency”. In the present computations the critical spacing, i.e. the limit between the two regimes, is about d/a=4. This is in agreement with the observations by (21) and the numerical results by (12). This is illustrated in figure (7) where the Strouhal number is plotted. Three frequencies may be identified according to (12) and (25) in connection with the bistable regime of the biased flow. It was the measurements of harmonic modes of vortex shedding that gave these three frequencies in (12) and (25). In the present results, only two frequencies appear clearly through the spectral analysis of the lift force signals. The analysis is difficult at d/a=4. For higher values of d/a, the Strouhal number is close to the experimental value (St≈0.19) for a single circle at Re=200 (see (24) p32).
1
The leading edge of the cavity will be assumed to be known, the rest of the surface SC(t) is to be determined. See section 6 for related discussion.
2
the exact limits of the interval of spacing (d/a) for each regime may vary with the Reynolds number
The computed drag coefficients are shown in figure SF. Below d/a=4, the numerical results do not reach a minimum as shown by the experimental data by (21) and (25) and also by the numerical data by (12). However there is a large difference in Reynolds numbers between the present numerical results and the experimental data reported by (21) and (25). Above d/a=6 the calculated drag coefficient approaches the experimental value of Cd≈1.5 for a single cylinder at Re=200 (see (24) p17).
The lift force is decomposed into a mean value and an oscillating component. The mean value is plotted in figure (9). The sign of the lift forces on the two cylinders are opposite and act in a repulsive manner. The amplitude of the oscillating component is plotted in figure (10) for d/a>4, where the spectral analysis of the lift force signal provides only one sharp peak.
CONCLUSIONS
A theoretical model for viscous flows around two cylinders is presented. This model is based on the Vortex-In-Cell Method combined with a Random Method. The interaction is taken into account by using a conformal mapping technique and the Green's Theorem. The numerical results are validated by studying two identical cylinders in tandem and side-by-side arrangements. The force coefficients are calculated for both steady and oscillating ambient flows and compared with experimental data. The agreement is generally good. The present model can handle two cylinders of arbitrary shape by using additional conformal transformations.
BIBLIOGRAPHIC REFERENCES
1. Christiansen J.P., “Numerical simulation of hydrodynamics by the method of point vortices. ”, Journal of Computational Physics 13, , 1973, pp 363–379.
2. Chorin A.J., “Numerical study of slightly viscous flow.”, Journal of Fluid Mechanics, Vol. 57, part 4, 1973, pp 785–796.
3. Scolan Y.M. and Faltinsen O.M., “Numerical prediction of vortex shedding around bodies with sharp corners at arbitrary KC-numbers” Osaka Colloquium '91, Japan, 1991.
4. Smith P.A., “Computation of viscous flows by the Vortex Method”, Ph.D. Thesis, University of Manchester, 1986.
5. Smith P.A. and Stansby P.K., “Impulsively started flow around a circular cylinder by the vortex method”, Journal of Fluid Mechanics Vol. 194, 1988, pp 45–77.
6. Smith P.A. and Stansby P.K., “An efficient surface algorithm for random—particle simulation of vorticity and heat transport”, Journal of Computational Physics, Vol. 81, nº 2, 1989.
7. Smith P.A. and Stansby P.K., “Viscous oscillatory flows around cylindrical bodies at low Keulegan —Carpenter numbers using the Vortex Method”, Journal of Fluid and Structures, Vol. 5, 1991, pp 339–361.
8. Stansby P.K. and Dixon A.G., “Simulation of flows around cylinders by a lagrangian vortex scheme ”, Applied Ocean Research, Vol. 5, Nº 3, 1983.
9. Turner J.T., “Measurement of the mean and fluctuating pressure levels around a circular cylinder in the wake of another.”, Report on an experimental study carried out for A/S Veritas Research, Simon Engineering Laboratories, University of Manchester, 1985.
10. Penoyre R. and Stansby P.K., “Pressure distribution on a circular cylinder in the wake of an upstream cylinder in a duct”, Internal report 1986, Simon Engineering laboratories, University of Manchester.
11. Vada T. and Skomedal N.G., “Simulation of supercritical viscous flow around two cylinders in various configurations” Technical Report nº 86–2020, A.S. Veritas Research, 1986.
12. Slaouti A. and Stansby P.K., “Flow around two circular cylinders by the Random-vortex method”, Journal of Fluid and Structures, Vol. 6, 1992, pp 641–670.
13. Lagally M., “The frictionless flow in a region around two circles”, Translated in NACA TM-626, 1929.

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14. Ives D.C., “A modern look on conformal mapping including multiply connected regions ”, AIAA Journal, Vol. 14, nº8, 1976, pp 1006–1011.
15. Suddhoo A., “Inviscid compressible flow past multi-element airfoils”, PhD thesis, University of Manchester, 1985.
16. Kober H., “Dictionnary of conformal representations”, Dover publications, inc. 1957.
17. Sortland B., “Force measurements and flow visualization on circular cylinders”, Internal report nº 86–0218, Marintek, 1987.
18. Ikeda Y., Horie H. & Tanaka N., “Viscous interference effect between two circular cylinders in harmonically oscillating flow”, Journal of the Kansai Society of Naval Architects , 1985.
19. Bearman P.W., “Vortex trajectories in oscillatory flow”, Proceedings of the International Symposium on Separated Flow Around Marine Structures, NTH, Trondheim, 1985.
20. Wang C.Y., “On high-frequency oscillating viscous flows”, Journal of Fluid Mechanics, Vol.32, 1968, pp 55–68.
21. Zdravkovich M.M., “The effects of interference between circular cylinders in cross flow ”, Journal of Fluid and Structures, Vol. 1, 1987, pp 239–261.
22. Tanida Y., Okajima A. and Watanabe Y., “Stability of a circular cylinder oscillating in uniform flow or in a wake”, Journal of Fluid Mechanics, Vol.61, part 4, 1973, pp 769–784.
23. Okajima A., “Flows around two tandem circular cylinders at very high Reynolds numbers”, Bulletin of the JSME, Vol. 22, nº166, 1979.
24. Schlichting H., “Boundary-Layer Theory”, McGraw-Hill Edt, 1979.
25. Williamson C.H.K., “Evolution of a single wake behind a pair of bluff bodies”, Journal of Fluid Mechanics, Vol. 159, 1985, pp 1–18.
Table (1): Tandem cylinders in oscillating ambient flow with the gap width d/a=4; variation of the numerically calculated force coefficients of the cylinder Nº1 with the Keulegan Carpenter number KC at β=534; CDf: drag coefficient due to skin friction; : drag coefficient due to pressure; CM: mass coefficient; σ: standard deviation of each coefficient through a total of Nc cycles of oscillation.
KC
Nc
CDf
CM
0.5
10
0.94
0.03
2.21
0.07
1.97
0.01
1.0
10
0.47
0.02
1.12
0.03
1.96
0.01
1.5
10
0.31
0.01
0.82
0.04
1.96
0.02
2.0
10
0.24
0.01
0.71
0.04
1.95
0.02
2.5
10
0.19
0.01
0.71
0.05
1.92
0.02
3.0
10
0.15
0.01
0.84
0.13
1.86
0.02
3.5
10
0.13
0.01
0.92
0.14
1.82
0.06
4.0
10
0.11
0.01
1.04
0.14
1.77
0.10
4.5
10
0.10
0.01
1.08
0.17
1.74
0.06
5.0
10
0.09
0.01
1.06
0.18
1.76
0.06
5.5
8
0.08
0.01
1.06
0.13
1.79
0.08
6.0
8
0.07
0.00
0.95
0.23
1.77
0.12
6.5
7
0.07
0.01
1.10
0.19
1.72
0.11
7.0
6
0.06
0.00
1.09
0.18
1.74
0.09
7.5
6
0.06
0.00
0.92
0.11
1.83
0.15
8.0
6
0.05
0.00
0.98
0.10
1.86
0.12
Table (2): Tandem cylinders in oscillating ambient flow with the gap width d/a=4; variation of the numerically calculated force coefficients of the cylinder Nº2 with the Keulegan Carpenter number KC at β=534; CDf: drag coefficient due to skin friction; : drag coefficient due to pressure; CM: mass coefficient; σ: standard deviation of each coefficient through a total of Nc cycles of oscillation.
KC
Nc
CDf
CM
0.5
10
0.93
0.02
2.32
0.10
1.97
0.01
1.0
10
0.47
0.01
1.22
0.07
1.97
0.02
1.5
10
0.31
0.01
0.87
0.04
1.96
0.01
2.0
10
0.23
0.01
0.72
0.04
1.95
0.02
2.5
10
0.18
0.01
0.74
0.07
1.92
0.02
3.0
10
0.15
0.01
0.82
0.13
1.87
0.03
3.5
10
0.13
0.01
0.94
0.12
1.83
0.04
4.0
10
0.12
0.01
1.03
0.15
1.84
0.03
4.5
10
0.10
0.01
1.02
0.12
1.80
0.04
5.0
10
0.09
0.00
1.05
0.16
1.78
0.05
5.5
8
0.08
0.01
1.08
0.37
1.74
0.15
6.0
8
0.07
0.01
0.91
0.42
1.90
0.14
6.5
7
0.07
0.00
1.03
0.13
1.81
0.09
7.0
6
0.06
0.00
0.97
0.14
1.85
0.11
7.5
6
0.06
0.00
1.16
0.20
1.75
0.09
8.0
6
0.06
0.00
1.06
0.10
1.76
0.14

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Table (3): Tandem cylinders in oscillating ambient flow with the gap width d/a=8; variation of the numerically calculated force coefficients of the cylinder Nº1 with the Keulegan Carpenter number KC at β=534; CDf: drag coefficient due to skin friction; : drag coefficient due to pressure; CM: mass coefficient; σ: standard deviation of each coefficient through a total of Nc cycles of oscillation.
KC
Nc
CDf
CM
0.5
10
0.98
0.02
2.61
0.12
2.06
0.01
1.0
10
0.49
0.02
1.38
0.09
2.06
0.02
1.5
10
0.33
0.01
0.98
0.06
2.05
0.02
2.0
10
0.25
0.01
0.83
0.04
2.04
0.02
2.5
10
0.20
0.01
0.91
0.08
1.99
0.01
3.0
10
0.16
0.00
1.08
0.17
1.92
0.02
3.5
10
0.14
0.01
1.11
0.20
1.94
0.04
4.0
10
0.12
0.01
1.22
0.17
1.94
0.06
4.5
10
0.11
0.01
1.14
0.17
1.94
0.10
5.0
9
0.10
0.00
1.17
0.17
1.91
0.08
5.5
8
0.09
0.00
1.19
0.14
1.88
0.05
6.0
7
0.08
0.00
1.20
0.12
1.89
0.08
6.5
7
0.07
0.00
1.22
0.21
1.92
0.12
7.0
6
0.07
0.00
1.20
0.14
1.93
0.19
7.5
6
0.06
0.00
1.16
0.08
1.81
0.03
8.0
5
0.06
0.00
1.17
0.04
1.84
0.09
Table (4): Tandem cylinders in oscillating ambient flow with the gap width d/a=8; variation of the numerically calculated force coefficients of the cylinder Nº2 with the Keulegan Carpenter number KC at β=534; CDf: drag coefficient due to skin friction; : drag coefficient due to pressure; CM: mass coefficient; σ: standard deviation of each coefficient through a total of Nc cycles of oscillation.
KC
Nc
CDf
CM
0.5
10
0.97
0.04
2.63
0.11
2.07
0.02
1.0
10
0.49
0.02
1.34
0.07
2.06
0.01
1.5
10
0.33
0.01
0.98
0.05
2.05
0.02
2.0
10
0.25
0.01
0.83
0.03
2.04
0.02
2.5
10
0.20
0.01
0.89
0.05
1.99
0.01
3.0
10
0.16
0.00
0.98
0.11
1.94
0.02
3.5
10
0.14
0.01
1.11
0.16
1.92
0.03
4.0
10
0.12
0.01
1.14
0.21
1.94
0.06
4.5
10
0.11
0.00
1.13
0.11
1.94
0.04
5.0
9
0.09
0.00
1.17
0.10
1.94
0.09
5.5
8
. 0.09
0.01
1.18
0.11
1.84
0.08
6.0
7
0.08
0.00
1.18
0.11
1.87
0.08
6.5
7
0.07
0.00
1.12
0.13
1.86
0.08
7.0
6
0.07
0.00
1.14
0.20
1.87
0.14
7.5
6
0.06
0.00
1.10
0.19
1.95
0.23
8.0
5
0.06
0.00
1.08
0.26
1.92
0.18
Figure 1: Conformal Mapping for two cylin-ders—Notations in the physical and the transformed planes

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Figure 2: Unseparated flow around two staggered cylinders. The direction of the ambient flow is indicated by the big arrow. The arrow plot shows the velocity field calculated by equation (13). The start point of each arrow in the physical plane is the image of one mark (•) in the transformed plane. The image of the physical origin and infinity are marked with (○) and (□), respectively.
Figure 3: Force coefficients of two tandem circles at β=534 with the gap width d/a=4; drag and mass coefficients respectively in figure (a) and (b); (——): cylinder Nº1; (– – –): cylinder Nº2; (Δ): experimental data from (17); (□): drag due to friction (present method); (◇): drag due to pressure (present method); (○): total drag (present method); (∇): mass coefficient (present method); (–·–·–): experimental data from (18) at Re=2.5 · 104 and d/a=3 and 5.

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Figure 4: Force coefficients of two tandem circles at β=534 with the gap width d/a=8; drag and mass coefficients respectively in figure (a) and (b); (——): cylinder Nº1; (– – –): cylinder Nº2; (Δ): experimental data from (17); (□): drag due to friction (present method); (◇): drag due to pressure (present method); (○): total drag (present method); (▽): mass coefficient (present method); (–·–·–): experimental data from (18) at Re=2.5 · 104 and d/a=8.
Figure 5: Drag force signal (Fx) of two tandem circles with spacing d/a=6 in a steady ambient flow at Re=100; (——): upstream cylinder; (-------): dowstream cylinder.
Figure 6: Drag coefficients CD for two tandem cylinders with the spacing d/a=6; the empty marks are connected to the downstream cylinder; (– – –): experimental data from (24) for a single cylinder; (▽): experimental data from (22); (◇): experimental data from (18); (Δ): numerical results from (12); (□): present numerical results

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Figure 7: Strouhal number of two side--by-side cylinders in a steady incident flow at Re=200; (×): experimental data reported by (21) and (25); (Δ): upper cylinder (present numerical results); (▽): lower cylinder (present numerical results); (○): numerical results from (12) for Re=200; (——): experimental data for a single cylinder reported by (24) for Re=200.
Figure 8: Drag coefficients of two side-by--side cylinders in a steady incident flow at Re=200; (Δ): upper cylinder (present numerical results); (▽): lower cylinder (present numerical results); (○): numerical results from (12) for Re=200; (+, ×, *): experimental data reported by (21) and (25) at Re=8 · 103, 2.5 · 104 and 6 · 104, respectively; (——-): experimental data for a single cylinder reported by (24) for Re=200.
Figure 9: Mean value of the lift coefficients of two side-by-side cylinders in a steady incident flow at Re=200; (Δ): upper cylinder (present numerical results); (▽): lower cylinder (present numerical results); (○): numerical results from (12); (+, ×, *): experimental data reported by (21) and (25) at Re=8 · 103, 2.5 · 104 and 6 · 104, respectively.
Figure 10: Amplitude of the oscillatory component of the lift coefficients of two side-by--side cylinders in a steady incident flow at Re=200; (Δ): upper cylinder (present numerical results); (▽): lower cylinder (present numerical results); (○): numerical results from (12).

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