Numerical investigation of flow field over four in-line circular cylinders for different spacing ratio

The objective of this paper is to study the effect of spacing ratio on the wake dynamics in an inline arrangement of four cylinders. Different spacing ratios, defined as ratio of distance between center of two cylinders and diameter of cylinder, are considered while. Reynolds number (Re) kept constant. At the chosen Re, far wake is expected to transition to turbulence, and hence κ-ω SST turbulence model is used for turbulence closure. The flow field and associated surface properties such as drag and lift force are strong function of spacing ratio. In the numerical solutions it is observed that, two counter rotating vortices are formed in the gap between the cylinders. These two vortices found to move towards downstream as the spacing ratio is increased. Also, the size of the vortices found increased. The two counter rotating vortices appears symmetric near cylinders, and hence lift coefficient observed to be very low for the first three cylinders. Lift coefficient observed on fourth cylinder found to one order of magnitude higher. The drag coefficient for all cylinders found to be positive except for the second cylinder which has negative drag for all spacing ratios.


Introduction
Bluff bodies have been subjected to extensive research especially flow over circular cylinders due to many practical applications such as risers, off shore structures, bridge pillars which show the phenomenon of vortex shedding whose frequency might match the natural frequency of these structures resulting in potential failure of these structures. The shear layers which originate from flow separation point roll up in wake and finally sheds in cylinder downstream. The flow over circular cylinders have been subjected to lots of research over the last few decades. Mainly these studies have been done to find the relationship between the flow parameters such as velocity, pressure distribution, hydrodynamic forces & Strouhal number. Many of the researches have performed numerical studies on flow over single & multi cylinders in different arrangements. The important types being tandem and staggered arrangements to understand the effect of this geometry on various flow parameters. Also there have been extensive researches performed to find out the effect of variation of non-dimensional parameter Reynolds number and the spacing ratio (S/D) between the cylinders on the hydrodynamic force coefficients i.e. drag and lift coefficients and also a non-dimensional parameter called Strouhal number used to quantify the vortex shedding occurring in the wake of the cylinder the frequency of which if matches the natural frequency of the structures then resonance might damage the structures which is why these studies are important and been performed.
Tong et.al. [1] investigated different cylinders in square arrangement at Re 100 to 500 at a fixed spacing of L/D = 2.0. From Reynolds number of 100 to 220, the array of four cylinders behave as a single cylinder and their wake has a single Karman street. Wake flow behind two tandem cylinders was studied numerically by Dekhordi et al. [2]. The variation of spacing ratio, drag and lift coefficients and Strouhal number are investigated. Reynolds number in the range between 100 to 200 are in laminar regime and higher values like 2.2 x 10 4 are consider as turbulent. If the spacing ratio is less than 3, then drag coefficient of second cylinder found to be negative, and vortices are formed downstream cylinder only. Sinhamahapatra et.al [3] studied the influence of spacing ratio in two tandem cylinders arrangement. Flow development is reported as a function of Reynolds number. For Re below 40, flow achieves steady state for any spacing ratio. When Reynolds number is 70, and spacing ratio is greater than 2, an unsteady flow is observed. For Reynolds number greater than 100, for all spacing ratio, flow found to be unsteady. At lower gaps the vortices formed by the front cylinder impinges over the rear cylinder which definitely influences its hydrodynamic forces and causing the drag and lift to oscillate at a higher frequency than upstream cylinder. Strouhal numbers are not much different for the two cylinders because of the uniform and in-sync vortex shedding from the cylinders.
Harichandan et. al [4] have investigated the simulation on multiple cylinders by creating unstructured mesh. The cylinders were placed in tandem as well as in-line. Reynolds number considered are 100 and 200, and spacing ratio chosen are 1.5 D and 3 D for side by side; 2 D and 5 D for tandem arrangement. It was found that the increased spacing reduces the interference effect between the two cylinders. For tandem arrangement, it is clear from the vorticity contours that vortex from upstream cylinder influences the hydrodynamic force coefficients of the downstream cylinders. When the spacing increases to 5 D, the upstream cylinder having no influence on downstream. Also, non-dimensional frequency was constant in any arrangement. Wake dynamics of the array of cylinders at different spacing ratio both in tandem and side by side arrangement was studied by Zdravkovich [5]. It was observed that different wake patterns at different spacing when the cylinders were placed in tandem and in side by side arrangement. Further, he observed that at certain spacing the downstream cylinder was interfering with wake of upstream cylinder and hence the shedding of upstream was affected. Sumner et.al [6] has studied Strouhal number variation for differed P/D ratio (pitch ratio) of two staggered cylinders of equal diameter, angle of inclination between upstream and downstream cylinders was varied from 0° to 90°. They observed reattachment and gap flows were observed at smaller and higher spacing ratio. Vortex dynamics is much more complex for array of cylinders. The relationship between the forces and the phases of the vortex shedding has been studied numerically by Alama et al. [6].
The different spacing was investigated for Reynolds number 200 with tandem arrangement. They have used SIMPLE algorithm with second order upwind scheme in their numerical technique. The analysis was extended for dependence of the force coefficients on the phases of the shedding of the vortices. They observed lift coefficient fluctuation is higher in amplitude for cylinder 2 compared to rest cylinder. The value of drag coefficient is highest when they are in phase, minimum for antiphase. The effect of the phase lag is thus much less or negligible on lift coefficient of the downstream cylinder. Masami et al. [7] numerically investigated for the vortex formation (von Karman vortices) phenomenon in the wake of a cylinder and its effect on the oscillations caused on the cylinder. For the Reynolds number 50, the flow separated on the cylinder wall surface at about 130 degrees from clockwise direction. Increase in Reynolds number increases the length of vortex region and higher Reynolds number causes to form the vortices that last for a shorter duration which consequently increased the distance between them. With the increase in Reynolds number over 1000 the flow in the wake becomes very irregular. The drag force was plotted with respect to time and it was observed that with low Reynolds number the fluid force just reached steady state quickly but as the Reynolds number became higher and the vortices begin to form the forces were observed to oscillate and became harmonic after a period of time. Sayers [8] has carried out experiments on cylinders with constant spacing cylinders at Reynolds number of 3x10 4 in range of spacing from L/D 1.5 to 5 and at inclination angle (α) 0° to 180°. They observed that for four cylinders Strouhal number was higher than the single cylinder, due to array of cylinders behave as a single cylinder and at higher spacing wake of cylinder array is quite complex. Meneghini et al. [9] did numerical study on cylinder with different gap. They observed that up to the three times cylinder diameter downstream length, both cylinders behave as a single cylinder and perform shedding. After this gap, both cylinders shed independently. Ding et al. [10] studied the cylinders in tandem and side arrangement. They observed that up to gap ratio 2.5, flow is steady but with higher gap ratio flow becomes unsteady.
All the numerical studies have been carried out using different numerical schemes, many researchers have adopted different numerical models for solving the Navier-Stokes equation. Abaqus was used by Masami et al. [1] and a 3-D element called FC3D8 was used to model the fluid and a decoupled method was adopted to solve the Navier-Stokes Equation. RNG k-ε was used Dekhordi et al. [2] to model turbulence and a staggered grid was used which was non-uniform in particular and the refinement was done in the zones closer to the cylinders. Harichandhan et al. [4] developed an unsteady viscous two-dimensional finite volume Navier stokes solver using a flux reconstruction technique. Alama et al. [6]. have used SIMPLE algorithm with second order upwind scheme in their numerical technique. Their grid is structured mesh and they have used the commercial code ANSYS-FLUENT package. Ding et al. [10] have employed finite difference method to carry out their numerical work whereas Meneghini et al. [9] performed a numerical study using finite element method.
In the present numerical work, four cylinders are taken in series with different spacing ratio for Re 200 which is not explored in past. Pressure coefficient, drag and lift coefficient are measured around each cylinder. Flow physics are explained with help of velocity contour.

Simulation Methodology:
ANSYS -Fluent is used for simulating the flow field over the array of cylinders. Simulations are performed on unstructured girds, which are generated using ANSYS Workbench meshing tool. The incompressible RANS equation is solved with pressure-based algorithm under steady state condition. κ-ω SST model is used to close the turbulence terms in RANS equation. This model is used to simulate the problem because of the satisfactory results it gives when it comes to capturing the viscous near-wall region and stream wise pressure gradients as was studied, suggested and reported by Rosetti et al. [13]. At inlet, uniform velocity is imposed. At outlet, zero-gauge pressure is applied. No-slip boundary condition is applied on cylinder walls. Figure 1 shows the nomenclature of cylinders and flow direction. Figure 2 shows a plane view of the gird in XY plane. Fine mesh is used at the near wall region of the cylinder wall. The girds are progressively stretched towards outlet boundary so that a coarse grid can be employed. The first gird point in the wall-normal direction is placed at a distance of 1.24e-07 mm from wall. The y+ value in majority portion of the wall is below 5. The total number of cells in various girds are shown in Table  1. Figure 1 shows geometry of the array of cylinders, this arrangement has the four cylinders in the C1, C2, C3, C4 in tandem respectively in the same order.   Figure 3 shows reduction in the predicted drag coefficient (Cd) as a function of the iterations for a typical simulation on logarithmic scale. There is a rapid drop in Cd from 0 to 100 iterations. In the sub sequent iterations, the reduction in the Cd is slower. The % change in Cd from 8000 to 10000iterations is less than 0.1%. As the drag coefficient (Cd) variation is negligible beyond 10000 iterations, solution is treated as converged.

Validation of CFD results
To validate the CFD results we compared the pressure distribution profile for flow over a single cylinder of our simulation with Rajani et al. [11] and Thom et al. [12] figure 4 and the profile was found out to be in pretty good agreement with little variation that is likely due to the steady state simulation.

Grid Convergence
To assess the gird point density to resolve flow gradients, a gird convergence study is performed for spacing ratio of 2. Simulations are performed by increasing the number of points by 50% from Gird 1 to Gird 2 and by 50% from Gird 2 to Gird 3. The near wall spacing is kept constant in the all the three grids. In the circumferential direction, the cylinder is divided into 100 divisions, 160 divisions and 200 divisions for grid 1, grid 2 and grid 3 respectively. Predicted coefficient of pressure (Cp) obtained from converged solutions on the grid 1, grid 2 and grid 3 are shown in Figure 5. For cylinder C1, there is not much change in the shape of the pressure distribution with the refinement in the grid which shows that the cylinder is facing the free stream so there is not much effect of grid refinement. For cylinders 2,3 and 4, the solution changes for entire 360 deg as the grid is refined. As these three cylinders faces the separated flow from upstream cylinders, the flow field is more susceptible to grid refinement. The solution obtained from coarse grid has varies a little bit from that of medium and finer grids. In case of medium and fine grids give a smoother variation of Cp. Again, the solution obtained from medium and finer grids are very closely identical. Over all, further refining the mesh from gird 2 to gird 3 does not yields any significant changes. Hence a gird 3 of 200 number of divisions is used for simulation. Enlarged views near the cylinders reveal the effect of spacing ratio on local flow features. The shear layer originated from cylinder C1, passes over cylinder C2. The extent of impingement of shear layer onto C2 is noticeable but not significant. A further enlarged view of the vector flow field around C2 is shown in Fig.7. This clearly shows formation of two counter rotating vortices in the space between C1 and C2. These vortices deflect the shear layer upward so that impingement of upstream shear layer from C1 onto downstream cylinder C2 in minimal. The Cylinder C2 will not face freestream, but it is encountered by the vortex flow from C1. A part of the vortex flow deflects towards C2, and moves along the surface of C2. However, the flow direction is opposite to that of freestream. This bifurcated flow marches towards base stagnation point of C1 and circulates around the counter rotating vortices. The gap flow between cylinder C2 and C3; C3 and C4 are similar.

Variation of Velocity magnitude for different spacing ratio
Downstream of C4, a complex behavior of flow can be observed. The two counter rotating vortices persist downstream of C4 also. But the global shear layer layers from top and bottom side of the cylinders will converge farther downstream of C4. This results in formation of another vortex. This vortex is placed in between the counter rotating vortices and merging point of shear layer. The center of this vortex lies close to the central axis of the cylinders.
As the spacing ratio increases, the two counter rotating vortices are prominent, and size of the vortices increases. The center of vortex also shifts towards downstream cylinder. The shape, size and orientation of single vortex that is formed downstream of C4 also changes with respect to the spacing ration. For spacing ratio of 1.5, orientation of vortex in downward and it is regular circular shape, whereas for spacing ratio of 3.5, the orientation is towards upward and it is in the form of triangle.

Pressure coefficient (Cp) variations with spacing ratio
Coefficient of pressure (Cp) for C1 is presented in Figure 8(a). As expected, at the nose stagnation point (theta =0), Cp value obtained from all solutions is nearly same. This indicates the Cp value at nose stagnation is not influenced by the spacing between the cylinders. The flow separation region is marked by the low values of Cp, and it spans approximately from 60 deg to 270 deg. The flow separation angle is reducing with increase in the spacing between the cylinders this is essentially due to the fact that the shear layer impingement of the upstream cylinders is reducing with the increase in spacing so it's affecting the separation of the boundary layers in the downstream cylinders. The effect of reduction in flow interaction or interference can also be seen in the graph as the increase in the range of Cp values with increased spacing. The pressure distribution curve shifts higher showing a reduced effect of interference.   The Cp values over entire circumference of the cylinder are lower compared to those observed for C1. This is due to the fact that, the cylinder C2 placed in the wake of C1. So C2 is exposed to separated flow in its entire circumference. Also same is the cases for C3, C4 which are kept in the wake or low-pressure region of their respective upstream cylinders we can see the plot to be different from that of the C1. The trends observed for C3 and C4 are different though they also show an increase in values with increase in spacing. Apart from the first cylinder C1, the nose stagnation point is not occurring at theta=0 that's because they are not facing the free stream but facing the low-pressure wake of the previous cylinders.

Comparison of Drag and Lift Coefficients
Comparison of drag and lift coefficients for different spacing ratio is presented in Table1. For all the spacing ratios, drag coefficient for C2 is found out to be negative which is in agreement with the literature works which says that the drag value will become positive above a critical value of S/D=3. This negative value is due to the C2 being placed in the low pressure region of the C1 cylinder the difference in the pressure between upstream and downstream is negative which ultimately results in negative drag which also means that this cylinder feels a force of attraction towards the C1 unlike other cylinders which are being forced in the direction of the free stream. The drag experienced by C1 in all the cases is higher than C2, C3, C4 since it's facing the free stream on its upstream side. The drag value for C3 changes to positive and it further is greater for C4 as can be seen in the reported values in the table. The magnitude of drag is continuously increasing with S/D 2, 2.5,3 which is obviously due to the reduction in the interference effects between the four cylinders. The lift coefficients reported in the table depict that the lift generated is only considerable for the extreme downstream cylinder i.e. C4. In fact, this observation is true for all the four spacing ratios. And the coefficients of lift reported for other three cylinders are much lower as compared to the C4 which shows that the vortex shedding phenomenon is only occurring from the this cylinder since the other cylinder such as C1, C2, C3 do not have sufficient space for the shedding of vortices by separation of the shear layers which is made pretty much clear in the vector images of these cylinder in figure 5 they are depicted by two contra rotating vortices in their downstream region.

Conclusions
The CFD Analysis of the flow-field indicates that the spacing ratio has dominant effect on the flow pattern and surface properties such as drag and lift coefficients. The vortex dynamics and the hydrodynamic force coefficients were reported and studied and it was found out that the C2 experiences a negative drag but all other cylinders are having positive drag coefficient. The wake of the cylinders C1, C2, C3 are having stable contra rotating vortices which is clearly having an impact on the distribution of pressure over the cylinder surfaces which has also been plotted and analyzed with respect to the angle measured clockwise direction. The interference effects is more dominant for S/D=1.5 & 2whereas the effect clearly has impact over the drag and lift forces and also the von Karman street is formed in the far wake of the extreme downstream cylinder in all the four spacing ratios, which sheds alternating vortices from its surface which is the reason for fluctuating forces.