Keywords

1 Introduction

From life science to engineering applications, turbulent flows are found everywhere in daily life. The practical application of flow past a bluff body in the domain of thermal and structure system is huge, some of the related such applications can be observed in [1,2,3,4]. A flow field is considered either laminar or turbulent based on Reynolds number, which is the ratio of the inertia force to the viscous force. With the change of Reynolds number, the random motion of fluid develops a variation in velocity and pressure [5]. Consequently, the momentum, the mass, and the energy of the flowing fluids are exchanged. The largest turbulent eddies interact with the mean flow and extract energy from it through a process known as vortex stretching. The angular momentum of the larger eddies is conserved during the process of vortex stretching, which maintains turbulence in the flow. Over time these large eddies become unstable and break into further smaller eddies. In this process, there is a cascading of energy from small to the smallest eddies. The energy cascading stops when the Reynolds number of the smallest eddies becomes unity. At this point, the viscosity dominates, and the energy dissipation occurs [6] and the intensity of the turbulence becomes more acute.

The turbulence created around the bridge pier loosens the sediment particles around the pier and the sediments are carried away from the vicinity of the pier, resulting in scour formation. The scour is considered as one of the major reasons behind the failure of bridges. The recent studies dealing with the scour are but not limited to [7,8,9]. To analysis this, fluid engineers require a potent tool for simulating the turbulence around the pier. Generally, Reynolds Averaged Navier Stokes (RANS), Large Eddy Simulation (LES), Direct Numerical Simulation (DNS) are used to model the turbulence phenomenon. In RANS model, the mean flow and the impact of the turbulence on the mean flow are the main features, which need a closure rule. The Boussinesq hypothesis are the classical closure method, which were formulated in 1872. Also in the RANS model, no care is taken when smoothing out the model equation, meaning the whole spectrum of the fluctuations are averaged out, and no special preference is given to any mode or scale. A different procedure for modeling more strong eddies is the LES. In LES, a filter is added to the vector fields which remove the specific fluctuating modes. Thus LES captures larger eddies by considering space filtering of the Navier Stokes equations in spite of time averaging.

In this paper, a comparative study of the flow hydrodynamics around tandem cylinders have been carried out using OpenFOAM toolbox for circular cylinders. The RANS and LES models are applied to observe flow hydrodynamics around tandem cylinders. The authors found that the study of flow around the tandem cylinder for both RANS and LES in the same study in an open channel flow is limited. Here, the authors have presented a detailed study of the flow physics around two tandem cylinder mainly highlighting the behavior of force coefficients and the vorticity with the variation of Reynolds number and gap ratios. This study can be further enhanced by performing laboratory experiment for further confirmation.

2 Numerical Model

The simulations presented in this paper are carried out in OpenFOAM5.0, which is a freely available source of C++ codes [10]. PisoFoam, an editable solver in OpenFOAM, was chosen for the purpose of simulation, which solves three-dimensional Navier–Stokes equations (NSE) for a transient, incompressible and turbulent flow. The solver is based on a finite-volume method and PISO (pressure-implicit with the splitting of operators) algorithm for pressure–velocity coupling. For the convective terms in the Navier Stokes equation, the central difference scheme is used; for pressure gradient and inviscid conditions, fourth-order accuracy is applied; for time integration, second-order implicit Euler method is applied. The volume of fluid (VOF) is used to capture the air–water interface. Further, a fraction function, α, is defined as the volume fraction of water in each cell. α is 0 means the cell is filled with air, and α is 1 means the cell is occupied by water; otherwise, 0 < α < 1 means cells are bisected by the free water surface.

The continuity and the Navier Stokes equation, in their simplified form, adopted in OpenFOAM are given by Eqs. (1) and (2).

$$\frac{\partial {u}_{i}}{\partial {x}_{i}}=0$$
(1)
$$\frac{\partial {u}_{i}}{\partial t}+{u}_{j}\frac{\partial {u}_{i}}{\partial {x}_{j}}=-\frac{1}{\rho }\frac{\partial p}{\partial {x}_{i}}+\frac{\partial }{\partial {x}_{j}}\left[\left(\nu +{\nu }_{t}\right)\left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}\right)\right]+{g}_{i}$$
(2)

where u is time-averaged velocity, p is pressure component, v is eddy viscosity, vt is kinematic eddy viscosity and gi is the body force.

2.1 Rans

RANS reduces the computational cost by averaging the flow quantity. In RANS equations the main focus is given to model the Reynolds stress −ρuiuj. The turbulence closure is provided by k–ε [11] model as given by Eqs. (3) and (4).

$$\frac{\partial k}{\partial t}+u\frac{\partial k}{\partial x}+v\frac{\partial k}{\partial y}+w\frac{\partial k}{\partial z}=\frac{\partial }{\partial z}\left(\frac{{\nu }_{t}}{{\sigma }_{k}}\frac{\partial k}{\partial z}\right)+{v}_{t}\left({\left(\frac{\partial u}{\partial z}\right)}^{2}+{\left(\frac{\partial v}{\partial z}\right)}^{2}\right)-\varepsilon $$
(3)
$$\frac{\partial \varepsilon }{\partial t}+u\frac{\partial \varepsilon }{\partial x}+v\frac{\partial \varepsilon }{\partial y}+w\frac{\partial \varepsilon }{\partial z}=\frac{\partial }{\partial z}\left(\frac{{\nu }_{t}}{{\sigma }_{\varepsilon }}\frac{\partial \varepsilon }{\partial z}\right)+{C}_{\varepsilon 1}\frac{\varepsilon }{k}{v}_{t}\left({\left(\frac{\partial u}{\partial z}\right)}^{2}+{\left(\frac{\partial v}{\partial z}\right)}^{2}\right)-{C}_{\varepsilon 2}\frac{{\varepsilon }^{2}}{k}$$
(4)

where k is the turbulent kinetic energy and ε is the turbulent dissipation rate. The eddy viscosity is expressed by Eq. (5).

$${v}_{t}={C}_{1}\frac{{k}^{2}}{\varepsilon }$$
(5)

For the non-linear k-ε model, the following Boussinesq hypothesis is assumed to model the Reynolds stress (τi,j)

$${\tau }_{i,j}=2{v}_{t}{S}_{i,j}-\frac{2}{3}k{\partial }_{i,j}$$
(6)

where Si,j is the average strain rate tensor. Constants for Eqs. (3), (4) and (5) have been used from [12], i.e., C1,Cε1, Cε2, σk, σε = 0.09, 1.44, 1.92, 1.0 and 1.3, respectively. First, the model geometry is defined, and meshes are constructed in blockMesh, with all the necessary boundary conditions in the respective directions. After setting all other parameters, simulation is started by the command PisoFOAM (the name of the solver in openFOAM is treated as a simulating command). The final results are viewed in third-party software paraview. The density of water is taken as 1000 kg/m3 and the dynamic viscosity of the water is considered as 1.002 × 10–3 kg m−1 s−1 and kept constant for all of the cases.

2.2 Les

The LES has the capability to solve the 3D turbulence flow condition. The small eddies created in the flow field are modelled by sub-grid scale model, whereas the large scale eddies are calculated directly in the LES. To make this demarcation between small scale and large scale eddies, a filter function is used in NSE. The main equations in LES are obtained by filtering the NSE, as given below-

$$\frac{\partial \overline{{u}_{i}}}{\partial {x}_{i}}=0$$
(7)
$$\frac{\partial \overline{{u}_{i}}}{\partial t}+\overline{{u}_{j}}\frac{\partial \overline{{u}_{i}}}{\partial {x}_{j}}=-\frac{1}{\rho }\frac{\partial \overline{p}}{\partial {x}_{i}}+\frac{\partial }{\partial {x}_{j}}\left[\nu \left(\frac{\partial \overline{{u}_{i}}}{\partial {x}_{j}}+\frac{\partial \overline{{u}_{j}}}{\partial {x}_{i}}\right)\right]-\frac{\partial {\tau }_{i,j}}{\partial {x}_{j}}$$
(8)

where ūi,j = velocity component along the flow direction of the resolved scale and p = corresponding pressure. Here, the Reynolds stress is considered as SGS tensor, which is expressed as following-

$${\tau }_{i,j}=\overline{{u}_{i,j}{u}_{i,j}}-\overline{{u}_{i,j}}\overline{{u}_{i,j}}$$
(9)

For the Smagoinsky-Lilly model, the SGS eddy viscosity is modelled as

$${\mu }_{t}={\rho L}_{s}^{2}\sqrt{2{\overline{S}}_{i,j}{\overline{S}}_{i,j}}$$
(10)

where Ls = min (kd, Cs(∆x∆y∆z)1/3) = mixing length for subgride scale; d = distance between the nearest walls; κ = von-Karman constant, and Cs = Smagorinsky coefficients.

3 Mesh Convergence Study

Mesh convergence studies are performed to lower the cost of the simulation without affecting the accuracy of the results. In this study meshes are divided into three categories based on the number of elements, viz., coarse (A, D), medium (B, E), and fine (C, F) (Tables 1 and 2). Coefficient of drag (Cd) and coefficient of lift (Cl) are considered for the convergence test, which are calculated as Cd = 2Fx/ρu2D and Cl = 2Fy/ρv2D, where u and Fx denote stream wise velocity and force; v and Fy denote spanwise velocity and force.

Table 1 Mesh convergence study for RANS
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Table 2 Mesh convergence study for LES
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From Tables 1 and 2 it can be observed that the difference between the coefficient of lift for coarse and medium meshes, for both RANS and LES, are insignificant (0.14% between A and B; 0.42% between D and E). However, the observed difference between the coefficient of drag for coarse and medium are quite high (10.28% between A and B; 11.11% between D and E). Further investigating, the difference between the coefficient of lift as well as the difference between the coefficient of drag for medium and fine meshes, for both RANS and LES, are negligible (the difference of Cl between B and C is 2.44%, the difference of Cl between E and F is 2.7%, the difference of Cd between B and C is 1.52%, and the difference of Cd between E and F is 1.65%, respectively). Hence, mesh B for RANS and E for LES are considered in this study.

4 Validation

To validate the model, the coefficients of drag and lift obtained in this paper are compared with the coefficients of drag and lift presented by previous researchers. For the validation, coefficients of drag and lift in tandem cylinders are calculated based on Reynolds number, Re = 25,348.

At gap ratio (S/D) = 4 and Reynolds number Re = 25,348, the values of Cd1 = 1.1 and Cd2 = 0.45 for circular cylinder in tandeare obtained in this study, respectively. Similarly, At the gap ratio (S/D) = 4 and Reynolds number Re = 25,348, the values of Cl1 = 0.2 and Cl2 = 0.7 for circular cylinder in tandem are obtained in this study, respectively (Table 3). This result agree well with the results of [13, 14]

Table 3 Comparative results for tandem cylinders. Cd1 and Cd2 are the drag coefficients at the upstream and the downstream of the cylinder, Cl1 and Cl2 are the lift coefficients at the upstream and the downstream of the cylinder, respectively
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5 Results and Discussions

In this section, all of the configurations for different Reynolds numbers and gap ratios are analyzed. For all of the cases, smallest grid size is chosen as 1.2, 1.5 and 0.30 mm along x, y and z directions, respectively. Time-steps ∆t = 0.0001, 0.0002 and 0.00001 s are used in the simulations and the Courant number is maintained lesser than unity. Melville and Chiew [15] concluded that for getting equilibrium scour depth in a small-scale laboratory experiment, the experiment should be done for several days. However, the main objective of this study is to compare the flow hydrodynamics around twin piers in between RANS and LES; hence, the test duration is not considered very important.

The flow hydrodynamics around single cylinder are different compared to the flow hydrodynamics around two cylinders. The reason is that the flow at the downstream becomes unsteady and produces different vortex shedding zones in the middle of two cylinders and a discontinuity in the flow pattern is observed. This discontinuity depends on Reynolds number and center to center distance between two cylinders [16]. Zdravkovich [17] described the wake interference in tandem cylinder, where the upstream cylinder behaves as a single body and the wake created around the downstream cylinder is strongly influenced by the upstream cylinder.

Here, simulations have been carried out around two circular cylinders placed in tandem for Reynolds number, Re = 25,348 and 126,746 with gap ratio, S/D = 2, 3 and 4. The velocity at the inlet is considered as 0.5 m/s and 2.5 m/s. A comparison of flow hydrodynamics simulated by Reynolds Average Navier Stoke simulations (RANS) and Large Eddy Simulations (LES) have been made. In RANS k-ε turbulence model and in LES Smagorinsky-Lilly turbulence model is used. Results of velocity distributions simulated in RANS and LES are displayed in Fig. 1 for Reynolds numbers, Re = 126,746.

Fig. 1
figure 1

Velocity variation around tandem cylinders simulated in RANS and LES for circular piers. Upper row is for RANS and lower row is for LES. S = center to center distance of piers, D = diameter of piers. Re = 126,746

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As can be seen from Fig. 1 that at gap ratio, S/D = 2 and Reynold number, Re = 126,746, vortices do not shed in the gap in RANS but it starts to form at gap ratio S/D = 2 in LES. This process of developing vorticity keeps on moving alternately with the increase of gap ratios, S/D = 3 to 4. However, the downstream cylinder interference interacts with this increasing vortex and hence modifying the forces experienced by the two cylinders. For smaller spacing (S/D = 2), vortices are not observed in the gap between the cylinders, although, the interaction with the surrounding (bottom or top side) of the downstream cylinder is observed. Whereas, at a higher gap ratio (S/D = 3, 4), vortices are affected more by the surrounding (bottom, top and front sides) of the cylinders, which leads to form a chaotic flow in the flow field.

On the other hand, when fluid particles past by the cylinders, then slow-moving fluid layers create boundary layers. Due to the presence of strong viscosity resistance, flow momentum becomes less in boundary layers. Further increasing the pressure in the flow direction, fluid particles have to overcome this increasing pressure with the addition of high viscosity resistance. Hence, fluid particles either stop or reverse. In this way, fluid layers detached from the solid surface. These separated layers create a high amount of turbulence at the rear side of the cylinder, which is called a wake vortex. It is observed that narrow and wide wakes are generated in the gap region and moved from one cylinder to another cylinder. Streamline plot for Reynolds numbers, Re = 25,348 and 126,746 in RANS and LES are shown in Fig. 2, which showed that the flow intensity in the gap region is more complicated than the surroundings of the cylinders. At Reynolds numbers, Re = 253 (laminar flow), a symmetric pattern of streamlines is noticed (not shown here). At Reynolds number, Re = 25,348, a series of unsteady pattern is observed in LES. At Reynolds number, Re = 126,746, present more comparative results between RANS and LES (Fig. 2). The formation of recirculation zones is noticed earlier in LES than RANS. The random pattern is explained as the irregular breakup of vortices..

Fig. 2
figure 2

Evolution of velocity contour for Re = 25,348 and 126,746 at the gap S = 3D

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The Reynolds number plays a vital role on the wake of two cylinders. Igarashi [18] reported six types of flow pattern in tandem cylinders for Re = (0.87–5.2) × 104. Similar kind of flow patterns were noticed by [17] at Re = 6.0 × 104.

The only difference in between their observation is that the former found two asymmetric and symmetric vortices in the gap in the cylinders. To study the effect of Re on the force coefficients in tandem cylinders, simulated in the LES and RANS, a plot of Re versus Cd and Re versus Cl is shown in Fig. 3. At low Re, the variation in the values of Cd and Cl are more in LES than in RANS is observed, whereas at higher Re the variation in the values of Cd and Cl become negligible. Also, for same Re the values of the coefficients Cd and Cl increase as the gap between the cylinders increases until a critical point (S = 4D) is reached, whereafter a fixed value of Cd and Cl are obtained. The force coefficients are calculated in this study as Cd = 2Fx/ρu2D and Cl = 2Fy/ρu2D, where Fx is stream-wise force and Fy is spanwise force. The average drag coefficients on the upstream cylinder are about 0.86 in RANS and 0.88 in LES and in the downstream cylinder it is 0.28 in RANS and 0.31 in the LES for S/D = 2 and Re = 126,746, respectively.

Fig. 3
figure 3

Cd versus Re for tandem circular cylinders simulated in RANS and LES

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The reason is that the downstream cylinder is affected by the wake of the upstream cylinder, resulting in lower mean drag concerning the upstream cylinder at smaller spacing. Further increasing of spacing, the average drag coefficients on the upstream cylinder becomes about 0.89 in RANS and 1.05 in LES and in the downstream cylinder it is 0.29 in RANS and 0.32 in the LES for S/D = 4 and Re = 126,746, respectively. These results are compatible with the existing experimental data [14].

It is noticed that for smaller spacing (S/D = 2), the drag on the first cylinder is almost unaltered in RANS simulation, but in LES alteration in the value is observed. However, the drag on the downstream cylinder is found periodic for both RANS and LES. The present result agreed well with previously existing data. When the spacing in between the cylinder is large (S/D ≥ 4), vortex shedding from each of the cylinder dominates. This is confirmed by [19]. At smaller spacing, stronger interaction is noticed. The mean value of lift coefficients for a large number of cycles is found to be almost zero at higher spacing. The fluctuation of lift coefficients in RANS is more symmetric than in the LES is also observed. At lower spacing, there is higher pressure in the middle of the cylinders compared to the higher spacing. This makes lower drag on the downstream cylinder at smaller spacing.