1 Introduction

Laminar flow past tandem circular cylinders has been the subject of many studies due to the wide range of its applications. This typical arrangement can be widely applied in engineering problems, which are not limited to flow over buildings, heat exchangers and cooling systems, marine risers, pipelines lay on the sea-bed, offshore structures, and even electronic chips on the mainboard.

Useful correlations between vortex frequency and aerodynamic instabilities are available in the literature for tandem circular cylinders. Nevertheless, a satisfactory study of the flow structure concerning the fluidic parameters of two tandem elliptical cylinders has not yet been reached in detail. In this study, numerical investigations are performed over tandem elliptical cylinders. The flow structure, the location of the separation on the upstream cylinder, the characteristics of the flow such as the drag and lift coefficients, and the Strouhal number are deeply investigated.

Notably, it is considered how the frequency of the vortices alters and affects the wake structure of the downstream cylinder. It has been shown that the flow around tandem cylinders highly depends on not only the spacing ratio between the centers of the cylinders [1,2,3,4,5,6,7] but also the cross section shape of the cylinder [8,9,10,11,12,13,14,15]. Zdravkovich [3] observed three regimes based on the spacing between the cylinders’ centers ratio (here, for two circular cylinders, \(L^{*} = L/D_{1}\) = \(L/D_{2}\)). The first regime or overshoot was identified when the spacing between the centers of the cylinders was \(L^{*}\) < 1.2. Because of a small spacing ratio, the generated positive and negative upstream vortices are stagnant between the cylinders. While \(L^{*}\) was set to 1.2 < \(L^{*}\) < 3.5, the second regime known as the reattachment regime can be effortlessly developed. In this condition, the generated upstream vortices have a chance to reattach the downstream cylinder. Finally, Zdravkovich [3] found by setting centers of the cylinder in 3.5 < \(L^{*}\) < 4, the co-shedding regime can be created, in which the upstream vortices are only rolled into the gap between the cylinders. Later, these regimes were identified, confirmed, and even subcategorized in more regimes by other scholars [4, 16,17,18,19,20,21,22]. Numerical results of Alam [22] indicated that when the spacing ratio is set to 3.75 \(\le\) \(L^{*}\), the frequency of vortices increases and the Strouhal number (St) sharply jumps from 0.12 to 0.175. The spacing ratio \(L^{*}\) = 4 plays a significant role not only in the dynamic behavior of the upstream vortices but also in generating different flow regimes. Therefore, \(L^{*}\) = 4 is chosen in this study to investigate the flow structure around the tandem elliptical cylinders.

Considering the cross section of the cylinders, in general, the geometry of cylindrical structures vary, and they can be aerodynamically categorized into three primary geometry [14, 23]: (1) circular and elliptical cylinders with a rounded surface, in which the flow separation may oscillate over a segment of the cylinder’s surface, and it is not fixed; (2) sharp-edged surfaces with a fixed flow separation point; (3) combination of both geometries (1) and (2), such as D-shape cross-sectional structures. Though circular cylinders [24,25,26,27,28,29,30,31] and sharp-edged cylinders [12, 15, 32,33,34,35,36,37,38,39,40] have received much attention in the literature, the studies of flow structure over elliptical [11, 41,42,43,44,45,46,47] and D-shape structures [9, 48, 49] are limited to fewer examinations.

The flow past tandem cylinders with sharp edges such as square have been studied in great detail due to their relevance to a wide range of engineering applications [50,51,52] and less attention has been received for tandem rectangular cylinders [53, 54]. It has been shown by Derakhshandeh and Alam [14] and Chatterjee and Amiroudine [52] that the vortex structure in the wake of the cylinders with sharp edges behaves differently as compared with tandem circular cylinders. One of the main reasons is related to the fact that the cylinders with sharp edges have a fixed stagnation point causing differences in critical regimes. The separation mechanism in tandem square cylinders alters the vortex frequency at a range of Re = 55–60 due to the change of flow pattern in the wake of the tandem cylinders. The numerical studies performed by Sohankar and Etminan [51] showed that by increasing Re, the drag coefficient of the upstream square cylinder remains constant while that of the downstream cylinder gradually increases due to the altering the flow structure in spacing between the cylinders.

Using elliptical structures, on the other hand, is becoming more common these days due to their structural strength [45]. Since the flow over the elliptical cylinders can be affected by geometrical parameters such as Aspect Ratio (AR = \(a_{1}\)/\(b_{1}\) see Fig. 1) and the angle of attack of the elliptical cylinder, the wake structure of vortices become more complex as compared with tandem circular cylinders. Here, \(a_{1}\) and \(b_{1}\) are defined as the parallel and normal radiuses of the cross section of the elliptic cylinder relative to the freestream velocity.

Fig. 1
figure 1

Schematic of the flow past a tandem elliptical cylinders, including the computational domain and the boundary conditions

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Flow over a normal and parallel elliptic cylinder to the flow was studied by Johnson et al. [41] and Raman et al. [46], respectively. The selected AR in both studies was 0 < AR < 1.0. While Johnson et al. [41] selected a smaller range of Reynolds numbers, e.g., 30 < Re < 200 in their studies, Raman et al. [46] focused on a relatively wider range of Re (50 < Re < 500). Here, Re = \(\rho Ud_{h} /\mu\), where \(\rho\) and U are the density and velocity of the fluid, \(d_{{\text{h}}}\) stands for the hydraulic diameter and \(\mu\) represents the dynamic viscosity of the fluid. Johnson et al. [41] in their numerical studies found that for the Normal Elliptic Cylinder (NEC or with 90° incidence) by decreasing AR, the vortex shedding in the wake of the elliptical cylinder occurs differently as compared with von Karman Street (VKS) of a single circular cylinder. Two distinct regions behind the elliptic cylinder were identified. The first region appeared with two expanded rows of vortices, where the vortices rolling up from each side of the cylinder’s surface. The vortices were placed around the second region with fairly dead dynamic vortices in between when AR = 0.5 at Re = 40. It was also observed that the drag coefficient increases by increasing the Re for all aspect ratios except AR = 1 [11]. Since the viscous drag acts parallel to the bluff body, they showed that when Re increases, the pressure drag also increases up to 25% depends on AR. The results revealed that for all AR of test cases, the viscous drag is smaller than that of a circular cylinder since it acts normal to the surface. In the meantime, it was found that the pressure drag is more significant due to its action, which is normal to the surface of the NEC. Even though impressive results were found in this study, the hydraulic diameter of the elliptical cylinder was approximated by the vertical diameter of the ellipse (2 \(b_{1}\)), which was not precise enough.

On the other hand, the flow over a parallel elliptic cylinder (PEC or with 0° incidence) was investigated by Raman et al. [46]. The results showed that by increasing AR, the drag coefficient increases while it decreases with increasing Re. The vortex frequency values also decrease with increasing AR; however, these vortex shedding frequencies can increase from 0.165 to 0.23 when the Re number grows from 100 to 500. Though the Re was relatively higher than the selected range of Re by Johnson et al. [11], a 2D numerical in-house code was used to investigate the flow features. As the 3D vortex structure can be generated for Re > 200, using a 2D code might correlate with some errors [33, 55]. In addition, the details of the evaluation of the length scale of the elliptical cylinder have been not provided in their study [46]. The dynamic response of the elastically mounted elliptical cylinder was studied at different angles of attack by Leontini et al. [56]. Though different flow regimes were observed by authors once the angle of attack changed, the hydraulic diameter of the elliptical cylinder was simply chosen as the length scale for evaluating Re [56].

The effect of AR on the flow past an elliptical cylinder was also examined by Faruquee et al. [42] at zero incidences, where the major axis of the elliptic cylinder was parallel to the flow stream. A series of test cases were chosen and examined with 0.3 < AR < 1 at Re < 40. The numerical results showed that at a very low range of Re (e.g., 5 < Re < 10), a pair of steady vortices generates in the wake of the cylinder when the critical aspect ratio was AR = 0.34. For smaller AR, no significant vortices were shed in the wake of the cylinder. They also observed that when AR < 0.6, the viscous forces become more dominant, while for AR > 0.6, the pressure forces become more influential.

Later, Vijay et al. [57] numerically studied the effect of AR on flow-induced vibration of the elliptical cylinder with a constant mass ratio of 10 when Re = 100. The authors showed that a synchronization regime is directly correlated with aspect ratio, and found that lock in response occurs for small aspect ratios and the magnitude of oscillation of elliptical cylinder with AR = 0.1 is twice that of a circular cylinder Vijay et al. [57]. While 2 single vortices (2S mode) were observed in this study for all examined aspect ratios, the Reynolds number was set based on the major axis with the elliptical cylinder.

2D numerical simulations were performed by Nair et al. [58] to study the influence of a wall on the wake structure of circular and elliptical cylinders. Nair et al. [58] examined different gaps between cylinder and wall, which was set between 0.1 and 1.0 with Re = 40–1000. In this study, the diameters of circular and elliptical cylinders were considered for evaluating Re. Thus, the results of circular and elliptical cylinders did not show significant changes due to the geometrical parameters, when an identical gap and diameter were used in simulations [58].

The literature review given above demonstrates that both the spacing between the centers of the tandem cylinders and the cross section area of the cylinders are two critical parameters that remarkably affect the Karman vortex structure and imposed forces on the cylinders. To the authors’ knowledge, there has been no prior study on the flow over tandem elliptical cylinders to examine the effect of the aspect ratio of the upstream cylinder on the wake structure. Besides, in the majority of the previous studies, the Reynolds number length scale was only identifiable with \(D_{1} = 2b_{1}\) (see Fig. 1). The previously published studies neither considered the wet perimeter of the elliptical cylinder nor provided the details of explanation for evaluating accurate hydraulic diameter.

According to the above gap in the literature, in this study, we attempt to evaluate and use the precise hydraulic diameter of the elliptical cylinder using the Ramanujan approximation theorem, which has not been used before in literature. Based on this theorem and to better understand the flow instabilities in the wake and around the cylinders, we numerically investigate the vortex shedding frequency, forces (lift and drag) coefficients of the cylinders once the aspect ratio of the upstream cylinder alters from VEC (with 90° incidence) to PEC (with 0° incidence).

2 Mathematical equations and numerical model

The Navier–Stokes (NS) equations in two dimensions are used for incompressible laminar viscous flow. The conservation of mass can be given as:

$$\frac{{\partial u_{i} }}{{\partial x_{j} }} = 0.$$
(1)

The conservation of the momentum can be written as:

$$\frac{{\partial u_{i} }}{\partial t} + u_{j} \frac{{\partial u_{i} }}{{\partial x_{j} }} = - \frac{1}{\rho }\frac{P}{{x_{i} }} + \frac{\partial }{{x_{j} }} \left( {\vartheta \frac{{\partial u_{i} }}{{\partial x_{j} }}} \right).$$
(2)

In the above equation, u (x, y, t) and P (x, y, t) are the velocity and pressure fields of the fluid in “i” and “j” directions. Considering the Eulerian solution of the NS equations, the finite volume method (FVM) was used in ANSYS Fluent software to simulate the test cases. The governing equations of fluid flow, thus, were integrated and solved iteratively based on the conservation of mass and momentum on each control volume. Since FVM can be used for the structured and unstructured meshes, the structured mesh was created for all simulations. The numerical discretization was based on the second-order upwind differencing scheme to increase the accuracy of the solution. The computational domain around the cylinders is shown in Fig. 1. Water properties as a Newtonian fluid were chosen and set in the ANSYS Fluent database with a density of \(\rho\) = 998.2 (kg/m3) and the dynamic viscosity of \(\mu\) = 0.001003 (Pa. s). In the simulations, the flow on the surfaces of the cylinders was set with zero velocity and no-slip.

The aspect ratio of the upstream cylinder varies from AR = 0.25 to 2.0. Therefore, the maximum and minimum gap ratio of G* = G/\(D_{2}\) = 3.375 and 2.5 were produced, respectively, and accordingly, eight configurations were modeled and studied as shown in Fig. 2. The advantage of this selection is examining both NEC and PEC in a tandem arrangement of the cylinders. Figure 2 highlights the variation of AR of the upstream cylinder in pink dash-lines. The thicker pink dash-lines highlight the minimum and maximum aspect ratios, whereas the blue dash-line shows the elliptic cylinder with AR = 1.0 or a simple circular cylinder. The hydraulic diameter of the upstream cylinder is considered for evaluating the Re. By using different AR, the hydraulic diameter of the upstream cylinder changes due to the alteration of area and perimeter. Consequently, it is important to note that Re should be evaluated and set at Re = 200 separately for each model based on the new hydraulic diameter (\(d_{{\text{h}}}\)) of the upstream cylinder. For each test case, \(d_{{\text{h}}}\) is calculated using the following formula:

$$d_{{\text{h}}} = \frac{4A}{{P_{{\text{w}}} }}.$$
(3)

Here A represents the cross-sectional area of the elliptical upstream cylinder and \(P_{{\text{w}}}\) stands for the wetted perimeter of the cylinder. Though the area of the elliptical cylinder can be calculated accurately based on the major and minor axes of the ellipse, e.g., A =  \(\pi a_{1} b_{1}\), the perimeter of the elliptical cylinder is challenging to be accurately calculated [59]. There are many theorems and formulas to estimate the cross-sectional perimeter of the elliptic cylinder, such as the Ramanujan approximation theorem and Maclaurin expansion [59, 60]. Ramanujan approximation theorem with high accuracy is one of the acceptable theorem, which can be used with an excellent satisfactory error (e < 10−2%) to estimate the perimeter of the elliptic cylinder. The Ramanujan approximation is given by:

$$P_{{\text{w}}} = \pi \left[ {\left( {a_{1} + b_{1} } \right) + \frac{{3(a_{1} - b_{1} )^{2} }}{{10\left( {a_{1} + b_{1} } \right) + \sqrt {a_{1}^{2} + 14a_{1} b_{1} + b_{1}^{2} } }} + e} \right]$$
(4)
Fig. 2
figure 2

Schematic of tandem elliptical cylinders showing the variation of the aspect ratio of the upstream cylinder in dash lines. The blue dash-line highlights the AR = 1 (simple circular cylinder). The spacing ratio between the centers of the cylinders is \(L^{*}\), while \(G^{*}\) is the gap ratio between the cylinders; the AR of the downstream cylinder was kept constant in all test cases. The thicker pink dash-lines of the upstream cylinder highlights the minimum and maximum AR corresponding to the normal and parallel elliptic cylinders to the flow, respectively

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Table 1 summarizes the geometrical parameters of the upstream elliptical cylinder, comprising the parallel and normal radiuses, and the area and perimeter and the hydraulic diameter of the upstream cylinder. In 2D modeling, the length of the cylinder is set to one in all simulations. It is observed that by changing the aspect ratio, both the cross section area and wetted perimeter of the upstream cylinder change. These two aspects directly affect the hydraulic diameter of the cylinder. Hence, the last column of the table shows the hydraulic diameter of the cylinder, which is used for evaluating Re.

Table 1 The geometrical parameters of the upstream elliptic cylinder to evaluate the hydraulic diameter of the elliptical cylinder
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It is worth estimating the percentage of the discrepancy (or error) for the hydraulic diameter if the Reynolds number length scale was identical with a diameter of a circular cylinder. Considering the hydraulic diameters of a circle and an ellipse, (e.g., \(D_{{\text{h,circle}}}\) and \(d_{{\text{h,ellipse}}}\), respectively), the accurate amount of Re for the circular cylinder (\({\text{Re}}_{{\text{c}}}\)) can be estimated according to the following equation:

$${\text{Re}}_{{\text{c}}} = {\text{Re}}_{{\text{e}}} \times \frac{{D_{{\text{h,circle}}} }}{{d_{{\text{h,ellipse}}} }}$$
(5)

Table 2 comprises two Reynolds numbers that can be calculated based on the precise hydraulic diameter of the elliptic cylinder using the Ramanujan formula or an estimated diameter of a simple circular cylinder. The last column of the table displays the percentage of the error between the accurate value and estimation one. The results show that nonlinear error is obtained for calculating Re resulting in a larger error for the NEC as compared with the PEC. To provide a clearer prospect, the amount of error versus AR is shown in Fig. 3, which is associated with Table 2. It is found that even for a relatively low Re, the maximum error for evaluating Re, approximately reaches 178% and 25% for NEC and PEC, respectively. The amount of error is calculated based on the following equation:

$${\text{Error}} = \frac{{\left( {D_{{\text{h,circle}}} - d_{{\text{h,ellipse}}} } \right)}}{{d_{{\text{h,ellipse}}} }} \times 100(\% )$$
(6)
Table 2 Estimation of the error by assuming that the hydraulic diameter of the elliptic cylinder is identical with the diameter of a circular cylinder
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Fig. 3
figure 3

Variation of the percentage of error as a function of AR at Re = 200 when the hydraulic diameter of the elliptic cylinder is evaluated by the diameter of a circular cylinder. The data are associated with Table 2

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Therefore, to avoid these errors and study the flow structure of elliptical cylinders with high accuracy and reliability, here the Ramanujan approximation is used to evaluate the perimeter of the elliptical cylinder, and consequently, hydraulic diameter of the cylinder.

In numerical simulations, to avoid the blockage effects, four computational widths were selected and set at H/\(D_{1}\) = 12, 15, 18, and 24 for a typical tandem circular cylinders (when AR = 1 for both cylinders) at Re = 200. Figure 4 shows the effect of blockage on the mean drag coefficient as well as fluctuating lift coefficient and Strouhal number of the upstream cylinder. Though it is seen that there is about 2% discrepancy between H/\(D_{1}\) = 15 and 18, the results reveal that by increasing the width to 18, the trends of \(\overline{C}_{{\text{D}}}\), \(C_{{{\text{lf}}}}\), and St remain constant. Thus, the computational width with H/D = 18 is chosen for the rest of the simulations.

Fig. 4
figure 4

Effects of blockage on the mean-drag coefficient, the fluctuating lift coefficient and the Strouhal number of the upstream cylinder arranged in a tandem elliptical cylinders with AR = 1 (circular cylinder) at Re = 200

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The upstream and downstream lengths of 10\(D_{1}\) and 30\(D_{1}\) are also selected for the total length of the domain. These distances are long enough to guarantee that the inlet and outlet effects are eliminated [61,62,63,64]. Here, \(D_{1}\) is the vertical length of the upstream cylinder and is kept constant and equal to \(D_{1}\) = 2\(b_{1}\) = 2\(b_{2}\) = \(D_{2}\) in all test cases. Consequently, the AR was varied only by altering the horizontal axis of the cylinder (e.g., \(a_{1}\)).

3 Results and discussion

In this section, the numerical simulation is initially validated. Then, the flow structure in the wake of the cylinders and fluidic parameters such as pressure coefficients, and the lift and drag coefficients are analyzed in detail. Finally, the effect of aspect ratio on the Strouhal number is studied, and a relationship between AR and St is derived.

3.1 Verification

To test the grid independence, three structured mesh resolution namely: \({\text{MR}}_{1}\), \({\text{MR}}_{2}\) and \({\text{MR}}_{3}\) were examined for the flow over tandem circular cylinders. The results of each test are summarized and compared in Table 3, comprising the mean-time drag coefficient of the cylinders (\(\overline{C}_{{{\text{D}}\left( 1 \right)}}\) and \(\overline{C}_{{{\text{D}}\left( 2 \right)}}\)) the fluctuating lift coefficient of the cylinders (\(C_{{{\text{lf}}\left( 1 \right)}}\), and \(C_{{{\text{Dlf}}\left( 2 \right)}}\)) and St. A typically generated mesh in the flow domain and around tandem elliptical cylinders are shown in Fig. 5, when the aspect ratio of the upstream and downstream cylinders are set at AR = 0.25 and 1.0, respectively.

Table 3 Mesh resolution independence results at Re = 200, including the mean-time drag coefficients and the fluctuating lift coefficients of the cylinders and the Strouhal number of the upstream cylinder at AR = 1
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Fig. 5
figure 5

A typically structured mesh around tandem elliptic cylinders, including a closer view of the grids around the cylinders, when the aspect ratio of the upstream cylinder is AR = 0.25

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It is worth noting that similar to the previously published data [22], the Strouhal number was identical for both circular cylinders, and therefore, one value is only presented in Table 3. The minimum mesh size in the finest mesh resolution (e.g., \({\text{MR}}_{3}\)) was approximately 0.02. Based on this size, the time step of \(\Delta t\) = 0.025 (s) was chosen, which was adequately small to satisfy the Courant–Friedrichs–Lewy number (CFL = \(U\Delta t/\Delta x\) < 1). The results of Table 3 highlight that \({\text{MR}}_{2}\) is an acceptable mesh resolution as compared with \({\text{MR}}_{3}\). While the maximum error of \(e_{{ C_{{{\text{D}}\left( 1 \right)}} }} \approx 4\%\) was obtained for the drag coefficient of the upstream cylinder, other parameters show less than 2% errors. Therefore, \({\text{MR}}_{2}\) was selected for the rest of the numerical simulations.

Further validations were also conducted for flow over tandem circular cylinders at Re = 200. Table 4 compares the obtained numerical results including \(\overline{C}_{{{\text{D}}\left( 1 \right)}}\), \(\overline{C}_{{{\text{D}}\left( 2 \right)}}\), \(C_{{{\text{lf}}\left( 1 \right)}}\), \(C_{{{\text{Dlf}}\left( 2 \right)}}\), and St with those in literature. The subscripts ‘1’ and ‘2’ in Table 3 refer to the upstream and downstream cylinders, respectively. It is seen that the present results are in good agreement with published data. While the percentage of deviation between the results is acceptable, the best match is seen between the present study and the data offered by Mahir and Altac [37] for all parameters with less than 5%.

Table 4 Comparison between the fluidic parameters comprising drag coefficients, fluctuating lift coefficients, and the Strouhal number of tandem circular cylinders with AR = 1 at Re = 200
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3.2 Separation points and reattachment locations

Highlighting the details of flow separation can gain more insight into the flow structure in the wake of the upstream elliptical cylinder. The separation points also directly can affect the reattachment of vortices on the downstream cylinder. Both of these locations can influence the imposed pressure on the cylinders. Samples of instantaneous vorticity contours are shown in Fig. 6a–c at Re = 200 for three aspect ratios, minimum (AR = 0.25), identical (AR = 1.00), and maximum (AR = 2.00). The figure shows the separation points, separation angles, and the locations of reattachments. As is well known, the boundary layer equation is a valid approximation for a wide range of Re until separation begins to intervene. In a laminar flow, such a feature corresponds to disappearing wall shear stress when the flow is broken away from the surface of the bluff body. Accordingly, the exact separation angle (e.g., \(\theta_{{\text{S}}}\)) can be extracted from numerical results, where the wall shear stress vanishes on the surface of the upstream elliptical cylinder.

Fig. 6
figure 6

Separation and reattachment points of the upstream and downstream cylinders for test cases with a minimum aspect ratio, AR = 0.25, b identical aspect ratio, AR = 1.00 and c maximum aspect ratio, AR = 2.00 at Re = 200

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Figure 6 highlights that by increasing AR, the separation points shift to the trailing surface of the upstream cylinder and the separation angle gradually increases. On the other hand, due to the increment of separation angle at larger AR, the spacing ratio between cylinders reduces; thus, the reattachment of the vortices on the downstream cylinder also takes place at larger angles as compared to the smaller AR.

The position of separation points can be understood by choosing the peak of the distribution of the fluctuating pressure coefficient around the cylinder [67, 68]. Accordingly, the RMS fluctuation pressure coefficients of the upstream and downstream cylinders are plotted in Fig. 7 for all AR.

Fig. 7
figure 7

Root-mean-square pressure fluctuations of a upstream and b downstream cylinders for tested cases of AR = 0.25 to 2.00 at Re = 200

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Figure 7 shows that the peak of \(C_{{{\text{Pf}}\left( 1 \right)}}\) occurs at about 90° for AR = 0.25. The sinusoidal trend, then, is seen after a peak of \(C_{{{\text{Pf}}\left( 1 \right)}}\). By increasing AR, the magnitude of \(C_{{{\text{Pf}}\left( 1 \right)}}\) gradually decreases. It is also observed that the peaks shift to larger angles (larger than 90°) and relocates on the trailing surface of the upstream cylinder, which again confirms the results of vorticity contours in Fig. 6. It is observed that the sinusoidal trend of \(C_{{{\text{Pf}}\left( 1 \right)}}\), gradually alters to one oscillation for 0.25 < AR < 1.25 and for AR \(\ge\) 1.25, the curves change to a relatively flat trend. On the other hand and in general, the maximum fluctuating pressure of the downstream cylinder is \(C_{{{\text{Pf}}\left( 2 \right)}}\) < 1 for all AR. The peaks of \(C_{{{\text{Pf}}\left( 2 \right)}}\) of AR = 0.25 and 0.5 occur at 68° and 60°, respectively, which are larger than that one with higher aspect ratios except AR = 2.0. By increasing AR to 0.75, the peak of \(C_{{{\text{Pf}}\left( 2 \right)}}\) approximately 30% increases as compared to that of AR = 0.5; however, the angle of reattachment at AR = 0.75 shifts to a smaller angle of 46°. For AR = 1.0 till 1.75, the peaks of \(C_{{{\text{Pf}}\left( 2 \right)}}\) are found to be about 43°, 40°, 39.5°, and 39°, respectively. For test cases with AR > 1, it is observed that the magnitude of \(C_{{{\text{Pf}}\left( 2 \right)}}\) gradually decreases. This means the influence of vortices becomes smaller on the downstream cylinder as compared to those with AR < 1. The pressure fluctuations become very low or minimum around the upstream and downstream cylinders when AR = 2.0. Therefore, it is expected the imposed forces on both cylinders with AR = 2.0 become minimum.

To gain a better understanding and insight about the pressure distribution of tandem cylinders, first, the time-averaged pressure distribution (\(\overline{C}_{{\text{P}}}\)), and RMS fluctuating pressure distribution (\(C^{\prime}_{{\text{P}}}\)) along with the single circular cylinder at Re = 200 are both shown in Fig. 8a, b. Figure 8c also depicts the time-averaged contour plots of the pressure coefficient of the cylinder. It is seen that the maximum pressure on the single cylinder occurs at around 82°, which is consistent with the literature [68].

Fig. 8
figure 8

Pressure coefficient distribution over the surface of a single cylinder at Re = 200, a time-averaged pressure coefficient distribution (\(\overline{C}_{{\text{P}}}\)), b RMS fluctuating pressure coefficient (\(C^{\prime}_{{\text{P}}}\)), c time-averaged contour plot of pressure coefficient

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In Fig. 9, the distribution of the time-averaged pressure coefficients (\(\overline{C}_{{\text{P}}}\)) along the surfaces of the upstream (left column) and downstream (middle column) cylinders are presented, respectively. The right column of Fig. 9 also shows the time-averaged contour plots of the pressure coefficient in the wake area of the cylinders. On comparing Figs. 8 and 9, it is observed that except for two test cases, AR = 0.25 and 0.5, the trend of \(\overline{C}_{{{\text{P}}\left( 1 \right)}}\) for the rest of the models is very similar to that around a single circular cylinder. It is seen that the pressure distribution on the upstream cylinder includes positive and negative values and alters as a function of AR; however, the downstream cylinder is always in a negative region apart from the aspect ratio of the cylinders.

Fig. 9
figure 9

Time-averaged pressure coefficient distribution along the surface (left and middle columns) and in the wake of the tandem elliptical cylinders (right column) with spacing L* = 4 at Re = 200 as a function of aspect ratio of the upstream cylinder, a AR = 0.25, b AR = 0.50, c AR = 0.75, d AR = 1.00, e AR = 1.25, f AR = 1.50, g AR = 1.75, h AR = 2.00

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The positive \(\overline{C}_{{{\text{P}}\left( 1 \right)}}\) on the upstream elliptical cylinder occurs at \(\theta\) \(\le\) 79° for AR = 0.25. By increasing AR, the angle of positive values of \(\overline{C}_{{{\text{P}}\left( 1 \right)}}\) on the leading edge of the elliptical cylinder gradually decreases, and therefore, \(\overline{C}_{{{\text{P}}\left( 1 \right)}}\) = 0 steadily shifts to smaller angles as shown in Fig. 9. Comparison of \(\overline{C}_{{{\text{P}}\left( 1 \right)}}\) reveals that the maximum negative magnitude of pressure coefficients of the upstream cylinder gradually decreases by increasing AR. This decrement is very sensible when AR alters from 0.25 to 0.50, in which \(\overline{C}_{{{\text{P}}\left( 1 \right)}}\) shows 25% less pressure (e.g., \(\overline{C}_{{{\text{P}}\left( 1 \right)}}\) changes from − 2.4 at \(\theta\) = 180° to − 1.8 at \(\theta\) = 86°). Since, \(\overline{C}_{{{\text{P}}\left( 1 \right)}}\) at the rear surface of the upstream cylinder for AR = 0.25 and 0.50 is much larger as compared with other test cases, it is expected to generate a reduced drag force on the upstream cylinder for these two test cases. This concept will be explained in the next section.

On the other hand, the results of \(\overline{C}_{{{\text{P}}2}}\) show that the time-averaged pressure distribution along the downstream cylinder is not similar to that around a single nor upstream one. The values of \(\overline{C}_{{{\text{P}}\left( 2 \right)}}\), in general, are negative and smaller than that of the upstream cylinder (Fig. 9) or the single cylinder (Fig. 8a). It is found that, for AR > 0.75, in which the gap between cylinders reduces, the maximum negative magnitude of \(\overline{C}_{{{\text{P}}\left( 2 \right)}}\) occurs after \(\theta\) = 90° as it was expected earlier due to the reattachment. The contour plots in the wake and around the cylinders confirm that at AR = 0.25 and 0.5, the pressure distribution corresponds to larger negative values of \(\overline{C}_{{\text{P}}}\) on the lower and upper sides of both cylinders and in the wake of the upstream cylinder.

The contour plots of Fig. 9 also show that the minimum time-averaged pressure coefficient over the surface of the cylinder occurs at smallest AR = 0.25 with the magnitude of \(\overline{C}_{{\text{P}}}\) = 3.7. Further increasing in AR = 0.5, the magnitude of \(\overline{C}_{{\text{P}}}\) shows a sudden drop to 1.92 on the cylinder surface and \(\overline{C}_{{\text{P}}}\) regularly decreases to the minimum magnitude of \(\overline{C}_{{\text{P}}}\) = 0.82 for the largest AR = 2.00. To better understand the discrepancy of the \(\overline{C}_{{\text{P}}}\) in the wake of tandem cylinders, the variation of \(\overline{C}_{{{\text{P}},\min }}\) versus to the AR is plotted in Fig. 10.

Fig. 10
figure 10

a Different angles of upstream elliptical cylinder where the magnitude of \(\overline{C}_{{\text{P}}}\) becomes maximum and zero, b variation of \(\overline{C}_{{{\text{P}},\min }}\) as a function of AR for the upstream cylinder extracted from contour plots of Fig. 9 (\(a_{3}\)–\(h_{3}\))

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3.3 Lift and drag coefficients

Figure 11 shows the time history of the lift coefficients of the upstream elliptical cylinder and downstream circular cylinder at different AR (left column). The figure also represents the Fast Fourier transform (FFT) function corresponding to the peaks of the lift coefficient of the upstream cylinder over the total convergence time (right column).

Fig. 11
figure 11

Time history of the lift coefficients of the upstream elliptic cylinder and downstream circular cylinder with different AR (left column) and the magnitude of FFT of the lift coefficient showing the Strouhal number (right column)

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The time history of the lift coefficients can be considered in terms of the amplitude of the forces and phase lags between the forces. As expected in the previous section, the magnitude of the lift coefficient shows the maximum value when AR = 0.25, and it reaches a minimum value at AR = 2.0. The results also reveal that the phase lag between the sinusoidal lift coefficients of the cylinders varies by changing AR and it gradually reduces. The phase lag reaches a minimum at AR = 1.5 and it increases again by an increment of AR to 2.0. When comparing the FFT figures for each AR it becomes readily apparent that the frequency of vortices also varies. The minimum St occurs at AR = 0.25, and it is regularly increased to the maximum value at AR = 1.75. The results show that St becomes close to the vortex frequency formation of a single cylinder when 1.25 \(\le\) AR \(\le\) 1.75 (e.g., St \(\approx\) 0.2). At this range of AR (e.g., 1.25 \(\le\) AR \(\le\) 1.75) the phase lags between lift forces are 14.08°, 9.98° and 27.9°, respectively. When AR increases to 2.0, vortex shedding frequency with 15% drops reaches to St = 0.17. The reason is due to the minimum gap ratio between the cylinders, which generates the overshoot regime, and hence, the vortices cannot be produced in the wake of the upstream cylinder and accelerated to increase St. It is emphasized that the overshot regime generates once the G* < 1.2, where the separated shear layers of the upstream cylinder overshoot the downstream one and rolling in the wake of the downstream cylinder without reattaching [31]. The FFT plots of the lift coefficients also demonstrate that the magnitude of the lift forces is different and it varies by AR. Considering the magnitude of the FFT plots, it is confirmed that the maximum and minimum magnitude of the lift forces occur at AR = 0.25 and 2.0, respectively.

The exact values of fluctuating lift coefficients (\(C_{{{\text{lf}}}}\)) and the time-averaged drag coefficients (\(\overline{C}_{{\text{D}}}\)) are shown in Table 5 for each test case. To better understand the trend of variation of the imposed forces on the cylinder, the fluctuating lift, and drag coefficients are plotted in Fig. 12. The results at AR = 0.75 and 1 are well matched with those measured and reported by Alam et al. [16]. In general, the results of the lift coefficients reveal that the upstream cylinder undergoes a smaller \(C_{{{\text{lf}}}}\) as compared with the downstream cylinder. The fluctuating lift coefficient of the cylinders follows similar trends as the spacing is increased. The fluctuations are remarkably large at AR = 0.25. The fluctuations of elliptical upstream cylinder remain extremely high, when G* = 3.375 (e.g., AR = 0.25). As this gap ratio, \(C_{{{\text{lf}}\left( 1 \right)}}\) and \(C_{{{\text{lf}}\left( 2 \right)}}\) are 2.15 and 1.11 (\(C_{{{\text{lf}}\left( 1 \right)}}\) 30 times higher than that of minimum G* (test case 8), respectively (see Table 5). Although the fluctuations of the lift coefficient of both cylinders become low as the gap ratio decreases, the trend of the reduction is not similar. The variation of the \(C_{{{\text{lf}}\left( 1 \right)}}\) is sinusoidal with a significant reduction, in particular, when AR increases to 0.5. AR = 1.0 can be nominated as a critical aspect ratio (\(AR_{{\text{C}}}\)), in which the fluctuating lift and drag coefficients of the downstream cylinder shows a peak. The results confirm that both \(C_{{{\text{lf}}}}\) downstream and \(C_{{{\text{Df}}}}\) upstream cylinders are very sensitive when AR < 1.

Table 5 Lift and drag coefficients of tandem elliptic cylinders at Re = 200
Full size table
Fig. 12
figure 12

Variation of the fluctuating of lift (rms) and time-averaged drag coefficient as a function of the aspect ratio for the upstream and downstream cylinders at Re = 200

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The results of the time-averaged drag coefficients of the cylinders are shown in Fig. 13. In general, the results reveal that the upstream elliptical cylinder undergoes a larger \(\overline{C}_{{\text{D}}}\) as compared with the downstream cylinder. It is found that the time-averaged drag coefficients of the upstream cylinder show a significant drop when AR increases. Besides, the amplitude of the fluctuating drag coefficient is larger for the upstream cylinder as compared with that of the downstream cylinder. For the downstream cylinder, it is seen that the time-averaged drag is negative at AR = 0.25 and 0.5 meaning that the downstream cylinder is pushed forwards by the fluid. The drag, then, gently increases to positive values by further increasing AR. Indeed, \(\overline{C}_{{\text{D}}}\) of the downstream cylinder neither approaches that of the upstream cylinder nor a single circular cylinder, even for maximum G*.

Fig. 13
figure 13

Variation of the mean-drag coefficient as a function of the aspect ratio for the upstream and downstream cylinders at Re = 200

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Interestingly, the presence of the downstream cylinder causes a significant reduction of \(\overline{C}_{{\text{D}}}\) of the upstream cylinder. The reason for this reduction is the increment of the pressure in the separated wake behind the upstream cylinder. This increment of the pressure occurs when the vortices of the upstream and downstream cylinders are in in-phase mode. This concept has been proved earlier for two and three tandem circular cylinders at Re = 200 [22, 64, 69,70,71,72].

3.4 Relationship between St and AR

The variation of St as a function of AR is shown in Fig. 14. It is seen that St significantly increases by growing AR as expected. The variation of St starts from St = 0.065 at AR = 0.25 and continues to the maximum value of St = 0.211 at AR = 1.75. This trend can be formulated by showing the relationship between St and AR to facilitate the estimation of St at other AR, which is arranged between the test cases studied in this paper. Therefore, a curve is fitted on the trend of Strouhal number versus the aspect ratio. As a result, an equation is driven and identified as shown in Fig. 14. The equation can be written as follow:

$${\text{St}} = -\, 0.11 AR^{2} + 0.3 AR + 0.0053$$
(7)
Fig. 14
figure 14

Variation of the Strouhal number as a function of aspect ratio, including a fitted curve for the obtained numerical results

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Evaluating the residuals is very valuable to understand the accuracy of the fitted curve on the trend of St versus the AR. It is found that the residuals approximate positive and negative errors randomly with less than 1.5% differentiation. Thus, Eq. (7) fits the data very well and can be applied with a 1.8% error.

4 Conclusions

In the present paper, a 2D laminar flow over elliptical tandem cylinders was studied numerically. The previously published studies of flow over an elliptic cylinder mostly estimated the hydraulic diameter of the elliptic cylinder by a diameter of a circular cylinder. In the case of using the hydraulic diameter, in some studies, the details of the calculation are missing, and the details have not been provided. In this study, there was an attempt to evaluate the correct hydraulic diameter due to its importance on all the fluidic parameters, using the Ramanujan theorem.

The percentage of the errors was highlighted considering the precise hydraulic diameter of the elliptical cylinder. Once using NEC the maximum error of evaluating Re approximately reaches 180%. It was also found that this error reaches 25% for PEC.

The results show that the peak of pressure fluctuation of the upstream cylinder (\(C_{{{\text{Pf}}\left( 1 \right)}}\)) occurs at about 90° for AR = 0.25. The sinusoidal trend, then, is generated after a peak. The peaks of pressure fluctuation of the downstream cylinders with AR = 0.25 and 0.5 occur at 68° and 60°, respectively. These peaks are larger than those with higher aspect ratios except AR = 2.0. It was found that further increasing AR to 0.75, the peak of \(C_{{{\text{Pf}}\left( 2 \right)}}\) approximately 30% rises as compared to pressure fluctuation of AR = 0.5. The pressure fluctuations become very low around both cylinders for the test case with AR = 2.0. Thus, the imposed forces on both cylinders with AR = 2.0 become minimum.

Based on the imposed pressure on the cylinders, it was observed that by altering the aspect ratio, the phase lag between the sinusoidal lift coefficients of the cylinders varies and it steadily reduces. The phase lag reaches the minimum at AR = 1.5 and it slightly increases again by the increment of AR to 2.0.

The results reveal that the upstream elliptical cylinder undergoes a larger \(\overline{C}_{{\text{D}}}\) as compared with the downstream cylinder. The time-averaged drag coefficients of the upstream cylinder significantly drop by increasing AR. Besides, the amplitude of the fluctuating drag coefficient is larger for the upstream cylinder as compared with that of the downstream cylinder. For the downstream cylinder, it is seen that the time-averaged drag is negative at AR = 0.25 and 0.5. This means that the downstream cylinder is pushed forwards by the fluid. By further increasing AR, the drag slightly increases to positive values.

Based on the generated phase lag between forces and accordingly vortices, the Strouhal number varies. The minimum St = 0.065 occur at AR = 0.25, and it is regularly increased to the maximum value of St = 0.211 at AR = 1.75. The St of tandem elliptic cylinders becomes close to the vortex frequency formation of a single cylinder when 1.25 \(\le\) AR \(\le\) 1.75 (e.g., St \(\approx\) 0.2). When AR increases to 2.0, vortex shedding frequency with 15% drops reaches St = 0.17 due to the minimum gap ratio between the cylinders. As a result, the relationship between the Strouhal number and AR was deduced and identified with high accuracy and a reasonable error of 1.8%.