Keywords

1 Introduction

Bluff body wakes have been of interest in fluid mechanics since a long time. Flow past an elliptic cylinder is the generalization of flow past a circular cylinder which involves a new variable: the aspect ratio (Ar). In the past, there have been several attempts to investigate this flow [5, 12,13,14, 20, 24, 26]. The dynamics of the wake of an elliptic cylinder is much more rich compared to the same for a circular cylinder. As we change the aspect ratio of the cylinder, the flow becomes more and more complex. For example, flow past an elliptic cylinder is characterized by the presence of a near wake vortex street (von Kármán vortex street) followed by the diffusion of vortices to form two parallel shear layers. As we go downstream in the wake, the oscillations in the shear layers intensify, finally causing the secondary vortex street to develop [19]. With a decrease in aspect ratio, the extent of the von Kármán vortex street reduces in the wake and the onset location of the secondary vortex street moves close to the cylinder. For low aspect ratios, the shedding pattern is very complex and chaotic with the appearance of vortex pairing and long wavelength wake oscillations [24].

The flow transition between steady to unsteady occurs at Re which depends on Ar of the cylinder. Several researchers [12, 17, 24] in the past have conducted linear stability analysis of steady flow past elliptic cylinders in order to investigate the transition phenomenon. Based on such studies, it is known that the cause and appearance of the Kármán vortex street could be related to the first instability of the wake. The first instability of the wake is related to the growth of the unstable mode, sometimes referred to as the Kármán mode. In the present work, we refer to it as PWM. This is therefore seen as the fundamental cause of unsteadiness in the flow. In the literature, the nature and characteristics of this mode is known for more than forty years. For flow past a circular cylinder, Barkley and Henderson [2] concluded that there is only one mode which becomes unstable leading to the primary wake instability in a two-dimensional flow before the three-dimensionality appears. In contrast, Boppana and Gajjar [3] reported the presence of a second unstable mode before the onset of three-dimensionality. We refer this mode as SWM. According to them, the second unstable mode loses stability somewhere between Re = 125 and 150. In the same year, Verma and Mittal [25] conducted a linear stability investigation of two-dimensional flow past a circular cylinder. They reported Re = 110.8 for the onset of instability of the SWM. They found that structure of SWM is very similar to PWM. However, SWM has a lower growth rate and higher St than PWM.

In this work, we share the discovery of yet another mode which is unstable in flow past elliptic cylinders, apart from the two already known. We refer this new mode as TWM. Knowledge of the nature and characteristics of these modes is of great relevance in understanding the wake flows. The present work examines the characteristics of these modes. We conduct this study for varying Ar of the cylinder and different Re in the range 30–200.

2 Methodology

The governing equations for the fluid flow are the incompressible form of the Navier–Stokes equations. In this work, we use the stabilized finite element formulation to solve the Navier–Stokes equations as well as the eigenvalue problems emanating from the linear stability equations. The stabilized finite element (streamline-upwind/Petrov–Galerkin and pressure-stabilizing/Petrov–Galerkin) method has been developed and used by several researchers in the past [4, 10, 11, 16, 23]. Mittal and Kumar [16] briefly describe the formulation and the numerical method employed here.

Applying the finite element formulation to linear stability equations leads to a generalized matrix eigenvalue problem, and its solution results in the eigenpair. Since the numerical dimension of the problem is usually very large, we look for the few modes which have the largest real part.

Figure 1 illustrates the flow configuration, the computational domain, as well as the boundary conditions. The elliptic cylinder is placed symmetrically in a rectangular domain with its major axis (2a) aligned with the flow direction. The cylinder center is considered the origin of the Cartesian coordinate system. Semi-major axis of the ellipse is taken as the length scale for presenting the results. The length of the minor axis is denoted by 2b, and b/a represents Ar of the cylinder. Nine different Ar values (= 1, 0.75, 0.5, 0.375, 0.25, 0.125, 0.0625, 0.03, 0.01) are used in this study. In the present work, the domain boundaries are located far enough so that there is no significant effect on the overall flow field. The distance of the downstream, the upstream, and the cross-stream boundaries are 700, 200, and 100 from the center of the cylinder. The finite element mesh employed in this work contains 249,165 nodes and 247,654 elements in the computational domain. Close to the cylinder surface and in the wake region, the mesh is kept fine enough to capture the separating shear layer, the boundary layer, the wake bubble of the steady flow, and the linear stability modes accurately. In the direction away from the cylinder, the mesh gradually becomes coarser. The details about the finite element mesh can be found in our recent article [15].

Fig. 1
a. A geometric representation of the flow configuration of the computational domain. It has a rectangular box with a height of 2 H. An ellipse is located on the left side of the box with a diameter of 2 a. b. An ellipsis with a height of 2 a and a diameter of 2 b is drawn on the x y plane.

Problem description: a computational domain and boundary conditions, where Ld and Lu represent the downstream and upstream boundaries location, and H denotes the half width of the cross-stream dimension. b elliptic cylinder geometry. The cylinder center is used as an origin to measure all the linear distances

Full size image

The following boundary conditions are used to compute the steady base flow. The velocity at the upstream boundary is given a free-stream value. In the downstream boundary, a boundary condition of Neumann type is imposed on the velocity, which corresponds to a zero stress vector. The components of velocity normal to and stress vector along the lower and upper boundaries are assigned zero values. The cylinder surface is a no-slip boundary. Linear stability computations are based on the same boundary conditions except at the inflow border, where the velocity components are set to zero.

The mesh convergence study is comprehensively discussed in one of our recent publications. Tables I and II of Kumar and Kumar [15] provide a detailed discussion of the mesh convergence study, and based on these results, it can be safely concluded that the mesh considered here is appropriate for performing the present study. Tables III and IV of Kumar and Kumar [15] compare the steady flow characteristics with the published data. The comparison reveals that the present results match very well with the earlier investigations on flow past elliptic cylinders.

3 Results and Discussion

3.1 Steady Flow Results

Here, we display results for steady flow past elliptic cylinders for (nine) different Ar and for Re in the range 30–200. As of now, it is well established that, beyond a particular Re, the flow separates from the cylinder surface [5, 20], which causes the formation of the wake bubble. Figure 2 demonstrates the description of the separation wake bubble. The bubble length (LB) is characterized as the distance between the rear stagnation point and the wake stagnation point along the wake centerline of the cylinder. The bubble width (WB) is measured as the maximum distance between the lower and upper separation streamlines.

Fig. 2
A diagrammatic representation of the wake bubble. A horizontal ellipsis is drawn with a height of W B and a length of L B. A small vertical ellipsis is drawn along the left side of the horizontal ellipsis.

Description of the symmetric wake bubble: LB denotes the bubble length, while WB represents the bubble width

Full size image
$$\left. {\begin{array}{*{20}l} {L_{B} = - 0.7155 + 0.1301{\text{Re}} } \hfill \\ {W_{B} = 0.4788 + 0.2444{\text{Re}}^{0.5} } \hfill \\ {C_{D} = 0.2938 + 7.6937{\text{Re}}^{ - 0.5} } \hfill \\ {\omega_{\max } = - 2.9870 + 2.3980{\text{Re}}^{0.35} } \hfill \\ \end{array} } \right\}$$
(1)

Figure 3 displays the variation of the bubble parameters with Ar and Re. For each Ar, it has been found that LB grows practically linearly as Re increases. The decrease in Ar causes bubble to substantially grow. A non-linear growth is observed for WB as Re increases while Ar decreases. The drag coefficient (CD) shows a monotonic decrease with increase in Re and Ar. The maximum vorticity on the cylinder surface (ωmax) increases with decrease in Ar and increase in Re. The least-square fit of the steady flow data shown in Fig. 3 indicates that for Ar = 1.0, LB, WB, ωmax, and CD vary linearly as a function of Re, Re0.5, Re0.35, and Re−0.5, respectively. These results agree with the characteristics of steady flow past a circular cylinder reported by several investigators in the past [6,7,8, 21, 22]. Theoretical study by Smith [22] showed that LB grows linearly with Re while WB increases like O(Re1/2). Fornberg [6] calculated the steady flow numerically up to Re = 600. He observed that LB increases linearly with Re, whereas WB grows as O(Re1/2) up to Re = 300 and increases linearly after that. Recently, Sen et al. [21] reported the empirical relations for steady flow past Ar = 1.0 at low Re in the range 6–40. They found that LB, CD, and ωmax vary with Reynolds number as Re, Re−0.5, and Re0.5, respectively. The variation of ωmax is accurate for low Re. For Re > 30, we find Re0.35 gives better results.

Fig. 3
Four 3 D surface plane graphs plot L B, W B, C D, and omega max versus R e and A r. a and b. The plane follows an increasing trend in the R e and A r axes. c. The plane follows a decreasing trend in the C D and R e axes. d. The plane follows an increasing trend in the omega max and A r axes.

Steady flow characteristics: effect of Re and Ar on a LB, b WB, c CD, d ωmax on the cylinder surface

Full size image

The empirical equations obtained for the curve fit for Ar = 1.0 are given in Eq. 1 for Re in the range 30–200. In general, with the decrease in Ar, it is observed that, the characteristics of the wake bubble vary in the same fashion as for Ar = 1.0, except that a higher-order polynomial fit is required. This shows that the non-linearity in the bubble properties increases with decrease in Ar. However, CD is one exception to this change. It shows linear variation with Re−0.5 even for smaller aspect ratios. The maximum percentage error between the actual value and the data obtained from the given equations are 0.88 for LB, 2.22 for WB, 0.31 for CD, and 0.53 for ωmax. The trends represented by the equations therefore are a good representation of the variation of the flow properties.

4 Linear Stability Analysis

In the past, linear stability analysis of flow past an elliptic cylinder has been carried out by several researchers [12, 17, 24]. However, the results are very scanty and do not give a comprehensive view of the flow behavior. Here, we provide a detailed analysis of the related flow problem. The linear stability study is carried out for cylinders with different Ar and different flow Re. Our results show the presence of three different kinds of eigenmodes in the flow which become unstable with increase in Reynolds number. We refer to them as PWM, SWM, and TWM. These are identified based on their growth rate and modal structure in the flow field. The growth rate (λr) of PWM is the highest. It loses stability at relatively low Re and is responsible for shedding of von Kármán vortices. Upon increasing the Re further, first the SWM and then the TWM loses stability. However, TWM is not observed to become unstable for Ar = 0.75 and 1.0.

The parameters like Ar and Re significantly affect the flow behavior, and hence, their effect is studied in more detail. Figure 4 illustrates the variation of λr and St with Re for PWM, SWM, and TWM. For PWM, the λr curves for Ar = 1.0 and 0.75 show a monotonic increase with Re. For other aspect ratios, it first increases and then passes through a maximum. It is found from these results that flow past a cylinder with low Ar exhibits higher growth rate at lower Re. Consequently, the critical Reynolds number (Rec) for the onset of vortex shedding decreases for cylinder with smaller Ar [13, 24]. Similar observations can be made for SWM and TWM. For SWM, the λr curve for Ar = 1.0, 0.75, and 0.5 increases monotonically with Re. For other aspect ratios, it shows a non-monotonic increase passing through a maximum. For TWM, the monotonic increase in λr is observed for Ar down to 0.25. Below this, the non-monotonic increase similar to that found with the other modes is observed. For the considered Re range, TWM remains stable for Ar = 1.0 and 0.75. It becomes marginally unstable for Ar = 0.5 at Re = 200. And for lower Ar, it becomes unstable over a range of Re. Bottom row of Fig. 4 displays the variation of St with Re for the three modes. For PWM, St curves show a non-monotonic variation for higher aspect ratio cylinders (Ar = 1.0, 0.75, 0.5). Similar observation has been reported for Ar = 1.0 by several researchers in the past [1, 9, 18]. For lower Ar, St of PWM shows a non-monotonic decrease with increase in Re. SWM and TWM exhibit a monotonic decrease in St with increase in Re, for each Ar. There is however one difference in the St curves for TWM. Overall, the slope of the St curves, for TWM, is smaller than the other two modes, and therefore, the minimum which they reach, in the range of Re shown, is higher than that achieved by the other two modes.

Fig. 4
Top. 3 multi-line graphs plot lambda r versus R e for P W M, S W M, and T W M. 9 lines follow an increasing trend. Bottom. 3 multi-line graphs plot S t versus R e for P W M, S W M, and T W M. 9 lines follow a decreasing trend between 0 and 0.12.

Variation of λr (top) and St (bottom) of PWM (left), SWM (middle), and TWM (right) with increasing Re for different Ar

Full size image

We determine the critical parameters (Rec and Stc) for PWM, SWM, and TWM, corresponding to their zero growth rate.

Figures 5 and 6 show that the value of Rec and Stc for each eigenmode decreases as we decrease the cylinder aspect ratio. The possible reason for the decrease in Rec is because the flow past a cylinder with low Ar exhibits higher growth rate at lower Re (see Fig. 4). Our calculations show that TWM does not become unstable for Ar = 1.0 and 0.75 below Re = 200. Thompson et al. [24] performed linear stability analysis of flow past elliptic cylinders with varying Ar. The critical parameters reported in their work for the onset of instability correspond to the same for PWM in our case. For Ar = 1.0, 0.75, 0.5, and 0.25, they reported Rec value as 47.2, 42.6, 38.8, and 35.6, respectively. In the present study, we find Rec value for Ar = 1.0, 0.75, 0.5, and 0.25 as 46.9, 42.1, 38.3, and 35.1, respectively. Similarly, they reported the corresponding Stc value for Ar = 1.0, 0.75, 0.5, and 0.25 as 0.1163, 0.1144, 0.1120, and 0.1074, respectively. We find Stc for the same values of Ar as 0.1167, 0.1150, 0.1126, and 0.1082, respectively. The comparison shows that the results are in agreement.

Fig. 5
A multi-line graph plots R e subscript c versus A r. 3 lines for P W M, S W M, and T W M follow an increasing trend between 0 and 200. The T W M line has the highest value, and the line P W M has the lowest value.

Variation of Rec with Ar for PWM, SWM, and TWM

Full size image
Fig. 6
A multi-line graph plots S t subscript c versus A r. 3 lines for P W M, S W M, and T W M follow an increasing trend between 0.06 and 0.12. The S W M line has the highest value, and the line T W M has the lowest value.

Variation of Stc with Ar for PWM, SWM, and TWM

Full size image

The work done by Boppana and Gajjar [3] on flow past a cascade of circular cylinder showed the presence of second unstable pair (SWM in the present nomenclature) which becomes unstable between Re = 125 and 150. The critical value was however not reported. Verma and Mittal [25] were the first to determine Rec value for the SWM for flow past a circular cylinder. Rec value reported by them is 110.8 and Stc though not mentioned explicitly is close to 0.12. The data presented in Fig. 5 depict a close match with these results. The difference in Rec value with the prediction of Boppana and Gajjar [3] could be attributed to the use of different lateral width of the domain. Table 1 gives the least-square curve fit equations for Rec and Stc as a function of Ar. The maximum percentage error between the data obtained from the given equations of Rec and the actual value are 0.26 for PWM, 0.47 for SWM, and 0.19 for TWM.

Table 1 Least-square fit equations for Rec and Stc for PWM, SWM, and TWM, obtained as a function of Ar
Full size table

Similarly for Stc, the maximum percentage errors are 0.07 for PWM, 0.41 for SWM, and 0.12 for TWM.

Figure 7 presents the vorticity field for the real parts of PWM, SWM, and TWM for flow past a cylinder with Ar = 0.25 at Re = 180. The mode structures shown are typical and are observed to have similar features at other Reynolds numbers as well. The vorticity field of these modes is symmetric about the wake centerline. This is opposite to that of the base flow. From the figure, we can see that the strength and extent of the three modes vary in the flow field. Here, it should be noted that the range of the vorticity was kept same for all plots. PWM appears more prominent in the near wake and decays rapidly as we move downstream. Comparatively, SWM extends to a much larger distance away from the cylinder. Its strength seems to increase as we move downstream; however, there is a decrease in strength in the far wake. TWM is not so prominent in the near wake, but has the longest range of presence and extends all the way up to the downstream boundary. In general, it is found that the modes with larger growth rates come close to the cylinder and are shorter in their streamwise length compared to the modes which have lower growth rates. In other words, stronger modes tend to approach the cylinder and also contract, whereas the modes which are weaker tend to recede and elongate. Also, the size of the vortical structure in the modes decreases with increase in their oscillation frequency. These observations are in agreement with the wide range of calculations which we have conducted and can also be seen to agree with the images shown in Fig. 7.

Fig. 7
A schematic diagram exhibits the structure of eigenmodes of vorticity fields for P W M, S W M, and T W M. The field ranges from negative 0.0002 to 0.0002. The field for P W M ranges from 0 to 170. The field for S W M ranges from 20 to 350. The field for T W M ranges from 50 to 400.

Structure of the eigenmodes: vorticity fields (real parts) of PWM, SWM, and TWM obtained for Re = 180 flow past a cylinder with Ar = 0.25

Full size image

5 Conclusion

The linear stability analysis of two-dimensional steady base flow past elliptic cylinders of varying aspect ratios (0.01 ≤ Ar ≤ 1.0) and Reynolds number (30 ≤ Re ≤ 200) has been investigated. The steady flow parameters, like LB, WB, ωmax on the cylinder surface, and CD, have been presented, and their trends are discussed. Linear stability analysis of steady flow past elliptic cylinders yields the presence of three different eigenmodes (PWM, SWM, and TWM). It is observed that these modes show a non-monotonic variation of growth rate with Re for low Ar cylinders. This may also be true for higher Ar cylinders for a larger range of Re. Low Ar cylinders exhibit higher growth rate at lower Re. As a result, Rec value for the onset of vortex shedding decreases for smaller Ar. For PWM, the variation of St with Re is non-monotonic for higher Ar. For SWM and TWM, St decreases monotonically with Re for all Ar. An interesting correlation exists between the structure of the eigenmodes and their growth rate and St. Modes which have larger growth rate reach close to the cylinder surface and have comparatively shorter length in the wake. Additionally, the size of the vortical structures present in a mode is large if its St is small and vice versa. The difference in the structure of PWM, SWM, and TWM among themselves can be related to the corresponding change in their growth rate and St.