Effects of Streamlining a Bluff Body in the Laminar Vortex Shedding Regime

Two-dimensional flow over bluff bodies is studied in the unsteady laminar flow regime using numerical simulations. In previous investigations, lift and drag forces have been studied over different cross section shapes like circles, squares and ellipses. We aim to extend the previous research ∗Address all correspondence for to this author. FE-19-1169 Skote 1 Journal of Fluids Engineering by studying the variation of hydrodynamic forces as the shape of the body changes from a circular cylinder to a more streamlined or a bluffer body. The different body shapes are created by modifying the downstream circular arc of a circular cylinder into an ellipse, hence elongating or compressing the rear part of the body. The precise geometry of the body is quantified by defining a shape factor. Two distinct ranges of shape factors with fundamentally different behavior of lift and drag are identified. The geometry constituting the limit is where the rear part ellipse has a semi minor axis of half the radius of the original circle, independent of the Reynolds number. On the other hand, the vortex shedding frequency decreases linearly over the whole range of shape factors. Furthermore, the variation of the forces and frequency with Reynolds number, and how the relations vary with the shape factor are reported.


INTRODUCTION
Flow around bluff bodies over a range of Reynolds numbers gives rise to a well-known periodic flow phenomenon of vortex shedding in the wake, more familiarly known as the von Kármán vortex street [1][2][3]. This kind of flow has been of academic interest for years and is also relevant for many practical applications in the designing of buildings, bridges, vortex flow-meters, towers etc., as vortex shedding results in periodically varying lift and drag forces on the body. The nature and periodicity of the fluctuating forces and vortex street depends upon the Reynolds number Re = V D/ν, based on the freestream velocity V , dimension of bluff body D and kinematic viscosity of fluid ν. From previous work it has been established that the flow over circular cylinders is laminar, two-dimensional (2D), and displays vortex shedding in the range of Re from approximately 50 to 170 [4][5][6]. At lower values of Re than this, the flow is steady and at higher values it starts to become three-dimensional [1,[7][8][9].
In previous studies, lift and drag forces have been studied over different cross section shapes like circles [10][11][12][13][14][15], squares [16,17], ellipses [18][19][20] and half-circles [21][22][23][24][25]. We aim to extend the previous research by studying the variation of hydrodynamic forces and vortex shedding frequency as the shape of the body changes from a circular cylinder to a more streamlined or a more bluff body. The different body shapes are created by modifying the downstream circular arc of a circular cylinder into an ellipse.
Applications may include novel designs of heat exchangers or risers in the offshore oil-gas industry, apart from the more obvious engineering problems listed earlier. The present study points out that when shape changes are introduced in the design and optimization process, extrapolation of the forces from the original geometries are unreliable. The simulations presented here forms a guideline on which changes in the forces can be expected when streamlining a shape or transforming it into a more bluff body.

PROBLEM FORMULATION AND NUMERICAL SIMULATION
The two parameters which are varied in the present study are Re and shape factor (SF ), which is defined as the ratio of length of body downstream of point of maximum thickness to half of its FE-19-1169 Skote 3 Journal of Fluids Engineering maximum thickness (Fig. 1). For a circular cylinder, half of maximum thickness is the radius, while its length downstream the point of maximum thickness is also the radius. Therefore, for a circular cylinder, the SF is unity. The semi-cylinder (with the rounded section facing the flow), on the other hand, has a SF of zero according to this definition. Other bluff bodies retain the same thickness and are created by elongating or squeezing the downstream section of circular cylinder by ellipse of appropriate dimensions. For an elongated body, the downstream circular arc is replaced by an ellipse of semi major axis greater than the radius. In contrast, for a squeezed body, the downstream circular arc is replaced by an ellipse of semi minor axis lesser than the radius (Fig. 2). Accordingly, SF for the bodies in the present study is equal to the ratio of semi For generating the mesh on this computational domain, it was divided into various sections as indicated in Fig. 3. In the present mesh, X = 15D, M = 0.5D. Note that the body is illustrated with an enlarged solid circle in order to clarify the grid structure near the surface. Please refer to Fig. 4 for a drawing in correct scale.

Journal of Fluids Engineering
Most effort was spent on refinement of region M which is essential for capturing the boundary layer and its separation. The number of divisions on M were increased until the values for maximum lift coefficient, mean drag coefficient and Strouhal number were no longer varying at a Re = 100. Details are given in Tab. 1. The final number of divisions on M was determined to be 40 as further refinement did not alter the results. Furthermore, the results did not change with more refinement of grid in the outer regions.
Side O is divided into 80 control volumes, and the whole circumference of the circle is divided into 400 equally spaced segments and each side of the square containing the body has 100 divisions each (i.e, side X is divided into 50 control volumes). As mentioned above, side M (shown in red in Fig and where F L and F D are the lift and resistance forces of the body, respectively, and ρ is the density of the fluid.
The non-dimensional frequency of lift and drag forces is expressed by is the frequency of variation of lift, while the frequency for the drag force is twice as large. We define C Lmax as the maximum extreme value of C L , while the average value of C L is zero due to symmetry.
Re is varied from 60 to 160 in order to keep the flow in the 2D unsteady regime of vortex shedding [9,15]. In our numerical simulations, the flow for bodies with SF = 1.75 and 2.0 are found to be steady for Re = 60 and are hence excluded from the analysis.

Lift coefficient
It is known that for a circular cylinder the maximum value of the lift coefficient (C Lmax ) increases with Re. In the present study, bodies with SF other than unity behave similarly. However, C Lmax Journal of Fluids Engineering

Drag coefficient
Fluctuations in the drag coefficient due to vortex shedding are very weak as compared to its mean value. Therefore, only the changes in mean drag with Re and SF are considered here. The mean drag coefficient (C Dmean ) for a circular cylinder in the unsteady laminar flow regime was found to become nearly constant above Re = 100 as shown in Fig. 6. This result is consistent with the previous results of Park [15] and Rajani [8].
However, the variation of C Dmean with Re for SF > 1.0 is the opposite from bodies with SF < 1.0 (see Fig. 6); decreasing for the former, while increasing for the latter. Thus, the limiting case is the circular cylinder (SF = 1).
Although C Dmean varies little with Re for a particular body, it changes dramatically when varying SF for a particular Re as shown in Fig. 7, where results for Re = 100 is given. The variation of mean drag with SF for SF ≥ 0.5 can be best described in terms of a power law of the form FE-19-1169 Skote 7

Journal of Fluids Engineering
The agreement between Eqn. (4) with the data points obtained from the numerical simulations (circles) is shown for Re = 100 in Fig. 7. The values for A and n obtained for the other Reynolds numbers are given in Tab. 3, which demonstrates that the value of coefficient A remains almost constant while n falls with Re. This indicates that C Dmean falls more steeply with SF at higher Re.
The bodies with SF < 0.5 are presented with stars in Fig. 7. The logarithmic behavior according to Eqn. (4) is clearly not valid for SF < 0.5. Instead, a linear relationship is revealed as seen in the inset of Fig. 7 where only the data for SF = 0.01 − 0.5 is shown. The linear behavior is also confirmed for other Re and the coefficients are given in Tab. 3 for the expression, The value of the intercept C Do in the expression (5)  The mean drag on a bluff body comprises of mean pressure drag and mean skin friction drag expressed by the coefficients C Dp and C Dsf , which are defined as in Eq. 2 with F D replaced by the forces obtained by integrating the pressure difference and the wall shear stress over the surface, respectively.
A comparison of the coefficients C Dp and C Dsf with previous studies of a circular cylinder is shown in Fig. 8, where it can be seen that they agree well with the present results for SF = 1.
C Dsf variation with Re remains identical for all SF , which is natural since the attached flow on the front of the cylinder is not affected by the SF . Hence, it can be concluded that the difference in mean drag (shown in Fig. 6) is due to the change in pressure drag caused by the streamlining of FE-19-1169 Skote 8 Journal of Fluids Engineering the body. Bodies which are more bluff have higher pressure drag as compared to bodies which are elongated and more streamlined. Due to the contribution to C Dmean by C Dsf , which is independent on SF , the C Dp variation with Re for the different bodies is very similar to the mean pressure in Fig. 6. However, for the pressure drag the limiting case (for which C Dp is constant with varying Re) is the body with SF = 1.25.

Strouhal number
Classical curve fit for variation of St with Re for a circular cylinder were given by Roshko [26] and Williamson [27] and the present results agree well, see Fig. 9. Similar results were obtained for all SF varying between 0.01 and 2.0, where the St was also found to increase with Re. However, for bodies with lower SF , St was higher while it was lower for bodies with higher SF , as shown in Fig. 10. Furthermore, for all Re, the St decreases linearly with increasing SF .
Similar to lift and drag, the sensitivity of St to SF was also studied. Journal of Fluids Engineering

CONCLUSION
In contrast to the geometries considered previously, the present study provides a detailed description of forces acting on the bluff body and shedding frequency as the circular cylinder is elongated (SF > 1.0) to make it more streamlined, or squeezed (SF < 1.0) so that it approaches a semi-circular geometry (SF = 0).
Variation of C Lmax (decreasing) with increasing SF was found to be linear for SF ≥ 0.5, while a non-linear relation exists for SF ≤ 0.5. On the other hand, the variation of C Dmean (decreasing) was described by a power law for SF ≥ 0.5, whereas a linear relation was found for SF ≤ 0.5.
Hence, both C Dmean and C Lmax changes behavior for SF ≤ 0.5.
While C Dmean is increasing or decreasing with Re depending on whether SF is less than or larger than unity (circular cylinder), the pressure drag coefficient behaves similarly but with

8
Comparison of pressure drag (C Dp ) and skin friction drag (C Dsf ) in present study for circular cylinder with previous results reported by Park [15] and Rajani [8]. 9 Comparison of St in the present study for a circular cylinder with classical curve fit from Roshko [26] and Williamson [27].