Separation delay through contoured transverse grooves on a 2D boat-tailed bluff body: effects on drag reduction and wake flow features A. Mariottia,∗, G. Burestia , M.V. Salvettia a
Dipartimento di Ingegneria Civile e Industriale, Universit` a di Pisa, Via G. Caruso 8, 56122 Pisa, Italia
Abstract The effectiveness of properly contoured transverse grooves in delaying the flow separation occurring on a two-dimensional boat-tailed bluff body is assessed through numerical simulations. The body has a cross-section with a 3:1 elliptical forebody and a rectangular main part followed by a circulararc boat tail. Three-dimensional Variational Multiscale Large Eddy Simulations are carried out at Re = Du∞ /ν = 9.6 × 104 , using a mixed finitevolume/finite-element method. The introduction of one contoured groove on each of the boat-tail lateral surfaces produces a significant delay of flow separation and a consequent increase of the base pressure, with a global drag reduction of the order of 9.7%. The wake dynamical structure remains qualitatively similar to the one typical of blunt-based two-dimensional bodies, with quantitative variations that are consistent with the reduction in wake width caused by boat tailing and ∗
Corresponding author Email addresses:
[email protected] (A. Mariotti),
[email protected] (G. Buresti),
[email protected] (M.V. Salvetti)
Preprint submitted to Elsevier
October 25, 2018
by the grooves. The introduction of the grooves leads also to a regularization of the vortex shedding downstream of the body, which is more correlated in the spanwise direction. Finally, a few supplementary simulations show that the effect of the grooves is also robust to the variation of the geometrical parameters defining their location and shape. Keywords: Two-dimensional bluff bodies, flow separation control, drag reduction, contoured grooves 1. Introduction The delay of boundary layer separation is one among the most important objectives of flow control. It implies positive practical consequences such as, for instance, the reduction of the drag of bluff bodies. Flow separation delay techniques have been widely described in the literature (see, e.g., Gad-el-Hak, 2000 or Choi et al., 2008 and the references therein) and the present work describes a passive strategy consisting in the introduction of small contoured grooves transverse to the flow in the surface where the boundary layer is developing. The introduction of cavities or grooves transverse to the main flow, with different shapes and characteristic dimensions, has already been proposed for some applications. It has been found that the formation of one or more subsequent small recirculation regions next to the solid surface may produce on the adjacent flow an effect that may be considered as a local relaxation of the no-slip boundary condition. As for internal flows, an improvement of the performance of a plane diffuser was found by Migay (1962) when subsequent small cavities were introduced over its diverging curved wall; similarly,
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transverse rectangular grooves were able to increase the pressure recovery of wide-angle diffusers (Stull and Velkoff, 1972). The use of rectangular grooves may cause a delay of separation also in external flows, as shown by Howard et al. (1983) and Howard and Goodman (1985) for boat-tailed axisymmetric bodies and by Selby et al. (1990) and Lin (1992) for a backward facing ramp. Rectangular and sinusoidal grooves, inspired by the dolphin skin, were successfully used by Jones (2013) and Lang et al. (2017) to delay the separation of a turbulent boundary layer over a flat plate in an adverse pressure gradient. Finally, cavities have also been used to obtain the so-called “trapped vortices”, which are large recirculating flow structures with almost constant vorticity, embedded inside a cavity and separated from the outer flow by a thin and strong shear layer. These structures cause a higher momentum to be present in the downstream boundary layer, which is thus more resistant to separation (Ringleb, 1961). However, in that case the cavities are typically much larger than the incoming boundary layer thickness and an active control, for instance through blowing and suction, is almost always necessary to obtain stable vortices inside the cavity (see, e.g., Iollo and Zannetti, 2001; Tutty et al., 2013). An increase of velocity fluctuations above grooves or cavities was found in many of the above-cited investigations. Self-sustained oscillations may indeed be present in the flow within and around a cavity, generally caused by shear layer instability, pressure wave propagation and acoustic resonance (see, e.g., Rossiter, 1964; Rockwell and Naudascher, 1978; Rowley et al., 2002). The geometry of the cavities and the characteristics of the incoming boundary layer have a significant effect on the occurrence and magnitude
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of these self-sustained oscillations. In particular, the presence of the rear sharp edge in the rectangular cavities is one of the main sources of instability and a considerable reduction of the pressure and velocity fluctuations may be obtained by changing the downstream vertical wall to a slanted or rounded ramp (see, e.g., Rockwell and Naudascher, 1978; Pey et al., 2014). Furthermore, the fluctuations significantly decrease when the boundary layer thickness becomes of the same order of the cavity depth (Rossiter, 1964); in fact, for the open-type cavities − i.e. cavities having a length to depth ratio below 10 − the oscillatory phenomena are not observed above a limit value of the incoming boundary layer thickness (Sarohia, 1977). In Mariotti et al. (2017) suitably-shaped contoured grooves transverse to the oncoming flow were successfully proposed as a strategy to delay boundary layer separation on curved surfaces and to reduce drag for external flows. In particular, this flow control technique was applied to axisymmetric boattailed bluff bodies. Boat-tailing consists in a gradual reduction of the body cross-section before a sharp-edged base, causing a pressure recovery over the final portion of the lateral surface and over the base (Maull and Hoole, 1967; Mair, 1969). Given the boat tail length, its drag-reducing performance increases with decreasing ratio between the final and initial dimensions d/D. However, when this ratio becomes too small boundary layer separation may occur before the base, thus limiting the pressure recovery. Delaying separation in such boat tails is expected to lead to a higher reduction of the total drag. In Mariotti et al. (2017) the effectiveness of the grooves was assessed through a synergic use of experiments and large eddy simulations at a Reynolds number, based on the body diameter and free-stream velocity,
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Re = Du∞ /ν = 9.6 × 104 . The steady small local recirculations embedded in the grooves allowed the reduction of the momentum losses in the near-wall region and, consequently, the presence of a higher momentum in the downstream boundary layers. The consequent delay of the flow separation was found to cause an increase of the base pressure and boat-tail drag reductions up to 29%. The shape and dimensions of the grooves were suitably chosen in order to ensure the steadiness and stability of the flow recirculating within them and to obtain an authentically passive control method. In particular, the considered grooves had a depth that was definitely smaller than the thickness of the upstream boundary layer and ended with a gently curved surface. The effectiveness of the grooves in delaying separation was also verified in internal flows. In particular, they were able to improve the pressure recovery in plane diffusers both in laminar and in turbulent flow conditions, with significant increases in the diffuser efficiency (Mariotti et al., 2013, 2014, 2015b). In the present work, we appraise the capability of the contoured transverse grooves in delaying flow separation and reducing drag in a two-dimensional boat-tailed bluff body. This is a challenging test case due to the higher wake fluctuations caused by the presence of a strong alternate vortex shedding. Furthermore, the stronger adverse pressure gradient present in twodimensional boat-tailed bodies generally causes an earlier flow separation. Therefore, the effectiveness of the grooves found in the axisymmetric case could not be assumed a priori for the two-dimensional case. Moreover, the steadiness of the flow in the cavity region in presence of a strong vortex shedding should also be carefully checked even for small grooves. Finally,
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the two-dimensional problem is paradigmatic of different possible practical applications, where the delay of flow separation may lead to a reduction of the aerodynamic drag and an increase in the maximum cross-flow load, as is requested for aerodynamic bodies or for the rear extractor of high performance cars. The effectiveness of the strategy is assessed through high-fidelity numerical simulations. In particular, the results of three-dimensional Variational MultiScale Large Eddy Simulations (VMS-LES) are presented and discussed. In the next section, the geometries of the analysed body and of the grooves are described, followed by a summary of the numerical methodology and of the simulation set-up in section 3. The main results for the boat tail with and without groove, as regards the drag, the mean pressure and the velocity fields, are reported in section 4. In the same section, the fluid dynamical mechanism of the groove is investigated by examining the mean and fluctuating velocity fields. Furthermore, the effects of the grooves on the wake dynamics and on the spanwise correlation of the vortex shedding are described in section 5, while in section 6 a preliminary analysis of the robustness of the groove performance to the variation of some of its geometrical parameters is described. Concluding remarks and possible future developments are the subject of section 7. 2. Geometry definition The two-dimensional boat-tailed body considered in the present investigation has the same cross-section as the axisymmetric boat-tailed body already analysed experimentally and numerically in Mariotti et al. (2017). It has a 6
3:1 elliptical forebody and a rectangular main part followed by a circulararc boat tail (see Fig. 1, where the reference axes, with origin at the centre of the base, are also shown). The ratio between the main body cross-flow dimension, D, and its overall streamwise length, L, is D/L = 0.175. The boat-tail length is D/2 and the ratio between its final dimension, d, and that of the main body is d/D = 0.791, as for one of the axisymmetric boat-tail configurations investigated in Mariotti et al. (2017). The spanwise dimension of the body, l, is chosen to be equal to L, in analogy with previous numerical investigations on elongated bluff bodies (Bruno et al. (2014)). At the considered Reynolds number Re = D · u∞ /ν = 9.6 × 104 , when no groove is present the boat-tail geometry implies early flow separation along its lateral surface. One suitably contoured transverse groove is then introduced in each of the boat-tail lateral surfaces. The groove has the same geometry as in Mariotti et al. (2017), i.e. it starts with a sharp edge, has an upstream part with a semi-elliptical shape and ends with a spline tangential to the boat tail lateral surface (see Fig. 2). The parameters defining the geometry of the groove are its distance from the start of the boat tail, s, its total length, t, its depth h and the length of the ellipse x-axis, a. The choice of the values of the parameters defining the reference groove is based on the results of the investigation on the axisymmetric case of Mariotti et al. (2017). The sensitivity to small modifications of these parameters will be addressed in Section 6. The initial sharp edge is introduced to fix the start of a local recirculation region embedded in the lower part of the boundary layer. In Mariotti et al. (2017) it has been found that the recirculation should be upstream of the flow
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separation and, thus, in the present case the groove has s/D = 0.065 and t/D = 0.129. The groove depth is chosen to be significantly smaller than the incoming boundary layer thickness to avoid the appearance of phenomena of cavity oscillations and to assure that the small recirculation region inside the groove is steady. As in Mariotti et al. (2017), h is thus constrained to be smaller than 40% of the thickness of the boundary layer present upstream of the groove. Table 1 summarizes the characteristic boundary layer thicknesses at x/D = −0.7, just upstream of the boat tail. The chosen value of the groove depth is h/D = 0.024, which corresponds to h/δx=0.7 = 0.366, so that the ratio between the groove depth and the boat-tail boundary layer thicknesses at the groove starting section is h/δs = 0.40. Indeed, δs < δx=0.7 due to the presence of pressure suctions at the location where the groove is positioned. Considering that the length of the ellipse x-axis, a, has been found in Mariotti et al. (2013) to have a small influence on the results, it was decided to connect its value to the depth of the groove, h, by setting a/h = 2/3. Finally, the gentle slope of the rear face produced by the spline also contributes to the generation of a steady flow; anyway, the steadiness of the recirculation region inside the groove will be checked a posteriori.
3. Simulation set-up and numerical methodology Variational Multiscale Large Eddy Simulations of the considered configuration have been carried out using AERO, a proprietary compressible flow solver that had already been successfully used for the simulation of bluff-body flows (see, e.g., Camarri et al., 2004; Mariotti et al., 2015a). The code is based 8
on a mixed finite-volume/finite-element method, applicable to unstructured grids for space discretization, and on linearized implicit time advancing. The accuracy of the numerical method is second order both in space and time. The Smagorinsky model is used as subgrid scale model in order to close the VMS-LES equations. The computational domain is rectangular, with a cross section of 5.71D × 150D (in the z and y directions, respectively) and a length of 50D (30D being the distance of the outlet from the body base). The blockage factor is about 0.66%. Periodic boundary conditions are imposed in the spanwise direction, characteristic-based boundary conditions (see e.g. Camarri et al., 2004) are used at the inflow, outflow and upper and lower surfaces of the computational domain, while no slip is imposed at the body surface. The computational domain is discretized with an unstructured grid with approximately 2 × 106 nodes. The grid resolution around the body and in the near wake is comparable to the one of the previous LES simulations of the axisymmetric body with the same cross-section, documented in Mariotti et al. (2015a), for which a grid sensitivity analysis was carried out. The grid is particularly refined near the body surface, where the wall y + is lower than 1, and in the near wake. Moreover, the groove is discretized with approximately 3 × 105 nodes. Simulations are carried out at a Reynolds number Re = u∞ D/ν = 9.6 × 104 and laminar free-stream conditions. At the inflow, a uniform and timeconstant velocity is assumed and the free-stream Mach number is equal to 0.1 in order to avoid compressibility effects. The adopted dimensionless time step is ∆T = ∆t(u∞ /D) ≈ 4.3×10−3 , which is more than two orders of magnitude smaller than the period of the wake
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vortex shedding. The adopted time step corresponds to a variable CFL − defined in each cell as CFL = ∆t(u/∆x) − with a maximum value at each time step approximately equal to 20. This value is well below the stability limits of the adopted implicit time advancing scheme. All the flow statistics showed in the following are computed by neglecting an initial numerical transient and then by using a time interval of 160T , being T = t(u/D) the dimensionless time. This interval corresponds to at least 50 vortex-shedding cycles in the wake. In section 4, the statistics are computed by also averaging in the spanwise direction. As in Mariotti et al. (2015a, 2017), a convergence analysis of the statistics of different quantities, such as the mean values of the boundary layer characteristic thicknesses or of the pressure coefficient on the body base, was carried out. As an example of convergence, for the boat-tailed body without the groove differences of less than 1% are found by comparing the mean values of these quantities computed over the above-mentioned time interval and over another one approximately 30% shorter .
4. Main results bt The total boat-tail mean drag coefficient, CD,tot , is defined as the sum of bt the total boat-tail mean pressure drag coefficient, CD,p , and the total boatbt tail mean viscous drag coefficient, CD,v . The boat-tail mean drag coefficients
obtained for the boat tail with and without the groove are reported in Table 2, together with the ones in the last 0.5D long portion of a blunt-based body without boat tail, in order to highlight the significant drag-reducing 10
performance of boat tailing. Besides the total boat-tail mean pressure drag bt coefficient, CD,p , the table contains also the contributions of the lateral surls base face, CD,p , and of the base, CD,p . The reference area for the evaluation of
the drag coefficient is S = D · l. The mean base pressure coefficient, Cp, base , averaged in time and over the base, is also given in the last column. The bt introduction of the groove leads to a reduction of CD,tot of the order of 9.7%,
mainly due to a reduction of the pressure drag. The pressure drag is largely reduced on the base, while it is almost unchanged on the boat-tail lateral surface. Moreover, the viscous contribution to drag in practically negligible.
The analysis of the mean pressure and velocity fields over the body and in the near wake, averaged in time and in the spanwise direction, allows to better understand the behaviour of the different contributions to the mean pressure drag. The mean pressure distributions over the boat-tail lateral surface and base are shown in Fig. 3. The left-hand part of the figure shows the Cp values on the last portion of the lateral surface of the body, in which the start of the boat tail is at x/D = −0.5. The rightmost part provides the pressure variation along the vertical coordinate of the base, starting from its border, in continuation to the values over the lateral surface. Two vertical lines and a sketch indicating the position of the groove are also shown. Additionally, the mean streamwise and vertical velocity components and the mean pressure field are globally shown in Fig. 4, together with the mean flow streamlines. The main feature evident from Fig. 3 is the significant increase of the pressure on the grooved-body base, which is due to a delay of flow separation, as can
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be seen by comparing, for instance, the mean flow streamlines in Fig. 4(a) and Fig. 4(b). In detail, the separation points may be derived quantitatively by analysing the contribution in the x direction of the mean friction coeffi2 ), where τw is the tangential wall stress, t and cient Cf,x = τw (t·ex )/(1 /2 ρu∞
ex are, respectively, the tangential and axial unit vectors. Figure 5 shows the distribution of Cf,x along the lateral surface of the boat tails; two vertical lines and a sketch indicate the position of the groove in the figure. The friction is negative within the groove due to the flow recirculation, then the flow reattaches and the separation point coincides with the more downstream position where the friction becomes negative. As can be seen, the separation point is delayed from xsep /D = −0.325 and ysep /D = ±0.487 in the boattailed body without groove to xsep /D = −0.269 and ysep /D = ±0.478 in the boat-tail with groove. The delay of flow separation has a significant effect on the increase of the curvature of the outer contours of the boundary layer and causes a greater pressure recovery in the last part of the body, which leads to a higher base pressure, as can be observed in figure 3, and, more qualitatively, by comparing Fig. 4(e) and Fig. 4(f). Moreover, from the pressure coefficient distribution in Fig. 3 it is evident that the presence of the groove causes an enhanced decrease of the pressure upstream and at the beginning of the boat tail; more downstream, a pressure peak is found at the reattachment region after the recirculation inside the groove, followed by another decrease due to the subsequent flow acceleration. Finally, the greater pressure recovery in the last part of the boat tail lateral surface corresponds to a much steeper flow deceleration, leading thus to a significantly higher pressure at separation and over the whole base.
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The mechanism at the basis of the separation delay caused by the groove introduction in the two-dimensional configuration is the same already described in Mariotti et al. (2017) for the axisymmetric case. It can be explained by considering the velocity field of three representative boundary layers, viz. those placed just upstream, in the mid and immediately downstream of the groove. For these positions, the profiles of the mean value and of the standard deviation of the streamwise velocity are shown in Fig. 6, where the coordinate yw corresponds to the local vertical position of the plain boat-tail surface. First, the velocity profile measured upstream of the groove has a higher mean near-wall momentum compared to that of the profile corresponding to the same x-coordinate for the plain boat tail (see Fig. 7a)). This is consistent with the higher negative pressure gradient present at the considered x-coordinate when the groove is introduced (see Fig. 3). Then, the recirculation inside the groove causes a significant slip velocity over the groove (of the order of 0.4u∞ ) at the vertical coordinate corresponding to the surface of the boat tail without grooves y = yw (see Fig. 6(b)). Downstream of its reattachment, the boundary layer is thinner and has higher near-wall momentum than in the case without groove (Fig. 7c)). Indeed, the boundary layer thickness decreases from δ/D = 0.065 to δ/D = 0.058 while the shape factor is reduced from H = 2.22 to H = 1.94, with a consequent increase of the distance from the separation condition. Thus, the effective relaxation of the no-slip condition along the outer boundary of the recirculation region reduces the momentum losses near the wall and leads to a delayed boundary layer separation.
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Moreover, for all the considered positions, a reduction of the velocity fluctuations within the boundary layer is found when the groove is present (see Figs. 6(d-f)). Particularly remarkable is the decrease of the near-wall velocity fluctuations after the groove, but a reduction of the fluctuations is present even at the upper edge of the boundary layer. This can represent an additional indication of the steadiness of the recirculation region inside the groove. The distribution of the pressure standard deviation over the lateral surface and the base of the boat tail, shown in Fig. 7, confirms the remarkable reduction of the pressure fluctuations inside the groove region compared to those present over the plain boat tail. Furthermore, no increase of the pressure fluctuations is seen to occur at the location x/D = −0.35, which corresponds to the reattachment of the streamline bounding the recirculation region present inside the groove. 5. Wake dynamics We now investigate on the effect of the introduction of the groove on the near-wake dynamics. At the Reynolds number of the present investigation, the wake flow of a two-dimensional body is characterized by the periodic alternate shedding of vortices, as shown in Fig. 8 by the instantaneous zcomponent of the vorticity on the plane z/D = 0. In particular, a complete cycle of vortex shedding is shown in the figure for the plain boat tail (left side) and for the boat tail with groove (right side). The images are separated by a time interval ∆t = τ /5, where τ is the characteristic period of the flow dynamics. A slight reduction of the wake width may be observed when the groove is present; this produces an increase of the Strouhal number, defined 14
as St = D/τ u∞ , from St = 0.304 for the body with the plain boat tail to St = 0.319 for the one with groove. The reduction of the wake width can better be appreciated from the analysis of the velocity fluctuations averaged in the spanwise direction, shown in Fig. 9 together with a sketch of the mean recirculation region downstream of the body. The standard deviation in time of the x-velocity in the near wake presents two points of maximum fluctuation, which are symmetric respect to the plane y/D = 0 (see Fig. 9(a,b)). These points are connected with the vortex formation length (Bearman, 1965; Griffin and Ramberg, 1974). As can be seen in Table 3, the introduction of the groove produces a slight reduction of the streamwise coordinate of these points and a narrowing in the cross-flow direction. The vortex shedding causes also a maximum in the y-velocity fluctuations downstream of the body along the plane y/D = 0 (see Fig. 9(c,d)). Consistently with the slight narrowing and shortening of the mean recirculation downstream of the boat tail with groove, seemingly caused by the delay of the flow separation, also the point with maximum y-velocity fluctuations is more upstream for the grooved boat tail. It has been observed in Rowe et al. (2001); Mariotti and Buresti (2013); Mariotti et al. (2015a); Barros et al. (2016); Mariotti (2018) that the base pressure is an increasing function of the length of the recirculation region for different blunt-based bodies without boat tail. This relation is no longer evident in the case of boat-tailed bodies, for which the streamlines outside the separating shear layers are generally characterized by a concave curvature (see Fig.4 and the results in Mariotti et al. (2017)). Thus, the mean recirculation region length is now a non-simple function of the lateral position and inclination
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of the boundary layer separation location and of the curvature of the outer streamlines in that position. The three-dimensional iso-surfaces of the vortex indicator λ2 (Jeong and Hussain (1995)) are shown in Fig. 10 for the same time instants as in Fig. 8. It is evident that the vortical structures shedding from the body tend to be more two-dimensional in the case with groove, indicating that an increase in the spanwise correlation of the near-wake vortical structures is produced by the groove. This may be quantitatively assessed through the evaluation of the correlation coefficient in the z-direction of the velocity components in the near wake and of the pressure field on the boat tail. The correlation coefficient is defined as: ρ(q) =
q(x, y, z)q(x, y, z + ∆z) , σq(x,y,z) σq(x,y,z+∆z)
(1)
where q(x, y, z)q(x, y, z + ∆z) is the covariance of the quantity of interest q, σq(x,y,z) and σq(x,y,z+∆z) are the relevant standard deviations in time. The cross sections x/D = 1 and x/D = 2 are considered for the analysis of the wake velocity correlation. The standard deviations of the velocity components are shown in Fig. 11(a,b,d,e) and Fig. 12(a,b,d,e) and for each cross section the correlation coefficient ρ(q) is evaluated along a line in the spanwise direction at a y-coordinate 0.07D outside the wake boundaries (these lines are sketched with a dashed line in Fig. 11(a,b,d,e) and Fig. 12(a,b,d,e)). The correlation coefficients, evaluated from the symmetry plane z/D = 0, are shown in Fig. 11(c,f) and Fig. 12(c,f). Along all the considered cross-lines a significant increase in the spanwise correlation of the two velocity components is found for the boat tail with groove. The magnitude of the velocity fluctuations at each spanwise cross-section does not significantly change with 16
the introduction of the groove. However, the increased regularity of the vortex shedding probably leads to a slight increase in their spanwise-averaged maximum values summarized in Table 3. The increased correlation of the vortex shedding is probably connected with the straighter flow separation line in the spanwise direction induced by the presence of the groove. Indeed, as may be seen by comparing Fig.13(a) with Fig.13(b), the increase in pressure fluctuations, which is associated with the flow separation, is significantly more definite in the spanwise direction for the boat tail with groove. In turn, this leads to a more correlated velocity dynamics in the near wake and also to more correlated pressure fluctuations on the boat tail lateral surface and base (compare Fig.13(c) with Fig.13(d)).
Finally, it should be noted that, in principle, the enhanced correlation of the vortex shedding obtained with the introduction of the grooves is negative in terms of drag reduction, since it leads to a more energetic wake. However, the positive effect of the grooves in delaying the flow separation along the boat tail lateral surface is more significant and leads to an overall decrease of the drag. 6. Robustness analysis The effects of the variation of the distance of the groove from the start of the boat tail, s, and of its depth, h, have been analysed by carrying out simulations with an additional value of each parameter and keeping the groove length constant to the reference value t/D = 0.129 (see Table 4). Considering first the position of the groove, the reference value s/D = 0.065 17
corresponds to a groove placed practically just upstream of the flow separation point in the plain boat tail. The supplementary value is chosen to be equal to s/D = 0.015, corresponding to an upstream displacement of the groove. The total pressure drag for the boat tail with the groove starting more upstream is compared with that for the boat tail with the reference groove in Table 4, reporting also the contributions to drag of the lateral surface and of the base of the boat tail, together with the base-averaged pressure coefficient. The groove at s/D = 0.015 is less effective and the total pressure drag slightly increases, even if it remains significantly lower than the one of the boat tail without groove (see Table 2). As can be seen from the pressure distributions for the boat tails with different grooves that are compared in Fig. 14, the groove placed at s/D = 0.015 produces a slightly lower base pressure compared to the groove at s/D = 0.065. Lower pressures are present also on the rear part of the lateral side of the boat tail, with a slightly increased contribution to the pressure drag. The separation point, reported in Table 5, is more upstream, and this also leads to a slightly wider wake and to a smaller increase of the vortex shedding Strouhal number. As for the influence of the depth of the groove, for both the previously considered groove positions we analyse the effect of a 33% decrease of h/D. We prefer not to further increase the groove depth to avoid reaching values of h of the same order of the boundary layer thickness. Thus, the supplementary depth corresponds to h/D = 0.016. The variations induced by these new geometries on the total, lateral and base pressure drags may again be appreciated in Table 4. The decrease of h/D causes a slight reduction of the
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effectiveness of the groove and a small upstream movement of the separation point (see Table 5). Thus, a weaker pressure recovery on the boat-tail base and lateral surface is obtained and, consequently, a smaller drag reduction (see Fig. 14). However, the differences are small and this confirms the substantial robustness of the contoured transverse grooves as a flow separation delay device. 7. Conclusions In the present work the performance of contoured transverse grooves as a method to delay flow separation on a two-dimensional boat-tailed bluff body was successfully assessed through VMS-LES simulations. The method, already validated for axisymmetric boat-tailed bluff bodies in Mariotti et al. (2017), has now been applied to a two-dimensional boat-tailed body having a wake with high fluctuations due to the presence of a strong alternate vortex shedding, which is not present in the axisymmetric case. The considered groove is introduced on both sides of the boat tail lateral surface, transversely to the incoming flow. The shape of the groove starts with a sharp edge and ends with a spline leading to a point on the original boat-tail contour, with a gentle slope of the rear face. Furthermore, the depth of the groove is constrained to be lower than the 40% of the thickness of the incoming boundary layer. These shape characteristics and size limitations are necessary to avoid the self-sustained oscillations that are often found over small cavities with rear sharp edges and the instabilities that occur in vortices trapped inside transverse cavities that are much larger than the incoming boundary layers. It is thus possible to assure that an intrinsi19
cally steady and stable flow is present in the recirculation regions produced by the introduction of the grooves and to achieve a really passive method of flow control. The grooves are indeed able to significantly delay the flow separation, leading to a boat-tail drag reduction of the order of 9.7%. The physical mechanism leading to the success of the proposed strategy is the same as previously found in Mariotti et al. (2017), i.e. the formation of a local recirculation region embedded in the lower part of the boundary layer, which causes an increased near-wall momentum in the downstream boundary layers and, consequently, a higher resistance to separation. This leads to a narrower wake, to a higher pressure recovery over the rear part of the boat tail lateral surface and thus to a higher base pressure. As for the wake dynamics, vortex shedding is present also after the introduction of the grooves in the boat-tailed body. However, as could be expected, the shedding frequency of the vorticity structures is a function of the wake width, which, in turn, depends on the position of the boundary layer separation. Therefore, the introduction of the grooves is found to produce an increase of the wake dominating frequency. The other significant effect is an increase of the spanwise correlation of the vortex shedding. Indeed, compared to the plain boat tail, a more definite and straighter separation line is found downstream of the groove region and, consistently, the induced pressure and velocity fluctuations are more correlated. However, the more regular vortex shedding caused by the grooves does not significantly affect their drag reducing performance. Indeed, the small increase of drag that may be caused by the regularized vortex shedding is more than counterbalanced by the positive effect of the grooves in delaying the flow separation along the boat tail lateral
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surface. A few supplementary simulations have also been carried out to investigate on the robustness of the effect of the grooves to variations of their position and depth. In general, the device was found to be robust to small variations of the considered parameters. As regards the groove position, one may infer that the drag reduction is more efficient when the groove is placed just upstream of the flow separation, provided the flow is able to reattach downstream of the groove. Furthermore, the reduction of the groove depth causes a slight decrease of its effectiveness. In any case, an upper limit exists for the maximum groove depth, which must always satisfy the requirement of being sufficiently smaller than the incoming boundary layer thickness to avoid the possible onset of instabilities in the flow recirculating inside the groove. In the present work, the body geometry and flow conditions were such that the boundary layer upstream of the groove was in transitional conditions. However, the previous numerical and experimental investigations on plane diffusers and axisymmetric boat-tailed bodies (Mariotti et al., 2014, 2015b, 2017) showed that the grooves are effective as a separation delay technique both for laminar and turbulent boundary layers. Therefore, the influence on the present results of a turbulent inflow condition or of the variation of the Reynolds number may be predicted by considering the ensuing variations in boundary layer thickness, which controls the maximum allowable groove depth. Thus, the main effect of increasingly turbulent inflow conditions may be expected to be an anticipation of transition, with a consequent increase of the boundary layer thickness before the groove. As for the effect of the Reynolds number, for a given shape of the body and considering dimensional
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values, the implications are different if an increase in Reynolds number is associated with an increase in body dimension or in reference velocity. Indeed, if the increase in Reynolds number is due to an increased dimension with constant velocity, in all cases the boundary layer thickness immediately before the groove would increase, together with the maximum allowable groove depth. Conversely, an increase in velocity with unaltered dimension may imply a decrease or an increase of the boundary layer thickness depending on the upstream displacement of the transition point. Summarizing, it can be inferred that the proposed control strategy remains effective for varying freestream turbulence and Reynolds number by adjusting the groove depth according to the changes in the boundary layer thickness. In conclusion, the present investigation confirms the effectiveness of contoured transverse grooves in delaying flow separation through the introduction of small local recirculations adjacent to the wall. Practical applications of the technique may also concern configurations in which delaying separation would have further beneficial effects besides drag reduction, such as, for instance, increasing the maximum cross-flow load and the efficiency of aerodynamic bodies or of the rear extractor of high performance cars. Finally, from a more basic point of view, possible further investigations might be carried out to obtain a clearer identification of the connection between the depth of the grooves and the appearance of flow instabilities. Acknowledgments The authors are grateful to Giulia Piancastelli for her precious contribution in carrying out the numerical simulations.
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References Barros, D., Bor´ee, J., Noack, B. R., Spohn, A., Ruiz, T., 2016. Bluff body drag manipulation using pulsed jets and coanda effect. J. Fluid Mech. 805, 422–459. Bearman, P. W., 1965. Investigation of the flow behind a two-dimensional model with blunt trailing edge and fitted with splitter plates. J Fluid Mech 21, 241–255. Bruno, L., Salvetti, M., Ricciardelli, F., 2014. Benchmark on the aerodynamics of a rectangular 5:1 cylinder: and overview after the first four years of activity. J. Wind Eng. Ind. Aerodyn. 126, 87–106. Camarri, S., Salvetti, M. V., Koobus, B., Dervieux, A., 2004. A low-diffusion MUSCL scheme for LES on unstructured grids. Comput. Fluids 33, 1101– 1129. Choi, H., Jeon, W. P., Kim, J., 2008. Control of flow over a bluff body. Annu. Rev. Fluid Mech. 40, 113–139. Gad-el-Hak, M., 2000. Flow control: passive, active, and reactive flow management. Cambridge University Press. Griffin, O. M., Ramberg, S. E., 1974. The vortex-street wakes of vibrating cylinders. J Fluid Mech 66, 553–576. Howard, F., Goodman, W., 1985. Axisymmetric bluff-body drag reduction through geometrical modification. J. Aircraft 22, 516–522.
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Howard, F. G., Goodman, W. L., Walsh, M. J., 1983. Axisymmetric bluffbody drag reduction using circumferential grooves. AIAA Paper 83-1788. Iollo, A., Zannetti, L., 2001. Trapped vortex optimal control by suction and blowing at the wall. Eur. J. Mech. B-Fluid 20, 7–24. Jeong, J., Hussain, F., 1995. On the identification of a vortex. J. Fluid Mech. 285, 69–94. Jones, E. M., 2013. An experimental study of flow separation over a flat plate with 2D transverse grooves. M.S. Thesis, University of Alabama, Tuscaloosa, AL, USA. Lang, A. W., Jones, E. M., Afroz, F., 2017. Separation control over a grooved surface inspired by dolphin skin. Bioinspir. Biomim. 12, 026005. Lin, J., 1992. Control of low-speed turbulent separated flow over a backwardfacing ramp. PhD Thesis, Old Dominion University, Norfolk, VA. Also NASA-TM-109740. Mair, W. A., 1969. Reduction of base drag by boat-tailed afterbodies in low-speed flow. Aeronaut. Quart. XX, 307–320. Mariotti, A., 2018. Axisymmetric bodies with fixed and free separation: Basepressure and near-wake fluctuations. J. Wind Eng. Ind. Aerod. 176, 21–31. Mariotti, A., Buresti, G., 2013. Experimental investigation on the influence of boundary layer thickness on the base pressure and near-wake flow features of an axisymmetric blunt-based body. Exp. Fluids 54 (11): 1612.
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Mariotti, A., Buresti, G., Gaggini, G., Salvetti, M. V., 2017. Separation control and drag reduction for boat-tailed axisymmetric bodies through contoured transverse grooves. J. Fluid Mech. 832, 514–549. Mariotti, A., Buresti, G., Salvetti, M. V., 2014. Control of the turbulent flow in a plane diffuser through optimized contoured cavities. Eur. J. Mech. B-Fluid 48, 254–265. Mariotti, A., Buresti, G., Salvetti, M. V., 2015a. Connection between base drag, separating boundary layer characteristics and wake mean recirculation length of an axisymmetric blunt-based body. J. Fluid Struct. 55, 191–203. Mariotti, A., Buresti, G., Salvetti, M. V., 2015b. Use of multiple local recirculations to increase the efficiency in diffusers. Eur. J. Mech. B-Fluid 50, 27–37. Mariotti, A., Grozescu, A. N., Buresti, G., Salvetti, M. V., 2013. Separation control and efficiency improvement in a 2D diffuser by means of contoured cavities. Eur. J. Mech. B-Fluid 41, 138–149. Maull, D., Hoole, B., 1967. The effect of boat-tailing on the flow round a two-dimensional blunt-based aerofoil at zero incidence. Journal of the Royal Aeronautical Society 71, 854–858. Migay, V. K., 1962. The efficiency of a cross-ribbed curvilinear diffuser. Energomashinostroenie 1, 45–46 (English translation FTD–TT–62–1151). Pey, Y., Chua, L., Siauw, W., 2014. Effect of trailing edge ramp on cavity flow and pressure drag. Int. J. Heat Fluid Fl. 45, 53–71. 25
Ringleb, F., 1961. Separation control by trapped vortices. In: Lachmann, G.V. (Ed.), Boundary layer and flow control 1, 265–294, Pergamon Press, Oxford. Rockwell, D., Naudascher, E., 1978. Review − self-sustaining oscillations of flow past cavities. J. Fluid Eng. - T. ASME 100, 152–165. Rossiter, J. E., 1964. Wind-tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. Aero. Res. Counc. R&M, No. 3438. Rowe, A., Fry, A. L. A., Motallebi, F., 2001. Influence of boundary-layer thickness on base pressure and vortex shedding frequency. AIAA J 39, 754–756. Rowley, C., Colonius, T., Basu, A., 2002. On self-sustained oscillations in two-dimensional compressible flow over rectangular cavities. J. Fluid Mech. 455, 315–346. Sarohia, V., 1977. Experimental investigation of oscillations in flows over shallow cavities. AIAA J. 15, 984–991. Selby, G., Lin, J., Howard, F., 1990. Turbulent flow separation control over a backward-facing ramp via transverse and swept grooves. J. Fluid Eng. T. ASME 112, 238–240. Stull, F., Velkoff, H., 1972. Effect of transverse ribs on pressure recovery in two-dimensional subsonic diffusers. AIAA Paper 72-1141.
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Figure 1: Sketch of the body geometry and reference system.
Figure 2: Sketch of the groove geometry and main parameters.
δ/D Boat tail
δ ∗ /D
θ/D
H
0.0656 0.0134 0.0069 1.94
Boat tail with groove 0.0642 0.0119 0.0063 1.89 Table 1: Boundary layer characteristics upstream of the boat tail (x/D = −0.7)
.
28
Table 2: Pressure and viscous contributions to drag coefficient and average base pressure coefficient, Cp ,base . bt CD,tot
bt CD,p
bt CD,v
ls CD,p
base CD,p
Cp ,base
Body without boat tail
0.6224
//
0.0012
//
0.6212
-0.620
Boat tail
0.4191
0.4183
0.0008
0.0853
0.3330
-0.421
Boat tail with groove
0.3783
0.3778
0.0005
0.0835
0.2942
-0.372
-0.2
Cp
-0.3
-0.4
-0.5 Boat tail Boat tail with groove
-0.6 -1
-0.75
-0.5
-0.25
x/D
0
0.25
0.5
( yb - y ) / D
Figure 3: Pressure coefficient over the lateral surface and the base of the boat tail, averaged in time and in the spanwise direction.
29
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4: Numerical fields and velocity streamlines averaged in time and in the spanwise direction. (a,b) x-velocity, (c,d) y-velocity, (e,f) pressure coefficient. (a,c,e) Boat tail, (b,d,f) boat tail with groove.
30
10-3
3
Cf,x
2
1
0 Boat tail Boat tail with groove
-1 -1
-0.75
-0.5
-0.25
0
x/D
Figure 5: Streamwise wall shear stress over the lateral surface of the boat tail, averaged in time and in the spanwise direction.
Table 3: Intensity and position of the points of maximum velocity fluctuations in the near wake.
Boat tail
Boat tail with groove
Value of max(std(ux ))
0.374
0.386
x/Dmax(std(ux ))
0.654
0.644
y/Dmax(std(ux ))
0.219
0.189
Value of max(std(uy ))
0.527
0.592
x/Dmax(std(uy ))
1.105
1.068
y/Dmax(std(uy ))
0
0
31
0.1
0.1 No groove With groove
0.08
0.04 0.02 0
0.06 0.04 0.02 0
-0.02 0.4
0.8
0
0.8
1.2
0
0.04 0.02 0
0.08 0.06 0.04 0.02
std (ux / u )
(d)
No groove With groove
0.08 0.06 0.04 0.02 0
-0.02 0.1
1.2
0.1 No groove With groove
0
-0.02
0.8
(c)
(y - yw) / D
0.06
0.4
ux / u
0.1 No groove With groove
(y - yw) / D
(y - yw) / D
0.4
(b)
0.1
0.05
0.02
ux / u
(a)
0
0.04
-0.02
1.2
ux / u
0.08
0.06
0
-0.02 0
No groove With groove
0.08
(y - yw) / D
0.06
(y - yw) / D
(y - yw) / D
0.08
0.1 No groove With groove
-0.02 0
0.05
0.1
0
0.05
0.1
std (ux / u )
std (ux / u )
(e)
(f)
Figure 6: Boundary layer x-velocity profiles averaged in the spanwise direction: mean value at (a) x/D = −0.45, (b) x/D = −0.41 and (c) x/D = −0.33; standard deviation at (d) x/D = −0.45, (e) x/D = −0.41 and (f) x/D = −0.33.
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0.15 Boat tail Boat tail with groove
std (C p)
0.1
0.05
0 -1
-0.75
-0.5
-0.25
0
x/D
0.25
0.5
( yb - y ) / D
Figure 7: Standard deviation of the pressure coefficient over the lateral surface and the base of the boat tail, averaged in the spanwise direction.
Table 4: Pressure and viscous contributions to drag coefficient and average base pressure coefficient, Cp ,base . Sensitivity analysis to groove position and depth. The values for the reference groove are written in bold.
s/D
h/D
bt CD,tot
bt CD,p
bt CD,v
ls CD,p
base CD,p
Cp ,base
0.065
0.024
0.3783
0.3778
0.0005
0.0835
0.2942
-0.372
0.015
0.024
0.3877
0.3872
0.0005
0.0843
0.3029
-0.383
0.065
0.016
0.3867
0.3861
0.0006
0.0839
0.3021
-0.382
0.015
0.016
0.3901
0.3895
0.0006
0.0850
0.3045
-0.385
33
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(l)
(m)
(n)
Figure 8: z-vorticity on the plane z/D = 0 during a vortex-shedding cycle for the boat tail (left side) and the boat tail with groove (right side). Considered times: (a,b) t/τ = 0, (c,d) t/τ = 0.2, (e,f) t/τ = 0.4, (g,h) t/τ = 0.6, (i,l) t/τ = 0.8, (m,n) t/τ = 1.
34
(a)
(b)
(c)
(d)
Figure 9: Standard deviation of the x-velocity averaged in spanwise direction: (a) plain boat tail, (b) boat tail with groove. Standard deviation of the y-velocity averaged in spanwise direction: (c) plain boat tail, (d) boat tail with groove. The continuous lines indicate the mean recirculation regions.
Table 5: Separation points and Strouhal numbers for different grooves. Sensitivity analysis to groove position and depth. The values for the reference groove are written in bold.
s/D
h/D
xsep /D
ysep /D
St
0.065
0.024
-0.269
0.478
0.319
0.015
0.024
-0.273
0.479
0.315
0.065
0.016
-0.270
0.479
0.317
0.015
0.016
-0.276
0.480
0.313
35
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(l)
(m)
(n)
Figure 10: Iso-surfaces of the λ2 criterion (Jeong and Hussain (1995)) during a vortexshedding cycle for the boat tail (left side)36and the boat tail with groove (right side). Considered times: (a,b) t/τ = 0, (c,d) t/τ = 0.2, (e,f) t/τ = 0.4, (g,h) t/τ = 0.6, (i,l) t/τ = 0.8, (m,n) t/τ = 1.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 11: Standard deviation of the x-velocity in the cross sections for the plain boat tail at (a) x/D = 1, (d) x/D = 2 and for the boat tail with groove at (b) x/D = 1, (e) x/D = 2. Cross correlations of the x-velocity evaluated 0.07D outside the wake boundary at (c) x/D = 1, (f) x/D = 2. The dashed lines in figures (a,b,d,e) are those along which the cross-correlations are evaluated.
37
(a)
(b)
(c)
(d)
(e)
(f)
Figure 12: Standard deviation of the y-velocity in the cross sections for the plain boat tail at (a) x/D = 1, (d) x/D = 2 and for the boat tail with groove at (b) x/D = 1, (e) x/D = 2. Cross correlations of the y-velocity evaluated 0.07D outside the wake boundary at (c) x/D = 1, (f) x/D = 2. The dashed lines in figures (a,b,d,e) are those along which the cross-correlations are evaluated.
38
(a)
(b)
(c)
(d)
Figure 13: Standard deviation of the pressure coefficient over the rear part of the lateral surface and the base of the body for (a) plain boat tail and (b) boat tail with groove. Correlation coefficient of the pressure over the 39 rear part of the lateral surface and the base of the body for (c) plain boat tail and (d) boat tail with groove.
-0.2
Cp
-0.3
-0.4 s/D = 0.065 s/D = 0.015 s/D = 0.065 s/D = 0.015
-0.5
h/D = 0.024 h/D = 0.024 h/D = 0.016 h/D = 0.016
-0.6 -1
-0.75
-0.5
-0.25
0
x/D
0.25
0.5
( yb - y ) / D
Figure 14: Pressure coefficient over the lateral surface and the base of the boat tail, averaged in time and in spanwise direction. Sensitivity analysis to groove position and depth. The thickest line corresponds to the reference groove.
40
