Aerodynamic forces and three-dimensional flow structures in the mean wake of a surface-mounted finite-height square prism

https://doi.org/10.1016/j.ijheatfluidflow.2020.108569Get rights and content

Highlights

  • The normal force coefficient has a significant contribution from skin-friction.

  • High-vorticity regions, like the corner and tip vortices, are explained.

  • The “tip” vortices in the near wake are shown to be due to the bending of the flow.

  • The mean near-wake structure is composed of two main sections of different origin.

  • A flow model is proposed for the time-averaged wake of finite prisms.

Abstract

Different flow models have been proposed for the flow around surface-mounted finite-height square prisms, but there is still a lack of consensus about the origin and connection of the streamwise tip vortices with the other elements of the wake. This numerical study was performed to address this gap, in addition to clarifying the relationship of the near-wake structures with the far wake and the near-wall flow, which is associated with the fluid forces. A large-eddy simulation approach was adopted to solve the flow around a surface-mounted finite-height square prism with an aspect ratio of AR = 3 and a Reynolds number Re = 500. The mean drag and normal forces and the bending moment for the prism were quantitatively compared in terms of skin-friction and pressure contributions, and related to the near-wall flow. Both three-dimensional visualizations and planar projections of the time-averaged flow field were used to identify, qualitatively, the main structures of the wake, including the horseshoe vortex, corner vortices and regions of high streamwise vorticity in the upper part of the wake. These features showed the same qualitative behavior as reported in high Reynolds number studies. It was found that some regions of high streamwise vorticity magnitude, like the tip vortices, are associated with the three-dimensional bending of the flow, and the tip vortices did not continuously extend to the free end of the prism. The three-dimensional flow analysis, which integrated different observations of the flow field around surface-mounted finite-height square prisms, also revealed that the mean near-wake structure is composed of two sections of different origin and location of dominance.

Introduction

The cross-flow around surface-mounted finite-height obstacles has been a field of intense research in recent decades, with engineering applications ranging from the study of wind loadings on buildings and dispersion of pollution and contaminants in urban environments to the cooling of electronic devices and enhancement of heat exchangers. This type of flow is composed of strongly three-dimensional structures. These structures differ in their vortex wake formation mechanisms and interactions when compared to the classic alternate vortex shedding past two-dimensional (2D or infinite) obstacles such as cylinders or square prisms (Martinuzzi and Havel, 2004; Ozgoren, 2006), which are the most commonly found cross-section geometries in engineering applications.

The three-dimensionality of the flow field becomes even more relevant for surface-mounted prisms of small aspect ratio (AR), defined here as AR = H/D, where H is the height of the obstacle and D is its width. Below a critical aspect ratio, different flow behaviors and trends for the fluid forces and vortex shedding frequency are observed (Sakamoto and Arie, 1983; Okamoto et al., 1990; Sumner et al., 2004; Wang et al., 2004; Wang and Zhou, 2009; Kawai et al., 2012; Sumner, 2013; McClean and Sumner, 2014; Sumner et al., 2015, 2017; Beitel et al., 2019). The critical AR value is dependent on the relative boundary layer thickness of the ground plane measured at the position of the prism (δ/H or δ/D). It typically lies between AR = 2–5 for many studies of prisms of circular or square cross-section (McClean and Sumner, 2014; Sumner et al., 2017). Fig. 1 presents a schematic of some of the mean flow features around a surface-mounted square prism partially immersed in a flat-plate boundary layer, where U is the free-stream velocity and the prism is oriented with its front face normal to the incident flow.

From this time-averaged perspective, the main flow structures that can be found for a generic surface-mounted finite square prism with its front face normal to the incident flow (angle of incidence α = 0°) include the horseshoe vortex system, the tip and base counter-rotating pairs of streamwise vorticity (the tip vortices and base vortices) and a recirculation region downstream of the prism (Martinuzzi and Tropea, 1993; Krajnović and Davidson, 2001; Yakhot et al., 2006; Wang and Zhou, 2009; McClean and Sumner, 2014; Sumner et al., 2017; Zhang et al., 2017). The mean flow field in the vertical (xy) symmetry plane (Fig. 1b) typically presents a main vortex with its focus point downstream of the trailing edge, referred to as vortex Bt by Krajnović (2011) and Sumner et al. (2017). It may also contain a second vortex near the junction of the prism with the ground surface denominated vortex Nw. A saddle point demarks the zones of influence of these two vortices in the recirculation region, and of the downwash and upwash of the flow away from the prism. The base streamwise vorticity pair is associated with a strong upwash, but is not found in every study as its occurrence depends on the prism aspect ratio and on the relative boundary layer thickness. For thin boundary layers the base vortices tend to be absent, characterizing a dipole wake type, but for thick boundary layers the base vortices are present and define a quadrupole wake type (Wang et al., 2006; Hosseini et al., 2013; Sumner et al., 2017). A transition state between dipole and quadrupole wakes was identified and designated a “six-vortices type” wake by Zhang et al. (2017).

In early flow models, the vortex structures in Fig. 1 were thought to be mostly independent. One of the first illustrations of this model with the inclusion of the tip (or trailing) vortices was made by Kawamura et al. (1984), in this case for a circular cylinder. From an instantaneous perspective, the near wake of the prism was proposed to present antisymmetric von Kármán vortex shedding along its span for aspect ratios above the critical threshold, except for regions close to the free end or the prism-wall junction where it was inferred that the tip and base vortices would suppress this mode (Wang et al., 2004). Below the critical AR, an arch-type or hairpin vortex was proposed to prevail, which consists of two spanwise “legs” connected near the free end and shed in a contiguous manner from the top and lateral surfaces (Sakamoto and Arie, 1983; Sakamoto, 1985; Kawai et al., 2012). Wang et al. (2004) described the flow field for low AR as being dominated by the downwash promoted by the tip vortices, which interacted with the base vortices.

This model did not, however, satisfactorily explain the formation of the streamwise tip and base vortices and their interaction with the instantaneous flow structures, as evidence such as a uniform frequency peak along the prism span suggested that these flow structures are not isolated (Wang et al., 2004; Wang and Zhou, 2009; Sattari et al., 2012; Hosseini et al., 2013; McClean and Sumner, 2014) (at least for AR <9~10 (Moreau and Doolan, 2013; Porteous et al., 2017)). A different model was proposed by Wang and Zhou (2009) based on instantaneous particle image velocimetry (PIV) measurements, which consisted of an arch-type structure with its connection, near the free end, and the bottom of the “legs”, near the ground surface, bent upstream by the downwash and upwash flows, respectively. The structure was also identified by Kawai et al. (2012) for a prism with AR = 2.7. In this model, the origin of the tip and base vortices was attributed to the streamwise projection of the arch structure in the yz plane (following the axes of Fig. 1). This structure was observed by Wang and Zhou (2009) throughout their considered range of AR = 3–7, with two different symmetric and antisymmetric modes. The antisymmetric vortices were found to be more probable at the mid-span, while the downwash and upwash flows promoted the symmetric mode and increased its probability near the ends of the prism. The critical AR would then be associated with a higher probability of the symmetric mode for lower AR and antisymmetric mode for higher AR, which is supported by the early observations of Sakamoto and Arie (1983).

The model of Wang and Zhou (2009) was later contested by Bourgeois et al. (2011) and Sattari et al. (2012). These latter studies observed that the phase-averaged wake of a prism with AR = 4 was always antisymmetric. There were occurrences of a co-existing symmetric vortex pair in the formation region, but these were followed by the alternate shedding of vortices at the end of this region. For their case, the time-averaged wake was a dipole-type and a new half-loop structure topological model was proposed based on a phase-average analysis. The half-loop vortical structure consists of a nearly vertical leg at the ground surface, which is the principal core and resembles a Kármán vortex, but with a predominantly streamwise connector strand at the top of it that links it to the adjacent half-loops, as they are shed alternately. For this model, the streamwise tip vortices were explained to be a result of the time-averaging of the connector strands (Bourgeois et al., 2011, 2012). Hosseini et al. (2013) extended the model for a square prism with a quadrupole wake, describing a full-loop structure connected to the neighboring structures at both ends.

The relationship of the critical AR with this new flow model pertaining to flow structure changes is not yet clear. However, as was mentioned previously, recent studies carried out for surface-mounted square prisms of higher AR found a second and even a third critical AR that caused changes in the frequency along the prism height. Moreau and Doolan (2013) were one of the first to identify more than one spectral peak along the prism span for AR > 8.7, accompanied by a change in the trend of the overall sound pressure level for AR > 6.8. This behavior was later confirmed by Porteous et al. (2017), who found additional spectral peaks at even higher AR. The authors classified the flow into four regimes according to the number of acoustic peaks present in the spectrum (0–3), and identified the dominant structures responsible for each frequency based on the phase coherence between the signals of a hot-wire anemometer and a microphone. The R0 regime was found for AR < 2, the primary critical AR, which is the same value reported by Sakamoto and Arie (1983) for a surface-mounted square prism. The flow in this regime had no acoustic peak in the spectrum, and the dominant flow structure was described as mostly vertical vortex filaments, shed from the lateral surfaces of the prism, that are inclined downstream near the free end. The RI regime occurred for 2 < AR < 10, with the sole peak in the spectrum caused by the half-loop structures. The RII regime presented a second peak of higher frequency for 10 < AR < 18. In this regime, the first peak remained, but with a lower frequency. The authors showed that the low frequency peak was caused mostly by vortical structures near the tip of the prism, deformed upstream near the free end, and the high frequency peak was caused by vortex filaments close to the mid-span. These same structures occur at regime RIII for AR > 18, with the addition of a third peak at a frequency intermediate to the other two, and associated with downstream-inclined vortex filaments near the base of the prism. Note that the critical AR thresholds are expected to be influenced by the relative boundary layer thickness.

Although significant effort has been made to improve the understanding of the near-wake structures of the flow based on time-average, phase-average or instantaneous analyses, little attention has been given to their relationship with the near-wall flow on the prism and ground plane surfaces. The near-wall flow was experimentally investigated by means of oil-film visualization, but usually with emphasis on the horseshoe vortex on the ground surface or in the occurrences of separation and reattachment on the surfaces of the prism (Martinuzzi and Tropea, 1993; Nakamura et al., 2001; El Hassan et al., 2015). Near-wall flow investigations for surface-mounted square prisms were also performed with particle image velocimetry (PIV) by Depardon et al. (2005) and Sumner et al. (2017). The use of computational fluid dynamics (CFD) techniques facilitates the analysis of the relationship between near-wall and wake flows by providing the complete flow field around the obstacle. Although simulations are usually restricted to low Reynolds numbers Re = DU/ν (where ν is the fluid kinematic viscosity and the characteristic length D is adopted as the surface-mounted finite-height prism width), they typically present the same behaviors observed at higher Reynolds numbers (Behera and Saha, 2019). This point is also mentioned by Rastan et al. (2017). Sau et al. (2003) investigated vorticity cross-cancelation effects by doing a direct numerical simulation (DNS) of the flow around a surface-mounted prism with AR = 1.7 and Re = 225–500, showing the wake and surface flow topologies. Saha (2013) performed DNS for the flow past surface-mounted square prisms with AR = 2–5 and Re = 250 to evaluate the influence of AR on the mean and instantaneous flow field. The near-wall flow on the prism with AR = 4 was presented and the spanwise variation of the drag coefficient was evaluated, but it was not directly linked to the flow structures. Zhang et al. (2017) studied the effects of different values of Re = 50–1000 (and, as a consequence, different δ/H = 0.08–0.29) for the flow around a surface-mounted square prism with AR = 4. The topological structure of the mean streamwise vortices and the instantaneous structures were highlighted, while the mean near-wall flow on the prism surfaces was compared for their different cases. A more detailed investigation of the near-wall flow was reported by Cao et al. (2019), using an implicit large-eddy simulation (LES) approach for prisms with AR = 3 and 4 and Re = 5 × 104. Krajnović and Davidson (2001, 2002) and Yakhot et al. (2006) provided a more thorough description of the flow structures, but limited to a surface-mounted cube, using LES with Re = 40,000 and DNS with Re = 1870, respectively.

Considering the information in the literature about the flow around surface-mounted finite-height square prisms, two main points motivated the present study. The first is the lack of agreement and consistency on the origin of the tip vortices, their development, and their relationship with the other structures in the prism wake. A second motivation lies in the connection between the three-dimensional structures observed in the near and far wake with the near-wall flow on the prism and ground plane surfaces. This issue has not yet been clearly addressed in the literature, even though it is of great importance in engineering design to relate the flow field with the fluid forces on the prism. The present study aims to describe the time-averaged three-dimensional wake flow structures around a surface-mounted finite-height square prism with AR = 3 and Re = 500, including their connection with the near-wall flow on the ground plane and prism surfaces. In addition, the forces caused by the flow around the prism will be accounted for, and both two-dimensional projections and three-dimensional visualizations will be used to make a comparison with the planar projections of the mean flow field, that are commonly used in the literature.

Section snippets

Computational models and methods

A large-eddy simulation (LES) turbulence approach was adopted for the present numerical analysis. The filtered Navier-Stokes equations which describe the large-scale and incompressible flow, for a Newtonian fluid, are given byu¯ixi=0andu¯it+xj(u¯iu¯j)=xj[ν(u¯ixj+u¯jxi)]1ρp¯xiτijSGSxj,where ui¯ or (u, v, w) are the filtered velocity components along the Cartesian coordinates xi or (x, y, z), ρ is the fluid density and p¯ is the filtered pressure. The subgrid-scale (SGS) stress

Results and discussion

In this section, the fluid force and bending moment coefficients of the surface-mounted finite-height square prism (Re = 500, AR = 3, δ/H = 0.13) will first be evaluated, followed by the mean flow field discussions. These will be based on a planar projection analysis including the near-wall flow in Section 3.2, and on three-dimensional visualizations in Section 3.3.

Conclusions

The flow around a surface-mounted finite-height square prism of AR = 3 with a low Reynolds number Re = 500, plus a very thin boundary layer at the location of the prism of δ/H = 0.13, was numerically investigated using large eddy simulation. A time-averaged approach was adopted to describe the very near-wall flow on the surfaces of the prism and on the ground plane, and its relationship with the near-wake and far-wake structures. Aerodynamic drag and normal force coefficients were evaluated for

CRediT authorship contribution statement

Barbara L. da Silva: Validation, Formal analysis, Writing - original draft, Visualization. Rajat Chakravarty: Conceptualization, Methodology, Software, Investigation. David Sumner: Conceptualization, Writing - review & editing, Supervision, Funding acquisition. Donald J. Bergstrom: Conceptualization, Software, Resources, Writing - review & editing, Supervision, Funding acquisition.

Declaration of Competing Interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The financial support of the Natural Sciences and Engineering Research Council of Canada (Discovery Grants Program numbers RGPIN 05103-2016 and 2018-03760) and the contributions of N. Moazamigoodarzi in setting-up the simulation are gratefully acknowledged.

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