# Flow characteristics around a wall-mounted spherical obstacle in a thin boundary layer

## Abstract

The mean wake structures and turbulent flow fields of a wall-mounted spherical obstacle placed in a thin laminar boundary layer, of thickness 14 % of the sphere diameter, were investigated at a Reynolds number of 17,800. Digital particle image velocimetry was used to interrogate the flow in the vicinity of the obstacle, and thermal anemometry measurements were performed to characterize unsteadiness in the wake. Streamwise features observed in the mean wake flow included counter-rotating tip vortices inducing downwash, horseshoe vortices, weak vortices inducing upwash at the top of the near wake, and counter-rotating ‘lobe’ vortices formed by the somewhat unique convex geometry of the sphere near the base. Under the present flow conditions, the sphere base geometry also prevented the roll-up of a horseshoe system upstream of the obstacle. The wake flow field, including lobe structures, is consistent with a vortex skeleton model developed to describe the simpler wakes of low-aspect-ratio wall-mounted semi-ellipsoidal obstacles, with modifications due to the unique junction geometry. Point velocity measurements in the wake identified a weak dominant frequency close to the bed. Cross-spectral analysis of these data at symmetrically located points revealed that, on average, flow oscillations were in phase. The turbulent stress distribution in the wake of the sphere showed a region of high magnitude near the bed not observed for other geometries, and spatially consistent with the lobe structures.

### List of symbols

- \(f\)
Frequency of velocity fluctuations in the sphere wake (Hz)

- \(D\)
Sphere diameter (mm)

- \(U_{\infty }\)
Free-stream velocity (m/s)

- \(d\)
Flow depth (mm)

- \(x\), \(y\), \(z\)
Streamwise, transverse, and spanwise (wall-normal) directions

- \(\omega \)
Vorticity (\(\hbox {s}^{-1}\))

- \(\omega ^*\)
Dimensionless vorticity, \(\omega ^*= \omega D / U_{\infty }\)

- \(\varGamma \)
Circulation (mm

^{2}/s)- \(\varGamma ^* = \varGamma /(U_{\infty }D)\)
Dimensionless circulation

- \(\tau _{t,p}\)
Turbulent projected shear stress (Pa)

- \(Re_D = \frac{U_{\infty } D}{\nu }\)
Reynolds number

- \(\nu \)
Kinematic viscosity (\(\hbox {m}^{2}/\hbox {s}\))

- \(fD/U_{\infty }\)
Dimensionless frequency

- \(St=f_{0}D/U_{\infty }\)
Strouhal number (\( f_{0}\) = dominant frequency)

### References

- Achenbach E (1972) Experiments on the flow past spheres at very high reynolds numbers. J Fluid Mech 62:565–575CrossRefGoogle Scholar
- Achenbach E (1974) Vortex shedding from spheres. J Fluid Mech 62:209–221CrossRefGoogle Scholar
- Baker CJ (1979) The laminar horseshoe vortex. J Fluid Mech 95(2):347–367CrossRefGoogle Scholar
- Baker CJ (1980) The turbulent horseshoe vortex. J Wind Eng Ind Aerodyn 6:9–23CrossRefGoogle Scholar
- Ballio F, Bettoni C, Franzetti S (1998) A survey of time-averaged characteristics of laminar and turbulent horseshoe vortices. ASME J Fluids Eng 120:233–242CrossRefGoogle Scholar
- Bourgeois JA, Sattari P, Martinuzzi RJ (2011) Alternating half-loop shedding in the turbulent wake of a finite surface-mounted square cylinder with a thin boundary layer. Phys Fluids 23(095101):1–14Google Scholar
- Castro IP, Rogers P (2002) Vortex shedding from tapered plates. Exp Fluids 33:66–74CrossRefGoogle Scholar
- Castro IP, Watson L (2004) Vortex shedding from tapered, triangular plates: taper and aspect ration effects. Exp Fluids 37:159–167CrossRefGoogle Scholar
- Chyu M, Natarajan V (1996) Heat transfer on the base of threedimensional protruding elements. Int J Heat Mass Transfer 39:2925–2935CrossRefMATHGoogle Scholar
- Constantinescu GS, Squires KD (2003) Les and des investigations of turbulent flow over a sphere at re = 10000. Flow Turbulence Combust 70:267–298CrossRefMATHGoogle Scholar
- Constantinescu GS, Squires KD (2004) Numerical investigations of flow over a sphere in the subcritical and supercritical regimes. Phys Fluids 16:1449–1466CrossRefGoogle Scholar
- Dey S, Sarkar S, Bose S, Tait S, Castro-Orgaz O (2011) Wall-wake flows downstream of a sphere placed on a plane rough-wall. J Hydraul Eng 137:1173–1189CrossRefGoogle Scholar
- Hajimirzaie SM, Buchholz JHJ (2013) Flow dynamics in the wakes of low-aspect-ratio wall-mounted obstacles. Exp Fluids 54:1616–1630CrossRefGoogle Scholar
- Hajimirzaie SM, Wojcik CJ, Buchholz JHJ (2012) The role of shape and relative submergence on the structure of wakes of low-aspect-ratio wall-mounted bodies. Exp Fluids 53:1943–1962CrossRefGoogle Scholar
- Hosseini Z, Bourgeois J, Martinuzzi R (2013) Large-scale structures in dipole and quadrupole wakes of a wall-mounted finite rectangular cylinder. Exp Fluids 54:1595–1611CrossRefGoogle Scholar
- Jang Y, Lee S (2008) Piv analysis of near-wake behind a sphere at a subcritical reynolds number. Exp Fluids 44:905–914CrossRefGoogle Scholar
- Johnson TA, Patel VC (1999) Flow past a sphere up to a reynolds number of 300. J Fluid Mech 378:19–70CrossRefGoogle Scholar
- Kim H, Durbin P (1988) Observations of the frequencies in a sphere wake and of drag increase by acoustic excitation. Phys Fluids 31:3260–3265CrossRefGoogle Scholar
- Krajnović S (2008) Large eddy simulation of the flow over a three-dimensional hill. Flow Turbulence Combust 81:189–204CrossRefMATHGoogle Scholar
- Krajnović S (2011) Flow around a tall finite cylinder explored by large eddy simulation. J Fluid Mech 676:294–317CrossRefMATHMathSciNetGoogle Scholar
- Lacey RJ, Roy AG (2008) Fine-scale characterization of the turbulent shear layer of an instream pebble cluster. J Hydraulic Eng 134:925–936CrossRefGoogle Scholar
- Lacey RWJ, Rennie CD (2012) Laboratory investigation of turbulent flow structure around a bed-mounted cube at multiple flow stages. J Hydraulic Eng 138:71–84CrossRefGoogle Scholar
- Martinuzzi R (2008) Dual vortex structure shedding from low aspect ratio, surface-mounted pyramids. J Turbulence 9:1–16CrossRefGoogle Scholar
- Martinuzzi RJ, AbuOmar M (2003) Study of the flow around surface-mounted pyramids. Exp Fluids 34:379–389CrossRefGoogle Scholar
- Mason PJ, Morton BR (1987) Trailing vortices in the wakes of surface-mounted obstacles. J Fluid Mech 175:247–293CrossRefGoogle Scholar
- Mittal R, Najjar FM (1999) Flow around a sphere in a plane turbulent boundary layer. In: 39th AIAA Fluid Dynamics Conference, San Antonio, TX, AIAA Paper 2009–4030.Google Scholar
- Narasimha R, Prasad SN (1994) Leading edge shape for flat plate boundary layer studies. Exp Fluids 17:358–360CrossRefGoogle Scholar
- Okamoto S (1980) Turbulent shear flow behind a sphere placed on a plane boundary. In: Bradbury LJS, Durst F, Launder BE, Schmidt FW, Whitelaw JH (eds) Turbulent shear flows 2, Springer, Berlin, pp 246–256.Google Scholar
- Okamoto S (1982) Turbulent shear flow behind hemisphere-cylinder placed on ground plane. In: Bradbury LJS, Durst F, Launder BE, Schmidt FW, Whitelaw JH (eds) Turbulent Shear Flows 2. Springer, Berlin, pp 171–185CrossRefGoogle Scholar
- Okamoto S, Sunabashiri Y (1992) Vortex shedding from a circular cylinder of finite length placed on a ground plane. ASME J Fluids Eng 112:512–521CrossRefGoogle Scholar
- Okamoto S, Uemura N (1991) Effect of rounding side-corners on aerodynamic forces and turbulent wake of a cube placed on a ground plane. Exp Fluids 11:58–64CrossRefGoogle Scholar
- Ozgoren M, Okbaz A, Dogan S, Sahin B, Akilli H (2013) Investigation of flow characteristics around a sphere placed in a boundary layer over a flat plate. Exp Therm Fluid Sci 44:62–74CrossRefGoogle Scholar
- Papanicolaou AN, Dermisis D, Elhakeem M (2011) Investigating the role of clasts on the movement of sand in gravel bed rivers. J Hydraulic Eng 137(9):871–883CrossRefGoogle Scholar
- Papanicolaou AN, Kramer C, Tsakiris A, Stoesser T, Bomminayuni S, Chen Z (2012) Effects of a fully submerged boulder within a boulder array on the mean and turbulent flow fields: implications to bedload transport. Acta Geophysica pp 1–45, doi:10.2478/s11600-012-0044-6
- Pattenden RJ, Turnock SR, Zhang Z (2005) Measurements of the flow over a low-aspect-ratio cylinder mounted on a ground plane. Exp Fluids 39:10–21CrossRefGoogle Scholar
- Sadeque MAF, Rajaratnam N, Loewen MR (2009) Effects of bed roughness on flow around bed-mounted cylinders in open channels. J Eng Mech 135(2):100–110CrossRefGoogle Scholar
- Sakamoto H, Haniu H (1990) A study on vortex shedding from sphere in uniform flow. Trans ASME J Fluid Eng 112:386–392CrossRefGoogle Scholar
- Savory E, Toy N (1986) Hemispheres and hemisphere-cylinders in turbulent boundary layers. J Wind Eng Ind Aerodyn 23:345–364CrossRefGoogle Scholar
- Seal CV, Smit CR, Akin O, Rockwell D (1995) Quantitative characteristics of a laminar, unsteady necklace vortex system at a rectangular block: flat plate juncture. J Fluid Mech 286:117–135CrossRefGoogle Scholar
- Shamloo H, Rajaratman N, Katopodis C (2001) Hydraulics of simple habitat structures. J Hydraulic Res 39(4):351–366CrossRefGoogle Scholar
- Simpson RL (2001) Junction flow. Annu Rev Fluid Mech 33:415–443CrossRefGoogle Scholar
- Strom K, Papanicolaou A (2007) ADV measurements around a cluster microform in a shallow mountain stream. J Hydraulic Eng 133(12):1379–1389CrossRefGoogle Scholar
- Strom K, Papanicolaou A, Constantinescu G (2007) Flow heterogeneity over 3d cluster microform: Laboratory and numerical investigation. J Hydraulic Eng 133(3):273–287CrossRefGoogle Scholar
- Taneda S (1978) Visual observations of the flow past a sphere at Reynolds numbers between 10
^{4}and 10^{6}. J Fluid Mech 85(1):187–192CrossRefGoogle Scholar - Tsutsui T (2008) Flow around a sphere in a plane turbulent boundary layer. J Wind Eng Ind Aerodyn 96:779–792CrossRefGoogle Scholar
- Uematsu Y, Yamada M, Ishii K (1990) Some effects of free-stream turbulence on the flow past a cantilevered circular cylinder. J Wind Eng Ind Aerodyn 33:43–52CrossRefGoogle Scholar
- Wang HF, Zhou Y (2009) The finite-length square cylinder near wake. J Fluid Mech 638:453–490CrossRefMATHGoogle Scholar
- Westerweel J (2000) Theoretical analysis of the measurement precision in particle image velocimetry. Exp Fluids 29:S3–S12CrossRefGoogle Scholar
- Yager E, Kirchner J, Dietrich W (2007) Calculating bed load transport in steep boulder bed channels. Water Resour Res 43(7):1–24Google Scholar