# Turbulent secondary flows in wall turbulence: vortex forcing, scaling arguments, and similarity solution

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## Abstract

Spanwise surface heterogeneity beneath high-Reynolds number, fully-rough wall turbulence is known to induce a mean secondary flow in the form of counter-rotating streamwise vortices—this arrangement is prevalent, for example, in open-channel flows relevant to hydraulic engineering. These counter-rotating vortices flank regions of predominant excess(deficit) in mean streamwise velocity and downwelling(upwelling) in mean vertical velocity. The secondary flows have been definitively attributed to the lower surface conditions, and are now known to be a manifestation of Prandtl’s secondary flow of the second kind—driven and sustained by spatial heterogeneity of components of the turbulent (Reynolds averaged) stress tensor (Anderson et al. J Fluid Mech 768:316–347, 2015). The spacing between adjacent surface heterogeneities serves as a control on the spatial extent of the counter-rotating cells, while their intensity is controlled by the spanwise gradient in imposed drag (where larger gradients associated with more dramatic transitions in roughness induce stronger cells). In this work, we have performed an order of magnitude analysis of the mean (Reynolds averaged) transport equation for streamwise vorticity, which has revealed the scaling dependence of streamwise circulation intensity upon characteristics of the problem. The scaling arguments are supported by a recent numerical parametric study on the effect of spacing. Then, we demonstrate that mean streamwise velocity can be predicted a priori via a similarity solution to the mean streamwise vorticity transport equation. A vortex forcing term has been used to represent the effects of spanwise topographic heterogeneity within the flow. Efficacy of the vortex forcing term was established with a series of large-eddy simulation cases wherein vortex forcing model parameters were altered to capture different values of spanwise spacing, all of which demonstrate that the model can impose the effects of spanwise topographic heterogeneity (absent the need to actually model roughness elements); these results also justify use of the vortex forcing model in the similarity solution.

## Keywords

Turbulence Vortex forcing model Streamfunction## Notes

### Acknowledgements

This work was supported by the U.S. Air Force Office of Scientific Research, Grant # FA9550-14-1-0394 (WA, JY) and Grant # FA9550-14-1-0101 (WA, AA), and by the Texas General Land Office, Contract # 16-019-0009283 (WA, KS).

## References

- 1.Raupach M, Antonia R, Rajagopalan S (1991) Rough-wall turbulent boundary layers. Appl Mech Rev 44:1CrossRefGoogle Scholar
- 2.Jimenez J (2004) Turbulent flow over rough wall. Annu Rev Fluid Mech 36:173CrossRefGoogle Scholar
- 3.Castro I (2007) Rough-wall boundary layers: mean flow universality. J Fluid Mech 585:469CrossRefGoogle Scholar
- 4.Grass A (1971) Structural features of turbulent flow over smooth and rough boundaries. J Fluid Mech 50:233CrossRefGoogle Scholar
- 5.Ghisalberti M (2009) Obstructed shear flows: similarities across systems and scales. J Fluid Mech 641:51CrossRefGoogle Scholar
- 6.Anderson W, Li Q, Bou-Zeid E (2015) Numerical simulation of flow over urban-like topographies and evaluation of turbulence temporal attributes. J Turbul 16(9):809CrossRefGoogle Scholar
- 7.Pan Y, Chamecki M (2016) A scaling law for the shear-production range of second-order structure functions. J Fluid Mech 801:459CrossRefGoogle Scholar
- 8.Marusic I, Mathis R, Hutchins N (2010) Predictive model for wall-bounded turbulent flow. Science 329:193CrossRefGoogle Scholar
- 9.Mathis R, Hutchins N, Marusic I (2009) Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J Fluid Mech 628:311CrossRefGoogle Scholar
- 10.Anderson W (2016) Conditionally averaged large-scale motions in the neutral atmospheric boundary layer: insights for Aeolian processes. J Fluid Mech 789:567CrossRefGoogle Scholar
- 11.Townsend A (1976) The structure of turbulent shear flow. Cambridge University Press, CambridgeGoogle Scholar
- 12.Volino R, Schultz M, Flack K (2007) Turbulence structure in rough- and smooth-wall boundary layers. J Fluid Mech 592:263CrossRefGoogle Scholar
- 13.Alfredsson P, Örlu R (2010) The diagnostic plot-a litmus test for wall-bounded turbulence data. Eur J Mech B Fluids 29:403CrossRefGoogle Scholar
- 14.Wu Y, Christensen KT (2010) Spatial structure of a turbulent boundary layer with irregular surface roughness. J Fluid Mech 655:380CrossRefGoogle Scholar
- 15.Alfredsson P, Segalini A, Örlu R (2011) A new scaling for the streamwise turbulence intensity in wall-bounded turbulent flows and what it tells us about the “outer” peak. Phys Fluids 23:702CrossRefGoogle Scholar
- 16.Hong J, Katz J, Meneveau C, Schultz M (2012) Coherent structures and associated subgrid-scale energy transfer in a rough-wall channel flow. J Fluid Mech 712:92CrossRefGoogle Scholar
- 17.Bou-Zeid E, Meneveau C, Parlange M (2004) Large-eddy simulation of neutral atmospheric boundary layer flow over heterogeneous surfaces: Blending height and effective surface roughness. Water Resour Res 40:W02505CrossRefGoogle Scholar
- 18.Yang X (2016) On the mean flow behaviour in the presence of regional-scale surface roughness heterogeneity. Bound-Layer Met 161:127CrossRefGoogle Scholar
- 19.Macdonald R, Griffiths R, Hall D (1998) An improved method for the estimation of surface roughness of obstacle arrays. Atmospheric Environment
**32**(11):1857Google Scholar - 20.Wood D (1981) The growth of internal layer following a step change in surface roughness. Report T.N. – FM 57, Dept. of Mech. Eng., Univ. of Newcastle, AustraliaGoogle Scholar
- 21.Andreopoulos J, Wood D (1982) The response of a turbulent boundary layer to a short length of surface roughness. J Fluid Mech 118:143CrossRefGoogle Scholar
- 22.Antonia R, Luxton R (1971) The response of a turbulent boundary layer to a step change in surface roughness Part I. Smooth to rough. J Fluid Mech 48:721CrossRefGoogle Scholar
- 23.Garratt J (1990) The internal boundary layer: a review. Bound-Layer Meteorol 40:171CrossRefGoogle Scholar
- 24.Wang ZQ, Cheng NS (2005) Secondary flows over artificial bed strips. Adv. Water Resour 28:441CrossRefGoogle Scholar
- 25.Mejia-Alvarez R, Christensen K (2013) Wall-parallel stereo PIV measurements in the roughness sublayer of turbulent flow overlying highly-irregular roughness. Phys Fluids 25:115109CrossRefGoogle Scholar
- 26.Vermaas D, Uijttewall W, Hoitink A (2011) Lateral transfer of streamwise momentum caused by a roughness transition across a shallow channel. Water Resour Res 47:W02530CrossRefGoogle Scholar
- 27.Willingham D, Anderson W, Christensen KT, Barros J (2013) Turbulent boundary layer flow over transverse aerodynamic roughness transitions: induced mixing and flow characterization. Phys Fluids 26:025111CrossRefGoogle Scholar
- 28.Nugroho B, Hutchins N, Monty J (2013) Large-scale spanwise periodicity in a turbulent boundary layer induced by highly ordered and direction surface roughness. Int J Heat Fluid Flow 41:90CrossRefGoogle Scholar
- 29.Barros J, Christensen K (2014) Observations of turbulent secondary flows in a rough-wall boundary layer. J Fluid Mech 748:1CrossRefGoogle Scholar
- 30.Vanderwel C, Ganapathisubramani B (2015) Effects of spanwise spacing on large-scale secondary flows in rough-wall turbulent boundary layers. J. Fluid Mech. https://doi.org/10.1017/jfm.2015.292 Google Scholar
- 31.Anderson W, Barros J, Christensen K, Awasthi A (2015) Numerical and experimental study of mechanisms responsible for turbulent secondary flows in boundary layer flows over spanwise heterogeneous roughness. J Fluid Mech 768:316CrossRefGoogle Scholar
- 32.Yang J, Anderson W (2017) Turbulent channel flow over surfaces with variable spanwise heterogeneity: establishing conditions for outer-layer similarity. Flow Turbul Combust. https://doi.org/10.1007/s10,494-017-9839-5 Google Scholar
- 33.Medjnoun T, Vanderwel C, Ganapathisubramani B (2018) Characteristics of turbulent boundary layers over smooth surfaces with spanwise heterogeneities. J Fluid Mech 838:516CrossRefGoogle Scholar
- 34.Awasthi A, Anderson W (2018) Numerical study of turbulent channel flow perturbed by spanwise topographic heterogeneity: amplitude and frequency modulation within low-and high-momentum pathways. Phys Rev Fluids 3:044602CrossRefGoogle Scholar
- 35.Bou-Zeid E, Parlange M, Meneveau C (2007) On the parameterization of surface roughness at regional scales. J Atmos Sci 64:216CrossRefGoogle Scholar
- 36.Nezu I, Nakagawa H (1993) Turbulence in open-channel flows. Balkema Publishers, RotterdamGoogle Scholar
- 37.Prandtl L (1952) Essentials of fluid dynamics. Blackie and Son, LondonGoogle Scholar
- 38.Hoagland L (1960) Fully developed turbulent flow in straight rectangular ducts—secondary flow, its cause and effect on the primary flow. Ph.D. thesis, Massachusetts Inst. of TechGoogle Scholar
- 39.Brundrett E, Baines WD (1964) The production and diffusion of vorticity in duct flow. J Fluid Mech 19:375CrossRefGoogle Scholar
- 40.Perkins H (1970) The formation of streamwise vorticity in turbulent flow. J Fluid Mech 44:721CrossRefGoogle Scholar
- 41.Gessner F (1973) The origin of secondary flow in turbulent flow along a corner. J Fluid Mech 58:1CrossRefGoogle Scholar
- 42.Bradshaw P (1987) Turbulent secondary flows. Ann Rev Fluid Mech 19:53CrossRefGoogle Scholar
- 43.Madabhushi R, Vanka S (1991) Large eddy simulation of turbulence-driven secondary flow in a square duct. Phys Fluids A 3:2734CrossRefGoogle Scholar
- 44.Leibovich S (1977) Convective instability of stably stratified water in the ocean. J Fluid Mech 82:561CrossRefGoogle Scholar
- 45.Leibovich S (1980) On wave-current interaction theories of Langmuir circulations. J Fluid Mech 99:715724CrossRefGoogle Scholar
- 46.Leibovich S (1983) The form and dynamics of Langmuir circulation. Annu Rev Fluid Mech 15:391CrossRefGoogle Scholar
- 47.Craik A (1985) Wave interactions and fluid flows. Cambridge University Press, CambridgeGoogle Scholar
- 48.McWilliams J, Sullivan P, Moeng CH (1997) Langmuir turbulence in the ocean. J Fluid Mech 334:1CrossRefGoogle Scholar
- 49.Yang D, Chen B, Chamecki M, Meneveau C (2015) Oil plumes and dispersion in Langmuir, upper-ocean turbulence: Large-eddy simulations and K-profile parameterization. J. Geophys. Research: Oceans 120:4729CrossRefGoogle Scholar
- 50.Shrestha K, Anderson W, Kuehl J (2018) Langmuir turbulence in coastal zones: structure and length scales. J Phys Oceanogr. https://doi.org/10.1175/JPO-D-17-0067.1 Google Scholar
- 51.Stokes G (1847) On the theory of oscillatory waves. Trans Camb Philos Soc 8:441Google Scholar
- 52.Mansfield J, Knio O, Meneveau C (1998) A dynamic LES scheme for the vorticity transport equation: formulation and a priori tests. J Comp Phys 145:693CrossRefGoogle Scholar
- 53.Mansfield J, Knio O, Meneveau C (1999) Dynamic LES of colliding vortex rings using a 3D vortex method. J Comp Phys 152:305CrossRefGoogle Scholar
- 54.Gayme D, McKeon B, Papachristodoulou A, Bamieh B, Doyle J (2010) A streamwise constant model of turbulence in plane Couette flow. J Fluid Mech 665:99CrossRefGoogle Scholar
- 55.Pope S (2000) Turbulent flows. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- 56.Reynolds R, Hayden P, Castro I, Robins A (2007) Spanwise variations in nominally two-dimensional rough-wall boundary layers. Exp Fluids 42:311CrossRefGoogle Scholar
- 57.Fishpool G, Lardeau S, Leschziner M (2009) Persistent Non-Homogeneous Features in Periodic Channel-Flow Simulations. Flow Turbulence Combust 83:823CrossRefGoogle Scholar
- 58.Arbogast L (1800) Du calcul des dérivation. Levrault, StrasbourgGoogle Scholar
- 59.Goursat E (1902) Cours d‘analyse mathématique. Gauthier-Villars, Paris, p 1Google Scholar
- 60.Albertson J, Parlange M (1999) Surface length scales and shear stress: implications for land-atmosphere interaction over complex terrain. Water Resour Res 35:2121CrossRefGoogle Scholar
- 61.Anderson W, Meneveau C (2010) A large-eddy simulation model for boundary-layer flow over surfaces with horizontally resolved but vertically unresolved roughness elements. Bound-Layer Meteorol 137:397CrossRefGoogle Scholar
- 62.Wilczek M, Stevens R, Meneveau C (2015) Spatio-temporal spectra in the logarithmic layer of wall turbulence: large-eddy simulations and simple models. J Fluid Mech 769:R1CrossRefGoogle Scholar
- 63.Bou-Zeid E, Meneveau C, Parlange M (2005) A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows. Phys Fluids 17:025105CrossRefGoogle Scholar
- 64.Orszag S (1970) Transform method for calculation of vector coupled sums: application to the spectral form of the vorticity equation. J Atmos Sci 27:890CrossRefGoogle Scholar
- 65.Deardorff J (1970) A numerical study of 3 dimensional turbulent channel flow at large Reynolds numbers. J Fluid Mech 41:453CrossRefGoogle Scholar
- 66.Piomelli U, Balaras E (2002) Wall-layer models for large-eddy simulation. Annu Rev Fluid Mech 34:349CrossRefGoogle Scholar
- 67.Adrian R, Christensen K, Liu ZC (2000) Vortex organization in the outer region of the turbulent boundary layer. Exp Fluids 29:275CrossRefGoogle Scholar
- 68.Anderson W (2012) An immersed boundary method wall model for high-Reynolds number channel flow over complex topography. Int J Numer Methods Fluids 71:1588CrossRefGoogle Scholar
- 69.Flack K, Schultz M (2010) Review of hydraulic roughness scales in the fully rough regime. J Fluids Eng 132(4):041203CrossRefGoogle Scholar