Turbulent secondary flows in wall turbulence: vortex forcing, scaling arguments, and similarity solution
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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.
KeywordsTurbulence Vortex forcing model Streamfunction
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).
- 11.Townsend A (1976) The structure of turbulent shear flow. Cambridge University Press, CambridgeGoogle 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
- 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
- 47.Craik A (1985) Wave interactions and fluid flows. Cambridge University Press, CambridgeGoogle Scholar
- 51.Stokes G (1847) On the theory of oscillatory waves. Trans Camb Philos Soc 8:441Google 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