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.
Similar content being viewed by others
References
Raupach M, Antonia R, Rajagopalan S (1991) Rough-wall turbulent boundary layers. Appl Mech Rev 44:1
Jimenez J (2004) Turbulent flow over rough wall. Annu Rev Fluid Mech 36:173
Castro I (2007) Rough-wall boundary layers: mean flow universality. J Fluid Mech 585:469
Grass A (1971) Structural features of turbulent flow over smooth and rough boundaries. J Fluid Mech 50:233
Ghisalberti M (2009) Obstructed shear flows: similarities across systems and scales. J Fluid Mech 641:51
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):809
Pan Y, Chamecki M (2016) A scaling law for the shear-production range of second-order structure functions. J Fluid Mech 801:459
Marusic I, Mathis R, Hutchins N (2010) Predictive model for wall-bounded turbulent flow. Science 329:193
Mathis R, Hutchins N, Marusic I (2009) Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J Fluid Mech 628:311
Anderson W (2016) Conditionally averaged large-scale motions in the neutral atmospheric boundary layer: insights for Aeolian processes. J Fluid Mech 789:567
Townsend A (1976) The structure of turbulent shear flow. Cambridge University Press, Cambridge
Volino R, Schultz M, Flack K (2007) Turbulence structure in rough- and smooth-wall boundary layers. J Fluid Mech 592:263
Alfredsson P, Örlu R (2010) The diagnostic plot-a litmus test for wall-bounded turbulence data. Eur J Mech B Fluids 29:403
Wu Y, Christensen KT (2010) Spatial structure of a turbulent boundary layer with irregular surface roughness. J Fluid Mech 655:380
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 23:702
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 712:92
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:W02505
Yang X (2016) On the mean flow behaviour in the presence of regional-scale surface roughness heterogeneity. Bound-Layer Met 161:127
Macdonald R, Griffiths R, Hall D (1998) An improved method for the estimation of surface roughness of obstacle arrays. Atmospheric Environment 32(11):1857
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, Australia
Andreopoulos J, Wood D (1982) The response of a turbulent boundary layer to a short length of surface roughness. J Fluid Mech 118:143
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 48:721
Garratt J (1990) The internal boundary layer: a review. Bound-Layer Meteorol 40:171
Wang ZQ, Cheng NS (2005) Secondary flows over artificial bed strips. Adv. Water Resour 28:441
Mejia-Alvarez R, Christensen K (2013) Wall-parallel stereo PIV measurements in the roughness sublayer of turbulent flow overlying highly-irregular roughness. Phys Fluids 25:115109
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:W02530
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 26:025111
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 41:90
Barros J, Christensen K (2014) Observations of turbulent secondary flows in a rough-wall boundary layer. J Fluid Mech 748:1
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
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 768:316
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
Medjnoun T, Vanderwel C, Ganapathisubramani B (2018) Characteristics of turbulent boundary layers over smooth surfaces with spanwise heterogeneities. J Fluid Mech 838:516
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 3:044602
Bou-Zeid E, Parlange M, Meneveau C (2007) On the parameterization of surface roughness at regional scales. J Atmos Sci 64:216
Nezu I, Nakagawa H (1993) Turbulence in open-channel flows. Balkema Publishers, Rotterdam
Prandtl L (1952) Essentials of fluid dynamics. Blackie and Son, London
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 Tech
Brundrett E, Baines WD (1964) The production and diffusion of vorticity in duct flow. J Fluid Mech 19:375
Perkins H (1970) The formation of streamwise vorticity in turbulent flow. J Fluid Mech 44:721
Gessner F (1973) The origin of secondary flow in turbulent flow along a corner. J Fluid Mech 58:1
Bradshaw P (1987) Turbulent secondary flows. Ann Rev Fluid Mech 19:53
Madabhushi R, Vanka S (1991) Large eddy simulation of turbulence-driven secondary flow in a square duct. Phys Fluids A 3:2734
Leibovich S (1977) Convective instability of stably stratified water in the ocean. J Fluid Mech 82:561
Leibovich S (1980) On wave-current interaction theories of Langmuir circulations. J Fluid Mech 99:715724
Leibovich S (1983) The form and dynamics of Langmuir circulation. Annu Rev Fluid Mech 15:391
Craik A (1985) Wave interactions and fluid flows. Cambridge University Press, Cambridge
McWilliams J, Sullivan P, Moeng CH (1997) Langmuir turbulence in the ocean. J Fluid Mech 334:1
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 120:4729
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
Stokes G (1847) On the theory of oscillatory waves. Trans Camb Philos Soc 8:441
Mansfield J, Knio O, Meneveau C (1998) A dynamic LES scheme for the vorticity transport equation: formulation and a priori tests. J Comp Phys 145:693
Mansfield J, Knio O, Meneveau C (1999) Dynamic LES of colliding vortex rings using a 3D vortex method. J Comp Phys 152:305
Gayme D, McKeon B, Papachristodoulou A, Bamieh B, Doyle J (2010) A streamwise constant model of turbulence in plane Couette flow. J Fluid Mech 665:99
Pope S (2000) Turbulent flows. Cambridge University Press, Cambridge
Reynolds R, Hayden P, Castro I, Robins A (2007) Spanwise variations in nominally two-dimensional rough-wall boundary layers. Exp Fluids 42:311
Fishpool G, Lardeau S, Leschziner M (2009) Persistent Non-Homogeneous Features in Periodic Channel-Flow Simulations. Flow Turbulence Combust 83:823
Arbogast L (1800) Du calcul des dérivation. Levrault, Strasbourg
Goursat E (1902) Cours d‘analyse mathématique. Gauthier-Villars, Paris, p 1
Albertson J, Parlange M (1999) Surface length scales and shear stress: implications for land-atmosphere interaction over complex terrain. Water Resour Res 35:2121
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 137:397
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:R1
Bou-Zeid E, Meneveau C, Parlange M (2005) A scale-dependent Lagrangian dynamic model for large eddy simulation of complex turbulent flows. Phys Fluids 17:025105
Orszag S (1970) Transform method for calculation of vector coupled sums: application to the spectral form of the vorticity equation. J Atmos Sci 27:890
Deardorff J (1970) A numerical study of 3 dimensional turbulent channel flow at large Reynolds numbers. J Fluid Mech 41:453
Piomelli U, Balaras E (2002) Wall-layer models for large-eddy simulation. Annu Rev Fluid Mech 34:349
Adrian R, Christensen K, Liu ZC (2000) Vortex organization in the outer region of the turbulent boundary layer. Exp Fluids 29:275
Anderson W (2012) An immersed boundary method wall model for high-Reynolds number channel flow over complex topography. Int J Numer Methods Fluids 71:1588
Flack K, Schultz M (2010) Review of hydraulic roughness scales in the fully rough regime. J Fluids Eng 132(4):041203
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).
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix A: Datapoints from Yang and Anderson (2017)
Figure 1 showed datapoints of compensated circulation from Yang and Anderson [32]. For that study, LES with an IBM was used to model flow over a series of topographies composed of streamwise-aligned, vertically-truncated pyramid obstacles. For the simulations, we used the LES code already described in Sect. 6 (while the IBM has been outlined in many previous articles [68]). Figure 9a shows a sample arrangement, while Fig. 9b is a close-up sketch of the elements. Two sets of simulations were considered: (a) Set 1 featured elements with \(H/h = 15\); (b) Set 2 featured elements with \(H/h = 20\).
For Set 1, we considered \(s_2/H = \{ 0.1, 0.2, 0.32, 0.46, 0.53, 0.64, 0.8, 1.0, \frac{1}{2} \pi , 2 \pi \}\), while for Set 2 we considered \(s_2/H = \{ 0.32, 0.46, 0.64, 1.0, \pi , 2 \pi \}\). Figure 9b shows detailed attributes of the elements; for all cases, \(w_b/H = 0.0756\), \(l_b/H = 0.0756\), \(w_t/H = 0.025\), \(l_t/H = 0.025\), and \(s_x/H = 0.0756\). For Set 1 and Set 2, \(\theta = 69.30\) and 58.52, respectively, though we stress that \(\theta \) is not expected to have a material affect on conclusions of this study, given the fully rough flow conditions [2, 69].
Appendix B: Faá di Bruno Formula
The univariate Faá di Bruno formula:
where the summation is over all possible solutions with non-negative integer input arguments to the relations, \(b_1 + 2 b_2 + \ldots + n b_n = n\) and \(k = \displaystyle \sum\nolimits _{i=1}^{n} b_i\). With this, the first-order derivatives in \(x_2\) and \(x_3\) are:
and
where the summation is made over a single partition: (1) \(b_1 = 1\), and \(k = 1\). The second-order derivatives in \(x_2\) and \(x_3\) are:
and
where the summation is made over two partitions: (1) \(b_1 = 2\), \(b_2 = 0\), and \(k = 2\); and (2) \(b_1 = 0\), \(b_2 = 1\), and \(k = 1\). The third-order derivatives in \(x_2\) and \(x_3\) are:
and
where the summation is made over three partitions: (1) \(b_1 = 3\), \(b_2 = 0\), \(b_3 = 0\), and \(k = 3\); (2) \(b_1 = 0\), \(b_2 = 0\), \(b_3 = 1\), and \(k = 1\); and (3) \(b_1 = 1\), \(b_2 = 1\), \(b_3 = 0\), and \(k = 2\). The fourth-order derivatives in \(x_2\) and \(x_3\) are:
and
where the summation is made over four partitions: (1) \(b_1 = 4\), \(b_2 = 0\), \(b_3 = 0\), \(b_4 = 0\), and \(k = 4\); (2) \(b_1 = 0\), \(b_2 = 0\), \(b_3 = 0\), \(b_4 = 1\), and \(k = 1\); (3) \(b_1 = 1\), \(b_2 = 0\), \(b_3 = 1\), \(b_4 = 0\), and \(k = 2\); and (4) \(b_1 = 2\), \(b_2 = 1\), \(b_3 = 0\), \(b_4 = 0\), and \(k = 3\). Finally, a multivariate version of Eq. 49 is needed for terms in Eq. 25. The multivariate form of Eq. 49 yields,
and
where it has been presumed throughout that the partial differential operators commute. Note that the efficacy of Eqs. 58 and 59 can be quickly established by setting index 3 to 2 or setting index 2 to 3 in the former and latter, respectively, and comparing against Eqs. 54 and 55. In addition, the denominators of prefactors A to F (Eqs. 30 to 35) make use of the spatial derivatives of the Eq. 3 Stokes drift function, which are defined here for completeness:
and,
Rights and permissions
About this article
Cite this article
Anderson, W., Yang, J., Shrestha, K. et al. Turbulent secondary flows in wall turbulence: vortex forcing, scaling arguments, and similarity solution. Environ Fluid Mech 18, 1351–1378 (2018). https://doi.org/10.1007/s10652-018-9596-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10652-018-9596-6