# Space time evolution of sand bed topography and associated flow turbulence: experiments with statistical analysis

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## Abstract

A series of flume experiments were conducted with varying the flow discharges at the Fluvial Mechanics Laboratory of Indian Statistical Institute (Kolkata) to understand the co-evolution patterns of generating bed forms and the corresponding flow turbulence. Instantaneous bed elevations and velocity components were recorded continuously for sufficient time using high resolution instruments, such as, Ultrasonic Ranging System and acoustic Doppler velocimeter, at some spatial location over the deformed bed for each flow discharge. Increase in mean bed elevations and bed-slopes was found to be increased in discharge. Heavy-tailed nature of the probability density functions of magnitude of bed elevation increments, magnitude of single continuous bed elevation increments and instantaneous Reynolds shear stresses along three planes were analyzed using Pareto and truncated Pareto distributions. The spectral analysis of bed elevations revealed that the slope of log–log linearity increased with increase in flow discharge. Wavelet cross-correlations depicted strong dependence of bed form evolution on the corresponding instantaneous Reynolds shear stress along *xz*-plane. A Gram–Charlier type of distribution was used to estimate the probability density function of fluctuating velocity components, instantaneous Reynolds shear stresses along three planes and the joint probability density functions of the fluctuating velocity components, which showed good fit with the experimental data.

### Keywords

Turbulence Bed form evolution Pareto and truncated Pareto distributions Gram–Charlier distribution Acoustic Doppler velocimeter (ADV) 5-MHz Ultrasonic Ranging System (URS)## Notes

### Acknowledgements

Authors would like to acknowledge the Editor-in-Chief and two anonymous reviewers for their fruitful comments and suggestions for improvement of the paper. Authors also acknowledge Professor Fotis Sotiropoulos for providing some of their important papers on this topic during revision.

### References

- Aban IB, Meerschaert MM, Panorska AK (2006) Parameter estimation for the truncated Pareto distribution. J Am Stat Assoc 101(473):270–277CrossRefGoogle Scholar
- Ager DV (1973) The nature of the stratigraphic record. Wiley, New YorkGoogle Scholar
- Batchelor GK, Townsend AA (1949) The nature of turbulent motion at large wave-numbers. Proc R Soc Lond Ser A Math Phys Sci. doi: 10.1098/rspa.1949.0136 Google Scholar
- Bernardara P, Schertzer D, Sauquet E, Tchiguirinskaia I, Lang M (2008) The flood probability distribution tail: how heavy is it? Stoch Environ Res Risk Assess 22(1):107–122. doi: 10.1007/s00477-006-0101-2 CrossRefGoogle Scholar
- Best J (2005) The fluid dynamics of river dunes: a review and some future research directions. J Geophys Res 110:F04S02. doi: 10.1029/2004JF000218 CrossRefGoogle Scholar
- Bruno R, Sorriso-Valvo L, Carbone V, Bavassano B (2004) A possible truncated-Levy-flight statistics recovered from interplanetary solar-wind velocity and magnetic-field fluctuations. Europhys Lett 66(1):146–152CrossRefGoogle Scholar
- Chou YJ, Fringer OB (2010) A model for the simulation of coupled flow-bed form evolution in turbulent flows. J Geophys Res 115:C10041. doi: 10.1029/2010JC006103 CrossRefGoogle Scholar
- Clauset A, Shalizi CR, Newman MEJ (2009) Power law distributions in empirical data. SIAM Rev 51(4):661–703CrossRefGoogle Scholar
- Debnath L (2002) Wavelets and signal processing. Birkhauser, Berlin. ISBN 0-8176-4235-8Google Scholar
- Esfahani FS, Keshavarzi AR (2011) Effect of different meander curvatures on spatial variation of coherent turbulent flow structure inside ingoing multi-bend river meanders. J Stoch Environ Res Risk Assess 25(7):913–928. doi: 10.1007/s00477-011-0506-4 CrossRefGoogle Scholar
- Ganti V, Straub KM, Foufoula-Georgiou E, Paola C (2011) Space-time dynamics of depositional systems: Experimental evidence and theoretical modeling of heavy-tailed statistics. J Geophys Res 116:F02011. doi: 10.1029/2010JF001893 CrossRefGoogle Scholar
- Hardy RJ, Best JL, Lane SN, Carbonneau P (2009) Coherent flow structures in a depth-limited flow over a gravel surface: the role of near-bed turbulence and influence of Reynolds number. J Geophys Res 114:F01003CrossRefGoogle Scholar
- Hardy RJ, Best JL, Lane SN, Carbonneau P (2010) Coherent flow structures in a depth-limited flow over a gravel surface: the influence of surface roughness. J Geophys Res 115:F03006CrossRefGoogle Scholar
- Hersen PP (2005) Flow effects on the morphology and dynamics of aeolian and subaqueous barchans dunes. J Geophys Res 110:F04S07. doi: 10.1029/2004JF000185 CrossRefGoogle Scholar
- Hill B (1975) A simple general approach to inference about the tail of a distribution. Ann Stat 3:1163–1173CrossRefGoogle Scholar
- Keshavarzy AR, Ball JE (1995) Instantaneous shear stress on the bed in a turbulent open channel flow. In: Proceedings of XXVI IAHR Congress, LondonGoogle Scholar
- Khosronejad A, Sotiropoulos F (2014) Numerical simulation of sand waves in a turbulent open channel flow. J Fluid Mech 753:150–216CrossRefGoogle Scholar
- Khosronejad A, Kang S, Borazjani I, Sotiropoulos F (2011) Curvilinear immersed boundary method for simulating coupled flow and bed morphodynamic interactions due to sediment transport phenomena. Adv Water Resour 34(7):829–843CrossRefGoogle Scholar
- Khosronejad A, Kozarek JL, Palmsten ML, Sotiropoulos F (2015) Numerical simulation of large dunes in meandering streams and rivers with in-stream rock structures. Adv Water Resour 81:45–61CrossRefGoogle Scholar
- Kolmogorov AN (1941) The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl Akad Nauk SSSR 30(4):299Google Scholar
- Kolmogorov AN (1951) Solution of a problem in probability theory connected with the problem of the mechanism of stratification. Trans Am Math Soc 53:171–177Google Scholar
- Lau KM, Weng H (1995) Climate signal detection using wavelet transform: how to make a time series sing. Bull Am Meteorol Soc 76:2391–2402. doi: 10.1175/1520-0477(1995) 076<2391:CSDUWT>2.0.CO;2 CrossRefGoogle Scholar
- Lu SS, Willmarth WW (1973) Measurements of the structure of the Reynolds stress in a turbulent boundary layer. J Fluid Mech 60:481–512CrossRefGoogle Scholar
- Mackenzie D (2002) Wavelets: seeing the forest and the trees. National Academy of Sciences (U.S.), WashingtonGoogle Scholar
- Mazumder BS, Pal DK, Ghoshal K, Ojha SP (2009) Turbulence statistics of flow over isolated scalene and isosceles triangular-shaped bedforms. J Hydraul Res 47(5):626–637CrossRefGoogle Scholar
- Maity H, Mazumder BS (2014) Experimental investigation of the impacts of coherent flow structures upon turbulence properties in regions of crescentic scour. Earth Surf Process Landf 39(8):995–1013. doi: 10.1002/esp.3496 CrossRefGoogle Scholar
- McElroy B, Mohrig D (2009) Nature of deformation of sandy bed forms. J Geophys Res Earth Surf 114:2009. doi: 10.1029/2008JF001220 CrossRefGoogle Scholar
- Mianaei SJ, Keshavarzi AR (2008) Spatio-temporal variation of transition probability of bursting events over the ripples at the bed of open channel. Stoch Environ Res Risk Assess 22:257–264. doi: 10.1007/s00477-007-0114-5 CrossRefGoogle Scholar
- Mianaei SJ, Keshavarzi AR (2010) Study of near bed stochastic turbulence and sediment entrainment over the ripples at the bed of open channel using image processing technique. Stoch Environ Res Risk Assess 24(5):591–598. doi: 10.1007/s00477-009-0346-7 CrossRefGoogle Scholar
- Nakagawa H, Nezu I (1977) Prediction of the contributions to the Reynolds stress from bursting events in open-channel flows. J Fluid Mech 80(1):99–128CrossRefGoogle Scholar
- Nakagawa H, Nezu I (1981) Structure of space-time correlations of bursting phenomena in an open-channel flow. J Fluid Mech 104:1–43CrossRefGoogle Scholar
- Nelson PA, Smith JA, Miller AJ (2006) Evolution of channel morphology and hydrologic response in an urbanizing drainage basin. Earth Surf Proc Land. doi: 10.1002/esp.1308 Google Scholar
- Nikora V (2008) Hydrodynamics of gravelbed rivers: scale issues. In: Habersack H, Piegay H, Rinaldi M (eds) Gravel-bed rivers VI: from process understanding to river restoration. Elsevier, New York, p 6181Google Scholar
- Nikora VI, Goring DG (2000) Eddy convection velocity and Taylor’s hypothesis of ’frozen’ turbulence in a rough-bed open-channel flow. J Hydrosci Hydraul Eng 18(2):75–91Google Scholar
- Niu J, Sivakumar B (2013) Scale-dependent synthetic streamflow generation using a continuous wavelet transform. J Hydrol 496:71–78CrossRefGoogle Scholar
- Ojha SP, Mazumder BS (2008) Turbulence characteristics of flow region over a series of 2-D dune shaped structures. Adv Water Resour 31:561–576CrossRefGoogle Scholar
- Paola C (2000) Quantitative models of sedimentary basin filling. Sedimentology 47(Suppl 1):121–178. doi: 10.1046/j.1365-3091.2000.00006.x CrossRefGoogle Scholar
- Roy S, Debnath K, Mazumder BS (2017) Distribution of eddy scales for wave current combined flow. Appl Ocean Res 63:170–183CrossRefGoogle Scholar
- Sarkar K, Chakraborty C, Mazumder BS (2016) Variations of bed elevations due to turbulence around submerged cylinders in sand bed. Environ Fluid Mech 16(3):659–693. doi: 10.1007/s10652-016-9449-0 CrossRefGoogle Scholar
- Sengupta S (1966) Studies on orientation and imbrication of pebbles with respect to cross-stratification. J Sediment Petrol 36(2):362–369Google Scholar
- Sengupta S (2007) Introduction to sedimentology. CBS Publications and Distributors, New DelhiGoogle Scholar
- Schindler RJ, Robert A (2004) Suspended sediment concentration and the ripple-dune transition. Hydrol Process. doi: 10.1002/hyp.1505 Google Scholar
- Singh A, Lanzoni S, Foufoula-Georgiou E (2009) Nonlinearity and complexity in gravel bed dynamics. Stoch Environ Res Risk Assess 23(7):967–975. doi: 10.1007/s00477-008-0269-8 CrossRefGoogle Scholar
- Singh A, Porte-Agel F, Foufoula-Georgiou E (2010) On the influence of gravel bed dynamics on velocity power spectra. Water Resour Res 46:1–10Google Scholar
- Singh A, Lanzoni S, Wilcock PR, Foufoula-Georgiou E (2011) Multiscale statistical characterization of migrating bed forms in gravel and sand bed rivers. Water Resour Res 47:W12526. doi: 10.1029/2010WR010122 CrossRefGoogle Scholar
- Singh A, Foufoula-Georgiou E, Porte-Agel F, Wilcock PR (2012) Coupled dynamics of the co-evolution of gravel bed topography, flow turbulence and sediment transport in an experimental channel. J Geophys Res 117:F04016. doi: 10.1029/2011JF002323 Google Scholar
- Sotiropoulos F, Khosronejad A (2016) Sand waves in environmental flows: insights gained by coupling large-eddy simulation with morphodynamics. Phys Fluids 28:021301CrossRefGoogle Scholar
- Venditti JG, Church MA, Benneth SJ (2005a) Bed form initiation from a flat sand bed. J Geophys Res 110:F01009. doi: 10.1029/2004JF000149 CrossRefGoogle Scholar
- Venditti JG, Church MA, Bennett SJ (2005b) Morphodynamics of small-scale superimposed sand waves over migrating dune bed forms. Water Resour Res 41:W10423. doi: 10.1029/2004WR003461 CrossRefGoogle Scholar
- Wilbers AWE, Ten Brinke WBM (2003) The response of subaqueous dunes to floods in sand and gravel bed reaches of the Dutch Rhine. Sedimentology 50:1013–1034CrossRefGoogle Scholar
- Wu Y, Christensen KT (2006) Reynolds-stress enhancement associated with a short fetch of roughness in wall turbulence. AIAA J 44(12):3098–3106CrossRefGoogle Scholar
- Zhang Y, Benson DA, Baeumer B (2007) Predicting the tails of breakthrough curves in regional-scale alluvial systems. Ground Water 45(4):473–484CrossRefGoogle Scholar