Morphology of a stream flowing down an inclined plane. Part 2. Meandering

Abstract

A stream of fluid flowing down a partially wetting inclined plane usually meanders, unless the volume flow rate is maintained at a highly constant value. Here we investigate whether the meandering of this stream is an inherent instability. In our experiment, we eliminate meandering on several partially wetting substrates by reducing perturbations entering the flow. By re-introducing controlled fluctuations, we show that they are responsible for the onset of the meandering. We derive a theoretical model for the stream shape, %from first principles which includes stream dynamics and forcing by external noise. The deviation h(x) from a straight linear stream h(x)=0 shows considerable variability as a function of downstream distance x. However, for an ensemble average of stream shapes acquired at different times, the power spectrum S(k) as a function of wavenumber k has a power-law scaling S(k) ~ k5/2. Moreover, the area A(x) swept by the stream at the distance x grows as A(x) ~ x1.75.

Meandering of a particle-laden rivulet

“…Moreover, our theory predicts how the value of Hack's law exponent is related to the scaling of the meandering power spectrum. If that scaling is -5/2, then Hack's law exponent should be 4/7 [8]. Surprisingly, 4/7 approximately equals 0.57, the same value we see in laboratory rivulets and in real rivers.…”

“…We averaged out velocity variations in the stream cross-section. We replaced the actual (variable) value of the contact angle between the fluid, the air, and the surface, with another average [8]. Finally, we radically simplified one very important feature of the stream.…”

“…As the flow rate changes in an alluvial system, scour or deposition will occur, further affecting meandering patterns [11]. Sediment transport was believed to be necessary for flow meanders [12], however, experiments without sediment transport also manifest meandering [8,10,13]. While the notion of meandering as an inherent global flow instability that arose in the 1980s [13] has been experimentally disproven [6,7], transverse shear stress distribution along the wetted perimeter of the meandering stream [14] likely still plays a role in meandering.…”

Numerical Modelling of Braiding and Meandering Instabilities in Gravity‐Driven Liquid Rivulets

“…To apply the Richardson extrapolation, the results need to fall within the asymptotic range of convergence. This condition is expressed as GCI 23 r p GCI 12 1 (8) Fig. 2 shows the results obtained for the grid convergence study of the interface area for both cases tested (Case 1 and Case 3).…”

“…Other authors point out the substrate microstructure and contact angle hysteresis as the main source , . Birnir et al stated that meandering happens because of upstream disturbances in the liquid flow rate.…”