The dam-break problem for viscous fluids in the high-capillary-number limit

Abstract

Experiments were undertaken to investigate dam-break flows where a finite volume of highly viscous fluid (glucose with viscosity μ ≈ 350 Pa s) maintained behind a lock gate was released into a horizontal or inclined flume. The resulting sequence of flow-depth profiles was tracked using a three-dimensional visualization system. In the low-Reynolds-number and high-capillary-number limits, analytical solutions can be obtained from the Navier-Stokes equations using lubrication theory and matched asymptotic expansions. At shallow slopes, similarity solutions can also be worked out. While the variation in the front position scaled with time as predicted by theory for both horizontal and sloping flumes, there was a systematic delay in the front position observed. Moreover, taking a closer look at the experimental flowdepth profiles shows that they were similar, but they noticeably deviated from the theoretical similarity form for horizontal planes. For sloping beds, the flow-depth profile is correctly predicted provided that different scalings are used at shallow and large slopes.

Granular suspension avalanches. II. Plastic regime

“…For the sake of comparison, the data were nondimensionalized and plotted on a log-log diagram. At short times (t < 0.1), the front position closely followed the theoretical solutionx f = (9t/4) 1/3 representing the evolution of the front position for a homogeneous Newtonian fluid 13,15 (macro-viscous regime). At later times, there was a sudden transition to another regime, which was reflected by a kink in thê x f (t) curve.…”

“…In particular, the front position varied with time as t 1/3 , a scaling consistent with the theory of thin elongating Newtonian flows down a sloping bed. 13 For φ ≥ 0.61, the flow came rapidly to rest. In the 0.56-0.61 range, we observed three distinct regimes: at early times, the suspension moved downstream like a viscous avalanche in agreement with observations made, for instance, by Bonnoit et al 14 For this reason, we refer to this regime as the macro-viscous regime, studied in detail in Paper I.…”

Competition between kinematic and dynamic waves in floods on steep slopes

“…Subsequently, reliable estimates of Φ in (2.17) can be derived from the characteristic time for the present scenario: substituting the values {Φ = 0.3769,Ã = 57.31 m 2 , f = 0.0223, θ = 1 • } into (2.15), (2.17) and (2.18), and taking into account (2.5), yields t c ≈ 45.81; however, as shown in figure 2(c), we have to wait until t 45 for the deviation between the numerical and asymptotic solution (2.16) to drop to zero; in order to establish a criteria for the election of Φ, we impose that the relative deviation between the slope of the asymptotic and numerical velocity profiles at t c should be lower than 4.5 %; in the present numerical simulation this deviation occurs at t c ≈ 4000, and by setting Φ ≈ 4.25 × 10 −3 (instead of Φ = 0.3769) one obtains approximately such value of t c . So it is found that the numerical solution converges rather slowly towards the outer solution (as for the dam-break flow for viscous fluids in the high-capillary-number limit for non-zero bed slopes; see Ancey et al 2009). However, the main feature of real flood waves on steep inclines in relation to the predictions of the asymptotic solutions is the appearance of roll waves.…”

“…To be consistent with volume conservation (see Ancey, Cochard & Andreini 2009), we select the aspect-ratio number as Higher-order terms of the outer expansion (2.8)-(2.9) can be integrated recursively along the family of characteristic curvesx/t = constant (see Appendix A), with V j and H j (1 6 j ) vanishing ast → ∞, whilst V 0 and H 0 remain constant. Furthermore, the leading-order terms in (2.8) and (2.9) that satisfy the kinematic-wave equations (2.10) and (2.11) are exact solutions to the full shallow-water equations (2.6) and (2.7) when f = 2 tan θ.…”

Granular suspension avalanches. I. Macro-viscous behavior

“…Note that these profiles were obtained by stitching images from different cameras, which explains why the point density was not uniform along the flume; stitching gave rise to profiles that were locally smoother than in reality whenever the overlap region was not sufficiently long. For the theoretical flow depth, we used the composite solution given by Ancey, Cochard, and Andreini 28 (Eq. (A9) is merely the outer solution).…”