<p>The aim of the study is to carry out two-phase gravity driven flows of large light particles on submerged steep slopes and, from their characteristics, identify the criteria that govern the transition between a purely dense and a mixed (dense-suspended) regime. For that, volumes of large light particles are released without any initial velocity at the top of a 2D flume immersed in a 20&#xa0;m<sup>3</sup> tank filled with tap water. The particles are spherical, monodisperse, with two different diameters: 10.6 and 14.4 mm. The volume expansion <i>β</i>, the height <i>H</i> and the front velocity <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10652_2025_10049_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(U_f\)</EquationSource> </InlineEquation> of the heavy flows are investigated varying the volume released between 0.2 and 3&#xa0;L, and the tilt angle of the flume (θ) between 30 and 60°. Two different regimes are observed, one where all the particles move in close contact with each other (dense regime), and as the volume and/or flume angle increases, another where part of the particles are suspended (mixed regime). We find that the overall dynamics of the flow is governed by a buoyancy/drag equilibrium ruled by the densimetric Froude number <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10652_2025_10049_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="176" /> </InlineMediaObject> <EquationSource Format="TEX">\(Fr=U_f/\sqrt{(\Delta \rho g/\rho _w) H}\)</EquationSource> </InlineEquation>, with <i>g</i> the gravity acceleration and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10652_2025_10049_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Delta \rho =\rho -\rho _w\)</EquationSource> </InlineEquation>. <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10652_2025_10049_Article_IEq4.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\rho \)</EquationSource> </InlineEquation> is the density of the flow and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10652_2025_10049_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">\(\rho _w\)</EquationSource> </InlineEquation> that of the ambient fluid: water. The corresponding drag coefficient exerted on the particles volume is found to vary as <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10652_2025_10049_Article_IEq6.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">\(C_d\approx 2{Fr^{-1.6}}\)</EquationSource> </InlineEquation>. The key parameter for the onset of the flow of some of the particles in suspension and the transition to the mixed regime proves to be <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10652_2025_10049_Article_IEq7.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="200" /> </InlineMediaObject> <EquationSource Format="TEX">\(St_\theta =St\cos \theta /(1-St\sin \theta )\)</EquationSource> </InlineEquation>, with <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10652_2025_10049_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(St=v_s/U_f\)</EquationSource> </InlineEquation> the Stokes number comparing the settling velocity of the particles to the flow front velocity. While pressure remains hydrostatic within the flow in the dense regime (<InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10652_2025_10049_Article_IEq9.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">\(St_\theta &gt;0.9\)</EquationSource> </InlineEquation>), it increases as suspension occurs in the mixed case and <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10652_2025_10049_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">\(St_\theta \)</EquationSource> </InlineEquation> decreases. The dynamic pressure at the forehead of the volume then evolves as <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10652_2025_10049_Article_IEq11.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="215" /> </InlineMediaObject> <EquationSource Format="TEX">\( P_{s} \approx \Delta \rho gH\cos (\theta )(1.9 - St_{\theta } )^{2} \)</EquationSource> </InlineEquation>.</p>

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Flow dynamics of volumes of large light particles released on submerged steep slopes

  • Marie Rastello,
  • Jean-Louis Marié,
  • Brivaël Collin,
  • Hervé Bellot,
  • Florence Naaim

摘要

The aim of the study is to carry out two-phase gravity driven flows of large light particles on submerged steep slopes and, from their characteristics, identify the criteria that govern the transition between a purely dense and a mixed (dense-suspended) regime. For that, volumes of large light particles are released without any initial velocity at the top of a 2D flume immersed in a 20 m3 tank filled with tap water. The particles are spherical, monodisperse, with two different diameters: 10.6 and 14.4 mm. The volume expansion β, the height H and the front velocity \(U_f\) of the heavy flows are investigated varying the volume released between 0.2 and 3 L, and the tilt angle of the flume (θ) between 30 and 60°. Two different regimes are observed, one where all the particles move in close contact with each other (dense regime), and as the volume and/or flume angle increases, another where part of the particles are suspended (mixed regime). We find that the overall dynamics of the flow is governed by a buoyancy/drag equilibrium ruled by the densimetric Froude number \(Fr=U_f/\sqrt{(\Delta \rho g/\rho _w) H}\) , with g the gravity acceleration and \(\Delta \rho =\rho -\rho _w\) . \(\rho \) is the density of the flow and \(\rho _w\) that of the ambient fluid: water. The corresponding drag coefficient exerted on the particles volume is found to vary as \(C_d\approx 2{Fr^{-1.6}}\) . The key parameter for the onset of the flow of some of the particles in suspension and the transition to the mixed regime proves to be \(St_\theta =St\cos \theta /(1-St\sin \theta )\) , with \(St=v_s/U_f\) the Stokes number comparing the settling velocity of the particles to the flow front velocity. While pressure remains hydrostatic within the flow in the dense regime ( \(St_\theta >0.9\) ), it increases as suspension occurs in the mixed case and \(St_\theta \) decreases. The dynamic pressure at the forehead of the volume then evolves as \( P_{s} \approx \Delta \rho gH\cos (\theta )(1.9 - St_{\theta } )^{2} \) .