Interaction of viscous free-surface flows with topography

The interaction of gravitationally driven, free-surface flows of viscous fluid with topographic features is investigated theoretically. The motion is studied in the regime where the depth of the flow is much smaller than the streamwise extent of the topography. A lubrication model of the motion is developed, integrated numerically and analysed asymptotically. For small mounds, it is shown that the flow surmounts the obstacles, but for larger mounds the flow is deflected around it and can form dry zones in its wake into which fluid does not flow, as well as forming deeper ponded regions upstream. Which of these phenomena prevails is shown to depend upon the amplitude of the mound height and the thickness of the oncoming flow relative to the streamwise length scale over which the topography varies. By using numerical and asymptotic results, we demonstrate that relatively wide mounds lead to the development of deep ponds of material upstream, which may lead to flow overtopping if the mound is not sufficiently high. These insights can be used to inform the design of barriers that defend built infrastructures from lava flows, and it is shown how this model can also provide an upper bound on the force exerted by the flow on them.

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A fully coupled solver for two- and three-dimensional incompressible and free surface flows in viscous fluid

Fifteenth International Conference on Numerical Methods in Fluid Dynamics - Lecture Notes in Physics ◽

10.1007/bfb0107100 ◽

2007 ◽

pp. 189-194

Author(s):

L. Gentaz ◽

B. Alessandrini ◽

G. Delhommeau

Keyword(s):

Free Surface ◽

Viscous Fluid ◽

Three Dimensional ◽

Free Surface Flows ◽

Surface Flows ◽

Fully Coupled

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Viscous banding instabilities: non-porous viscous fingering

Journal of Fluid Mechanics ◽

10.1017/jfm.2021.660 ◽

2021 ◽

Vol 926 ◽

Author(s):

Katarzyna N. Kowal

Keyword(s):

Free Surface ◽

Viscous Fluid ◽

Pressure Gradient ◽

Dynamic Pressure ◽

Viscous Fingering ◽

Free Surface Flows ◽

Surface Flows ◽

Viscosity Contrast ◽

Slope Discontinuity ◽

Rigid Walls

We demonstrate a novel instability found within unconfined viscous bands/rims, or free-surface flows involving a longitudinal viscosity contrast. Such instabilities may be described as viscous banding instabilities, non-porous viscous fingering instabilities or unconfined viscous fingering instabilities of free-surface flows involving the intrusion of a less viscous fluid into a band of more viscous fluid. A consequence of this work is that viscous fingering instabilities, widely known to occur in porous media following the seminal work of Saffman & Taylor (Proc. R. Soc. Lond. A, vol. 245, 1958, pp. 312–329), also occur in non-porous environments. Although the mechanism of the viscous banding instability is characteristically different from that of the Saffman–Taylor instability, there are important similarities between the two. The main similarity is that a viscosity contrast leads to instability. A distinguishing feature is that confinement, such as the rigid walls of a Hele-Shaw cell, is not necessary for viscous banding instabilities to occur. More precisely, Saffman–Taylor instabilities are driven by a jump in dynamic pressure gradient, whereas viscous banding instabilities, or non-porous viscous fingering instabilities, are driven by a jump in hydrostatic pressure gradient, directly related to a slope discontinuity across the intrusion front. We examine the onset of instability within viscous bands down an inclined plane, determine conditions under which viscous banding instabilities occur and map out a range of behaviours in parameter space in terms of two dimensionless parameters: the viscosity ratio and the volume of fluid ahead of the intrusion front.

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