(199f) Dynamical Scaling of Jet-Induced Crater Formation in a Granular Bed

BACKGROUND

A jet of gas impinging normally on the free surface of a granular material creates an axisymmetric crater in the surface.  Selected aspects of the cratering have been studied previously [Rajaratnam and Beltaos 1977, and Rajaratnam 1982].  It was found in those studies that for typical subsonic jet parameters the crater depth and width grow linearly with the logarithm of time for the first 100 to 1000 seconds.  Eventually the crater approaches some asymptotic size limits and so the growth departs from the logarithmic trend.  The major focus of those previous studies was to scale the asymptotic (final) crater dimensions as a function of the physical parameters.  In particular, the asymptotic sizes were related to the densimetric Froude number Fr and the Erosion Parameter E_c, which are defined by

where ρ_g and ρ_s are the densities of the gas and sand, respectively, v is the jet velocity, d is the particle diameter, g is the gravitational acceleration, D is the jet diameter at the nozzle's exit plane, and H is the height of the exit plane above the original surface of the sand.  However, these studies did not determine the crater's dynamical scaling?the scaling of its growth rate as a function of time prior to the asymptotic state.  It is important to understand the dynamical behaviors of the crater in order to predict and control processes that involve jets whose characteristics are constantly changing (as when a rocket lands) or that occur over shorter periods of time.

EXPERIMENTAL OBSERVATIONS

At the
Kennedy Space Center's Launch Effects Test Facility (LETF), we have performed cratering tests in sand using subsonic gas jets from a straight, circular pipe.  Various gases were used (Helium, Neon, Nitrogen, Argon, and Carbon Dioxide) with carefully controlled mass flow rates.  The cratering processes were observed by splitting the sandbox and jet in half with a transparent, sharp-edged, vertical wall, allowing us to see into the subsurface during the cratering event.   The processes are surprisingly complex as illustrated in Figure 1.

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UNIVERSAL FUNCTIONAL DEPENDENCE

Considering the complex phenomenology, the simplicity of a logarithmic growth rate as shown in Figure 2 is surprising. 

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A logarithm requires three fitting parameters, a to scale length, b^-1 to scale time, and t₀ to determine the valid region of the fit (since log(0) =  ? ∞ is outside that region):

Fitting a, b, and t₀ to our experimental results, we find that b∙t₀ = 1 for all cases so that we have

This function implies that the cratering excavation process is described by the differential equation

which can be related to the mass flow rate out of the crater and the physics of the erosion process.  Apparently the physics of crater growth are dominated by a few very simple features that blur over the messy complexities.  A key to understanding crater growth, then, is to identify these dominant, organizing features.

DYNAMICAL SCALING RELATIONSHIPS

Our on-going research is aimed at determining the dependence of a and b upon the physical parameters of the experiment.  For example, we have found that varying gas density and velocity do not affect the value of a as shown in Figure 3.  Furthermore, we have found that b varies linearly with (ρ_gv²), the kinetic energy density of the jet, as shown in Figure 4.  These and further results, including the width, volume and aspect ratios of the crater versus time, will be discussed along with their physical interpretation relating the simple scaling to the complex phenomenology.

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