Slumping regime in lock-release turbidity currents

Bottom slope and particle settling

C. Gadal, M. Mercier and L. Lacaze

Institut de Mécanique des Fluides de Toulouse (IMFT), France

Turbidity currents

  • gravity driven flow
  • excess density = suspended particles (maybe combined to temperature, salinity or humidity differences)

Snow avalanche, Zinal, Switzerland @Zacharie Grossen

Pyroclastic flow, Sinabung, Indonesia @Jean-Guillaume Feignon

Dust storm, Phoenix, USA @Alan Stark

Lock-release systems

Particles: glass beads (\(d \sim 120 \mu\)m). Ambient: fresh water. Inclination: \(\theta = 7^\circ\)

Front position

Rottman et al. 1983, saline water, no slope (\(\theta=0^\circ\))

  • Scales:
    • length: \(L_{0}\) (lock length)
    • velocity: \(u_{0} = \sqrt{h_{0} g'}\), \(g' = \frac{\delta\rho}{\rho}g\)
    • time: \(t_{0} = L_{0}/u_{0}\)
  • \(u_{\rm c} = \mathcal{F}_{r} u_{0}\), \(t_{\rm end} = \tau t_{0}\)
  • constant prefactors: \(\mathcal{F}_{r} = 0.5\), \(\tau \simeq 20\)
    • (Boussinesq, full depth-release)
  • What happens for an inclined tank ?
  • Influence of particles ?

\[ \mathcal{F}_{r} = f(?), \, \tau = f(?)\]

Experimental set-up and parameter range (170 runs)

Parameter space:

  • volume fraction, \(\phi \in [0.5, 15]~\%\)
    • excess density, \(\delta\rho \in [2, 500]~\textrm{kg}~\textrm{m}^{-3}\)
    • velocity scale, \(u_{0} \in [5, 10^{2}]~\textrm{cm}~\textrm{s}^{-1}\)
  • bottom slope, \(\theta \in [0, 7]^\circ\)
  • silica particles, \(d \in [60, 250]~\mu\)m
    • settling velocity, \(v_{\rm s} \in [0.3, 3]~\textrm{cm}~\textrm{s}^{-1}\)
    • saline currents (same \(\delta\rho\))

Dimensionless control parameters:

  • Reynolds, \(\mathcal{R}_{e} = u_{0}h_{0}/\nu \in [10^{4}, 10^{5}]\)
  • Settling, \(\mathcal{S} = v_{\rm s}/u_{0} \in [4.10^{-3}, 10^{-1}]\)
  • Froude, \(\mathcal{F}_{r}^{*} = u_{0}/\sqrt{g'h_{0}} \equiv 1\)
  • \(\theta \in [0, 7]^\circ\)
  • \(\phi\)
  • \(\delta\rho/\rho\)

Experimental set-up and parameter range (170 runs)

Parameter space:

  • volume fraction, \(\phi \in [0.5, 15]~\%\)
    • excess density, \(\delta\rho \in [2, 500]~\textrm{kg}~\textrm{m}^{-3}\)
    • velocity scale, \(u_{0} \in [5, 10^{2}]~\textrm{cm}~\textrm{s}^{-1}\)
  • bottom slope, \(\theta \in [0, 7]^\circ\)
  • silica particles, \(d \in [60, 250]~\mu\)m
    • settling velocity, \(v_{\rm s} \in [0.3, 3]~\textrm{cm}~\textrm{s}^{-1}\)
    • saline currents (same \(\delta\rho\))

Dimensionless control parameters:

  • Reynolds, \(\mathcal{R}_{e} = u_{0}h_{0}/\nu \in [2.10^{4}, 4.10^{5}]\)
  • Settling, \(\mathcal{S} = v_{\rm s}/u_{0} \in [4.10^{-3}, 10^{-1}]\)
  • Froude, \(\require{cancel} \xcancel{\mathcal{F}_{r}^{*} = u_{0}/\sqrt{g'h_{0}} \equiv 1}\)
  • \(\theta \in [0, 7]^\circ\)
  • \(\require{cancel} \xcancel{\phi}\) \(\rightarrow\) \(\phi < \phi_{\rm c} \simeq 0.45~\%\), no interparticle interactions
  • \(\require{cancel} \xcancel{\delta\rho/\rho}\) \(\rightarrow\) Boussinesq approx.

Front dynamics

Dimensionless front velocity \(\mathcal{F}_{r}\)

\(\mathcal{F}_{r} = f(\mathcal{R}_{e}, \mathcal{S}, \theta)\)

\(\require{cancel} \mathcal{F}_{r} = f(\xcancel{\mathcal{R}_{e}}, \mathcal{S}, \theta)\)

\(\require{cancel} \mathcal{F}_{r} = f(\xcancel{\mathcal{R}_{e}}, \xcancel{\mathcal{S}}, \theta)\)

\(\require{cancel} \mathcal{F}_{r} = f(\xcancel{\mathcal{R}_{e}}, \xcancel{\mathcal{S}}, \color{orange}{\theta})\)

Dimensionless front velocity \(\mathcal{F}_{r}(\theta)\)

\(\bullet\) Global increasing trend, but scatter across datasets \(\rightarrow\) influence of other parameters ? \(h_{0}/L_{0}\) ?

Dimensionless duration \(\tau\) and influence of \(\mathcal{S}\)

\(\bullet\) \(\theta=7^\circ\), \(\mathcal{R}_{e} \simeq 6{\times}10^{4}\)

\(d \sim 60~\mu\)m, \(\mathcal{S} = 0.01\)

\(d \sim 135~\mu\)m, \(\mathcal{S} = 0.04\)

\(d \sim 250~\mu\)m

\(\mathcal{S} = 0.1\)

Dimensionless duration \(\tau\) and influence of \(\mathcal{S}\)

\(\require{cancel} \tau = f(\xcancel{\mathcal{R}_{e}}, \mathcal{S}, \theta)\)

\(\require{cancel} \tau = f(\xcancel{\mathcal{R}_{e}},\color{orange}{\mathcal{S}}, \theta)\)

\(\bullet\) settling dominated: \(\tau \propto (\mathcal{S}/a)^{-1} \iff t_{\rm end} \propto h_{0}/v_{\rm s}\)

Dimensionless duration \(\tau\) and influence of \(\mathcal{S}\)

\(\require{cancel}\tau = f(\xcancel{\mathcal{R}_{e}}, \color{orange}{\mathcal{S}}, \theta\rightarrow?)\)

\(\bullet\) settling dominated: \(\tau \propto (\mathcal{S}/a)^{-1} \iff t_{\rm end} \propto h_{0}/v_{\rm s}\)

Mixing and water entrainment

from Wilson et al. 2016


Entrainment coefficient, \(E = \displaystyle\frac{w_{\rm e}}{U}\)

Entrainment coefficient, \(E = f(\mathcal{R}_{e})\)

In a nutshell

Slumping regime:

  • dimensionless velocity \(\mathcal{F}_{r}\) increases with \(\theta\)
  • dimensioless duration \(\tau\):
    • for \(\mathcal{S} < 2.10^{-2}\) \(\rightarrow\) \(\tau \simeq 30\)
    • for \(\mathcal{S} > 2.10^{-2}\) \(\rightarrow\) \(\tau \simeq 0.8 (\mathcal{S}/a)^{-1}\)
  • entrainment increases with Reynolds as \(E \propto \mathcal{R}_{e}\)

More in: Slumping regime in lock-release turbidity currents. Gadal et al. 2023, in review in JFM. preprint:arXiv2301.00192

What’s next ?

  • Comparison with depth-averaged models
  • Influence of lock aspect-ratio \(h_{0}/L_{0}\) ?
  • \(\tau(\theta)\), \(E(\theta, \mathcal{S})\) ?


  • Moving towards steady influx turbidity currents

Steady injection of a water/plastic suspension inside fresh water.