Turbidity (particle-laden gravity) currents

Relevant processes and numbers

Cyril Gadal

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

Short personal presentation

PhD on sand dunes (IPGP/PMMH, 2017–2020)

with Clément Narteau & Philippe Claudin

A brief introduction of myself

PhD on sand dunes (IPGP/PMMH, 2017–2020)

with Clément Narteau & Philippe Claudin

PostDoc on turbidity currents (IMFT, 2021–2022)

with Laurent Lacaze & Matthieu Mercier

Short personal presentation

PhD on sand dunes (IPGP/PMMH, 2017–2020)

with Clément Narteau & Philippe Claudin

PostDoc on turbidity currents (IMFT, 2021–2022)

with Laurent Lacaze & Matthieu Mercier

PostDoc on the cloggind of riverbeds (IMFT, 2023)

with Laurent Lacaze & Matthieu Mercier

Short personal presentation

PhD on sand dunes (IPGP/PMMH, 2017–2020)

with Clément Narteau & Philippe Claudin

PostDoc on turbidity currents (IMFT, 2021–2022)

with Laurent Lacaze & Matthieu Mercier

PostDoc on the cloggind of riverbeds (IMFT, 2023)

with Laurent Lacaze & Matthieu Mercier

PostDoc on the self-organization of cohesive granular flows (Manchester)

with Nico Gray & Chris Johnson

Short personal presentation

PhD on sand dunes (IPGP/PMMH, 2017–2020)

with Clément Narteau & Philippe Claudin

PostDoc on turbidity currents (IMFT, 2021–2022)

with Laurent Lacaze & Matthieu Mercier

PostDoc on the cloggind of riverbeds (IMFT, 2023)

with Laurent Lacaze & Matthieu Mercier

PostDoc on the self-organization of cohesive granular flows (Manchester)

with Nico Gray & Chris Johnson

Particle-laden (turbidity) currents

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

Snow avalanche, Zinal, Switzerland @Zacharie Grossen

Pyroclastic flow, Sinabung, Indonesia @Jean-Guillaume Feignon

Dust storm, Phoenix, USA @Alan Stark

Particle-laden (turbidity) currents

Almost always destructive natural hazards.

Power transmission overturned by a powder snow avalanche, Switzerland. Caviezel et al. 2021

Dammages by pyroclastic flow after the Merapi eruption, Indonesia. @Noer Cholik

Dust storm, Phoenix, USA. @Mike Olbinski

Hence

  • they need reliable modelling
    • shallow flows -> depth-averaged (shallow water) modelling for hazard assessment
    • closure in large-scale models (meteorological, oceanic, sediment balance, …)

\(\rightarrow\) determination of relevant processes and corresponding parameterization

But in-situ measurements are difficult !!!

  • hard to predict
  • destructive
  • many processes involved

A simple laboratory system: lock-release devices

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

A simple laboratory system: lock-release devices

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

Using \(x_{\rm f}(t)\) and \(h(x, t)\) as proxy for the current morphodynamics

An extensive literature, especially on saline currents

Rottman and Simpson. 1984, saline water, no slope (\(\theta=0^\circ\))

Front position and slumping regime

Rottman and Simpson. 1984, saline water, no slope (\(\theta=0^\circ\))

Rottman and Simpson. 1984, saline water, no slope (\(\theta=0^\circ\))
  • Scales:
    • length: \(l_{0}\) (lock length)
    • velocity: \(u_{0} = \sqrt{h_{0} g'}\), \(g' = \frac{\rho_{0} - \rho_{a}}{\rho_{0}}g\)
    • time: \(t_{0} = l_{0}/u_{0}\)

Slumping (constant-velocity) regime

  • Equilibrium: inertia - pressure gradient at the front: \(u_{\rm c} = \mathcal{F}_{r} u_{0}\)
  • Duration: until bore comes back: \(t_{\rm bore} = \tau t_{0}\)
  • constant prefactors: \(\mathcal{F}_{r} = 0.5\), \(\tau \simeq 20\)

Towards more complex situations

Slumping (constant-velocity) regime

  • Equilibrium: inertia - pressure gradient at the front: \(u_{\rm c} = \mathcal{F}_{r} u_{0}\)
  • Duration: until bore comes back: \(t_{\rm end} = \tau t_{0}\)
  • constant prefactors: \(\mathcal{F}_{r} = 0.5\), \(\tau \simeq 20\)

Horizontal tank, homogenous fluid or particle but low volume fraction/small settling !

Now what happens in more complex systems ?

  • inclined bottom: \(\theta\)
  • particulate currents
    • high volume fractions: \(\phi\)
    • strong settling: \(v_{\rm s}\)
    • unusual particles (hydrogel beads): ??
  • triphasic currents: \(\delta_{\rho}\)

Experimental setups and datasets (ANR PALAGRAM)

C. Gadal, M. Mercier & L. Lacaze

J. Schneider, S. Kiesgen de Richter, Y. Dossmann

Marie Rastello

\(+\) two-fluids simulations (SedFoam)

Julien Chauchat, Cyrille Bonamy

Dimensional analysis

Experimental parameters:

  • Slope \(\alpha\)
  • Lock geometry \(h_{0}\), \(l_{0}\)

  • Interstitial fluid density \(\rho_{\rm f}\)

  • Ambient fluid density \(\rho_{\rm a}\)

  • Volume fraction \(\phi\)

  • Particle diameter \(d\)

  • Particle density \(\rho_{\rm p}\)

\(\rightarrow \rho_{0} = \phi \rho_{p} + (1-\phi \rho_{\rm f})\)

Dimensional analysis

Dimensionless numbers as control parameters:

  • Slope \(\alpha\)

  • Aspect ratio \(a = \displaystyle\frac{h_{0}}{l_{0}}\)

  • Reynolds number \(\mathcal{R}e = \displaystyle\frac{u_{0}h_{0}}{\nu}\)

  • Volume fraction \(\phi\)

  • Stokes number \(\mathcal{S}t = \displaystyle\frac{v_{\rm s}}{u_{0}}\frac{l_{0}}{h_{0}}\)

  • Atwood number \(\require{cancel} \xcancel{\mathcal{A}t = \displaystyle\frac{\rho_{0} - \rho_{a}}{\rho_{a}}}\) \(\rightarrow\) Boussinesq approx.

Scales:

  • length: \(l_{0}\) (lock length)
  • velocity: \(u_{0} = \sqrt{h_{0} g'}\), \(g' = \frac{\rho_{0} - \rho_{a}}{\rho_{0}}g\)
  • time: \(t_{0} = l_{0}/u_{0}\)

Dimensionless observables:

  • \(\displaystyle\frac{x_{\rm f}}{l_{0}}\)
  • \(\left(\displaystyle\frac{h}{h_{0}}\right)\)

Parameter space

Bulk analysis - 280 experiments

\(\displaystyle\frac{x_{\rm f}}{l_{0}} = \mathcal{F}r\left[\displaystyle\frac{t}{t_{0}} - \displaystyle\frac{1}{\tau}\left(\displaystyle\frac{t}{t_{0}}\right)^{2} +\,...\right]\)

\(\displaystyle\frac{x_{\rm f}}{l_{0}} = \color{orange}{\mathcal{F}r}\left[\displaystyle\frac{t}{t_{0}} - \displaystyle\frac{1}{\color{orange}{\tau}}\left(\displaystyle\frac{t}{t_{0}}\right)^{2} +\,...\right]\)

Influence of the slope

\(\bullet\) PMMA particles, \(\phi \sim 1~\%\)

\(\alpha = 0^\circ\)

@Marie Rastello, LEGI

\(\alpha = 45^\circ\)

@Marie Rastello, LEGI

Influence of the slope

Slumping dimensionless velocity increase with slope !

Influence of the slope – A non-trivial increase

A first interpretation: slope-induced acceleration increases velocity

But

  • Issue 1: constant velocity \(\leftrightarrow\) equilibrium (inertia / pressure gradient)

  • Issue 2: slope-induced acceleration takes a time \(t_{\theta} \sim 4\displaystyle\frac{t_{0}}{\sin\theta}\) to be significant (Birman et al. 2007), so usually \(t_{\theta} \gg \tau t_{0}\)

Hypothesis (unverified): slope acts during the early transient regime

Adapted from Cantero et al. 2007

Adapted from Cantero et al. 2007

Influence of the slope – A simple model

An energetic balance between the initial state and the end of the transient phase:

\[ \underbrace{C \Delta\rho g \cos\alpha h_{0}}_{\textrm{initial}} -\underbrace{\left[\frac{1}{2}\rho_{0}u_{\text c}^{2} + A \Delta\rho g \cos\alpha h_{0} - B \Delta\rho g \sin\alpha L\right]}_{\textrm{final}} = \underbrace{\frac{1}{2}c_{\text d}\rho_{0}u_{\text c}^{2} \frac{L}{h_{0}}}_{\textrm{dissipation}} \]

Adapted from Cantero et al. 2007

\[ \mathcal{F}r = \frac{u_{\text c}}{u_0} = \frac{Fr_0}{\sqrt{1 + C_{\text D}}} \sqrt{1 + \frac{\tan\alpha}{S}} \]

\[ \mathcal{F}r = \frac{u_{\text c}}{u_0} = \frac{\color{orange}{Fr_0}}{\sqrt{1 + \color{orange}{C_{\text D}}}} \sqrt{1 + \frac{\tan\alpha}{\color{orange}{S}}} \]

Influence of the slope – A simple model

Influence of the volume fraction (\(\alpha\sim 0^\circ\))

Slumping dimensionless velocity decreases with \(\phi\) !

Influence of the volume fraction (\(\alpha\sim 0^\circ\)) – A non-trivial increase

A first interpretation: particle-induced dissipation decreases velocity

But

  • Issue 1: constant velocity \(\rightarrow\) equilibrium (inertia / pressure gradient)

  • Issue 2: dissipation takes a time \(t_{\nu}/t_{0} \gg \tau\) (Huppert & Simpson 1980, Bonnecaze et al. 1993)

Hypothesis (unverified): dissipation acts during the early transient regime

Adapted from Cantero et al. 2007

Adapted from Cantero et al. 2007

Influence of the volume fraction (\(\alpha\sim 0^\circ\)) – A simple model

An energetic balance between the initial state and the end of the transient phase:

  • \(\underbrace{C \Delta\rho g \cos\alpha h_{0}}_{\textrm{initial}} -\underbrace{\left[\frac{1}{2}\rho_{0}u_{\text c}^{2} + A \Delta\rho g \cos\alpha h_{0} - B \Delta\rho g \sin\alpha L\right]}_{\textrm{final}} = \underbrace{\frac{1}{2}c_{\text d}\rho_{0}u_{\text c}^{2} \frac{L}{h_{0}}}_{\textrm{dissipation}}\)

  • \(c_{\rm d} \equiv c_{\text d}\left(1 + \frac{E}{\mathcal{R}e}\frac{u_0}{u_{\text c}}\frac{\eta_{\text eff}}{\eta_{\text f}} \right)\), \(\eta_{\rm eff}(\phi)\) effective viscosity (i.e Krieger & Dougherty 1959, Boyer et al. 2011, …)

Adapted from Cantero et al. 2007

\[ \mathcal{F}r = \frac{1}{1 + C_{\text D}} \left[-\frac{Re_{\text c}}{\mathcal{R}e}\frac{\eta_{\text eff}}{\eta_{\text f}} +\sqrt{\left(\frac{Re_{\text c}}{\mathcal{R}e}\frac{\eta_{\text eff}}{\eta_{\text f}} \right)^{2} + Fr_0^{2} \left(1 + \left[1 + C_{\text D}\right]\frac{\tan\alpha}{S} \right)} \right] \]

\[ \mathcal{F}r = \frac{1}{1 + C_{\text D}} \left[-\frac{\color{orange}{Re_{\text c}}}{\mathcal{R}e}\frac{\eta_{\text eff}}{\eta_{\text f}} +\sqrt{\left(\frac{\color{orange}{Re_{\text c}}}{\mathcal{R}e}\frac{\eta_{\text eff}}{\eta_{\text f}} \right)^{2} + Fr_0^{2} \left(1 + \left[1 + C_{\text D}\right]\frac{\tan\alpha}{S} \right)} \right] \]

Influence of \(\phi\) (\(\alpha\sim 0^\circ\)) – A simple model

Influence of \(\mathcal{S}\)t (settling)

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

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

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

\(d \sim 250~\mu\)m, \(\mathcal{S} = 0.1\)

Influence of \(\mathcal{S}\)t (settling)

  • “Bore” dominated regime: \(\tau \propto \frac{t_{\rm bore}}{t_{0}}= cste\)
  • Settling dominated regime: \(\tau \propto \frac{t_{\rm sed}}{t_{0}} = \frac{h_{0}}{v_{\rm s}} \frac{u_{0}}{l_{0}} = \mathcal{S}t\)

In a nutshell, \(x_{\rm f}(t)\) during the slumping regime:

Slumping velocity \(\mathcal{F}r\) increases with \(\alpha\)

Slumping velocity \(\mathcal{F}r\) decreases with \(\phi\)

For \(\mathcal{S}t > 10^{-2}\), slumping duration \(\tau \propto \mathcal{S}t^{-1}\)

What about current shapes \(h(x, t)\) ?

What about current shapes \(h(x, t)\) ?

All current shapes during slumping regime superimposed.

All current shapes during slumping regime superimposed.

What about current shapes \(h(x, t)\) ?

Multiple average shapes superimposed.

Multiple average shapes superimposed.

Multiple average shapes superimposed.
  • influence of \(\mathcal{S}t\) ?
  • influence of \(\alpha\) ?
  • influence of \(\phi\), \(\mathcal{R}e\) ?

What about current internal structure \(\vec{u}(z)\), \(\phi(z)\) ?

  • Is \(u_{\rm f}\), \(h(x)\) enough to describe the dynamics ?

\(\rightarrow\) Use of steady influx set-ups !

Steady injection of (saline water + polystyrene beads) into fresh water.

What about current internal structure \(\vec{u}(z)\), \(\phi(z)\) ?

  • 1st round: 3 saline density currents with \(\rho_{\rm c} \in [1015, 1050, 1080]~\textrm{kg}~\textrm{m}^{-3}\)
  • 2nd round: same at first round + adding \(3~\%\) of particles at \(\rho_{\rm p} = 1050~\textrm{kg}~\textrm{m}^{-3}\).

What about unusual beads ?

What about unusual beads ?

Hydrogel beads (\(d \sim 1~\textrm{cm}\)) down a slope (\(\alpha = 45^\circ\)) from a non-suspended initial condition (@Marie Rastello, LEGI).