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The Granular Blasius Problem Boundary layers in granular flows Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge Friday 17 September 2017
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  • The Granular Blasius ProblemBoundary layers in granular flows

    Jonathan Michael Foonlan Tsang ([email protected]),Stuart B. Dalziel, Nathalie M. Vriend

    DAMTP, University of Cambridge

    Friday 17 September 2017

  • My research: Granular currents

    Modelling granular currents is important

    > 7, 600 deaths from landslides annually (Perkins 2012)

    Usually in developing countries

    Common models are depth-averaged (‘shallow water’)

    Ad hoc description of depthwise velocity profile

    Want to understand internal dynamics better

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • Depth-averaged models

    Shallow water equations on a slope

    ∂h

    ∂t+∂(hu)

    ∂t= 0

    ∂(hu)

    ∂t+

    ∂t

    (1

    2hu2 +

    1

    2gh2 cos θ

    )= gh sin θ

    Depth h, depth-averaged velocity u

    Closure relation u2 = χu2 for shape factor χ ≥ 1

    Shape factor characterises depthwise velocity profile

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • Depthwise velocity profile

    χ = u2/u2

    Usually assume constant χ, e.g. plug flow, χ = 1

    Reasonable assumption over long lengthscales

    But χ is not constant when topography is present

    Difficult to measure velocity profile experimentally

    Can be measured in DPM simulations

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • Granular Blasius problem

    x > 0bumpy surface

    flow introducedupstream

    current flows ofend of surface

    depth profile of a steady flow

    z

    x

    HU

    h(x)

    x < 0smooth surface

    (possibly frictional)

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • Granular boundary layer problem

    smooth bumpy

    Model of increasing topographical resistance

    x < 0: Smooth, slip allowed

    x > 0: No-slip condition creates boundary layer

    BL grows and eventually takes over

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • Granular vs. classical Blasius problems

    blade (no-slip)

    smooth bumpy

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • From classical to granular:

    blade (no-slip)

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • From classical to granular: Free surface, finite depth

    sliding plate static plate

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • From classical to granular: Slope

    slip allowed no slip

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • From classical to granular: Granular rheology

    smooth bumpy

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • From classical to granular: Free surface, finite depth

    sliding plate static plate

    BL induces flow in outer layer, which affects BL

    Behaviour as Re→∞ depends on Fr

    Tsang et al. submitted to JFM Rapids

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • From classical to granular: Slope

    slip allowed no slip

    Evolution towards far-field profile

    Nusselt film for laminar Newtonian fluid

    Bagnold profile for granular flow

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • From classical to granular: Granular rheology

    smooth bumpy

    µ(I ) rheology (Jop et al. 2006)

    high γ̇ =⇒ high I in BL

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • The BL equation has the same structure

    Classical:

    u∂u

    ∂x+ w

    ∂u

    ∂z= − ∂p

    ∂x+

    1

    Re

    ∂2u

    ∂z2

    Under µ(I ):

    u∂u

    ∂x+ w

    ∂u

    ∂z= sin θ +

    ∂z

    (µ(I )p

    )∼ sin θ + µ ∂p

    ∂z+ p

    dI

    ∂ I

    ∂z

    ∼ (· · · ) + (· · · ) ∂2u

    ∂z2

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • Analysing the granular BL equation

    Solutions depend on behaviour of µ(I ) as I →∞

    µ(I ) ∼ µ1 +µ2 − µ1I0/I + 1

    Generalise µ(I )

    µ(I ) ∼ µ2 −m

    α− 1

    (I0I

    )α−1

    u∂u

    ∂x+ w

    ∂u

    ∂z∼ ∂

    2u/∂z2

    (∂u/∂z)α

    Problems with well-posedness for high I ? (Barker et al. 2017)

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • Analysing the granular BL equation

    Approximate similarity solutions

    u ∝ f ′(z/β(x)), f ′′′ + u1+αs

    2− αff ′′1+α = 0

    2 4 6 8 10ζ

    0.2

    0.4

    0.6

    0.8

    1.0

    f'(ζ)Similarity solutions for the granular boundary layer profile

    α = 0 (classical)

    α = 1

    α = 1.25

    α = 1.5

    α = 1.75

    Singular behaviour as α→ 2−?

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • Realisation in DPM (MercuryDPM)

    0

    0.1

    0.2

    -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

    z -

    perp

    endic

    ula

    r coord

    inate

    x - downstream coordinate

    Topography : ior-ballotini-slope16-run2 : v200-h020 : 150

    0

    0.5

    1

    1.5

    2

    2.5

    speed

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • Realisation in DPM (MercuryDPM)

    0

    0.1

    0.2

    -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

    z -

    perp

    endic

    ula

    r coord

    inate

    x - downstream coordinate

    Topography : ior-ballotini-slope16-run2 : v200-h020 : 150

    0

    0.5

    1

    1.5

    2

    2.5

    speed

    Is the no-slip condition realistic?Rolling resistanceRestitution coefficient. . .

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows

  • Summary

    Study how granular currents respond to topography

    Similar to classical Blasius problem

    BLs dynamics governed by high I

    Generalisations of µ(I )

    DPM realisation has some subtleties

    Jonathan Michael Foonlan Tsang ([email protected]), Stuart B. Dalziel, Nathalie M. Vriend DAMTP, University of Cambridge

    The Granular Blasius Problem Boundary layers in granular flows


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