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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

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:

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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