the granular blasius problem boundary layers in granular...
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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
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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
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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
gθ
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
gθ
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
gθ
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
gθ
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
gθ
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
gθ
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
gθ
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
dµ
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