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
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