Debris Flows - Environmental Geomechanics - page 21

vertical velocities, the timescale, the typical pressure, and the order of magnitude of
bulk viscosity are referred to as U ∗ , V ∗ , T ∗ , P ∗ and η ∗ , respectively. Moreover, in
addition to the lengthscale ratio , we introduce the following dimensionless numbers
that characterize free-surface, gravity-driven flows: the flow Reynolds number and the
Froude number

Re = ρU ∗ H ∗
η ∗

and Fr = U ∗ √ gH ∗ cos θ .

The following dimensionless variables will be used in this section:
u = u
U ∗

, v = v
V ∗

, x = x
L ∗

, y = y
H ∗

t = t , and ˆ

T ∗ .
A natural choice for T ∗ is T ∗ = L ∗ /U ∗ . The stresses are scaled as follows:
σ xx = η ∗ U ∗
L ∗

σ xx , σ xy = η ∗ U ∗

H ∗ σ xy , σ yy = η ∗ U ∗

σ yy , and p = p

P ∗ ,
where σ xx , σ xy , and σ yy are the normal stress in the x direction, the shear stress, and
the normal stress in the x direction, respectively. Here, we are interested in free-surface
flows. This leads us to set P ∗ = ρgH ∗ cos θ , since we expect that, to leading order, the
pressure adopts a hydrostatic distribution (see below). If we define the vertical velocity
scale as V ∗ = U ∗ , the mass balance equation [1.2] takes the following dimensionless
form

L ∗

∂u
∂x

+ ∂v
∂y

=0 . [1.5]

Substituting the dimensionless variables into the momentum balance equation [1.3]
leads to
Re d

1 + 2 ∂σ xx

∂x

d ˆ

= Re
Fr 2

∂p
∂x

+ ∂σ xy
∂y

tan θ − , [1.6]

t

+ 2 ∂σ xy
∂x

3 Re d

d ˆ

= Re
Fr 2

∂p
∂y

+ 2 ∂σ yy
∂y

− 1 − . [1.7]

t

The momentum balance equation expresses a balance between gravity
acceleration, inertial terms, pressure gradient, and viscous dissipation, whose order
of magnitude is ρg sin θ , ρU ∗ /L ∗ , P ∗ /L ∗ , and η ∗ U ∗ /H ∗ , respectively. Depending on
the values considered for the characteristic scales, different types of flow regime occur.
At least four regimes, where two contributions prevail compared to the others, could
be achieved in principle
- Inertial regime , where inertial and pressure-gradient terms are of the same
magnitude. We obtain

U ∗ = gH ∗ cos θ.
The order of magnitude of the shear stress is ∂σ xy /∂y = ρg O ( − 1 Re − 1 ). This

regime occurs when: Re 1 and Fr = O (1).