IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
A Mathematical and NumericalStudy of Roll Waves
Serge D’Alessio1 and J.P. Pascal2
1Department of Applied MathematicsUniversity of Waterloo
2Department of MathematicsRyerson University
AFM 2008
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Outline
1 Introduction
2 Mathematical Formulation
3 Stability Analysis
4 Numerical Solution Procedure
5 Simulations
6 Summary
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Unstable flow down an inclineUnstable Flow Down an Incline
θ
Unstableuniform flow
θ
Roll waves
Critical conditions for the onset of Instability.Structure of Roll WavesInvestigate the effect of bottom topography
J.P. Pascal and S.J.D. D’Alessio Instability of Laminar Flow Down an Uneven Inclined Plane
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Chapter 1. Introduction 5
Figure 1.1: Spillway from Llyn Brianne Dam, Wales [115]. For an idea of the scale, the
width of the spillway is about 75 feet.
The spillway from the LlynBrianne Dam in Wales
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Chapter 2. Dynamics of roll waves 32
PSfrag replacements
t (sec)
h (mm)
0 2 4 6 8 10−12
−10
−8
−6
−4
−2
0
2
0 2 4 6 8 10
−20
−15
−10
−5
0
5
PSfrag replacements
t (sec)
t (sec)
h(m
m)
h(m
m)
Figure 2.1: The picture on the left shows a laboratory experiment in which roll waves
appear on water flowing down an inclined channel. The fluid is about 7 mm deep and the
channel is 10 cm wide and 18 m long; the flow speed is roughly 65 cm/sec. Time series
of the free-surface displacements at four locations are plotted in the pictures on the
right. In the upper, right-hand panel, small random perturbations at the inlet seed the
growth of roll waves whose profiles develop downstream (the observing stations are 3 m,
6 m, 9 m and 12 m from the inlet and the signals are not contemporaneous). The lower
right-hand picture shows a similar plot for an experiment in which a periodic train was
generated by moving a paddle at the inlet; as that wavetrain develops downstream, the
wave profiles become less periodic and there is a suggestion of subharmonic instability.
Experiment taken from Balmforth & Mandre (JFM, 2004)
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Coordinate system
θ
g
x, u
z, w
h(x, t)
ζ(x)
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Equations of motion
∂u∂x
+∂w∂z
= 0
ρ
(∂u∂t
+ u∂u∂x
+ w∂u∂z
)= −∂p
∂x+ gρ sin θ + µ
(∂2u∂x2 +
∂2u∂z2
)1ρ
∂p∂z
+ g cos θ − µ
ρ
∂2w∂z2 = 0
Assumed Re ∼ O(1) and neglected terms O(δ2) and higherwhere δ = H/L is the aspect ratio
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Interface conditions
Free surface conditions:
p − 2µ∂w∂z
= 0
µ∂u∂z
= 0
w =∂h∂t
+ u∂h∂x
+ uζ ′(x)
at z = ζ(x) + h(x , t)
Bottom boundary conditions:
u + ζ ′(x)w = 0 and ζ ′(x)u − w = 0 at z = ζ(x)
⇒ u = w = 0 at z = ζ(x)
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Integral boundary layer (IBL) method
Depth-integrate equations and introduce flow variables
h(x , t) and q(x , t) =
∫ ζ+h
ζudz
To convert terms∫ ζ+h
ζu2dz ,
µ
ρ
∂u∂z
∣∣∣∣z=ζ
assume the parabolic velocity profile:
u(x , z, t) =3q2h3
[2(h + ζ)z − z2 − (ζ + 2h)ζ
]
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Dimensionless equations
In terms of h, q the dimensionless equations become∂h∂t
+∂q∂x
= 0
∂q∂t
+65
∂
∂x
(q2
h
)=
1Fr2
(h − h
∂h∂x
− ζ ′(x)h − qh2
)+
3Fr2
Re2
[72
∂2q∂x2 −
9h
∂q∂x
∂h∂x
+9qh2
(∂h∂x
)2
− 9q2h
∂2h∂x2
−6ζ ′(x)
h∂q∂x
+6ζ ′(x)q
h2∂h∂x
− 3ζ ′′(x)qh
− 6 (ζ ′(x))2 qh2
]
where Fr2 =Re
3 cot θ, Re =
ρQµ
and ζ(x) = ab cos(kbx)
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Linear stability: ab = 0 case
The steady-state flow is: qs = hs = 1Imposing disturbances on this steady flow and linearizing yieldsthe dispersion equation
Fr2σ2 +
(21Fr4
2Re2 k2 + 1 + i125
Fr2k)
σ +
(1− 6
5Fr2)
k2
+i(
3k +27Fr4
2Re2 k3)
= 0
where σ is the growth rate and k is the wavenumber of thedisturbance
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Linear stability results for ab = 0
The flow is stable if Fr <1√3
while for Fr >1√3
instability
occurs for wavenumbers k < kmax where
kmax =10Re√30Fr2
√3Fr2 − 1
3Fr2 + 35 + 12Fr√
6Fr2 + 25
For large Fr the asymptotic behaviour is
kmax ∼10Re
Fr2√
30(1 + 4√
6)
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Neutral stability curves for ab = 0
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
k
Fr
STABLE
Re=20
Re=10
UNSTABLE
Re=5
Neutral stability curveshave a maximum atFr ≈ 0.76286(independent of Re)
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Linear stability: ab 6= 0 case
The steady state solution is qs = 1 and hs(x) satisfies
3β[hsh′′s − 2(h′s)2] + (2αh3
s − 4βζ ′ − 125
)h′s
+2βζ ′′hs − 2α(1− ζ ′)h3s = −2α− 4β(ζ ′)2
where α =1
Fr2 and β = 9(
FrRe
)2
An approximate solution can be constructed in the form
hs(x) = 1 + (abkb)h(1)s (x) + (abkb)2h(2)
s (x) + · · ·
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Periodic steady state solution
0 0.2 0.4 0.6 0.8 10.9
0.95
1
1.05
1.1
1.15
x
h
Analytical (1 term)Analytical (2 terms)Numerical
Fr = 1, Re = 10,ab = 0.1, kb = 2π
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Linear stability: ab 6= 0 case
To study how small disturbances will evolve, introduceperturbations h, q superimposed on the steady-state solutionand linearize equations using
h = hs(x) + h , q = 1 + q
For an uneven bottom, the coefficients in the linearizedequations are periodic functions; hence apply Floquet-Blochtheory to conduct the stability analysis and represent theperturbations as Bloch-type functions having the form
h = eσteiKx∞∑
n=−∞hneinkbx , q = eσteiKx
∞∑n=−∞
qneinkbx
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Numerical linear stability results for ab 6= 0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
0.58
0.6
0.62
0.64
0.66
0.68
ab
Frcrit
kb=π
kb=2π
kb=2.5π
Critical Froude numberas a function of bottomamplitude with Re = 10
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Numerical linear stability results for ab 6= 0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.5
0.55
0.6
0.65
0.7
0.75
ab
Frcrit
Re=3
Re=8
Re=5
Re=10
Re=20
Re=7
Critical Froude numberas a function of bottomamplitude with kb = 2π
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Numerical linear stability results for ab 6= 0
0
0.5
1
1.5
2
2.5
3
3.5
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
K
Fr
0
0.5
1
1.5
2
2.5
3
0.5 0.6 0.7 0.8 0.9 1
K
Fr
UNSTABLE
STABLE
ab=0.1
ab=0.2
ab=0
STABLE
ab=0.2
UNSTABLE
ab=0.1ab=0
Neutral stability curvesfor the case withRe = 10 and kb = 2π
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Numerical linear stability results for ab 6= 0
0
0.5
1
1.5
2
2.5
3
3.5
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
K
Fr
0
0.5
1
1.5
2
2.5
3
0.5 0.6 0.7 0.8 0.9 1
K
Fr
STABLE
kb=0
UNSTABLE
kb=2.5π
kb=2π
kb=π
STABLE
UNSTABLEkb=0kb=π
kb=2.5π
kb=2π
Neutral stability curvesfor the case withRe = 10 and ab = 0.2
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Begin by expressing equations in the form
∂h∂t
+∂q∂x
= 0
∂q∂t
+∂
∂x
(65
q2
h+
α
2h2)
= Ψ(h, q)+χ
(x , h, q,
∂h∂x
,∂q∂x
,∂2h∂x2 ,
∂2q∂x2
)where Ψ = α
(h − q
h2
)and χ = −αζ ′h − 2βζ ′
(ζ ′ − ∂h
∂x
)qh2 − βζ ′′
qh− 2β
ζ ′
h∂q∂x
+β
(76
∂2q∂x2 −
32
qh
∂2h∂x2 −
3h
∂q∂x
∂h∂x
+ 3qh2
(∂h∂x
)2)
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Fractional-step method (LeVeque, 2002)
Decouple the advective and diffusive components, first solve
∂h∂t
+∂q∂x
= 0
∂q∂t
+∂
∂x
(65
q2
h+
α
2h2)
= Ψ(h, q)
over a time step ∆t , and then solve
∂q∂t
= χ
(x , h, q,
∂h∂x
,∂q∂x
,∂2h∂x2 ,
∂2q∂x2
)using the solution obtained from the first step as an initialcondition for the second step; the second step returns thesolution for q at the new time t + ∆t
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
First step
This involves solving a nonlinear system of hyperbolicconservation laws; express in vector form
∂U∂t
+∂F(U)
∂x= b(U)
where U =
[hq
], F(U) =
[q
65
q2
h + α2 h2
], b(U) =
[0Ψ
]Utilize MacCormack’s method to solve this system; this is aconservative second-order accurate finite difference schemewhich correctly captures discontinuities and converges to thephysical weak solution of the problem
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
First step
LeVeque & Yee (JCP, 1990) extended MacCormack’s method toinclude source terms; this explicit predictor-corrector schemetakes the form
U∗j = Un
j −∆t∆x
[F(Un
j+1)− F(Unj )]
+ ∆t b(Unj )
Un+1j =
12
(Un
j + U∗j
)− ∆t
2∆x
[F(U∗
j )− F(U∗j−1)
]+
∆t2
b(U∗j )
where the notation Unj ≡ U(xj , tn) was adopted, ∆x is the grid
spacing and ∆t is the time step; second-order accuracy isachieved by first forward differencing and then backwarddifferencing
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Second step
This reduces to solving the generalized one-dimensional lineardiffusion equation given by:
∂q∂t
=7β
6∂2q∂x2 + S1
∂q∂x
+ S0q + S
where S = −αζ ′h and
S0 = −βζ ′′
h− 2β
ζ ′
h2
(ζ ′ − ∂h
∂x
)− 3
2β
h∂2h∂x2 + 3
β
h2
(∂h∂x
)2
and S1 = −2βζ ′
h− 3
β
h∂h∂x
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Computational parameters
The problem is completely specified by Fr , Re, ab and kb;typical computational parameters used were:Computational Domain: 0 ≤ x ≤ L
with λb ≤ L ≤ 300λb , λb =2π
kbGrid Spacing: ∆x = .01Time Step: ∆t = .002
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Evolution of flow rate
0 5 10 15 20 25 30 35 40 450.5
1
1.5
2
2.5
3
3.5
4
x
q
t=400
t=495
t=264
t=450
Parameters:ab = 0.1, kb = 2π,Re = 10, Fr = 0.7A subharmonicinstability known aswave coarseningoccurs for L = 45λb
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Evolution of flow rate
0 5 10 15 20 25 300.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
x
q
t=450
t=260
Parameters:ab = 0.1, kb = 2π,Re = 10, Fr = 0.7Interruption in wavecoarsening occursfor L = 30λb
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Wave spawning
0 50 100 150 200 250 3000
1
2
3
4
5
6
7
8
9
10
x
q
t=700
t=400 Parameters:ab = 0.1, kb = 2π,Re = 10, Fr = 0.7A wave-spawninginstability occurs forL = 300λb
Serge D’Alessio and J.P. Pascal Study of Roll Waves
IntroductionMathematical Formulation
Stability AnalysisNumerical Solution Procedure
SimulationsSummary
Concluding remarks
A mathematical model of roll waves along with a numericalmethod to solve the model were presentedInvestigated the effect of sinusoidal bottom topography onthe formation of roll wavesBottom topography has a stabilizing effect on the flow forsmall to moderate waviness parametersFuture work includes repeating the analysis for a porouswavy bottom and to include surface tension
Serge D’Alessio and J.P. Pascal Study of Roll Waves