Simple analytical models for flow over an erodable bed

P.-Y. LAGREELab. de Modelisation en Mecanique, Universite Paris 6.

K.K.J. Kouakou, D. Lhuillier, Josserand (LMM), H. Caps (GRASP Liege)

• applications in the Nature

• simple point of view

• simple models for the fluid

• simple models for the erodable bed

• stability analysis/ numerical computation

The coupled problem

- for a given soil f(x, t)- ...

The coupled problem

- for a given soil f(x, t)- we have to compute the flow (u(x, y, t)).

The coupled problem

- for a given soil f(x, t)- we have to compute the flow (u(x, y, t)).

- the flow erodes the soil.

The coupled problem

- for a given soil f(x, t)- we have to compute the flow (u(x, y, t)).

- the flow erodes the soil.

The coupled problem

- for a given soil f(x, t)- we have to compute the flow (u(x, y, t)).

- the flow erodes the soil.- which changes the soil.

The coupled problem

- for a given soil f(x, t)- we have to compute the flow (u(x, y, t)).

- the flow erodes the soil.- which changes the soil.- etc

The coupled problem

- for a given soil f(x, t)- we have to compute the flow (u(x, y, t)).

- the flow erodes the soil.- which changes the soil.- etc

The coupled problem

- for a given soil f(x, t)- we have to compute the flow (u(x, y, t)).

- the flow erodes the soil.- which changes the soil.- etc

we aim to present a simple description for the flow andobtain simple model equations to describe the interaction.

The fluid

Numerical resolution of Navier Stokes equations.In real applications: viscosity changed... turbulence...

here we will present some simplifications:

• Steady flow• Linearized solutions• ideal fluid at Re = ∞• creeping flow at Re = 0• asymptotic solution of N.S.: Triple Deck: laminar viscous theory at Re = ∞

The erodable bed

Mass conservation for the sediments:

∂f

∂t= −∂q

∂x.

Problem :What is the relationship between q and the flow?hint: the larger u the larger the erosion, the larger q

q seems to be proportional to the skin frictionu

q

The erodable bed: relations between q and u

∂f

∂t+

∂q

∂x= 0

In the literature one founds Charru /Izumi & Parker / Yang / Blondeau

q = EH(τ − τs)((τ − τs)a)

or with a slope correction for the threshold value:

τs + λ∂f

∂x,

with H Heaviside function, a,E coefficients

Continuous theory

... leads to q = τ3

First example: Basic case, at Re = ∞

Uniform flow over a topography at large Reynolds number

Starting from an initial shape, the ideal fluid flow is computed (in the Small PerturbationTheory):

f(x, t) gives u = (1 + 1π

∫∞−∞

f ′

x−ξdξ)

This is a very good approximationBut problems arise in the decelerated region (we will see this in the next section).

Second example: Basic case, at Re = 0

Shear flow over a topography f(x, t) at small Reynolds number

Starting from an initial shape, the creeping flow is computed (in the Small PerturbationTheory), we obtain after some algebra:

τ = 1 +2π

∫ ∞

−∞

f ′

x− ξdξ

perturbation of a shear flow Re = 0

-2

-1

0

1

2

3

4

5

-3 -2 -1 0 1 2

Fro

ttem

ent

LineaireCASTEM,h=0.05

CASTEM,h=0.1CASTEM,h=0.2CASTEM,h=0.3CASTEM,h=0.4

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0

0.5

1

1.5

2

2.5

3

3.5

4

-3 -2 -1 0 1 2 3

frot

tem

ent

x

Tau FOURIERTau FREEFEM

L: flow over a gaussian bump, comparisons linear theory/ computationsperturbation of skin friction computed with CASTEM 1

h0

∂u∂y for 0.05 < h0 < 0.4 (bump

size) and Re = 1R: perturbation of skin friction computed with FreeFem.

Linking q and u

assuming that q is proportional to u− 1 or q proportional to τ − 1without threshold this gives the same relation in the two cases (2!):

∂f

∂t= −1

π

∂

∂x

∫f ′

x− ξdξ.

we recognize the linear Benjamin -Ono equation.

Supposed Evolution

The ideal fluid theory has been introduced by Exner.

Issued from Yang (1995) reproduced from Exner (1925?).”wave” inspiration in the dune evolution

Computed Evolution

Numerical resolution: finite differences, explicitTested on complete Benjamin - Ono: RHS+ 4f∂f/∂x gives the soliton 1/(1 + x2)

But here we observe the dispersion of the bump...

-5 0 5 10 15 20 25

x

t=0

t=1

t=2

t=3

t=4

t=5

animation

Remark: linear KDV equation

The linear KDV equation reads ∂f∂t = ∂3f

∂x3 , with selfsimilar solutions, η = xt−1/3:

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0.2

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0.8

1

1.2

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-0.4

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0.2

0.4

0.6

0.8

1

”Mascaret” solution: f(x, t) =∫∞3−1/3η

Ai(ξ)dξ; Airy solution: f(x, t) = t−1/3Ai( η

31/3).

animation

asymptotic solution of L.B.O.

L.B.O.∂f

∂t= −1

π

∂

∂x

∫f ′

x− ξdξ.

Selfsimilar variable η = xt−1/2, self similar solution f(x, t) = t−1/2φ(xt−1/2).

In the Fourier space exp(−ikx) gives, in the RHS, −i|k|kexp(−ikx), so:

−1π

∂

∂x

∫f ′

x− ξdξ ' i

∂2f

∂x2

The self similar problem is approximated by:

−12

(φ(η) + ηφ′(η)) ' iφ′′(η).

whose exact solution is φ(η) = exp(i(η/2)2)

asymptotic solution of L.B.O.

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-1

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0

0.5

1

1.5

-5 0 5 10 15

phi(e

ta)

eta

calculprediction

Plot of the numerical solution t1/2f(x, t) function of xt−1/2

the exact solution of the approximated problem cos(1 + (η/2)2).

Conclusions for those two special cases L.B.O.

When Re = 0 or Re = ∞ (in a simple ideal flow theory), the topography is stable,there is a dispersive wave

There is an approximate solution (non linearities...?)

Stability arises because of the fact that skin friction is ”in phase” with the bump crest

We have to introduce viscous effects, but small

It will change the the fluid response: skin friction will be ”out of phase” (in advance)with the bump crest

Asymptotic solution of the flow over a bump; double deck theory

We guess that viscous effects are important near the wallPerturbation of a shear flow

∂u

∂y|0 = 1 + αFT−1[(3Ai(0))(−ik)1/3FT [f ]] + O(α2).

Asymptotic solution of the flow over a bump;Linear/ Non Linear double deck theory

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0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5

y

x

lin.0.100.501.002.002.252.50

Reduced wall shear (∂u∂y0

− 1)/α

function of x

for the bump αe−πx2

with α = 0.10, α = 0.5, α = 1.0,α = 2, α = 2.25, α = 2.50.The plain curve (”lin.”) is the linear prediction , othercurves come from the non linear numerical solution.

Notice the numerical oscillations in the case of separated

flow (separation is for α > 2.1)

Comparison with Navier Stokes

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1

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Re grand

BosseLineaire

D-DCASTEM, Re=100CASTEM, Re=500

CASTEM, Re=1000

0.1

1

10

1 10 100 1000

hauteur de separation2.1*Re**(-1/3)

good!

conclusion: Perturbation of shear flow is in advance compared to the bump crest.

Other asymptotic solution of the flow over a bump; triple decktheory

d

y

xLower Deck

Main Deck

Upper Deck

Boun

dary

Lay

erId

eal F

luid

Lay

er*

*

U0

f (x ,t )* * *

-A(x)

FT [τ ] = (−ik)2/3

Ai′(0) Ai(0) FT [f ]β∗−βpf

, with β∗ = (3Ai′(0))−1(−ik)1/3

Simplification of mass transport

qAr

-Vf

qAr

-Vf

c

∂

∂xq + V q = β(H(τ − τs − λ

∂f

∂x)(τ − τs − λ

∂f

∂x))γ.

- total flux of convected sediments q (left figure).- threshold effect τs

- ”slope” effect λ∂f∂x

- H: Heaviside function, γ, V , β...

Simple instability mechanism

fHx,tLqHx,tL

fHx,t+Delta tL

Stability analysis

• If pure shear flow λ = 0, V = 0, Re(σ) > for any kλ 6= 0 or V 6= 0 stabilises high frequencies

• Infinite depth case (Hilbert case). The real part of σ for β = V = γ = 1 as functionof the wave length k:- on the left figure λ = 0: there is no slope effect- on the right figure, we focus on the small k which are amplified when λ = 0, butare damped for λ > 0 (following the arrow, from up to down λ = 0, λ = 0.1, λ = 0.2, λ = 0.3, λ = 0.316 and λ = 0.4).

0.2 0.4 0.6 0.8 1

0

0.02

0.04

0.06

k

S

0.1 0.2 0.3 0.4 0.5-0.004-0.002

00.0020.0040.0060.0080.01

k

S

Time Evolution

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0

0.005

0.01

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y

x

’Hilbert’’p=A’’A=0’

The topography at time t = 100 for the three unstable regimes (β = V = γ = 1, λ = 0,Lx = 64 τs = −0.1). At time t = 0, a random noise of level 0.001 was introduced.The spatial frequency kM giving the larger Re(σ) in the band 0 < kM < km has beenselected.

Slope effect: influence of λ

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0

0.5

1

1.5

2

2.5

3

-8 -6 -4 -2 0 2 4 6 8

y

x

l=0.0l=0.1l=0.2

Bump shape t = 500, (4 bumps coexist with β = 1, γ = 1, V = 1, τs = −0.05),λ = 0, λ = 0.1 and λ = 0.2 (the curves are shifted to place the maximum at the origin)

Coarsening process, pure shear flow case

0

10

20

30

40

50

60

70

80

90

100

10 100 1000

y

x

Lx=12.5Lx=25.0Lx=50.0

The wave length 2π/k of themaximum of the bump spectrumversus time, correspondingmostly to the number of bumpspresent in the domain, is plottedas function of time (log scale),here in the case A = 0. As timeincreases, there is less and lessbumps present in the domain,finally a single bump fills it:2π/kfinal = 2Lx. Here,Lx = 12.5, 25 and 50. The longwave are unstable in such a waythat the final length of thebump is the size of thecomputational domain.

Coarsening process, pure shear flow case

10 20 30 40 50 60

2

4

6

8

10

a b

L

h

The maximum of the final bump height hmax plotted as a function of half the domainsize Lx in the case A = 0. The case τs = −0.1, V = 1, λ = 0 is the upper curve. Thelower curves correspond to τs = −0.05, V = 1, the arrow is directed to the increasingvalues of λ (λ = 0, 0.1 and 0.2). The subcritical case gives qualitatively the sameresults.

Coarsening process,Hilbert case

10

20

30

40

50

60

70

10 100 1000 10000 100000

L

t

l=0l=0.2

l=0.31

Examples of long time evolution of 2π/k thewave length value maximizing the bumpspectrum (corresponding mostly to thenumber of bumps present in the domain).This is an infinite depth case for a domain oflength 2Lx. If λ = 0, there is finally only onebump of size 2Lx (the largest possible). Ifλ < 0.316, two bumps (of size Lx) arepresent, the larger are damped. If λ isincreased, there is no dune anymore aspredicted by the linearized theory. HereV = β = 1, Lx = 32, τs = −0.25. Noticethat several bumps may live during a verylong time: here in the case λ = 0.31, during avery long time (10 < t < 25000) three bumpsare present.

Conclusion

avalanches...

Equations sans dimension

q =R2

2E =

R3

3h(x, t) = Z(x, t) + R(x, t)

∂h

∂t+ R

∂R

∂x= 0

∂R

∂t+ R

∂R

∂x= − 1

1 + (∂Z∂x )2

(∂Z

∂x+ µI) + ν

∂2R

∂x2

Cas BRdG 98

h(x, t) = Z(x, t) + R(x, t)∂h

∂t+

∂R

∂x= 0

∂R

∂t+

∂R

∂x= − 1

1 + (∂Z∂x )2

(∂Z

∂x+ µI) + ν

∂2R

∂x2

ActeI

ActeIetII

ActeIetIIetIII

Cas DAD 99 modifie

q =R2

2E =

R3

3h(x, t) = Z(x, t) + R(x, t)

∂h

∂t+ R

∂R

∂x= 0

∂R

∂t+ R

∂R

∂x= − 1

1 + (∂Z∂x )2

(∂Z

∂x+ µI) + ν

∂2R

∂x2

relation µI(R,Z ′, ...)

Exemples• sans frein:

si |∂Z∂x | > µs alors µI = µs

Exemples• sans frein:

si |∂Z∂x | > µs alors µI = µs

• simpleµI = µs

RR+d

si |∂Z∂x | < µs alors µI = −∂Z

∂x

Exemples• sans frein:

si |∂Z∂x | > µs alors µI = µs

• simpleµI = µs

RR+d

si |∂Z∂x | < µs alors µI = −∂Z

∂x

• DAD 99 simp.:si |∂Z

∂x | > µs alors µI = µs

si |∂Z∂x | < µs alors µI = µs − (∂Z

∂x + µs)exp(−R/R0)

Exemples• sans frein:

si |∂Z∂x | > µs alors µI = µs

• simpleµI = µs

RR+d

si |∂Z∂x | < µs alors µI = −∂Z

∂x

• DAD 99 simp.:si |∂Z

∂x | > µs alors µI = µs

si |∂Z∂x | < µs alors µI = µs − (∂Z

∂x + µs)exp(−R/R0)

• DAD 99 modif.:

si |∂Z∂x | > µs alors µI = µs

si |∂Z∂x | < µs alors µI = µs − (∂Z

∂x + µs)exp(− R∂Z∂x +µs

)

un cas lin\’eairedt=.0001tmax=10dx=0.01nx=1000nu=0.05mu=0.3modelmu=4tas=-1theta=0.35K=0Nlin=0v0=1

dt=.001

tmax=350dx=0.01nx=1000nu=0.05mu=0.3modelmu=4tas=-1theta=0.35K=0Nlin=1v0=0

Non Linear velocity: The Acts

Non Linear velocity: The Acts

0

0.05

0.1

0.15

0.2

0 0.5 1 1.5 2 2.5

R

x

acte I

0

0.05

0.1

0.15

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0 0.5 1 1.5 2 2.5

R

x

acte I

0

0.01

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0 5 10 15 20

Q

t

linquad

Linear velocity: The Acts

Linear velocity: The Acts

0

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0 0.5 1 1.5 2 2.5

R

x

acte I

0

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0 0.5 1 1.5 2 2.5

R

x

acte III

0

0.01

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0.03

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0.05

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0.07

0 0.5 1 1.5 2 2.5

R

x

acte II

0

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0.02

0.03

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0.05

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0.07

0 1 2 3 4 5 6 7

R

t

R(L,t)

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1

1.2

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tau

x

Bumpcase A=0

case Hilbert

Figure 1: The linear solution (in the Triple Deck scales) for the perturbationof the wall shear function of x, in the A = 0 case, and in the Hilbert casep = −π−1

∫(x − ξ)−1(−A′)dξ. The bump perturbation is here e−πx2

. The caseA = 0 leads to no upstream influence, the Hilbert case leads to a small upstreaminfluence: the skin friction anticipates the bump. The skin friction is extreme before themaximum of the bump. Skin friction s larger in the wind side than in the lee side.

-1

-0.5

0

0.5

1

1.5

2

2.5

-4 -2 0 2 4

tau

x

Bumpcase p= Acase p=-A

Figure 2: The linear solution (in the Triple Deck scales) for the perturbation of thewall shear function of x, in the p = −A case, and in the p = A. The bump perturbation

is here e−πx2. The case p = A (subcritical) leads to no upstream influence, the case

p = −A (supercritical) leads to a strong upstream influence: the skin friction anticipatesthe bump. The skin friction is extreme before the maximum of the bump only in thefluvial case.

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0

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1

1.2

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y

x

f(x,t), Lb=3Lb=1Lb=2Lb=3Lb=4Lb=5Lb=6

Figure 3: Influence of the initial width Lb of a bump exp(−π(x/Lb)2), the maximumof the bump is plotted for Lb = 1, 2, 3, 4 and 5 for t < 100; f(x, t) is plotted as well(for t = 0, 2, 4, 6, . . . , 100 with Lb = 3). The larger the bump, the smaller its velocity;β = 1, γ = 1, V = 1, λ = 0 and τs = 0.

0

0.2

0.4

0.6

0.8

1

-5 0 5 10 15 20

y

x

f(x,t)

Figure 4: Destruction of a bump exp(−π(x/6)2) in the supercritical regime, withβ = 1, γ = 1, V = 1, λ = 0 and τs = 0; the maximum of the bump is plotted fort < 100, it is moving upstream; f(x, t) is plotted as well (for t = 0, 2, 4, 6, . . . , 100).

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