On the road to Navier-Stokes Yohan Penel Team ANGE (CEREMA, Inria, - - PowerPoint PPT Presentation

On the road to Navier-Stokes

Yohan Penel

Team ANGE (CEREMA, Inria, UPMC, CNRS) Project leader: Cindy Guichard (UPMC)

COmplex Rheology SURface Flows

Carg` ese – June, 1st. 2017

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Outline

1

Project

2

Settings

3

Derivation of the hierarchy of models

4

Rheology

5

Conclusion

Yohan Penel (ANGE) CORSURF

• 2 / 25

/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Description

Project leader: Cindy Guichard Funding: 8000e Members: ❧ Marie-Odile Bristeau (Inria Paris) ❧ Enrique Fern´ andez-Nieto (Spain, Univ. Sevilla) ❧ Anne Mangeney (IPGP) ❧ Bernard di Martino (Univ. Corse, ANGE) ❧ Martin Parisot (Inria) ❧ Yohan Penel (CEREMA) ❧ Jacques Sainte-Marie (CEREMA)

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Achievements

Workshop: ❧ Complex rheology of granular flows: barriers, challenges and deadlocks ❧ October, 13th-14th 2016 ❧ 30 participants, 4 speakers (C. Ancey, E. Lemaire, G. Ovarlez, J. Weiss) Articles: M.-O. Bristeau, C. Guichard, B. di Martino, J. Sainte-Marie, Layer-averaged Euler and Navier-Stokes equations (Comm. Math. Sci., to appear)

• E. Fern´

andez-Nieto, M. Parisot, Y. Penel, J. Sainte-Marie, A hierarchy of non-hydrostatic layer-averaged approximations of Euler equations for free surface flows (submitted)

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Literature about free-surface flows

Free-surface incompressible Euler equations

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Literature about free-surface flows

Free-surface incompressible Euler equations Shallow water equations Hydrostatic pressure Homogeneous velocity

Shallow water assumption

• A. Barr´

e de Saint-Venant, Th´ eorie du mouvement non permanent des eaux, avec application aux crues des rivi` eres et ` a l’introduction des mar´ ees dans leurs lits (C. R. Acad. Sci. 73, 1871) J.-F. Gerbeau, B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation (Discrete Contin. Dyn. Syst. Ser. B 1(1), 2001)

• S. Ferrari, F. Saleri, A new two-dimensional Shallow Water model including pressure effects and slow varying bottom topography

(Math. Model. Numer. Anal. 38(2), 2004)

• F. Marche, Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects

(Eur. J. Mech. B Fluids 26(1), 2007) Yohan Penel (ANGE) CORSURF

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/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Literature about free-surface flows

Free-surface incompressible Euler equations Shallow water equations

❤❤❤❤❤❤❤❤ ❤

Hydrostatic pressure Homogeneous velocity

Shallow water assumption

• F. Serre, Contribution `

a l’´ etude des ´ ecoulements permanents et variables dans les canaux (La Houille Blanche 6, 1953) A.E. Green, P.M. Naghdi, A derivation of equations for wave propagation in water of variable depth (J. Fluid Mech. 78(2), 1976) M.-O. Bristeau, J. Sainte-Marie, Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems (Discrete Contin. Dyn. Syst. Ser. B 10(4), 2008)

• D. Lannes, P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation (Phys. Fluids

21(1), 2009) Peregrine ’67, Madsen et al. ’91 ’96 ’03 ’06, Nwogu ’93, Casulli et al. ’95 ’99, Yamazaki et al. ’09, . . . Yohan Penel (ANGE) CORSURF

• 5 / 25

/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Literature about free-surface flows

Free-surface incompressible Euler equations Shallow water equations Hydrostatic pressure

❤❤❤❤❤❤❤❤ ❤

Homogeneous velocity

❤❤❤❤❤❤❤❤❤ ❤

Shallow water assumption

• E. Audusse, M.-O. Bristeau, B. Perthame, J. Sainte-Marie, A multilayer Saint-Venant system with mass exchanges for Shallow Water flows.

Derivation and numerical validation (Math. Model. Numer. Anal. 45(1), 2011)

• F. Bouchut, V. Zeitlin, A robust well-balanced scheme for multi-layer shallow water equations (Discrete Contin. Dyn. Syst. Ser. B 13(4), 2010)

E.D. Fern´ andez-Nieto, E.H. Kon´ e, T. Morales de Luna, R. B¨ urger, A multilayer shallow water system for polydisperse sedimentation (J. Comput.

• Phys. 238, 2013)

Castro et al. ’01 ’04 ’10, Narbona et al. ’09 ’13, . . . Yohan Penel (ANGE) CORSURF

• 5 / 25

/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Literature about free-surface flows

Free-surface incompressible Euler equations Shallow water equations

❤❤❤❤❤❤❤❤ ❤

Hydrostatic pressure

❤❤❤❤❤❤❤❤ ❤

Homogeneous velocity

❤❤❤❤❤❤❤❤❤ ❤

Shallow water assumption

Derivation of multilayer non-hydrostatic models

• M. Zijlema, G.S. Stelling, Further experiences with computing non-hydrostatic free-surface flows involving water waves (Int. J. Numer. Methods

Fluids 48(2), 2005)

• Y. Bai, K.F. Cheung, Dispersion and nonlinearity of multi-layer non-hydrostatic free-surface flow (J. Fluid Mech. 726, 2013)

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Fluid domain

z = zb(x) z = η(t, x) H(t, x) u = (u, w) x z

Water height: H(t, x) = η(t, x) − zb(x)

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Euler equations

Model      ∂xu + ∂zw = 0 ∂tu + ∂x(u2 + p) + ∂z(uw) = 0 ∂tw + ∂x(uw) + ∂z(w 2 + p) = −g set in the domain Ω(t) =

• (x, z) ∈ R2

zb(x) ≤ z ≤ η(t, x)

• Boundary conditions

∂tη(t, x) + u

• t, x, η(t, x)
• ∂xη(t, x) − w
• t, x, η(t, x)
• = 0

p

• t, x, η(t, x)
• = patm(t, x)

u

• t, x, zb(x)
• z′

b(x) − w

• t, x, zb(x)
• = 0

together with well-prepared initial conditions Pressure fields p(t, x, z) = patm(t, x) + g

• η(t, x) − z
• + q(t, x, z)

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Euler equations

Model      ∂xu + ∂zw = 0 ∂tu + ∂x(u2 + q) + ∂z(uw) = −∂x(gη + patm) ∂tw + ∂x(uw) + ∂z(w 2 + q) = 0 set in the domain Ω(t) =

• (x, z) ∈ R2

zb(x) ≤ z ≤ η(t, x)

• Boundary conditions

∂tη(t, x) + u

• t, x, η(t, x)
• ∂xη(t, x) − w
• t, x, η(t, x)
• = 0

q

• t, x, η(t, x)
• = 0

u

• t, x, zb(x)
• z′

b(x) − w

• t, x, zb(x)
• = 0

together with well-prepared initial conditions

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Multilayer framework

z = zb(x) = z1/2(x) z = η(t, x) = zL+1/2(t, x) z = zα−1/2(t, x) z = zα+1/2(t, x) H(t, x) hα(t, x) x z

Height decomposition: hα(t, x) = ℓαH(t, x) with ℓα ∈ (0, 1) and L

α=1 ℓα = 1

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Multilayer framework

z = zb(x) = z1/2(x) z = η(t, x) = zL+1/2(t, x) z = zα−1/2(t, x) z = zα+1/2(t, x) H(t, x) hα(t, x) x z

Height decomposition: hα(t, x) = ℓαH(t, x) with ℓα ∈ (0, 1) and L

α=1 ℓα = 1

Homogeneous mesh: ℓα = 1 L

Yohan Penel (ANGE) CORSURF

• 8 / 25

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Multilayer framework

z = zb(x) = z1/2(x) z = η(t, x) = zL+1/2(t, x) z = zα−1/2(t, x) z = zα+1/2(t, x) H(t, x) hα(t, x) x z

Notations [ [f ] ]α+1/2 = f +

α+1/2 − f − α+1/2,

• fα+1/2 = γα+1/2f −

α+1/2 + (1 − γα+1/2)f + α+1/2

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Multilayer framework

z = zb(x) = z1/2(x) z = η(t, x) = zL+1/2(t, x) z = zα−1/2(t, x) z = zα+1/2(t, x) H(t, x) hα(t, x) x z

Notations f α(t, x) = 1 hα(t, x) zα+1/2(t,x)

zα−1/2(t,x)

f (t, x, z) dz

Yohan Penel (ANGE) CORSURF

• 8 / 25

/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Multilayer framework

z = zb(x) = z1/2(x) z = η(t, x) = zL+1/2(t, x) z = zα−1/2(t, x) z = zα+1/2(t, x) H(t, x) hα(t, x) x z

Notations nα+1/2 = (−∂xzα+1/2, 1)T

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Discontinuous Galerkin framework

Let us be given a velocity field satisfying [ [u] ]α+1/2 · nα+1/2 = 0 ⇐ ⇒ [ [w] ]α+1/2 = [ [u] ]α+1/2∂xzα+1/2 Toy model ∂tR + ∂x(uR + P) + ∂z(wR + Q) = S (1) where R, P, Q and S take values in Rp Semi-discrete formulation over each layer Lα = (zα+1/2, zα−1/2) ∂t(hαRα) + ∂x(hα[uRα + Pα]) + F R

α+1/2 − F R α−1/2 = hαS α

where F R

α+1/2 = Γα+1/2

Rα+1/2 − Pα+1/2∂xzα+1/2 + Qα+1/2 Γα+1/2 = wα−1/2∂tzα+1/2 − uα+1/2∂xzα+1/2

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Spaces of approximation

qα−1/2, uα−1/2, wα−1/2 qα+1/2, uα+1/2, wα+1/2 z = zα−1/2 z = zα z = zα+1/2 w+

α−1/2

w−

α+1/2

wα uα, qα

u(t, x) =

L

• α=1

uα(t, x)1{Lα(t,x)}(z) + EL w(t, x) =

L

• α=1
• wα(t, x) −
• z − zα(t, x)
• ∂xuα(t, x)
• 1{Lα(t,x)}(z) + E′

L

q continuous over the water column

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Core of the models

Applying the previous semi-discretisation to the Euler equations leads to        ∂thα + ∂x(hαuα) + Γα+1/2 − Γα−1/2 = 0 ∂t(hαuα) + ∂x

• hαu2

α + hαqα

• + Uα+1/2 − Uα−1/2 = −hα∂x(gη + patm)

∂t(hαwα) + ∂x (hαuαwα) + Wα+1/2 − Wα−1/2 = 0 with    Uα+1/2 = uα+1/2Γα+1/2 − ∂xzα+1/2qα+1/2 Wα+1/2 = wα+1/2Γα+1/2 + qα+1/2 and Γα+1/2 = wα+1/2 − ∂tzα+1/2 − uα+1/2∂xzα+1/2

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Core of the models

Applying the previous semi-discretisation to the Euler equations leads to        ∂thα + ∂x(hαuα) + Γα+1/2 − Γα−1/2 = 0 ∂t(hαuα) + ∂x

• hαu2

α + hαqα

• + Uα+1/2 − Uα−1/2 = −hα∂x(gη + patm)

∂t(hαwα) + ∂x (hαuαwα) + Wα+1/2 − Wα−1/2 = 0 with    Uα+1/2 = uα+1/2Γα+1/2 − ∂xzα+1/2qα+1/2 Wα+1/2 = wα+1/2Γα+1/2 + qα+1/2 and Γα+1/2 =

L

• β=α+1

∂x (hβ [uβ − u]) where u =

L

• α=1

ℓαuα

Yohan Penel (ANGE) CORSURF

• 11 / 25

/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Core of the models

Applying the previous semi-discretisation to the Euler equations leads to        ∂tH + ∂x (Hu) = 0 ∂t(hαuα) + ∂x

• hαu2

α + hαqα

• + Uα+1/2 − Uα−1/2 = −hα∂x(gη + patm)

∂t(hαwα) + ∂x (hαuαwα) + Wα+1/2 − Wα−1/2 = 0 with    Uα+1/2 = uα+1/2Γα+1/2 − ∂xzα+1/2qα+1/2 Wα+1/2 = wα+1/2Γα+1/2 + qα+1/2 and Γα+1/2 =

L

• β=α+1

∂x (hβ [uβ − u]) where u =

L

• α=1

ℓαuα

Yohan Penel (ANGE) CORSURF

• 11 / 25

/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Requirements for the P1 choice

Additional equation ∂t(zw) + ∂x(zuw) + ∂z

• z(w 2 + q)
• = w 2 + q.

which is discretised as ∂t(hαzwα) + ∂x(hαuαzwα) + F zw

α+1/2 − F zw α−1/2 = hα

• w 2 + q
• α

Yohan Penel (ANGE) CORSURF

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Requirements for the P1 choice

Additional equation ∂t(zw) + ∂x(zuw) + ∂z

• z(w 2 + q)
• = w 2 + q.

which is discretised as ∂t(hαzwα) + ∂x(hαuαzwα) + F zw

α+1/2 − F zw α−1/2 = hα

• w 2 + q
• α

Let us introduce the signed standard deviation σα = − hα∂xuα

2 √ 3

such that

• w 2

α = w 2 α + σ2 α,

zwα = zαwα + hασα 2 √ 3

Yohan Penel (ANGE) CORSURF

• 12 / 25

/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Requirements for the P1 choice

Additional equation ∂t(zw) + ∂x(zuw) + ∂z

• z(w 2 + q)
• = w 2 + q.

which is discretised as ∂t(hαzwα) + ∂x(hαuαzwα) + F zw

α+1/2 − F zw α−1/2 = hα

• w 2 + q
• α

Then ∂t(hαzαwα) + ∂x(hαzαuαwα) + ∂t h2

ασα

2 √ 3

• + ∂x

h2

ασαuα

2 √ 3

• + zα+1/2(

wα+1/2Γα+1/2 + qα+1/2) − zα−1/2( wα−1/2Γα−1/2 + qα−1/2) = hα

• w 2

α + σ2 α + qα

• Yohan Penel (ANGE)

CORSURF

• 12 / 25

/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Requirements for the P1 choice

Additional equation ∂t(zw) + ∂x(zuw) + ∂z

• z(w 2 + q)
• = w 2 + q.

which is discretised as ∂t(hαzwα) + ∂x(hαuαzwα) + F zw

α+1/2 − F zw α−1/2 = hα

• w 2 + q
• α

Then ∂t(hασα) + ∂x(hασαuα) = 2 √ 3

• qα − qα+1/2 + qα−1/2

2 − Γα+1/2 hα∂xuα 12 + wα+1/2 − wα 2

• +Γα−1/2

hα∂xuα 12 + wα − wα−1/2 2

• Yohan Penel (ANGE)

CORSURF

• 12 / 25

/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Requirements for the P1 choice

Additional equation ∂t(zw) + ∂x(zuw) + ∂z

• z(w 2 + q)
• = w 2 + q.

which is discretised as ∂t(hαzwα) + ∂x(hαuαzwα) + F zw

α+1/2 − F zw α−1/2 = hα

• w 2 + q
• α

Rather using a Hermite interpolation leads to zw|Lα ≈ zαwα + (z − zα)(wα − zα∂xuα), w 2

|Lα ≈ w 2 α − 2(z − zα)wα∂xuα

Yohan Penel (ANGE) CORSURF

• 12 / 25

/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Requirements for the P1 choice

Additional equation ∂t(zw) + ∂x(zuw) + ∂z

• z(w 2 + q)
• = w 2 + q.

which is discretised as ∂t(hαzwα) + ∂x(hαuαzwα) + F zw

α+1/2 − F zw α−1/2 = hα

• w 2 + q
• α

Then ∂t(hαzαwα) + ∂x(hαzαuαwα) + zα+1/2qα+1/2 − zα−1/2qα−1/2 + Γα+1/2

• zα+1/2

wα+1/2 + H2 4L2 (∂xu)α+1/2

• − Γα−1/2
• zα−1/2

wα−1/2 + H2 4L2 (∂xu)α−1/2

• = hα
• w 2

α + qα

• Yohan Penel (ANGE)

CORSURF

• 12 / 25

/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Requirements for the P1 choice

Additional equation ∂t(zw) + ∂x(zuw) + ∂z

• z(w 2 + q)
• = w 2 + q.

which is discretised as ∂t(hαzwα) + ∂x(hαuαzwα) + F zw

α+1/2 − F zw α−1/2 = hα

• w 2 + q
• α

Then qα = qα+1/2 + qα−1/2 2 + Γα+1/2 H 4L

• (∂xu)α+1/2 +

wα+1/2 − wα 2

• − Γα−1/2

H 4L

• (∂xu)α−1/2 + wα −

wα−1/2 2

• Yohan Penel (ANGE)

CORSURF

• 12 / 25

/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

NHML2 model: u|Lα ∈ P0, w|Lα ∈ P1, q|Lα ∈ P2, E|Lα ∈ P2

∂tH + ∂x (Hu) = 0 ∂t(hαuα) + ∂x

• hαu2

α + hαqα

• + Uα+1/2 − Uα−1/2 = −hα∂x(gη + patm)

∂t(hαwα) + ∂x (hαuαwα) + Wα+1/2 − Wα−1/2 = 0 ∂t(hασα) + ∂x(hασαuα) = 2 √ 3

• qα − qα+1/2 + qα−1/2

2 −Γα+1/2 hα∂xuα 12 + wα+1/2 − wα 2

• + Γα−1/2

hα∂xuα 12 + wα − wα−1/2 2

• ∂xuα +

w −

α+1/2 − wα

hα/2 = 0 σα = −hα∂xuα 2 √ 3 w +

α−1/2 − uα∂xzα−1/2 + α−1

• β=1

∂x(hβuβ) = 0

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• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

NHML1 model: u|Lα ∈ P0, w|Lα ∈ P1, q|Lα ∈ P2, E|Lα ∈ P0

∂tH + ∂x (Hu) = 0 ∂t(hαuα) + ∂x

• hαu2

α + hαqα

• + Uα+1/2 − Uα−1/2 = −hα∂x(gη + patm)

∂t(hαwα) + ∂x (hαuαwα) + Wα+1/2 − Wα−1/2 = 0 qα = qα+1/2 + qα−1/2 2 + Γα+1/2 H 4L

• (∂xu)α+1/2 +

wα+1/2 − wα 2

• − Γα−1/2

H 4L

• (∂xu)α−1/2 + wα −

wα−1/2 2

• ∂xuα +

w −

α+1/2 − wα

hα/2 = 0 σα = −hα∂xuα 2 √ 3 w +

α−1/2 − uα∂xzα−1/2 + α−1

• β=1

∂x(hβuβ) = 0

Yohan Penel (ANGE) CORSURF

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/ / :

• 1. Project
• 2. Settings
• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

NHML0 model: u|Lα ∈ P0, w|Lα ∈ P0, q|Lα ∈ P1, E|Lα ∈ P0

∂tH + ∂x (Hu) = 0 ∂t(hαuα) + ∂x

• hαu2

α + hαqα

• + Uα+1/2 − Uα−1/2 = −hα∂x(gη + patm)

∂t(hαwα) + ∂x (hαuαwα) + Wα+1/2 − Wα−1/2 = 0 qα = qα+1/2 + qα−1/2 2 wα − uα∂xzα +

α−1

• β=1

∂x(hβuβ) + 1 2∂x(hαuα) = 0

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• 4. Rheology
• 5. Conclusion

Energy inequality

Denoting K = u2+w 2

2

, smooth solutions to the Euler equations satisfy ∂t η

zb

• K + g η + zb

2 + patm

• dz
• + ∂x

η

zb

u(K + q + gη + patm) dz

• = H∂tpatm.

Proposition Let us assume that

• γα+1/2 − 1

2

• Γα+1/2 ≥ 0. If (H, uα, wα, qα) are smooth

solutions to the NHML2-model, then with Kα = u2

α+w 2 α+σ2 α

2

∂t L

• α=1

hα

• Kα + gzα + patm
• + ∂x

L

• α=1

hαuα

• Kα + qα + gη + patm
• ≤ H∂tpatm.

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• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Comment

Constraint

• γα+1/2 − 1

2

• Γα+1/2 ≥ 0 is equivalent to taking

γα+1/2 = 1 2

• 1 + λ sign(Γα+1/2)
• for any λ ≥ 0, which gives
• Rα+1/2Γα+1/2 =

R+

α+1/2 + R− α+1/2

2 Γα+1/2 − λ 2 |Γα+1/2|

• R+

α+1/2 − R− α+1/2

• .

The energy inequality is satisfied in particular for γα+1/2 =

1 2 (λ = 0) and for

γα+1/2 = 1{Γα+1/2≥0} (λ = 1).

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• 4. Rheology
• 5. Conclusion

Linear dispersion relation

Let us linearise around the so-called lake-at-rest steady state (H0, 0, 0, 0). Proposition There exists a plane wave solution

• ˆ

H, ˆ uα, ˆ wα, ˆ qα

• ei(kx−ωt) to the linearised

NHMLk system provided the following dispersion relation holds c2

L(kH0) =

ω2 k2gH0 = PL(kH0) QL(kH0) where PL and QL are explicit polynomials. Moreover when the number of layers L goes to infinity, c2

L tends to

c2

Airy(kH0) = tanh(kH0)

kH0 .

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• 4. Rheology
• 5. Conclusion

NHML2 for L ∈ {1, 2, 3}

L PL QL 1 1 1 + x2

3

2 1 + x2

12

1 + 5x2

12 + 7x4 576

3 1 + x2

9 + 5x4 2916

1 + 4x2

9 + 19x4 972 + 13x6 78732

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• 4. Rheology
• 5. Conclusion

NHML1 for L ∈ {1, 2, 3}

L PL QL 1 1 1 + x2

4

2 1 + x2

16

1 + 3x2

8 + x4 256

3 1 + 5x2

54 + x4 1296

1 + 5x2

12 + 5x4 432 + x6 46656

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• 4. Rheology
• 5. Conclusion

L ≥ 4 (λ = 3 for NHML2 and λ = 2 for NHML1/0)

PL(x) = 1 L

• 1 −

x2 2λL2 L−1 + ξL−4

• 1 −

x2 2λL2 2 − ξL−3

• 1 + 2λ − 1

2λ x2 L2

• QL(x) =
• 1 −

x2 2λL2 L−1 1 + λ − 1 2λ x2 L2

• +
• 1 −

x2 2λL2 2 x2ξL−4 2L2 −

• 3 + 2λ − 3

2λ x2 L2 x2ξL−3 2L2 ξk = L2 x2

• 1 −

x2 2λL2 k+2 + Ξe

• 0≤2m≤k

k 2m 1 + λ − 1 2λ x2 L2 k−2m x2m−1 L2m−1

• 1 + λ − 2

4λ x2 L2 m + Ξo

• 0≤2m+1≤k
• k

2m + 1 1 + λ − 1 2λ x2 L2 k−2m−1 x2m+1 L2m+1

• 1 + λ − 2

4λ x2 L2 m where Ξe = −1+ 1−3λ

λ x2 L2 + −1+6λ−4λ2 4λ2 x4 L4 and Ξo = − 5 2 + 5(1−λ) 2λ x2 L2 + − 5

2 +5λ−2λ2

4λ2 x4 L4 .

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• 3. Derivation of the hierarchy of models
• 4. Rheology
• 5. Conclusion

Phase velocity

0.97 0.98 0.99 1.00 1.01 1.02 1.03

c M

φ

c E

φ

(E) Euler (SW) Shallow Water (LDNH2) (LDNH0,1)

10-2 10-1 100 101 102 103 104

kH0

10-15 10-13 10-11 10-9 10-7 10-5 10-3 10-1 101

|c M

φ

−c E

φ |

c E

φ

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• 5. Conclusion

Navier-Stokes for a hydrostatic flow

Model                ∂xu + ∂zw = 0 ∂tu + ∂x(u2) + ∂z(uw) + ∂xp = ∂xΣxx + ∂zΣxz ∂zp = −g + ∂xΣzx + ∂zΣzz

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• 4. Rheology
• 5. Conclusion

Navier-Stokes for a hydrostatic flow

Model                  ∂xu + ∂zw = 0 ∂tu + ∂x(u2) + ∂z(uw) + g∂xη = ∂xΣxx + ∂zΣxz + ∂2

xx

η(t,x)

z

Σzx

• − ∂xΣzz

p = g(η(t, x) − z) − ∂x η(t,x)

z

Σzx

• + Σzz

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• 4. Rheology
• 5. Conclusion

Navier-Stokes for a hydrostatic flow

Model                  ∂xu + ∂zw = 0 ∂tu + ∂x(u2) + ∂z(uw) + g∂xη = ∂xΣxx + ∂zΣxz + ∂2

xx

η(t,x)

z

Σzx

• − ∂xΣzz

p = g(η(t, x) − z) − ∂x η(t,x)

z

Σzx

• + Σzz

Layer-averaging              ∂tH + ∂x(Hu) = 0 ∂t(hαuα) + ∂x

• hαu2

α + g h2

α

2ℓα

• +

uα+1/2Γα+1/2 − uα−1/2Γα−1/2 = −ghα∂xzb + zα+1/2

zα−1/2

• ∂xΣxx + ∂zΣxz + ∂2

xx

η(t,x)

z

Σzx

• − ∂xΣzz
• Yohan Penel (ANGE)

CORSURF

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• 4. Rheology
• 5. Conclusion

Discretisation of viscous terms

zα+1/2

zα−1/2

• ∂xΣxx − ∂xΣzz + ∂2

xx

η(t,x)

z

Σzx

• + ∂zΣxz
• ≈

∂x

• hα(Σxx,α − Σzz,α)
• − ∂xzα+1/2(Σxx,α+1/2 − Σzz,α+1/2)

+ ∂xzα−1/2(Σxx,α−1/2 − Σzz,α−1/2) + ∂2

xx(hαzαΣzx,α) + zα+1/2∂2 xx

 

L

• β=α+1

hβΣzx,β   − Σzx,α+1/2∂2

xxzα+1/2

− zα−1/2∂2

xx

 

L

• β=α

hβΣzx,β   + Σzx,α−1/2∂2

xxzα−1/2

+ Σxz,α+1/2 − Σxz,α−1/2

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• 4. Rheology
• 5. Conclusion

Conclusion

❧ Derivation of a class of multilayer non-hydrostatic models as semi-discretisations of the Euler equations ❧ Analysis of physical properties (energy, hydrodynamic balances, dispersive effects) ❧ Study of the discretisation of viscous terms in accordance with energy estimates ❧ On-going works

➠ Numerical strategies for each model ➠ Incorporation of viscous effects in the non-hydrostatic framework

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Thank you for your attention . . . . . . and see you in November at Inria Paris

Workshop: An overview on free surface flows