a pressurized model for compressible pipe flows: derivation including friction. · 2014-06-13 · a...
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![Page 1: A pressurized model for compressible pipe flows: derivation including friction. · 2014-06-13 · A pressurized model for compressible pipe ows: derivation including friction. M](https://reader034.vdocument.in/reader034/viewer/2022042202/5ea2ad7ca194c26c2b4af559/html5/thumbnails/1.jpg)
A pressurized model for compressiblepipe flows: derivation including friction.
M. Ersoy, IMATH, ToulonMTM workshop
Bilbao,June 12-13, 2014
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Outline of the talkOutline of the talk
1 Physical background, Mathematical motivation andprevious works
2 Derivation of the model including friction
3 Numerical experiment and concluding remarks
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 2 / 23
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OutlineOutline
1 Physical background, Mathematical motivation andprevious works
2 Derivation of the model including friction
3 Numerical experiment and concluding remarks
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 3 / 23
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Pressurized flows : overviewSimulation of pressurized flows
plays an important role in many engineering applications such asI storm sewersI wasteI or supply pipes in hydroelectric installations, . . . .
“geyser” effect −→ pressure can reach severe values and may causeirreversible damage !
requiring efficient mathematical models and accurate numerical schemes
(a) Orange-Fish tunnel (b) Sewers . . . in Paris (c) Forced pipe
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 4 / 23
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Pressurized flows : overviewSimulation of pressurized flows
plays an important role in many engineering applications such asI storm sewersI wasteI or supply pipes in hydroelectric installations, . . . .
“geyser” effect −→ pressure can reach severe values and may causeirreversible damage !
requiring efficient mathematical models and accurate numerical schemes
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 4 / 23
![Page 6: A pressurized model for compressible pipe flows: derivation including friction. · 2014-06-13 · A pressurized model for compressible pipe ows: derivation including friction. M](https://reader034.vdocument.in/reader034/viewer/2022042202/5ea2ad7ca194c26c2b4af559/html5/thumbnails/6.jpg)
Pressurized flows : overviewSimulation of pressurized flows
plays an important role in many engineering applications such asI storm sewersI wasteI or supply pipes in hydroelectric installations, . . . .
“geyser” effect −→ pressure can reach severe values and may causeirreversible damage !
requiring efficient mathematical models and accurate numerical schemes
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 4 / 23
![Page 7: A pressurized model for compressible pipe flows: derivation including friction. · 2014-06-13 · A pressurized model for compressible pipe ows: derivation including friction. M](https://reader034.vdocument.in/reader034/viewer/2022042202/5ea2ad7ca194c26c2b4af559/html5/thumbnails/7.jpg)
Pipe friction : definition and applications
Friction law
F (u) = −k(uτ )uτ , uτ : tangential fluid flow
tangential constraint
σ(u)n · τ = ρk(uτ )uτ , ρ : density, σ : total stress tensor
k can be writtenk(uτ ) = Cl + Ct|uτ |.
Cl and Ct are the so-called friction factor given byI empirical laws depending
F on the fluid flow : laminar, transient, turbulentF on the material (roughness, geometry, hydraulic radius, . . . )
I approximated and not always applicable
Hydraulic engineering applications : canal, irrigation, dam-break, sedimenttransport, geyser, energy loss, failure pumping, fluid blockage, boundary layer,. . .
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 5 / 23
![Page 8: A pressurized model for compressible pipe flows: derivation including friction. · 2014-06-13 · A pressurized model for compressible pipe ows: derivation including friction. M](https://reader034.vdocument.in/reader034/viewer/2022042202/5ea2ad7ca194c26c2b4af559/html5/thumbnails/8.jpg)
Pipe friction : definition and applications
Friction law
F (u) = −k(uτ )uτ , uτ : tangential fluid flow
tangential constraint
σ(u)n · τ = ρk(uτ )uτ , ρ : density, σ : total stress tensor
k can be writtenk(uτ ) = Cl + Ct|uτ |.
Cl and Ct are the so-called friction factor given by
I empirical laws dependingF on the fluid flow : laminar, transient, turbulentF on the material (roughness, geometry, hydraulic radius, . . . )
I approximated and not always applicable
Hydraulic engineering applications : canal, irrigation, dam-break, sedimenttransport, geyser, energy loss, failure pumping, fluid blockage, boundary layer,. . .
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 5 / 23
![Page 9: A pressurized model for compressible pipe flows: derivation including friction. · 2014-06-13 · A pressurized model for compressible pipe ows: derivation including friction. M](https://reader034.vdocument.in/reader034/viewer/2022042202/5ea2ad7ca194c26c2b4af559/html5/thumbnails/9.jpg)
Pipe friction : definition and applications
Friction law
F (u) = −k(uτ )uτ , uτ : tangential fluid flow
tangential constraint
σ(u)n · τ = ρk(uτ )uτ , ρ : density, σ : total stress tensor
k can be writtenk(uτ ) = Cl + Ct|uτ |.
Cl and Ct are the so-called friction factor given byI empirical laws depending
F on the fluid flow : laminar, transient, turbulentF on the material (roughness, geometry, hydraulic radius, . . . )
I approximated and not always applicable
Hydraulic engineering applications : canal, irrigation, dam-break, sedimenttransport, geyser, energy loss, failure pumping, fluid blockage, boundary layer,. . .
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 5 / 23
![Page 10: A pressurized model for compressible pipe flows: derivation including friction. · 2014-06-13 · A pressurized model for compressible pipe ows: derivation including friction. M](https://reader034.vdocument.in/reader034/viewer/2022042202/5ea2ad7ca194c26c2b4af559/html5/thumbnails/10.jpg)
Pipe friction : definition and applications
Friction law
F (u) = −k(uτ )uτ , uτ : tangential fluid flow
tangential constraint
σ(u)n · τ = ρk(uτ )uτ , ρ : density, σ : total stress tensor
k can be writtenk(uτ ) = Cl + Ct|uτ |.
Cl and Ct are the so-called friction factor given byI empirical laws depending
F on the fluid flow : laminar, transient, turbulentF on the material (roughness, geometry, hydraulic radius, . . . )
I approximated and not always applicable
Hydraulic engineering applications : canal, irrigation, dam-break, sedimenttransport, geyser, energy loss, failure pumping, fluid blockage, boundary layer,. . .
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 5 / 23
![Page 11: A pressurized model for compressible pipe flows: derivation including friction. · 2014-06-13 · A pressurized model for compressible pipe ows: derivation including friction. M](https://reader034.vdocument.in/reader034/viewer/2022042202/5ea2ad7ca194c26c2b4af559/html5/thumbnails/11.jpg)
Pipe friction : definition and applications
Friction law
F (u) = −k(uτ )uτ , uτ : tangential fluid flow
tangential constraint
σ(u)n · τ = ρk(uτ )uτ , ρ : density, σ : total stress tensor
k can be writtenk(uτ ) = Cl + Ct|uτ |.
Cl and Ct are the so-called friction factor given byI empirical laws depending
F on the fluid flow : laminar, transient, turbulentF on the material (roughness, geometry, hydraulic radius, . . . )
I approximated and not always applicable
Hydraulic engineering applications : canal, irrigation, dam-break, sedimenttransport, geyser, energy loss, failure pumping, fluid blockage, boundary layer,. . .
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 5 / 23
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How to determine the friction factorThe friction factor is called Fanning friction factor (whenever it is related to theshear stress) or Darcy friction factor (whenever it is related to the head loss = 4×Fanning friction factor) and
C = C(Re, δ, Rh, . . .) .
Examples :
laminar flows
I Cl =C0
Retransient flows
I Colebrook (1939) formula :1√C
= −2 log10
(δ
αRh+
β
Re√C
)I or approximated Colebrook formula : Blasius, Haaland, Swamee-Jain,. . .
turbulent flows
I Chezy (1776), Manning (1891), Strickler (1923) : Ct =1
K2sRh(S(x))4/3
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 6 / 23
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How to determine the friction factorThe friction factor is called Fanning friction factor (whenever it is related to theshear stress) or Darcy friction factor (whenever it is related to the head loss = 4×Fanning friction factor) and
C = C(Re, δ, Rh, . . .) .
Examples :
laminar flows
I Cl =C0
Re
transient flows
I Colebrook (1939) formula :1√C
= −2 log10
(δ
αRh+
β
Re√C
)I or approximated Colebrook formula : Blasius, Haaland, Swamee-Jain,. . .
turbulent flows
I Chezy (1776), Manning (1891), Strickler (1923) : Ct =1
K2sRh(S(x))4/3
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 6 / 23
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How to determine the friction factorThe friction factor is called Fanning friction factor (whenever it is related to theshear stress) or Darcy friction factor (whenever it is related to the head loss = 4×Fanning friction factor) and
C = C(Re, δ, Rh, . . .) .
Examples :
laminar flows
I Cl =C0
Retransient flows
I Colebrook (1939) formula :1√C
= −2 log10
(δ
αRh+
β
Re√C
)I or approximated Colebrook formula : Blasius, Haaland, Swamee-Jain,. . .
turbulent flows
I Chezy (1776), Manning (1891), Strickler (1923) : Ct =1
K2sRh(S(x))4/3
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 6 / 23
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How to determine the friction factorThe friction factor is called Fanning friction factor (whenever it is related to theshear stress) or Darcy friction factor (whenever it is related to the head loss = 4×Fanning friction factor) and
C = C(Re, δ, Rh, . . .) .
Examples :
laminar flows
I Cl =C0
Retransient flows
I Colebrook (1939) formula :1√C
= −2 log10
(δ
αRh+
β
Re√C
)I or approximated Colebrook formula : Blasius, Haaland, Swamee-Jain,. . .
turbulent flows
I Chezy (1776), Manning (1891), Strickler (1923) : Ct =1
K2sRh(S(x))4/3
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 6 / 23
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How to determine the friction factorThe friction factor is called Fanning friction factor (whenever it is related to theshear stress) or Darcy friction factor (whenever it is related to the head loss = 4×Fanning friction factor) and
C = C(Re, δ, Rh, . . .) .
These coefficients are determined through the Moody diagram.
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 6 / 23
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Schematic : circular pipe
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 7 / 23
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Mathematical motivations : thin-layer approximationMathematical motivations
to reduce Viscous Compressible 3D NS p(ρ) = cρ → inviscid compressible 1DSW-like model
to obtain the “motion by slices” through a Neumann problem
I u(t, x, y, z) = u(t, x) + ˜u(t, x, y, z),
I
∫Ω
˜u(t, x, y, z) dy dz = 0,
I ˜u(t, x, y, z) = O(ε) where ε is the aspect-ratio.
I u(t, x)2 ≈ u(t, x)2
to include the friction with its geometrical dependency as well as othergeometrical source terms
general barotropic law p(ρ) = cργ , γ 6= 1
ργ ≈ ργ
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) :339–365, 2001.
C. Bourdarias, M. Ersoy, S. Gerbi,
A model for unsteady mixed flows in non uniform closed water pipes : a Full Kinetic Approach.Accepted in Numerische Mathematik, 2014.
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 8 / 23
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Mathematical motivations : thin-layer approximationMathematical motivations
to reduce Viscous Compressible 3D NS p(ρ) = cρ → inviscid compressible 1DSW-like model
to obtain the “motion by slices” through a Neumann problem
I u(t, x, y, z) = u(t, x) + ˜u(t, x, y, z),
I
∫Ω
˜u(t, x, y, z) dy dz = 0,
I ˜u(t, x, y, z) = O(ε) where ε is the aspect-ratio.
I u(t, x)2 ≈ u(t, x)2
to include the friction with its geometrical dependency as well as othergeometrical source terms
general barotropic law p(ρ) = cργ , γ 6= 1
ργ ≈ ργ
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) :339–365, 2001.
C. Bourdarias, M. Ersoy, S. Gerbi,
A model for unsteady mixed flows in non uniform closed water pipes : a Full Kinetic Approach.Accepted in Numerische Mathematik, 2014.
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 8 / 23
![Page 20: A pressurized model for compressible pipe flows: derivation including friction. · 2014-06-13 · A pressurized model for compressible pipe ows: derivation including friction. M](https://reader034.vdocument.in/reader034/viewer/2022042202/5ea2ad7ca194c26c2b4af559/html5/thumbnails/20.jpg)
Mathematical motivations : thin-layer approximationMathematical motivations
to reduce Viscous Compressible 3D NS p(ρ) = cρ → inviscid compressible 1DSW-like model
to obtain the “motion by slices” through a Neumann problem
I u(t, x, y, z) = u(t, x) + ˜u(t, x, y, z),
I
∫Ω
˜u(t, x, y, z) dy dz = 0,
I ˜u(t, x, y, z) = O(ε) where ε is the aspect-ratio.
I u(t, x)2 ≈ u(t, x)2
to include the friction with its geometrical dependency as well as othergeometrical source terms
general barotropic law p(ρ) = cργ , γ 6= 1
ργ ≈ ργ
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) :339–365, 2001.
C. Bourdarias, M. Ersoy, S. Gerbi,
A model for unsteady mixed flows in non uniform closed water pipes : a Full Kinetic Approach.Accepted in Numerische Mathematik, 2014.
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 8 / 23
![Page 21: A pressurized model for compressible pipe flows: derivation including friction. · 2014-06-13 · A pressurized model for compressible pipe ows: derivation including friction. M](https://reader034.vdocument.in/reader034/viewer/2022042202/5ea2ad7ca194c26c2b4af559/html5/thumbnails/21.jpg)
Mathematical motivations : thin-layer approximationMathematical motivations
to reduce Viscous Compressible 3D NS p(ρ) = cρ → inviscid compressible 1DSW-like model
to obtain the “motion by slices” through a Neumann problem
I u(t, x, y, z) = u(t, x) + ˜u(t, x, y, z),
I
∫Ω
˜u(t, x, y, z) dy dz = 0,
I ˜u(t, x, y, z) = O(ε) where ε is the aspect-ratio.
I u(t, x)2 ≈ u(t, x)2
to include the friction with its geometrical dependency as well as othergeometrical source terms
general barotropic law p(ρ) = cργ , γ 6= 1
ργ ≈ ργ
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) :339–365, 2001.
C. Bourdarias, M. Ersoy, S. Gerbi,
A model for unsteady mixed flows in non uniform closed water pipes : a Full Kinetic Approach.Accepted in Numerische Mathematik, 2014.
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 8 / 23
![Page 22: A pressurized model for compressible pipe flows: derivation including friction. · 2014-06-13 · A pressurized model for compressible pipe ows: derivation including friction. M](https://reader034.vdocument.in/reader034/viewer/2022042202/5ea2ad7ca194c26c2b4af559/html5/thumbnails/22.jpg)
OutlineOutline
1 Physical background, Mathematical motivation andprevious works
2 Derivation of the model including friction
3 Numerical experiment and concluding remarks
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 9 / 23
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SettingsLet us consider a compressible fluid confined in a three dimensional domain P, anon deformable pipe of length L oriented following the i vector,
P :=
(x, y, z) ∈ R3; x ∈ [0, L], (y, z) ∈ Ω(x)
where the section Ω(x), x ∈ [0, L], is
Ω(x) = (y, z) ∈ R2; y ∈ [α(x, z), β(x, z)], z ∈ [−R(x), R(x)]
(d) Configuration (e) Ω-plane
Figure : Geometric characteristics of the pipe
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 10 / 23
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The Compressible Navier-Stokes equations ∂tρ+ div(ρu) = 0 ,∂t(ρu) + div(ρu⊗ u)− divσ − ρF = 0 ,
p = p(ρ) = cργ with γ = 1 ,
velocity : u =
(uv
),
density : ρ,
gravity : F = g
sin θ(x)0
− cos θ(x)
,
tensor : σ =
(−p+ λdiv(u) + 2µ∂xu R(u)t
R(u) −pI2 + λdiv(u)I2 + 2µDy,z(v)
),
dynamical viscosity : µ,volume viscosity : λ,
and R(u) = µ (∇y,zu+ ∂xv) , ∇y,zu =
(∂yu∂zu
), Dy,z(v) = ∇y,zv +∇ty,zv
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 11 / 23
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The Compressible Navier-Stokes equations ∂tρ+ div(ρu) = 0 ,∂t(ρu) + div(ρu⊗ u)− divσ − ρF = 0 ,
p = p(ρ) = cργ with γ = 1 ,
velocity : u =
(uv
),
density : ρ,
gravity : F = g
sin θ(x)0
− cos θ(x)
,
tensor : σ =
(−p+ λdiv(u) + 2µ∂xu R(u)t
R(u) −pI2 + λdiv(u)I2 + 2µDy,z(v)
),
dynamical viscosity : µ,volume viscosity : λ,
and R(u) = µ (∇y,zu+ ∂xv) , ∇y,zu =
(∂yu∂zu
), Dy,z(v) = ∇y,zv +∇ty,zv
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 11 / 23
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Boundary conditions : inner wall ∂Ω(x), ∀x ∈ (0, L)
wall-law condition including a general friction law k :
(σ(u)nb) · τbi = (ρk(u)u) · τbi , x ∈ (0, L), (y, z) ∈ ∂Ω(x)
where τbi is the ith vector of the tangential basis. with
nb =1√
(∂xϕ)2 + n · n
(−∂xϕ
n
)where n =
(−∂yϕ
1
)is the outward normal vector in the Ω-plane.
completed with a no-penetration condition :
u · nb = 0, x ∈ (0, L), (y, z) ∈ ∂Ω(x)
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 12 / 23
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Boundary conditions : inner wall ∂Ω(x), ∀x ∈ (0, L)
wall-law condition including a general friction law k :
(σ(u)nb) · τbi = (ρk(u)u) · τbi , x ∈ (0, L), (y, z) ∈ ∂Ω(x)
where τbi is the ith vector of the tangential basis. with
nb =1√
(∂xϕ)2 + n · n
(−∂xϕ
n
)where n =
(−∂yϕ
1
)is the outward normal vector in the Ω-plane.
completed with a no-penetration condition :
u · nb = 0, x ∈ (0, L), (y, z) ∈ ∂Ω(x)
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 12 / 23
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Boundary conditions : inner wall ∂Ω(x), ∀x ∈ (0, L)
wall-law condition including a general friction law k :
(σ(u)nb) · τbi = (ρk(u)u) · τbi , x ∈ (0, L), (y, z) ∈ ∂Ω(x)
where τbi is the ith vector of the tangential basis. with
nb =1√
(∂xϕ)2 + n · n
(−∂xϕ
n
)where n =
(−∂yϕ
1
)is the outward normal vector in the Ω-plane.
completed with a no-penetration condition :
u · nb = 0, x ∈ (0, L), (y, z) ∈ ∂Ω(x)
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thin-layer assumption and asymptotic ordering
“thin-layer” assumption : ε =D
L=W
U=V
U 1 and T =
L
U
dimensionless quantities :
t, (x, y, z), (u, v, w), ρ
non-dimensional numbers :
Fr, FL, Rµ, Rλ,Ma, C
asymptotic ordering :
R−1λ = ελ0, R−1
µ = εµ0, K = εK0 .
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thin-layer assumption and asymptotic ordering
“thin-layer” assumption : ε =D
L=W
U=V
U 1 and T =
L
Udimensionless quantities :
I time t =t
T,
I coordinate (x, y, z) =( xL,y
D,z
D
)I velocity field (u, v, w) =
( uU,v
W,w
W
)I density ρ =
ρ
ρ0
t, (x, y, z), (u, v, w), ρ
non-dimensional numbers :
Fr, FL, Rµ, Rλ,Ma, C
asymptotic ordering :
R−1λ = ελ0, R−1
µ = εµ0, K = εK0 .
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thin-layer assumption and asymptotic ordering
“thin-layer” assumption : ε =D
L=W
U=V
U 1 and T =
L
U
dimensionless quantities : t, (x, y, z), (u, v, w), ρ
non-dimensional numbers :
Fr, FL, Rµ, Rλ,Ma, C
Fr Froude number following the Ω-plane : Fr =U√gD
,
FL Froude number following the i-direction : FL =U√gL
,
Rµ Reynolds numbers with respect to µ : Rµ =ρ0UL
µ,
Rλ Reynolds numbers with respect to λ : Rλ =ρ0UL
λ,
Ma Mach number : Ma =U
c,
C Oser number : C =Ma
Fr=
√gD
c.
asymptotic ordering :
R−1λ = ελ0, R−1
µ = εµ0, K = εK0 .
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thin-layer assumption and asymptotic ordering
“thin-layer” assumption : ε =D
L=W
U=V
U 1 and T =
L
U
dimensionless quantities : t, (x, y, z), (u, v, w), ρ
non-dimensional numbers : Fr, FL, Rµ, Rλ,Ma, C
asymptotic ordering :
R−1λ = ελ0, R−1
µ = εµ0, K = εK0 .
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The non-dimensional systemDropping the ·, the system becomes :
∂tρ+ ∂x(ρu) + divy,z(ρv) = 0 ,
∂t(ρu) + ∂x(ρu2) + divy,z(ρuv) + ∂xρ
M2a
= −ρ sin θ(x)
F 2L
+Gρu
+divy,z
(R−1µ
ε2∇y,zu
)
ε2 (∂t(ρv) + ∂x(ρuv) + divy,z(ρv⊗ v)) +∇y,zρ
M2a
=
0
−ρ cos θ(x)
F 2r
+Gρv ,
where the source terms are
Gρu = divy,z(R−1µ ∂xv
)+ ∂x
(2R−1
µ ∂xu+R−1λ div(u)
),
Gρv = ∂x (εRε(u)) + divy,z(R−1λ div(u) + 2R−1
µ Dy,z(v)).
keeping in mind :R−1λ = ελ0, R−1
µ = εµ0
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The non-dimensional systemDropping the ·, the system becomes :
∂tρ+ ∂x(ρu) + divy,z(ρv) = 0 ,
∂t(ρu) + ∂x(ρu2) + divy,z(ρuv) + ∂xρ
M2a
= −ρ sin θ(x)
F 2L
+Gρu
+divy,z
(R−1µ
ε2∇y,zu
)
ε2 (∂t(ρv) + ∂x(ρuv) + divy,z(ρv⊗ v)) +∇y,zρ
M2a
=
0
−ρ cos θ(x)
F 2r
+Gρv ,
where the source terms are
Gρu = divy,z(R−1µ ∂xv
)+ ∂x
(2R−1
µ ∂xu+R−1λ div(u)
),
Gρv = ∂x (εRε(u)) + divy,z(R−1λ div(u) + 2R−1
µ Dy,z(v)).
keeping in mind :R−1λ = ελ0, R−1
µ = εµ0
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The non-dimensional systemThe system becomes :
∂tρ+ ∂x(ρu) + divy,z(ρv) = 0 ,
∂t(ρu) + ∂x(ρu2) + divy,z(ρuv) + ∂xρ
M2a
= −ρ sin θ(x)
F 2L
+Gρu
+divy,z(µ0
ε
ε2∇y,zu
)∇y,z
ρ
M2a
=
0
−ρ cos θ(x)
F 2r
+Gρv ,
where the source terms are
Gρu = +divy,z (µ0ε∂xv) + ∂x (2µ0ε∂xu+ λ0εdiv(u)) ,
Gρv = ∂x (εRε(u)) + divy,z (λ0εdiv(u) + 2µ0εDy,z(v)) + O(ε2) .
keeping in mind :R−1λ = ελ0, R−1
µ = εµ0
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The non-dimensional systemThe system becomes :
∂tρ+ ∂x(ρu) + divy,z(ρv) = 0 ,
∂t(ρu) + ∂x(ρu2) + divy,z(ρuv) + ∂xρ
M2a
= −ρ sin θ(x)
F 2L
+Gρu
+divy,z(µ0
ε∇y,zu
)∇y,z
ρ
M2a
=
0
−ρ cos θ(x)
F 2r
+Gρv ,
where the source terms are
Gρu = O(ε)
Gρv = O(ε)
keeping in mind :R−1λ = ελ0, R−1
µ = εµ0
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The first order approximation
Formally, dropping all terms of order O(ε), we obtain the so-called hydrostaticapproximation :
∂tρε + ∂x(ρεuε) + divy,z(ρεvε) = 0
∂t(ρεuε) + ∂x(ρεu2ε) + divy,z(ρεuεvε) +
1
M2a
∂xρε = −ρεsin θ(x)
F 2L
+divy,z(µ0
ε∇y,zuε
)1
M2a
∇y,zρε =
(0
−ρε cos θ(x)F 2r
)
Remark
Let us emphasize that even if this system results from a formal limit of Equationsas ε goes to 0, we note its solution (ρε, uε,vε) due to the explicit dependency onε.
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The first order approximation
Formally, dropping all terms of order O(ε), we obtain the so-called hydrostaticapproximation :
∂tρε + ∂x(ρεuε) + divy,z(ρεvε) = 0
∂t(ρεuε) + ∂x(ρεu2ε) + divy,z(ρεuεvε) +
1
M2a
∂xρε = −ρεsin θ(x)
F 2L
+ divy,z(µ0
ε∇y,zuε
)1
M2a
∇y,zρε =
(0
−ρε cos θ(x)F 2r
)
Remark
Let us emphasize that even if this system results from a formal limit of Equationsas ε goes to 0, we note its solution (ρε, uε,vε) due to the explicit dependency onε.
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The boundary conditions
& the Neumann problem
Boundary conditions : ∀x ∈ (0, L), (y, z) ∈ ∂Ω(x) :
µ0
ε∇y,zuε · n = ρεK0(u) +O(ε) and µ0∇y,zuε = O(ε) .
Neumann problemdivy,z (µ0∇y,zuε) = O(ε) , (y, z) ∈ Ω(x)µ0∂nuε = O(ε) , (y, z) ∈ ∂Ω(x)
.
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The boundary conditions
& the Neumann problem
Boundary conditions : ∀x ∈ (0, L), (y, z) ∈ ∂Ω(x) :
µ0
ε∇y,zuε · n = ρεK0(u) +O(ε) and µ0∇y,zuε = O(ε) .
Momentum equation on ρεuε :
∂t(ρεuε) + ∂x(ρεu2ε) + divy,z(ρεuεvε) +
1
M2a
∂xρε = −ρεsin θ(x)
F 2L
+divy,z(µ0
ε∇y,zuε
)
Neumann problemdivy,z (µ0∇y,zuε) = O(ε) , (y, z) ∈ Ω(x)µ0∂nuε = O(ε) , (y, z) ∈ ∂Ω(x)
.
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The boundary conditions
& the Neumann problem
Boundary conditions : ∀x ∈ (0, L), (y, z) ∈ ∂Ω(x) :
µ0
ε∇y,zuε · n = ρεK0(u) +O(ε) and µ0∇y,zuε = O(ε) .
Momentum equation on ρεuε :
∂t(ρεuε) + ∂x(ρεu2ε) + divy,z(ρεuεvε) +
1
M2a
∂xρε = −ρεsin θ(x)
F 2L
+divy,z(µ0
ε∇y,zuε
)Order
1
ε:
divy,z (µ0∇y,zuε) = O(ε)
Neumann problemdivy,z (µ0∇y,zuε) = O(ε) , (y, z) ∈ Ω(x)µ0∂nuε = O(ε) , (y, z) ∈ ∂Ω(x)
.
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The boundary conditions
& the Neumann problem
Boundary conditions : ∀x ∈ (0, L), (y, z) ∈ ∂Ω(x) :
µ0
ε∇y,zuε · n = ρεK0(u) +O(ε) and µ0∇y,zuε = O(ε) .
Momentum equation on ρεuε :
∂t(ρεuε) + ∂x(ρεu2ε) + divy,z(ρεuεvε) +
1
M2a
∂xρε = −ρεsin θ(x)
F 2L
+divy,z(µ0
ε∇y,zuε
)Order
1
ε:
divy,z (µ0∇y,zuε) = O(ε)
Neumann condition
µ0
ε∇y,zuε · n = ρεK0(u) +O(ε) −→ µ0∇y,zuε · n = O(ε)
Neumann problemdivy,z (µ0∇y,zuε) = O(ε) , (y, z) ∈ Ω(x)µ0∂nuε = O(ε) , (y, z) ∈ ∂Ω(x)
.
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The boundary conditions & the Neumann problem
Boundary conditions : ∀x ∈ (0, L), (y, z) ∈ ∂Ω(x) :
µ0
ε∇y,zuε · n = ρεK0(u) +O(ε) and µ0∇y,zuε = O(ε) .
Neumann problemdivy,z (µ0∇y,zuε) = O(ε) , (y, z) ∈ Ω(x)µ0∂nuε = O(ε) , (y, z) ∈ ∂Ω(x)
.
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The boundary conditions & the Neumann problem
Boundary conditions : ∀x ∈ (0, L), (y, z) ∈ ∂Ω(x) :
µ0
ε∇y,zuε · n = ρεK0(u) +O(ε) and µ0∇y,zuε = O(ε) .
Neumann problemdivy,z (µ0∇y,zuε) = O(ε) , (y, z) ∈ Ω(x)µ0∂nuε = O(ε) , (y, z) ∈ ∂Ω(x)
.
⇓
“motion by slices”
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Consequence : first order approximation
“motion by slices”uε(t, x, y, z) = uε(t, x) +O(ε) =⇒ uε(t, x, y, z) = uε(t, x) .
non-linearity : u2ε = uε
2 .
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Consequence : first order approximation
“motion by slices”uε(t, x, y, z) = uε(t, x) +O(ε) =⇒ uε(t, x, y, z) = uε(t, x) .
non-linearity : u2ε = uε
2 .
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Consequence : first order approximation
“motion by slices”uε(t, x, y, z) = uε(t, x) +O(ε) =⇒ uε(t, x, y, z) = uε(t, x) .
non-linearity : u2ε = uε
2 .
stratified structure of the density :
1
M2a
∇y,zρε =
0
−ρε cos θ(x)
F 2r
⇐⇒ (∂yρε∂zρε
)=
(0
−ρεC2 cos θ(x)
)⇓
ρε(t, x, y, z) = ξε(t, x) exp(−C2 cos θ(x)z
)for some positive function ξε
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Consequence : first order approximation
“motion by slices”uε(t, x, y, z) = uε(t, x) +O(ε) =⇒ uε(t, x, y, z) = uε(t, x) .
non-linearity : u2ε = uε
2 .
stratified structure of the density :
1
M2a
∇y,zρε =
0
−ρε cos θ(x)
F 2r
⇐⇒ (∂yρε∂zρε
)=
(0
−ρεC2 cos θ(x)
)⇓
ρε(t, x, y, z) = ξε(t, x) exp(−C2 cos θ(x)z
)for some positive function ξε
⇓
ρε(t, x) =ξε(t, x)Ψ(x)
S(x)
Ψ(x) =
∫Ω(x)
exp(−C2 cos θ(x)z) dy dz : weighted pipe section ,
S(x) =
∫Ω(t,x)
dydz : physical pipe section .
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Consequence : first order approximation
“motion by slices”uε(t, x, y, z) = uε(t, x) +O(ε) =⇒ uε(t, x, y, z) = uε(t, x) .
non-linearity : u2ε = uε
2 .
ρε(t, x, y, z) = ξε(t, x) exp(−C2 cos θ(x)z
)for some positive function ξε
Momentum :
ρεuε =1
S
∫Ω
ρεuε dydz =ξεΨ
Suε = ρε uε
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Consequence : first order approximation
“motion by slices”uε(t, x, y, z) = uε(t, x) +O(ε) =⇒ uε(t, x, y, z) = uε(t, x) .
non-linearity : u2ε = uε
2 .
ρε(t, x, y, z) = ξε(t, x) exp(−C2 cos θ(x)z
)for some positive function ξε
ρεuε = ρε uε
ρεu2ε = ρε u2
ε
= ρε uε2
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Consequence : first order approximation
“motion by slices”uε(t, x, y, z) = uε(t, x) +O(ε) =⇒ uε(t, x, y, z) = uε(t, x) .
non-linearity : u2ε = uε
2 .
ρε(t, x, y, z) = ξε(t, x) exp(−C2 cos θ(x)z
)for some positive function ξε
ρεuε = ρε uε
ρεu2ε = ρε u2
ε = ρε uε2
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The averaged model : P-model
Integration of the hydrostatic equations over the cross-section Ω :
∂t(ρεS) + ∂x(ρεSuε) =
∫∂Ω(x)
ρε (uε∂xm− vε) · n ds
∂t(ρεSuε) + ∂x
(ρεSuε
2 +1
M2a
ρεS
)= −ρεS
sin θ(x)
F 2L
+1
M2a
ρεSdS
dx
+
∫∂Ω(x)
ρεuε (uε∂xm− v) · n ds
−∫∂Ω(x)
µ0
ε∇y,zuε · n ds
I Using Leibniz FormulaI m = (y, ϕ(x, y)) ∈ ∂Ω(x) : the vector ωm
I n =m
|m| : the outward normal to ∂Ω(x) at m in the Ω-plane
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The averaged model : P-model
Integration of the hydrostatic equations over the cross-section Ω :
∂t(ρεS) + ∂x(ρεSuε) =
∫∂Ω(x)
ρε (uε∂xm− vε) · n ds
∂t(ρεSuε) + ∂x
(ρεSuε
2 +1
M2a
ρεS
)= −ρεS
sin θ(x)
F 2L
+1
M2a
ρεSdS
dx
+
∫∂Ω(x)
ρεuε (uε∂xm− v) · n ds
−∫∂Ω(x)
µ0
ε∇y,zuε · n ds
no-penetration condition =⇒ (uε∂xm− vε) · n = 0
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The averaged model : P-model
Integration of the hydrostatic equations over the cross-section Ω :
∂t(ρεS) + ∂x(ρεSuε) = 0
∂t(ρεSuε) + ∂x
(ρεSuε
2 +1
M2a
ρεS
)= −ρεS
sin θ(x)
F 2L
+1
M2a
ρεSdS
dx
−∫∂Ω(x)
µ0
ε∇y,zuε · n ds
Friction term :
∫∂Ω(x)
µ0
ε∇y,zuε · n ds =
∫∂Ω(x)
ρεK0(uε) ds =(ξεΨ(x)
S
)S
(K0(uε)
ψ(x)
Ψ(x)
)= ρεSK(x, uε)
ψ : the curvilinear integral of z → exp(−C2 cos θ(x)z) along ∂Ω(x) calledweighted wet perimeter.
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The averaged model : P-model
Integration of the hydrostatic equations over the cross-section Ω :
∂t(ρεS) + ∂x(ρεSuε) = 0
∂t(ρεSuε) + ∂x
(ρεSuε
2 +1
M2a
ρεS
)= −ρεS
sin θ(x)
F 2L
+1
M2a
ρεSdS
dx
−ρεSK (x, uε)
ψ : weighted wet perimeter of Ω =⇒(ψ(x)
Ψ(x)
)−1
: weighted hydraulic radius
I Meaning that the friction is also a function of the Oser numberI Neglected by engineers since ψ = wet perimeter.
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The averaged model : P-model
Integration of the hydrostatic equations over the cross-section Ω :
∂t(ρεS) + ∂x(ρεSuε) = 0
∂t(ρεSuε) + ∂x(ρεSuε
2 + c2ρεS)
= −gρεS sin θ(x) + c2ρεSdS
dx
−gρεSK (x, uε)
multiply Equations byρ0DU
2
L
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The averaged model : P-model
Integration of the hydrostatic equations over the cross-section Ω :
∂t(A) + ∂x(Auε) = 0
∂t(Auε) + ∂x(Auε
2 + c2A)
= −gA sin θ(x) + c2A
S
dS
dx
−gAK (x, uε)
multiply Equations byρ0DU
2
Lset A = ρεS : the wet area
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The averaged model : P-model
Integration of the hydrostatic equations over the cross-section Ω :
∂t(A) + ∂x(Q) = 0
∂t(Q) + ∂x
(Q2
A+ c2A
)= −gA sin θ(x) + c2
A
S
dS
dx
−gAK(x,Q
A
)
multiply Equations byρ0DU
2
Lset A = ρεS : the wet area
set Q = Auε : the discharge
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 18 / 23
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OutlineOutline
1 Physical background, Mathematical motivation andprevious works
2 Derivation of the model including friction
3 Numerical experiment and concluding remarks
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A “dam-break” like experiment (C=1)
Generalized kinetic scheme introduced by Bourdarias, Ersoy and Gerbi (2014)
Manning-Strickler friction law (Ks =1
M).
We consider :Horizontal circular pipe : L = 100 m, D = 1 m.
(a) M = 0 (b) M = 0.2
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A “dam-break” like experiment (C=1)
Figure : Influence of the friction
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 20 / 23
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Concluding remarks
the case p(ρ) = ργ , γ = 1 : second order approximation(ε = 10−3, C = 1)−→ paraboloid profile
IsoValue7.03551e-062.11065e-053.51775e-054.92485e-056.33196e-057.73906e-059.14616e-050.0001055330.0001196040.0001336750.0001477460.0001618170.0001758880.0001899590.000204030.0002181010.0002321720.0002462430.0002603140.000274385
IsoValue1.91188e-065.73565e-069.55941e-061.33832e-051.72069e-052.10307e-052.48545e-052.86782e-053.2502e-053.63258e-054.01495e-054.39733e-054.7797e-055.16208e-055.54446e-055.92683e-056.30921e-056.69159e-057.07396e-057.45634e-05
IsoValue2.49553e-067.48658e-061.24776e-051.74687e-052.24597e-052.74508e-053.24418e-053.74329e-054.2424e-054.7415e-055.24061e-055.73971e-056.23882e-056.73792e-057.23703e-057.73613e-058.23524e-058.73434e-059.23345e-059.73255e-05
the case p(ρ) = ργ , γ 6= 1
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Concluding remarks
the case p(ρ) = ργ , γ = 1 : second order approximation(ε = 10−3, C = 1)−→ paraboloid profile
IsoValue7.03551e-062.11065e-053.51775e-054.92485e-056.33196e-057.73906e-059.14616e-050.0001055330.0001196040.0001336750.0001477460.0001618170.0001758880.0001899590.000204030.0002181010.0002321720.0002462430.0002603140.000274385
IsoValue1.91188e-065.73565e-069.55941e-061.33832e-051.72069e-052.10307e-052.48545e-052.86782e-053.2502e-053.63258e-054.01495e-054.39733e-054.7797e-055.16208e-055.54446e-055.92683e-056.30921e-056.69159e-057.07396e-057.45634e-05
IsoValue2.49553e-067.48658e-061.24776e-051.74687e-052.24597e-052.74508e-053.24418e-053.74329e-054.2424e-054.7415e-055.24061e-055.73971e-056.23882e-056.73792e-057.23703e-057.73613e-058.23524e-058.73434e-059.23345e-059.73255e-05
the case p(ρ) = ργ , γ 6= 1
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 21 / 23
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Concluding remarks
the case p(ρ) = ργ , γ = 1 : second order approximation(ε = 10−3, C = 1)−→ paraboloid profile
IsoValue7.03551e-062.11065e-053.51775e-054.92485e-056.33196e-057.73906e-059.14616e-050.0001055330.0001196040.0001336750.0001477460.0001618170.0001758880.0001899590.000204030.0002181010.0002321720.0002462430.0002603140.000274385
IsoValue1.91188e-065.73565e-069.55941e-061.33832e-051.72069e-052.10307e-052.48545e-052.86782e-053.2502e-053.63258e-054.01495e-054.39733e-054.7797e-055.16208e-055.54446e-055.92683e-056.30921e-056.69159e-057.07396e-057.45634e-05
IsoValue2.49553e-067.48658e-061.24776e-051.74687e-052.24597e-052.74508e-053.24418e-053.74329e-054.2424e-054.7415e-055.24061e-055.73971e-056.23882e-056.73792e-057.23703e-057.73613e-058.23524e-058.73434e-059.23345e-059.73255e-05
the case p(ρ) = ργ , γ 6= 1
I hydrostatic equation −→ ρε(t, x, y, z) = ξε(t, x)N(t, x, z) where
N(t, x, z) =
(1 + zC2 cos θ(x)
1− γγξε(t, x)γ−1
) 1γ−1
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Concluding remarks
the case p(ρ) = ργ , γ = 1 : second order approximation(ε = 10−3, C = 1)−→ paraboloid profile
IsoValue7.03551e-062.11065e-053.51775e-054.92485e-056.33196e-057.73906e-059.14616e-050.0001055330.0001196040.0001336750.0001477460.0001618170.0001758880.0001899590.000204030.0002181010.0002321720.0002462430.0002603140.000274385
IsoValue1.91188e-065.73565e-069.55941e-061.33832e-051.72069e-052.10307e-052.48545e-052.86782e-053.2502e-053.63258e-054.01495e-054.39733e-054.7797e-055.16208e-055.54446e-055.92683e-056.30921e-056.69159e-057.07396e-057.45634e-05
IsoValue2.49553e-067.48658e-061.24776e-051.74687e-052.24597e-052.74508e-053.24418e-053.74329e-054.2424e-054.7415e-055.24061e-055.73971e-056.23882e-056.73792e-057.23703e-057.73613e-058.23524e-058.73434e-059.23345e-059.73255e-05
the case p(ρ) = ργ , γ 6= 1
I hydrostatic equation −→ ρε(t, x, y, z) = ξε(t, x)N(t, x, z) where
N(t, x, z) =
(1 + zC2 cos θ(x)
1− γγξε(t, x)γ−1
) 1γ−1
I the assumption ργ ≈ ργ is wrong ! ! !
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Concluding remarks
the case p(ρ) = ργ , γ = 1 : second order approximation(ε = 10−3, C = 1)−→ paraboloid profile
IsoValue7.03551e-062.11065e-053.51775e-054.92485e-056.33196e-057.73906e-059.14616e-050.0001055330.0001196040.0001336750.0001477460.0001618170.0001758880.0001899590.000204030.0002181010.0002321720.0002462430.0002603140.000274385
IsoValue1.91188e-065.73565e-069.55941e-061.33832e-051.72069e-052.10307e-052.48545e-052.86782e-053.2502e-053.63258e-054.01495e-054.39733e-054.7797e-055.16208e-055.54446e-055.92683e-056.30921e-056.69159e-057.07396e-057.45634e-05
IsoValue2.49553e-067.48658e-061.24776e-051.74687e-052.24597e-052.74508e-053.24418e-053.74329e-054.2424e-054.7415e-055.24061e-055.73971e-056.23882e-056.73792e-057.23703e-057.73613e-058.23524e-058.73434e-059.23345e-059.73255e-05
the case p(ρ) = ργ , γ 6= 1
I hydrostatic equation −→ ρε(t, x, y, z) = ξε(t, x)N(t, x, z) where
N(t, x, z) =
(1 + zC2 cos θ(x)
1− γγξε(t, x)γ−1
) 1γ−1
I the assumption ργ ≈ ργ is wrong ! ! !I except if the Oser number C 1 −→ a class of low Oser compressible γ
models. This occurs when the gravity has no influence.
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Concluding remarks
the case p(ρ) = ργ , γ = 1 : second order approximation(ε = 10−3, C = 1)−→ paraboloid profile
IsoValue7.03551e-062.11065e-053.51775e-054.92485e-056.33196e-057.73906e-059.14616e-050.0001055330.0001196040.0001336750.0001477460.0001618170.0001758880.0001899590.000204030.0002181010.0002321720.0002462430.0002603140.000274385
IsoValue1.91188e-065.73565e-069.55941e-061.33832e-051.72069e-052.10307e-052.48545e-052.86782e-053.2502e-053.63258e-054.01495e-054.39733e-054.7797e-055.16208e-055.54446e-055.92683e-056.30921e-056.69159e-057.07396e-057.45634e-05
IsoValue2.49553e-067.48658e-061.24776e-051.74687e-052.24597e-052.74508e-053.24418e-053.74329e-054.2424e-054.7415e-055.24061e-055.73971e-056.23882e-056.73792e-057.23703e-057.73613e-058.23524e-058.73434e-059.23345e-059.73255e-05
the case p(ρ) = ργ , γ 6= 1
I First order Pressurized γ model can be derived in a similar way :
∂t(ξεS) + ∂x(ξεSu) = 0
∂t(ξεSuε) + ∂x
(ξεSuε
2 +1
M2a
ξγεS
)= −ξεS
sin θ(x)
F 2L
+1
M2a
ξγεdS
dx−ξεK(x, uε)
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Concluding remarks
the case p(ρ) = ργ , γ = 1 : second order approximation(ε = 10−3, C = 1)−→ paraboloid profile
IsoValue7.03551e-062.11065e-053.51775e-054.92485e-056.33196e-057.73906e-059.14616e-050.0001055330.0001196040.0001336750.0001477460.0001618170.0001758880.0001899590.000204030.0002181010.0002321720.0002462430.0002603140.000274385
IsoValue1.91188e-065.73565e-069.55941e-061.33832e-051.72069e-052.10307e-052.48545e-052.86782e-053.2502e-053.63258e-054.01495e-054.39733e-054.7797e-055.16208e-055.54446e-055.92683e-056.30921e-056.69159e-057.07396e-057.45634e-05
IsoValue2.49553e-067.48658e-061.24776e-051.74687e-052.24597e-052.74508e-053.24418e-053.74329e-054.2424e-054.7415e-055.24061e-055.73971e-056.23882e-056.73792e-057.23703e-057.73613e-058.23524e-058.73434e-059.23345e-059.73255e-05
the case p(ρ) = ργ , γ 6= 1
I Second order approximation (ε = 10−3, C = 10−3) : paraboloid profile
IsoValue6.23256e-061.86977e-053.11628e-054.36279e-055.60931e-056.85582e-058.10233e-059.34884e-050.0001059540.0001184190.0001308840.0001433490.0001558140.0001682790.0001807440.0001932090.0002056750.000218140.0002306050.00024307
IsoValue1.84178e-065.52535e-069.20892e-061.28925e-051.65761e-052.02596e-052.39432e-052.76268e-053.13103e-053.49939e-053.86775e-054.2361e-054.60446e-054.97282e-055.34117e-055.70953e-056.07789e-056.44624e-056.8146e-057.18296e-05
IsoValue5.99634e-061.7989e-052.99817e-054.19744e-055.3967e-056.59597e-057.79524e-058.99451e-050.0001019380.000113930.0001259230.0001379160.0001499080.0001619010.0001738940.0001858860.0001978790.0002098720.0002218650.000233857
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PerspectivesMain objectives are
make the asymptotic analysis rigorous for γ > 0
applications dealing with the impact of sediment transport during floodingbased on
I Pressurised γ models for the hydrodynamicsI Exner like equations for the morphodynamics (derived from Vlasov equations)
to findI optimal pipe shapeI including variable rugosity
(a) what happen inside the pipe (b) This is not a river ! ! !
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PerspectivesMain objectives are
make the asymptotic analysis rigorous for γ > 0
applications dealing with the impact of sediment transport during floodingbased on
I Pressurised γ models for the hydrodynamicsI Exner like equations for the morphodynamics (derived from Vlasov equations)
to findI optimal pipe shapeI including variable rugosity
(c) what happen inside the pipe (d) This is not a river ! ! !
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 22 / 23
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PerspectivesMain objectives are
make the asymptotic analysis rigorous for γ > 0
applications dealing with the impact of sediment transport during floodingbased on
I Pressurised γ models for the hydrodynamicsI Exner like equations for the morphodynamics (derived from Vlasov equations)
to findI optimal pipe shapeI including variable rugosity
(e) what happen inside the pipe (f) This is not a river ! ! !
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 22 / 23
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Thank youThank you
for yourfor your
attentionattentio
n
M. Ersoy (IMATH) Compressible pipe flow including friction Bilbao, June 12-13, 2014 23 / 23