Air entrainment in transient flows in closed water pipes: A - - PowerPoint PPT Presentation
Air entrainment in transient flows in closed water pipes: A two-layer approach. Mehmet Ersoy 1 joint work with C. Bourdarias and S. Gerbi, LAMA, Chamb ery, France NTM, Porquerolles, June 2013 1. Mehmet.Ersoy@univ-tln.fr and http: // ersoy.
Air entrainment in transient flows in closed water pipes: A two-layer approach.
Mehmet Ersoy 1 joint work with C. Bourdarias and S. Gerbi, LAMA, Chamb´ ery, France NTM, Porquerolles, June 2013
Outline of the talk
Outline of the talk 1 Physical and mathematical motivations
Air entrainement Previous works
2 The two layer model
Fluid Layer : incompressible Euler’s Equations Air Layer : compressible Euler’s Equations The two-layer model Properties
3 Numerical approximation
The kinetic scheme A numerical experiment
Air entrainment in transient flows NTM 2 / 33
Outline
Outline 1 Physical and mathematical motivations
Air entrainement Previous works
2 The two layer model
Fluid Layer : incompressible Euler’s Equations Air Layer : compressible Euler’s Equations The two-layer model Properties
3 Numerical approximation
The kinetic scheme A numerical experiment
Air entrainment in transient flows NTM 3 / 33
Outline
Outline 1 Physical and mathematical motivations
Air entrainement Previous works
2 The two layer model
Fluid Layer : incompressible Euler’s Equations Air Layer : compressible Euler’s Equations The two-layer model Properties
3 Numerical approximation
The kinetic scheme A numerical experiment
Air entrainment in transient flows NTM 4 / 33
Air entrainment
The air entrainment appears in the transient flow in closed pipes not completely filled : the liquid flow (as well as the air flow) is free surface.
(a) Settings
Air entrainment in transient flows NTM 5 / 33
Air entrainment
The air entrainment appears in the transient flow in closed pipes not completely filled : the liquid flow (as well as the air flow) is free surface. may lead to two-phase flows for transition : free surface flows → pressurized flows.
(e) Settings (f) Sewers . . . in Paris (g) Forced pipe
Air entrainment in transient flows NTM 5 / 33
Air entrainment
The air entrainment appears in the transient flow in closed pipes not completely filled : the liquid flow (as well as the air flow) is free surface. may lead to two-phase flows for transition : free surface flows → pressurized flows. may cause severe damage due to the pressure surge.
(i) Settings (j) Sewers . . . in Paris (k) Forced pipe (l) . . . at Minne- sota http://www. sewerhistory.org/ grfx/misc/disaster.htm
Air entrainment in transient flows NTM 5 / 33
Outline
Outline 1 Physical and mathematical motivations
Air entrainement Previous works
2 The two layer model
Fluid Layer : incompressible Euler’s Equations Air Layer : compressible Euler’s Equations The two-layer model Properties
3 Numerical approximation
The kinetic scheme A numerical experiment
Air entrainment in transient flows NTM 6 / 33
Previous works
the homogeneous model : a single fluid is considered where sound speed depends on the fraction of air
Air entrainment in transient flows NTM 7 / 33
Previous works
the homogeneous model : a single fluid is considered where sound speed depends on the fraction of air
the drift-flux model : the velocity fields are expressed in terms of the mixture center-of-mass velocity and the drift velocity of the vapor phase Ishii et al. 2003, Faille and Heintze 1999.
Air entrainment in transient flows NTM 7 / 33
Previous works
the homogeneous model : a single fluid is considered where sound speed depends on the fraction of air
the drift-flux model : the velocity fields are expressed in terms of the mixture center-of-mass velocity and the drift velocity of the vapor phase Ishii et al. 2003, Faille and Heintze 1999. the two-fluid model : a compressible and incompressible model are coupled via the
Tiselj, Petelin et al. 1997, 2001.
Air entrainment in transient flows NTM 7 / 33
Previous works
the homogeneous model : a single fluid is considered where sound speed depends on the fraction of air
the drift-flux model : the velocity fields are expressed in terms of the mixture center-of-mass velocity and the drift velocity of the vapor phase Ishii et al. 2003, Faille and Heintze 1999. the two-fluid model : a compressible and incompressible model are coupled via the
Tiselj, Petelin et al. 1997, 2001. the rigid water column : Hamam and McCorquodale 1982, Zhou, Hicks et al 2002.
Air entrainment in transient flows NTM 7 / 33
Previous works
the homogeneous model : a single fluid is considered where sound speed depends on the fraction of air
the drift-flux model : the velocity fields are expressed in terms of the mixture center-of-mass velocity and the drift velocity of the vapor phase Ishii et al. 2003, Faille and Heintze 1999. the two-fluid model : a compressible and incompressible model are coupled via the
Tiselj, Petelin et al. 1997, 2001. the rigid water column : Hamam and McCorquodale 1982, Zhou, Hicks et al 2002. the PFS equations (Ersoy et al, IJFV 2009, JSC 2011).
Air entrainment in transient flows NTM 7 / 33
Previous works
the homogeneous model : a single fluid is considered where sound speed depends on the fraction of air
the drift-flux model : the velocity fields are expressed in terms of the mixture center-of-mass velocity and the drift velocity of the vapor phase Ishii et al. 2003, Faille and Heintze 1999. the two-fluid model : a compressible and incompressible model are coupled via the
Tiselj, Petelin et al. 1997, 2001. the rigid water column : Hamam and McCorquodale 1982, Zhou, Hicks et al 2002. the PFS equations (Ersoy et al, IJFV 2009, JSC 2011). the two layer model (Ersoy et al M2AN 2013).
Air entrainment in transient flows NTM 7 / 33
Mathematical problems
Almost all previous models introduce several mathematical and numerical difficulties such as
◮ the ill-posedness ◮ the presence of discontinuous fluxes ◮ the loss of hyperbolicity (eigenvalues may become complex)
Air entrainment in transient flows NTM 8 / 33
Mathematical and numerical problems
Almost all previous models introduce several mathematical and numerical difficulties such as
◮ the ill-posedness ◮ the presence of discontinuous fluxes ◮ the loss of hyperbolicity (eigenvalues may become complex)
The last one is the problem analyzed here for a two-layer problem :
◮ any consistent finite difference scheme is unconditionally unstable ◮ any consistent finite volume scheme (based on eigenvalues) is useless
Air entrainment in transient flows NTM 8 / 33
Mathematical and numerical problems
Almost all previous models introduce several mathematical and numerical difficulties such as
◮ the ill-posedness ◮ the presence of discontinuous fluxes ◮ the loss of hyperbolicity (eigenvalues may become complex)
The last one is the problem analyzed here for a two-layer problem :
◮ any consistent finite difference scheme is unconditionally unstable ◮ any consistent finite volume scheme (based on eigenvalues) is useless
Air entrainment in transient flows NTM 8 / 33
Mathematical and numerical problems
Almost all previous models introduce several mathematical and numerical difficulties such as
◮ the ill-posedness ◮ the presence of discontinuous fluxes ◮ the loss of hyperbolicity (eigenvalues may become complex)
The last one is the problem analyzed here for a two-layer problem :
◮ any consistent finite difference scheme is unconditionally unstable ◮ any consistent finite volume scheme (based on eigenvalues) is useless
Air entrainment in transient flows NTM 8 / 33
Mathematical and numerical problems
Almost all previous models introduce several mathematical and numerical difficulties such as
◮ the ill-posedness ◮ the presence of discontinuous fluxes ◮ the loss of hyperbolicity (eigenvalues may become complex)
The last one is the problem analyzed here for a two-layer problem :
◮ any consistent finite difference scheme is unconditionally unstable ◮ any consistent finite volume scheme (based on eigenvalues) is useless
⇒ Kelvin-Helmholtz instability, for which the two-layer model is not a priori suitable :
Figure: Kelvin-Helmholtz instability (source : wiki Kelvin-Helmholtz instability)
Air entrainment in transient flows NTM 8 / 33
Outline
Outline 1 Physical and mathematical motivations
Air entrainement Previous works
2 The two layer model
Fluid Layer : incompressible Euler’s Equations Air Layer : compressible Euler’s Equations The two-layer model Properties
3 Numerical approximation
The kinetic scheme A numerical experiment
Air entrainment in transient flows NTM 9 / 33
Settings Figure: Geometric characteristics of the domain.
We have then the first natural coupling : Hw(t, x) + Ha(t, x) = 2R(x) .
Air entrainment in transient flows NTM 10 / 33
Outline
Outline 1 Physical and mathematical motivations
Air entrainement Previous works
2 The two layer model
Fluid Layer : incompressible Euler’s Equations Air Layer : compressible Euler’s Equations The two-layer model Properties
3 Numerical approximation
The kinetic scheme A numerical experiment
Air entrainment in transient flows NTM 11 / 33
Fluid layer : incompressible Euler’s Equations (Gerbeau, Perthame, 2001)
Incompressible Euler’s equations
div(ρ0Uw) = 0,
∂t(ρ0Uw) + div(ρ0Uw ⊗ Uw) + ∇Pw = ρ0F,
where Uw(t, x, y, z) = (Uw, Vw, Ww) the velocity, Pw(t, x, y, z) the pressure, F the gravity strength.
Figure: Cross-section of the domain
Air entrainment in transient flows NTM 12 / 33
Fluid layer : incompressible Euler’s Equations (Gerbeau, Perthame, 2001)
Incompressible Euler’s equations
div(ρ0Uw) = 0,
∂t(ρ0Uw) + div(ρ0Uw ⊗ Uw) + ∇Pw = ρ0F,
where Uw(t, x, y, z) = (Uw, Vw, Ww) the velocity, Pw(t, x, y, z) the pressure, F the gravity strength. Write non dimensional form of Euler equations using the parameter ǫ = H/L ≪ 1 and takes ǫ = 0. Equality of the pressure of air and water Pa = Pw at the free surface interface. Section averaging ρU ≈ ρU and U 2 ≈ U U. Introduce A(t, x) =
dydz, u(t, x) = 1 A(t, x)
Uw(t, x, y, z) dydz, and Q(t, x) = A(t, x)u(t, x).
Figure: Cross-section of the domain
Air entrainment in transient flows NTM 12 / 33
Fluid layer : incompressible Euler’s Equations (Gerbeau, Perthame, 2001)
Incompressible Euler’s equations
div(ρ0Uw) = 0,
∂t(ρ0Uw) + div(ρ0Uw ⊗ Uw) + ∇Pw = ρ0F,
where Uw(t, x, y, z) = (Uw, Vw, Ww) the velocity, Pw(t, x, y, z) the pressure, F the gravity strength. Write non dimensional form of Euler equations using the parameter ǫ = H/L ≪ 1 and takes ǫ = 0. Equality of the pressure of air and water Pa = Pw at the free surface interface. Section averaging ρU ≈ ρU and U 2 ≈ U U. Introduce A(t, x) =
dydz, u(t, x) = 1 A(t, x)
Uw(t, x, y, z) dydz, and Q(t, x) = A(t, x)u(t, x).
Figure: Cross-section of the domain
Air entrainment in transient flows NTM 12 / 33
Fluid layer : incompressible Euler’s Equations (Gerbeau, Perthame, 2001)
Incompressible Euler’s equations
div(ρ0Uw) = 0,
∂t(ρ0Uw) + div(ρ0Uw ⊗ Uw) + ∇Pw = ρ0F,
where Uw(t, x, y, z) = (Uw, Vw, Ww) the velocity, Pw(t, x, y, z) the pressure, F the gravity strength. Write non dimensional form of Euler equations using the parameter ǫ = H/L ≪ 1 and takes ǫ = 0. Equality of the pressure of air and water Pa = Pw at the free surface interface. Section averaging ρU ≈ ρU and U 2 ≈ U U. Introduce A(t, x) =
dydz, u(t, x) = 1 A(t, x)
Uw(t, x, y, z) dydz, and Q(t, x) = A(t, x)u(t, x).
Figure: Cross-section of the domain
Air entrainment in transient flows NTM 12 / 33
Fluid layer : incompressible Euler’s Equations (Gerbeau, Perthame, 2001)
Incompressible Euler’s equations
div(ρ0Uw) = 0,
∂t(ρ0Uw) + div(ρ0Uw ⊗ Uw) + ∇Pw = ρ0F,
where Uw(t, x, y, z) = (Uw, Vw, Ww) the velocity, Pw(t, x, y, z) the pressure, F the gravity strength. Write non dimensional form of Euler equations using the parameter ǫ = H/L ≪ 1 and takes ǫ = 0. Equality of the pressure of air and water Pa = Pw at the free surface interface. Section averaging ρU ≈ ρU and U 2 ≈ U U. Introduce A(t, x) =
dydz, u(t, x) = 1 A(t, x)
Uw(t, x, y, z) dydz, and Q(t, x) = A(t, x)u(t, x).
Figure: Cross-section of the domain
Air entrainment in transient flows NTM 12 / 33
Fluid layer model
Fluid layer model
∂tA + ∂xQ = ∂tQ + ∂x Q2 A + APa(ρ)/ρ0 + gI1(x, A) cos θ
−gA∂xZ +gI2(x, A) cos θ +Pa(ρ)/ρ0 ∂xA where the hydrostatic pressure : I1(x, A) = hw
−R
(hw − z)σ(x, z) dz, the pressure source term : I2(x, A) = hw
−R
(hw − z)∂xσ(x, z) dz , the air pressure : Pa.
Air entrainment in transient flows NTM 13 / 33
Outline
Outline 1 Physical and mathematical motivations
Air entrainement Previous works
2 The two layer model
Fluid Layer : incompressible Euler’s Equations Air Layer : compressible Euler’s Equations The two-layer model Properties
3 Numerical approximation
The kinetic scheme A numerical experiment
Air entrainment in transient flows NTM 14 / 33
Air Layer : compressible Euler’s Equations
Compressible Euler’s equations
∂tρa + div(ρaUa) = 0,
∂t(ρaUa) + div(ρaUa ⊗ Ua) + ∇Pa = 0,
where Ua(t, x, y, z) = (Ua, Va, Wa) the velocity, Pa(t, x, y, z) the pressure, ρa(t, x, y, z) the density.
Air entrainment in transient flows NTM 15 / 33
Air Layer : compressible Euler’s Equations
Compressible Euler’s equations
∂tρa + div(ρaUa) = 0,
∂t(ρaUa) + div(ρaUa ⊗ Ua) + ∇Pa = 0,
with Pa(ρ) = k ργ with k = pa ργ
a where γ is set to 7/5.
where Ua(t, x, y, z) = (Ua, Va, Wa) the velocity, Pa(t, x, y, z) the pressure, ρa(t, x, y, z) the density.
Air entrainment in transient flows NTM 15 / 33
Air Layer : compressible Euler’s Equations
Compressible Euler’s equations
∂tρa + div(ρaUa) = 0,
∂t(ρaUa) + div(ρaUa ⊗ Ua) + ∇Pa = 0,
with Pa(ρ) = k ργ with k = pa ργ
a where γ is set to 7/5.
where Ua(t, x, y, z) = (Ua, Va, Wa) the velocity, Pa(t, x, y, z) the pressure, ρa(t, x, y, z) the density. Write non dimensional form of Euler equations using the parameter ǫ = H/L ≪ 1 and takes ǫ = 0. Equality of the pressure of air and water Pa = Pw at the free surface interface. Section averaging Averaged nonlinearity ∼ Nonlinearity of the averaged. Introduce A(t, x) =
dydz, uv(t, x) = 1 A(t, x)
Ua(t, x, y, z) dydz, M = ρ/ρ0A, D = Mv and c2
a = ∂p
∂ρ = kγ ρ0M A γ−1 .
Air entrainment in transient flows NTM 15 / 33
Air Layer : compressible Euler’s Equations
Compressible Euler’s equations
∂tρa + div(ρaUa) = 0,
∂t(ρaUa) + div(ρaUa ⊗ Ua) + ∇Pa = 0,
with Pa(ρ) = k ργ with k = pa ργ
a where γ is set to 7/5.
where Ua(t, x, y, z) = (Ua, Va, Wa) the velocity, Pa(t, x, y, z) the pressure, ρa(t, x, y, z) the density. Write non dimensional form of Euler equations using the parameter ǫ = H/L ≪ 1 and takes ǫ = 0. Equality of the pressure of air and water Pa = Pw at the free surface interface. Section averaging Averaged nonlinearity ∼ Nonlinearity of the averaged. Introduce A(t, x) =
dydz, uv(t, x) = 1 A(t, x)
Ua(t, x, y, z) dydz, M = ρ/ρ0A, D = Mv and c2
a = ∂p
∂ρ = kγ ρ0M A γ−1 .
Air entrainment in transient flows NTM 15 / 33
Air layer model : mean value on Ωa
Air layer model
∂tM + ∂xD = ∂tD + ∂x D2 M + M γ c2
a
M γ c2
a ∂x(A)
. where the γ pressure : M γ c2
a,
the pressure source term : M γ c2
a ∂x(A).
Air entrainment in transient flows NTM 16 / 33
Outline
Outline 1 Physical and mathematical motivations
Air entrainement Previous works
2 The two layer model
Fluid Layer : incompressible Euler’s Equations Air Layer : compressible Euler’s Equations The two-layer model Properties
3 Numerical approximation
The kinetic scheme A numerical experiment
Air entrainment in transient flows NTM 17 / 33
The two-layer model
A + A = S where S = S(x) denotes the pipe section
Two-layer model
∂tM + ∂xD = ∂tD + ∂x D2 M + M γ c2
a
M γ c2
a ∂x(S − A)
∂tA + ∂xQ = ∂tQ + ∂x Q2 A + gI1(x, A) cos θ + A (S − A) M γ c2
a
−gA∂xZ +gI2(x, A) cos θ + A (S − A) M γ c2
a ∂xA
.
Air entrainment in transient flows NTM 18 / 33
Outline
Outline 1 Physical and mathematical motivations
Air entrainement Previous works
2 The two layer model
Fluid Layer : incompressible Euler’s Equations Air Layer : compressible Euler’s Equations The two-layer model Properties
3 Numerical approximation
The kinetic scheme A numerical experiment
Air entrainment in transient flows NTM 19 / 33
Mathematical entropy and energetically closed system
1 Energies
Ea = Mv2 2 + c2
a M
γ(γ − 1) and Ew = Au2 2 + gA(hw − I1(x, A)/A) cos θ + gAZ satisfy the following entropy flux equalities : ∂tEa + ∂xHa = c2
a M
γ(S − A) ∂tA and ∂tEw + ∂xHw = − c2
a M
γ(S − A) ∂tA where Ha =
a M
γ
a M
(S − A)
Air entrainment in transient flows NTM 20 / 33
Mathematical entropy and energetically closed system
1 Energies
Ea = Mv2 2 + c2
a M
γ(γ − 1) and Ew = Au2 2 + gA(hw − I1(x, A)/A) cos θ + gAZ satisfy the following entropy flux equalities : ∂tEa + ∂xHa = c2
a M
γ(S − A) ∂tA and ∂tEw + ∂xHw = − c2
a M
γ(S − A) ∂tA where Ha =
a M
γ
a M
(S − A)
2 The total energy satisfies
∂tE + ∂xH = 0 .
Air entrainment in transient flows NTM 20 / 33
A conditionally hyperbolic system : eigenstructure
Quasi-linear form : W = (M, D, A, Q)t ∂tW + D(x, W)∂XW = 0 with D = 1 c2
a − v2
2v M S − Ac2
a
1 A (S − A)c2
a
c2
w +
AM (S − A)2 c2
a − u2
2u where cm := c2
w +
AM (S − A)2 c2
a : water sound speed under the air effect.
Air entrainment in transient flows NTM 21 / 33
A conditionally hyperbolic system : eigenstructure
Quasi-linear form : W = (M, D, A, Q)t ∂tW + D(x, W)∂XW = 0 with D = 1 c2
a − v2
2v M S − Ac2
a
1 A (S − A)c2
a
c2
w +
AM (S − A)2 c2
a − u2
2u where cm := c2
w +
AM (S − A)2 c2
a : water sound speed under the air effect.
Writing F = v − u cm , √ H = ca cm , cm =
w + sc2 a with s =
AM (S − A)2 0 , the characteristic polynom reads P(x = λ/cm) = x4 − 2(2 + F)x3 + ((1 + F)(5 + F) − H) x2 + 2
x − sH2 where λ stands for an eigenvalue of D.
Air entrainment in transient flows NTM 21 / 33
Conditional hyperbolicity of the two-layer system
Theorem (Fuller, IEEE Trans. Automat. Control, 81)
All the root of Equation P are real if and only if one of the following conditions holds : (i) ∆3 > 0, ∆5 > 0 and ∆7 0, (ii) ∆3 0, ∆5 = 0 and ∆7 = 0 where ∆3, ∆5, ∆7 are the inner determinant of the discriminant of P.
Air entrainment in transient flows NTM 22 / 33
Conditional hyperbolicity of the two-layer system
Theorem (Fuller, IEEE Trans. Automat. Control, 81)
All the root of Equation P are real if and only if one of the following conditions holds : (i) ∆3 > 0, ∆5 > 0 and ∆7 0, (ii) ∆3 0, ∆5 = 0 and ∆7 = 0 where ∆3, ∆5, ∆7 are the inner determinant of the discriminant of P. From physical consideration, ∆3 > 0 and ∆5 > 0 = ⇒ hyperbolic ⇐ ⇒ ∆7 0.
Air entrainment in transient flows NTM 22 / 33
Conditional hyperbolicity of the two-layer system
Theorem (Fuller, IEEE Trans. Automat. Control, 81)
All the root of Equation P are real if and only if one of the following conditions holds : (i) ∆3 > 0, ∆5 > 0 and ∆7 0, (ii) ∆3 0, ∆5 = 0 and ∆7 = 0 where ∆3, ∆5, ∆7 are the inner determinant of the discriminant of P. From physical consideration, ∆3 > 0 and ∆5 > 0 = ⇒ hyperbolic ⇐ ⇒ ∆7 0. i.e.
(a) ymin < 0 (b) ymin > 0
Figure: Behavior of the polynomial ∆7(y)
Air entrainment in transient flows NTM 22 / 33
Conditional hyperbolicity of the two-layer system
Theorem (Fuller, IEEE Trans. Automat. Control, 81)
All the root of Equation P are real if and only if one of the following conditions holds : (i) ∆3 > 0, ∆5 > 0 and ∆7 0, (ii) ∆3 0, ∆5 = 0 and ∆7 = 0 where ∆3, ∆5, ∆7 are the inner determinant of the discriminant of P. From physical consideration, ∆3 > 0 and ∆5 > 0 = ⇒ hyperbolic ⇐ ⇒ ∆7 0. Then,
Theorem (Ersoy et al., M2AN,13)
The two-layer system is strictly hyperbolic if
◮ y = F 2 F 2
max := y2 max ⇐
⇒ large relative speed
◮ ymin > 0 and 0 F 2 ymin = F 2
min ⇐
⇒ small relative speed
where s = ρ ρ0 A S − A ∼ ρ2 ρ1 < 1 as in the two-layer shallows water equations.
Air entrainment in transient flows NTM 22 / 33
Hyperbolic region : examples
(a) large relative speed (ρ = 1000 (air density), F x-axis and A y- axis) (b) Small relative speed(Zoom
axis) (c) ρ = 10, F x-axis and Ay-axis
Figure: black grey = non hyperbolic region
Air entrainment in transient flows NTM 23 / 33
Hyperbolic region : examples
(a) large relative speed (ρ = 1000 (air density), F x-axis and A y- axis) (b) Small relative speed(Zoom
axis) (c) ρ = 10, F x-axis and Ay-axis
Figure: black grey = non hyperbolic region
As a consequence system may lose its hyperbolicity. solver based on the computation of eigenvalues are useless.
Air entrainment in transient flows NTM 23 / 33
Hyperbolic region : examples
(a) large relative speed (ρ = 1000 (air density), F x-axis and A y- axis) (b) Small relative speed(Zoom
axis) (c) ρ = 10, F x-axis and Ay-axis
Figure: black grey = non hyperbolic region
As a consequence system may lose its hyperbolicity. solver based on the computation of eigenvalues are useless.
Air entrainment in transient flows NTM 23 / 33
Outline
Outline 1 Physical and mathematical motivations
Air entrainement Previous works
2 The two layer model
Fluid Layer : incompressible Euler’s Equations Air Layer : compressible Euler’s Equations The two-layer model Properties
3 Numerical approximation
The kinetic scheme A numerical experiment
Air entrainment in transient flows NTM 24 / 33
Outline
Outline 1 Physical and mathematical motivations
Air entrainement Previous works
2 The two layer model
Fluid Layer : incompressible Euler’s Equations Air Layer : compressible Euler’s Equations The two-layer model Properties
3 Numerical approximation
The kinetic scheme A numerical experiment
Air entrainment in transient flows NTM 25 / 33
Kinetic interpretation
As in gas theory, Describe the macroscopic behavior from particle motions, here, assumed fictitious by introducing a χ density function and a M(t, x, ξ; χ) maxwellian function (or a Gibbs equilibrium)
Air entrainment in transient flows NTM 26 / 33
Kinetic interpretation
As in gas theory, Describe the macroscopic behavior from particle motions, here, assumed fictitious by introducing a χ density function and a M(t, x, ξ; χ) maxwellian function (or a Gibbs equilibrium) i.e., transform the nonlinear system into a kinetic transport equation on M.
Air entrainment in transient flows NTM 26 / 33
Kinetic interpretation
As in gas theory, Describe the macroscopic behavior from particle motions, here, assumed fictitious by introducing a χ density function and a M(t, x, ξ; χ) maxwellian function (or a Gibbs equilibrium) i.e., transform the nonlinear system into a kinetic transport equation on M. Thus, to be able to define the numerical macroscopic fluxes from the microscopic one. ...Faire d’une pierre deux coups...
Air entrainment in transient flows NTM 26 / 33
Principle
Density function
We introduce χ(ω) = χ(−ω) ≥ 0 ,
χ(ω)dω = 1,
ω2χ(ω)dω = 1 ,
Air entrainment in transient flows NTM 27 / 33
Principle
Gibbs Equilibrium or Maxwellian
We introduce χ(ω) = χ(−ω) ≥ 0 ,
χ(ω)dω = 1,
ω2χ(ω)dω = 1 , then we define the Gibbs equilibrium by M(t, x, ξ) = A b χ ξ − u b
Air entrainment in transient flows NTM 27 / 33
Principle
Gibbs Equilibrium or Maxwellian
We introduce χ(ω) = χ(−ω) ≥ 0 ,
χ(ω)dω = 1,
ω2χ(ω)dω = 1 , then we define the Gibbs equilibrium by M(t, x, ξ) = A b χ ξ − u b
micro-macroscopic relations
A =
M(t, x, ξ) dξ Au =
ξM(t, x, ξ) dξ Au2 + Ab2 =
ξ2M(t, x, ξ) dξ
Air entrainment in transient flows NTM 27 / 33
Kinetic interpretation The kinetic formulation [Perthame, Oxford Lect. Ser. in Math. and its Applic., 02]
(A, Q) is solution of the (air or water) system if and only if M satisfies the transport equation : ∂tM + ξ · ∂xM − gΦ ∂ξM = K(t, x, ξ) where K(t, x, ξ) is a collision kernel satisfying a.e. (t, x)
K dξ = 0 ,
ξ Kd ξ = 0 and Φ are the source terms.
Air entrainment in transient flows NTM 28 / 33
Kinetic interpretation The kinetic formulation [Perthame, Oxford Lect. Ser. in Math. and its Applic., 02]
(A, Q) is solution of the (air or water) system if and only if M satisfies the transport equation : ∂tM + ξ · ∂xM − gΦ ∂ξM = K(t, x, ξ) where K(t, x, ξ) is a collision kernel satisfying a.e. (t, x)
K dξ = 0 ,
ξ Kd ξ = 0 and Φ are the source terms. General form of the source terms : Φ =
conservative
d dxZ +
non conservative
dxW +
friction
A2 conservative term : classical upwind non conservative term : mid point rule (DLM, 95) friction : dynamic topography (Ersoy, Ph.D.)
Air entrainment in transient flows NTM 28 / 33
Discretization of source terms (for the sake of simplicity, consider only Z)
Recalling that Z is constant per cell
Air entrainment in transient flows NTM 29 / 33
Discretization of source terms (for the sake of simplicity, consider only Z)
Recalling that Z is constant per cell Then ∀(t, x) ∈ [tn, tn+1[×
Z′(x) = 0
Air entrainment in transient flows NTM 29 / 33
Simplification of the transport equation
Recalling that Z is constant per cell Then ∀(t, x) ∈ [tn, tn+1[×
Z′(x) = 0 = ⇒ ∂tM + ξ · ∂xM = K(t, x, ξ)
Air entrainment in transient flows NTM 29 / 33
Simplification of the transport equation
Recalling that Z is constant per cell Then ∀(t, x) ∈ [tn, tn+1[×
Z′(x) = 0 = ⇒ ∂tf + ξ · ∂xf = f(tn, x, ξ) = M(tn, x, ξ)
def
:=
Air entrainment in transient flows NTM 29 / 33
Simplification of the transport equation
Recalling that Z is constant per cell Then ∀(t, x) ∈ [tn, tn+1[×
Z′(x) = 0 = ⇒ ∂tf + ξ · ∂xf = f(tn, x, ξ) = M(tn, x, ξ)
def
:=
i.e. f n+1
i
(ξ) = Mn
i (ξ) + ξ ∆tn
∆x
i+ 1
2 (ξ) − M+
i− 1
2 (ξ)
Air entrainment in transient flows NTM 29 / 33
Simplification of the transport equation
Recalling that Z is constant per cell Then ∀(t, x) ∈ [tn, tn+1[×
Z′(x) = 0 = ⇒ ∂tf + ξ · ∂xf = f(tn, x, ξ) = M(tn, x, ξ)
def
:=
i.e. f n+1
i
(ξ) = Mn
i (ξ) + ξ ∆tn
∆x
i+ 1
2 (ξ) − M+
i− 1
2 (ξ)
Un+1
i
= An+1
i
Qn+1
i
:=
1 ξ
i
(ξ) dξ
Air entrainment in transient flows NTM 29 / 33
Simplification of the transport equation
Recalling that Z is constant per cell Then ∀(t, x) ∈ [tn, tn+1[×
Z′(x) = 0 = ⇒ ∂tf + ξ · ∂xf = f(tn, x, ξ) = M(tn, x, ξ)
def
:=
i.e. f n+1
i
(ξ) = Mn
i (ξ) + ξ ∆tn
∆x
i+ 1
2 (ξ) − M+
i− 1
2 (ξ)
Un+1
i
= Un
i − ∆tn
∆x
i+1/2 − F + i−1/2
i± 1
2 =
ξ 1 ξ
i± 1
2 (ξ) dξ.
Air entrainment in transient flows NTM 29 / 33
Outline
Outline 1 Physical and mathematical motivations
Air entrainement Previous works
2 The two layer model
Fluid Layer : incompressible Euler’s Equations Air Layer : compressible Euler’s Equations The two-layer model Properties
3 Numerical approximation
The kinetic scheme A numerical experiment
Air entrainment in transient flows NTM 30 / 33
De l’air dans les tuyaux Figure: “Dam break in presence of air in a closed water pipe.”
Air entrainment in transient flows NTM 31 / 33
Conclusion
Conclusion
Existence of a convex entropy function ⇒ admissible weak solutions
Air entrainment in transient flows NTM 32 / 33
Conclusion
Conclusion
Existence of a convex entropy function ⇒ admissible weak solutions System is hyperbolic even for large relative speed
Air entrainment in transient flows NTM 32 / 33
Conclusion
Conclusion
Existence of a convex entropy function ⇒ admissible weak solutions System is hyperbolic even for large relative speed Advantages of the kinetic scheme :
Air entrainment in transient flows NTM 32 / 33
Conclusion
Conclusion
Existence of a convex entropy function ⇒ admissible weak solutions System is hyperbolic even for large relative speed Advantages of the kinetic scheme :
◮ easy implementation
Air entrainment in transient flows NTM 32 / 33
Conclusion
Conclusion
Existence of a convex entropy function ⇒ admissible weak solutions System is hyperbolic even for large relative speed Advantages of the kinetic scheme :
◮ easy implementation ◮ no use of eigenvalues ⇒ computation in non hyperbolic region
Air entrainment in transient flows NTM 32 / 33
Conclusion
Conclusion
Existence of a convex entropy function ⇒ admissible weak solutions System is hyperbolic even for large relative speed Advantages of the kinetic scheme :
◮ easy implementation ◮ no use of eigenvalues ⇒ computation in non hyperbolic region ◮ apparition of vaccum, drying and flooding are obtained
Air entrainment in transient flows NTM 32 / 33
Conclusion
Conclusion
Existence of a convex entropy function ⇒ admissible weak solutions System is hyperbolic even for large relative speed Advantages of the kinetic scheme :
◮ easy implementation ◮ no use of eigenvalues ⇒ computation in non hyperbolic region ◮ apparition of vaccum, drying and flooding are obtained ◮ equilibrium states are well-approximated
Air entrainment in transient flows NTM 32 / 33
Conclusion & Perspectives
Conclusion
Existence of a convex entropy function ⇒ admissible weak solutions System is hyperbolic even for large relative speed Advantages of the kinetic scheme :
◮ easy implementation ◮ no use of eigenvalues ⇒ computation in non hyperbolic region ◮ apparition of vaccum, drying and flooding are obtained ◮ equilibrium states are well-approximated
a lot of things . . . to do
air entrapment and mixed flows more realistic models based on
Air entrainment in transient flows NTM 32 / 33
Air entrainment in transient flows NTM 33 / 33
inner determinant
∆3 = a0 a1 a2 4a0 3a1 4a0 3a1 2a2 , ∆5 = a0 a1 a2 a3 a4 a0 a1 a2 a3 4a0 3a1 2a2 4a0 3a1 2a2 a3 4a0 3a1 2a2 a3 , ∆7 = a0 a1 a2 a3 a4 a0 a1 a2 a3 a4 a0 a1 a2 a3 a4 4a0 3a1 2a2 a3 4a0 3a1 2a2 a3 4a0 3a1 2a2 a3 4a0 3a1 2a2 a3 .
Air entrainment in transient flows NTM 33 / 33
Total head
Theorem
For smooth solutions, the velocities u and v satisfy ∂tv + ∂x v2 2 + c2
a
γ − 1
0 , ∂tu + ∂x u2 2 + ghw(A) cos θ + gZ + c2
a M
γ(S − A)
0 .
Air entrainment in transient flows NTM 33 / 33
The microscopic fluxes
Interpretation : potential bareer
M−
i+1/2(ξ) = positive transmission
i (ξ)
+ 1{ξ<0, ξ2−2g∆Φn
i+1/2>0}Mn
i+1
i+1/2
∆Φi+1/2 := ∆Zi+1/2 = Zi+1 − Zi
Air entrainment in transient flows NTM 33 / 33
The microscopic fluxes
Interpretation : potential bareer
M−
i+1/2(ξ) = positive transmission
i (ξ)
+
reflection
i+1/2<0}Mn
i (−ξ)
+ 1{ξ<0, ξ2−2g∆Φn
i+1/2>0}Mn
i+1
i+1/2
∆Φi+1/2 := ∆Zi+1/2 = Zi+1 − Zi
Air entrainment in transient flows NTM 33 / 33
The microscopic fluxes
Interpretation : potential bareer
M−
i+1/2(ξ) = positive transmission
i (ξ)
+
reflection
i+1/2<0}Mn
i (−ξ)
+ 1{ξ<0, ξ2−2g∆Φn
i+1/2>0}Mn
i+1
i+1/2
∆Φi+1/2 := ∆Zi+1/2 = Zi+1 − Zi
Air entrainment in transient flows NTM 33 / 33