Introduction
ARBITRARY HIGH ORDER STAGGERED SEMI-IMPLICIT DISCONTINUOUS GALERKIN SCHEMES FOR WEAKLY COMPRESSIBLE FLOWS IN ELASTIC PIPES Matteo Ioriatti - Michael Dumbser Universit` a degli studi di Trento Dipartimento di Ingegneria Civile, Ambientale e Meccanica Robert Bosch GmbH Future Mechanical and Fluid Components (CR-ARF3) Renningen - Stuttgart (Germany)
May 24, 2016
Matteo Ioriatti - Michael Dumbser
Shark FV 2016
May 24, 2016
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Introduction
Overview 1
Introduction
2
Derivation of the semi-implicit staggered DG for the 1D model Equations of the model Numerical scheme Solution algorihtm Explicit terms
3
Numerical tests for the 1D model Hagen-Poiseuille profile Steady flow in an elastic pipe Womersley test for unsteady flow Frequency domain
4
The semi-implicit DG scheme for the 2Dxr model Radial DG Theoretical notions Test for the DG 2Dxr
5
Conclusions Matteo Ioriatti - Michael Dumbser
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Introduction
Motivations Investigate compressible fluids in rigid and flexible tubes Recent developements in low order semi-implicit scheme for pressurized flows in flexible pipe:
2Dxr model for blood flows in arteries and veins (Casulli et al 2011); Scalar transport with local time stepping (Tavelli et al 2013); Semi-implicit method for non-hydrostatic flow (Fambri et al. 2013) Semi-implicit method for axially symmetric weakly compressible flows (Dumbser et al. 2014) Arbitrary high order of accuracy using the staggered DG method
Figure: Casulli, Dumbser and Toro 2011 Matteo Ioriatti - Michael Dumbser
Figure: Fambri, Dumbser and Casulli 2013 Shark FV 2016
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Derivation of the semi-implicit staggered DG for the 1D model
Equations of the model
Equations of the model Hypothesis
Simplification of the Navier-Stokes equation in cylindrical coordinates L >> R (Hydrostatic equilibrium) Barotropic fluid PDEs of the 1D Model ∂ ∂ ρA + ρAU = 0 ∂t ∂x ∂ ∂ ∂p ρAU + ρAU 2 + A = −2πRτw ∂t ∂x ∂x Closure - Equation of state for weakly compressible fluid p − pv ρ(p) = ρ0 + c02
(2)
Closure - Tube laws A = A(p) Hooke law s
Laplace Law RL (p) = R0 +
p − p0 β
RH (p) = R0
1+
2Wp E∞
Closure - Friction models τw = τw ,s + τw ,us Matteo Ioriatti - Michael Dumbser
Shark FV 2016
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Derivation of the semi-implicit staggered DG for the 1D model
Numerical scheme
Computational grids
pi(x) pi-1(x)
pi+1(x)
ui+1/2(x)
ui-1/2(x)
xi-1
xi
xi+1
Figure: Staggered one-dimensional meshes
Velocity grid
Pressure grid ph,i (x, t) = φh (x)ˆ ph,i (t) φh (x) = ϕ(ξ)
ˆ h,i+1/2 (t) U(x, t)h,i+1/2 = ψh (x)U ψh (x) = ϕ(ξ)
x = xi + ξ∆x
Matteo Ioriatti - Michael Dumbser
Shark FV 2016
x = xi+1/2 + ξ∆x
May 24, 2016
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Derivation of the semi-implicit staggered DG for the 1D model
Numerical scheme
Integration of the continuity equation xi+1/2
Z
φk xi−1/2
h∂ i ∂ (ρA)i,h + (ρAU)h dx = 0 ∂t ∂x
(5)
ˆ i,h Time derivative: (ρA)i,h = φh (ρA) Z
n+1
xi+1/2
φk φh dx xi−1/2
n
ˆ ˆ (ρA) ∂ ˆ i,h − (ρA)i,h (ρA)i,h = Mekh ∆x ∂t ∆t
1
Z Mehk =
ϕh (ξ)ϕk (ξ)dξ
(6)
0
ˆ i+1/2,h Spatial derivative: (ρAU)i+1/2,h = ψh (ρAU) Z
xi+1/2
φk xi−1/2
h i xi+1/2 ∂ (ρAU)h dx = φk (ρAU)h − xi−1/2 ∂x
Z
xi xi−1/2
∂φk (ρAU)h dx − ∂x
xi+1/2
Z
xi
∂φk (ρAU)h dx (7) ∂x
Universal tensors Z
1
Rukh = ϕk (1)ϕh (1/2) − 1/2
dϕk ϕk (ξ − 1/2)dξ dξ
1/2
Z Lukh = ϕk (0)ϕh (1/2) + 0
dϕ(ξ)k ϕk (ξ + 1/2)dξ dξ (8)
Discrete form with θ-method ρAU n+θ = θρAU n+1 + (1 − θ)ρAU n h i h i ˆ n+1 − (ρA) ˆ n + ∆t Rukh (ρAU) ˆ n+θ ˆ n+θ Mekh (ρA) i,h i,h h,i+1/2 − Lukh (ρAU)h,i−1/2 = 0 ∆x
Matteo Ioriatti - Michael Dumbser
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Derivation of the semi-implicit staggered DG for the 1D model
Numerical scheme
Integration of the momentum equation - Part 1 Left hand side of the momentum equation Z xi+1 ψk xi
hd i ∂ (ρAU)h + Aq ph dx dt ∂x
(10)
ˆ i+1/2,h Time derivative: (ρAU)i+1/2,h = ψh (ρAU) xi+1
Z
ψk ψh dx xi
n ˆ n+1 ˆ (ρAU) d i+1/2,h − (F ρAU)i+1/2,h ˆ (ρAU) i+1/2,h = Mekh ∆x dt ∆t
ˆn Spatial Derivative: Aq = ψq A , ph = φh p ˆh i+1/2,q Z xi+1 h i ˆ q ∂ φh p ˆ ni+1/2 p ˆ ni+1/2 p ψk ψq A ˆh dx = RpA ˆn+1 − LpA ˆn+1 kh h,i+1 kh h,i ∂x xi
(11)
(12)
Universal tensors 1
Z Rpkqh =
ϕk (ξ)ϕq (ξ) 1/2
Z Lpkqh = −
1/2
ϕk (ξ)ϕq (ξ) 0
Matteo Ioriatti - Michael Dumbser
∂ϕh (ξ − 1/2) dξ + ϕk (1/2)ϕq (1/2)ϕh (0) ∂ξ
(13)
∂ϕh (ξ + 1/2) dξ + ϕk (1/2)ϕq (1/2)ϕh (1) ∂ξ
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Derivation of the semi-implicit staggered DG for the 1D model
Numerical scheme
Integration of the momentum equation - Part 2 Right hand side of the momentum equation Z xi+1 h i ψk τs,i+1/2,h + τus,i+1/2,h dx
(14)
xi
Darcy Weisbach model τs = λρ
U 2
λ=
64 Re
γ=
2πλReρU 8
→
τs,h = γq (ρAU)h
(15)
Steady friction term Z xi+1 ˆ h dx = Mekhq γ ˆ n+1 ˆ n+1 ∆x ψk ψq γ ˆq ψh (ρAU) ˆi+1/2,q ψh (ρAU) = Γi+1/2,hk (ρAU) i+1/2,h i+1/2,h xi
(16) Unsteady friction term Z
xi+1 xi
n n ψk ψh τˆus,i+1/2,h dx = Mehk τˆus,i+1/2,h ∆x
(17)
Discrete form Z xi+1 h i n ˆ n+1 ψk τs,i+1/2,h + τus,i+1/2,h dx = Γi+1/2,hk (ρAU) + Mehk τˆus,i+1/2,h ∆x (18) i+1/2,h xi
Matteo Ioriatti - Michael Dumbser
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Derivation of the semi-implicit staggered DG for the 1D model
Numerical scheme
Convolution integral unsteady friction models Zielke Model
t
∂U(ˆt ) tν W (t − ˆt )d ˆt ˆt = 2 ∂t R P −ni ˆt Use an approximated weighted function of the form W = i=N i=1 mi e τw ,us =
τ =
N X
τi
τi =
i=1
2µ R
Z
2µ R
Z
(19)
0
0
t
n ν dU(ˆt ) − i (t−ˆt ) d ˆt mi e R 2 dt
i = 1, 2, ...N
Time derivative + Leibiniz Rule Z d 2µ t dU(ˆt ) ni ν − ni 2ν (t−ˆt ) 2µ dU(t) τi = − mi 2 e R d ˆt + mi dt R 0 dt R R dt
(20)
i = 1, 2, ...N
(21)
The ODE model (Ioriatti - Dumbser - Iben) d ni ν 2µ dU(t) τi = − 2 τi + mi dt R R dt
(22)
Numerical discretization: Implicit Euler for stability reasons τin+1 = Matteo Ioriatti - Michael Dumbser
τin +
2µ mi (U n+1 R 1 + nRi ν2 ∆t Shark FV 2016
− Un)
i = 1, 2, ...N
(23)
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Derivation of the semi-implicit staggered DG for the 1D model
Solution algorihtm
Coupling the equations Final form of the mometum equation h i−1 i ∆t h ˆ n n+1 n+1 n ˆ n+1 ˆn ˆ ni+1/2 p ρAU MeG RpAi+1/2 p ˆh,i+1 −LpA ˆh,i (24) i+1/2,h = Me+∆tΓi+1/2 i+1/2,h −θ ∆x ∆t n n n ˆn ˆ n ˆ ni+1/2 p ˆ ni+1/2 p G ˆus,h ∆t − (1 − θ) Me−1 [RpA ˆh,i+1 − LpA ˆh,i ] i+1/2,h = F ρAU i+1/2,h − τ ∆x
(25)
Continuity equation + Momentum equation n+1 MeρA(ˆ ph,i )
h i−1 ∆t 2 n+1 ˆ ni+1/2 p Ru Me + ∆tΓni+1/2 RpA ˆh,i+1 ∆x 2 h i−1 i−1 ∆t 2 ∆t 2 h n+1 n+1 ˆ ni+1/2 p ˆ ni−1/2 p Ru Me + ∆tΓni+1/2 Lu Me + ∆tΓni−1/2 LpA ˆh,i + θ2 RpA ˆh,i + θ2 ∆x 2 ∆x 2 i−1 ∆t 2 h n+1 ˆ ni−1/2 p − θ2 Lu Me + ∆tΓni−1/2 LpA ˆh,i−1 = ∆x 2 h i h i−1 −1 ∆t n ˆn ˆn ˆ n −θ Ru Me + ∆tΓni+1/2 MeG MeG MeρA h,i i+1/2,h − Lu Me + ∆tΓi−1/2 i−1/2,h ∆x i ∆t h ˆ n ˆ n − (1 − θ) Ru(ρAU) h,i+1/2 − Lu(ρAU)h,i−1/2 ∆x (26) − θ2
Matteo Ioriatti - Michael Dumbser
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Derivation of the semi-implicit staggered DG for the 1D model
Solution algorihtm
Solution algorithm Mildly non-linear system solved using the Newton method MeρA(ˆ p n+1 ) + T · p ˆn+1 = b(pn ) f(pn+1 ) = Me ρA(ˆ pn+1 ) + T · p ˆn+1 − bn
(27)
Newton step df(ˆ pn+1 ) = Me ρA0 (ˆ pn+1 ) + T dˆ pn+1 h df(pˆ n+1 ) i−1 k f(pˆk n+1 ) p ˆn+1 ˆn+1 − k+1 = p k dˆ pn+1 Mass flow and velocity update
(28) (29)
h i−1 i ∆t h ˆ n n+1 n+1 n ˆ n+1 ˆn ˆ ni+1/2 p ρAU MeG RpAi+1/2 p ˆh,i+1 − LpA ˆh,i i+1/2,h = Me + ∆tΓi+1/2 i+1/2,h − θ ∆x (30) Arbitrary high order of accuracy in space ˆ = ρAU) ˆ Scheme unconditionally stable for 0.5 ≤ θ ≤ 1 (when F ρAU θ = 1 first order scheme (Backward Euler method) θ = 0.5 second order scheme (Crank-Nicolson type method) Matteo Ioriatti - Michael Dumbser
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Derivation of the semi-implicit staggered DG for the 1D model
Explicit terms
Non linear convective terms PDE Formulation ∂ρAU ∂ρAU 2 dρAU = + dt ∂t ∂x DG approach Z
xi+1
ψk ψh dx xi
Me∆x
ˆ h,i+1/2 dQ dt
Z
∆t
dQ ∂Q ∂f = + dt ∂t ∂x
xi+1
=
ˆ n+1 ˆ nh,i+1/2 Q − FQ h,i+1/2
→
ψk ψh dx xi
= Me∆x
ˆ h,i+1/2 ∂Q ∂t
Q = ρAU
h i xi+1 Z + ψk · fhn − xi
ˆ n+1 ˆn Q −Q h,i+1/2 h,i+1/2 ∆t
with
xi+1
xi
f = uQ
(31)
∂φk n φh dx · fˆh,i+1/2 ∂x (32)
Z n +[ϕk (1)·fi+1 −ϕk (0)·fi n ]−
1 0
∂ϕ(ξ)k n ϕ(ξ)h dξ·fˆh,i+1/2 ∂ξ (33)
Numerical computation with Rusanov flux h ∆t n ˆ nh,i+1/2 = Q ˆn FQ Me−1 ϕk (1) · fi+1 − ϕk (0) · fi n − h,i+1/2 − ∆x
fi+1 =
Z
1
0
i ∂ϕ(ξ)k n ϕ(ξ)h dξ · fˆh,i+1/2 (34) ∂ξ
i 1 h i 1h n ˆ h,i+3/2 −ϕ(1)·Q ˆ h,i+1/2 ϕ(0)·fˆh,i+3/2 +ϕ(1)·fˆh,i+1/2 − Smax,i+1 ϕ(0)·Q 2 2
h i i 1 n ˆ h,i+1/2 −ϕ(1)·Q ˆ h,i−1/2 ϕ(0)·Q fi = ϕ(0)·fˆh,i+1/2 +ϕ(1)·fˆh,i−1/2 − Smax,i 2 2 1h
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n,+ n,− n Smax,i+1 = max(2|ui+1 |, 2|ui+1 |)
(35)
n Smax,i = max(2|uin,+ |, 2|uin,− |)
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Numerical tests for the 1D model
Hagen-Poiseuille profile
Hagen-Poiseuille solution for rigid pipe µ = 10−3 Pa · s
Laminar and steady flow (Re = 1000 Rigid pipe (R0 = 4 ·
10−3 m
β=
Incompressible flow (c0 = 1030 u(z) =
ρ0 = 998.2kg /m3 );
∆p (R 2 − z 2 ) 4µL
U=
Z
1 A
u(z)dA = A
(37)
0.16 0.14 0.08
0.10
0.12
U [m/s]
500040 500020
p [Pa]
R2 ∆p 8µL
Velocity − Nx=10 P=5
500060
Pressure − Nx=10 P=5
500000
L = 1)
1030 )
0.0
0.2
0.4
0.6
0.8
1.0
0.0
x [m]
0.4
0.6
0.8
1.0
x [m]
Figure: Analytical solution (
Matteo Ioriatti - Michael Dumbser
0.2
) and numerical data •
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Numerical tests for the 1D model
Steady flow in an elastic pipe
Steady flow in an elastic pipe ν = 10−3 Pa · s
Laminar and steady flow (Q = 1.875m3 /s Incompressible flow (c0 =
L = 1)
1030 );
Elastic pipe - Laplace law(R0 = 0.025m
2Q (R 2 (x) − z 2 ) uex (x, t) = πR 4 x
β = 2500) s
Rex (x) =
5
40νQ x πβ
R05 −
pex = p0 + β(Rex (x) − R0 ) (38)
Velocity − Nx=10 P=5
3.0 2.0
2.5
U [m/s]
−15 −20
1.0
−30
1.5
−25
p [Pa]
−10
3.5
−5
4.0
0
Pressure − Nx=10 P=5
0.0
0.2
0.4
0.6
0.8
1.0
0.0
x [m]
0.4
0.6
0.8
1.0
x [m]
Figure: Analytical solution (
Matteo Ioriatti - Michael Dumbser
0.2
) and numerical data •
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Numerical tests for the 1D model
Womersley test for unsteady flow
Womersley test: general information Oscillating pressure gradient pin (t) =
