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A multiphase approach to the construction of pod-rom for flows induced by rotating solids
Antoine Falaize, Erwan Liberge, Aziz Hamdouni
To cite this version:
Antoine Falaize, Erwan Liberge, Aziz Hamdouni. A multiphase approach to the construction of pod- rom for flows induced by rotating solids. Workshop CSMA Junior, Mar 2018, Gif-sur-Yvette, France.
�hal-01736872�
A MULTIPHASE APPROACH TO THE CONSTRUCTION OF POD-ROM FOR FLOWS INDUCED BY ROTATING SOLIDS
Antoine Falaize, Erwan Liberge and Aziz Hamdouni,
Team M2N, LaSIE, CNRS UMR 7356, Universit´ e de la Rochelle Avenue Michel Cr´ epeau 17042 La Rochelle cedex 1
Introduction
• Objective: Construct reduced order models (ROM) for the simulation of turbomachinery with imposed rotation velocity by proper orthogonal decomposition (POD).
• Difficulty: The POD yields a spatial basis from temporal correlations (here of the velocity).
• Approach:
1. Extend the Navier-Stokes equations to the solid (rotor) domain by the multiphase ap- proach. The body velocity is enforced via distributed Lagrange multipliers.
2. Build a single POD basis for the multiphase velocity and project the governing equations.
References
[1] Glowinski, R., Pan, T. W., Hesla, T. I., & Joseph, D. D. (1999). A distributed Lagrange multiplier/fictitious domain method for par- ticulate flows. International Journal of Multiphase Flow, 25(5), 755-794.
[2] Liberge, E., & Hamdouni, A. (2010). Reduced order modelling method via proper orthogonal decomposition (POD) for flow around an oscillating cylinder. Journal of fluids and structures, 26(2), 292-311.
[3] Mosquera, R. , Hamdouni, A., El Hamidi, A., Allery, C. (2018).
POD Basis Interpolation via Inverse Distance Weighting on Grass- mann Manifolds. Manuscript submitted to AIMS Journals.
1. Multiphase approach
Level-set Signed distance to the fluid/solid in- terface Γ
I(t):
χ(x, t) =
+d
x, Γ
I(t)
if x ∈ Ω
S(t) ∪ Γ
I(t),
−d
x, Γ
I(t)
if x ∈ Ω
F(t)
Smoothed Heaviside (immersion depth )
h
(•) = 1 2
1 + tanh π •
.
Membership function for Ω
S(t) 1
ΩS(t)(x) = h
χ(x, t)
.
Multiphase quantities
Velocity field over Ω = Ω
S(t) ∪ Ω
F(t):
u(x, t) = 1
ΩS(t)(x) u
S(x, t) + I − 1
ΩS(t)(x)
u
F(x, t).
Material properties (density and viscosity):
ρ(x, t) = 1
ΩS(t)(x) ρ
S+ I − 1
ΩS(t)(x)
ρ
Fν (x, t) = 1
ΩS(t)(x) ν
S+ I − 1
ΩS(t)(x)
ν
F.
χ(x, t) 1
ΩS(t)(x)
2. Full order model [1]
Denoting u
ωthe rota- tion velocity, λ the La- grange multiplier and µ the test function as- sociated with the ro- tation constraint, the weak form of the cou- pled problem is
0 =
Z
Ω
ρ
∂ u
∂t + ∇u · u
· v dx − Z
Ω
f · v dx + R
Ω
2 ν Tr (D (u) · D (v )) dx
− R
Ω
p ∇ · v dx + R
ΩS(t)
λ · v dx + R
Ω
q ∇ · u dx + R
ΩS(t)
µ · (u − u
ω) dx, with an appropriate standard functional setting.
3. Standard POD-ROM [2]
Mean field u(x) = 1 N
TNT
X
n=1
u(x, t
n)
Fluctuating filed u(x, t) = e u(x, t) − u(x) Data matrix U
mn≡ u(x e
m, t
n)
POD basis: left singular vectors of U
Φ =
φ
i(x)
1≤i≤NΦ
, N
Φ<< N
T<< N
XDecomposition
u(x, t) ' u(x) +
NΦ
X
i=1
φ
i(x) α
i(t)
Galerkin projection over the POD basis
A(t) · dα(t)
dt = B(t) · α(t)+ C(t) : α(t) ⊗ α(t)+ F(t),
⇒ Full projection at each timestep (cost ∼ N
X).
Aij(t) =
Z
Ω
ρ(x, t) I dx, Bij(t) =
Z
Ω
ρ(x, t) bρ
i,j(x) dx + Z
Ω
ν(x, t) bν
i,j(x) dx, Cijk(t) =
Z
Ω
ρ(x, t) ci,j,k(x) dx, Fi(t) =
Z
Ω
ρ(x, t) fρ
i (x) dx + Z
Ω
ν(x, t) fν
i (x) dx +
Z
ΩS(t)
fλ
i (x) dx.
4. Proposed POD-ROM
⇒ POD of the membership function POD basis Λ =
Λ
i(x)
1≤i≤NΛ
, N
Λ<< N
X. Decomposition
1
ΩS(t)(x) ' P
NΛi=1
Λ
i(x) γ
i(t) Periodicity
Coefficients γ
i(t) → b γ
i(θ ) determined a priori.
Insertion in the standard POD-ROM
A(t) b · dα(t)
dt = B(t) b · α(t) + C(t) : b α(t) ⊗ α(t)+ F(t), b
⇒ Matrices evaluation at each timestep (cost ∼ N
Λ).
Aijb (θ) = ai,j + PNΛ
k=1 ai,j,k γke (θ), Bijb (θ) = bi,j + PNΛ
k=1 ebi,j,k γk(θ), Cijkb (θ) = ci,j,k + PNΛ
k=1 ci,j,k,l γke (θ), Fib (θ) = f i + PNΛ
j=1 fi,j γke (θ) + λi.
5. Results
Φ: POD modes for the velocity
Coeff α
i(t): full order Vs. standard POD-ROM.
Λ: POD modes for the membership function
Coeff α
i(t): full order Vs. proposed POD-ROM.
Conclusion
Contributions
• Efficient reconstruction of the velocity in both the fluid and solid domains, while sub- stantially reducing the computational cost.
• Very general: Any simulation code for the incompressible Navier-Stokes eq. can be used to generate the data (u(x, t
n)
1≤n≤NT
