Development and calibration of a Reduced Order Model for cavity flows
Development and calibration of a Reduced Order Model for cavity flows Kaushik Kumar NAGARAJAN .
Laurent CORDIER, Christophe AIRIAU Azeddine KOURTA IMFT, France
24 November 2008 Marie Curie EST action AeroTraNet (www.imft.fr/aerotranet)
Development and calibration of a Reduced Order Model for cavity flows Outline
Outline 1
Introduction
2
Reduced Order Modelling
3
Cavity flow configuration
4
Current work and perspectives
Development and calibration of a Reduced Order Model for cavity flows Introduction Objectives
Objectives To reduce the cost of computation involved with the classical CFD tools like DNS, URANS, LES To identify the most essential features of fluid flow and be able to approximate the original dynamics of the more complicated N–S equations To perform optimal flow control to use in unsteady aerodynamics
Development and calibration of a Reduced Order Model for cavity flows ROM Introduction to POD based Reduced Order Modelling
Introduction to POD based Reduced Order Modelling
POD is a static decomposition of flow fields into uncorrelated energy ranked coherent structures Let uk (x) = u(x, tk ) denote the snapshot taken at time tk set obtained from a DNS computation. (uk )M k =1 ⊂ H be an ensemble of M ∈ N snapshots We seek an expression for the velocity field of the form u(x, t) =
M X i=0
ai (t)ui (x)
Development and calibration of a Reduced Order Model for cavity flows ROM Introduction to POD based Reduced Order Modelling
Introduction to POD based Reduced Order Modelling Among all the subspaces S ⊂ H of a given dimension N < M the POD bases minimises the average projection error of J(S) = Ekuj − Ps uj k2 A M dimensional Dynamical Representation is obtained by a Galerkin projection of the governing equations onto the subspace S : a˙ = F(a(t), t)
Development and calibration of a Reduced Order Model for cavity flows ROM ROM for cavity flows
Governing equations for cavity flows (Rowley, 2001) Isentropic flow equations with u, v and c (sound velocity)
1 2 1 ccx = (uxx + uyy ) Re M2 γ − 1 2 1 1 vt + uvx + vvy + 2 ccy = (vxx + vyy ) Re M γ−1 γ−1 c(ux + vy ) = 0 ct + ucx + vcy + 2
ut + uux + vuy +
with q = (u, v, c) : q˙ =
1 1 L(q) + 2 Q(q, q) Re M
where uxx + uyy L(q) = vxx + vyy ,
0
2 c1 c2 u1 u2x + v1 u2y + γ− x 1 2 1 2 1 2 1 2 u vx + v vy + γ−1 c cy γ−1 1 1 2 1 2 2 2 u cx + v cy + 2 c (ux + vy )
Q(q1 , q2 )
=
Development and calibration of a Reduced Order Model for cavity flows ROM ROM for cavity flows
Inner product preserving energy norm Z
hq1 , q2 i =
V
(u1 u2 + v1 v2 +
2α γ−1
c1 c2 )dV
q decomposition with the POD modes : q(x, t) = q(x) +
n X
aj (t)φj (x)
j=1
ROM obtained by Galerkin projection are X 1 1 b + b2k + Re k n
a˙ k =
i=1
n X 1 1 Lik + L2ik ai + Qijk ai aj Re i,j=1
where b1k = hL(q), φk i, L1ik = hL(φ), φk i,
L2ik = hQ(q, φi ) + Q(φi , q), φk i,
b2k = hQ(q, q), φk i Qijk = hQ(φi , φj ), φk i
Development and calibration of a Reduced Order Model for cavity flows Cavity flow configuration Eigen Spectrum
Cavity flow, DNS (P. Comte), L/D = 2.0, Mach = 0.6, ReD = 1500 (Rowley, 2001) 0 −1
log(singular Value)
−2 −3 −4 −5 −6 −7 0
10
20 30 Mode Number
40
50
1
The eigen modes occur in pairs with almost the same amplitude but with a phase shift
2
The first 4 modes capture 99.99% of the total energy
Development and calibration of a Reduced Order Model for cavity flows Cavity flow configuration Time Traces
0.2 a1(t) a2(t) a3(t) a4(t)
0.15
amplitude
0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 0
5
10
15
20
25
30
35
40
45
non−dimensional time
Figure: Time traces of the first 4 POD coefficients
50
Development and calibration of a Reduced Order Model for cavity flows Cavity flow configuration Spatial POD mode - u
Development and calibration of a Reduced Order Model for cavity flows Cavity flow configuration Spatial POD mode - v
Development and calibration of a Reduced Order Model for cavity flows Cavity flow configuration Time traces of ROM non calibrated case
0.5 a2(t)
a1(t)
0.5 0 −0.5
0
1
2
3
4
0 −0.5
5
0
1
2
Time a3(t)
a4(t) 0
1
2
3
4
5
0
1
2
Time a5(t)
a (t)
6
0
1
2
3
4
5
0
1
2
Time
3
4
5
3
4
5
Time a (t)
a7(t)
5
0.02 0
8
0 0
1
2
3
4
5
−0.02
0
1
Time
2 Time
0.2 a10(t)
0.1 a9(t)
4
0 −0.02
0.02
0 −0.1
3 Time
0.02
0
−0.02
5
0 −0.1
0.05
−0.05
4
0.1
0 −0.05
3 Time
0.05
0
1
2
3 Time
4
5
0 −0.2
0
1
2 3 4 5 Résultats Time POD interpolés Prédiction POD ROM non calibré
Figure: Evolution of the first 10 coefficients
Development and calibration of a Reduced Order Model for cavity flows Cavity flow configuration Calibration of the ROM
ROM prediction differs from the DNS prediction ⇒ need for calibration a˙ k =
n n X X 1 1 1 1 bk + b2k + Lik + L2ik ai + Qijk ai aj Re Re i=1
i,j=1
Minimise E=
Nt h K X X i=1 n=1
∂E ∂L1,2 ik
= 0,
∂E ∂b1,2 i
i2
a˙ DNS (tn ) − a˙ ROM (tn ) i i
=0
Galletti (2004)
Development and calibration of a Reduced Order Model for cavity flows Cavity flow configuration Time traces of ROM calibrated case
0.2 a2(t)
a1(t)
0.2 0 −0.2
0
1
2
3
4
0 −0.2
5
0
1
2
Time
0 0
1
2
3
4
0
1
2
a5(t)
a (t) 1
2
3
4
−0.02
5
0
1
2
4
5
3
4
5
0.01 a (t)
a7(t)
3 Time
0.01
0
8
0 0
1
2
3
4
5
−0.01
0
1
Time −3
a10(t)
x 10
0 0
2 Time
−3
a9(t)
5
0
6
0
Time
−5
4
0.02
0
5
3 Time
0.05
−0.01
5
0 −0.1
5
Time
−0.05
4
0.1 a4(t)
a3(t)
0.05 −0.05
3 Time
1
2
3 Time
4
5
5
x 10
0 −5
0
1
2 3 4 Résultats POD interpolés Time Prédiction POD ROM calibré
Figure: Comparison after calibration (L. CORDIER)
5
Development and calibration of a Reduced Order Model for cavity flows Cavity flow configuration Calibration Error Erreurs e1 par mode. Mean square procedure (constant and linear terms)
−1
10
−2
10
−3
10
−4
10
−5
10
−6
10
1
1.5
2
2.5
3
3.5 4 4.5 POD ROM non calibr� Modes POD POD ROM calibr�
5
5.5
Figure: Comparison of errors before and after calibration
6
Development and calibration of a Reduced Order Model for cavity flows Current work and perspectives
Current Work Sensitivity analysis and optimal control of cavity instabilities Applications to higher Reynolds number flow Feedback control of non linear dynamical systems
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