Eigenmodes in swirling flow in an annular duct

Eigenmodes in swirling flow in an annular duct - recent developments James R. Mathews1 and Nigel Peake1 with Vianney Masson2 , Stéphane Moreau2 , Hélène Posson3 , Ed Brambley4 1 Department

of Engineering/DAMTP, University of Cambridge

2 D´ epartement 3 Airbus 4 Warwick

de G´ enie M´ ecanique, Universit´ e de Sherbrooke

Acoustic Department, Airbus Commercial Aircraft Mathematics Insitute/WMG, University of Warwick

October 4, 2017

Mathews et al

Eigenmodes in swirling flow

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Motivation

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New UHBR turbofan designs have shorter inlet and exhaust.

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Primarily interesting in interstage region with significant swirling flow.

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Key modelling assumptions I I I I I

Assume infinite annular duct h < r < 1. The duct walls can either be hard, or have an acoustic liner with impedance Z . Assume cylindrical coordinate system (x, r , θ). Consider linearised Euler equations for compressible, non-heat conducting, perfect gas. Assume radially varying swirling base flow of the form (Ux (r ), 0, Uθ (r ))

Look for eigenmodes k such that perturbations Z XZ {u, v , w , p}(r , x, θ, t) = {U(r ), V (r ), W (r ), P(r )}e ikx dke imθ e −iωt dω,

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m

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solve linearised Euler equations and boundary conditions. Each eigenmode provides a contribution to the Green’s function of the swirling acoustic analogy as in Posson & Peake JFM 2013. Mathews et al


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Other considerations I I I I

We can consider the eigenmodes numerically or analytically. We can consider homentropic or isentropic flow. We consider how to model the boundary condition with lining: Myers vs corrected Myers vs Modified Myers boundary condition. We can look at the acoustic, hydrodynamic or surface modes: Im(k)

Hydrodynamic modes Acoustic modes Surface modes Critical layer Integration contour

Re(k)

Figure: Schematic of eigenmodes with acoustic liner

All the numerical calculations are calculated in Matlab using my own code, “GreenSwirl”, which relies on open source package Chebfun. Mathews et al


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Numerical modes vs analytical modes (1) As in Mathews & Peake, JSV 2017 (Feb). We consider for now, the acoustic modes in homentropic flow with the Myers boundary condition. I

The eigenmodes are found numerically by solving the eigenvalue problem A(iU, V , iW , P)T = k(iU, V , iW , P)T with Myers boundary condition for standard function matrix A which depends on base flow.

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The analytical method involves using a high-frequency limit, which allows the calculation of the eigenfunction P(r ) using the WKB method and Airy functions.

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Applying the boundary condition then gives an analytical dispersion relation for the eigenmodes, which must then in general be solved numerically. Mathews et al


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Numerical modes vs analytical modes (2) We get excellent agreement for the eigenmodes even in complicated swirling flows:

Im(k)

20

0

−20 −50

−40

−30

−20

−10 Re(k)

0

10

20

Asymptotic −45.802 − 0.062i −43.562 − 0.153i −40.427 − 0.398i −35.267 − 0.823i −26.156 − 1.956i 1.396 + 1.930i 10.056 + 0.863i 14.760 + 0.438i? 17.718 + 0.225i? 20.263 + 0.210i

Numerical −45.803 − 0.062i −43.572 − 0.153i −40.406 − 0.391i −35.258 − 0.820i −26.152 − 1.954i 1.409 + 1.930i 10.078 + 0.862i 14.786 + 0.439i 17.741 + 0.225i 20.274 + 0.210i

Error 0.003% 0.024% 0.055% 0.026% 0.017% 0.535% 0.215% 0.175% 0.131% 0.054%

Figure: Plot of the asymptotic (green squares) and numerical eigenmodes (red crosses) and corresponding table. The parameters are ω = 25, n = −20, h = 0.5, Ux = 0.2 + 0.4r 2 − 0.3r 3 , Uθ (r ) = 0.1r + 0.2/r + 0.3r 2 with lined walls of impedance Zj = 1 − 2i.

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Numerical modes vs analytical modes (3) We also get excellent agreement for the downstream Green’s function numerically and analytically (for a different flow): 1

0.1

1

0.1

0.8

0.08

0.8

0.08

0.6

0.06

0.6

0.06

0.4

0.04

0.4

0.04

0.2

0.02

0.2

0.02

0

0

0

0

−0.2

−0.02

−0.2

−0.02

−0.4

−0.04

−0.4

−0.04

−0.6

−0.06

−0.6

−0.06

−0.8

−0.08

−0.8

−1 −1 −0.8 −0.6 −0.4 −0.2

0

0.2

0.4

0.6

0.8

1

−0.1

−1 −1 −0.8 −0.6 −0.4 −0.2

−0.08 0

0.2

0.4

0.6

0.8

1

−0.1

Figure: Colour plot of the Green’s function for Uθ (r ) = 0.1r + 0.1/r , Ux = 0.5, hard walls, ω = 25 and h = 0.6, x − x0 = 0.5 (a) Real part of numerical acoustic Green’s function A (x |x ); (b) real part of asymptotic acoustic Green’s function p A (x |x ). bω pbω 0 0

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Effect of entropy (1) As in Mathews & Peake, JSV 2017 (Nov). We now consider the modes in isentropic flow with the Myers boundary condition. I

We now have a base flow entropy s0 (r ) which is not constant, and this affects the base flow speed of sound c0 (r ) and density ρ0 (r ).

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The eigenmodes are found numerically in the same way, except we have a fifth variable S (the Fourier transform of the entropy perturbation), a fifth equation and a couple of extra terms.

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The analytical method is unchanged, with the only differences coming from the base flow speed of sound and density varying with base flow entropy.

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We can derive a new acoustic analogy to take into account the isentropic flow, which reduces to the one in Posson and Peake 2013 JFM for homentropic flow. Mathews et al


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Effect of entropy (2) Effect of entropy can be quite significant on the eigenmodes. It can cause: I The number of cut-on modes to change. I The line of cut-off modes to move. I Surface modes to appear. We still get very good agreement between the numerics and analytics: 150

50

= −3 = −2 = −1 =0 =1

U

Cut-on

100

β β β β β

−50 D

−150 −150

S-E

−100

Cut-off

0

−100

−50

0

β = −3 Numerical Asymptotic −91.69 − 0.877i −91.72 − 0.8769i −69.29 − 0.7952i −69.32 − 0.7951i 4.582 + 0.9333i 4.597 + 0.9313i 9.415 + 0.4665i 9.411 + 0.4613i 11.56 + 0.1457i 11.54 + 0.1404i −19.84 + 18.81i −19.82 + 18.77i −26.92 − 19.15i −26.89 − 19.10i −20.45 + 33.94i −20.44 + 33.91i −25.96 − 35.57i −25.96 − 35.53i −7.613 + 55.25i −7.631 + 55.24i 25.49 − 94.02i 25.49 − 94.02i

Figure: Parameters are ω = 25, n = 12, Ux (r ) = 0.3 + 0.2r 2 , Uθ (r ) = 0.2r + 0.1/r , h = 0.6, s0 (r ) = − log(r β ) and lined walls of impedance Zj = 1 − 2i Mathews et al


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Effect of entropy (3)

The entropy has a very significant effect on the Green’s function as well (different flow, no surface modes).

0.8

0.6

0 −1 0.5

1

1.5

2 x

2.5

3

(b) β = 0

1

·10−2

1

3.5

4

(c) β = 0.3

1

·10−2

1 r

r

·10−2

0.8

0.6

0 −1 0.5

1

1.5

2 x

2.5

3

3.5

4

1 r

(a) β = −0.3

1

0.8

0.6

0 −1 0.5

1

1.5

2 x

2.5

3

3.5

4

Figure: Acoustic downstream Green’s function for a source at x0 = 0 and r0 = 0.8. The parameters of the flow are Ux (r ) = 0.3 + 0.2r 2 , Uθ (r ) = 0.1r + 0.1/r , s0 (r ) = − log(r β ), n = 32, ω = 50, h = 0.6 and hard walls.

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Effect of entropy (4) We now consider the hydrodynamic modes which accumulate at the ends of the critical layer. Heaton & Peake JFM 2006 showed there were three regimes, depending on the swirling base flow. 1. Exponential accumulation on the real line (stable). 2. Algebraic accumulation on the real line (stable). 3. Algebraic accumulation in the complex plane (unstable). These cases still hold in isentropic flow, but the conditions change. Changing the entropy can change regime, eg from unstable to stable. 1 β = −0.3 β=0 β = 0.3 β = 0.6

0.5 0 −0.5

19

19.5

20

20.5

21

21.5

−1 13

k

13.2

13.4 k

13.6

13.8

Figure: Plot of the hydrodynamic modes for different flows. Orange diamonds: β = 0.6, green crosses : β = 0.3, blue triangles: β = 0 and red circles: β = −0.3. (a) Ux (r ) = 0.5 + 0.1r 4 , Uθ (r ) = 0.5r 4 , h = 0.5, n = −3, ω = 10. Case 2. (b) Ux (r ) = 0.7 − 0.5r 2 , Uθ (r ) = 0.1r + 0.25/r , h = 0.5, n = −5, ω = 3. Case 2/3. Mathews et al


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Boundary conditions and surface modes (1) Joint work with Masson, Moreau, Posson & Brambley. I We now consider the modes in homentropic flow with lined walls, modelled with different boundary conditions. I Want to consider boundary conditions which avoid having to consider boundary layers in the swirling base flow. (Ingard-)Myers Deriving by matching fluid and solid displacement at the boundary (+ outer wall, − inner wall): P ω ± = Zr, V ω − kUx (r ) − mUθ (r ) r

Infinitely thin BL In [Masson et al, AIAA 2017] it was shown that (Corrected Myers) 2 P ω r ,? r ,? r iρ0 (r )Uθ (r ) ± = Z , Z = Z ∓ , V rω ω − kUx (r ) − mUθ (r ) r

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Extra term related to centrifugal force which causes spring like term in mass-spring-damper. Biggest difference at low frequency due to 1/ω term. Mathews et al


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Boundary conditions and surface modes (2) I

We now consider how to model a boundary layer of finite thickness δ using the Modified Myers BC (derived in [Mathews et al AIAA 2017])

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Let Λ = kUx (r ) + ±

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mUθ (r ) r

− ω, then

P ωZ r ,? + Λ(r )δI2r + Z r Λ(r )δJ2r =− , V Λ(r ) (1 − δI1r − Z r δJ1r )

where δI , δJ are O(δ) integrals involving base flow.

In the limit of δ = 0 we get the corrected Myers boundary condition.

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In the limit of no swirl Uθ = 0 we recover the result from Brambley AIAA 2011.

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For a linear boundary layer we no longer get the boundary condition is a rational polynomial in k (except special cases such as hard walls, high frequency or no swirl), which makes eigenvalue problem non-linear. Mathews et al


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Boundary conditions and surface modes (3) (a) Eigenmodes (b) Upstream Green’s function with source (r0 , x − x0 ) = (0.75, −1)

60

·10−5

40

3 20 =(k)

2 0

1 −20

0

−40 −60 −20

−1 −15

−10

−5

0

5

10

0.5

0.6

0.7

0.8

0.9

1

r