[PPT] - Amplification and saturation of the thermoacoustic instability in a PowerPoint Presentation

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Experiments Theory Simulations of transient regimes Conclusion

Amplification and saturation of the thermoacoustic instability in a standing-wave prime mover.

Matthieu Gu´ edra(a), Thibaut Devaux, Guillaume Penelet, Pierrick Lotton

Laboratoire d’Acoustique de l’Universit´ e du Maine, UMR CNRS 6613 Avenue Olivier Messiaen 72085 Le Mans Cedex 9, FRANCE (a) matthieu.guedra.etu@univ-lemans.fr 1 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion

Framework

1

Experiments description of the thermoacoustic device transient regimes measurements

2

Theory description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

3

Simulations of transient regimes

4

Conclusion

2 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of the thermoacoustic device transient regimes measurements 1

Experiments description of the thermoacoustic device transient regimes measurements

2

Theory description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

3

Simulations of transient regimes

4

Conclusion

3 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of the thermoacoustic device transient regimes measurements

Experiments

description of the thermoacoustic device

Figure: (a) Photograph of the experimental

apparatus. (b) Photograph of the hot end of the

stack.

Q

xs xh Lx microphone

Figure: Schematic drawing of the standing-wave prime mover.

600 CPSI glass tube ceramic stack L 59 cm ls 4.8 cm R 2.6 cm rs 0.45 mm Microphone Bruel & Kjaer – 1/4 inch Acquisition PC SoundCard

4 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of the thermoacoustic device transient regimes measurements

Experiments

description of the thermoacoustic device

Figure: (a) Photograph of the experimental

apparatus. (b) Photograph of the hot end of the

stack.

Q

xs xh Lx microphone

Figure: Schematic drawing of the standing-wave prime mover.

10 20 30 40 50 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Qonset (W ) xs (m)

Figure: Stability curve as function of the location xs of the stack.

4 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of the thermoacoustic device transient regimes measurements

Experiments

transient regimes measurements

Figure: Q(t = 0) = 16W (slightly below Qonset = 16.9W ). (a) ∆Q/Q = 16%, (b) ∆Q/Q = 34% ,(c) ∆Q/Q = 53%.

31.6cm

Q xs xh L x microphone

Figure: Q(t = 0) = 18W (slightly below Qonset = 19.6W ). (a) ∆Q/Q = 16%, (b) ∆Q/Q = 24% ,(c) ∆Q/Q = 30%.

21.6cm

Q xs xh L x microphone 5 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime 1

Experiments description of the thermoacoustic device transient regimes measurements

2

Theory description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

3

Simulations of transient regimes

4

Conclusion

6 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

description of acoustic propagation using transfer matrices

Harmonic plane wave assumption

p1(x, t) = ℜ

˜

p1(x)e−iωt and ξ1(x, y, t) = ℜ

˜

ξ1(x, y)e−iωt , (1) where ξ1 = v1,x, ρ1, τ1, s1.

xs

xh L x Tc Th

˜

p1(L) ˜ u1,x(L)

= Mw × Ms × M1 ×
˜

p1(0) ˜ u1,x(0)

M1 =
cos(kxs)

iZc sin(kxs) iZ−1 c sin(kxs) cos(kxs)

Ms and Mw derived from the linear thermoacoustic propagation equation transformed into a Volterra integral equation of the

second kind [Penelet et al., Acust. Acta Acust. (2005)]. 7 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

description of acoustic propagation using transfer matrices

Harmonic plane wave assumption

p1(x, t) = ℜ

˜

p1(x)e−iωt and ξ1(x, y, t) = ℜ

˜

ξ1(x, y)e−iωt , (1) where ξ1 = v1,x, ρ1, τ1, s1.

xs

xh L x Tc Th

˜

p1(L) ˜ u1,x(L)

=
Mpp (ω, T (x))

Mpu (ω, T (x)) Mup (ω, T (x)) Muu (ω, T (x))

×
˜

p1(0) ˜ u1,x(0)

Appropriate boundary conditions

rigid wall : ˜ u1,x(L) = 0 no radiation : ˜ p1(0) = 0

= ⇒ Muu (ω, T(x)) = 0.

7 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

amplification/attenuation of the acoustic wave

Muu (ω, T(x)) = 0. (2) A solution (ω, T) of Eq. (2) represents an operating point of the system. In the Fourier domain (ω ∈ R), it describes an equilibrium point : either unstable (onset threshold),

r stable (steady state),

corresponding to an acoustic wave which is neither amplified, nor attenuated in both cases.

8 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

amplification/attenuation of the acoustic wave

Muu (ω, T(x)) = 0. (2) A solution (ω, T) of Eq. (2) represents an operating point of the system. In the Fourier domain (ω ∈ R), it describes an equilibrium point : either unstable (onset threshold),

r stable (steady state),

corresponding to an acoustic wave which is neither amplified, nor attenuated in both cases. “quasi-steady” state assumption ω = Ω + iǫg ⇒ p1(x, t) = eǫgtℜ ˜ p1(x)e−iΩt , ǫg << Ω on the time scale of few acoustic periods. For a fixed temperature distribution T(x), the solution of Muu (Ω, ǫg) = 0 gives the angular frequency of the oscillations and the amplification rate.

8 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

determination of the onset threshold

ǫg (T(x)) = 0. (3)

Qonset (W ) xs (m) 10 20 30 40 50 0.1 0.15 0.2 0.25 0.3 0.35 0.4

T (◦C) T (◦C) T (◦C) xs xs xs xh xh xh L L L 40 80 120 100 200 200 400

Figure: Stability curve as function of the location xs of the stack.

9 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

Ordinary differential equation for the acoustic pressure amplitude

dP1 dt − ǫg (T (x, t)) P1(t) = 0, with P1(t) = |p1(L, t)|. (4)

10 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

Ordinary differential equation for the acoustic pressure amplitude

dP1 dt − ǫg (T (x, t)) P1(t) = 0, with P1(t) = |p1(L, t)|. (4)

ne-dimensional heat diffusion in the prime mover
xs

xh L x

10 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

Ordinary differential equation for the acoustic pressure amplitude

dP1 dt − ǫg (T (x, t)) P1(t) = 0, with P1(t) = |p1(L, t)|. (4)

ne-dimensional heat diffusion in the prime mover
xs

xh L x Tc Tc Tc

Q πR2

10 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

Ordinary differential equation for the acoustic pressure amplitude

dP1 dt − ǫg (T (x, t)) P1(t) = 0, with P1(t) = |p1(L, t)|. (4)

ne-dimensional heat diffusion in the prime mover
xs

xh L x Tc Tc Tc

Q πR2

ϕac

10 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

thermoacoustic heat flux along the stack

x x

xs xs xh xh L L 0.2 0.2 0.4 0.4 0.6 0.6 0.8 0.8 1 1 |˜ p1/˜ p1(L)| |˜ v1,x/˜ v1,x(L)| Φp,v (rad) 1.56 1.58 1.6 1.62

Figure: Acoustic fields in the thermoacoustic prime mover at onset threshold, for xs = 30 cm and for an arbitrary acoustic pressure amplitude |˜ p1(L)| = 1 P a.

ϕac = 1 2ρ0c0ℜ

˜

s1˜ v∗

1,x

,
scillating part of entropy :

˜ s1 = −˜ p1 ρ0T Fκ(y) − i Cp ω ∂xT T ˜ v1,x 1 − fν ×

1 −

σFν (y) − Fκ(y) σ − 1

11

Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

acoustic streaming

L x 12 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

acoustic streaming

L xs xh x 12 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

acoustic streaming

L xs xh x 12 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

acoustic streaming

L xs xh x

Axial component of the streaming velocity : V2,x = v2,x + ρ1v1,x ρ0 , where v2,x is the second-order time-averaged Eu- lerian velocity [Bailliet et al., J. Acoust. Soc.

Am. (2001)].

V2x (m.s−1) ×1.10−9 r/R

4
2

2 4 1

1 √ 2

1

−1 √ 2

12 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

acoustic streaming

L xs xh x −v(f)

str(x)

v(f)

str(x)

inner zone (i)

uter zone (o)

r/R 1

1 √ 2

1

−1 √ 2

Averaged streaming velocity :

v(f)

str(x)=

2π πR2 R

V2,x
rdr,

v(s)

str(x)= 2π

πr2

s

rs

V2,x
rdr.

V2x (m.s−1) ×1.10−9 r/R

4
2

2 4 1

1 √ 2

1

−1 √ 2

12 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

acoustic streaming

−v(f)

str(x)

v(f)

str(x)

inner zone (i)

uter zone (o)

r/R 1

1 √ 2 heat convection at x = xh

13 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

acoustic streaming

−v(f)

str(x)

v(f)

str(x)

inner zone (i)

uter zone (o)

r/R 1

1 √ 2 heat convection at x = xh

Tc Tc Th Th ϕconv ϕconv

13 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

acoustic streaming

−v(f)

str(x)

v(f)

str(x)

inner zone (i)

uter zone (o)

r/R 1

1 √ 2 heat convection at x = xh

Estimation of the convection flux taken away from the hot end stack by the mass flow : ϕconv(xh) ≃ ρ0Cpv(f)

str(xh) (T(xh) − Tc)

Tc Tc Th Th ϕconv ϕconv

13 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

acoustic streaming Computation of the temporal evolution of acoustic streaming in the resonator :

1

First estimation of averaged streaming velocity Γ(f)

v

(x) = v(f)

str(P1 = 1 P a, Tonset(x)), 2

First order differential equation for the acoustic streaming : τ (f) dv(f)

str

dt + v(f)

str = Γ(f) v

P 2

1 ,

where τ (f) = 4R2

π2ν is a characteristic time for stabilization of acoustic streaming [Amari et

al., Acust. Acta Acust. (2003)].

ϕconv and τ (f) are estimated in a very simplified way and can constitute adjusting parameters for the model.

14 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

ne-dimensional heat diffusion in the prime mover
xs

xh L x

15 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

ne-dimensional heat diffusion in the prime mover
xs

xh L x

(ρ0Cp)(i,o) DT (i,o)

Dt = ∂ ∂x  λ(i,o) ∂T (i,o) ∂x   − h(i,o)

T (i,o) − Tc
,

with DT (i,o) Dt = ∂T (i,o) ∂t ∓v(f) str ∂T (i,o) ∂x , 15 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 29

Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

ne-dimensional heat diffusion in the prime mover
xs

xh L x

(ρ0Cp)(i,o) DT (i,o)

Dt = ∂ ∂x  λ(i,o) ∂T (i,o) ∂x   − h(i,o)

T (i,o) − Tc
,

with DT (i,o) Dt = ∂T (i,o) ∂t ∓v(f) str ∂T (i,o) ∂x ,

(ρsCs)(i,o) DT (i,o)

Dt = ∂ ∂x  λ(i,o) s ∂T (i,o) ∂x   − h(i,o) s

T (i,o) − Tc
−

∂ϕac ∂x , with DT (i,o) Dt = ∂T (i,o) ∂t ±v(s) str ∂T (i,o) ∂x 15 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 30

Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

ne-dimensional heat diffusion in the prime mover
xs

xh L x

(ρ0Cp)(i,o) DT (i,o)

Dt = ∂ ∂x  λ(i,o) ∂T (i,o) ∂x   − h(i,o)

T (i,o) − Tc
,

with DT (i,o) Dt = ∂T (i,o) ∂t ∓v(f) str ∂T (i,o) ∂x ,

(ρsCs)(i,o) DT (i,o)

Dt = ∂ ∂x  λ(i,o) s ∂T (i,o) ∂x   − h(i,o) s

T (i,o) − Tc
−

∂ϕac ∂x , with DT (i,o) Dt = ∂T (i,o) ∂t ±v(s) str ∂T (i,o) ∂x

T (xs) = T (i)(L) = T (o)(L) = Tc,

15 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

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Experiments Theory Simulations of transient regimes Conclusion description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

Theory

transient regime

ne-dimensional heat diffusion in the prime mover
xs

xh L x

(ρ0Cp)(i,o) DT (i,o)

Dt = ∂ ∂x  λ(i,o) ∂T (i,o) ∂x   − h(i,o)

T (i,o) − Tc
,

with DT (i,o) Dt = ∂T (i,o) ∂t ∓v(f) str ∂T (i,o) ∂x ,

(ρsCs)(i,o) DT (i,o)

Dt = ∂ ∂x  λ(i,o) s ∂T (i,o) ∂x   − h(i,o) s

T (i,o) − Tc
−

∂ϕac ∂x , with DT (i,o) Dt = ∂T (i,o) ∂t ±v(s) str ∂T (i,o) ∂x

T (xs) = T (i)(L) = T (o)(L) = Tc,
T (i)(xh) − T (o)(xh) = 0,
λs ∂xT |

x+ h − λf ∂xT | x− h +ϕconv(xh) − ϕac(xh) = Q πR2 , 15 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 32

Experiments Theory Simulations of transient regimes Conclusion 1

Experiments description of the thermoacoustic device transient regimes measurements

2

Theory description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

3

Simulations of transient regimes

4

Conclusion

16 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 33

Experiments Theory Simulations of transient regimes Conclusion

Simulations of transient regimes

50 100 150 200 250 300 50 100 150 200 250 300 350 400 450 t (s) p (Pa) x

s=10cm, Q=Q

nset(x

s), ∆Q/Q=1%

thermoacoustic heat flux acoustic streaming both effects

50 100 150 200 250 300 50 100 150 200 250 300 350 400 450 t (s) p (Pa) xs=20cm, Q=Qonset(xs), ∆Q/Q=1% 100 200 300 400 500 600 700 800 900 1000 200 400 600 800 1000 1200 1400 1600 1800 t (s) p (Pa) xs=40cm, Q=Qonset(xs), ∆Q/Q=1%

17 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 34

Experiments Theory Simulations of transient regimes Conclusion

Simulations of transient regimes

P1(L, t) (P a) t (s) τ (f) = 0 s τ (f) = 5 s τ (f) = 10 s τ (f) = 18 s [Amari et al., (2003)] 200 400 600 800 500 1000 1000 1500 2000 2500 3000 3500 4000

Figure: At t = 0, Q = Qonset, ∆Q

Q

= 1%. xs = 10 cm and ϕac = 0.

P1(L, t) (P a) t (s) 200 400 600 800 500 1000 1000 1500 2000 2500 3000 3500 4000

Figure: xs = 20 cm.

P1(L, t) (P a) t (s) 200 400 600 800 1000 1000 2000 3000 4000 5000

Figure: xs = 40 cm.

18 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 35

Experiments Theory Simulations of transient regimes Conclusion 1

Experiments description of the thermoacoustic device transient regimes measurements

2

Theory description of acoustic propagation using transfer matrices amplification/attenuation of the acoustic wave determination of the onset threshold transient regime

3

Simulations of transient regimes

4

Conclusion

19 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 36

Experiments Theory Simulations of transient regimes Conclusion

Conclusion

A quarter-wavelength thermoacoustic prime mover has been studied and measurements of transient regimes show an “overshoot” before stabilization and periodic “on-off” of the acoustic wave for particular exciting conditions. A model describing amplification and saturation of the acoustic wave has been developped and applied to this prime mover, including two non-linear effects : thermoacoustic heat flux in the stack, and acoustic streaming in the heated part

f the resonator.

The “overshoot” is reproduced by both effects. Saturation seems to be mainly due to thermoacoustic heat pumping in the stack Complete switch-off of the wave would be linked to different time scales (stabilization time of acoustic streaming seems to play an important role).

20 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 37

Appendix

ther experimental results

Figure: xs = 34.1 cm, Qonset = 17 W Figure: xs = 26.6 cm, Qonset = 16.9 W

21 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 38

Appendix

amplification rate and onset threshold 50 100 150 200 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Qonset (W ) xs (m)

nset mode 1
nset mode 2

Figure: Stability curve as function of the location xs of the stack.

22 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 39

Appendix

amplification rate and onset threshold 1 1 1 1.4 1.4 1.6 1.6 1.8 1.8 2 2 2.2 2.2 2.4 2.4 2.6 2.6

100
50

50 100 0.9 1.1 1.2 1.2 1.2

Th Tc Th Tc

ǫg

Ω Ωres

Figure: Amplification rate ǫg and normalised frequency

Ω Ωres in function of temperature for 4

locations of the stack.

22 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 40

Appendix

thermoacoustic flux in the stack 0.001

0.001
0.002
0.003
0.004
0.005

0.3 0.31 0.32 0.33 0.34 ϕac ϕac −λac ∂T

∂x

ℑ{g}J − ℜ{g}I x (m)

ϕac = ℑ{g}J − ℜ{g}I − λac ∂T ∂x , with

I = 1 2 ℜ{˜ p1˜ v∗

1,x}

J = 1 2 ℑ{˜ p1˜ v∗

1,x}

g = fκ − f ∗

ν

(σ + 1)(1 − fν) λac = ρ0Cp ℑ{σfν − fκ} (σ2 − 1)|1 − fν|2 |˜ v1,x|2 2ω

23 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 41

Appendix

spatial distributions of acoustic streaming

1e−8

−3 −2 −1 1 2 3 vs 1 0.5 −0.5 −1 0.31 0.32 0.33 0.34 0.3 eta x

V

1
1

1 1 2 3 1e-8 r/rs

1 √ 2 −1 √ 2

V

2 4 6

2
4
6

1e-9 r/R

1

1 √ 2 −1 √ 2

1 24 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 42

Appendix

taking into account heat convection at interface

25 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 43

Appendix

ther simulations of transient regimes

50 100 150 200 250 300 200 400 600 800 1000 1200 1400 1600 t (s) p (Pa) xs = 31.6cm 16% 34% 53% 50 100 150 200 250 300 100 200 300 400 500 600 700 t (s) p (Pa) xs = 21.6cm 16% 24% 30%

26 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine

SLIDE 44

Appendix

ther simulations of transient regimes

P1(L, t) (P a) t (s) xs = 10 cm xs = 15 cm xs = 20 cm xs = 30 cm xs = 40 cm

50 100 150 200 200 250 300 400 600 800 1000 1200 1400

Figure: At t = 0, Q = Qonset, ∆Q

Q

= 1%. — : v(f)

str = 0.

−− : both non-linear effects are taken into account.

27 Matthieu Gu´ edra et al. Thermoacoustic instability in a standing-wave engine