16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23: Liquid Motors: Stability (High Frequency); Acoustics
Combustion Instability: High Frequency
Methods of Analysis for High Frequency Instabilities
Prior to the advent of large-scale computations, the most successful
theoretical development in this area was the “sensitive time lag theory” of L. Crocco
[26]. More than a detailed physical theory, this was a model in which a few basic
parameters were introduced from intuitive considerations, and then used to correlate
experimental observations on stability thresholds. The principal parameter was the
sensitive time lag, during which the various rates which eventually resulted in
vaporization at the total time lag τ T after injection were assumed to vary with
pressure, velocity, stoichiometry, etc. This variation was characterized by means of
other important parameter, the “sensitivity index”. For pressure sensitivity, this is
n=
∂ ln ( Rates )
∂ ln P
(1)
and the definition of τ is such that the variations in gas generation rate due to this
sensitivity are given locally by
•
•
m− m
•
m
=−
∂τ
P '(t ) − P '(t − τ )
=n
∂t
P
(2)
Similar sensitivity indices can be introduced for velocity, etc. Once this parameterization
is accepted, it is only a matter of mathematical modeling to obtain the stability limits of
a given acoustic wave or cavity. This modeling could be linear or even allow for nonlinearities in the gas dynamics. It is one of the strengths of this theory that the acoustic
part of the problem, namely, combustor geometry, steady state combustion and heat
release, etc. are separated from the unsteady combustion effects, which allows for
generalization of test results and accumulation of meaningful stability data.
The results of calculations using the linear sensitive time lag theory are
displayed as shown in Fig. 1 (Ref. 26).
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23
Page 1 of 21
Fig 1. An n - τ diagram, showing instability zones for various modes.
Here are shown the loci of marginal stability for severe modes of a combustor,
on a map of interaction index n vs. sensitive lag τ . Each point on one of the lines
corresponds to a particular oscillation frequency, and these frequencies are found to
be within ± 10% of the undisturbed acoustic frequency of the mode. The goal of the
designer is to manipulate the factors influencing n and τ in order to place the
operating point outside all the stable regions of the various modes.
The parameters τ and n (into which is lumped the modifying effects of
velocity or other sensitivities) are basically empirical, and a large data base has been
laboriously accumulated on their dependencies upon many design factors. As an
example, Figs. 2(a) and 2(b) (Ref. 26) show data on τ for coaxial injectors, for
which n ≅ 0.5 throughout.
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23
Page 2 of 21
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23
Page 3 of 21
Starting with the early of Priem and Guentert [29], these methods have been
progressively replaced by detailed numerical time-dependent simulations of the
combustion process. The complexity of these processes is such that, even now these
simulations still contain large elements of modeling and approximation, and are
generally limited to one or two selected spatial dimension plus time.
A recent example of this approach is described in Ref. 30. Here the emphasis
is on the tangential acoustic modes. A detailed 2-D fluid mechanical model (in the
transverse plane) is coupled to a series of empirically derived droplet vaporization
laws for UDMH and N2O4. The flow is turbulent, modeled using a K- ε approximation,
and the drops are allowed to slip, exerting drag forces which are computed from
empirical drag coefficients, and also modifying the vaporization rates due to
convective heat transfer. The drop-heating transients are ignored. The computations
yield detailed time histories of all the fluid parameters, and comparisons to limited
test data on parametric effects of pressure and injector type are found to be
favorable.
A similar computation, but for longitudinal modes only, is described in Ref. 27.
Here the spatial dependence is on one dimension only (axial), but the droplet
interactions are calculated in somewhat more detail, including drop thermal inertia.
These calculations show strongest instability when the ratio of the acoustic period to
the droplet vaporization time is 0.15, which, as noted before, can be interpreted as
indicating a “sensitive” time lag which is a fraction (0.1-0.2) of the total vaporization
time. The calculations also show cases of entropy wave excitation, for which the
frequency corresponds closely to the convective time in the chamber.
In what follows, we provide a simplified analysis of the acoustic effects of
several combustion-related phenomena (heat release, mass addition, etc.), and then
use this in conjunction with Crocco’s theory for an assessment of stability in a simple
1-D situation. Some general conclusions about stability and destability effects are
also drawn from the acoustic analysis.
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23
Page 4 of 21
Acoustic equations with heat and mass addition, fluid forces, and molecular mass
changes.
∂
∂
=
=0
∂z
∂y
1.
∂ρ ∂ ( ρ u )
+
=m
∂t
∂x
2.
ρ
3.
⎛ ∂ (1 / p )
∂ (1 / p ) ⎞
∂T ⎞ q
⎛ ∂T
+u
=
− p⎜
+u
cv ⎜
⎟⎟
⎟
⎜
∂x ⎠ p
∂t
∂x
⎝ ∂t
⎝
⎠
4.
p
ρ
(mass addition p.u. volume, p.u. time)
∂u
∂u ∂p
+ ρu
+
= f
∂t
∂x ∂x
=
(force p.u. volume)
(q=heating rate p.u. volume)
R
T
M
Assume small perturbations about a uniform steady background, (with µ, m ,f, q):
ρ = ρ + ρ ' , u = u + u' , p = p + p' , T = T + T ' , µ = µ + µ '
u =0
(5)
(6)
0
Because of (6),
D()
∂ ()
∂ ()
∂ ()'
∂ ()'
≡
+ u
=
+ (u + u ')
Dt
∂t
∂x
∂t
∂x
2nd order
∂ ()'
∂t
(7)
Linearizing then,
∂ρ '
∂u '
+ρ
=m
∂t
∂x
(8)
∂u ' ∂p '
+
=f
∂t
∂x
(9)
ρ
ρ cv
p'
p
−
∂T '
p ∂p '
=q+
∂t
ρ ∂t
(10)
ρ' T ' M'
=
+
T
M
ρ
(11)
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
∂ 2u '
∂2 p '
∂f
+
=
2
∂x ∂t
∂x
∂x
(12)
⎛ p' ρ ' µ ' ⎞
−
+
T ' = T ⎜⎜
⎟
ρ
µ ⎟⎠
⎝ p
(13)
ρ
Lecture 23
Page 5 of 21
(13) into (10):
⎛ 1 ∂p ' 1 ∂ρ '
1 ∂M ' ⎞
p ∂ρ '
−
+
⎟⎟ = q +
M ∂t ⎠
ρ ∂t
ρ ∂t
⎝ p ∂t
ρ c v T ⎜⎜
p'
p
(
c v ∂p '
c v ∂ M '/ M
⎛R
⎞ ∂ρ '
T⎜
+ cv ⎟
= −q +
+p
∂t
R g ∂t
Rg
⎝M
⎠ ∂t
=
R
T = Rg T
M
)
cp
(
)
⎡
⎤
∂ρ '
p ∂ M '/ M ⎥
1 ⎢ 1 ∂p '
=
−q+
⎥
∂t
γ −1
∂t
c p T ⎢ γ − 1 ∂t
⎣
⎦
(14)
Sub. Into (8)
(
)
⎡
⎤
1 ⎢ 1 ∂p '
p ∂ M '/ M ⎥
∂u '
−q+
+ρ
=m
⎢
⎥
γ −1
∂t
∂x
c p T γ − 1 ∂t
⎣
⎦
(15)
Differentiate w.r.t. t:
(
)
2
⎡
⎤
1 ⎢ 1 ∂ 2 p ' ∂q
p ∂ M '/ M ⎥
∂ 2u '
∂m
−
+
+
=
ρ
2
2
⎢
⎥
∂t
γ −1
∂ x ∂t
∂t
∂t
c p T γ − 1 ∂t
⎣
⎦
but, from 12,
ρ
∂ 2u '
∂ 2 p ' ∂f
= −
+
∂x ∂t
∂x
∂x 2
(
substitute in (16)
)
2
⎡
⎤
1 ⎢ 1 ∂ 2 p ' ∂q
p ∂ M '/ M ⎥ ∂ 2 p ' ∂f
∂m
−
+
−
+
=
2
2
2
⎢
⎥
∂t
γ −1
∂x
∂t
∂t
∂x
c p T γ − 1 ∂t
⎣
⎦
Now ( γ − 1 ) c p T = γ R g T = c
2
( c =speed of sound)
(
)
⎡
2
∂ µ '/ µ ⎤
2 ∂ p'
∂2 p '
∂ ⎢ 2
⎥ − c 2 ∂f
−
c
=
c
m
+
γ
−
1
q
−
p
(
)
2
2
⎢
⎥
∂t
∂t
∂x
∂t
∂x
⎣
⎦
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
(16)
(17)
(18)
(19)
Lecture 23
Page 6 of 21
( γ − 1) C p T ×
∂ (15 )
∂x
∂(9)
∂t
2
→
c ρ
→
ρ
(
∂2 µ ' µ
2 ∂m
∂2u '
∂2 p '
∂q
+
=
γ
−
1
+
−
c
p
(
)
∂x ∂t
∂x
∂x
∂t ∂x
∂x 2
∂ 2u '
∂2 p '
∂f
+
=
2
∂x ∂t
∂t
∂t
)
(20)
(21)
Subtract, divide by ρ :
(
)
⎡
2
∂ M' M ⎤
2 ∂ u'
1 ∂f
1 ∂ ⎢ 2
∂ 2u '
⎥
−
=
−
+
γ
−
−
1
c
c
m
q
p
(
)
⎥
∂t
∂t 2
∂x 2
ρ ∂t ρ ∂x ⎢
⎣
⎦
2
Note the combination c m + ( γ − 1 ) q − p
(
∂ M' M
(22)
)
∂t
Adding heat, adding mass, or having a decrease rate of molecular mass, all are (up
to factors) equivalent acoustic disturbances.
To close the problem, one needs to relate the perturbations (m, q, f, ∂µ ' ) to
∂t
the state variables ( p ', ρ ', u ',T ' ), by looking at the particular mechanisms
(vaporization, combustion, etc), and how they depend on pressure, velocity, and so
on.
General Conditions for Instability
Looking at Eq. (19), we see that the effects of gas mass generation m, heat
∂µ '/ µ
are similar. All of them have the
generation q and molecular mass change −
∂t
effect of increasing the local volume, and we suspect therefore that when these
quantities (acting together) peak when the pressure also peaks, we will have
unstable condition.
To examine this, define the quantity
2
Q = c m + ( γ − 1) q − p
(
∂ M' M
)
∂t
(23)
and ignore for now the local disturbing forces f. Then
2
2 ∂ p'
∂2 p '
∂Q
c
−
=
2
2
∂t
∂t
∂x
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
(24)
Lecture 23
Page 7 of 21
For a sinusoidal wave of the type
⎡ ∧ i ω t − kx ) ⎤
p '(t ) = Re ⎢ p e (
⎥
⎣
⎦
(25)
with ω complex and k real, we then have
∧
2 ∧
∧
(26)
− ω 2 p + k 2 c p = iω Q
Define
ν ≡
ω
kc
;
and re-write (26) as
h=
∧
1
iQ
2k c
∧
(27)
p
(28)
ν 2 + 2hν − 1 = 0
which has the complex solution
(29)
ν = − h ± 1 + h2
We now ask what form h should have for stability. First, we note that at the
stability threshold, ν is real, and so h = (1 − ν 2 ) / 2ν must also be real. From (27),
∧
∧
this means that i Q must be in phase (or counter-phase) with p , namely, “volume
addition rate” must be 90o ahead of or behind pressure oscillations:
More generally, re-write (28) as ν + 2h − 1 = 0 , and put explicitly h = hR + ihI ,
ν
ν = ν R + iν I :
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23
Page 8 of 21
νR − iνI
=0
ν R2 + i ν I2
Separate real and imaginary parts:
(30)
ν R + iν I + 2 hR + 2 i hI −
ν R + 2hR −
ν I + 2hI +
From (31b),
νR
=0
ν R2 + ν I2
(31)
νI
ν R2 + ν I2
hI = −
=0
⎤
1 ⎡
1
ν I ⎢1 + 2
2 ⎥
2 ⎣
ν R + iν I ⎦
(32)
This shows that whenever hI > 0 , ν I < 0 . Since p ' ∼ e iωt = e −ω t e iω t ,ν I < 0
I
⎛
∧
R
⎞
implies instability. Also, from (27b), hI > 0 implies ⎜ Q∧ ⎟ > 0 , which means that the
⎜ ⎟
⎝ p ⎠R
“volume addition rate” Q(t) must have a positive projection on pressure p’(t),
namely, a part in-phase with it. This is a confirmation of the physical intuition that
releasing “volume” when pressure is high must be de-stabilizing. We repeat that this
may mean heat addition, gas addition (vaporization) or molecular mass reduction
rate (decomposition of complex molecules).
Let us now consider briefly the effect of body forces f. Returning to (19) and
defining the complex quantity.
∧
ρ =
if
(33)
∧
kp
we can see that (28) is expanded to the form
ν 2 + 2hν (1 − ρ ) = 0
(34)
For neutral conditions (stability threshold), we must have real ν (ν I = 0) , so,
taking the imaginary part of (34),
(35)
2ν hI + ρ I = 0
In the absence of forces, we found that hI > 0 would lead to instability. We see now
that if ρ I = −2ν hI , which is negative, then the process stabilizes at least to the point
⎛
∧
⎞
of neutral stability. From (33), the conclusion is that ⎜ f∧ ⎟ < 0 is stabilizing, i.e., the
⎜ ⎟
⎝ p ⎠R
body forces should be in the backwards direction (against H velocity) when pressure
is high, and vice-versa.
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23
Page 9 of 21
Looking at one instant of time vs. distance:
So, the body forces are in quadrature with the pressure gradient force ( −∇ P ), and
are taking energy away from the wave motion.
The “Sensitive time lag” theory (Harrje and Reardon, p.1)
Basic scenario:
-
Drops injected at t − τ T complete evaporation and combustion at
-
However, from t − τ T to t − τ , no complete evaporation occurs, only
“precursor processes”. τ ≡ ”sensitive lag.”
-
The rate of evaporation + combustion during τ is sensitive to pressure and/or
velocity. For p,
∂ln ( rates )
∂ln P
≡ n
(1)
(n= “sensitivity index”).
-
The duration τ of actual evap. + comb. changes in time, in response to these
rate changes. However, the total mass burnt is that of the drop, and it is
assumed their mass (and their number) are independent of P, υ in the
chamber.
Say, R is the relevant rate. Under “quiet” or “mean” conditions, R = R . When P ≠ P ,
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23
Page 10 of 21
R
⎛
p − p⎞
R ⎜1 + n
⎟
⎜
p ⎟⎠
⎝
(2)
Express that the total mass burnt in ( t − τ , t ) is always the same, equal to
that burnt under mean condition
t
⎛
t−
⎝
p − p⎞
⎟ dt1 =
p ⎟⎠
∫τ R ⎜⎜ 1 + n
t −τ
∫τ
t
R dt1 + R n
t−
∫
P −P
t −τ
P
t
∫
(3)
R d t1
t −τ
dt1 = 0
here replacing τ by τ is ok.
(2nd order error)
τ − τ = −n
t
P −P
∫
P
t −τ
(4)
dt1
Let now, specifically, m be the rate of gas generation from liquid (local, per
unit volume). The liquid injection rate is constant, equal to m . The gas generated in
(t, t+td) is mdt. This gas originates from liquid that reached its “maturity” for
vaporization between t − τ and ( t − τ )+d ( t − τ ), and since liquid arrives at m ,
m dt = m d ( t − τ )
m−m
m
From (4)
and so
= −
dτ
dt
(5)
P (t ) − P (t − τ )
dτ
= −n
dt
P
m−m
m
=n
P (t ) − P (t − τ )
or, in terms of perturbations,
(6)
P
m'
m
=n
P ' (t ) − P ' (t − τ )
P
(6’)
NOTE: Eq. (3) is wrong in Harrje and Reardon.
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23
Page 11 of 21
Control of Instabilities
Despite all efforts at avoidance through design, the need remains for devices
that will help damp the many potential modes of instability in any given rocket
combustor. A new technology for active control is now evolving [31], in which
feedback controlled acoustic generators are used to cancel unstable waves. Since the
growth can be detected at small amplitude, it may not be necessary to inject very
large acoustic powers for this purpose. Nevertheless, the mainstay of current
practice is based on passive damping methods. A good description of these methods,
with design guidelines, is given in Ref. 32.
The most important high frequency stabilization devices are injector head
baffles and acoustic absorbers. Baffles are radial or circumferential barriers attached
to the injector head and extending 0.1-0.2 diameters in the axial direction. An
example of a baffled injector is that of the SSME (Fig. 3, from Ref. 2). The exact
mechanism by which baffles enhance stability is not well understood, which has led
to some divergence in design. It appears that the effect is related to the sensitivity
of the droplet velocity cross-over point, which occurs quite close to the injector face,
and it may involve disruption of the tangential gas motion associated with tangential
modes, or shifting of the local acoustic frequencies to values above the characteristic
drop vaporization frequencies.
Acoustic absorbers are cavities on the chamber walls with relatively narrow
connecting channels to the chamber, so as to dissipate power during pressure
oscillations in their vicinity. Their action is much better understood than that of
baffles, and designers can proceed with some confidence, using methods described,
for example in Ref. 32. Absorbers are often located on the cylindrical walls, near the
injector, or as “corner slots” between injector and cylinder. They can also take the
form of a continuous double wall with an array of holes periodically arranged to
connect to the chamber. The absorption coefficient of a well-designed absorber can
be high over a relatively wide frequency band, so as to contribute damping to the
most prevalent modes. Sometimes several different absorbers are used, each tuned
to a different frequency. Fig. 4 (Ref. 32) shows a baffled injector with corner
absorbers, and Fig. 5 (Ref. 32) shows an extended acoustic liner.
A simplified analysis of an acoustic absorber is described next.
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23
Page 12 of 21
Acoustic Absorbers: Resonators
The schematic shows a device
which is a variation on the
Helmholtz Resonator concept.
Consider first the halfcycle during which flow is leaving
•
the resonator ( m >0). The
pressure inside the resonator is
PR, while at the entrance to the
narrow inlet duct, it is
PR' = PR −
1
ρ v 2 , due to the
2
(subsonic) acceleration towards
the inlet. The density changes δρ
can be equated to δρ / c2 (c =
speed of sound in the resonator).
The mass balance is then
•
VR dPR
= −m
2
c dt
(1)
and the momentum balance in the duct is
d
( ρ Ai Lv ) = Ai PR' − Pc
dt
(
)
•
or, using m = ρ A i v ,
•
• 2
⎛
⎞
d m Ai ⎜
m ⎟
P
P
=
−
−
R
C
dt
L ⎜⎜
2 ρ A i2 ⎟⎟
⎝
⎠
(2)
•
Consider next the other half-cycle, when the cavity is filling ( m <0). The
pressure at the chamber-side of the duct, which is now the flow inlet is
Pc' = Pc −
1
ρ −v 2 , whereas that at the exit of the duct is now just PR. Eq. (1) still
2
(
)
holds, whereas the momentum balance is now
•
d m Ai
=
dt
L
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
• 2
⎛
⎞
m ⎟
⎜
P
P
−
+
R
C
⎜
2 ρ A i2 ⎟⎟
⎜
⎝
⎠
(3)
Lecture 23
Page 13 of 21
We now differentiate (2) and (3), and unify the two half-cycle momentum equations
as
•
Ai
d2 m
=
L
dt 2
•
⎛
•
m
⎜ dP
d m dPc
⎜ R −
−
dt
⎜ dt
ρ A i2 dt
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎠
Substituting Eq. (1) here,
•
•
m
•
Ai •
A i dPc
d m
dm
+
+ c2
m=−
2
ρ L A i dt
LVR
L dt
dt
2
(4)
•
This equation is non-linear in m , but we can obtain reasonable results for steady
•
state operation if we replace m by its time average over one cycle. The analysis
•
reduces them to that of a forced linear oscillator, except that m needs to be
calculated self-consistently rather than being a prescribed quantity.
Assume solutions of the form
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23
Page 14 of 21
⎡ ∧•
⎤
m = Re ⎢ m e iω t ⎥ ,
⎢
⎥
⎣
⎦
⎡ ∧
⎤
δPR = R e ⎢ δPR e iω t ⎥
⎣
⎦
•
,etc.
(5)
Eq. (4) becomes
•
⎛
m
⎜
Ai
⎜ 2
2
⎜ −ω + ρ L A i ω + c L V
i
R
⎜
⎜
⎝
⎞
⎟ ∧
Ai ∧
⎟ •
⎟ m = − i ω L δ pc
⎟
⎟
⎠
(6)
while Eq. (1) gives
∧
∧
•
V
i ω R2 δ PR = − m
c
(7)
and solving (6) and (7) together,
c2 A i
∧
δ PR
L VR
=
∧
δ Pc
(8)
•
c2 A i
L VR
m
+
ρ L Ai
i ω − ω2
From this, the natural frequency of the oscillations is seen to be ωn = c
Ai
L VR
(9)
and the damaging factor ζ is given by
•
•
m
2 ζ ωn =
ρ L Ai
m
,
or
ζ =
2 ρ L Ai c
L VR
Ai
(10)
∧
•
•
•
For the “rectified sine wave” m (t), we can see that m = 2 m , and from (7) and (8),
π
c2 A i
•
m =
2
π
ω
∧
VR
V
2
δ PR =
ω R2
2
π
c
c
L VR
•
c2 A i
L VR
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
∧
δ Pc
(11)
m
+
ρ L Ai
i ω − ω2
Lecture 23
Page 15 of 21
•
⎛
⎜ m
2
⎛ c Ai
⎞
⎜
− ω2 ⎟ + ⎜
⎜
⎜ L VR
⎟
ρLA
⎝
⎠
⎜
⎜
⎝
•
2
Using c A i − ω 2 +
L VR
m
ρ L Ai
2
2
iω =
i
⎞
⎟
⎟ 2 , this yields a bi-quadratic
⎟ ω
⎟
⎟
⎠
∧
•
equation for m in terms of the amplitude δ Pc of the pressure fluctuations in the
chamber. The algebra is simpler in non-dimensional terms. Define
ν ≡
ω
ω
=
ωn
c
LVR
Ai
(12)
∧
ρ ≡
2
δ Pc
VR
(13)
π L A i 1 ρ c2
2
2
•
⎛ ρ c Ai
We then obtain m = 1 ⎜
2 ⎜⎝ ω VR
2
2
2
⎞ ⎡
⎟ ⎢− 1 − ν 2
⎟ ⎣
⎠
(
)
2
+
(1 − ν )
2
4
⎤
+ p2ν 4 ⎥ , and substituting in
⎦
(10), the damping ratio is
ζ2 =
p2ν 2 / 2
(
1 −ν2
)
2
+
(
1 −ν2
)
4
(14)
+ p2ν 4
Several points can now be made:
(a) For any frequency, the damping increases with pressure fluctuation intensity.
This is a favorable circumstance because we need the damping most when
combustion is rough. Mathematically, this is a consequence of the nonlinearity of the equation. Physically, energy is dissipated both during
aspiration and during expulsion of gas from the cavity, (by the mixing out of
the jet kinetic energy), and it is clear that more energy is dissipated when the
driving pressure differences are stronger.
(b) Although the algebra is still tedious, differentiation of (14) shows that ζ is
maximum at ν = 1 , i.e., when the cavity is tuned in resonance with the
pressure fluctuations (by selecting parameters so that ωn = ω ). Putting ν = 1
in (14) gives
∧
ζ MAX =
p /2 =
VR
π L Ai
δ Pc
1
2
ρc2
(15)
which again shows ζ increasing with fluctuation intensity.
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23
Page 16 of 21
(c) Since we should design for resonance for the given ω , the dimensions should
Ai
ω2
= 2 . Eq. (15) can then be put in the two equivalent forms
satisfy
LVR
c
∧
ζ MAX =
1
2
ωL
π
δ Pc
ρ
∧
= ω
VR
Ai
2
π
δ Pc
(16)
ρ
This indicates we should use short inlets, large cavity volumes and
Ai
ω2
= 2 ).
small inlet areas (subject to
LVR
c
As an indication of what is possible, consider 1% fluctuations on a
P
P
8.3
× 3000 = 1.25 × 106 m2 / s2 . The
pressure Pc such that c = RgTc is c =
0.02
ρ
ρ
acoustic frequency to be damped is at f = 2000 Hz, or ω = 2 π f = 12 , 5000 m/s.
Then from the first of Eqs. (16), we can obtain critical damping ( ζ MAX = 1 ) by
choosing an inlet length
∧
L=
1
2
ω ζ MAX
π
δ Pc
ρ
=
1
12 , 500
2
π
(1.25 × 10 )
4
= 7.2 × 10−3 m = 7.2 mm
which appears reasonable. To ensure resonance, then, we need to select.
γ Rc Tc
VR
c2
= 2 =
Ai
ω L
ω2 L
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
1.2 × 1.25 × 106
(12 , 500)
2
× 7.2 × 10−3
= 1.34 m
Lecture 23
Page 17 of 21
A possible choice of geometry is then to take an inlet diameter of, say, 5mm, and a
cavity volume
VR = 1.34
π
4
0.0052 = 2.63 × 10−5 m3 = 26.3 cm2
If this is shaped as a cubic cavity, its side is about 3 cm. These dimensions are
sketched (roughly to scale) below.
As a final note, remember that the
“lumped parameter” idealization
used in the theory may not be very
precise. Only semi-quantitative
accuracy is to be expected, but the
trends should be correct and the
analysis can be used for preliminary
design, to be refined through
numerical simulation or physical
testing.
How many acoustic dampers:
δξ R =
•
1
δ m v 2 + ρ δ PR VR =
2
d (δξ R )
dt
•
•
dδ PR
1
δ m v δv +
VR
=
2
dt
m
v =
ρ Ai
•
δm
δv =
ρ Ai
With no dissipation
•
d m Ai
=
dt
L
•
m=−
•
( PR
− PC )
VR dPR
c 2 dt
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
d δ m Ai
=
δ PR
dt
L
•
L
δm
Ai
•
d δ PR
c2
=−
δm
dt
VR
VR
δ PR
c2
Lecture 23
Page 18 of 21
d
dt
•
⎛ L
δP
δ m + VR 2R
⎜⎜
A
c
⎝ i
⎞
⎟⎟ = 0
⎠
d
dt
× c2
⎛ L 2 ⎛ • ⎞2
2⎞
c ⎜ δ m ⎟ + VR (δ PR ) ⎟ = 0
⎜
⎜A
⎟
⎝
⎠
⎝ i
⎠
2
ζR
2
L 2⎛ • ⎞
c ⎜ δ m ⎟ + VR (δ PR )
Ai
⎝
⎠
L
=
=
2
2 ρ Ai
2ρc
2
VR
2
⎛ • ⎞
δ PR )
⎜δ m⎟ +
2 (
2ρc
⎝
⎠
With dissipation
•
• ⎞
⎛
m δ m⎟
⎜
dδ m A i ⎜
⎟
δ PR −
=
2 ⎟
⎜
dt
L
ρAi
⎜
⎟
⎜
⎟
⎝
⎠
•
d (δ PR )
dt
d
dt
=−
•
c2 L
δm
Ai
•
c2
δm
VR
VR δ PR
⎡ L 2 ⎛ • ⎞2
•
2⎤
c ⎜ δ m ⎟ + VR (δ PR ) ⎥
⎢
m
2
A
⎝
⎠
⎢⎣ i
⎛ • ⎞
⎦⎥ =
m
δ
⎜
⎟
2 ρ c2
2 ρ 2 A i2 ⎝
⎠
•
•
εR
Average
=
m =
εR
1
•
m =
2 ρ 2 A2i
∧
•
•
3
m
At resonance, m = −
•
Ai
L
π
•
∧
•
m
= −
m
2
2 ρ Ai
π
2
ρ L Ai
∧
δ Pc
•
m
∧
•
δ Pc
m =
3/2
∧
⎛2
⎞
ρ A2i δ Pc ⎟
=
2
2 ⎜
2 ρ Ai ⎝ π
⎠
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
(energy dissipation rate) ζ R
3
1 ⎛2⎞
= ⎜ ⎟
2 ⎝π ⎠
2
Ai
ρ
ρ A2i
•
∧
δ Pc
m
2
π
∧
∧
ρ A2i δ Pc
3/2
δ Pc
Lecture 23
Page 19 of 21
2
∧
ξc = Vc
Average chamber acoustic energy
•
( ∆ωI )ch.
contribution of ωI of
chamber, due to N
resonators
∆ζ c =
3
1 ζR
1 N ⎛2⎞
=N
=
2 ξc
2 2 ⎜⎝ π ⎟⎠
Also, (ωn )c = (ωn )R = c
( ∆ωI )c
ωn
=
1
N Ai
2π 3
Vc
Example ω = 2π × 2000 Hz
L = 7.2 × 10−3 m
c2
Ai
2
ρ
δ Pc
ρ c2
ρ c2
=
∧
Vc δ Pc
NA i
2
2π
3/2
Vc
ρ
c2
∧
δ Pc
Ai
L VR
ρ
L VR
=
Ai
1
δ Pc c
1
2 π3
N
Vc
L VR A i
ρ c2
∧
δ Pc
δ Pc
1.25 × 104
1
=
=
2
6
120
ρc
1.2 × 1.25 × 10
Ai =
VR = 2.63 × 10−5 m3
π
4
25 × 10−6 m2
and say Vc = 0.1 m3
∆ζ c =
1
N
2π ∋ 0.1
7.2 × 10−3 × 2.63 × 10−5 × 25
π
4
× 10−6
120 = 2.7 × 10−5 N
So,for ∆ζ c = 0.1 , need 3730 dampers.
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23
Page 20 of 21
Say l = 4 cm
D = 40 cm
N=
π D lT
l
2
lT =
N l 2 3730 × 16
=
= 475 cm
πD
π × 40
Not enough room
Can get ∼ ∆ζ
0.02 in lT = 1m
16.512, Rocket Propulsion
Prof. Manuel Martinez-Sanchez
Lecture 23
Page 21 of 21