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Effect of flame-induced geometrical straining on turbulence levels in explosions and common burner configurations
#20-7225192 $5.00+ 0.00 Copy&hi @ 1992PergamonPressLtd
int. J. EngirgSci. Vol. 30, No. 8, pp. 983-1002, 1992 Printedin Great Britain. All rightsreserved
EFFECT OF FLAME-INDUCED GEOMETRICAL STRAINING ON TURBULENCE LEVELS IN EXPLOSIONS AND COMMON BURNER CONFIGURATIONS T. C. CHEW Department of Mechanical and Production Engineering, National University of Singapore, Kent Ridge 0511, Republic of Singapore
R. E. BRITTER Engineering Department, Cambridge University, Trumpington Street, Cambridge CB2 lPZ, U.K. Abstract-The modification of turbulence ahead of the flame front in a point-ignited explosion developing in a homogeneous turbulent flow is investigated. The spherical flame front may be freely expanding or confined within a vessel. In the former, geometrieal considerations predict that the total strain on a fluid element just before its engulfment by the flame is independent of the initial position of the element. This results in a uniform change of the turbulence. For example, the turbulence intensity generally increases by about 50%. In the case of confined explosions, the distortion of fluid elements leads to increasingly larger turbulence amplifications, starting from a value approximately equal to that of the unconfined case to up to several hundred per cent at the final stages of combustion. Rapid Distortion Theory is used for the analysis. The modified energy spectra are also evaluated. Available experimental data lend qualitative support to the results. The range of validity of this analysis is discussed. The inde~ndent variable is cast in terms of a pressure ratio and straining factors, thus permitting the appli~tion of the data to a wide range of ~~~ratio~ without the need to delve into the complicated mathematics each time. The conclusions are expected to be moderated but not negated by taking into account the process of nonlinear turbulence decay. Finally, the implications on turbulent flame velocity measurements in various configurations are briefly explored.
INTRODUCTION
pre-mixed gas explosions present a simple and practical situation where vorticity dynamics indicate that the turbulence in the unreacted region may be modified significantly by the expansion of the burnt products. A curved, advancing flame front causes stretching of the vorticity in the unburned gases; a similar effect is achieved if the unburned mixture was compressed (e.g. Wong and Hoult [l]). Th e instantaneous and local turbulence field at the flame front can then be quite different to that present before combustion. This has implications on burning velocity measurement techniques using expanding turbulent flames [2-41, as well as in industrial gas explosions [5] where the rate of pressure rise depends on the speed of flame propagation (which normally increases with turbulence intensity). This study examines first an unconfined explosion, and then a centrally-ignited explosion in a spherical vessel. The Rapid Distortion Theory (RDT) is used to estimate the extent of turbulence enhancement by the mechanism of vortex stretching. The results are discussed in the context of the validity of the RDT. These results for sphe~ca~y~~an~ng flames are then used to examine similar effects in some common burner conjurations. Expanding
PURPOSE
The purpose of this paper is to first point out the said modification of turbulence, and then by providing charted results based on simple accessible parameters, show how the effects can be assessed in different configurations without resorting to the complex mathematical formulations each time. The section on RDT formulation, which is fairly well established, is included for continuity and compIeteness, but may be omitted at the first reading of the paper. 983
984
T. C. CHEW
STRAIN
OF CONCENTRIC AN
and R. E. BRIl’TER
ANNULAR ELEMENTS EXPLOSION
ENVELOPING
When a combustible mixture is ignited at a point a spherically-expanding flame front results, which we shall call a fireball for conciseness. The fireball grows as a result of the consumption of unburnt gas, as well as the expansion due to heat release. Unreacted material around the flame is pushed away from the centre until the expanding flame surface overtakes it. If the flame is unconfined, the material is then burnt and remains stationary. The presure drop across the flame front is negligible and there is no compression of the unburnt material. If the explosion took place in a sealed vessel, the burnt gas will be accelerated back towards the centre because of the expansion of the gases burned subsequently [6]. We evaluate the strains? imposed on the unreacted fluid by the growth of the fireball. Following the notation of Lewis and von EIbe [6], the speed of propagation of the flame front relative to the laboratory frame is called the flame speed S,,, while the speed of advance of the flame relative to the unburned mixture is the burning velocity S,. As is well known, the values of both S, and S, are dependent on the turbulence levels in the burning mixture.
CASE
OF UNCONFINED
EXPLOSION
Let r, represent the position of a particle originally at ri, and r, = position of the flame front. When the flame catches up with the particle, r, = r, [Fig. l(a)]. By conservation of mass,
113
rb/ri =
(p,/p,,) =f,
is sometimes
(>
called the isenthalpic
Fig. 1. (a) Unconfined _._...__P._-_.._~
explosion;
fi PI-I
expansion
(b) deformation ~~
t In this paper, the word strain is used rather loosely length.
,
ratio. If we now consider a small
of an element
to mean extension
on a spherical
ratio of a line element
shell.
or, new length/initial
Flame-induced
geometrical
straining
985
element on the thin, annular shell originally of radius ri,
direction 1 is taken to be normal to the surface while 2 and 3 lie on the surface of the spherical shell (Fig. 1). The off-diagonal elements are zero because the chosen axes coincide with the directions of the principal strains. Also e11e22e33= 1 as the flow is incompressible. If we assume negligible heat loss, pU/pb is constant for a given gas mixture. Therefore, the amount of distortion undergone by a fluid element before it is burned is independent of its original distance from the centre of ignition. Besides simplifying matters considerably, this conclusion is important for two reasons. Firstly, it is contrary to the reasoning in some previous studies (such as [l]) which stressed that the straining is most severe when the flame ball is small. In fact, it is the case that during the early stages, the distortion, while of similar magnitude to later stages, is incapable of affecting turbulence scales that are larger than the size of the flame ball, and as a result the possible influence on the turbulence field is less. Secondly, since the geometrical distortion is constant for fluid elements at all distances from the point of ignition, any effect on the turbulence cannot be inferred from trends in the mass burning rate during the explosion. Direct measurement of the turbulence levels just ahead of the flame front is necessary. While the total distortion is ultimately the same, the time history of distortion will obviously be different for elements at different distances from the point of ignition. To calculate the rate of distortion, we note that the position of a particle in the unburnt mixture at a time t is given where
by
[61, r,=
[r~+r;(1-~)]1’3.
If the flame speed &, is assumed constant, then rb = &t and r,= [r?+s’,t(l-$)]lil
art_ SZ(l t-
- l/f’)? rr2
*
This shows that &,/at is close to zero when t is small, but + %(l - l/f3)‘” when t is large. Thus, a large proportion of the distortion occurs just before the element is consumed by the flame (Fig. 2). As will become apparent, this enhances the applicability of the RDT.
THE CASE OF A CONFINED
EXPLOSION
Consider a centrally-ignited mixture contained in a spherical vessel. The propagation of the flame wave is now accompanied by a continuous rise in pressure which is spatially uniform when S, is much smaller than the speed of sound. The temperature of the unburned gas rises as a result of the compression (which can be considered adiabatic). The burning velocity, which is dependent on both the pressure and the initial temperature, is expected to increase. The continuous compression of the burnt gases results in a temperature gradient from the centre to the flame front, the temperature being highest at the centre. Thus, on completion of
986
T. C. CHEW
and R. E. BRITTER
Time
Fig. 2. Growth
of r,. Straight lines are the corresponding
flame positions.
combustion, there exists a difference in temperature between the centre and the vessel walls of up to several hundred degrees centrigrade. When this gradient relaxes, there will be a slight decrease in pressure. The detailed mathematical relationships between burnt fraction, flame position and the state variables of reactants and products can be found in standard text (such as [6]). To simplify matters here, chemical kinetics is avoided by assuming that the final pressure, P,, is known, and it is not critical for the present analysis whether this pressure is measured before or after the aforementioned temperature gradient has been allowed to relax. First, we shall derive the total distortion of a fluid element from the moment of ignition to the time it is caught by the flame. Referring to Fig. 3(a), let R = radius of spherical vessel, r,, = radius of flame ball, ri = initial radius of burnt mixture and n = burnt fraction = (~i/R)3. Geometrical considerations yield Surface strain = rb/r, Normal strain = (z)‘($-$j). Therefore
the distortion tensor is given by
ei, =
(;,‘(EA, ... ... . .. rb/ri *** ...
...
(2)
b/c I.
TO find Th/Ti,it can be shown that
ri/R = [l - (1 - ~z)(P,/P)“‘~]““, where Pi and P are the initial and instantaneous pressures, respectively. Also, the pressure rise is approximately proportional to the fraction burnt, i.e. n - (P - Pi)/(P, - P,). Substituting for
987
Flame-induced geometrical straining (b)
(b) surface and nonnaf %rains.
Fig. 3. (a) Confined expkhn;
P/P, and putting R = 1 (i.e. non-dimo3nsionalizing rb and ;riwrt R), we get Ib-[l-(l-n)(~)m.]m where
z
(3)
&--r?, =T-
The variable 2, which we shall call the pressure ratio, is easily measured for most systems. The value of 2 depends on the amount of the heat release and the specific heat capacity of the burnt products, providing the effect of molecular dissociation is neglected. Typically, 2 ranges from 4 to 8.
i
0
0.10
I
i
0.20
0.30
radius-u~burfled
Fig.
4. Position
1
1
0.40
0.50
(i.e.
at
I 0.60
initiol pressure
I
I 0.70
0.80
i 0.90
l
3
i
a
0
7
0
f3
+
5
x
2=4
I 1 .oo
and temp.)
at whi& a fluid element is caught by the flame compared with its origintlr position.
988
T. C. CHEW
and R. E. BRIITER
0 80
0 20
0
I
I
I
I
I
I
I
I
I
_-s
0 10
0 20
0 30
0 40
0 50
0 60
0 70
0 60
0 90
1 00
Fraction
burned
Fig. 5. Flame radius as a function of fraction burned for a confined
explosion.
Equation (3) relates r, to n, and hence ri, for given values of Z and yU. Plots of rb/ri, and extension (or compression) ratios wrt to rb and mass fraction burnt n, are given in Figs 4-7. In general, a 20% burnt fraction will result in an rb of almost 0.8Rso that initially, straining is the confinement by the vessel restricts surface predominantly geometrical. Subsequently, strain, and normal straining is caused mainly by comparison. In the initial stages too, the rate of distortion will resemble that of the unconfined explosion. In the later stages the predominantly compressive straining is governed by the rate of conversion of unburnt material into burnt products. Even if the flame speed remains constant, more material is consumed per unit time as the surface area of the flame ball grows. The result
Pressure 1 50
c_ 0 L
ratio2
.
9
l
8
0
7
0
6
+
5
x
4
1 00
t;
0
50
0
0 10
0 20
0 30
0 40
0 50 Flame
0 60
0 70
0 80
radius
Fig. 6. Surface strain as a function of flame radius.
” B”
2 il”
989
Flame-induced geometrical straining
.c F
0.60
-
0
-
50
0.40
-
0.30
-
0.20
-
0.10
-
Pressure
ratio
2
;;
0
1
0.10
I
I
I
I
I
I
I
I
0.20
0.30
0.40
0.50
0.60
0.70
0.60
0.90
Flame
I 1.00
radius
Fig. 7. Normal strain as a function of flame radius.
quite fluid element evaluate the only for the and pressure
is then
similar to the case of the unconfined explosion, so that much of the distortion of a occurs just before the element is consumed by the flame. We shall not attempt to actual rate of distortion here because in fact S, is unlikely to remain constant, if fact that the laminar burning velocity is enhanced by the increase in temperature in the vessel.
ANALYSIS
BY
RDT
The RDT has been used with success by many researchers to predict the effect on turbulence of a distortion in a fluid taking place so rapidly that non-linear effects can be neglected [e.g. 1, 7, 9 and 111. Batchelor and Proudman [9] calculated the changes produced in a homogeneous turbulent motion when the fluid is subjected to a superimposed uniform distortion. In particular the asymptotic case of a large symmetrical contraction (as commonly occurs in the wind tunnel) was examined in detail. It was found that the energy for the longitudinal component of velocity is reduced, while the energy for the lateral velocity components is increased. The total energy is always increased. Qualitatively, the process can be explained in terms of vorticity stretching. When the vortex is pulled out along its axis of rotation, its outlying elements are drawn closer to the centre, and conservation of angular momentum requires that it now spins faster. The velocity components in a plane parallel to the axis of rotation are increased. Similarly, if a vortex is caused to expand so that the rotating fluid elements are drawn away from the axis of rotation, the velocities will be slowed down. Again only the velocity components in a plane parallel to the axis will be affected. The components in a plane perpendicular to the axis will not be affected by the distortion of this vortex. Isotropic turbulence consists of vortices oriented at random throughout the fluid. The effect of an arbitrary distortion is to increase the energy in some directions while decreasing it in others. The net result is always an increase, with the additional energy input coming from the distortion. Hunt [8] formalized much of the mathematics and also examined the applicability of the assumptions to various regimes of turbulence conditions. The analysis used distortion by bluff
990
T. C. CHEW and R. E. BRIIITER
bodies as a basis to examine the effect of varying the length scale ratios between the turbulence scales and the scale of the distortions. Britter et al. [7] showed, with careful experimental measurements of the grid generated turbulence flow past a circular cylinder, that the amplification and reduction of the three components of turbulence can indeed be explained by the distortion of the mean flow and the blocking effect caused by the turbulence impinging on the cylinder surface. Good agreement was found with predictions by Hunt [B]. Insofar as applications to cornbusting flows are concerned, only a few publications are available in the literature. Wong et al. [lo] used RDT to calculate the effect of compression by the piston in an internal combustion engine. Wong and Hoult [l] looked at the distortion of the turbulence during combustion in the engine cylinder. The flame was assumed to be ignited on the axis of a disk-shaped combustion chamber and which then propagates radially outwards. Enhancement of turbulence was predicted which can be used to explain the increased burning rates in some engines. Experimental measurements by Witze [15,16] revealed increases in turbulence levels during combustion in an engine cylinder with qualitative agreement with the predictions of [l]. Other applications of RDT include, among others Townsend [13] and Stretch [12]. Townsend applied the theory to evaluate the structure in turbulent shear flows with extensive comparisons with experimental results. Stretch studied the dispersion of slightly dense contaminants in turbulent boundary layers. That the theory is extremely useful in making both quantitative and trend predictions in turbulent flows under some limiting conditions is without doubt. Quantitative predictions are confirmed when the theory is applicable, e.g. Britter et al. [7]. The ideas and methods are still continuously being developed. A review of current applications of RDT is given by Savill [ 141.
MATHEMATICAL
ANALYSIS
We present below a brief outline of our analysis of the distortion of follow closely that of [9], except for Starting from Cauchy’s equations,
OF A DISTORTION
ON TURBULENCE
of the mathematical formulation which will form the basis the turbulence field ahead of explosions. The main ideas the inclusion of density changes. the vorticity of a fluid element is given by
2
f OJX) = + 0
oj(a) I
where a and x are the respective positions of the fluid element before and after distortion and p. and p are the original and new densities respectively. Writing dXi - = the distortion tensor eij,
&j
and PIPoeij
=
&j,
gives Oi(X) =
SijWj
(a).
The prime (‘) on the vorticity denotes its pre-distortion expressed as velocity gradients:
eiik is the unit alternating
(4) value. The vorticity components
can be
tensor, taking the value of 1 or -1 when i, j and k are in cyclic or
991
Flame-induced geometrical straining
reverse cyclic order respectively, and zero otherwise. If K is a wave-number vector of the original energy spectrum, we choose the new wave-number vector to be 2 such that
The velocity distribution at any time during the distortion can be written as a Fourier integral eiY”dZ(K)
u(a) =
and the energy spectrum tensor, at the initial instant, is given by [9]
‘$;{K)
&j(K)
=Kl
dZ;(w)
drc
.
where d2 is the complex conjugate of dZ. #b is the Fourier transform of the (initial) correlation tensor, i.e.
Making use of the result derived by [9] in identical formulation, the distortion is found to be
Choosing the axes to coincide turbulence initially, i.e.
&?2 + $33 = (f-)‘S
with the axes of principal
(2k:(K$+
Kg)+
(Kg+
strain,
velocity
the energy spectrum after
and assuming
isotropic
&)(;K;+$K:)
W)
For the spherically symmetric configurations
being considered in this study, we can write
e2=e3=c-ln
and e, = cc whence w2e3
= t
g is the factor by which fluid has been compressed,
5;=
i.e.
($)-’
992
T. C. CHEW and R. E. BRIITER
Substituting into equations (5)) h(x)
= 5-“%
(6)
(d + 4)
$22(x) + @33(X)= 5-2=4JcX4K2(c2cX4 + ?-‘K:K2)
(7)
For the case of an unconfined explosion f=l and equations (6) and (7) reduce to the same form as in [9].
EFFECT
ON THE
TURBULENCE
INTENSITY
The ratio of the energy due to the velocity components flame surface) to that before distortion is
Similarly, the effect on the turbulence
in the 1 direction
energies in the 2 and 3 directions is given by
+22(x) + #33(x) dzt Y2 = P3 = 442(K) + 443(K) dK = j- c-2 $$$
(g2cx-4 + E(K) -
45GK4
c-2C-3K:K2)
(K2 + K:)
dX
dK
Noting that dX -=dK
1 e,e2e3
. ie
. .
dX= c-‘du,
and putting K, = K COS 8,
(K;
+ K:)
= (K
Sin 6)”
(following [9]), we get p*+-3
=
* sin3 8 de n sin2 8 d0 I0 [( 5;~)~”cos2 0 + c sin2 @I2/I 0 f-Q-2$
(normal to the
B’ - I tan-l /j + p-2 P3
1
where p”=(c-‘c-“-l)
note::::
993
Flame-induced geometrical straining
and g-3-
E(K) K2x4
( c2Cx4
+
c-2C-3K;K2)
Kf)
= &g-l
+ g-‘C-5
L (
dK
dK
tan-’ /3 -
B’
1 B’(B” + 1) > *
(8)
RESULTS
The foregoing analyses allow for distortion resulting from a compression as well as pure geometrical straining. We can now use the distortion tensors evaluated in equations (1) and (2) to obtain the effects on the turbulence for both the cases of confined and unconfined explosion. Unconfined explosion
The vorticity distortion tensor Sij = p/poeii. From equation (l),
Since the fluid element (pulpbY3,
does not undergo
volumetric
compression
here,
c = 1 and rb/ri =
giving
...
(z>l/3 P"
Sij
=
...
...
... ( >1’3 it3 , Pll
As emphasized earlier, the total distortion is independent of the initial position of the fluid element. Figure 8 shows the variation of the energy ratios p1 and ,u~as evaluated from equation (8). It is seen that the energy due to velocity fluctuations in the normal direction (direction 1) is always amplified. p2 and p3 decrease slightly at first, but are also amplified for a compression factor (in the normal direction) of 0.4 or less. The overall turbulence energy is always amplified. For an isenthalpic expansion ratio of say 8, p1 is about 2.5. Figure 9 shows plots of
p1 and ~2 vs It is concluded that the turbulence as seen by the travelling flame front is significantly different from that present initially. This is especially true for the early stages of the development of the explosion, and for large flame speeds, where the time scale is too small for viscous effects to be relevant, and also allowing nonlinear momentum transfer process to be neglected. The time just after ignition, however, may not be amenable to this analysis because the flame ball is probably too small to affect the integral scales of turbulence in the way (PJPLJ.
ES 30:8-C
994
T. C. CHEW and R. E. BRI’ITER
I 0
I
I
I
I
10
20
30
40
Isentholpic
expansion
ratio
Fig. 8. Effect of isenthaipic expansion ratio on the intensity for an unconfined explosion.
described, distortion.
i.e. when the integral scale of turbulence
is large compared
with the scale of the
Confined explosion We have
-rb
...
C
-rb
ri ~
Using values of rb and ri evaluated previously, the energy ratios pl and pz (= ,IA~)are found as shown in Figs 10 and 11. In the initial stages of the explosion, corresponding to the prepressure phase, the effect on the turbulence is similar to that for the unconfined case. However, as the flame progresses, the turbulence is amplified further. Examination of the terms in Sii shows that this is because the distortion resulting from the compression of the unbu~t gas more than com~nsates for the restraint on the growth of rb by the vessel walls. For a typical pressure ratio (P, - PI/pi) of say 6, a ten-fold increase in the normal component energy is predicted and a five-fold increase in the surface component near the last stages of the explosion. When the flame front is midway across the vessel, the numbers are about 2.5 and 1.5, respectively. The increase in the turbulent velocity fluctuations is proportional to the square root of these values. This again represents a marked deviation from the turbulence field which existed before ignition. From Figs 10, 11, and 7, it is obvious that several factors affect the extent of the
Flame-induced geometrical straining 4.00
-
3.50
-
300
-
995
.-s z s 4 ET 50
250-
a z 0
zoo-
iii w G
1.50
-
1.00
-
6 z 2 z 5
0.50
-
fz I 0
0.10
I
I
0.20
I
0.30
I
0.40
Compression
I
I
0.50
0.60
factor
0 70
I 0.80
I 0 90
I 100
c
Fig. 9. Effect of compression factor c on the turbulence energy for an unconfined explosion.
vorticity distortion mechanism, the most important of which are the specific heat capacities, expansion ratios, and mass fraction burnt (or stage of explosion). One concludes that the use of measurements in the initial turbulence to characterize the flow during an explosion may be unwise. A correction factor, easily derived from the results presented here, could perhaps be applied to obtain a more realistic representation of the turbulence field instantaneous and local to the flame. Spectra The one-dimensional spectrum tensor &((x
spectrum, &(x1) may be found by integrating the three-dimensional for all values of x2 and x3. We assume, following Townsend [13], that
Pressure
I 0.10
I 0.20
ratio
.
9
l
a
a
7
0
6
+
5
x
4
I 0.30
2
I 0.40
I 0.50
Radius-
I 0 60
1 0.70
I
0.90
I
I
0 90
1 .oo
burned
Fig. 10. The amplification of the normal component of turbulence for a confined explosion.
994
T. C. CHEW and R. E. BRITI’ER
*
0.10
0.20
a
0.30
0 40
0 50
060
0 70
080
0.90
4 00
Rodws -burned
Fig. 11. ‘Ike amplification of the lateral (surface) component explosion.
of turbulence
energy for a confined
the initial isotropic spectrum function is given by E(K)=
(2,)-1f2K4~&-1/2f%Q
where LO is the original integral length scale of turbulence. Hence, from (6)
where as before
The
one-dimensional
spectrum is then given by F,,(x1) = A i,,, KgK~r,LZe-“2~~Cx:
+ II-:>drGf3&
where A = ~-*~2~)-“‘t4~)-~.
(9)
From equation (9), the spectra for both confined and unconfined cases may be evaluated with previously calculated values of 5 and c. However, we shall first normalize all initial wave numbers by putting k1 = LOKl etc. where E(K) becomes ~~~(2~)-1~~4e-1~~* and
997
Flame-induced geometrical straining
Equation (9) becomes R4e-‘nn2(R2k3) {t-%-Q: To facilitate integration,
+ c(kZ + k:)}2
dRz dk3.
we substitute
giving 2n
(2k3j2 + fi” + n:)e -1/2fi* rl drl de, ]a; + q212
(10)
The right-hand side of (10) is now numerically integrated and the results for one case (2 = 4) of the confined explosion shown in Fig. 12. The increased turbulence energy is reflected in the larger areas under the curves for higher compression ratios. The pronounced difference in the shapes of the curves are due to normalization with respect to initial values of ugLo. If the individual values of u2L were used to normalize each of the curves, then in fact they would all look very similar, although the respective energy contents will then not be immediately obvious, but instead the integral length scale can be deduced from the intercept with the y-axis. It has been shown [l] that the integral length scale in these situations do not vary significantly from the initial value. (Strictly speaking, the turbulence after distortion is non-isotropic, and therefore up to nine different length scales are necessary to describe the correlations for all velocity components in all directions. The length scale deduced from the F’,hl) spectrum is for the direction normal to the advancing flame front.) Applicability of the RDT For the RDT much faster than and furthermore, larger than those The conditions
to apply, one important requirement is that the distortion must take place there is time for viscous and non-linear transfer effects to become significant; displacements between particles resulting from the distortion must be much caused by the turbulence fluctuations. can be reduced approximately to the expression (Batchelor [28]),
qE+Ol Flame
rodus .
1 00
*
0.96
.
0 92
.
0 96
l
0.60
l
0.70
0
0 60
0
045
+
0.30
x
0 15
lE+OO Initial
stage !
L
\ 1 E-01
1 E-02
I IE-01
1 E+OO
Normalized
wave- number
Fig. 12. The energy spectrum at various stages of combustion, for the case of pressure ratio Z = 4.
998
T. C. CHEW
and R. E. BRITTER
where C- t’ is the distortion time and 1 is the length scale of energy-containing expanding spherical flame ball, t-t’=.-.-
rb - ri %
“<<[
@~I,2
.
eddies.
For an
(11)
For an unconfined explosion, & is constant and (11) is satisfied for large flame speeds or low turbulence levels; and also if Ar is small, i.e. if we consider fluid elements close to the ignition point. A similar argument applies to the confined explosion, although now S, is a function of flame position. Taking a typical case, say in the experiments for turbulent velocity measurements by Andrews et al. [4], the values for the various terms are U’ - l-3 m s-l I - lo-15 mm & - lo-20 m s-’
For distortions of size comparable with the turbulent length scale, so that Ar/l - 1, equation (11) is reasonably satisfied. On the other hand, similar work by Abdel-Gayed et al. [3] for high turbulence levels, u’ is of comparable magnitude or larger than &, and condition (11) is unlikely to hold, although it is likely that there will be a range of wave numbers fitting the requirements, and to this range only the RDT may be applied. Furthermore, as was shown earlier, much of the distortion occurs just before the flame arrives, so in fact t - t’ may indeed be shorter than Ar/S,, and therefore equation (11) can be considered an upper-bound test. It is possible to account for viscous and non-linear effects by various means, such as described by Savill [14]. These modifications will not be attempted in this study. However, we expect that their inclusion will modify quantitatively some of the results presented but will not negate the major conclusions that have been made. An appropriate area of further study is to include into the calculations only those length scales that are affected by the distortion, so that initially when the flame kernel is small, only proportionally small scales are affected by the distortion, while the larger scales only serve to advect the flame kernel about in the fluid. Experimental evidence
Direct comparison with experimental results cannot, at present, be made because of the lack of suitable data. As far as the authors are aware, there have not been measurements made of the turbulent levels just ahead of the leading edge of a freely-expanding spherical flame front. Indirect evidence, however, can be inferred from experiments of turbulent flow round bluff bodies (e.g. [7]), where the application of RDT is substantiated. In the case of a confined explosion, again the experimental evidence is indirect, but positive. Witze and co-workers have reported results from extensive LDV work on combustion in test engines, aimed largely at investigating the effect on turbulence by combustion, and compression by the piston.? Witze [15] Sine
this paper was prepared, further work by Witze and co-workers have been brought to our attention. In addition to those mentioned above, Foster and Witze [19], using a two-dimensional laser Doppler velocimeter to measure the turbulence in a fired combustion chamber found that the combustion has little effect on the normal turbulent stresses, but that the compression in front of the flame does appear to reduce the degree of correlation of the velocity fluctuations. The stresses normal to the flame surface were found to be enhanced at lower engine speeds (corresponding to low turbulence) while the results for the higher engine speeds did not display any discernible trends. A conclusion stated in [19], and also in Witze and Foster [20], was that the enhancement of turbulence by compression is significant only for low initial levels of turbulence. In the context of Rapid Distortion, we suggest that this observation can be rationalized as follows. For a low turbulence level, the characteristic time scales of the flow field are relatively large and the rate of dissipation is small. The criteria for “rapid” distortion more easily approached.For high turbulence levels, the dissipation of energy becomes significant even during the short time it takes for the geometrical distortion of the flow to occur. Hence, not only does the linearized formulations begin to falter, whatever enhancement of turbulence energy that might have still occurred would be much more quickly attenuated through nonlinear mechanisms which are not considered in this paper.
Flame-induced geometrical straining
999
and Witxe et al. [16] measured the fluid velocity in a disc-shaped spark ignition engine cylinder, conditioned by detecting flame arrival with an ionization probe. The result is that the velocity just ahead of the flame front is resolved. Substantial increases over the pre-ignition turbulence levels were recorded, and were attributed to the compression of the turbulence field by a similar mechanism to that presented in this paper. Martin et al. [17,18], using the pressure history for conditioning flame arrival, also reported a significant increase in the turbulence ahead of the flame, especially in the direction of flame propagation, i.e. normal to the flame surface. This is consistent with the results presented in Fig. 9. It must be stressed that comparison can only be qualitative here because Witze was primarily interested in the compression produced by the moving piston, and the geometry is cylindrical rather than spherical. (Furthermore, Martin et al. [17] have also suggested that part of the observed turbulence increase might possibly be attributed to the advection of the flow field by the flame.) Nevertheless, the inference is strong, and the many successes of rapid distortion theory in other fluid mechanics situations (see, e.g. [14]) strongly suggest that this analysis is relevant to many applied problems in turbulent combustion. The implication on flow configurations Having evaluated the effect of geometric straining in spherical expanding flame fronts on turbulence levels, we now examine briefly a few ~mmon burner config~ations. (a) Stirred bomb, e.g. Andrews et al. [4]. This corresponds to the configuration analysed above with the turbulence enhancements given by Figs 10 and 11, suggesting that using pre-ignition turbulence levels to characterize the situation may lead to difficulties if comparisons of burning velocity measurements are to be made with other burner configurations where the effect of turbulence modification is different. (b) Inverted cone, e.g. Heitor [21]. This causes straining, as shown in Fig. 13(a). The magnitude of circumferential stretch, rl/r,, depends on the degree of divergence of the streamlines in the unburnt flow; this is usually small. (c) Bunsen flame, urn-stabil~ed on a tube, e.g. Chew et af. [22]. As shown in Fig. 13(b), the streamlines are practically parallel in the unreacted flow, and hence no distortion results. Incidentally, the Bunsen flame also benefits from the burnt gases acting as a shield against the turbulence from the shear layer (see Yoshida [23]), and its therefore one experimental arrangement where the flame “sees” the turbulent field that was measured in the unreacted flow (suitably protected from the shear layer by, say, a concentric air curtain). (d) Stagnation flow geometry, e.g. Cho et al. [24] and Kostiuk et al. [25]. The geometry shown in Fig. 13(c), and that of two opposing jets, are popular in studies of strain rates in laminar flames. It is also increasingly used in turbulent flame studies because of the absence of a flame holder and also because the t~b~en~ intensity is uniform across the span of the flame. Classical potential flow theory for a ho-dimensional stagnation jet suggest a streamfunction of the form ly=kxy, giving u, = kX ur = -ky. The origin is sited at the stagnation point. The strain rate at any distance d from the stagnation plate 1dA 1 _--__i,,Adtx = k,
5x x independent
of x or d.
loo0
T. C. CHEW and R. E. BRITTER (a)
(b)
w Burned
Mean f lome position
Xreamlme
Annular fluid element
Unburned
Fig. 13. Schematic diagram of some burner configurations: (a) inverted cone; (b) Bunsen flame; (c) stagnation flow.
the rate of distortion is uniform over a transverse cross-section. The above result is for a two-dimensional geometry and is used here for simplicity. A similar conclusion, i.e. that the strain rate on a fluid element is independent of position can be derived for the case of cylindrical geometry shown in Fig. 13(c) (e.g. Mendes-Lopes [26]). Since x x l/y, the area of the fluid element increases as l/d so that the degree of distortion increases, and hence the enhancement of turbulence, as one approaches the stagnation plate. This is supported by experiments by Cho et al. [24], who found that the turbulence levels remain constant, instead of decaying along the jet axis, as would happen in the absence of the stagnation plate. The foregoing discussion highlights the importance of flow configuration on turbulence-flame correlations. In general, geometrical straining enhances the turbulence, but to varying degrees. Thus
1001
FIame-induced geometrical straining
Andrews et al. [4] have collated flame speed measurements of many researchers. The data is expectably large, but one firm conclusion is that data from similar geometries well. We suggest that the effect described here, namely the relationship between distortion and turbulence, is one of probably several factors responsible for the variation between geometries.
scatter of correlate flow-field observed
CONCLUSIONS The geometrical distortion of the turbulence field just ahead of an advancing curved flame front of a point-ignited explosion was evaluated. It was found that this distortion can increase turbulence levels substantially. The results obtained formed a basis from which it is possible to examine the implications on various flow configurations without having to go through detailed calculations for each case. The conclusions drawn have significant ramifications on attempts to correlate turbulent flame propagation with the initial turbulence levels in the unreacted flow.
REFERENCES [l] V. W. WONG and D. P. HOULT, SAE paper 790357 (1979). [2] R. G. ABDEL-GAYED and D. BRADLEY, 16th Symp. (Znt.) Comb., p. 1725 (1977). [3] R. G. ABDEL-GAYED, K. J. AL-KHISHALI and D. BRADLEY, Proc. R. Sot. Land. A, p. 391 (1984). [4] G. E. ANDREWS, D. BRADLEY and S. B. LWAKABAMBA, 15th Symp. (Int.) Comb., p. 655 (1975). (51 P. H. TAYLOR and S. J. BIMSON, Proc. Znd Symp. (Znt.) Comb. (1988). (61 B. LEWIS and G. VON ELBE, Combustion, Flames and Explosions of Gases. Cambridge Univ. Press, Cambridge (1961). [7] R. E. BRITTER, J. C. R. HUNT and J. C. J. MUMFORD, J. Fluid Mech. 92, 269 (1979). [8] J. C. R. HUNT, J. Fluid Mech. 61, 625 (1973). [9] G. K. BATCHELOR and I. P. PROUDMAN, Q. J. Mech. Appl. Mech. VII (1954). ilO1 V. W. WONG, V. P. HOULT and J. C. R. HUNT, unpublished oaoer (1978). ill] D. P. HOULT and V. W. WONG, in Combustion-Modelling’ h Rkcipkating Engines (Edited by J. N. MATI’AVI and C. A. AMANN) Plenum Press. New York 11980). [12] D. D. STRETCH, Ph.D. thesis,Cambridge Univ. (1986). ’ ’ (131 A. A. TOWNSEND, Turbulent Shear Flow. Cambridge Univ. Press, Cambridge (1976). [14] A. M. SAVILL, A. Rev. Fluid Mech. 19, 531 (1987). [15] P. 0. WITZE, Combust. Sci. Technol. 36, 301 (1984). [16] P. 0. WITZE, J. K. MARTIN and C. BORGNAKKE, SAE paper 840377 (1984). [17] J. K. MARTIN, P. 0. WITZE and C. BORGNAKKE, 2&h Symp. (Int.) Comb. p. 29 (1984). [18] J. K. MARTIN, P. 0. WITZE and C. BORGNAKKE, SAE paper 850122 (1985). [19] D. E. FOSTER and P. 0. WITZE, Cornbust. Sci. Technol. 59,85 (1988). [20] D. E. FOSTER and P. 0. WITZE, Proc. IMECHE, CSl/sS, p. 225 (1988). [21] M. V. HEITOR, Ph.D. thesis, Imperial College, London (1986). [22] T. C. CHEW, R. E. BRITTER and K. N. C. BRAY, Cornbust. Flame 75, 165 (1989). [23] A. YOSHIDA and H. TSUJI, 17th Symp. Comb., p. 945 (1979). [24] P. CHO, C. K. LAW, J. R. HERTZBERG and R. K. CHENG, 3fst Symp. (Int.) Comb. (1986). [25] L. W. KOSTIUK, K. N. C. BRAY and T. C. CHEW, Combust. Sci. Technol. 64,233 (1989). [26] J. M. C. MENDES-LOPES, Ph.D. thesis, Cambridge Univ. (1984). [27] G. E. ANDREWS, D. BRADLEY and S. B. LWAKABAMA, Cornbust. Flame 24, 285 (1975). [28] G. K. BATCHELOR, Theory of Homogeneous Turbulence. Cambridge Univ. Press, Cambridge (1953). (Revision received and accepted 18 December 1991)
NOMENCLATURE a = position of fluid element before distortion A = area A = coefficient c = normal strain due to geometrical straining eij = distortion tensor e, = principal strains E = energy spectrum function f = compression ratio for unconfined explosion F,, = one-dimensional energy spectrum
L, L, = n= P= r= R= Sij = S, =
integral scale burnt fraction pressure flame radius radius of spherical vessel distortion tensor flame speed relative to laboratory frame of reference S, = burning velocity relative to unburnt mixture
T. C. CHEW and R. E. BRI’ITER
loo2 r= u’ = U= u= x= Z= etjk = p = yU= Yz K=
time turbulence intensity mean velocity velocity position of fluid element after distortion pressure ratio unit alternating tensor density specific heat ratio of unreacted mixtures wave number vector before distortion magnitude of K 7&= stream function (p, = three dimensional spectrum tensor
X= x = t = 0 =
wave number vector after distortion magnitude of x compression factor = e, e,e, vorticity
Subscripts 1, 2, 3 = principal directions
i = initial properties (e.g. position, pressure,
etc.) e = properties at the completion of comb~tion u = related to the unreacted state b = reiated to the burnt state t = at time t
int. J. EngirgSci. Vol. 30, No. 8, pp. 983-1002, 1992 Printedin Great Britain. All rightsreserved
EFFECT OF FLAME-INDUCED GEOMETRICAL STRAINING ON TURBULENCE LEVELS IN EXPLOSIONS AND COMMON BURNER CONFIGURATIONS T. C. CHEW Department of Mechanical and Production Engineering, National University of Singapore, Kent Ridge 0511, Republic of Singapore
R. E. BRITTER Engineering Department, Cambridge University, Trumpington Street, Cambridge CB2 lPZ, U.K. Abstract-The modification of turbulence ahead of the flame front in a point-ignited explosion developing in a homogeneous turbulent flow is investigated. The spherical flame front may be freely expanding or confined within a vessel. In the former, geometrieal considerations predict that the total strain on a fluid element just before its engulfment by the flame is independent of the initial position of the element. This results in a uniform change of the turbulence. For example, the turbulence intensity generally increases by about 50%. In the case of confined explosions, the distortion of fluid elements leads to increasingly larger turbulence amplifications, starting from a value approximately equal to that of the unconfined case to up to several hundred per cent at the final stages of combustion. Rapid Distortion Theory is used for the analysis. The modified energy spectra are also evaluated. Available experimental data lend qualitative support to the results. The range of validity of this analysis is discussed. The inde~ndent variable is cast in terms of a pressure ratio and straining factors, thus permitting the appli~tion of the data to a wide range of ~~~ratio~ without the need to delve into the complicated mathematics each time. The conclusions are expected to be moderated but not negated by taking into account the process of nonlinear turbulence decay. Finally, the implications on turbulent flame velocity measurements in various configurations are briefly explored.
INTRODUCTION
pre-mixed gas explosions present a simple and practical situation where vorticity dynamics indicate that the turbulence in the unreacted region may be modified significantly by the expansion of the burnt products. A curved, advancing flame front causes stretching of the vorticity in the unburned gases; a similar effect is achieved if the unburned mixture was compressed (e.g. Wong and Hoult [l]). Th e instantaneous and local turbulence field at the flame front can then be quite different to that present before combustion. This has implications on burning velocity measurement techniques using expanding turbulent flames [2-41, as well as in industrial gas explosions [5] where the rate of pressure rise depends on the speed of flame propagation (which normally increases with turbulence intensity). This study examines first an unconfined explosion, and then a centrally-ignited explosion in a spherical vessel. The Rapid Distortion Theory (RDT) is used to estimate the extent of turbulence enhancement by the mechanism of vortex stretching. The results are discussed in the context of the validity of the RDT. These results for sphe~ca~y~~an~ng flames are then used to examine similar effects in some common burner conjurations. Expanding
PURPOSE
The purpose of this paper is to first point out the said modification of turbulence, and then by providing charted results based on simple accessible parameters, show how the effects can be assessed in different configurations without resorting to the complex mathematical formulations each time. The section on RDT formulation, which is fairly well established, is included for continuity and compIeteness, but may be omitted at the first reading of the paper. 983
984
T. C. CHEW
STRAIN
OF CONCENTRIC AN
and R. E. BRIl’TER
ANNULAR ELEMENTS EXPLOSION
ENVELOPING
When a combustible mixture is ignited at a point a spherically-expanding flame front results, which we shall call a fireball for conciseness. The fireball grows as a result of the consumption of unburnt gas, as well as the expansion due to heat release. Unreacted material around the flame is pushed away from the centre until the expanding flame surface overtakes it. If the flame is unconfined, the material is then burnt and remains stationary. The presure drop across the flame front is negligible and there is no compression of the unburnt material. If the explosion took place in a sealed vessel, the burnt gas will be accelerated back towards the centre because of the expansion of the gases burned subsequently [6]. We evaluate the strains? imposed on the unreacted fluid by the growth of the fireball. Following the notation of Lewis and von EIbe [6], the speed of propagation of the flame front relative to the laboratory frame is called the flame speed S,,, while the speed of advance of the flame relative to the unburned mixture is the burning velocity S,. As is well known, the values of both S, and S, are dependent on the turbulence levels in the burning mixture.
CASE
OF UNCONFINED
EXPLOSION
Let r, represent the position of a particle originally at ri, and r, = position of the flame front. When the flame catches up with the particle, r, = r, [Fig. l(a)]. By conservation of mass,
113
rb/ri =
(p,/p,,) =f,
is sometimes
(>
called the isenthalpic
Fig. 1. (a) Unconfined _._...__P._-_.._~
explosion;
fi PI-I
expansion
(b) deformation ~~
t In this paper, the word strain is used rather loosely length.
,
ratio. If we now consider a small
of an element
to mean extension
on a spherical
ratio of a line element
shell.
or, new length/initial
Flame-induced
geometrical
straining
985
element on the thin, annular shell originally of radius ri,
direction 1 is taken to be normal to the surface while 2 and 3 lie on the surface of the spherical shell (Fig. 1). The off-diagonal elements are zero because the chosen axes coincide with the directions of the principal strains. Also e11e22e33= 1 as the flow is incompressible. If we assume negligible heat loss, pU/pb is constant for a given gas mixture. Therefore, the amount of distortion undergone by a fluid element before it is burned is independent of its original distance from the centre of ignition. Besides simplifying matters considerably, this conclusion is important for two reasons. Firstly, it is contrary to the reasoning in some previous studies (such as [l]) which stressed that the straining is most severe when the flame ball is small. In fact, it is the case that during the early stages, the distortion, while of similar magnitude to later stages, is incapable of affecting turbulence scales that are larger than the size of the flame ball, and as a result the possible influence on the turbulence field is less. Secondly, since the geometrical distortion is constant for fluid elements at all distances from the point of ignition, any effect on the turbulence cannot be inferred from trends in the mass burning rate during the explosion. Direct measurement of the turbulence levels just ahead of the flame front is necessary. While the total distortion is ultimately the same, the time history of distortion will obviously be different for elements at different distances from the point of ignition. To calculate the rate of distortion, we note that the position of a particle in the unburnt mixture at a time t is given where
by
[61, r,=
[r~+r;(1-~)]1’3.
If the flame speed &, is assumed constant, then rb = &t and r,= [r?+s’,t(l-$)]lil
art_ SZ(l t-
- l/f’)? rr2
*
This shows that &,/at is close to zero when t is small, but + %(l - l/f3)‘” when t is large. Thus, a large proportion of the distortion occurs just before the element is consumed by the flame (Fig. 2). As will become apparent, this enhances the applicability of the RDT.
THE CASE OF A CONFINED
EXPLOSION
Consider a centrally-ignited mixture contained in a spherical vessel. The propagation of the flame wave is now accompanied by a continuous rise in pressure which is spatially uniform when S, is much smaller than the speed of sound. The temperature of the unburned gas rises as a result of the compression (which can be considered adiabatic). The burning velocity, which is dependent on both the pressure and the initial temperature, is expected to increase. The continuous compression of the burnt gases results in a temperature gradient from the centre to the flame front, the temperature being highest at the centre. Thus, on completion of
986
T. C. CHEW
and R. E. BRITTER
Time
Fig. 2. Growth
of r,. Straight lines are the corresponding
flame positions.
combustion, there exists a difference in temperature between the centre and the vessel walls of up to several hundred degrees centrigrade. When this gradient relaxes, there will be a slight decrease in pressure. The detailed mathematical relationships between burnt fraction, flame position and the state variables of reactants and products can be found in standard text (such as [6]). To simplify matters here, chemical kinetics is avoided by assuming that the final pressure, P,, is known, and it is not critical for the present analysis whether this pressure is measured before or after the aforementioned temperature gradient has been allowed to relax. First, we shall derive the total distortion of a fluid element from the moment of ignition to the time it is caught by the flame. Referring to Fig. 3(a), let R = radius of spherical vessel, r,, = radius of flame ball, ri = initial radius of burnt mixture and n = burnt fraction = (~i/R)3. Geometrical considerations yield Surface strain = rb/r, Normal strain = (z)‘($-$j). Therefore
the distortion tensor is given by
ei, =
(;,‘(EA, ... ... . .. rb/ri *** ...
...
(2)
b/c I.
TO find Th/Ti,it can be shown that
ri/R = [l - (1 - ~z)(P,/P)“‘~]““, where Pi and P are the initial and instantaneous pressures, respectively. Also, the pressure rise is approximately proportional to the fraction burnt, i.e. n - (P - Pi)/(P, - P,). Substituting for
987
Flame-induced geometrical straining (b)
(b) surface and nonnaf %rains.
Fig. 3. (a) Confined expkhn;
P/P, and putting R = 1 (i.e. non-dimo3nsionalizing rb and ;riwrt R), we get Ib-[l-(l-n)(~)m.]m where
z
(3)
&--r?, =T-
The variable 2, which we shall call the pressure ratio, is easily measured for most systems. The value of 2 depends on the amount of the heat release and the specific heat capacity of the burnt products, providing the effect of molecular dissociation is neglected. Typically, 2 ranges from 4 to 8.
i
0
0.10
I
i
0.20
0.30
radius-u~burfled
Fig.
4. Position
1
1
0.40
0.50
(i.e.
at
I 0.60
initiol pressure
I
I 0.70
0.80
i 0.90
l
3
i
a
0
7
0
f3
+
5
x
2=4
I 1 .oo
and temp.)
at whi& a fluid element is caught by the flame compared with its origintlr position.
988
T. C. CHEW
and R. E. BRIITER
0 80
0 20
0
I
I
I
I
I
I
I
I
I
_-s
0 10
0 20
0 30
0 40
0 50
0 60
0 70
0 60
0 90
1 00
Fraction
burned
Fig. 5. Flame radius as a function of fraction burned for a confined
explosion.
Equation (3) relates r, to n, and hence ri, for given values of Z and yU. Plots of rb/ri, and extension (or compression) ratios wrt to rb and mass fraction burnt n, are given in Figs 4-7. In general, a 20% burnt fraction will result in an rb of almost 0.8Rso that initially, straining is the confinement by the vessel restricts surface predominantly geometrical. Subsequently, strain, and normal straining is caused mainly by comparison. In the initial stages too, the rate of distortion will resemble that of the unconfined explosion. In the later stages the predominantly compressive straining is governed by the rate of conversion of unburnt material into burnt products. Even if the flame speed remains constant, more material is consumed per unit time as the surface area of the flame ball grows. The result
Pressure 1 50
c_ 0 L
ratio2
.
9
l
8
0
7
0
6
+
5
x
4
1 00
t;
0
50
0
0 10
0 20
0 30
0 40
0 50 Flame
0 60
0 70
0 80
radius
Fig. 6. Surface strain as a function of flame radius.
” B”
2 il”
989
Flame-induced geometrical straining
.c F
0.60
-
0
-
50
0.40
-
0.30
-
0.20
-
0.10
-
Pressure
ratio
2
;;
0
1
0.10
I
I
I
I
I
I
I
I
0.20
0.30
0.40
0.50
0.60
0.70
0.60
0.90
Flame
I 1.00
radius
Fig. 7. Normal strain as a function of flame radius.
quite fluid element evaluate the only for the and pressure
is then
similar to the case of the unconfined explosion, so that much of the distortion of a occurs just before the element is consumed by the flame. We shall not attempt to actual rate of distortion here because in fact S, is unlikely to remain constant, if fact that the laminar burning velocity is enhanced by the increase in temperature in the vessel.
ANALYSIS
BY
RDT
The RDT has been used with success by many researchers to predict the effect on turbulence of a distortion in a fluid taking place so rapidly that non-linear effects can be neglected [e.g. 1, 7, 9 and 111. Batchelor and Proudman [9] calculated the changes produced in a homogeneous turbulent motion when the fluid is subjected to a superimposed uniform distortion. In particular the asymptotic case of a large symmetrical contraction (as commonly occurs in the wind tunnel) was examined in detail. It was found that the energy for the longitudinal component of velocity is reduced, while the energy for the lateral velocity components is increased. The total energy is always increased. Qualitatively, the process can be explained in terms of vorticity stretching. When the vortex is pulled out along its axis of rotation, its outlying elements are drawn closer to the centre, and conservation of angular momentum requires that it now spins faster. The velocity components in a plane parallel to the axis of rotation are increased. Similarly, if a vortex is caused to expand so that the rotating fluid elements are drawn away from the axis of rotation, the velocities will be slowed down. Again only the velocity components in a plane parallel to the axis will be affected. The components in a plane perpendicular to the axis will not be affected by the distortion of this vortex. Isotropic turbulence consists of vortices oriented at random throughout the fluid. The effect of an arbitrary distortion is to increase the energy in some directions while decreasing it in others. The net result is always an increase, with the additional energy input coming from the distortion. Hunt [8] formalized much of the mathematics and also examined the applicability of the assumptions to various regimes of turbulence conditions. The analysis used distortion by bluff
990
T. C. CHEW and R. E. BRIIITER
bodies as a basis to examine the effect of varying the length scale ratios between the turbulence scales and the scale of the distortions. Britter et al. [7] showed, with careful experimental measurements of the grid generated turbulence flow past a circular cylinder, that the amplification and reduction of the three components of turbulence can indeed be explained by the distortion of the mean flow and the blocking effect caused by the turbulence impinging on the cylinder surface. Good agreement was found with predictions by Hunt [B]. Insofar as applications to cornbusting flows are concerned, only a few publications are available in the literature. Wong et al. [lo] used RDT to calculate the effect of compression by the piston in an internal combustion engine. Wong and Hoult [l] looked at the distortion of the turbulence during combustion in the engine cylinder. The flame was assumed to be ignited on the axis of a disk-shaped combustion chamber and which then propagates radially outwards. Enhancement of turbulence was predicted which can be used to explain the increased burning rates in some engines. Experimental measurements by Witze [15,16] revealed increases in turbulence levels during combustion in an engine cylinder with qualitative agreement with the predictions of [l]. Other applications of RDT include, among others Townsend [13] and Stretch [12]. Townsend applied the theory to evaluate the structure in turbulent shear flows with extensive comparisons with experimental results. Stretch studied the dispersion of slightly dense contaminants in turbulent boundary layers. That the theory is extremely useful in making both quantitative and trend predictions in turbulent flows under some limiting conditions is without doubt. Quantitative predictions are confirmed when the theory is applicable, e.g. Britter et al. [7]. The ideas and methods are still continuously being developed. A review of current applications of RDT is given by Savill [ 141.
MATHEMATICAL
ANALYSIS
We present below a brief outline of our analysis of the distortion of follow closely that of [9], except for Starting from Cauchy’s equations,
OF A DISTORTION
ON TURBULENCE
of the mathematical formulation which will form the basis the turbulence field ahead of explosions. The main ideas the inclusion of density changes. the vorticity of a fluid element is given by
2
f OJX) = + 0
oj(a) I
where a and x are the respective positions of the fluid element before and after distortion and p. and p are the original and new densities respectively. Writing dXi - = the distortion tensor eij,
&j
and PIPoeij
=
&j,
gives Oi(X) =
SijWj
(a).
The prime (‘) on the vorticity denotes its pre-distortion expressed as velocity gradients:
eiik is the unit alternating
(4) value. The vorticity components
can be
tensor, taking the value of 1 or -1 when i, j and k are in cyclic or
991
Flame-induced geometrical straining
reverse cyclic order respectively, and zero otherwise. If K is a wave-number vector of the original energy spectrum, we choose the new wave-number vector to be 2 such that
The velocity distribution at any time during the distortion can be written as a Fourier integral eiY”dZ(K)
u(a) =
and the energy spectrum tensor, at the initial instant, is given by [9]
‘$;{K)
&j(K)
=Kl
dZ;(w)
drc
.
where d2 is the complex conjugate of dZ. #b is the Fourier transform of the (initial) correlation tensor, i.e.
Making use of the result derived by [9] in identical formulation, the distortion is found to be
Choosing the axes to coincide turbulence initially, i.e.
&?2 + $33 = (f-)‘S
with the axes of principal
(2k:(K$+
Kg)+
(Kg+
strain,
velocity
the energy spectrum after
and assuming
isotropic
&)(;K;+$K:)
W)
For the spherically symmetric configurations
being considered in this study, we can write
e2=e3=c-ln
and e, = cc whence w2e3
= t
g is the factor by which fluid has been compressed,
5;=
i.e.
($)-’
992
T. C. CHEW and R. E. BRIITER
Substituting into equations (5)) h(x)
= 5-“%
(6)
(d + 4)
$22(x) + @33(X)= 5-2=4JcX4K2(c2cX4 + ?-‘K:K2)
(7)
For the case of an unconfined explosion f=l and equations (6) and (7) reduce to the same form as in [9].
EFFECT
ON THE
TURBULENCE
INTENSITY
The ratio of the energy due to the velocity components flame surface) to that before distortion is
Similarly, the effect on the turbulence
in the 1 direction
energies in the 2 and 3 directions is given by
+22(x) + #33(x) dzt Y2 = P3 = 442(K) + 443(K) dK = j- c-2 $$$
(g2cx-4 + E(K) -
45GK4
c-2C-3K:K2)
(K2 + K:)
dX
dK
Noting that dX -=dK
1 e,e2e3
. ie
. .
dX= c-‘du,
and putting K, = K COS 8,
(K;
+ K:)
= (K
Sin 6)”
(following [9]), we get p*+-3
=
* sin3 8 de n sin2 8 d0 I0 [( 5;~)~”cos2 0 + c sin2 @I2/I 0 f-Q-2$
(normal to the
B’ - I tan-l /j + p-2 P3
1
where p”=(c-‘c-“-l)
note::::
993
Flame-induced geometrical straining
and g-3-
E(K) K2x4
( c2Cx4
+
c-2C-3K;K2)
Kf)
= &g-l
+ g-‘C-5
L (
dK
dK
tan-’ /3 -
B’
1 B’(B” + 1) > *
(8)
RESULTS
The foregoing analyses allow for distortion resulting from a compression as well as pure geometrical straining. We can now use the distortion tensors evaluated in equations (1) and (2) to obtain the effects on the turbulence for both the cases of confined and unconfined explosion. Unconfined explosion
The vorticity distortion tensor Sij = p/poeii. From equation (l),
Since the fluid element (pulpbY3,
does not undergo
volumetric
compression
here,
c = 1 and rb/ri =
giving
...
(z>l/3 P"
Sij
=
...
...
... ( >1’3 it3 , Pll
As emphasized earlier, the total distortion is independent of the initial position of the fluid element. Figure 8 shows the variation of the energy ratios p1 and ,u~as evaluated from equation (8). It is seen that the energy due to velocity fluctuations in the normal direction (direction 1) is always amplified. p2 and p3 decrease slightly at first, but are also amplified for a compression factor (in the normal direction) of 0.4 or less. The overall turbulence energy is always amplified. For an isenthalpic expansion ratio of say 8, p1 is about 2.5. Figure 9 shows plots of
p1 and ~2 vs It is concluded that the turbulence as seen by the travelling flame front is significantly different from that present initially. This is especially true for the early stages of the development of the explosion, and for large flame speeds, where the time scale is too small for viscous effects to be relevant, and also allowing nonlinear momentum transfer process to be neglected. The time just after ignition, however, may not be amenable to this analysis because the flame ball is probably too small to affect the integral scales of turbulence in the way (PJPLJ.
ES 30:8-C
994
T. C. CHEW and R. E. BRI’ITER
I 0
I
I
I
I
10
20
30
40
Isentholpic
expansion
ratio
Fig. 8. Effect of isenthaipic expansion ratio on the intensity for an unconfined explosion.
described, distortion.
i.e. when the integral scale of turbulence
is large compared
with the scale of the
Confined explosion We have
-rb
...
C
-rb
ri ~
Using values of rb and ri evaluated previously, the energy ratios pl and pz (= ,IA~)are found as shown in Figs 10 and 11. In the initial stages of the explosion, corresponding to the prepressure phase, the effect on the turbulence is similar to that for the unconfined case. However, as the flame progresses, the turbulence is amplified further. Examination of the terms in Sii shows that this is because the distortion resulting from the compression of the unbu~t gas more than com~nsates for the restraint on the growth of rb by the vessel walls. For a typical pressure ratio (P, - PI/pi) of say 6, a ten-fold increase in the normal component energy is predicted and a five-fold increase in the surface component near the last stages of the explosion. When the flame front is midway across the vessel, the numbers are about 2.5 and 1.5, respectively. The increase in the turbulent velocity fluctuations is proportional to the square root of these values. This again represents a marked deviation from the turbulence field which existed before ignition. From Figs 10, 11, and 7, it is obvious that several factors affect the extent of the
Flame-induced geometrical straining 4.00
-
3.50
-
300
-
995
.-s z s 4 ET 50
250-
a z 0
zoo-
iii w G
1.50
-
1.00
-
6 z 2 z 5
0.50
-
fz I 0
0.10
I
I
0.20
I
0.30
I
0.40
Compression
I
I
0.50
0.60
factor
0 70
I 0.80
I 0 90
I 100
c
Fig. 9. Effect of compression factor c on the turbulence energy for an unconfined explosion.
vorticity distortion mechanism, the most important of which are the specific heat capacities, expansion ratios, and mass fraction burnt (or stage of explosion). One concludes that the use of measurements in the initial turbulence to characterize the flow during an explosion may be unwise. A correction factor, easily derived from the results presented here, could perhaps be applied to obtain a more realistic representation of the turbulence field instantaneous and local to the flame. Spectra The one-dimensional spectrum tensor &((x
spectrum, &(x1) may be found by integrating the three-dimensional for all values of x2 and x3. We assume, following Townsend [13], that
Pressure
I 0.10
I 0.20
ratio
.
9
l
a
a
7
0
6
+
5
x
4
I 0.30
2
I 0.40
I 0.50
Radius-
I 0 60
1 0.70
I
0.90
I
I
0 90
1 .oo
burned
Fig. 10. The amplification of the normal component of turbulence for a confined explosion.
994
T. C. CHEW and R. E. BRITI’ER
*
0.10
0.20
a
0.30
0 40
0 50
060
0 70
080
0.90
4 00
Rodws -burned
Fig. 11. ‘Ike amplification of the lateral (surface) component explosion.
of turbulence
energy for a confined
the initial isotropic spectrum function is given by E(K)=
(2,)-1f2K4~&-1/2f%Q
where LO is the original integral length scale of turbulence. Hence, from (6)
where as before
The
one-dimensional
spectrum is then given by F,,(x1) = A i,,, KgK~r,LZe-“2~~Cx:
+ II-:>drGf3&
where A = ~-*~2~)-“‘t4~)-~.
(9)
From equation (9), the spectra for both confined and unconfined cases may be evaluated with previously calculated values of 5 and c. However, we shall first normalize all initial wave numbers by putting k1 = LOKl etc. where E(K) becomes ~~~(2~)-1~~4e-1~~* and
997
Flame-induced geometrical straining
Equation (9) becomes R4e-‘nn2(R2k3) {t-%-Q: To facilitate integration,
+ c(kZ + k:)}2
dRz dk3.
we substitute
giving 2n
(2k3j2 + fi” + n:)e -1/2fi* rl drl de, ]a; + q212
(10)
The right-hand side of (10) is now numerically integrated and the results for one case (2 = 4) of the confined explosion shown in Fig. 12. The increased turbulence energy is reflected in the larger areas under the curves for higher compression ratios. The pronounced difference in the shapes of the curves are due to normalization with respect to initial values of ugLo. If the individual values of u2L were used to normalize each of the curves, then in fact they would all look very similar, although the respective energy contents will then not be immediately obvious, but instead the integral length scale can be deduced from the intercept with the y-axis. It has been shown [l] that the integral length scale in these situations do not vary significantly from the initial value. (Strictly speaking, the turbulence after distortion is non-isotropic, and therefore up to nine different length scales are necessary to describe the correlations for all velocity components in all directions. The length scale deduced from the F’,hl) spectrum is for the direction normal to the advancing flame front.) Applicability of the RDT For the RDT much faster than and furthermore, larger than those The conditions
to apply, one important requirement is that the distortion must take place there is time for viscous and non-linear transfer effects to become significant; displacements between particles resulting from the distortion must be much caused by the turbulence fluctuations. can be reduced approximately to the expression (Batchelor [28]),
qE+Ol Flame
rodus .
1 00
*
0.96
.
0 92
.
0 96
l
0.60
l
0.70
0
0 60
0
045
+
0.30
x
0 15
lE+OO Initial
stage !
L
\ 1 E-01
1 E-02
I IE-01
1 E+OO
Normalized
wave- number
Fig. 12. The energy spectrum at various stages of combustion, for the case of pressure ratio Z = 4.
998
T. C. CHEW
and R. E. BRITTER
where C- t’ is the distortion time and 1 is the length scale of energy-containing expanding spherical flame ball, t-t’=.-.-
rb - ri %
“<<[
@~I,2
.
eddies.
For an
(11)
For an unconfined explosion, & is constant and (11) is satisfied for large flame speeds or low turbulence levels; and also if Ar is small, i.e. if we consider fluid elements close to the ignition point. A similar argument applies to the confined explosion, although now S, is a function of flame position. Taking a typical case, say in the experiments for turbulent velocity measurements by Andrews et al. [4], the values for the various terms are U’ - l-3 m s-l I - lo-15 mm & - lo-20 m s-’
For distortions of size comparable with the turbulent length scale, so that Ar/l - 1, equation (11) is reasonably satisfied. On the other hand, similar work by Abdel-Gayed et al. [3] for high turbulence levels, u’ is of comparable magnitude or larger than &, and condition (11) is unlikely to hold, although it is likely that there will be a range of wave numbers fitting the requirements, and to this range only the RDT may be applied. Furthermore, as was shown earlier, much of the distortion occurs just before the flame arrives, so in fact t - t’ may indeed be shorter than Ar/S,, and therefore equation (11) can be considered an upper-bound test. It is possible to account for viscous and non-linear effects by various means, such as described by Savill [14]. These modifications will not be attempted in this study. However, we expect that their inclusion will modify quantitatively some of the results presented but will not negate the major conclusions that have been made. An appropriate area of further study is to include into the calculations only those length scales that are affected by the distortion, so that initially when the flame kernel is small, only proportionally small scales are affected by the distortion, while the larger scales only serve to advect the flame kernel about in the fluid. Experimental evidence
Direct comparison with experimental results cannot, at present, be made because of the lack of suitable data. As far as the authors are aware, there have not been measurements made of the turbulent levels just ahead of the leading edge of a freely-expanding spherical flame front. Indirect evidence, however, can be inferred from experiments of turbulent flow round bluff bodies (e.g. [7]), where the application of RDT is substantiated. In the case of a confined explosion, again the experimental evidence is indirect, but positive. Witze and co-workers have reported results from extensive LDV work on combustion in test engines, aimed largely at investigating the effect on turbulence by combustion, and compression by the piston.? Witze [15] Sine
this paper was prepared, further work by Witze and co-workers have been brought to our attention. In addition to those mentioned above, Foster and Witze [19], using a two-dimensional laser Doppler velocimeter to measure the turbulence in a fired combustion chamber found that the combustion has little effect on the normal turbulent stresses, but that the compression in front of the flame does appear to reduce the degree of correlation of the velocity fluctuations. The stresses normal to the flame surface were found to be enhanced at lower engine speeds (corresponding to low turbulence) while the results for the higher engine speeds did not display any discernible trends. A conclusion stated in [19], and also in Witze and Foster [20], was that the enhancement of turbulence by compression is significant only for low initial levels of turbulence. In the context of Rapid Distortion, we suggest that this observation can be rationalized as follows. For a low turbulence level, the characteristic time scales of the flow field are relatively large and the rate of dissipation is small. The criteria for “rapid” distortion more easily approached.For high turbulence levels, the dissipation of energy becomes significant even during the short time it takes for the geometrical distortion of the flow to occur. Hence, not only does the linearized formulations begin to falter, whatever enhancement of turbulence energy that might have still occurred would be much more quickly attenuated through nonlinear mechanisms which are not considered in this paper.
Flame-induced geometrical straining
999
and Witxe et al. [16] measured the fluid velocity in a disc-shaped spark ignition engine cylinder, conditioned by detecting flame arrival with an ionization probe. The result is that the velocity just ahead of the flame front is resolved. Substantial increases over the pre-ignition turbulence levels were recorded, and were attributed to the compression of the turbulence field by a similar mechanism to that presented in this paper. Martin et al. [17,18], using the pressure history for conditioning flame arrival, also reported a significant increase in the turbulence ahead of the flame, especially in the direction of flame propagation, i.e. normal to the flame surface. This is consistent with the results presented in Fig. 9. It must be stressed that comparison can only be qualitative here because Witze was primarily interested in the compression produced by the moving piston, and the geometry is cylindrical rather than spherical. (Furthermore, Martin et al. [17] have also suggested that part of the observed turbulence increase might possibly be attributed to the advection of the flow field by the flame.) Nevertheless, the inference is strong, and the many successes of rapid distortion theory in other fluid mechanics situations (see, e.g. [14]) strongly suggest that this analysis is relevant to many applied problems in turbulent combustion. The implication on flow configurations Having evaluated the effect of geometric straining in spherical expanding flame fronts on turbulence levels, we now examine briefly a few ~mmon burner config~ations. (a) Stirred bomb, e.g. Andrews et al. [4]. This corresponds to the configuration analysed above with the turbulence enhancements given by Figs 10 and 11, suggesting that using pre-ignition turbulence levels to characterize the situation may lead to difficulties if comparisons of burning velocity measurements are to be made with other burner configurations where the effect of turbulence modification is different. (b) Inverted cone, e.g. Heitor [21]. This causes straining, as shown in Fig. 13(a). The magnitude of circumferential stretch, rl/r,, depends on the degree of divergence of the streamlines in the unburnt flow; this is usually small. (c) Bunsen flame, urn-stabil~ed on a tube, e.g. Chew et af. [22]. As shown in Fig. 13(b), the streamlines are practically parallel in the unreacted flow, and hence no distortion results. Incidentally, the Bunsen flame also benefits from the burnt gases acting as a shield against the turbulence from the shear layer (see Yoshida [23]), and its therefore one experimental arrangement where the flame “sees” the turbulent field that was measured in the unreacted flow (suitably protected from the shear layer by, say, a concentric air curtain). (d) Stagnation flow geometry, e.g. Cho et al. [24] and Kostiuk et al. [25]. The geometry shown in Fig. 13(c), and that of two opposing jets, are popular in studies of strain rates in laminar flames. It is also increasingly used in turbulent flame studies because of the absence of a flame holder and also because the t~b~en~ intensity is uniform across the span of the flame. Classical potential flow theory for a ho-dimensional stagnation jet suggest a streamfunction of the form ly=kxy, giving u, = kX ur = -ky. The origin is sited at the stagnation point. The strain rate at any distance d from the stagnation plate 1dA 1 _--__i,,Adtx = k,
5x x independent
of x or d.
loo0
T. C. CHEW and R. E. BRITTER (a)
(b)
w Burned
Mean f lome position
Xreamlme
Annular fluid element
Unburned
Fig. 13. Schematic diagram of some burner configurations: (a) inverted cone; (b) Bunsen flame; (c) stagnation flow.
the rate of distortion is uniform over a transverse cross-section. The above result is for a two-dimensional geometry and is used here for simplicity. A similar conclusion, i.e. that the strain rate on a fluid element is independent of position can be derived for the case of cylindrical geometry shown in Fig. 13(c) (e.g. Mendes-Lopes [26]). Since x x l/y, the area of the fluid element increases as l/d so that the degree of distortion increases, and hence the enhancement of turbulence, as one approaches the stagnation plate. This is supported by experiments by Cho et al. [24], who found that the turbulence levels remain constant, instead of decaying along the jet axis, as would happen in the absence of the stagnation plate. The foregoing discussion highlights the importance of flow configuration on turbulence-flame correlations. In general, geometrical straining enhances the turbulence, but to varying degrees. Thus
1001
FIame-induced geometrical straining
Andrews et al. [4] have collated flame speed measurements of many researchers. The data is expectably large, but one firm conclusion is that data from similar geometries well. We suggest that the effect described here, namely the relationship between distortion and turbulence, is one of probably several factors responsible for the variation between geometries.
scatter of correlate flow-field observed
CONCLUSIONS The geometrical distortion of the turbulence field just ahead of an advancing curved flame front of a point-ignited explosion was evaluated. It was found that this distortion can increase turbulence levels substantially. The results obtained formed a basis from which it is possible to examine the implications on various flow configurations without having to go through detailed calculations for each case. The conclusions drawn have significant ramifications on attempts to correlate turbulent flame propagation with the initial turbulence levels in the unreacted flow.
REFERENCES [l] V. W. WONG and D. P. HOULT, SAE paper 790357 (1979). [2] R. G. ABDEL-GAYED and D. BRADLEY, 16th Symp. (Znt.) Comb., p. 1725 (1977). [3] R. G. ABDEL-GAYED, K. J. AL-KHISHALI and D. BRADLEY, Proc. R. Sot. Land. A, p. 391 (1984). [4] G. E. ANDREWS, D. BRADLEY and S. B. LWAKABAMBA, 15th Symp. (Int.) Comb., p. 655 (1975). (51 P. H. TAYLOR and S. J. BIMSON, Proc. Znd Symp. (Znt.) Comb. (1988). (61 B. LEWIS and G. VON ELBE, Combustion, Flames and Explosions of Gases. Cambridge Univ. Press, Cambridge (1961). [7] R. E. BRITTER, J. C. R. HUNT and J. C. J. MUMFORD, J. Fluid Mech. 92, 269 (1979). [8] J. C. R. HUNT, J. Fluid Mech. 61, 625 (1973). [9] G. K. BATCHELOR and I. P. PROUDMAN, Q. J. Mech. Appl. Mech. VII (1954). ilO1 V. W. WONG, V. P. HOULT and J. C. R. HUNT, unpublished oaoer (1978). ill] D. P. HOULT and V. W. WONG, in Combustion-Modelling’ h Rkcipkating Engines (Edited by J. N. MATI’AVI and C. A. AMANN) Plenum Press. New York 11980). [12] D. D. STRETCH, Ph.D. thesis,Cambridge Univ. (1986). ’ ’ (131 A. A. TOWNSEND, Turbulent Shear Flow. Cambridge Univ. Press, Cambridge (1976). [14] A. M. SAVILL, A. Rev. Fluid Mech. 19, 531 (1987). [15] P. 0. WITZE, Combust. Sci. Technol. 36, 301 (1984). [16] P. 0. WITZE, J. K. MARTIN and C. BORGNAKKE, SAE paper 840377 (1984). [17] J. K. MARTIN, P. 0. WITZE and C. BORGNAKKE, 2&h Symp. (Int.) Comb. p. 29 (1984). [18] J. K. MARTIN, P. 0. WITZE and C. BORGNAKKE, SAE paper 850122 (1985). [19] D. E. FOSTER and P. 0. WITZE, Cornbust. Sci. Technol. 59,85 (1988). [20] D. E. FOSTER and P. 0. WITZE, Proc. IMECHE, CSl/sS, p. 225 (1988). [21] M. V. HEITOR, Ph.D. thesis, Imperial College, London (1986). [22] T. C. CHEW, R. E. BRITTER and K. N. C. BRAY, Cornbust. Flame 75, 165 (1989). [23] A. YOSHIDA and H. TSUJI, 17th Symp. Comb., p. 945 (1979). [24] P. CHO, C. K. LAW, J. R. HERTZBERG and R. K. CHENG, 3fst Symp. (Int.) Comb. (1986). [25] L. W. KOSTIUK, K. N. C. BRAY and T. C. CHEW, Combust. Sci. Technol. 64,233 (1989). [26] J. M. C. MENDES-LOPES, Ph.D. thesis, Cambridge Univ. (1984). [27] G. E. ANDREWS, D. BRADLEY and S. B. LWAKABAMA, Cornbust. Flame 24, 285 (1975). [28] G. K. BATCHELOR, Theory of Homogeneous Turbulence. Cambridge Univ. Press, Cambridge (1953). (Revision received and accepted 18 December 1991)
NOMENCLATURE a = position of fluid element before distortion A = area A = coefficient c = normal strain due to geometrical straining eij = distortion tensor e, = principal strains E = energy spectrum function f = compression ratio for unconfined explosion F,, = one-dimensional energy spectrum
L, L, = n= P= r= R= Sij = S, =
integral scale burnt fraction pressure flame radius radius of spherical vessel distortion tensor flame speed relative to laboratory frame of reference S, = burning velocity relative to unburnt mixture
T. C. CHEW and R. E. BRI’ITER
loo2 r= u’ = U= u= x= Z= etjk = p = yU= Yz K=
time turbulence intensity mean velocity velocity position of fluid element after distortion pressure ratio unit alternating tensor density specific heat ratio of unreacted mixtures wave number vector before distortion magnitude of K 7&= stream function (p, = three dimensional spectrum tensor
X= x = t = 0 =
wave number vector after distortion magnitude of x compression factor = e, e,e, vorticity
Subscripts 1, 2, 3 = principal directions
i = initial properties (e.g. position, pressure,
etc.) e = properties at the completion of comb~tion u = related to the unreacted state b = reiated to the burnt state t = at time t