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Diffusion-flame dynamics in Von Kármán boundary layers
Diffusion-Flame Dynamics in Von Ka ´rma ´n Boundary Layers V. NAYAGAM
National Center for Microgravity Research, NASA Lewis Research Center, Cleveland, OH 44135, USA
and F. A. WILLIAMS*
Center for Energy and Combustion Research, Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA Experimental and theoretical results are reported for flat diffusion flames adjacent to a spinning fuel disk in an oxidizing atmosphere. Disk-shaped flames are observed, the radii of which increase or decrease with time, depending on whether an appropriate Damko ¨hler number is large or small. Experimentally, initially large flame disks expand at sufficiently high Damko ¨hler numbers, while at sufficiently small Damko ¨hler numbers, initially small flame disks contract, as predicted by theory. A stability map is derived theoretically in a plane of the Damko ¨hler number and the flame-disk radius normalized by a diffusion length scale, showing regimes of disk growth and shrinkage. © 2001 by The Combustion Institute
INTRODUCTION The von Ka´rma´n swirling flow [1] is a classical viscous flow adjacent to a solid spinning disk, having radial and tangential velocity components proportional to the radius. A self-similar solution then exists with the normal velocity component and all properties dependent only on the normal distance z from the disk through the nondimensional variable 公⍀/v ⬁ 兰 z0 ( / ⬁ ) dz. Here ⍀ denotes the angular rotational velocity of the disk, v is the kinematic viscosity and the gas density, and the subscript ⬁ identifies properties evaluated in the ambient atmosphere. We have made measurements on polymethyl methacrylate (PMMA) disks in this flow configuration, ignited in air to produce a flat diffusion flame in the von Ka´rma´n boundary layer [2]. Experimental conditions can be established under which these flames burn steadily after growing to envelop the entire fuel disk [2], extinguish by contracting axisymmetrically, or form spiral patterns [3]. The present paper concerns conditions under which axisymmetric flame-disk growth or contraction occurs and addresses the dynamics of those processes. Besides results of experimental observations, a theoretical analysis is presented which builds on our previous [4] theory of diffusion-flame holes. We present the theoretical considerations *Corresponding author. E-mail: [email protected] 0010-2180/01/$–see front matter PII 0010-2180(00)00253-4
first and then show the experimental results for comparison with theory. The theory adopts the Buckmaster [5, 6] simplified edge-flame description because that facilitates analysis by reducing the problem to one space dimension while preserving all proper scalings. The role of the Buckmaster approximation and its relationship to more thorough analyses have been discussed previously [4 – 6] and need no further amplification. Embedding the Buckmaster theory into the von Ka´rma´n swirling flow is a new contribution of the present work, which adopts notation different from the earlier papers, seeking economy of presentation.
FORMULATION Generalizing Buckmaster’s formulation to unsteady axisymmetric conditions [4] and choosing for this flow the length and time units 公␣⬁/⍀ and 1/⍀ for forming the nondimensional radial coordinate r and non-dimensional time t, where the thermal diffusivity ␣ is of the same order as v, the Prandtl number being taken to be of order unity, we write the equation for conservation of thermal energy in the form
冉 冊
⭸ 1 ⭸ ⭸ ⫺ r ⭸t r ⭸r ⭸r
⫽ ␦(1⫺)2e共1⫺1/ 兲/ ⑀ ⫺ 共 ⫺ a兲
(1)
COMBUSTION AND FLAME 125:974 –981 (2001) © 2001 by The Combustion Institute Published by Elsevier Science Inc.
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where ⬅ T/T f is the ratio of the temperature to the flame temperature far from the edge, (conditions identified by the subscript f ), ⑀ ⬅ RT f /E is the reciprocal of the non-dimensional activation energy (E being the activation energy and R the universal gas constant), a is a representative constant nondimensional boundary temperature (e.g., T ⬁ /T f ), and ␦ is a (constant) Damko ¨hler number. The form of the reactionrate term applies to a one-step, Arrhenius process of first order with respect to both fuel and oxidizer. The factor (1 ⫺ )2 in that term arises from the product of fuel and oxidizer concentrations, each of which, to leading order in activation-energy asymptotics, is proportional to (1 ⫺ ) in the reaction zone for the diffusionflame regime [7]. The corresponding Damko ¨hler number is then
␦ ⫽ 关共 fB OQ FL FL O兲/共c pT fM F⍀兲兴e⫺E/共RT f兲
(2)
where B O is the pre-exponential factor in the Arrhenius expression for the molar rate of consumption of oxidizer, Q F is the heat released per unit mass of fuel consumed, c p is the (constant) specific heat at constant pressure, M F is the molecular weight of fuel, and L F and L O are the Lewis number (␣ divided by diffusion coefficient) of the fuel and oxidizer, respectively. The last term in Eq. 1 models the energy loss to the sides of the edge flame by heat conduction. An implicit approximation is that such transport to the side can be described keeping a constant everywhere, an assumption that is not exactly applicable for this or most other real problems. Nevertheless, it is qualitatively reasonable and does not affect scaling. The character of this approximation has been discussed in much greater detail by Buckmaster [6]. Although the side loss is essential, for most purposes the a dependence readily scales out of the problem, so that the value of a provides a small correction, the dominant physics being carried by the parameters ␦ and ⑀. The further approximations of constant thermal conductivity and thermal diffusivity and neglect of radial convection also are present in Eq. 1; otherwise, factors and terms accounting for these effects would appear on the left-hand side. The objective is to
obtain a model that is as simple as possible but yet retains most of the essential phenomena. The coupling with species conservation equations that occurs in the reaction term can be treated differently in detail through different formulations. If Lewis numbers are unity and boundary and initial conditions appropriate, then the linear dependence on temperature applies everywhere, independent of chemistry. It is often desirable to account for different Lewis numbers, necessitating suitable separate analyses of concentration fields in external zones with matching to reaction-zone solutions through activation-energy asymptotics. Equations for the concentration fields are not addressed here. Instead, results of this type of analysis [4] are included through Eq. 2, in which T f differs from the adiabatic flame temperature when Lewis numbers differ from unity. The advantage of the Buckmaster model is that, because zones on the sides of the reaction sheet are all implicit, there is only one space dimension in which zones arise. For a flame disk, there is a uniform flame zone in the region r ⬍ r e , an inert transport zone in the region r ⬎ r e and a reactive-diffusive flame-edge zone near r ⫽ r e , where r e (t) is the non-dimensional location of the flame edge. In the uniform flame zone the derivatives in Eq. 1 vanish, there being a balance between chemical reaction and side loss. For small ⑀ this balance exhibits ignition and extinction [5, 6], the latter, to leading order, in terms of ⬅ (1 ⫺ )/⑀, being tangency of 2e⫺ ⫽ (1 ⫺ a)/( ⑀ 2 ␦ ), namely ⫽ 2, ␦ ⫽ (1 ⫺ a)e2/(4⑀2). In the inert transport zone, Eq. 1 is a linear parabolic partial differential equation in r and t whose solution must have approach a as r approaches infinity and must match with the edge-region solution at r ⫽ r e . The edge-region solution is to be determined by stretching in r and matching to the uniform flame at ⫺⬁ and the inert solution at ⫹⬁.
SOLUTION The stretchings ⫽ (r ⫺ r e )/ ⑀ , ⫽ (1 ⫺ )/ ⑀ , in Eq. 1 produce at leading order the equation d2/d2 ⫽ ␦⑀32e⫺
(3)
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V. NAYAGAM AND F. A. WILLIAMS
the first integral of which, subject to approaching zero as approaches ⫺⬁, as is appropriate in the asymptotic context [6], is
冉 冕
d/d ⫽ 2␦⑀
3
冊
1/ 2
2 ⫺
e
0
d
(4)
which approaches 2公␦⑀3 as approaches infinity. The necessary slope matching for the solution in the inert zone at r ⫽ r e therefore is ⭸ /⭸r ⫽ ⫺2 公␦⑀ 3 . In the inert zone, it is convenient to work with the normalized temperature variable ⌿ ⬅ ( ⫺ a)/(1 ⫺ a). The problem in the inert zone, obtained from Eq. 1 and the matching and boundary conditions, then becomes
冉 冊
冧
⭸⌿ ⭸⌿ 1 ⭸ ⫽ r ⫺ ⌿, ⭸t r ⭸r ⭸r ⌿ ⫽ ⌿ 0共r兲 at t ⫽ 0, ⌿ ⫽ 0 at r ⫽ ⬁, ⌿ ⫽ 1 and ⭸⌿/⭸r ⫽ ⫺⌬ at r ⫽ re 共t兲,
(5)
˙ e/ 冑⍀ ␣ ⬁. V⫽R
where ⌬⬅ ⫽
册
4f BO QF LF LO R3 Tf4 1/ 2 ⫺E/共2 RT f兲 e MF cp 共Tf ⫺ T⬁ 兲2 E3 ⍀
(6)
and ⌿ 0 (r) is a specified initial distribution. The preceding restriction on existence of the uniform-flame solution, ␦ ⱖ (1 ⫺ a)e2/(4⑀2), translates into ⌬ ⱖ e公⑀ /(1 ⫺ a). With r e fixed and the boundary condition on the gradient removed, this is a well-posed fixed-boundary problem for the linear, parabolic partial differential equation. The additional boundary condition then serves to determine the motion of the boundary, r e (t), in the free-boundary problem defined by Eq. 5. The relevant parameter affecting this motion is thus seen to be the modified Damko ¨hler number ⌬ of Eq. 6. The initial distribution ⌿ 0 (r) is to have ⌿0 nonnegative and monotonic, approaching zero as r approaches infinity, ⌿0 ⬍ 1 for r ⬎  and ⌿0() ⫽ 1, where the initial nondimensional edge radius (r e ⫽  ), in terms of the initial dimensional edge radius R e , is
 ⫽ R e 冑⍀/ ␣ ⬁
(8)
It is convenient to work with ⫽ cr and u ⫽ ekr ⌿ as independent and dependent variables, respectively, obtaining the confluent hypergeometric equation [8],
2 冑␦⑀ 3 1⫺a
冋
according to the original non-dimensionalization. Solutions for stationary flame disks can be sought by putting ⭸⌿/⭸t ⫽ 0 in Eq. 5. The solution to the resulting ordinary differential equation with the boundary conditions stated previously will determine  as a function of ⌬, the radius as a function of the Damko ¨hler number. More generally, disks with edges expanding at a non-dimensional constant velocity V can be addressed by putting ⭸⌿/⭸t ⫽ ⫺Vd⌿/dr in Eq. 5. In the resulting differential equation,  will be neglected in comparison with r; this simplifying approximation has been thoroughly discussed previously and was shown (e.g., in an appendix) to give qualitatively correct results [4]. The non-dimensional velocity V is related to ˙ e according to the dimensional edge velocity R
(7)
d2 u/d2 ⫹ 共1 ⫺ 兲 du/d ⫺ bu ⫽ 0
(9)
where c ⫽ 冑4 ⫹ V 2,
k ⫽ 共c ⫹ V兲/ 2,
b ⫽ k/c (10)
requiring
冎
u ⫽ 0 at ⫽ ⬁, u ⫽ ek  at ⫽ c, du/d ⫽ ⫺ek  共k ⫹ ⌬兲/c at ⫽ c
(11)
according to the boundary conditions in Eq. 5. The first of these conditions selects a constant times the Kummer function U(b, 1, ) as the solution [8], and the second and third then yield, from the properties of U, a relationship determining V as a function of  and ⌬. The resulting predicted variation of edge velocity with flameedge radius of the flame disk applies for small magnitudes of V because otherwise, terms involving ⭸/⭸t remain in the transformed equation. From the solution for u, the properties of U,
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such as dU(b, a, )/d ⫽ ⫺bU(b ⫹ 1, a ⫹ 1, ), and the conditions in Eq. 11, the expression ⌬ ⫽ k关1 ⫹ U共1 ⫹ b, 2, c  兲/U共b, 1, c  兲兴
(12)
can be derived, which from Eq. 10 and expansions of U approaches the result of Buckmaster [5, 6], ⌬ ⫽ k (as it must), as c  approaches infinity, and approaches
冋冉冊 册 冒冋 冉 冊册
⌬ ⫽  ⫺1 ln ⬇1
 ln
1 ⫺ 共b兲 c
⫺1
⫹k
1 c
(13)
in which denotes the digamma function, as c  approaches zero. For large , the problem is readily reformulated as the straight-flame-edge problem of Buckmaster, curvature effects vanishing, whereas for small , from Eqs. 10 and 13, the approximation V ⫽ ⫺e1/(⌬)/ develops, which, because V ⫽ d/dt, reduces in the first approximation to dy/d ⫽ ⫺e1/y , where y ⬅  ⌬, ⬅ ⌬ 2 t. The solution to this differential equation subject to y being infinite when is zero obeys 共 y ⫺2 ⫹ 2y ⫺1 ⫹ 2兲e⫺1/y ⫽ 2 ⫺
(14)
which for small y leads approximately to
 ⫽ 兵⌬ ln关1/共2 ⫺ ⌬2 t兲兴其⫺1
(15)
stating in dimensional terms that as the flame disk shrinks to zero radius, 1/R e becomes proportional to the logarithm of the reciprocal of the time until its disappearance, all constants being independent of the rotational frequency ⍀ in this limit, as might be expected. Figure 1 is an experimental test of this last prediction, which proves reasonable even though the analysis clearly fails because V becomes large, necessitating reinstatement of ⭸/⭸t terms. As a further test, because from Eq. 8, and the preceding solution, it is clear that at large radii, where V becomes independent of , the ˙ e 兩, magnitude of the flame-edge velocity, 兩R should be proportional to 公⍀ at fixed ⌬, it is of ˙ e 兩 versus 公⍀ for the interest to plot the initial 兩R experiments. Figure 2 is such a plot from data ˙ e is negative and the flame eventufor which R ally dies. The spread in velocity arises from the different values of ⌬ established experimentally,
Fig. 1. Comparisons of the predicted and observed dependences of the radius of the flame disk on time for contracting flames near the time of flame disappearance.
there being a minimum ⌬ that needs to be exceeded to achieve ignition, corresponding approximately to the straight line through the highest data point. The extent of agreement in Fig. 2 is encouraging, because the different experimental runs have different heating times (resulting in different values of ⌬) and in view of the facts that buoyancy influences arise at small ⍀ and that the theory neglects radial convection of the gas which, contrary to Fig. 1, can become significant at the larger radii of Fig. 2. There
Fig. 2. Observed dependence of the velocity of flame-edge contraction at large flame radii on the square root of the fuel-disk rotation rate, with linearity predicted at constant Damko ¨hler number, although different experimental points have different Damko ¨hler numbers.
978
Fig. 3. The predicted dependence of the non-dimensional flame radius  on the modified Damko ¨hler number ⌬ for different values of the non-dimensional flame-edge velocity V.
thus exists some observational support for aspects of the solutions. RESULTS Use of Eq. 10 in Eq. 12 enables ⌬ to be calculated as a function of  for specified values of V. Figure 3 displays the results of that calculation in a graph of  as a function of ⌬ for different values of V, with ⌬ and  ranging from 0 to 5 and V from ⫺2 to 2. The curve for V ⫽ 0 identifies stationary flame disks and shows that the radius of the stationary disk decreases as the Damko ¨hler number increases. For V ⫽ 0, the use of recursion relations for confluent hypergeometric functions and relationships between hypergeometric functions and Bessel functions enables Eq. 12 to be written as ⌬ ⫽ K 1 (  )/ K 0 (  ), where K 0 and K 1 denote the modified Bessel functions of the second kind of order zero and one, respectively; this facilitates calculation of the curve in Fig. 3 for V ⫽ 0 and enables asymptotic expansions of Bessel functions to be used to demonstrate that ⌬ approaches 1/[ ln(1/)] as  approaches zero and 1 ⫹ 1/(2) as  approaches infinity for this curve. In particular, the vertical asymptote of the curve for V ⫽ 0 in Fig. 3 is at ⌬ ⫽ 1, well above the limit ⌬ ⫽ e公⑀/(1 ⫺ a) at which the edgeless flame would extinguish. The fact that V ⬎ 0 when  exceeds the value that it has for V ⫽ 0 and V ⬍ 0 when  is less than the value that it has for V ⫽ 0 in Fig. 3 implies that the stationary solution (V ⫽ 0) exhibits static insta-
V. NAYAGAM AND F. A. WILLIAMS
Fig. 4. A map of regimes of dynamics of flame disks and holes in a plane of the modified Damko ¨hler number ⌬ and the non-dimensional flame radius ; holes contract and disks expand in the shaded region on the right, holes expand and disks contract in the shaded region on the left, and the disk contraction velocity equals the hole contraction velocity along the dotted curve.
bility, that is, any perturbation will grow; flame disks larger than the stationary solution will spread outward, whereas those smaller will contract inward. At any fixed Damko ¨hler number, flame disks are unstable, as are the flame holes analyzed previously [4]. It is of interest to combine the present results for disks with the previous [4] results for holes to produce a map of the different regimes of dynamics of hole-disk combinations. Figure 4 is such a map in the plane of ⌬ and . The boundary shown in the figure for ⌬ ⬎ 1 is that obtained here with V ⫽ 0 for disks, while the boundary for ⌬ ⬍ 1 was obtained previously [4] with V ⫽ 0 for holes. In the shaded region at large ⌬ and  holes shrink and disks expand, so that the flame envelops the entire fuel disk. In the oppositely shaded region at small ⌬ and large , holes expand and disks shrink, so that the flame extinguishes irrespective of whether there is a hole or a disk or both. If a disk and hole both exist, then an annular flame ring is present at extinction in this region. In the unshaded region that extends to  ⫽ 0, holes and disks both shrink, under which conditions the question of whether an annular flame will extinguish before spreading to the center depends on whether the disk contraction velocity exceeds the hole contraction velocity. Those two velocities are equal along the dashed curve in the figure. To the right of this curve, the hole shrinks faster, so that the flame does reach the
DIFFUSION-FLAME DYNAMICS
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center before being extinguished by the contracting disk, whereas to the left of this curve the disk can overtake the hole, producing an annular extinction. EXPERIMENTS Details of these types of experiments have been described in full previously [2]. A planar diffusion-flame sheet was created by igniting a spinning PMMA fuel disk of 20 cm in diameter and 1.2 cm thickness in air. The burning surface of the fuel disk was oriented facing downward to avoid distortions of the flame by buoyancy. In this configuration the buoyancy forces aid the von Ka´rma´n flow generated by the rotation of the fuel disk, and the diffusion flame established within this boundary-layer flow remains flat. Ignition was achieved by plying a butane torch across the face of the fuel plate after the desired spin velocity was established by a stepper motor controlled by a laptop computer. The ignition duration of the butane torch was varied from a few seconds to a minute. From earlier results [2] it is known that the value of the modified Damko ¨hler number ⌬ depends on the ignition duration, increasing with increasing heating time. When long-duration ignition was employed, the fuel disk was heated more uniformly across its thickness, and a steadily burning flame covering the entire face of the fuel disk was established for all rotational frequencies ⍀. When short-duration ignition was employed, however, only a thin layer of the fuel surface was heated, and the in-depth heat loss from the fuel surface led to transient burning in the gas phase lasting several seconds. The dynamics of these transient, blue flames were recorded using and intensified-array video camera at a rate of 30 frames per second. Starting with a fresh fuel disk, several runs were made using the same disk. Each run consisted of igniting the fuel disk and recording the transient flames behavior. During earlier runs, when only a thin layer of the fuel disk was heated, flame disks of finite radius were found to form and then shrink rapidly. After the disk was ignited several times it eventually became sufficiently heated to have a high enough Damko ¨hler number ⌬ to achieve V ⬎ 0 in Fig.
Fig. 5. Measured variation of flame-disk radius with time for subsequent runs with a particular PMMA disk at a rotational rate of 8 rps.
3 and exhibited a growing flame disk; the flame edge thus is observed to experience a transition from a failure wave to an ignition wave, as predicted by the model. This sequence of heating affects all of the parameters in the model of Eq. 1, most strongly through its ultimate influence on the flame temperature T f . Because of the relatively slow heating compared with the flame evolution time, the variations of these parameters with time are neglected in the theory and comparison. Figure 5 shows representative experimental results for several consecutive runs with a particular PMMA disk at ⍀ ⫽ 8 rps. This figure serves to illustrate the general behavior and variability of the experiments. The heating time before the second run was shorter than that prior to the first so that, although R e initially was larger, ⌬ was smaller, and V, therefore, had a larger negative value. Subsequent runs had longer heating times and therefore negative initial values of V smaller in magnitude. The last run is seen to experience a transition from negative to positive V with the flame eventually growing to envelop the entire PMMA disk. Such a transition is necessarily a time-dependent phenomenon, with the time-derivative term important in Eq. 5, so that Fig. 3 does not apply, but it requires conditions such that ⌬ is sufficiently large for V to be positive in Fig. 3 over the entire range of flame-disk radius of this
980
V. NAYAGAM AND F. A. WILLIAMS straight according to theory if ⌬ is constant there. The experiments thus support the theory, within the accuracy of these fairly difficult experiments.
CONCLUSIONS
Fig. 6. Representative measured variations of flame-disk radius with time for different fuel disks at different rotation rates ⍀.
experimental run. The last two runs in Fig. 5 suggest a sharp demarkation between tests exhibiting failure and full envelopment in that, if the flame radius becomes small enough, then failure always occurs; the turn-around to flame expansion must set in before the flame becomes too small if envelopment is to occur. This observation is consistent with Fig. 3, which shows that at any given value of ⌬, if the flame becomes too small then V becomes negative, leading to continued flame contraction. Figure 6 shows the flame-disk radius as function of time for several different rotational speeds. This figure provided all of the data employed in Fig. 1 and much of that for Fig. 2. This figure verifies the general qualitative consistency of flame-disk histories for different values of ⍀ while also demonstrating typical variability observed during experimental runs. Edge velocities are seen to exhibit variations, reflected in different points in Fig. 2. These variations are caused by differences in initial conditions and differences in heating durations, leading to different values of the modified Damko ¨hler number ⌬. Only runs having ⌬ small enough for V to be negative in Fig. 3 are shown in Fig. 6, but the different curves still have different but unknown values of ⌬, so that a single curve in Fig. 2 is not to be expected. The different values of ⌬ produce a band of data extending to the line at the upper limit, which is
The simple Buckmaster model of edge flames, extended to cylindrical geometry, provides a qualitatively good general description of the dynamics of flame disks in von Ka´rma´n swirling flows. The effects of curvature of the edge of the flame modify the dynamics from that which would be associated with straight edges. For flame disks, as the radius decreases the enhanced heat loss to the inert zone outside the disk produces an increase in the value of the Damko ¨hler number required for an ignition front (outward propagation) rather than a failure front (inward propagation). Conversely, for flame holes, as the radius decreases, the enhanced heat supplied from the uniform flame outside the hole produces a decrease in the value of the Damko ¨hler number required for the edge to be an ignition front rather than a failure front. Stationary flame disks and flame holes are all unstable at constant Damko ¨hler number and become either ignition fronts, exhibiting a positive propagation velocity or failure fronts, exhibiting a negative propagation velocity. The analysis developed here has identified the appropriate modified Damko ¨hler number for diffusion flames in von Ka´rma´n swirling flows, Eq. 6. That Damko ¨hler number bears resemblance to a burning-rate eigenvalue in that it is proportional to the square root of the reaction rate. In the von Ka´rma´n flow, it is inversely proportional to the square root of the angular velocity of the fuel disk as well as to the temperature difference driving the heat loss. A map showing the different regimes of dynamics of flame disks and holes has been derived. The predictions of the model are in reasonable agreement with experiment. This work was supported by the NASA Microgravity Combustion Science Program. Stephen Hostler helped with the experiments.
DIFFUSION-FLAME DYNAMICS REFERENCES von Ka´rma´n, Th., Z. Angew. Math. Mech. 1:233 (1921). King, M. D., Nayagam, V., and Williams, F. A., Combust. Sci. Tech. 160:151 (2000). 3. Nayagam, V., and Williams, F. A., Phys. Rev. Lett. 84:479 (2000). 4. Nayagam, V., Balasubramaniam, R., and Ronney, P. D., Combust. Theory Modelling 3:727 (1999).
1. 2.
981 5. Buckmaster, J. D., J. Eng’g Math. 31:269 (1997). 6. Buckmaster, J. D., Combust. Sci. Tech. 115:41 (1996). 7. Lin ˜´an, A., Acta Astronautica 1:1007 (1974). 8. Abramowitz, M., and Stegun, A., Handbook of Mathematical Functions, Dover, New York, 1965, p. 504 –515. Received 25 April 2000; revised 14 November 2000; accepted 22 November 2000
National Center for Microgravity Research, NASA Lewis Research Center, Cleveland, OH 44135, USA
and F. A. WILLIAMS*
Center for Energy and Combustion Research, Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA Experimental and theoretical results are reported for flat diffusion flames adjacent to a spinning fuel disk in an oxidizing atmosphere. Disk-shaped flames are observed, the radii of which increase or decrease with time, depending on whether an appropriate Damko ¨hler number is large or small. Experimentally, initially large flame disks expand at sufficiently high Damko ¨hler numbers, while at sufficiently small Damko ¨hler numbers, initially small flame disks contract, as predicted by theory. A stability map is derived theoretically in a plane of the Damko ¨hler number and the flame-disk radius normalized by a diffusion length scale, showing regimes of disk growth and shrinkage. © 2001 by The Combustion Institute
INTRODUCTION The von Ka´rma´n swirling flow [1] is a classical viscous flow adjacent to a solid spinning disk, having radial and tangential velocity components proportional to the radius. A self-similar solution then exists with the normal velocity component and all properties dependent only on the normal distance z from the disk through the nondimensional variable 公⍀/v ⬁ 兰 z0 ( / ⬁ ) dz. Here ⍀ denotes the angular rotational velocity of the disk, v is the kinematic viscosity and the gas density, and the subscript ⬁ identifies properties evaluated in the ambient atmosphere. We have made measurements on polymethyl methacrylate (PMMA) disks in this flow configuration, ignited in air to produce a flat diffusion flame in the von Ka´rma´n boundary layer [2]. Experimental conditions can be established under which these flames burn steadily after growing to envelop the entire fuel disk [2], extinguish by contracting axisymmetrically, or form spiral patterns [3]. The present paper concerns conditions under which axisymmetric flame-disk growth or contraction occurs and addresses the dynamics of those processes. Besides results of experimental observations, a theoretical analysis is presented which builds on our previous [4] theory of diffusion-flame holes. We present the theoretical considerations *Corresponding author. E-mail: [email protected] 0010-2180/01/$–see front matter PII 0010-2180(00)00253-4
first and then show the experimental results for comparison with theory. The theory adopts the Buckmaster [5, 6] simplified edge-flame description because that facilitates analysis by reducing the problem to one space dimension while preserving all proper scalings. The role of the Buckmaster approximation and its relationship to more thorough analyses have been discussed previously [4 – 6] and need no further amplification. Embedding the Buckmaster theory into the von Ka´rma´n swirling flow is a new contribution of the present work, which adopts notation different from the earlier papers, seeking economy of presentation.
FORMULATION Generalizing Buckmaster’s formulation to unsteady axisymmetric conditions [4] and choosing for this flow the length and time units 公␣⬁/⍀ and 1/⍀ for forming the nondimensional radial coordinate r and non-dimensional time t, where the thermal diffusivity ␣ is of the same order as v, the Prandtl number being taken to be of order unity, we write the equation for conservation of thermal energy in the form
冉 冊
⭸ 1 ⭸ ⭸ ⫺ r ⭸t r ⭸r ⭸r
⫽ ␦(1⫺)2e共1⫺1/ 兲/ ⑀ ⫺ 共 ⫺ a兲
(1)
COMBUSTION AND FLAME 125:974 –981 (2001) © 2001 by The Combustion Institute Published by Elsevier Science Inc.
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where ⬅ T/T f is the ratio of the temperature to the flame temperature far from the edge, (conditions identified by the subscript f ), ⑀ ⬅ RT f /E is the reciprocal of the non-dimensional activation energy (E being the activation energy and R the universal gas constant), a is a representative constant nondimensional boundary temperature (e.g., T ⬁ /T f ), and ␦ is a (constant) Damko ¨hler number. The form of the reactionrate term applies to a one-step, Arrhenius process of first order with respect to both fuel and oxidizer. The factor (1 ⫺ )2 in that term arises from the product of fuel and oxidizer concentrations, each of which, to leading order in activation-energy asymptotics, is proportional to (1 ⫺ ) in the reaction zone for the diffusionflame regime [7]. The corresponding Damko ¨hler number is then
␦ ⫽ 关共 fB OQ FL FL O兲/共c pT fM F⍀兲兴e⫺E/共RT f兲
(2)
where B O is the pre-exponential factor in the Arrhenius expression for the molar rate of consumption of oxidizer, Q F is the heat released per unit mass of fuel consumed, c p is the (constant) specific heat at constant pressure, M F is the molecular weight of fuel, and L F and L O are the Lewis number (␣ divided by diffusion coefficient) of the fuel and oxidizer, respectively. The last term in Eq. 1 models the energy loss to the sides of the edge flame by heat conduction. An implicit approximation is that such transport to the side can be described keeping a constant everywhere, an assumption that is not exactly applicable for this or most other real problems. Nevertheless, it is qualitatively reasonable and does not affect scaling. The character of this approximation has been discussed in much greater detail by Buckmaster [6]. Although the side loss is essential, for most purposes the a dependence readily scales out of the problem, so that the value of a provides a small correction, the dominant physics being carried by the parameters ␦ and ⑀. The further approximations of constant thermal conductivity and thermal diffusivity and neglect of radial convection also are present in Eq. 1; otherwise, factors and terms accounting for these effects would appear on the left-hand side. The objective is to
obtain a model that is as simple as possible but yet retains most of the essential phenomena. The coupling with species conservation equations that occurs in the reaction term can be treated differently in detail through different formulations. If Lewis numbers are unity and boundary and initial conditions appropriate, then the linear dependence on temperature applies everywhere, independent of chemistry. It is often desirable to account for different Lewis numbers, necessitating suitable separate analyses of concentration fields in external zones with matching to reaction-zone solutions through activation-energy asymptotics. Equations for the concentration fields are not addressed here. Instead, results of this type of analysis [4] are included through Eq. 2, in which T f differs from the adiabatic flame temperature when Lewis numbers differ from unity. The advantage of the Buckmaster model is that, because zones on the sides of the reaction sheet are all implicit, there is only one space dimension in which zones arise. For a flame disk, there is a uniform flame zone in the region r ⬍ r e , an inert transport zone in the region r ⬎ r e and a reactive-diffusive flame-edge zone near r ⫽ r e , where r e (t) is the non-dimensional location of the flame edge. In the uniform flame zone the derivatives in Eq. 1 vanish, there being a balance between chemical reaction and side loss. For small ⑀ this balance exhibits ignition and extinction [5, 6], the latter, to leading order, in terms of ⬅ (1 ⫺ )/⑀, being tangency of 2e⫺ ⫽ (1 ⫺ a)/( ⑀ 2 ␦ ), namely ⫽ 2, ␦ ⫽ (1 ⫺ a)e2/(4⑀2). In the inert transport zone, Eq. 1 is a linear parabolic partial differential equation in r and t whose solution must have approach a as r approaches infinity and must match with the edge-region solution at r ⫽ r e . The edge-region solution is to be determined by stretching in r and matching to the uniform flame at ⫺⬁ and the inert solution at ⫹⬁.
SOLUTION The stretchings ⫽ (r ⫺ r e )/ ⑀ , ⫽ (1 ⫺ )/ ⑀ , in Eq. 1 produce at leading order the equation d2/d2 ⫽ ␦⑀32e⫺
(3)
976
V. NAYAGAM AND F. A. WILLIAMS
the first integral of which, subject to approaching zero as approaches ⫺⬁, as is appropriate in the asymptotic context [6], is
冉 冕
d/d ⫽ 2␦⑀
3
冊
1/ 2
2 ⫺
e
0
d
(4)
which approaches 2公␦⑀3 as approaches infinity. The necessary slope matching for the solution in the inert zone at r ⫽ r e therefore is ⭸ /⭸r ⫽ ⫺2 公␦⑀ 3 . In the inert zone, it is convenient to work with the normalized temperature variable ⌿ ⬅ ( ⫺ a)/(1 ⫺ a). The problem in the inert zone, obtained from Eq. 1 and the matching and boundary conditions, then becomes
冉 冊
冧
⭸⌿ ⭸⌿ 1 ⭸ ⫽ r ⫺ ⌿, ⭸t r ⭸r ⭸r ⌿ ⫽ ⌿ 0共r兲 at t ⫽ 0, ⌿ ⫽ 0 at r ⫽ ⬁, ⌿ ⫽ 1 and ⭸⌿/⭸r ⫽ ⫺⌬ at r ⫽ re 共t兲,
(5)
˙ e/ 冑⍀ ␣ ⬁. V⫽R
where ⌬⬅ ⫽
册
4f BO QF LF LO R3 Tf4 1/ 2 ⫺E/共2 RT f兲 e MF cp 共Tf ⫺ T⬁ 兲2 E3 ⍀
(6)
and ⌿ 0 (r) is a specified initial distribution. The preceding restriction on existence of the uniform-flame solution, ␦ ⱖ (1 ⫺ a)e2/(4⑀2), translates into ⌬ ⱖ e公⑀ /(1 ⫺ a). With r e fixed and the boundary condition on the gradient removed, this is a well-posed fixed-boundary problem for the linear, parabolic partial differential equation. The additional boundary condition then serves to determine the motion of the boundary, r e (t), in the free-boundary problem defined by Eq. 5. The relevant parameter affecting this motion is thus seen to be the modified Damko ¨hler number ⌬ of Eq. 6. The initial distribution ⌿ 0 (r) is to have ⌿0 nonnegative and monotonic, approaching zero as r approaches infinity, ⌿0 ⬍ 1 for r ⬎  and ⌿0() ⫽ 1, where the initial nondimensional edge radius (r e ⫽  ), in terms of the initial dimensional edge radius R e , is
 ⫽ R e 冑⍀/ ␣ ⬁
(8)
It is convenient to work with ⫽ cr and u ⫽ ekr ⌿ as independent and dependent variables, respectively, obtaining the confluent hypergeometric equation [8],
2 冑␦⑀ 3 1⫺a
冋
according to the original non-dimensionalization. Solutions for stationary flame disks can be sought by putting ⭸⌿/⭸t ⫽ 0 in Eq. 5. The solution to the resulting ordinary differential equation with the boundary conditions stated previously will determine  as a function of ⌬, the radius as a function of the Damko ¨hler number. More generally, disks with edges expanding at a non-dimensional constant velocity V can be addressed by putting ⭸⌿/⭸t ⫽ ⫺Vd⌿/dr in Eq. 5. In the resulting differential equation,  will be neglected in comparison with r; this simplifying approximation has been thoroughly discussed previously and was shown (e.g., in an appendix) to give qualitatively correct results [4]. The non-dimensional velocity V is related to ˙ e according to the dimensional edge velocity R
(7)
d2 u/d2 ⫹ 共1 ⫺ 兲 du/d ⫺ bu ⫽ 0
(9)
where c ⫽ 冑4 ⫹ V 2,
k ⫽ 共c ⫹ V兲/ 2,
b ⫽ k/c (10)
requiring
冎
u ⫽ 0 at ⫽ ⬁, u ⫽ ek  at ⫽ c, du/d ⫽ ⫺ek  共k ⫹ ⌬兲/c at ⫽ c
(11)
according to the boundary conditions in Eq. 5. The first of these conditions selects a constant times the Kummer function U(b, 1, ) as the solution [8], and the second and third then yield, from the properties of U, a relationship determining V as a function of  and ⌬. The resulting predicted variation of edge velocity with flameedge radius of the flame disk applies for small magnitudes of V because otherwise, terms involving ⭸/⭸t remain in the transformed equation. From the solution for u, the properties of U,
DIFFUSION-FLAME DYNAMICS
977
such as dU(b, a, )/d ⫽ ⫺bU(b ⫹ 1, a ⫹ 1, ), and the conditions in Eq. 11, the expression ⌬ ⫽ k关1 ⫹ U共1 ⫹ b, 2, c  兲/U共b, 1, c  兲兴
(12)
can be derived, which from Eq. 10 and expansions of U approaches the result of Buckmaster [5, 6], ⌬ ⫽ k (as it must), as c  approaches infinity, and approaches
冋冉冊 册 冒冋 冉 冊册
⌬ ⫽  ⫺1 ln ⬇1
 ln
1 ⫺ 共b兲 c
⫺1
⫹k
1 c
(13)
in which denotes the digamma function, as c  approaches zero. For large , the problem is readily reformulated as the straight-flame-edge problem of Buckmaster, curvature effects vanishing, whereas for small , from Eqs. 10 and 13, the approximation V ⫽ ⫺e1/(⌬)/ develops, which, because V ⫽ d/dt, reduces in the first approximation to dy/d ⫽ ⫺e1/y , where y ⬅  ⌬, ⬅ ⌬ 2 t. The solution to this differential equation subject to y being infinite when is zero obeys 共 y ⫺2 ⫹ 2y ⫺1 ⫹ 2兲e⫺1/y ⫽ 2 ⫺
(14)
which for small y leads approximately to
 ⫽ 兵⌬ ln关1/共2 ⫺ ⌬2 t兲兴其⫺1
(15)
stating in dimensional terms that as the flame disk shrinks to zero radius, 1/R e becomes proportional to the logarithm of the reciprocal of the time until its disappearance, all constants being independent of the rotational frequency ⍀ in this limit, as might be expected. Figure 1 is an experimental test of this last prediction, which proves reasonable even though the analysis clearly fails because V becomes large, necessitating reinstatement of ⭸/⭸t terms. As a further test, because from Eq. 8, and the preceding solution, it is clear that at large radii, where V becomes independent of , the ˙ e 兩, magnitude of the flame-edge velocity, 兩R should be proportional to 公⍀ at fixed ⌬, it is of ˙ e 兩 versus 公⍀ for the interest to plot the initial 兩R experiments. Figure 2 is such a plot from data ˙ e is negative and the flame eventufor which R ally dies. The spread in velocity arises from the different values of ⌬ established experimentally,
Fig. 1. Comparisons of the predicted and observed dependences of the radius of the flame disk on time for contracting flames near the time of flame disappearance.
there being a minimum ⌬ that needs to be exceeded to achieve ignition, corresponding approximately to the straight line through the highest data point. The extent of agreement in Fig. 2 is encouraging, because the different experimental runs have different heating times (resulting in different values of ⌬) and in view of the facts that buoyancy influences arise at small ⍀ and that the theory neglects radial convection of the gas which, contrary to Fig. 1, can become significant at the larger radii of Fig. 2. There
Fig. 2. Observed dependence of the velocity of flame-edge contraction at large flame radii on the square root of the fuel-disk rotation rate, with linearity predicted at constant Damko ¨hler number, although different experimental points have different Damko ¨hler numbers.
978
Fig. 3. The predicted dependence of the non-dimensional flame radius  on the modified Damko ¨hler number ⌬ for different values of the non-dimensional flame-edge velocity V.
thus exists some observational support for aspects of the solutions. RESULTS Use of Eq. 10 in Eq. 12 enables ⌬ to be calculated as a function of  for specified values of V. Figure 3 displays the results of that calculation in a graph of  as a function of ⌬ for different values of V, with ⌬ and  ranging from 0 to 5 and V from ⫺2 to 2. The curve for V ⫽ 0 identifies stationary flame disks and shows that the radius of the stationary disk decreases as the Damko ¨hler number increases. For V ⫽ 0, the use of recursion relations for confluent hypergeometric functions and relationships between hypergeometric functions and Bessel functions enables Eq. 12 to be written as ⌬ ⫽ K 1 (  )/ K 0 (  ), where K 0 and K 1 denote the modified Bessel functions of the second kind of order zero and one, respectively; this facilitates calculation of the curve in Fig. 3 for V ⫽ 0 and enables asymptotic expansions of Bessel functions to be used to demonstrate that ⌬ approaches 1/[ ln(1/)] as  approaches zero and 1 ⫹ 1/(2) as  approaches infinity for this curve. In particular, the vertical asymptote of the curve for V ⫽ 0 in Fig. 3 is at ⌬ ⫽ 1, well above the limit ⌬ ⫽ e公⑀/(1 ⫺ a) at which the edgeless flame would extinguish. The fact that V ⬎ 0 when  exceeds the value that it has for V ⫽ 0 and V ⬍ 0 when  is less than the value that it has for V ⫽ 0 in Fig. 3 implies that the stationary solution (V ⫽ 0) exhibits static insta-
V. NAYAGAM AND F. A. WILLIAMS
Fig. 4. A map of regimes of dynamics of flame disks and holes in a plane of the modified Damko ¨hler number ⌬ and the non-dimensional flame radius ; holes contract and disks expand in the shaded region on the right, holes expand and disks contract in the shaded region on the left, and the disk contraction velocity equals the hole contraction velocity along the dotted curve.
bility, that is, any perturbation will grow; flame disks larger than the stationary solution will spread outward, whereas those smaller will contract inward. At any fixed Damko ¨hler number, flame disks are unstable, as are the flame holes analyzed previously [4]. It is of interest to combine the present results for disks with the previous [4] results for holes to produce a map of the different regimes of dynamics of hole-disk combinations. Figure 4 is such a map in the plane of ⌬ and . The boundary shown in the figure for ⌬ ⬎ 1 is that obtained here with V ⫽ 0 for disks, while the boundary for ⌬ ⬍ 1 was obtained previously [4] with V ⫽ 0 for holes. In the shaded region at large ⌬ and  holes shrink and disks expand, so that the flame envelops the entire fuel disk. In the oppositely shaded region at small ⌬ and large , holes expand and disks shrink, so that the flame extinguishes irrespective of whether there is a hole or a disk or both. If a disk and hole both exist, then an annular flame ring is present at extinction in this region. In the unshaded region that extends to  ⫽ 0, holes and disks both shrink, under which conditions the question of whether an annular flame will extinguish before spreading to the center depends on whether the disk contraction velocity exceeds the hole contraction velocity. Those two velocities are equal along the dashed curve in the figure. To the right of this curve, the hole shrinks faster, so that the flame does reach the
DIFFUSION-FLAME DYNAMICS
979
center before being extinguished by the contracting disk, whereas to the left of this curve the disk can overtake the hole, producing an annular extinction. EXPERIMENTS Details of these types of experiments have been described in full previously [2]. A planar diffusion-flame sheet was created by igniting a spinning PMMA fuel disk of 20 cm in diameter and 1.2 cm thickness in air. The burning surface of the fuel disk was oriented facing downward to avoid distortions of the flame by buoyancy. In this configuration the buoyancy forces aid the von Ka´rma´n flow generated by the rotation of the fuel disk, and the diffusion flame established within this boundary-layer flow remains flat. Ignition was achieved by plying a butane torch across the face of the fuel plate after the desired spin velocity was established by a stepper motor controlled by a laptop computer. The ignition duration of the butane torch was varied from a few seconds to a minute. From earlier results [2] it is known that the value of the modified Damko ¨hler number ⌬ depends on the ignition duration, increasing with increasing heating time. When long-duration ignition was employed, the fuel disk was heated more uniformly across its thickness, and a steadily burning flame covering the entire face of the fuel disk was established for all rotational frequencies ⍀. When short-duration ignition was employed, however, only a thin layer of the fuel surface was heated, and the in-depth heat loss from the fuel surface led to transient burning in the gas phase lasting several seconds. The dynamics of these transient, blue flames were recorded using and intensified-array video camera at a rate of 30 frames per second. Starting with a fresh fuel disk, several runs were made using the same disk. Each run consisted of igniting the fuel disk and recording the transient flames behavior. During earlier runs, when only a thin layer of the fuel disk was heated, flame disks of finite radius were found to form and then shrink rapidly. After the disk was ignited several times it eventually became sufficiently heated to have a high enough Damko ¨hler number ⌬ to achieve V ⬎ 0 in Fig.
Fig. 5. Measured variation of flame-disk radius with time for subsequent runs with a particular PMMA disk at a rotational rate of 8 rps.
3 and exhibited a growing flame disk; the flame edge thus is observed to experience a transition from a failure wave to an ignition wave, as predicted by the model. This sequence of heating affects all of the parameters in the model of Eq. 1, most strongly through its ultimate influence on the flame temperature T f . Because of the relatively slow heating compared with the flame evolution time, the variations of these parameters with time are neglected in the theory and comparison. Figure 5 shows representative experimental results for several consecutive runs with a particular PMMA disk at ⍀ ⫽ 8 rps. This figure serves to illustrate the general behavior and variability of the experiments. The heating time before the second run was shorter than that prior to the first so that, although R e initially was larger, ⌬ was smaller, and V, therefore, had a larger negative value. Subsequent runs had longer heating times and therefore negative initial values of V smaller in magnitude. The last run is seen to experience a transition from negative to positive V with the flame eventually growing to envelop the entire PMMA disk. Such a transition is necessarily a time-dependent phenomenon, with the time-derivative term important in Eq. 5, so that Fig. 3 does not apply, but it requires conditions such that ⌬ is sufficiently large for V to be positive in Fig. 3 over the entire range of flame-disk radius of this
980
V. NAYAGAM AND F. A. WILLIAMS straight according to theory if ⌬ is constant there. The experiments thus support the theory, within the accuracy of these fairly difficult experiments.
CONCLUSIONS
Fig. 6. Representative measured variations of flame-disk radius with time for different fuel disks at different rotation rates ⍀.
experimental run. The last two runs in Fig. 5 suggest a sharp demarkation between tests exhibiting failure and full envelopment in that, if the flame radius becomes small enough, then failure always occurs; the turn-around to flame expansion must set in before the flame becomes too small if envelopment is to occur. This observation is consistent with Fig. 3, which shows that at any given value of ⌬, if the flame becomes too small then V becomes negative, leading to continued flame contraction. Figure 6 shows the flame-disk radius as function of time for several different rotational speeds. This figure provided all of the data employed in Fig. 1 and much of that for Fig. 2. This figure verifies the general qualitative consistency of flame-disk histories for different values of ⍀ while also demonstrating typical variability observed during experimental runs. Edge velocities are seen to exhibit variations, reflected in different points in Fig. 2. These variations are caused by differences in initial conditions and differences in heating durations, leading to different values of the modified Damko ¨hler number ⌬. Only runs having ⌬ small enough for V to be negative in Fig. 3 are shown in Fig. 6, but the different curves still have different but unknown values of ⌬, so that a single curve in Fig. 2 is not to be expected. The different values of ⌬ produce a band of data extending to the line at the upper limit, which is
The simple Buckmaster model of edge flames, extended to cylindrical geometry, provides a qualitatively good general description of the dynamics of flame disks in von Ka´rma´n swirling flows. The effects of curvature of the edge of the flame modify the dynamics from that which would be associated with straight edges. For flame disks, as the radius decreases the enhanced heat loss to the inert zone outside the disk produces an increase in the value of the Damko ¨hler number required for an ignition front (outward propagation) rather than a failure front (inward propagation). Conversely, for flame holes, as the radius decreases, the enhanced heat supplied from the uniform flame outside the hole produces a decrease in the value of the Damko ¨hler number required for the edge to be an ignition front rather than a failure front. Stationary flame disks and flame holes are all unstable at constant Damko ¨hler number and become either ignition fronts, exhibiting a positive propagation velocity or failure fronts, exhibiting a negative propagation velocity. The analysis developed here has identified the appropriate modified Damko ¨hler number for diffusion flames in von Ka´rma´n swirling flows, Eq. 6. That Damko ¨hler number bears resemblance to a burning-rate eigenvalue in that it is proportional to the square root of the reaction rate. In the von Ka´rma´n flow, it is inversely proportional to the square root of the angular velocity of the fuel disk as well as to the temperature difference driving the heat loss. A map showing the different regimes of dynamics of flame disks and holes has been derived. The predictions of the model are in reasonable agreement with experiment. This work was supported by the NASA Microgravity Combustion Science Program. Stephen Hostler helped with the experiments.
DIFFUSION-FLAME DYNAMICS REFERENCES von Ka´rma´n, Th., Z. Angew. Math. Mech. 1:233 (1921). King, M. D., Nayagam, V., and Williams, F. A., Combust. Sci. Tech. 160:151 (2000). 3. Nayagam, V., and Williams, F. A., Phys. Rev. Lett. 84:479 (2000). 4. Nayagam, V., Balasubramaniam, R., and Ronney, P. D., Combust. Theory Modelling 3:727 (1999).
1. 2.
981 5. Buckmaster, J. D., J. Eng’g Math. 31:269 (1997). 6. Buckmaster, J. D., Combust. Sci. Tech. 115:41 (1996). 7. Lin ˜´an, A., Acta Astronautica 1:1007 (1974). 8. Abramowitz, M., and Stegun, A., Handbook of Mathematical Functions, Dover, New York, 1965, p. 504 –515. Received 25 April 2000; revised 14 November 2000; accepted 22 November 2000