On the flame length in firewhirls with strong vorticity

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1 On the flame length in firewhirls with strong vorticity A. Y. Klimenko a, F. A. Williams b a The University of Queensland, SoMME, Qld, 472, Australia, corresponding author klimenko@mech.uq.edu.au b Dept. of Mechanical and Aerospace Engineering, University of California, San Diego, USA Abstract The influence of strong rotation on the flame length in firewhirls (normalized by the fuel supply rate) is investigated analytically. A new prediction for the length of the flame in firewhirls with strong rotation is obtained by introducing the compensating regime of vortical flows. The prediction is in a good agreement with experimental results. Keywords: Vortical flows, fire and combustion. 1. Introduction Firewhirls, which often occur in wild fires, can be very destructive. They develop from buoyantly rising flames in an atmosphere in which locally there is rotation of the ambient air about the base of the fire [1]. Through conservation of angular momentum, the rotational velocity increases as the air is entrained, and the resulting high velocities are associated with appreciable lengthening of the flames. Besides increasing radiant energy fluxes transmitted to the surroundings from the fire, this also intensifies the fire plume, enhancing distant spread by the development of spot fires. For these reasons, there is considerable interest in flame lengths in firewhirls. There are a number of different kinds of firewhirls, with different physical processes dominating flame lengths. Stationary firewhirls can be generated from an irregularly shaped flame base in a constant prevailing wind [2, 3] or from a symmetrical flame base in a horizontal wind shear [4, 5]. Firewhirls moving horizontally at constant velocities also have been observed [6], as have stationary firewhirls inclined from the vertical on the side of a hill [7]. Combust. Flame (212)

2 These last types must correspond to low Rossby numbers because the rotation must prevent buoyancy from causing the axis to be vertical. A buoyancycontrolled solution for velocity components near the axis was suggested by Battaglia et al. [8], but since this solution is inviscid and diffusion-free, it does not determine the length of the flame in firewhirls, and it does not apply at low Rossby numbers. The well-known Burke-Schumann formula [9, 1] for the flame length in a constant-density, constant-velocity, axisymmetric flow without rotation can be shown [7, 11] to also apply with rotation if the firewhirl is approximated as a constant-density Burgers vortex [12]. Measured flame lengths in laboratory scale-model vertical and inclined firewhirls at low Rossby numbers, however, exceed this prediction [7]. Since buoyant acceleration of the light, hot gases is not an appropriate explanation under these conditions, a mechanism may be the acceleration of the low-density burning gas, with respect to the cooler gas of higher density being entrained, by the axial pressure gradient caused by the entrainment [7]. Chuah et al. [11] also conducted a scaling analysis to determine major parameters controlling properties of the firewhirls. Our present work, focuses on the flame length normalized by the fuel injection rate and the other determining parameters. Kuwana et al. [13] demonstrated that changes in the velocity profile represent a different possible mechanism to lengthen the flame. We improve and extend this analysis by connecting flame characteristics to the strong-vortex approximation [14, 15, 16, 17] and its compensating regime [18], while accounting for the influence of the viscous core. An arbitrary exponent responsible for the flame elongation in the analysis of Kuwana et al. [13] becomes in our approach a specific value predicted theoretically. This value is tested against experimental data and demonstrates a good match. We explore the implications of two similarity approximations for the mixture-fraction field. In our analysis we employ the strong-vortex approximation [14, 15, 16, 17] and its compensating regime [18], which differs from the Burgers vortex but still retains its constant-density assumption. In relation to vortical flows with strong vorticity, we should also mention the family of self similar solutions, which was obtained by Long [19], generalised by Fernandez-Feria et al. [2] and dedicated to analysis of vortical flows with substantial axial vorticity. These swirling flows, whose instability has been investigated by Fernandez-Feria [21], are, however, not of the same type as the firewhirl-like flows. The suggested mechanism operates as if the density were constant, and 2

3 the large density change in the flame is a factor favoring overall stability of the mechanism considered, but this density change also may well have a significant additional influence on the flame. Section 2 presents major equations and introduces the compensating regime of the vortical flow. In section 3, the solution obtained by Chuah et al. [7] for Burgers vortex is generalized the power-laws of the compensating regime while a higher order correction is evaluated for the solution obtained by Kuwana et al. [13]. In Section 4, a similar solution is obtained for the far field of the strong vortex approximation, which is not necessarily represented by a radial power law. In Section 5, the formula for the normalized flame length is derived and compared with experimental data. The results are summarized in Section Governing equations We treat firewhirls as axisymmetric flows with rotation. Although the formulation addresses variable density, ultimately the assumption of constant density is used, in line with the classical Burke-Schumann model [9, 1], to simplify the analysis. Fires can involve a large variety of fuels, but to the extent that they are non-premixed flames their main parameters can be characterized in terms of a mixture fraction Z. For example, the flame length is usually defined assuming that the tip of the fire is the point where the mixture fraction reaches its stoichiometric value Z st [7]. In the model under consideration, this corresponds to Z = Z st at the axis of the flow. The transport equation for the mixture fraction is given by rρuz x + rρvz r = r ( Dρr Z r where x and r are axial and radial coordinates, and ρ is density, while u and v represent the corresponding velocity components. The tangential velocity w does not enter this equation as the flow is treated as axisymmetric. The diffusion coefficient D often is assumed constant, but it actually increases with temperature, reflecting the effects of molecular diffusion, and in modeling turbulent flows it is often replaced by a constant turbulent diffusivity, so that the same equation can be used for an average mixture fraction. The value Z = 1 corresponds to pure fuel while Z = corresponds to pure oxidizer (air), resulting in a stoichiometric value that typically is very small, 3 ) (1)

4 less than.1. The boundary conditions are not known exactly in real fires, but in typical experimental set-ups these conditions are represented by { } 1, r r Z = at x = x, r > r (2) where x represents axial location of the fuel source (a pan in experiments), and r is its radius, while Z is constrained by the boundary conditions Z, as r Z = at r = (3) r above the ground x > x. The main parameter of the problem is the flux of the mixture fraction F Z, which is preserved in steady flow, F Z = uzρrdr = ρ u r 2 2 where u is the average velocity and ρ is the average density at the source r r, x = x. The integral (4) is obtained by integrating equation (1) over cross-sections of x = const. In axisymmetric conditions, the velocity components can be expressed in terms of the stream function ψ, ψ ρru = ρ r, ρrv = ρ ψ (5) x where ρ is a characteristic value of the density. These velocity components satisfy the continuity equation rρu x + rρv = (6) r It is useful to consider power-law approximations of the stream function ψ xr α. The value α = 2 corresponds to the Burgers vortex, a vortex model which is widely used because of its relative simplicity [12]. However, a series of arguments has been put forward that α must fall below 2 in realistic strong vortices [tornadoes, hurricanes, etc. see 22]. The theoretical equilibrium value for α for the so-called compensating regime (with core of the flow controlling the circulation γ) is α = 4/3 [18]. In the present work we examine the far (large x) asymptotic solution of equation (1) for two kinds of approximations of the velocity field. We are interested in far asymptotes because of the small value of the stoichiometric value of Z st, which causes the flame to reach the axis at a large distance from the fuel source. (4) 4

5 3. Velocity field approximated by a radial power law In this section, as was done before [13], we assume that the stream function and velocity field are represented by power laws ψ = s(x)r α, u = αs(x)r α 2, v = s (x)r a 1 (7) where s(x) is an arbitrary function and s (x) is its derivative. We note that this approximation does not take into account viscous core and cannot be applied at the axis since u as r according to (7); the effect of viscous core is accounted for in the following sections. Assuming constant density ρ = ρ = ρ and diffusivity D, while taking into account the continuity equation we obtain from (1) ru Z x Z + rv r = D ( r Z ) r r Replacing variables x and r by new variables ξ = ξ(x) and η = η(r, x) defined by ξ = αd (x x 2ru 2 ), η = 2 ψ(r, x) ψ a (9) ru 2 where ψ a is the axial value of the stream function, we obtain the equation Z ξ = 1 ( η Z ) (1) η η η (8) with the initial conditions Z = { 1, η 1, η > 1 } at ξ = (11) The boundary-value problem specified by (1) and (11) corresponds to the classical Burke-Schumann solution [9, 1] which is given by Chuah et al. [7] Z = J 1 (ω)j (ωη) exp ( ω 2 ξ ) dω (12) where J and J 1 are the Bessel functions. The solution of the same problem was obtained previously [7] for α = 2 (that is for the Burgers vortex). Here, we have generalized this solution so that it will apply for arbitrary α. 5

6 Considering that J () = 1 and that J 1 (ω) ω/2 ω 3 / as ω, we obtain as the far-field (ξ 1) representation for the axial value of Z a = Z r= the expression Z a = 1 4ξ 1 32ξ = (13) 4 ξ + ξ 1 where ξ 1 = 1/8. After returning back to the original physical variables, we obtain Z a = Pe d (14) 8α x x + x 1 where the Peclet number is defined by Pe = d u D (15) and d 2r is the pan diameter. The average velocity u is defined in (4) and given by u = 2s r α 2. The contribution of the value x 1 = d Pe /(16α) to Z a is relatively small in the far field. Since the correction factor specified by x 1 and ξ 1 = 1/8 is quite small for small Z st, in the rest of the paper we follow previous works [7, 13] and neglect this correction. 4. Velocity field approximated by a strong vortex In this section we consider an alternative solution of the problem when the radial dependence of u and v is not restricted to the power law (7) but complies with the strong vortex approximation [14, 15, 16, 18]. This approximation is enforced by a strong rotation (i.e. small Rossby number) in the flow. According to the strong-vortex approximation, the stream function and velocity components in a constant-density flow are specified by ψ = xf(r), u = x f (r), v = f(r) r r If density ρ and diffusivity D are constant, the function Z = F ( Z/ρ xdi exp 1 r ) vdr D (16) (17) satisfies equation (1) with velocity specified by (16) and the overall flux F Z given by (4). Equation (17) is not compliant with the initial conditions (2) 6

7 but, since the correct flux F Z is preserved by this solution, it asymptotically represents the far field of the problem under consideration. Note that it is the far field that is needed to determine the flame length. The exact solution obtained in the previous section is based on the stream function given by the radial power-law and, at large x, is consistent with (17) when f r α as discussed further in the paper. The integral I, which is given by I = 1 Dx u exp ( 1 D r ) vdr rdr = 1D 2 ( 1 r ) v 2 exp vdr rdr D (18) is obtained from condition (4) to ensure the correct value of the flux F Z. Note that v = v(r) is negative in this equation. Integration by parts and use of the continuity equation (6) results in the replacement of u and v as shown. The velocity can be normalized by v = v V (R), R = r r, r = v D (19) where v and r represent characteristic scales of the vortex core. The integral (18) then becomes ( R ) I = V 2 exp V dr RdR (2) With the approximation for V (R) given by { } R α 2 1 R, R R V = 1 R α 1, R R 1 (21) in the inner core R R 1 the flow is similar to Burgers vortex α = 2, v r, while outside this core v r α 1 and, generally, α < 2. Substitution of (21) into (2) results in ( ) I = 2 (2 α) exp Rα 1 (22) 2 The value of I can vary between α and 2. It is easy to see that if the expression for the velocity is given by an exact power law (that is, V R α 1 in equation (2)), then I = α. Thus for velocity profiles that are more complex than the power laws, the value of I can be referred to as an effective exponent and denoted by α eff. 7

8 Note that if ρ and D are not constant, equation Z = u ( r ) F Z u a DIρ exp v D dr, (23) where u a = u r=, is quite similar to equation (17) but, unlike (17), (23) is an estimation. Equation (23) is based on the estimated balance of the convective and diffusive fluxes in radial direction and can be used as an approximate constraint for Z, ρ, u, v and D when the density is not constant. This, however, does not allow for direct evaluation of Z since velocities and diffusivity are dependent on ρ and Z and remain unknown in (23). 5. The flame length The solutions presented in the previous sections indicate that the equation for the axial value of Z a = Z(r = ) is Z a = F Z (x x )Dα eff (24) where α eff either coincides with α according to (14) or is determined by the integral I in (2). The solutions presented in the last two sections are, generally, different, but they have a significant overlap. Indeed if s(x) x in (7) and f(r) r a in (16) then the asymptotic form of (12) and (17) represent solutions for the same problem; both of the solutions have the same asymptote in the limit x. There are, however, some differences: equation (12) is an exact solution of the boundary-value problem, while equation (17) matches the flux F Z and approximates the far field. Although the power-law behavior is consistent with obtaining a more general analytical solution, extending the singularity for α < 2 to the flow axis violates the condition u/ r as r. The flow specified by (16) allows us to avoid this problem and obtain a more general expression for flame length based on α eff = I. The main outcome of the solutions that have been obtained is a link between the flame length and the effective value of the exponent α. The flame length L = x 2 x is determined by the condition Z a (x 2 ) = Z st, resulting in F Z L = = u d 2 Pe = d (25) α eff DZ st 8α eff DZ st 8α eff Z st 8

9 where d = 2r is used in (4), and the Peclet number is defined by (15). It can be seen that in the case of a Burgers vortex α eff = 2, and equation (25) coincides with the equation for the flame length obtained previously for Burke-Schumann flames [7]. Equation (25) is also similar to the equation for the flame length obtained by Kuwana et al. [13] but as demonstrated below gives a specific prediction for the exponents α eff. Our approach also takes into account the existance of the viscous core, resulting in deviations of α eff = I from α. As exponents α eff < 2 are expected in strong vortices [22], the normalized flame length L/F Z becomes longer in firewhirls. The physical mechanism of the flame elongation is that the lower values of α correspond to lower axial velocities at the periphery and to higher axial velocities near the centerline, which tend to stretch the flame. It should be stressed that here we refer to the flame length normalized by F Z. This distinction is important as F Z depends on the ground conditions while the intensification of fires by rotation tend to produce larger F Z and, obviously, longer flames L F Z in firewhirls. Considering complexity of the firewhirl phenomenon, it is essential to compare our findings with experimental data. Figure 1 shows experimental data [7] on flame length in laboratory-simulated inclined firewhirls. The symbols below the horizontal dotted line at L/d = 5 represent experiments without rotation, that is tangential velocity is zero in this flow w = while the other velocity components should be close to (7) with α = 2. As expected, these experiments are matched very well by the dashed line corresponding to α eff = 2 [see 7]. The vortical experiments are shown as solid symbols above the line L/d = 5. The dash-dotted line corresponds to the theoretical compensating-regime value α eff = 4/3 [18]. The solid line in this figure corresponds to α eff 1.43, obtained from equation (22) assuming R 1.43 this choice for R 1 is explained below. The open symbols above the line L/d = 5 represent experiments without rotation (shown below the line) recalculated for vortical conditions using α eff = 2 and α eff 1.43 for vortex-free and vortical flows, respectively and a Pe amplification factor. This factor is not constrained by the present theory and was evaluated from experimental data [7] to be 2.3. The parameter R 1 determines the size of the inner core, where the axial velocity u either is close to a constant or decreases towards the axis. In the outer core, which is located between R 1 and the scaled radius of maximal winds R RMW, the axial velocity decreases away from the axis. The radius of maximal winds r RMW = r R RMW is defined as the radius where the tangential 9

10 velocity w (r) takes its maximal value w(r RMW ); it is traditionally abbreviated as RMW and is conventionally used as the location of the core boundary. The parameter R 1 determines the radial velocity profile v(r) in (21), while v(r) determines tangential (rotational) velocity w = γ/r. Indeed, in the strongvortex approximation [14], the velocity v and circulation γ = wr do not depend on x at the leading order; in this case, ( r ) r Ω = Ω a exp vdr, γ = Ωrdr where Ω is the axial component of vorticity, and Ω a is its axial value. Note that v is negative. The value R 1.43 corresponds to R RMW Note that R RMW is 4 times greater than R 1 when R This choice of R 1 is based on experiments [23], simulating conditions in atmospheric vortices. For very similar conditions two cases of axial and tangential velocity profiles are reported [23]; one case has a suppressed axial velocity near the axis u a = u r= and R 1 R RMW while the second case has a high value of u a and R 1 R RMW /4. Although depressed (or even negative) axial velocities are quite typical for bifurcating conditions in the cores of strong atmospheric vortices (including dust devils, tornadoes and hurricanes) [24], the effect of the large density change strongly stimulates high axial velocities in firewhirls, and it seems unlikely to have depressed u a in these flows. Hence, the estimate of the second case R 1 R RMW /4 is more relevant to firewhirls and is used here. 6. Discussion and conclusions These results offer an interpretation of the physical reason that the flame lengths (normalized by the fuel injection rate and other parameters of the problem) are larger in strong firewhirls than in fire plumes without rotation. Basically, the entrained flow in the strong whirls of low Rossby number should approach the compensating regime, which is not described by a Burgers vortex. This change in axial and radial velocities stimulates higher values of the mixture fraction at the axis. The agreement shown in Figure 1 is consistent with this interpretation. As indicated in the introduction, there are many different aspects to firewhirls, and different physical phenomena can influence their flame lengths. The present contribution indicates a consistency of lengthening the flames 1

11 in the presence of strong rotation with the predictions of the compensating regime, independently of any effects of gas density variations. It would be of interest in the future to investigate influences of density variations and other related factors on predictions of flame lengths in firewhirls, which may be significant but are not considered in this short analytical work.. References [1] F. A. Williams, Prog. Energy Combust. Sci. 8 (1982) [2] K. Kuwana, K. Sekimoto, K. Saito, F. A. Williams, Y. Hayashi, H. Masuda, AIAA Journal 45 (27) [3] K. Kuwana, K. Sekimoto, K. Saito, F. A. Williams, Fire Safety Journal 43 (28) [4] H. W. Emmons, S. J. Ying, Proc. Combust. Inst 1 (1928) [5] B. R. Morton, Fire Research Abstracts and Reviews 12 (197) [6] K. Kuwana, K. Sekimoto, T. Minami, T. Tashiro, K. Saito, Proc. Combust. Inst 34 (212) to appear. [7] K. H. Chuah, K. Kuwana, K. Saito, F. A. Williams, Proc. Combust. Inst. 33 (211) [8] F. Battaglia, R. G. Rehm, H. R. Baum, Physics of Fluids 12 (2) [9] S. P. Burke, T. E. W. Schumann, Proc. Combust. Inst 1 (1928) [1] F. A. Williams, Combustion Theory, Addison-Wesley, Reading, MA, 2nd edition, 1985 pp [11] K. H. Chuah, K. Kuwana, K. Saito, Combustion and Flame 156 (29) [12] J. M. Burgers, Adv. Appl. Mech. 11 (1967) [13] K. Kuwana, S. Morishita, R. Dobashi, K. H. Chuah, K. Saito, Proc. Combust. Inst. 33 (211)

12 [14] H. A. Einstein, H. Li, Proc. Heat Trans. and Fluid Mech. Inst. 4 (1951) [15] W. S. Lewellen, J.Fluid Mech. 14 (1962) [16] T. S. Lundgren, J.Fluid Mech. 155 (1985) [17] A. Y. Klimenko, Physics of Fluids 13 (21) [18] A. Y. Klimenko, Theoretical and Computational Fluid Mechanics 14 (21) [19] R. R. Long, J.Fluid Mech. 11 (1961) [2] R. Fernandez-Feria, J. F. de la Mora, A. Barrero, J.Fluid Mech. 35 (1995) [21] R. Fernandez-Feria, J. Fluid Mech. 323 (1996) [22] A. Y. Klimenko, in: Proceedings of 18th Australian Fluid Mechanics Conference, 212), paper 349. [23] J. Dessens(Jr), Journal of Applied Meteorology 11 (1972) [24] W. S. Lewellen, in: Tornado: its structure, dynamics, prediction and hazards. Geophysical Monograph 79, Amer. Geophys. Union, 1993, pp

13 Figure 1: Normalized flame length plotted against normalized Peclet number. Solid symbols represent experimental data from Ref. [7]: methanol ( ), ethanol ( ), 2-propanol ( ). Symbols below horizontal line are for experiments without rotation while the symbols above the line are for experiments with vortex. Open symbols are vortex-free experiments recalculated for vortical conditions. The lines correspond to different values of α eff = 2 ( ), α eff = 1.43 ( ), α eff = 1.33 ( ). 13