Burning rates of liquid fuels in fire whirls

Combustion and Flame 159 (2012) 2104–2114

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Combustion and Flame

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Burning rates of liquid fuels in fire whirls

Jiao Lei a, Naian Liu a,⇑, Linhe Zhang a, Zhihua Deng a, Nelson Kudzo Akafuah b, Tianxiang Li b,Kozo Saito b, Kohyu Satoh a

a State Key Laboratory of Fire Science, University of Science and Technology of China, Hefei, Anhui 230026, PR Chinab Institute of Research for Technology Development, College of Engineering, University of Kentucky, Lexington, KY 40506-0503, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 July 2011Received in revised form 7 November 2011Accepted 23 January 2012Available online 1 March 2012

Keywords:Fire whirlBurning rateDiffusion flameBoundary layerFlame height

0010-2180/$ - see front matter � 2012 The Combustdoi:10.1016/j.combustflame.2012.01.019

⇑ Corresponding author. Fax: +86 551 3601669.E-mail address: [email protected] (N. Liu).

This paper presents semi-empirical investigations on the quasi-steady burning rates of laminar and tur-bulent fire whirls established over liquid fuel pools. The inflow boundary layer above the fuel surface con-sists of two regions: outer reactive region and inner non-reactive region. Based on the momentumboundary layer solutions with the applications of stagnant film model and Chilton–Colburn analogy,the burning rates are correlated with ambient circulation and pool size for laminar and turbulent firewhirls respectively. It is shown that in general pool fires the mass and heat transfers on the fuel surfaceare controlled by natural convection, while in fire whirls they are strongly enhanced by forced convec-tion. Fuel evaporation rate in the outer region is relatively larger than that in the inner region. The largeproportion of fuel evaporated from the outer region is mainly due to its larger area. The predictions agreewell with the data from the present experiments and the literature. Furthermore, the flame height is con-firmed to be proportional to the ambient circulation for both laminar and turbulent liquid fire whirls.

� 2012 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction

Fuel burning rate, which determines the heat release rate (HRR),is the most fundamental quantity in pool fire research. In general,the burning rate is approximately taken as the mass loss rate of thecondensed phase fuel, namely the evaporation rate of liquid fuel orpyrolysis rates of solid fuel respectively [1]. Fire whirls, which aregenerally induced by normal pool fires under ambient circulation(C), can be seen during intense fires in combustible building struc-tures or, more commonly, in forest or bush fires. It is well knownthat mass burning rates in fire whirls are much larger than thatin general buoyant pool fires of the same sizes up to medium scale,which implies that fire whirls are more vigorous and dangerous[2–4].

Over the last few decades, there have been fruitful studies onmass burning rates of general pool fires. However, limited experi-mental and theoretical results are available for burning rates of firewhirls. The earliest quantitative work on fire whirls was traced to1967 conducted by Emmons and Ying [4]. In their studies, the massloss rate of fire whirl was measured using an acetone fuel pan(d = 10 cm) surrounded by a rotating screen under various valuesof ambient circulations (C = 0.60–3.38 m2/s), and it was shownthat the mass loss rate increased steadily with ambient circulation,for which, however, no definite explanation was proposed.

ion Institute. Published by Elsevier

Muraszew et al. [5] performed experiments on fire whirls byusing a cylindrical channel (diameter: 0.91 m, length: 3.65 m) witha conical section at the top (length: 1.82 m, outlet diameter:0.46 m) and six tangential inlets located in the bottom of the sec-tion. Acetone and wood cribs were used as fuels. The authors ar-gued that in fire whirls the heat transfer coefficient on the fuelsurface consists of two parts, i.e. the heat transfer coefficient with-out swirl and the added convective heat transfer contribution dueto swirl. The authors also derived the ratio of burning rates withand without swirl based on the theory of fluid dynamics and heattransfer. However, the chemical reaction and the specific boundarylayer above the fuel surface were not considered. By carefulnumerical simulation, Snegirev et al. [6] found that the radiativeheat feedback fraction from the upper flame to the fuel surfacedecreases slightly with circulation. Thus they believed that the in-crease in burning rate is induced by the enhanced air entrainmentwhich induces more mixing of reactants in the boundary layer nearthe fuel surface. Recently, Chuah et al. [7] pointed out that the heatfeedback to the fuel surface increases inversely with vortex core ra-dius. However, they did not notice that the evaporation processtakes place in the boundary layer. In our previous work [3], wefound that a free burning fire whirl is a highly stable burning phe-nomenon with large quasi-steady periods and the quasi-steadyburning rate data of medium-scale fire whirls were correlated withpool diameters in a similar way as general pool fires.

A survey of the above literature reveals that there appears to beno interpretation on the physical mechanism of liquid fuelevaporation in the boundary layer of fire whirls, and also the

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Nomenclature

a amplitude coefficient of the radial velocity (–)B mass transfer number (–)B0 convective mass transfer number (–)cp specific heat at constant pressure (kJ/kg K)cf friction coefficient (–)d pool diameter (m)e boundary layer thicknesses ratio (dm/d) (–)f, f(g) dimensional radial velocity profile (–)F constant in Eq. (41) (–)g, g(g) dimensional radial velocity profile (–)g0 gravitational acceleration (m2/s)h specific enthalpy (kJ)_m00 fuel evaporation rate per unit area (kg/m2 s)_m fuel evaporation rate (kg/s)

M molecular weight (kg/kmol)n power exponent for tangential velocity profile in the ra-

dial direction (–)p pressure (Pa)Pr Prandtl number (–)R radius of the fuel pan (=d/2) (m)Re Reynolds number (–)R0 radius of the inner non-reactive region (m)R⁄ radius ratio of inner non-reactive region and the fuel

pan (–)St Stanton number (–)T temperature (K)Tb fuel boiling temperature (K)Ti fuel original temperature (K)Qc heat of combustion (kJ/mol)Qg effective heat of gasification (kJ/mol)Qv latent heat of vaporization (kJ/mol)_Q 00cond

_Q 00rad_Q 00conv conductive, radiative and convective heat fluxfeedback (kW/m2)

_Q 00loss heat loss flux of fuel surface (kW/m2)_Q 00corr downward heat transfer flux from the fuel surface (kW/

m2)u velocity component at radial direction (m/s)U resultant velocity at z = dm (m/s)v velocity component at tangential direction (m/s)V tangential velocity above the boundary layer (m/s)m0 stoichiometric coefficients (–)w velocity component at axial direction (m/s)Yj mass concentration of species j (–)zf flame height (m)Z� dimensionless flame height (–)

Greek symbolsa, b power exponent for flame position correlations in the

outer region (–)c stoichiometric coefficient (–)d boundary layer thicknesses (m)f Schvab–Zeldovich variables (–)g dimensionless vertical coordinate (–)l dynamic viscosity (kg/m s)m kinematic viscosity (m2/s)q density (kg/m3)s shear stress (kg/m s2)vr radiation heat feedback fraction (–)x parameter of tangential velocity distribution above the

boundary layer (m1-n/s)C circulation (m2/s)

Subscriptsr radial coordinateh azimuthal coordinatez vertical coordinateF fuelO oxygenT temperature/transferred gass fuel surfacef flameg gas stream far from the fuel surface1 free stream conditionm vertical position of maximum radial velocity0 radial position at r = R0

l laminart turbulento outer reactive regioni inner non-reactive regionil, it Inner non-reactive region of laminar/turbulent flowol, ot outer reactive region of laminar/turbulent flow

Superscripts00 per unit area000 per unit volume

Overheads� variable in incompressible coordinates. per unit time

J. Lei et al. / Combustion and Flame 159 (2012) 2104–2114 2105

burning rate ( _m) has not yet correlated versus ambient circulations(C) and pool sizes (R). In this paper, we propose semi-empiricalmodels which couple the burning rates with ambient circulationsand pool sizes for laminar and turbulent fire whirls with liquidfuels. The mechanism which induces the enhanced evaporationrate in the specific boundary layer is thus revealed. The results ob-tained by the new models are compared to the data in the presentwork and previous experiments. Relationship between flameheight and ambient circulation is also clarified.

2. Theoretical consideration

It is well known that fire whirl is a special concentrated vortexwith combustion. The flow in the fast rotational vortex core is incyclostrophic balance, which means the radial pressure gradientforce is approximately equal to the centrifugal force [8,9]. How-ever, near the end wall, the flow is retarded by wall friction and

can not sustain the radial pressure gradient any more. Then thefluid is accelerated and moves towards the core along a spiral tra-jectory. As a result, an intense laminar or turbulent Ekman-like in-flow boundary layer forms at the base of fire whirl [4,8]. Ascompared to the weak entrainment above the fuel surface in gen-eral pool fires without swirl, the intense air inflow in fire whirlsincreases the heat and mass transfer rates on the liquid fuel surfaceconsiderably. Then much higher burning rates are observed in firewhirls than those in general pool fires with the same pool sizes.

As shown in Figs. 1 and 2, near the pool edge, the flame locatesin the boundary layer. As the radius decreases, the flame is lifted updue to increasing fuel vapor and decreasing air in the inflowboundary layer. As soon as the fuel rich gas approaches a specificlocation (r = R0), the flame suddenly rises and becomes to be aspiral cylinder above the boundary layer with a fuel rich core in-side. Obviously, the end-wall boundary layer just above the fuelpan consists of two regions: an inner non-reactive region (r < R0)and an outer reactive region (R0 < r < R). As pointed out by Spalding

Fig. 1. Instantaneous picture of fire whirl at the base (d = 50 cm).

Fig. 2. Schematic of fire whirl illustrating boundary layer, radial flow and flame.

2106 J. Lei et al. / Combustion and Flame 159 (2012) 2104–2114

[10], the mechanisms of heat and mass transfers in the two regionsare of great difference. Compared to the inner region where there isno burning, the outer region involves flame close to the fuel sur-face, which helps enhance the heat transfer, inducing the increaseof the fuel evaporation rate. Therefore, it is obvious that the fuelevaporation should be investigated respectively for the tworegions.

2.1. Heat balance on the liquid fuel surface

In any pool fire, the fuel vapor is released from the fuel surfacewith an initial velocity ws, which depends on the net heat receivedby the fuel surface. The heat balance for fuel volume was formu-lated by Hamins et al. [11]:

_Q 00cond þ _Q 00rad þ _Q 00conv ¼ _m00Z Ts

Ti

cpdT þ Qv

!þ _Q 00loss þ _Q 00corr ð1Þ

where _Q 00cond;_Q 00rad;

_Q 00conv are local conductive, radiative and convec-tive heat flux feedback respectively. Qv is the latent heat of vapori-zation. _Q 00loss denotes the heat loss of fuel surface and _Q 00corr thedownward heat transfer from the fuel surface. _Q 00cond, _Q 00loss and _Q 00corr

are neglectable because they are much smaller than other quanti-ties. The surface temperature Ts is very close to, but slightly below,the fuel boiling point Tb [12]. In the present work, the fuel surface

temperature Ts is taken as Tb. Ti is the fuel original temperature._m00 is the fuel evaporation rate per unit area.

As shown in Fig. 2, the incident radiation _Q 00rad on the fuel surfaceis not only ascribed to the flame within the boundary layer, butalso due to the main cylindrical flame above the boundary layer,which serves as an external radiation source and is assumed tobe not affected by the thin flame in the boundary layer [1]. Further-more, we believe that the radiation from the thin flame within theboundary layer is much smaller and thus neglectable as comparedto the radiation from the major flame column. We define a radia-tive heat feedback fraction of vr as

vr ¼ _Q 00rad=ð _Q 00rad þ _Q 00convÞ ð2ÞThen the mass evaporation rate _m00can be expressed by

_m00 ¼ qsws ¼1

1� vr

_Q 00convQ g

ð3Þ

where Qg ¼R Ts

TicpdT þ Qv is the effective heat of gasification. Snegi-

rev et al. [6] showed that the total radiative feedback to the fuel sur-face does not change significantly with circulation. In this paper, wefurther assume that local radiative feedback is proportional to thelocal burning rate, which means vr is a constant for any specific fuel.The influence of this approximation will be discussed later.

2.1.1. Chilton–Colburn analogy for turbulent boundary layers in firewhirls

The convective heat transfer to the fuel surface can be ex-pressed by:

_Qconv ¼ StqsUcpDT ð4Þ

where St ¼ h=ðqscpUÞ is the Stanton number. U is the resultantvelocity in 3-D boundary layer. DT is the effective temperaturedifference.

In turbulent flow, the convective heat transfer on the fuelsurface can be related to the wall friction by the Chilton–Colburnanalogy [13], which is corrected by Rotta for the cases with non-unity turbulent Prandtl number as follows [14]:

St ¼ 12

cf 0nPr�2=3 ð5Þ

where cf 0; St and nPr�2=3 are respectively the friction coefficient,Stanton number and Rotta’s Reynolds analogy factor. For highReynolds number flow, nPr�2=3 ¼ Pr�1

ts , where Prts is the turbulentPrandtl number near the surface [14,15]. For near-wall flows, exist-ing theories and measurements indicate that nPr�2=3 < 1 [15–18].However, it is difficult to obtain the exact value of the analogyfactor nPr�2=3 here due to no experimental work for such boundarylayers with injection and combustion. In the present work, we takea value of 1.70 for the turbulent Prandtl number near the surface(Pr = 0.7) as suggested by Hammond [17]. Thus, we have nPr�2=3 ¼Pr�1

ts ¼ 0:59.Then the burning rate problem in turbulent boundary layer of

fire whirl comes down to obtaining the wall friction or the momen-tum boundary layer solutions.

2.2. Stagnant film theory for laminar boundary layer in fire whirls

We note that the Chilton–Colburn analogy is not suitable forlaminar boundary layer due to the large radial pressure gradient[13]. Thus we try to seek another way to obtain the fuelevaporation rate for laminar problems. As shown by Glassmanand Yetter [19], the evaporation rate per unit area ( _m00) for laminarconvective burning problems can be approximated by stagnantfilm theory, which can be written in the form

_m00 ¼ ld

lnð1þ BoÞ ð6Þ


where Bo ¼ YO1Qc=ðMOm0OQgÞ � hs=Qg is the so-called Spaldingtransfer number in the outer region, which is the most importantchemical parameter that controls the fire behaviors. hs ¼

R Ts

T1cpdT

is the specific enthalpy. The former term is the ratio of the heat re-lease of chemical reaction to the latent evaporation heat and ismuch larger than the latter term.

The radiation correction suggested by Fineman [20] is:

_m00 ¼ ld

lnð1þ B0oÞ ð7Þ

where B0o ¼ Bo=ð1� vrÞ is the convective mass transfer number inthe outer reactive region.

Assuming that the stagnant film model is also correct for innernon-reactive region, the results in the outer reactive region areapplicable for the inner non-reactive region by replacing B0o by B0i,which is the Spalding transfer number for non-combustion prob-lem and is defined by [10]:

Bi ¼YFs � YFT

YFs � 1; Bi ¼

cpDTQ g

ð8Þ

where YFs and YFT are respectively the mass fractions of fuel vaporson the fuel surface and in the transferred gas, DT is the temperaturedifference across the boundary layer. The two transfer numbers inthe inner non-reactive region are assumed to be equal and constant.Similar to the turbulent case, we need to solve the momentumboundary layer equations in order to derive the boundary layersolutions.

2.3. Solutions of momentum boundary layer

2.3.1. Governing equationsThe momentum boundary layer equations for a quasi-steady

axisymmetric (laminar or turbulent) fire whirl of compressiblefluid in cylindrical coordinates are as follows.

Continuity:

1r@qru@rþ @qw

@z¼ 0 ð9Þ

Axial momentum:

@p@zþ qg0 ¼ 0 ð10Þ

Radial momentum:

q w@u@zþ u

@u@r� v2

r

� �¼ � @p

@rþ @sr

@zð11Þ

Tangential momentum:

q w@v@zþ u

@v@rþ uv

r

� �¼ @sh

@zð12Þ

where variables u, v, and w are the velocity components and r, h, zare the radial, azimuthal and vertical coordinates respectively. q, p,g0 are respectively the air density, pressure and the gravitationalacceleration. The shear stresses are defined as sr = lou/oz andsh = lov/oz. For laminar flow, l is the dynamic viscosity. For turbu-lent flow l should be replaced by turbulent value which depends onthe actual flow conditions.

Above the inflow boundary layer, the small radial velocity is ne-glected and the tangential velocity is V(r) of the upper flow, whichis in cyclostrophic balance between centrifugal force and radialpressure gradient force.

1q@p@r¼ V2

rð13Þ

From Eq. (10), we can ignore the pressure variation across theboundary layer, thus p = p(r) in the boundary, which indicates that

buoyancy is not significant in the inflow boundary layer with forcedconvection as soon as the fire whirl forms.

2.3.2. Transformations from compressible to incompressible flowIn the foregoing discussion, the effects of variable fluid proper-

ties are not considered. One approach is to convert the boundarylayer equations with variable properties into incompressible form.Such mathematical transformations are all related to the Howarth–Dorodnitsyn transformation [21]:

~r ¼ r; ~h ¼ h;~z ¼Z z

0

qqs

dz ð14Þ

~u ¼ u; ~v ¼ v ;qs ~w ¼ qwþ uZ z

0

@q@r

dz ð15Þ

The transformations given by Eqs. (14) and (15) in Eqs. (9), (11), and(12) yield the following equations:

1~r@~r~u@~rþ @

~w@~z¼ 0 ð16Þ

~w@~u@~zþ ~u

@~u@~r�

~v2

~r¼ � 1

q@p@r

� �þ @

@zqlq2

s

@~u@~z

� �ð17Þ

~w@~v@~zþ ~u

@~v@~rþ

~u~v~r¼ @

@zqlq2

s

@~v@~z

� �ð18Þ

By assuming that ql = const, Eqs. (16)–(18) are identical to thecontrolling equations for incompressible flow.

2.3.3. Solutions of momentum boundary layersBy assuming reasonable velocity profiles, the system of equa-

tions can be solved analytically by Karman integral equations.The details for the derivations are shown in Appendix A. Theboundary layer solutions are as follows:

Laminar: a = 4.52

d ¼ ~rð1�nÞ=2 60ms

nax

� �1=2

ð19Þ

Turbulent:

a ¼ �L4

L5M þ L3 þ ð1� etÞðML2 þ L1Þ

� �1=2

ð20Þ

d ¼ C � ~rM ms

x

� m=ð1þmÞð21Þ

where a and d are the radial velocity amplitude and the boundarylayer thickness respectively. All the other parameters are shownin Appendix A.

2.4. Burning rates of laminar fire whirls

2.4.1. Burning rate in the outer reactive region of laminar fire whirlIn Fig. 2, the flame is established within the momentum bound-

ary layer in the outer reactive region. As pointed by Marxman andGilbert [23], the zone below the flame is the effective boundarylayer for heat transfer to the liquid fuel surface and the wholeboundary layer for momentum transfer. Thus the effectivethickness d in Eq. (7) should be replaced by the distance (df) be-tween the flame and fuel surface. Substituting Eq. (19) into Eq.(7), yields

_m00ol ¼ ~rðnol�1Þ=2qsg�1ol

msxolnola60

� 1=2lnð1þ B0olÞ ð22Þ

where gol = df/d. For the subscript ‘‘ol’’ used here, ‘‘o’’ denotes theouter reactive region, and ‘‘l’’ denotes laminar. In fire whirls, the


gaseous fuel enters the boundary layer due to evaporation at the li-quid fuel surface, while the oxygen is entrained into the boundarylayer by unbalanced pressure gradient. The flame locates at the po-sition where a suitable mixture ratio is achieved. In this work, it isdifficult to obtain the variation of flame location within the bound-ary layer due to the complexity of the boundary layer flow. Notethat gol equals to 0 and 1 at r = R and r = R0 respectively in Fig. 2.Thus we assume a simple correlation for gol as follows:

gol ¼ ½ðR� rÞ=ðR� R0Þ�a ð23Þ

where a > 0 and can be determined by experiments. Thus, we have

_m00ol ¼ ~rðnol�1Þ=2qsR� R0

R� r

� �a msxolnola60

� 1=2lnð1þ B0olÞ ð24Þ

We found that the local burning rate is a function of r, and may in-crease or decrease which depends on the actual values of nol and a.However, the local burning rate near r = R should reach a maximumvalue due to the singularities at this point.

The overall evaporation rate in the outer reactive region is

_mol ¼Z R

R0

2pr _m00old~r ¼ HolC1=2RDol ð25Þ

where R� = R0/R, C ¼ 2pVhRR ¼ 2pxolRnolþ1, Hol ¼ qsð2pmsnola=60Þ1=2

lnð1þ B0olÞ and

DolðR�Þ ¼ ð1� R�ÞaZ 1�R�

0x��að1� x�Þðnolþ1Þ=2dx� ð26Þ

where x� = 1 � r/R, Dol is a non-dimensional value.

2.4.2. Burning rate in the inner non-reactive region of laminar firewhirl

In the inner non-reactive region, the convective heat transferredto the fuel surface is all transported from the outer reactive region.If the local radiative heat feedback is a constant in the radial direc-tion, the axial temperature gradient at the fuel surface decreaseswith decreasing r, thus the local mass loss rate also decreases.The effective layer for convective heat transfer should be the tem-perature boundary layer (dT), thus

_m00il ¼ qsg�1il ~rðnil�1Þ=2 msxilnila

60

� 1=2lnð1þ B0ilÞ ð27Þ

where gil = dT/d, which should increase with decreasing r. For thesubscript ‘‘il’’ used here, ‘‘i’’ denotes the inner non-reactive region,and ‘‘l’’ denotes laminar.

We have

_mil ¼Z R0

02pr _m00ild~r ¼ HilC

1=20 R0Dil ð28Þ

where C0 ¼ 2pVhR0 R0 ¼ 2pxilRnilþ10 , Hil ¼ qsð2pmsnila=60Þ1=2

lnð1þ B0ilÞ, r� = r/R0, Dil ¼R 1

0 g�1il ðr�Þr�ðnilþ1Þ=2dr�.

At the boundary of the inner and outer regions (r = R0), thematches of the local burning rates gives

_m00iljr¼R0¼ _m00oljr¼R0

ð29Þ

We obtain

g�1il jr¼R0

¼ lnð1þ B0ilÞ=lnð1þ B0olÞ ð30Þ

In the present work, we assume that gil|r=0 = 1 and

g�1il ¼ ðg�1

il jr¼R0� 1Þr� þ 1 ð31Þ

In the later discussion, the mass loss rate in the inner region isproved to be much smaller than that in the outer region. Therefore,the linear treatment of gil will not induce significant errors.

2.5. Burning rate of turbulent fire whirls

2.5.1. Burning rate in the outer reactive region of turbulent fire whirlIn the outer reactive region, the flame is located in the velocity

boundary layer and the temperature gradient is positive below theflame and negative above the flame. The zone below the flame isthe effective boundary layer for convective heat transfer to the fuelsurface. Thus the Stanton number (St) should be defined at theflame,

_Qconv ¼ Stqf V f cpðTf � TsÞ ð32Þ

where Vf is the velocity at the flame, Tf is the flame temperaturewhich can be expressed by [24],

cpðTf � TsÞ ¼ Botð1� ff ÞQg ð33Þ

For the subscript ‘‘ot’’ used here, ‘‘o’’ denotes outer reactive region,and ‘‘t’’ denotes turbulent. Here ff is a constant value of the Schvab–Zeldovich variables at the flame [24], expressed by

ff ¼ ½ðBot þ 1Þ=Bot �½c=ðcþ 1Þ� ð34Þ

where c is the stoichiometric coefficient defined by

c ¼ YFT MOv 0O=ðYO1MFm 0FÞ ð35Þ

For liquid fuels, c is a small parameter and is much smaller than Bot

[24]. Then, ff also has a small value. As suggested by Marxman andGilbert [23], St at the flame can be related to the friction coefficientcf by introducing a parameter V/Vf as following:

St ¼ 12

cf nPr�2=3ðq1V2=qf V2f Þ ð36Þ

where the Chilton–Colburn analogy of Eq. (5) is assumed to holdacross the entire boundary layer. The St number is related to theflame position by the new parameter V/Vf.

Using Eqs. (32), (33), and (36) in (3) lead to

_m00ot ¼12

cf q1VnPr�2=3ð1� ff ÞB0otðV=Vf Þ ð37Þ

The empirical formulae of Eqs. (5) and (36) do not consider theblowing effect which reduces the skin friction due to fuel surfacemass injection. As suggested by Spalding [25], as first approxima-tion, this effect can be corrected by a factor of ln(1 + B0)/B0. Theresulting expression for cf with incident radiation on the fuel sur-face is

cf ¼ cf 0lnð1þ B0otÞ=B0ot ð38Þ

where cf0 is the friction coefficient without blowing on the fuel sur-face. With the use of turbulent boundary layer solutions, we have

_m00ot ¼ Gtq1mm=ð1þmÞs x1=ð1þmÞ

ot ~rðnot�mÞ=ð1þmÞð1� ff Þlnð1þ B0otÞ� ðV=Vf Þ ð39Þ

where

Gt ¼ Fe�8m=7t A�mC�mnPr�2=3 ð40Þ

The parameter V/Vf is given by the assumed velocity profiles inAppendix A.

V=Vf ¼ g�1=7ot 1þ a2ð1� gotÞ

2h i�1=2

ð41Þ

where got = df/d�got increases with decreasing r and got equals to 0and 1 at r = R and r = R0 respectively. Similar to the laminar case,the local burning rate may increase or decrease with r, which de-pends on the relative variation of boundary layer (not) and flamelocation (got). Anyway, the point at r = R should be a singularitywhere the local burning rate reach its maximum value.


By integrating Eq. (39) with respect to ~r from R0 to R, the massloss rate in the outer reactive region is

_mot ¼Z R

R0

2pr _m00otdr ¼ HotC1=ð1þmÞRDot ð42Þ

where C ¼ 2pxotRnotþ1;Hot ¼ q1ð2pmsÞm=ð1þmÞGtð1� ff Þlnð1þ B0otÞ

x� = 1 � r/R = 1 – r� and

Dot ¼Z 1�R�

0ð1� x�Þðnotþ1Þ=ð1þmÞV=Vf ðx�Þdx� ð43Þ

Due to the difficulty in obtaining the position of flame zone, we stillassume that got follows:

got ¼ ½ðR� rÞ=ðR� R0Þ�b ð44Þ

where b > 0.

2.5.2. Burning rate in the inner non-reactive region of turbulent firewhirl

The heat transfer to the fuel surface in the inner region is givenby

_Q conv ¼ StqsUcpDT ð45Þ

where DT is the temperature difference across the effective bound-ary layer. By using Eq. (8), we obtain

_Q conv ¼ StqsUBitQ g ð46Þ

For the subscript ‘‘it’’ used here, ‘‘i’’ denotes inner non-reactive re-gion, and ‘‘t’’ denotes turbulent. Then the mass loss rate per unitarea is

_m00it ¼12

cf 0q1nPr�2=3V lnð1þ B0itÞðV=VTÞ ð47Þ

where VT is the velocity at the upper temperature boundary layer.

_m00it ¼ Gtq1mm=ð1þmÞs x1=ð1þmÞ

it~rðnit�mÞ=ð1þmÞlnð1þ B0itÞðV=VTÞ ð48Þ

Integrating Eq. (48) with respect to ~r from 0 to R0 yields

_mit ¼Z R0

02pr _m00itdr ¼ HitC

1=ð1þmÞ0 R0Dit ð49Þ

where C0 ¼ 2pxitRnitþ10 , Hit ¼ q1ð2pmsÞm=ð1þmÞGtlnð1þ B0itÞ,

Dit ¼R 1

0 x�ðnitþ1Þ=ð1þmÞðV=VTÞdx�, x� = r/R0.By matching the local burning rate at the boundary of the inner

and outer regions, we have

V=VT jr¼R0¼ ð1� ff Þlnð1þ B0otÞ=lnð1þ B0itÞ ð50Þ

Similarly, we still assume that V/VT follows linear distribution,

V=VT ¼ ðV=VT jr¼R0� 1Þr� þ 1 ð51Þ

The error of the total burning rate introduced by this linear assump-tion should be small due to the small portion of mass loss rate in theinner region as shown below.

2.6. Total mass loss rate in fire whirls

The integration of the above analyses leads to the total fuelevaporation rate of the fuel pan, which is the sum of the rate valuesat in the inner and outer regions.

_ml ¼ HlC1=2R ð52Þ

_mt ¼ HtC1=ð1þmÞR ð53Þ

Here C is the circulation at the pool edge (r = R) and

Hl ¼ HilDilR�ðnolþ3Þ=2 þ HolDol ð54Þ

Ht ¼ HitDitR�ðnotþmþ2Þ=ð1þmÞ þ HotDot ð55Þ

where Hl and Ht can determined exactly by previous formulae,while the complete expressions are very complex. In order to showthat Eqs. (54) and (55) are closed, here we present the qualitativeexpressions for Hl and Ht.

Hl ¼ Hlðel; a;nil;nol;qs; ms;R�;Bol;Bil;vr;aÞ ð56Þ

Ht ¼ Htðet ; a; L1; L2;nit ;not ;qs; ms;R�;Bot ;Bit ; F;m; ff ;vr; nPr�2=3;bÞ

ð57Þ

For laminar (turbulent) case, el and a (et, a, L1 and L2) are knownwith prescribed f(g) and g(g). nil and nol (nit and not) are obtainedfrom tangential velocity measurement above the boundary layer.qs and ms are known since the fuel surface temperature is assumedto be equal to the boiling point. R� is derived from high-speed pho-tography. Bol (Bot) is known for specific liquid fuel and Bil (Bit) isdetermined from numerical simulations as shown below. Due tolimited experimental data, the feedback heat friction (vr) is un-known and is assumed as constant for specific fuel with reasonablevalues. ff is a small constant and neglected. F and m are constant forturbulent case with smooth surface. nPr�2/3is approximately 0.59.The flame position indicated by a and b can be obtained by carefulexperiments or reasonable assumption.

A general correlation

_m ¼ HC1=ð1þmÞR ð58Þ

is applicable for laminar (m = 1) and turbulent burning respectively.

3. Experimental

As shown in Fig. 4, the medium-scale fire whirl facility was asquare enclosure made of tempered glasses, with dimension of2 m � 2 m � 15 m. The channel was open at the top. Each channelwall had a uniform 20 cm wide vertical gap at its corner so that theentrained air induced by the burning flame could enter into thechannel, thus imparting a rotational flow necessary for fire whirlformation. The base table (2 m � 2 m) was made of pine wood witha round hole (60 cm in diameter and 10 cm in depth) in the center.Wood rings with different sizes were embedded into the hole so asto fit different sizes of fuel pans (with diameters d of 10–55 cmwith step of 5 cm, and 10 cm in depth), whose rims were flush withthe top of the base table. The initial height of the liquid fuel (n-hep-tane, 97%) surface was 6 cm. A water layer was used in the larger

Fig. 3. Boundary layer velocity profiles at fixed radius.

Fig. 4. Schematic for fire whirl experiments.

Table 1Transfer numbers of various liquid fuels in air.a

Bob ln(1 + Bo) Bo ln(1 + Bo)

n-Pentane 8.15 2.21 Ethanol 3.25 1.45n-Hexane 6.70 2.04 Gasoline 4.98 1.79n-Heptane 5.82 1.92 Kerosene 3.86 1.58n-Octane 5.24 1.83 Light diesel 3.96 1.60i-Octane 5.56 1.88 Medium diesel 3.94 1.60n-Decane 4.34 1.68 Heavy diesel 3.91 1.59n-Dodecane 4.00 1.61 Acetone 5.10 1.81Octene 5.64 1.89 Toluene 6.06 1.95Benzene 6.05 1.95 Xylene 5.76 1.91Methanol 2.70 1.31

a T1 = 20 �C, YO1 = 0.232.b Bo ¼ YO1Qc=ðMOv0OQgÞ � hs=Qg .


fuel pans in order to control the fuel depth and also insure safety.The mass of fuel variation versus time was recorded using an elec-tronic balance with a precision of 0.1 g. In Fig. 1, the flame attachedto the fuel surface closely near the pool edge and thus the reverseflow induced by the lip was strongly suppressed due to the largepositive radial pressure gradient in the boundary layer. This wasconsistent with the conclusion that lip height had little effect onthe burning rates of fire whirls in our previous work [3].

The tangential velocities were measured at several heights (0.5,1.2, 1.9, 2.6, and 3.0 m) by calibrated temperature-compensated Pi-tot tubes and type K thermocouples (diameter: 0.4 mm) arrangedin radial direction (0–25 cm, with step of 5 cm). The circulationwas obtained by multiplying tangential velocity by correspondingradius outside the flame where the tangential velocity decreasedwith radius slowly. The average values of the steady circulationsin the radial direction at the above heights were taken as the ambi-ent circulations. Then the error of final ambient circulation was notmore than 10% in present tests. Flame radius was monitored byhigh-speed camera. The large test hall was kept closed and thesmoke exhaust system was turned off to reduce turbulence duringtests.

4. Results and discussion

4.1. Spalding transfer numbers in the inner and outer regions

Two transfer numbers Bi and Bo are defined in the inner and out-er regions respectively. They are the driving forces of mass transferin the boundary layer similar to temperature difference in heattransfer problems.

Transfer numbers Bo of various liquids in air were given byGlassman and Yetter [19] and are listed in Table 1. Hydrocarbonfuels have values of Bo a factor of 2–3 greater than alcohol fuels.However, the mass burning rate does not vary greatly since thetransfer number enters the burning rate expression in a term ofln(1 + Bo). Correspondingly, it is more sensitive to the diffusivitiesand gas density [19].

On the other hand, Bi is unknown due to the uncertainty of YFs

and YFT which relate to the actual flow and boundary conditions.However, a rather rough calculation, based on the results of com-bustion on a flat plate, is presented here. YFs for laminar flame isin the vicinity of 0.80 (Ali et al. [26], YFT = 0.84 for methanol;

Andreuss [27], YFs = 0.70–0.75 for ethyl alcohol). By assumingYFs = 0.8 and YFT = 0.3–0.4, Bi has a typical value of 2–2.5. Moreover,if we assume the temperature difference across the boundary layeris 850 K, Bi defined by cpDT/Qg are about 1.6 and 1.8 for acetoneand n-heptane respectively, which are also smaller than Bo.

As shown in the foregoing discussion, the convective transfernumber was introduced due to the strong external radiation onthe fuel surface from the long cylindrical flame above the boundarylayer. Furthermore, if radiation feedback from the thin flame in theboundary layer is neglected and the combustion efficiency is as-sumed as unity in the outer reactive region of boundary layer,the convective transfer number defined by Delichatsios [28] isactually reduced to B0o in the present work. Due to limited experi-mental data, vr is assumed to be constant for any fuel with speci-fied appropriate value and the influence should be small due to thelnð1þ B0oÞ term in final correlations.

4.2. Tangential velocity distributions in the radial direction

The burning rate correlations in the two regions depend on thetangential velocity profile in the radial direction above the bound-ary layer. We note that the present momentum boundary layersolution of laminar fire whirl only exists when n > 0 due to thesquare root of n in the solution. However, the solution for turbulentboundary layer is valid for n ranging from �(3 + 4m)/(4 + 5m) to 1for solid rotation.


We generally believe that the flow is in solid body rotation inthe inner region surrounded by cylindrical flame. However, theflow characteristics in the outer region are not revealed clearlydue to limited experimental data. To our knowledge, onlyMuraszew et al. [5], Hassan et al. [29] and Hayashi et al. [30]presented the radial profiles of tangential velocity (circulation)for small-scale fire whirls. It was found that tangential velocity in-creases almost linearly with radius in the whole pool or crib region,indicating that the flow above the whole fuel surface is in solidrotation (nil = nol = 1).

The radial distribution of tangential velocity for medium orlarge scale fire whirls has not been reported. In our previous exper-iments, the radial profiles of tangential velocity (d = 40 and 50 cm)were measured as shown in Fig. 5, in which the gases were approx-imated as air. The radial temperatures were not corrected for radi-ative losses. The non-zero tangential velocity at the geometricalaxis was mainly induced by the discrepancy of the flame axisand the geometrical centerline. The non-zero velocity has no sig-nificant influence on the present work due to that the profiles oftangential velocity were only used as a rough indication for thevelocity distribution. As shown in Fig. 5, the tangential velocity in-creases quickly in the inner fuel rich core, then remains nearly con-stant in the flame zone due to flame wander and high dissipation,and decreases outside the flame (Rf = 13.6 cm and 15.7 cm ford = 40 cm and 50 cm respectively). The turbulent flame thicknessis finite, being roughly one-tenth of the pan diameter [3]. Theboundary between the inner and outer regions should locate inthe inner boundary of cylindrical flame (Fig. 2). Therefore, n de-creases from zero to a negative value (n > �1) in the outer region.Here, we take n = 0 � �0.5 for the outer region in turbulent firewhirls.

4.3. Burning rates in the two regions

The circulations in Eqs. (52) and (53) are assumed to be equal tothe ambient circulations. If we take nil = nol = 1and gil = gol = 1 forlaminar fire whirls in both the inner and outer regions, the ratioof burning rates in the two regions is

_mol

_mil¼ Hol

Hil

1� R�2

R�2ð59Þ

By assuming nit = 1, not = 0 and V/Vf = V/VT = 1 with rough fuel sur-face (m = 0), the ratio for turbulent fire whirls is

_mot

_mit¼ Hot

Hit

ð1� R�2Þ=2R�3=3

ð60Þ

Fig. 5. Radial profiles of tangential velocity for medium scale fire whirls.

Obviously, the ratios are functions of R�, especially proportionalto the area of evaporating surface for laminar fire whirls. In ourexperiments, the area ratio between the inner and outer regionswas only about 0.1–0.3 with typical values of R� = 0.3–0.5. In addi-tion, Ho should be larger than Hi. Therefore, a large part of fuel isevaporated from the outer reactive region. Moreover, the totalburning rate increases when R� decreases, which is consistent withthe previous findings by Chuah et al. [7].

Although chemical reaction takes place in the outer region, alarge proportion of fuel vapor enters the inner non-reactive region,and then moves upward in the high speed convection column andcombusts gradually above the boundary layer.

4.4. Verification of burning rate correlations

4.4.1. Solutions of momentum boundary layerThe correlations for burning rates rely on the solutions of

momentum boundary layer. In the present work, the radial velocityamplitude (a) is assumed to be constant, which is only valid in cer-tain radial range. Here, we try to verify this in the turbulent flowwith smooth surface due to its recognized parameters withm = 1/4 and F = 0.0225. We found that a was about 1.10 in the out-er region with not = �0.5, which was very close to the numerical re-sults by Chi et al. [31]. Thus, this approach is reasonable and wouldnot introduce large errors in the correlations for burning rates.

4.4.2. Correlation for burning rates of laminar fire whirlsThe present experimental data for n-heptane in quasi-steady

state are shown in Table 2. The results are also illustrated inFig. 6 along with the data for acetone due to Emmons and Ying[4], with two kinds of pools conditions, i.e. one with and the otherwithout a flame holder. The pool conditions may affect the bottomboundary conditions and cause differences in burning rate. As indi-cated previously [3] for the continuous flame, the ambient circula-tion remains nearly constant in axial direction in both facilities.The variance can be attributed to the spiral structure of flow. Theambient circulations in Table 2 were calculated by the method in[3], i.e. averaging the circulations at several axial and radial posi-tions adjacent to the flame surface, where the circulations re-mained stable.

As shown in Fig. 6, the slopes of data from various sources differsignificantly, which shows the difference of flow conditions in theboundary layer. We found that the laminar theory correlated wellthe data of Emmons and Ying if the power a equals to 0.63 in Eq.(23). Moreover, the position given by Eq. (23) with a = 0.63 is con-sistent to the recent experiments by Hayashi et al. [30] in spite ofdifferent fuels and facilities. Therefore, the present theory is be-lieved to be reasonable for the available data. The property valuesand calculated a and H in the two regions are summarized in Table2 and Table 3 respectively. The maximum error is about 20%. Thedeviations of experimental data from theoretical assumptions

Table 2The experimental data in the present work.

d (cm) _m (g/s) _m=R (g/m s) Ca (m2/s)

10 0.50 10.00 1.4915 1.38 18.40 2.0020 2.33 23.30 2.6725 3.52 28.16 3.2130 5.38 35.87 3.7835 7.72 44.11 4.6040 9.91 49.55 5.0845 12.69 56.40 5.8350 15.03 60.12 6.3155 17.67 64.25 6.81

a The gas is assumed to be air outside the flame.

Fig. 6. Comparison of experimental and predicted burning rates.


due to the factors such as the burner edge above the ground andthe non-zero lip heights are considered to be responsible for the er-rors. These influences should be studied in future work.

4.4.3. Correlation for burning rates of turbulent fire whirlsSince the constant F was not very clear for turbulent flow with

rough surface, the data in the present work were firstly correlatedby turbulent theory for smooth surface. The empirical power b is2.20 for nPr�2/3 = 0.59 as shown in Table 3. The observations exhi-bit a steeper slope than the theoretical curve of smooth surface aspresented in Fig. 6.

Consistent with previous discussion, the turbulent theory forthe rough surface, with b = 2.20 and nPr�2/3 = 0.59 (identical tothe smooth surface results), provided a very good representationof the data in the present work. Although the present value for bseems reasonable, further experiments are demanded on the flameposition in the boundary layer of turbulent fire whirls to verify thepresent results.

In the above case, F has a relatively large value of 0.0082. Wenote that the power 1/7 in the velocity profiles is related tom = 1/4 by the equation m/(2 �m) [32]. Errors will be introducedwhen 1/7 power velocity profiles are also adopted for rough sur-face with m = 0. However, in the burning rate correlations, thepower of circulation [1/(1 + m)] stays the same. The burning of firewhirl with pool diameter of 10 cm is an exception to the turbulenttheory and will be discussed later.

4.4.4. Discussion of the turbulent burning rate correlationsAs shown above, the turbulent burning rate correlation for

rough surface, rather than that for smooth surface, is more consis-tent with the present experiments, which agrees well with theexperimental observations. We found that the fuel surface alwaysoscillated slightly due to flame wander and other unstable second-ary flow. Moreover, in relatively large fire whirls, circular

Table 3Radial velocity amplitude and mass transfer coefficients predicted by the present theories

m no F ao ai

Laminar (Acetone) b 1 1 x 4.52 4.Turbulent (n-heptane) c(Smooth surface) 1/4 �0.5 0.0225 1.14 2.Turbulent (n-heptane) c(Rough surface) 0 �0.5 0.0082 1.10 2.

a R�and vr are assumed to be unchanged for specific fuel.b qs = 1.065 kg/m3, ms = 18.81 � 10�6 m2/s, Bil = 2.00. The transport properties of air arc q1 = 1.19 kg/m3, ms = 23.23 � 10�6 m2/s, c = 0.066, Bit = 2.00. The transport propertie

ripples were generated and then they moved towards the centercontinuously on the liquid fuel surface due to strong inflow air inthe boundary layer. As the ripples were approaching the center,their height increased gradually. In few cases with relatively strongrotation, the waves with increasing height suddenly broke intosprays in the center which evaporated immediately in the hightemperature core. Although the detailed mechanisms are not clear,the oscillations, ripples and wave breaking are believed to increasethe fuel evaporation rate to some extent.

In addition, we note that wave heights and lengths increasewith surface wind speed (rotation) and wave-breaking should beimportant for fuel mass transfer in stronger and larger fire whirls[33]. If the rotation is too strong, the liquid fuel is directly suckedinto the vortex core with substantial increase in burning rate,which is similar to the real fire whirls with solid combustibles inforest fires. However, the survival time of solid fuels is much longerdue to slower pyrolysis rate as compared to evaporation rate. Thismay lead to spot fires. In the present work, since the rotation wassmall, the enhanced evaporation induced by the oscillations, rip-ples and wave breaking is limited and only convective controlledmass transfer is considered.

Furthermore, the liquid fuel surface is no longer smooth (m = 1/4) and should be considered as aerodynamically rough surface(m = 0). However, the friction coefficient F depends on the actualsurface and flow conditions. No simple law is available for liquidfuel surface with limited fetch [34]. In addition to the error inducedby the power 1/7 in the velocity profiles for rough surface, anothersource for the large value of F may be induced by the slightly en-hanced evaporation by the oscillations, ripples and wave breaking,if any.

4.5. Transition from laminar to turbulent in the boundary layer

The data of Emmons and Ying [4], correlated by laminar theory,show no appreciable change in slope with circulations which over-lap the circulations in our experiments with turbulent fire whirl.The smooth flame edges indicated that the flow in the boundarylayer above the fuel surface keeps in laminar within the parame-ters ranges [35].

The burning rate of fire whirl with d = 10 cm in our experimentsis much smaller than the predicted value by the turbulent theory,and the burning rate actually falls into the laminar regime. Corre-spondingly, we observed that the plume above the flame has verysmooth edges and could even maintain to the outlet (z = 13.5 m),which indicated low dissipation and laminar flow. As the pooldiameters increase, the mass transfer on the fuel surface rapidlybecomes turbulent convective controlled. The turbulent flamebrush is more obvious and the plume core decays quickly due toturbulent dissipation. Thus, the transition from laminar to turbu-lent occurs in the inflow boundary layer. Although the transitionmechanism is not clear, here we propose a possible explanation.The entrained air enters our facility from four organized slits andthe radial and tangential velocity components are comparable,while air is entrained from all radial directions and the radial

in the inner and outer regions.a

R�a vra nPr�2/3 a/b Ho � 10�3 Hi � 10�3 H � 10�3

52 0.5 0.3 x 0.63 6.72 4.29 8.5937 0.3 0.5 0.59 2.20 15.08 18.18 11.6130 0.3 0.5 0.59 2.20 13.82 9.27 9.52

e evaluated at Ts = 329 K.s of air are evaluated at T1 = 293 K, Ts = 371 K.


velocity is much smaller than the tangential velocity in Emmons’facility. Thus, the flow is more axisymmetric and the generatingeddy is more stable in the facility of Emmons, which is beneficialfor maintaining laminar flow. In addition, the flame of n-heptaneemits more radiation to the ground. The rising heated gas is favor-able for the formation of turbulence.

4.6. Flame heights of fire whirls

A correlation for flame heights of fire whirls was presented pre-viously [3].

Z� ¼ Kð _Q � � C�2Þs ð61Þ

where K is a comprehensive dimensionless quantity. Z� = zf/d,_Q � ¼ _Q=ðq1cp1T1g1=2d5=2Þ, C� = C/(g1/2d3/2) are the non-dimen-

sional flame height, heat release rate and circulation respectively.The values for the power s are 0.43 (Emmons without flame holder),0.39 (Emmons with flame holder), and 0.33 in our study. Note thatthe burning rate of liquid fuels is determined by circulation andpool diameter.

_Q � � _m=d5=2 � Cqd�3=2 ð62Þ

where q = 1/(1 + m) and q = 1/2 and 1 for laminar and turbulent firewhirls (rough fuel surface) respectively. Thus we have

Z� � C�sð2þqÞd3sðq�1Þ=2 ð63Þ

For laminar fire whirls, q and s can be taken as 0.5 and 0.4 respec-tively, then we have zf � Cd�0.8. While in turbulent fire whirls, Eq.(63) becomes Z⁄ � C⁄ with q = 1 and s = 0.33. As shown by Emmons[35], the flame height initially increased rapidly with small circula-tion, then decreased with too strong rotation due to vortexbreakdown at the top. Therefore, the above equations are onlyapplicable for fire whirls without vortex breakdown.

The above results differ greatly from those for general flameswithout rotation. For example, the flame height of a laminar jetis proportional to the mass burning rate, while the non-dimen-sional turbulent jet flame height (normalized by nozzle diameter)is a constant. In fire whirls, the rotation combined with radial den-sity gradient strongly impedes turbulent mixing and results in thelaminarization of turbulent flame [36]. Thus, the flame height ofturbulent fire whirls increases with circulation due to the reducedturbulent mixing and increased burning rate for liquid fuel pools.

5. Conclusions

The present work studied the quasi-steady burning rate of lam-inar and turbulent fire whirls with liquid fuels. Correlations wereestablished based on the boundary layer theory, film theory andanalogies between momentum and mass transfer. The major re-sults are summarized below:

(1) The increase of burning rates in fire whirls results from theenhanced convective heat and mass transfer in the inflowboundary layer above the liquid fuel surface.

(2) The outer and inner regions, with and without combustionin the inflow boundary layer, were distinguished. The evap-oration rate is relatively larger and the fuel is mainly evapo-rated in the outer region.

(3) A semi-empirical correlation _m ¼ HC1=ð1þmÞR couples themass loss rate to ambient circulation and pool size. The fac-tor m is definite for laminar (m = 1) and turbulent flows(m = 1/4 and 0 for smooth and rough surfaces respec-tively).The correlation agrees well with the data in literatureand in present work. The fuel surface is aerodynamical roughsurface for relatively large pools.

(4) The flame heights are both proportional to ambient circula-tions for laminar and turbulent fire whirls established overliquid fuel pools.

Acknowledgments

This work was sponsored by National Basic Research Program ofChina (973 Program, No. 2012CB719702), National Natural ScienceFoundation of China under Grants 51076148 and 51120165001,and National Key Project of Scientific and Technical SupportingPrograms(No. 2011BAK07B01). Jiao Lei was funded by the CAS Spe-cial Grant for Postgraduate Research, Innovation and Practice andChinese Scholarship Council.

Appendix A

A.1. General solutions of momentum boundary layers

As shown in Fig. 3, in the boundary layer, the tangential velocity(v) decreases from the value of the upper flow at z = d to zero on theground along the axial direction, and thus the radial pressure gra-dient force is in excess of the centrifugal force, which induces astrong radial air inflow to the vortex core. The radial velocity (u)first increases quickly from zero to its maximum value at z = dm

and then decreases to zero at the top of the boundary (z = d). Thenthe momentum equations will be solved by Karman integrationmethod with prescribed velocity profiles.

~u ¼ �aVð~rÞf ðgÞ ðA1Þ

~v ¼ Vð~rÞgðgÞ ðA2Þ

where a ¼ að~rÞ is the amplitude coefficient of the radial velocity.V ¼ x~rn is the tangential velocity above the boundary layer withn = �1 for free vortex and n = 1 for forced vortex. g ¼ ~z=d is thenon-dimensional height. The functions f(g) and g(g) are the velocityprofiles and satisfy the boundary conditions, then

f ð0Þ ¼ 0; f ð1Þ ¼ 0; fmax ¼ f ðeÞgð0Þ ¼ 0; gð1Þ ¼ 1

ðA3Þ

where e = dm/d and the radial velocity is maximum at ~z ¼ dm.Since we have considered the compressible effect induced byhigh temperature, it is reasonable to assume that chemical reac-tion does not change the velocity boundary layer structure signif-icantly. The vertical velocity is determined by the continuityequation of (16).

As indicated by Volchkov et al. [22], the radial velocity profile isinfluenced by the wall in the layer of thickness in which the radialvelocity increases from zero to the maximum value. Thus, Eqs. (11)and (12) are integrated once from ~z ¼ 0 to dm and d respectively. Inthe present work, a is taken as constant, which is the first orderapproximation. The integrated momentum equations withoutblowing are as follows:

ddd~rþ L1 þ L2=a2

L3

d~r¼ � 1

L3a2x2~r2nm@~u@~z

� �~z¼0

ðA4Þ

ddd~rþ L4

L5

d~r¼ � 1

L5ax2~r2nm@~v@~z

� �~z¼0

ðA5Þ

where

L1 ¼ �ðnþ 1ÞeI1I3 þ ð1þ 2nÞeI5; L2 ¼ eI6

L3 ¼ �eI1ðI3 � I1Þ þ eðI5 � I21Þ

L4 ¼ ðnþ 1ÞðI2 � 2I4Þ; L5 ¼ I2 � I4

ðA6Þ


where

I1 ¼ f ðeÞ; I2 ¼R 1

0 f ðgÞdg; I3 ¼R 1

0 f ðegÞdgI4 ¼

R 10 fgdg; I5 ¼

R 10 f 2ðegÞdg; I6 ¼

R 10 ½1� g2ðegÞ�dg

ðA7Þ

A.2. Laminar boundary layer solutions

The velocity profiles that satisfy the boundary conditions are as-sumed as

f ðgÞ ¼ gð1� gÞ2; gðgÞ ¼ gð2� gÞ ðA8Þ

Substituting Eqs. (A6), (A7), and (A8) into Eqs. (A4) and (A5), wehave a = 4.52 and

d ¼ ~rð1�nÞ=2 60ms

nax

� �1=2

ðA9Þ

A.3. Turbulent boundary layer solutions

The velocity profiles for turbulent boundary layer are written as

f ðgÞ ¼ g1=7ð1� gÞ; gðgÞ ¼ g1=7 ðA10Þ

where the ‘1/7’ power law for velocity distribution has been exten-sively adopted for turbulent boundary layer. The total wall frictionss in three-dimensional incompressible boundary layer is assumedto be the same as that on a plane without injection.

ss ¼12

cf 0qsU2 ¼ FRe�mqsU

2 ðA11Þ

where Re = Udm/ms, m = 1/4, F = 0.0225 for smooth surface and m = 0for rough surface. By using Eqs. (A1), (A2), and (A10), the resultantvelocity U at ~z ¼ dm is

U ¼ ð~u2m þ ~v2

mÞ2 ¼ x~rne1=7

t ½1þ a2ð1� etÞ2�1=2 ðA12Þ

where et = 1/8 for velocity profiles in Eq. (A10).Based on Prandtl’s hypothesis for three-dimensional boundary

layer, ss can be expressed by

ss ¼ ðs2s~r þ s2

s~hÞ1=2 ðA13Þ

Therefore, we have

ss~r ¼ �að1� etÞA�1ss; ss~h ¼ A�1ss ðA14Þ

where A = [1 + a2(1 � et)2]1/2.Substituting Eqs. (A10), (A11), (A12), (A13), (A14) into Eqs. (A4)

and (A5), we get a and d

a ¼ �L4

L5M þ L3 þ ð1� etÞðML2 þ L1Þ

� �1=2

ðA15Þ

d ¼ C � ~rM vS

x

� m=ð1þmÞðA16Þ

where M = (1 � nm)/(1 + m) and

C1þm ¼ � Feð2�8mÞ=7t A1�m

aðML2 þ L1ÞðA17Þ

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