10.1016 0045 7825(91)90178 9 a Numerical Study of Ignition in a Premixed Flame Burner

Computer Methods in Applied Mechanics and Engineering 90 (1991) 671-686 North-Holland

A numerical study of ignition in a premixed flame burner

Yu Song ~ Department of Mathematics, Tulane University, New Orleans, LA 70118, USA

Received 2 October 1990 Revised manuscript received 11 March 1991

This paper presents a random numerical method which combines a random vortex method and a random choice method. With the assumption of incompressibility, the equations governing the fluid motion can be uncoupled from the equations governing tlae chemical reaction. A hybrid random vortex method is used for solving the Navier-Stokes equation which governs the fluid motion. A combustion process is governed by the reaction-diffusion system for the conservation of energy and the various chemical species participating in reaction. A random choice method is used for modeling reaction- diffusion equations. The method is applied to an ignition of the Bunsen burner. Flame propagation profiles are obtained and have good agreement with experimental results.

1. Introduction

Consider a two-dimensional Navier-Stokes equation for a viscous, incompressible flow in a domain D with boundary dD:

0,u + ( u . V ) u = Re -~ Au -VP/p in D , ( l . l a )

div(u)=0 i n D , u = ( 0 , 0 ) o n 0 D , ( l . lb)

where u = (u, v) is the velocity vector, r = (x, y) is the position vector, t is time, A -V" is the Laplacian operator, P is the pressure, p is the density and Re is the Reynolds number associated with the flow.

Equations (1.1) are difficult to solve by the finite difference method, particularly at large Reynolds numbers. Chorin [1] has developed a grid-free method for modeling turbulent flow in 1973. In this, the random vortex method, the creation of vorticity along boundaries is modeled by creation of vortex blobs, discrete quantities of vorticity. The vortex blobs themselves are not vortices but elements whose unions form vortices. The motion of the fluid flow is modeled by considering the interaction of these blobs. The vortex sheet method, presented by Chorin [8] in 1978, is used near the boundary to solve the Prandtl boundary equations. This method is also a grid-free method and has the advantage that the interaction between vortex sheet elements is not singular.

Combustion is governed by a complicated interaction of process of advection, diffusion and reaction. For simplicity, we consider a single step reaction given by A----> B which describes a

1 Current address: Department of Mathematics, Indiana University, South Bend, IN 46634. USA.

0045-7825/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved

672 Y. Song, A numerical study of ignition

premixed combustible or an oxidation reaction where either fuel or oxidizer is present in small quantities. When written in dimensionless variables, the equations governing this combustion process are

O(O,Y A + ( u . V ) Y A ) = L e A' A Y , t - O K A Y a e - N A / T ,

p(O,T + ( u - V ) T ) = AT + PQAKAYA e -NAIT , (1.2)

where T is the temperature, p the density, YA the mass fraction of chemical spices A and u the velocity vector of fluid flow. Le A is the Lewis number associated with the mixture and Qa is the heat released by the reaction. The reaction rate is determined by N,a, the activation energy and K A , the pre-exponential factor. In this form, 0 <~ YA ~< 1 and in absence of external heat T u<~T~ < T ~ + e with T h = I + T , , Tu is the ambient temperature and T b the burning temperature, where e is to accommodate slight overshoot of T possible for a reaction which is much faster than diffusion.

To solve (1.2) numerically, difficulties arise due to the difference in time scales, especially in a fast reaction. A very small time step is necessary in order to maintain the stability of the solution [3]. The random choice method of Glimm [4] has successfully modeled the system of hyperbolic conservation laws [5, 6] and is suitable for solving differential equations which have steep gradients in the solution profiles. Sod [7] has shown how the random choice method can be extended to solve reaction-diffusion equations. The applications of the method can be found in [8, 9].

2. Vortex methods

Taking the curl (V x) of (1. la), we obtain the scalar vorticity transport equation

0,~ + (u.V)~: = Re -t A~, div(u) = 0 , (2.1)

where ~ = V x u = O,.v - Oyu is the vorticity. The vorticity field ~ gives rise to a velocity field u which transports it. A stream function ~b is introduced which satisfies

u -- ~y~0, v = - O ~ . (2.2)

Using the definition of vorticity, we obtain a relationship between ~ and ~:

A~ = -s~ in D . (2.3)

The vortex method is based upon this stream function-vorticity formulation. To solve (2.1), we first assume that the flow is inviscid, that is Re -~ = 0, and that there are no boundaries. Equation (2.1) reduces to the inviscid Euler equations

D~:/Ot = 0 , A~b= -~ : , (2.4)

where D / D t denotes the total derivative.

Y. Song, A numerical study of ignition 673

A c a certain time t = n At, where At denotes the time step and n is the nonnegative integer, suppose vorticity consists of N point vortices with strength c,, (,, . . . , &, centered at r1,r2,“‘7rN’ If the flow is started from rest there will be no vortices initially, otherwise the initial vorticity must be discretized. We may write the vorticity field in the form

(2.5)

where 6(r) is the Dirac delta function. The equation A+i = -@(r - ri) for the ith point vertex can be solved to obtain the stream

function

$j = -( &/27F)ln lir - ri/ . (2.6)

This is the fundamental solution to Laplace’s equation in the plane. From this the velocity field induced by the ith point vortex by (2.2) is

q(r) = - ti(Y - Yi) 5itx -4) 2nllr - ril12 ’ ‘Jr)= 27Fllr _ rill? ’ (2.7)

where r = (x, y), ri = (xi, y,). This velocity becomes unbounded as r+ ri. Chorin [l] proposed replacing the delta

function with an approximation which does not produce a singularity

(2.8)

where u is a cut-off value to be determined later. The vorticity field (2.5) is now replaced by

For llrll 2 a, the stream function could be taken from (2.6); for l\rll G 0, we need to solve

A#i = 1/(2nul)r - rill) ,

which has the solution +!+(r) = 1/(2na)llr - riII + Constant. Combining this solution with (2.6), we have

+0(r)z(-l/(2a)ln llr’rill, 1 l(2na))lr - r.11 + Constant , Ilr - rill < U ,

Ilr - riI[ 2 (T . (2.10)

The velocity field from (2.2) is


f ( - ( y - y~)/ (2~ro' l lr- r, ll), (x - x,)/(2"rrtrllr- r, l l)) , ui(r) r 2 2 [ ( - (Y - Yi)/(2~11 - rill ), (x - x i ) / (2 r r l l r - rill ))

IIr-rill< , IIr- r, II >t

(2.11)

The velocity field induced by the vorticity field ~(r) is then

N

u(r) = ~ ~iui(r). (2.12) i=l

The vorticity transport equation states that the vorticity is advected by this velocity field, i.e.,

d r J dt = u(ri) = ~ ¢/uj(ri). (2.13)

This can be approximated by Euler's method

X; '+l = X;' + A t ~ ~juj(r'i ') , y'/+' = y;' + At ~, ~:jvj(rT), (2.14) j ~ i j # i

II , tl where r,. = (xi, y~') denotes the position of the ith vortex blob at time n At. Higher order approximation for (2.13) can be found in [10], in which Sethian has used Heun's method to obtain second order accuracy in time.

To modify the vorticity field due to the effect of the diffusion, i.e. Re -! # 0, we consider the diffusion equation

0,~ = Re- ' A~:. (2.15)

A grid-free method for solving this is based on the connection between the diffusion (heat) equation and the random walk. In one dimension, this reduces to the diffusion equation

= Re-' (2.16)

The solution to (2.16) with initial data ~: = 8(x) is given by

¢(x, t) = V~4"rrt/Re exp(- x2/ (2 t /Re) ) . (2.17)

This is also the probability density function associated with a Gaussian distribution with mean 0 and variance 2t/Re. The effect of diffusion on the vortex blobs from t= n At to t= (n + 1) At is modeled by a random walk. Using operator splitting to write (2.15) as (2.16) in both x and y directions, random variables r/i, ~/2 are picked from a Gaussian distribution of mean 0 and variance 2 At/Re, and vortex blobs perform a random walk of the form

n + I n n + I _ n

Xi ~- Xi -~ 1~il ' Yi - Y i + 7qi2 • (2.18)

On the boundary, the normal condition u . n = 0, where n is an outward pointing unit normal to 0D, and the tangential condition u. s = 0, where s is the unit vector in the tangential


direction, have to be satisfied. The normal boundary condition is satisfied by adding a potential flow to the flow field induced by the vorticity field, which will be discussed in Section 4 for a specific application. The tangential boundary condition can be satisfied by creating vorticity on the boundary by the vortex sheet method; in this, the vortex sheets are introduced at points on OD, equally spaced a distance Ah apart, with vorticity u . Ah so that the induced velocity as the vortex sheets move on the OD will satisfy u. s = 0.

Combining these steps, the discrete approximation for a vortex blob motion is

n + l n x i = x i + At(ui + Up,) + ~q~t, n n y~ +1 = Y~ + At(v~ + Vp,) + r/j2, (2.19)

where (ui, o~) is from (2.11) and the (Up,, op,) potential flow given in Section 4. (7/l, r/2) is an independent Gaussian random variable with mean 0 and variance 2 At/Re.

The vortex sheet method, based on the fact that, within the boundary layer, the Navier- Stokes equation can be approximated by the Prandtl equation [11]

0,~ + (u . V ) g = Re -~ Ozyg, = -OyU, div(u) -- O,

(2.20a)

(2.20b,c)

where u = (u, o), u is tangential to 0D and o is normal to #D. The computational element vortex sheet S~ is parallel to the boundary of length Ah and centered at r,. = (x i, y~). The intensity sri of a vortex sheet is defined as

~ , = - ( u ( y + ) - u ( y [ ) ) ,

the jump of the velocity as y cross the vortex sheet Si. Equation (2.20b) can be integrated in the form

u(x, y ) = U®(x)- ~(x, "r) dr (2.21a)

and equation (2.20c) yields

v(x, y ) = -O x f f ~(x, ~')dr. (2.21b)

It can be readily seen that if ~ is known, u = (u, o) can be calculated by (2.21). Consider a collection of N segments S~ of vortex sheets of intensities ~:i and centered r~,

i = 1 , . . . , N. The velocity u i = (u~, oi) of S~ is obtained by discretizing (2.21):

where d r

u~= U®(x,)- ½~,- ~ ~jd~, (2.22)

= 1 - [xj - xi[ [Ah and the summation is taken over all j such that y] > YI' and d r > O.

v , = - ( l + - - l _ ) / A h , (2.23)

676

where

Y. Song, A numerical study of ignition

i: E. + * = _ ¢id-iyj ,

:g

d ~ = l - l x i +- ½ A h - x j l / A h , y~ =min(y~, y j ) .

The sum E+ (respectively, E ) is over Sj such that d~- ~< 1 (respectively, d~" ~< 1). From (2.22) and (2.23), the position of the vortex sheets are advanced according to

n + l n n + l n x~ =x~ + A t u i , y~ =y~ + A t e i+,l~,

where % is a Gaussian distribution with mean 0 and variance 2 A t / R e , which only applies to the y-component since the diffusion in the x-direction is neglected on the Prandtl boundary layer approximation.

3. Random choice method

Consider the reaction-diffusion system of equations

o,v + axF(v) = a Z,v + G(v) , v(x, O) =f (x) , (3.1)

where v = (v 1, v,, . . . . Vp), ~ = diag(v I, v2 , . . . , Vp), with v~ ~>0, i = 1 , . . . , p, G is a non-linear reaction or source term and the Jacobian of F, OF~/Ovj has real and distinct eigenvalues

l

If ~;---0 and G = 0, (3.1) reduces to a system of hyperbolic conservation laws

O,v + 0xF(v ) = 0 . (3.2)

A random choice scheme was proposed by Glimm [4] in order to construct solutions to (3.2) and prove existence under the condition that the total variation of the initial data is sufficiently small. The method was developed as a computational technique by Chorin [5, 12], Colella [13] and Sod [6]. For the numerical solution to (3.1), the general flow can be decomposed into elementary waves (3.2) and steady waves [7], it is in the same spirit as the method developed by Liu [14, 15] using the boundary value problem to obtain the steady waves. The steady waves reduce to the system of ordinary differential equations

o.,r(v) + G(v). (3.3)

Given the approximate solution u'i' to v(ih, nk), where h is the space step size and k is the time step length, firstly, (3.3) is solved as a two point boundary value problem in each interval [ih, (i + 1)h] to obtain a steady state solution v ~ with boundary conditions

tl v~(ih) = ui , v~((i + 1)h)= u",+l • (3.4)


The solution on [ih, (i + 1)h] is sampled at the midpoint to produce a piecewise constant approximation

n v(x, nk)=ui+,/2=vs((i+ ½)h), i h < x < ( i + l )h .

The conservation laws (3.2) are solved using Glimm's method for nk <~ t ~ (n + 1)k, using the piecewise constant states as initial conditions:

v(x, nk)=uT+t/2, i h < x < ( i + l )h .

Thus a sequence of Riemann problems is produced centered at the grid points x = ih. If the CFL condition is satisfied, that is, h/k <~ 1/21Ai(v)l, for all eigenvalues A;, i = 1, 2 . . . . , p of the Jacobian of F(v), the waves generated by the individual Riemann problem will not interact. Hence, each is solved separately and the solutions can be combined by superposition into a single exact solution, denoted by ve(x, t) defined for nk <<-t<~ (n + 1)k. Define the approximate solution for (3.1) at the next time level by

u~ '+' = ve((i + ~.)h,(n + 1 ) k ) ,

wher-, ~n is an equidistributed random variable in the interval ( -1 /2 , 1/2). By using the random walk solution to diffusion equation, Sod [7] showed that for a single

scalar coefficient equation

OtO + C OxO = P 02xO ,

the condition

k = h2/8u (3.5)

is necessary to provide the consistent amount of diffusion. For a system of equations, the extension of (3.12) requires that

k=h2[8Vl=. . .=h2/8up,

which is impossible to satisfy unless different grid spacing h i are used for each different value of v~, such that

k=h2/8pi , i = 1 , 2 , . . . , p .

The two-point boundary value problems can then be solved on the finest grid, with boundary values interpolated at that grid. The initial values for the Riemann problems are interpolated from the solution to (3.4) on the finest grid.

The computation required in solving the large number of boundary value problems could be


prohibitive, however, Sod [16] proposed the use of a dictionary. Equation (3.3) can be reduced by omitting the advection term

+ G ( v ) = 0 ;

this is independent of x and t and the solution is only needed at the midpoint. Therefore it is necessary to consider different possibilities for the boundary values (3.4) and the solution for other boundary values can be obtained by using dictionary and linear interpolations.

Let v = (YA, T) , then (1.2) can be written as

Otv + (u . V ) v = -~ Av + G ( v ) , (3.6)

where -~ = diag(Le~ I , 1), (3.6) is an equation of the form (3.1) except that it is not in conservation form; however, u is known at each grid point from the solution of the Navier-Stokes equation, so u can be treated as piecewise constant on intervals centered at the grid points. The CFL condition becomes

k / h ~ 1/2maxllull.

In the applications, a general boundary condition often considered is

av + c Or~On = b(t) on OD, (3.7)

where a and c are constants. To demonstrate how the boundary conditions are implemented, without loss of generality, consider the x-sweep with OD = {x Ix •0}. Condition (3.7) becomes

ao(O, t) + c OxV(O, t) ffi b(t) on OD .

Approximating O, by a centered difference, we have

aug + c(u' l - u" l) / 2 h = b" , (3.8)

with t = nk , b " = b(nk) . The grid point corresponding to i = - 1 lies outside the physical domain. We may solve (3.8) for u" ' - - l '

u"_ 1 = u'~ + (2h/c) (aug - b") . (3.9)

In order to find the approximate solution at the next time level at the boundary x = 0, we need to establish a left and right state for the Riemann problem associated with this point. The


inside the boundary layer, use vortex sheets

. . . . . . . . . . . . . . . vo .x ,ooo

vortex blobs

................................ iC)'~O')" .................. ; ~ ......................................................................... Centerline (imaginary)" ~]D2

eymmetrlo to up-half, no computation

Fig. I. Computational domain for the Bunsen burner.

appropriate two-point boundary value problem is solved on the interval [ -h , 0] with the left boundary condition given by (3.9) and the appropriate two-point boundary value problem is solved on the interval [0, h]. If c = 0, the solution is known at x = 0 and a Riemann problem is not needed at this point.

4. Application to the ignition of the Bunsen burner

The first laboratory premixed-flame burner was invented by Bunsen around 1855. Many experimental studies of premixed burner have been done since [17]. The difficulty of the numerical study of the ignition lies in modeling such an ignition laappening in a very short time.

For simplicity, we consider the cross section of a rectangular burner, so that the fluid motion is governed by two-dimensional Navier-Stokes equations, and hence can be modeled by the vortex method. Furthermore, we assume that the flow is symmetric about the centerline, so that we only need to study the flow motion of the upper half of the cross section thus saving almost half of the computations (see Fig. 1).

4.1. Hydrodynamics

We use the hybrid vortex method for hydrodynamics. Within the boundary layer the computational elements are vortex sheets, outside the boundary layer we use vortex blobs (see Fig. 1). The thickness of the boundary is assumed to be proportional to 1/V Re. Sheets moving out the boundary layer become blobs with ~bloh = Ah * ~shcct; however, sheets which cross the boundary are reflected back to preserve the vorticity.

On the boundary 0D I, the fluid must satisfy the no-slip condition and normal boundary condition u. n = 0, while on the boundary d D 2 the fluid only needs to satisfy the normal boundary condition since it is an imaginary boundary. For the normal boundary condition, we use a single layer potential since the two pieces of boundary are disconnected and ODI is finite in length while OD E is infinitely long.

The goal is to find the velocity field up = V4~, where


1 f Jo a(q)ln R(q) dq, (4.1) ~(r) = ~

where R(q)= ( x - xq)2+ (y _yq)2, q = (X¢, yq) is a point on the boundary OD, a(q) is a single-layer source and satisfies the integral equation

1 f~ ot(q') 0,,(In R(q')) dq'= -2u¢. n. (4.2) a(q) = ~ D

We divide the boundary OD into M segments of equal length Ah with centers q~ = (xq,, yq), i= 1 , . . . , M . The integral equation (4.2) can be approximated by a system of linear equations. A source of intensity 1 at qi induces at the point qj, j ~ i, a velocity field with components

1 Xqj -- Xqi 1 Yq/-- Y¢~ U,(i, j )= --2",r I[q/ - qil[ 2 ' U2(i' j) = 2"tr Ilq/- q,l[ 2" (4.3)

Approximate a(q) by the M.component vector a = ( a ( q l ) , . . . , a(qM)), which must satisfy the matrix equation

A a = b ,

where the components of b are the vector values of - u¢. n evaluated at the points q~, and the matrix A has the components

{ a,~ -~ u ~ ( i / ) n l + u 2 ( q ) n , , i ~ / , al, 1 /2Ah, i - - 1 , . . . , M .

where n = (n 1, n2) is the unit outward normal vector at q~ = (xq,, yq,) on 0D. One can then derive the discrete form of Up =Vd)using (4.1),

M

up(r)- ~ up(i),

where

f (1/2~)~(q,)r(q,)/llr(q,)ll ~ , II'<q,)ll ~ Ah/2, Up(i)= [ 1/(2 Ah)a(q,)n(q,), Ilr(q,)ll < Ah/2, (4.4)

where r(q~) is the vector joining the point r in D to the point q~ on OD. At the edge of the nozzle, a shear mixing layer with up stream velocity 0 is formed. Several

studies of a turbulent mixing layer have been made using the random vortex method. Simulation of a shear layer flow by Ashurst and Meiburg [18] showed good agreement with experimental results for mixing layer growth and Reynolds stress distribution. In the case of Bunsen burner, when x, t are large enough, two mixing layers meet on the centerline. We


consider that the thickness of the mixing layer is proportional to Vx-/Re and time t, and use average free stream velocity in the mixing area.

The numerical parameters are chosen to be M = 40, L = 10, Ah = 0.1, U= = 25.0, R = 0.25, which is the half-width of the nozzle and the cut-off value tr = Ah/~ [19]. The Reynolds numbers are chosen from 500 to 10000. The time step size of the vortex method is chosen as At =0 .1 .

Figure 2 depicts the streakline plot of blob motions at times t = 4.0, 6.0, 8.0 and 10.0 for Re = 5000, Figure 3 depicts the velocity profile for corresponding times, which is substracted by average free stream velocity. From Fig. 2 we can see how the vortices along the centerline are developed and grow with the time. The blobs just coming out of the boundary layer are forming a small vortex which grows fairly fast, and later becomes the large leading vortex. As the leading vortex grows and moves, smaller vortices are formed and moved with it. This result has good agreement with the experimental observations.

(a)

, ". . , .'~,~" .',.: (b)

(c)

- - - - • . " . . " • ~ b ~ % ~ . "~.

. . . ~ ~ . ' J ¢ . ~ ; ; ~ . : . ' 4 ~ . ~ ' , , . : : : ' ~ . ~ . " ~ . . . . . : . ' " • " . " '~ ' . ' / " . ~ " " " . . " ~ t . , , .~X: ' . . ' . '~ l . . '~ . . ' . p . . : . . . : .~ . . ' ' . . . ' . . , . . , . : . " : ; ~ : ' . - , : : : ~ , C ' . , ' . " ~ , ' . ' ° .

Fig. 2. Streakline plot of vortex biob motion in the Bunsen burner. (a) t = 4.0, (b) t = 6.0, (c) t = 8.0, (d) t = 10.0.


::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::

~ ! ! ! ~ i i ! !~ El ~ i i ! i i i i i i i i i i i i ! ! i i i ! ! i i i i ! ! i i i ! ! i i i ! ! ! ! ! ! i i ! ! ! ========================================================================== ~]~- -~ .~- : - : " l - i - ~ f - - _ : ======================================================== (a) ,.,'~:$=~,

::::::::::::::::::::::::::::::::::::: =-":=~'.. : ::::::::::::::::::::::::::::::::::::::

,.,.,~===,,..===="=== ..... : ....... : : •,, ......... ~... •,,,, ..................................

~=='=-~==~==.=:.-[==:~i!i:i ii ! ii i ! i i iF"-:iii!~iiiii~!!i~!!!~!!!!!!'!i!~!i!iii!iiii • ....,s,. o ~. ~..~., o .... o.ooo.,.oooo,, ..... o ........o ~a • ~,' p ~,, • ...............................

(b) ,==J'~,=,

~ ' : : : : = : : : : : : : : : : : : - - = - " = - = = - ~ - : : : : : : : ~ : - - ~ = L ~ . : : = : : : : = : : : : : : : : : " ~ , , , , . , . . . . . . . . . . . ' " . ' . . " : . . . . . . .

. . . . . . . . , . , . _ _ 7,._.-..,.,~..~ , , . , - ~ - - .,,,,,,,~, ~ . . , . . . . . . . . . . . . . .

(c) ,,,,_.,.,~, H#UKINL~N YKCTON

. . . . = = - = = = = = ~ = - - = = = = . : - = = = o . = = , : : : : : H =HHHHF..H"HHE='HHHHHHHH" ":-=HqH~:H~H~'HHHH~HH~ HHHHHHHH~HH~H "'Hii:

Fig. 3. Velocity profile in the Bunsen burner. (a) t = 4.0, (b) t = 6.0, (c) t = 8.0, (d) t = 10.0.

4.2. Combustion

For combustion, the problem is considered in the domain

D = {x, y I -l~<x~< 100R, ,0~<y ~ R} O {x, y 10~x~< 100R, R~<y ~< 10R},

with the boundaries (see Fig. 1)

Y. Song, A numerical study of ignition

ODo={x, ylx=-l,O~y<-R}, aD~={x, yl-l<-x<-O,y=R}, aD 2 = {x, y I - 1 ~<x~< 100R, y = 0 } .

A rectangular mesh with grid spacing h = 0.05 is set up, leading to the combustion time step

683

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JT I I I ) I I I I I t l I I t I ~ I I IT~ I I I I I I ~ "~ ' I ' I ' TT ' [ ' I ' ~T I ]T I I ~ F T T ' r T T ~ ~

(d)

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b

i [ i t l l i i l i l l l l l l lllltl11111111F 1 I ~TO~IR ~ l . i I[0 1.31~ ¢~IaTOLIR IHTI~Yil. Of l . i i l PT(i,$1, I.&3@47

Fig. 4. Flame propagation temperature contour in the Bunsen burner. (a) t=0.01250, (b) t =0 .0250 , (c) t = 0.03125, (d) t = 0.0500.


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, 4 . 5 . 0 . 7 , 8 . 9 1 . g

X

Fig. 5. F lame burning velocity in the Bunsen burner. (a) U~ = 25, (b) U~ = 50; with A: at t --- 0.012, B: at t -- 0.024, C: at t = 0.036, D: at t = 0.048.

Y. Song, A numerical study of igplition 685

k = 0.0003125 by (3.5). Other parameters in (1.2) are chosen to be K A

NA=4, T u = 0 . 2 a n d L e A = I . The initial conditions are

= 2.5 × 103, QA = 1,

Y A = 0 , T = T u = 0 . 2 , every where.

The premixed gas is ignited at the nozzle edge {x =0,0 ~< y ~< R} by raising the temperature to T = T b = 1.2 and holding at this temperature until t = 0.003125.

The boundary conditions are

Y A = I , T = I",, onOD o,

to keep the gas supplies and

OYa/On = 0, OT/On = 0 on all other boundaries.

Figure 4 depicts the flame propagation contours at times t = 0.0125, 0.025, 0.0375 and 0.05. After time t = 0.05, the flame is stabilized. The height 0fthe cone depends on the free stream velocity. The flame burning velocity is calculated by measuring the distance between the temperature profiles at same time interval. Figure 5 depicts the flame velocity for two different free stream velocities U~ = 25 and U~ = 50 at several different time level. The numerical results have good agreement with experimental results [17, 20]. The maximum velocity occurs at the center of the burner and at time t = 0.024. The velocity decreases to zero at the edge of the nozzle. As time reaches t = 0.48 the burning velocity becomes close to zero and the flame is stabilized.

Acknowledgment

The author wishes to thank Professor Gary A. Sod for his many helpful discussions and comments.

References

[1] A.J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973) 785-796. [21 A.J. Chorin, Vortex sheet approximation of boundary layers, J. Comput. Phys. 27 (1978) 428-442. [3] R.A. Dwyer and G.R. Otey, A numerical study of the interacti0a of fast chemistry and diffusion, AIAA J. 7

(1979) 606-617. [4] J. Glimm, Solution in large for nonlinear hyperbolic systems 0f equations, Commun. Pure Appl. Math. 18

(1965) 697-715. [5] A.J. Chorin, Random choice solution of hyperbolic system, J. C0mput. Phys. 22 (1976)517-533. [6] G.A. Sod, Numerical study of converging cylindrical shock, J. Fluid Mech. 83 (1977) 785-794. [7] G.A. Sod, A random choice method with application to reacti0~.diffusion system in combustion, Internat. J.

Comput. Math. Appl. 11 (1985) 129-144. [8] G.A. Sod, A dictionary approach to reaction-diffusion systems with nonlinear diffusion coefficients, Internat.

J. Comput. Math. Appl. 13 (1987) 771-783.


[9] Y. Song, Numerical simulation of flame propagations in circular cylinders, Appl. Numer. Math. (1991) to appear.

[10] J.A. Sethian, Turbulent combustion in open and closed vessels, J. Comput. Phys. 54 (1984) 425-456. [11] H. Schlichting, Boundary Layer Theory, 7th Edition (McGraw-Hill, New York, 1979). [12] A.J. Chorin, Random choice methods with application to reacting gas flow, J. Comput. Phys. 25 (1977)

253-272. [13] P. Colella, Glimm's method for gas dynamics, Siam J. Sci. Statist. Comput. 3 (1982) 76-110. [14] T.P. Liu, Quasilinear hyperbolic systems, Commum. Math. Phys. 68 (141) (1979). [15] T.P. Liu, Nonlinear stability and instability of transonic gas flow through a nozzle, Commun. Math. Phys. 83

(243) (1982). [16] G.A. Sod, A flame dictior,',ry approach to modeling unsteady combustion phenomena, Math. Appl, Comput.

3 (1984) 157-182, [17] B. Lewis and G. von Elbe, Combustion, Flames and Explosions of Gases, Third Edition (Academic Press,

New York, 1987). [18] W.T. Ashurst and E. Meiburg, Three-dimensional shear layers via vortex dynamics, J. Fluid Mech. 189 (1988)

87-116. ]191 A.Y. Cheer, Numerical study of incompressible slightly viscous flow past blunt bodies and airfoils, Si~am J. Sci.

Statist. Comput. 4 (1983) 81-92. [20] K.K. Kuo, Principles of Combustion (Wiley, New York, 1986). [21] C. Anderson, A method for local corrections for computing the velocity field held due to a distribution of

vortex blobs, J. Comput. Phys. 62 (1986) 111-123. [22] G.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge Univ. Press, Cambridge, 1967).

10.1016 0045 7825(91)90178 9 a Numerical Study of Ignition in a Premixed Flame Burner

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