REFERENCE

NBS

PUBLICATIONS

NBSIR 86-3334

AlllDS blb077

ependent Simulation of

>cale Turbulent Mixing and

NATL INST OF STANDARDS & TECH R.I.C.

A111 0261 6077

Baum, Howard R/Time-dependent simulation QC100 .U56 NO.86-3334 1986 V19 C.1 NBS-P Reaction

Howard R. Baum, Ronald G. Rehm

Daniel M. Corley and Daniel W. Lozier

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Applied Mathematics and

Center for Fire Research

Gaithersburg, MD 20899

February 1986

Issued November 1986

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

-QC

100

. 1)56

^ 86-3334

1986

NBS

RESEARCH

INFORMATION CENTER

NBSIR 86-3334

TIME-DEPENDENT SIMULATION OF

SMALL-SCALE TURBULENT MIXING AND

REACTION

Howard R. Baum, Ronald G. Rehm

Daniel M. Corley and Daniel W. Lozier

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Applied Mathematics and

Center for Fire Research

Gaithersburg, MD 20899

February 1986

Issued November 1986

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Director
-Hm

.'“ .

' '"' ZiSa,, ;

#

mm\

m.MOri^llM^ TH30H3,^M-3

JfM3jiwyr aj^off^jAMe

wMTOAan HP'-' II " ' WWfiijlliiL ' l l/i

m.

f , ^1 M ^ 'r«Q UfU .M

}m% fymkftmmSk^u

OM p, >i4£>

'Mr

•-'• M

lim.

.33MISMM<^^ TMaM7fCr't'33 JLU

'^UMwaa ,wMnA «•!.<• .•OAAWv^TT «U UAST^ il««OfTA«
ABSTRACT

A matheinati cal model of the local transient diffusion-controlled reaction

between initially unmixed species is presented. It is intended ultimately as

a computational "molecule" to be imbedded in direct simulations of larger

scale reacting flows. The model consists of an interacting three-dimensional

strain vortex field which exactly satisfies the Navi er-Stokes equation, an analytically determined Lagrangian representati on of the mixing process and

convection-diffusion equations for the reacting species in

Lagrangian coordinates. The length scale established by the stretching of the

vorticity field is shown to be directly relatable to the Kolmogoroff scale if

the local strain rate has a scale consistent with laminar boundary layer

theory coordinates. Results are shown for the flow pattern and the induced

mixing. An analytical solution to the convection-diffusion equation governing

the diffusion-controlled reaction is derived. The solution is valid for large

Schmidt number and describes the evolution of any initially two-dimensional

configuration of reactants. A special two-dimensional case of this model, in

which vortex strain is excluded and fuel and oxidizer initially occupy

adjacent half-spaces, is also analyzed. This problem was originally

formulated by F. Marble, who treated it in a very different manner from that

described here.

INTRODUCTION

The numerical simulation of turbulent reacting flows has become increasingly

viable in recent years. More powerful computers, advances in computational

techniques, and clearer physical insight, have combined to make possible the

calculation of at least some aspects of turbulent combustion directly from the

governing equations.

Despite this progress, many serious problems remain unsolved. Although large

scale features of turbulent flows seem amenable to computation, the prediction

of local reactant consumption and energy release rates requires a description

of the flow at length and time scales sufficiently small for molecular

diffusion to be effective. These scales are well below the resolution limit

of any finite difference simulation; particularly since the inherently threedimensional mechanism of vortex stretching must be accounted for in the

description of phenomena at this level of detail. In fact, it is the vortex

stretching mechanism that is responsible for the fact that the ratio of

dissipation scale to macroscopic length is proportional to (Re)~3/4 -j p fully

developed turbulent flow. Here Re is a Reynolds number based on macroscopic

velocity and length scales. The description of flow, mixing, and diffusion of

reactants at this scale is the subject of the present research.

The mathematical model presented here is intended ultimately as a

computational "molecule" to be imbedded in direct simulations of larger scale

reacting flows. In this spirit, it is assumed that a three-di mensi onal

time-dependent calculation capable of resolving the largest length and time
scales of the relevant phenomena, is being performed. The resolution limits

of this calculation are too coarse to directly calculate the small scale

mixing and diffusion which actually control the reaction. The large-scale

simulation does, however, define the environment in which the phenomena of

interest evolve.

Progress made this year has been reported at the 1985 AFOSR/ONR Contractors'

Meeting on Turbulent Combustion on 23-25 July in Pasadena, California in a

talk entitled "Time-Dependent Simulation of Turbulent Combustion" by

H. R. Baum and R. G. Rehm; and at the Eastern States Section/Combustion

Institute Fall 1985 Meeting on November 4-6, 1985 in Philadelphia,

Pennsylvania in a talk entitled, "Time-Dependent Simulation of Small-Scale

Turbulent Mixing and Reaction", by H. R. Baum, D. M. Corley and R. G. Rehm. A

paper based on this work entitled, "Time-Dependent Simulation of Small-Scale

Turbulent Mixing and Reaction", by H. R. Baum, D. M. Corley and R. G. Rehm,

has been submitted to The Twenty-First Internati onal Symposium on Combustion,

which is to be held in Munich, West Germany on August 3-8, 1986. A copy of

this paper appears as Section II. In Section III additional

analyti cal /numeri cal research on the special two-dimensional case described as

"the Marble problem" is presented; this is work in progress and will be

published later when it is completed.
SECTION II

TIME-DEPENDENT SIMULATION OF SMALL SCALE

TURBULENT MIXING AND REACTION

Howard R. Baum, Daniel M. Corley

and Ronald G. Rehm

National Bureau of Standards

Gaithersburg, MD 20899

(9) Fire

(11) Fluid Dynamics

(18) Modeling and Scaling
Time-Dependent Simulation of Small-Scale

Turbulent Mixing and Reaction

Howard R. Baum, Daniel M. Corley

and Ronald G. Rehm

National Bureau of Standards

Gaithersburg, MD 20899

ABSTRACT

A mathematical model of the local transient diffusion-controlled reaction

between initially unmixed species is presented. It is intended ultimately as a computational "molecule" to be imbedded in direct simulations of larger

scale reacting flows. The model consists of an interacting three-dimensional

strain vortex field which exactly satisfies the Navier-Stokes equation, an analytically determined Lagrangian representation of the mixing process and

convection-diffusion equations for the reacting species in

Lagrangian coordinates. The length scale established by the stretching of the

vorticity field is shown to be directly relatable to the Kolmogoroff scale if

the local strain rate has a scale consistent with laminar boundary layer

theory coordinates. Results are shown for the flow pattern and the induced

mixing. An analytical solution to the convection-diffusion equation governing

the diffusion-controlled reaction is derived. The solution is valid for large

Schmidt number and describes the evolution of any initially two-dimensional

configuration of reactants.

2
INTRODUCTION

The numerical simulation of turbulent reacting flows has become Increasingly

viable in recent years. More powerful computers, advances in computational

techniques, and clearer physical insight, have combined to make possible the

calculation of at least some aspects of turbulent combustion directly from the

governing equations. For example, finite difference simulations of inviscid

turbulent mixing layers in two dimensions and axially symmetric jets have

been performed at fairly high resolution. Less detailed calculations of

simple transport by buoyant plumes in enclosures have been carried out using

finite-difference and particle-tracking methods in both two and three

3 4 dimensions » . Finally, the random vortex method has been combined with a

flame front algorithm and finite difference techniques to produce elegant

simulations of preialxed turbulent combustion in two dimensions'*^.

Despite this progress, many serious problems remain unsolved. Although large

scale features of turbulent flows seem amenable to computation, the prediction

of local reactant consumption and energy release rates requires a description

of the flow at length and time scales sufficiently small for molecular

diffusion to be effective. These scales are well below the resolution limit

of any finite difference simulation; particularly since the inherently threedimensional mechanism of vortex stretching must be accounted for in the /

description of phenomena at this level of detail. In fact, it is the vortex

stretching mechanism that is responsible for the fact that the ratio of

dissipation scale to macroscopic length is proportional to (Re)~3/4 in fully

1 3 developed turbulent flow . Here Re is a Reynolds number based on macroscopic

velocity and length scales. The description of flow, mixing, and diffusion of

reactants at this scale is the subject of the present paper.

3
The mathematical model presented below is intended ultimately as a

computational "molecule" to be imbedded in direct simulations of larger scale

reacting flows. In this spirit, it is assumed that a three-dimensional

time-dependent calculation capable of resolving the largest length and time

scales of the relevant phenomena, is being performed. The resolution limits

of this calculation are too coarse to directly calculate the small scale

mixing and diffusion which actually control the reaction. The large-scale

simulation does, however, define the environment in which the phenomena of

interest evolve. This environment can be defined as follows: Classical

analyses of the kinematics of an arbitrary flow^ show that the velocity field

in the neighborhood of any point can be decomposed into a uniform

translational velocity, a rigid body rotation, and a pure strain. This

decomposition can be readily performed for most numerical simulations.

Similarly, the thermodynamic state expressed in terms of the pressure and

conserved scalar quantities, (e.g. Schvab-Zeldovich variables) are assumed

known. At high Reynolds numbers, the conserved scalars are primarily

convected at computationally resolveable scales. It is easy to track the

coordinates of a material point moving with the computationally resolveable

fluid motion > . We now choose a local coordinate system in this frame of

reference.

Viewed from the moving coordinate system, the ambient velocity field consists

of a linear strain field and a vorticity, with a specified pressure. There

are two possibilities for the conserved scalar fields. If the nominal

interface between reactants is macroscopically determined then at the

resolution limit the interface is taken to be initially planar. If It Is of

subgrid dimensions, then its initial shape is assumed known. Examples of both

4
types are shown below. The fundamental problem can now be defined as follows:

Given initial conditions for the "local ambient state" (i.e. strain rate,

vorticity, pressure, and conserved scalar configuration), determine their

subsequent local evolution in space and time.

This problem is broken down below into three subsidiary problems. First, the

construction of the local velocity field produced by the effect of the strain

on the vorticity field. Next, the calculation of a local Lagrangian

coordinate system so that small scale mixing of reactants can be separated

from molecular diffusion. Finally, solution of the evolution equation for

conserved scalars in the Lagrangian coordinates. In each case simplifications

in geometry and thermophysical properties are introduced in the Interest of

maintaining an essentially analytical approach yielding readily interpretable

results. The consequences of these simplifications are assessed in a

concluding section.

A related set of investigations has been performed by Karagozlan and

3 9 Marble » . However, both the general approach and the results obtained by

them are significantly different from those reported below. More detailed

comments are included in subsequent sections.

THE VELOCITY FIELD

Consider a flow in which the vorticity is locally aligned in the x-direction.

Fig. 1, and subjected to a uniform strain of strength a. Then, if v^, Vj-, and

V 0 are respectively the axial, radial, and swirl components of velocity, the

Navier Stokes equations for a constant-property incompressible fluid have

5
solutions of the form:

Vx = ax ; Vf = -ar /2

9v0

3t

Equation (1) is the swirl component of the momentum equation for the fluid

whose kinematic viscosity is v. The axial vorticity field o)(r,t) is defined

in terms of V0(r,t) by the expression

m ^ (rv0) . (2)

r 3r

The linearity of Eqs. (1) and (2) suggests seeking a solution satisfying the

initial condition

u)(r,0) = ro J^ 6 (r-ro) (3)

Here 6 denotes the Dirac delta function. The resulting solution for V 0 is in

effect a Greens function for Eq. (1), in that its convolution with any iniLiai

axial vorticity distribution generates the corresponding solution to the

Navier-Stokes equations. As such, it contains all the phenomena implied by

the equation; i.e. the convection, stretching, and diffusion of vorticity.

To proceed, it is convenient to introduce the following dimensionless

variables

AAA

V 0 = ro w(r,t)

A r = ro r (4)

t = (a)“l t

6
Then, define a new dependent variable v, and independent variables p,T in the

form

A A p = r exp (t/2)

A T = e[exp(t)-l] (5)

A A w = exp(t/2) v(p,t)

In terms of these variables, Eqs. (l)-(3) become:

_n = 1 ^ ( pax ] 9t p 9p 3p

X(p,0) = 6(p-l) (6)

where

X =

9 (pv)

P 9p

The variable p is the characteristic coordinate of the left hand side of

Eq.(l), while X is a reduced vorticity. The parameter (e)“^ is a local

Reynolds number based on the strain rate and the initial radius of the vortex

tube whose evolution is described by X. Note that the first of Eqs. (6) is the

heat conduction equation with cylindrical symmetry. The solution is readily

obtained by taking Laplace Transforms in The result is:

X = (2t) 1 exp [-(l+p2)/4T] Io(p/2t)

P V = (p)"l

/

X (Po,t) PodPo

0

(7)

Here, Iq is the modified Bessel function of order zero. Equations (5) and (7)

constitute the desired solution. They describe the evolution of an initial

shell of vorticity concentrated at radius ro into the classical steady state

stretched vortex, whose velocity is given by:

7
w(r,<») = (r) 1{ l-exp[-(r)2/4e ]

}

( 8 )

The dimensionless swirl velocity w defined by Eqs. (4), (5), and (7) is

plotted in Fig. 2 for e = .002, corresponding to a local Reynolds number of

500. Note that the stretching mechanism acts to steepen the velocity

gradients with time until the viscous effects become strong enough to provide

the steady-state balance described by Eq. (8). The internal length scale il In

these solutions is

I = (v/a)l/2 (9)

Thus, (e)l/2 is the ratio of the Internal length scale to rQ, that Imposed by

the largest scale flow. The actual magnitude of I depends on the imposed

strain rate ot. It is interesting to observe that in a laminar boundary or

free shear layer characterized by a macroscopic length L and velocity U, the

local strain rate a can be characterized by

a ~ U(UL/v)1/2/L (10)

Inserting the estimate given by Eq . (10) in Eq . (9), the Internal scale I can

then be related to macroscopic parameters as:

l/L ~ (Re)"3/4

( 11 ) Re = UL/v

Equation (11) shows that under circumstances where a fragment of laminar shear

layer can stretch a vortex, the resulting flow will quickly collapse to

the Kolmogoroff or dissipation scale Thus, the solution described here

offers a concrete dynamic realization of how such a small scale local flow can

develop.

The fact that the vorticlty evolution can be related to Lhe cylindrical heat

conduction equation under the symmetry conditions implied by Eqs. (1) has been

8
8 9 10 12

noted by several authors » > > . However, when solutions investigated are

8 9 12 restricted to Gaussian forms > » in the x,p coordinate system, a rich

variety of phenomena, including the collapse process Itself, is lost. This

has further Implications which arise in the study of mixing and diffusion,

which are considered next.

REACTANT MIXING

Assume that the fluid consists of reacting species "A" and "B", which are

initially separated on either the macroscopic scale L or the local scale ro.

For simplicity (and consistency with the flow field) the reaction is taken to

be Isothermal. Let n and M be the stoichiometric coefficient and molecular

mass of each species, and Y be the corresponding mass fraction. Then, if Y(0)

denotes the initial unmixed mass fraction of each species, it is possible to

define a normalized Shvab-Zeldovich variable Z as follows:

Z = 2f(YA/nAMA)-(YB/nBMB) - (1/2) [ (YA(0)/nAMA) - YB(0)/nBMB]}

•[(YA(0)/nAMA) +

( 12 )

The variable Z satisfies the following equation and initial condition:

az + u*VZ = DAZ (13)

at

at t = 0 Z = +1 (Species A) (14)

Z = -1 (Species B)

Here u is the velocity vector defined in Eq. (1), D is the diffusivity of each

species, and A the Laplacian operator. The Initial species distributions are

assumed to be independent of the axial coordinate x. This restriction is not

essential to the analysis which follows, and it greatly simplifies the algebra

9
required to obtain results of interest. Then, introducing the dimensionless

variables defined by Eq. (4), Eq, (13) can be written in the form:

3Z r 3Z w 3Z el 32z 1 3Z 1 32z I (15)

A A A IaaAA I 3t 2 3r a r 39 Sc | 6r^ r 6r r^ 602

|

Sc = v/D

The variable 9 in Eq. (15) is the azimuthal coordinate, increasing in the

A direction of the swirl velocity component w (see Fig. 1), while Sc is the

Schmidt number.

The left-hand side of Eq. (15) can be thought of as governing the small-scale

mixing of the reactants by the local velocity field, while the right-hand side

controls the true molecular diffusion. It is desirable to cast the solution

in a form which illustrates these two effects explicitly. This can be

accomplished by introducing Lagrangian coordinates x,p,(J). The variables t and

p are defined in Eq. (5), while (|) is defined as:

<j) = 9 - (51/ae) 4'(p ,t)

T (16)

iKp,t) = (p)“l

/ v(p,Tq) dTo

0

Equation (15) in Lagrangian coordinates becomes:

3Z , 3^Z 1 3Z 1 — = (Sc) 1 { —- + - — + [- + (Jl/ae)

3t 2 p 3p 2 3p p i-r]3p

- (Q/ae)

8il; 3^Z 3 3ii; 3Z 1 3i|; 3Z

[ — + — ( — —) + - — —1

}

3p 3p3(j) 3p 3p 34> P 3p 3(j>

(17)

10
Note that in this system of coordinates there is no longer any convection, as

the mixing has been incorporated in the transformation from Lagrangian to

Eulerian coordinates. Since the quantity v(p,r) is given analytically by Eqs.

(7), the transformation and any of its derivatives can be evaluated readily to

arbitrary accuracy. This transformation is easily generalized to three

dimensions, with little added complexity to Eq. (17).

In the absence of diffusion (Sc-»-«>), Eq. (17) reduces to the statement that the

initial configuration of the reactants does not change in Lagrangian

coordinates. Thus, the mixing process can be followed by mapping each point

in the Eulerian coordinates at a given time into the Lagrangian space and

determining which species occupied that point originally. This has been done

for two examples, shown in Fig. 3 and Fig. 4. Figure 3 shows the mixing of

two species initially occupying separate half spaces displaced one half the

initial vortex radius from the vortex center. This is an example of a

macroscoplcally resolveable interface. The mixing process in this case is

seen to be a relatively large scale engulfment followed by a continuous

elongation of the Interface between species. Figure 4 shows a "subgrid scale"

interface, a circular blob of one reactant displaced one half the vortex

radius with respect to the vortex center. This time the mixing seems to tear

apart the blob, with two highly distorted fragments connected by a slender

filament. In each case, the local Reynolds number (e)“^ was taken to be 500,

and the ratio of vorticity to strain (.Q/a) is 2. While the Infinite Schmidt

number limit is unrealistic for most fluids, it is an excellent approximation

for typical smoke aerosols.

11
MOLECULAR DIFFUSION EFFECTS

Now consider the solution of Eq. (17) for finite values of Sc. In the

Lagrangian representation, the vorticity acts to enhance the diffusivity in

the azimuthal direction in Langranglan coordinates. The degree of enhancement

varies greatly with p and T. Fig. 5 shows the variation of (J2/ae)i|; with r.

A At any Instant of time, however, r and p differ only by a multiplicative

constant, so each individual spatial profile has the same shape in both

systems of coordinates. The analogous plot in Lagrangian space would show

each profile centered at p = l. The result is that the gradient of i|; is very

large in an annular ring centered about p = l. Inside this ring, ij; vanishes

exponentially, while outside it decays algebraically (Fig. 6).

This distribution arises from the fact that the vorticity is confined to the

annular ring. Thus, there is no angular deformation inside the annulus, while

the deformation outside the ring is irrotational . The ring thickens with time

under the influence of viscosity, until the inner core disappears. Soon after

this ocurs; all the reactants present within the ring (now a disc) will be

consumed

.

Since the coefficients in Eq. (17) are independent of the angular variable <> ,

it is useful to express the solution in terms of a Fourier series.

Z = I Znc(p ,T)i;nc(P cos (n^)

n

+ I Zns(P ,T)Cns(P »T)sin (n<J>). n

(18)

The quantities and Z^s sre defined to be the Fourier coefficients that

would occur in the absence of vorticity. This corresponds to the solution for

diffusion in a uniform strain field, a problem which has already received

attentlon^'^ » ^5,

12
/ dpipi exp [-Sc(p2+p2j^)/4x] i^(Scppi/2t)| (19)

P 1 » 0)

vZns (P 1 > 0)/

Here, are the Modified Bessel functions of order n and Znc(Pl>0)> Zyis(pi, 0)

are the Fourier coefficients of the initial distribution of reactants.

To proceed further, it is necessary to either compute the remaining factors

?ns numerically or make an additional assumption to obtain analytical

results. In the present work, the latter choice is made and it is assumed

that the Schmidt number Sc is large. Under these circumstances, the diffusion

is relatively ineffective in changing the initial distribution except in the

A annular region. Since the whole vortex collapse process occurs in times t

0(1), then T<<1 for the times of interest. For small t, it can be shown that

^ has the form:

>1^ = (p)"2 T 4'(n,T)

n = (p-1)/2(t)1/2

( 20 )

The quantity 'I in Eq. (20) is a smooth bounded function of n which tends to

unity as n->^ and vanishes exponentially as

For convenience in what follows let the expansion given by Eq. (18) be

temporarily recast in complex exponential form: /

Z = ^ exp (in(}))Z„(p ,T)?n(P »t) (21) n

13
Substitution of Eq. (21) into Eq. (17) yields:

9?n 9^?n 3^n

Zn = (Sc)-l {Zn + (2

9t 3p2 6p

3ip 9 Zn?n

-in(S^/ae)[-- (— {ZnCnl + ) 9p 9p p

Zn 9^n 9if; + __ ) _ n2 (j^/ae)2 Z^Cn ( " )^

P 9p 9p

9 94^ + -- (Zn?n -- )]} (22)

9p 9p

The independent variables are now rescaled as follows:

T = [ae( Sc) ^/2 /q ]t^*

p-1 = [ae(Sc)l/2/s^]p*

Equation (20) then implies the scaling

dJp — = q = [ae(Sc)^/ 2

/j^] 1/2 q*(t*,p*)

9p

(23)

(24)

Since the reactants are initially assumed to be resolveable on a scale of

order one in p, then Z^ and its derivatives remain 0(1). When the scaling

laws given by Eqs. (23) and (24) are used in Eq. (22), the result Is:

Zn [ + n2(q*)2 = 0(Sc)“l/2 (25)

9t*

Thus, the leading terms in the expansion for large Schmidt number can be

written in terms of p and T as:

T 94^ ?nc = ^ns = exp {-n2(Sc)“l (Q/ae)^ f ( — )2 dx} (26)

0 9p

Equations (18), (19), and (26) then constitute the solution to Eq . (17) in the

annular region. They are in fact the large Schmidt number solution everywhere. To see this, note that inside the annulus, 94^/9p vanishes exponentially and Cnc “ ^ns ~ so the vortex free solution is recovered exactly.

For values of p in the region 1 < p < <», Eq. (22) is properly scaled In p.

14
In order to effect a balance for times t* =0(1) the wave number n must be

rescaled according to

n = [J2/ae(Sc) ]n* (27)

This is merely a Fourier representation of a boundary layer in the angular

variable 4>, When the rescaling is performed, the same terms are dominant.

This solution can be interpreted physically as saying that the large Schmidt

number prevents significant radial diffusion in the Lagrangian coordinates.

The intense vortlcity sufficiently enhances the azimuthal diffusion in the

annulus so that only relatively low wavenumber angular modes are required to

describe the solution. In the irrotational region, the diffusion is confined

to narrow regions near the interface between reactants and occurs

preferentially in the azimuthal direction. The implications for the

combustion of a fuel blob (see Figure (6)) are:

(1) The fuel in the annular region containing vortlcity is

very rapidly consumed.

(2) By the time the vortex collapses to its final steady state, the only

remaining fuel is in the Irrotational flow region.

(3) Any remaining fuel is consumed along the fuel-oxidizer interface in

the irrotational region over a period of time determined by the extent of

the fuel blob as well as the vortlcity and strain rate.

15
CONCLUDING REMARKS

A mathematical model of the local convection, mixing, and diffusion of

initially unmixed reactants in a turbulent flow has been presented. The model

is based directly on solutions of the Navier-Stokes equations in limiting

geometrical and parametric cases; no empirical parameters are introduced.

At present, the model cannot be regarded as complete. In particular, more

work is needed to extend the model to incorporate three-dimensional reactant

configurations, large temperature variations, and larger molecular

dif fusivities . It is currently more suited to the analysis of acid-base

reactions in turbulent flow than to combustion. Still, many of the phenomena

are similar in both reaction systems, and the study of acid-base reactions in

jets and mixing layers is an ongoing project in turbulent combustion

1 16 research > . Extended versions of this model, employed in conjunction with

large-scale flow simulations, offer the potential for predicting turbulent

reacting flows over a much wider range of scales than is possible with

supercomputers using conventional numerical methods. This statement Is likely

to hold for some time to come.

ACKNOWLEDGEMENT

This work was supported by the Air Force Office of Scientific Research under

contract Number AFOSR-ISSA-85-0026.

16
REFERENCES

1. Grinstein, F.F., Oran, E.S., and Boris, J.P.: "Numerical Simulations of

Asymmetric Mixing in Planar Shear Flows," Naval Research Laboratory Memorandum

Report 5621, 1985.

2. Boris, J.P., Oran, E.S., Gardner, J.H. , Grinstein, F.F., and Oswald, C.E.:

"Ninth International Conference on Numerical Methods in Fluid Dynamics"

(Soubbaramayer and J.P. Boujot, Eds.), p. 98, Springer-Verlag , 1985.

3. Baum, H.R., Rehm, R.G., and Mulholland, G.W. : "Nineteenth Symposium

(International) on Combustion," p. 921, The Combustion Institute, 1982.

4. Baum, H.R. and Rehm, R.G.: Combustion Science and Technology 40, p. 55

(1984).

5. Ghonlem, A.F., Chorin, A.J., and Oppenheim, A.K. : Phil. Trans. R. Soc.

London A 304, p.303, (1982).

6. Ghoniem, A.F., Marek, C.J., and Oppenheim, A.K. : "Modeling Interface

Motion of Combustion," NASA Technical Paper 2132, 1983.

7. Batchelor, G.K. : An Introduction to Fluid Dynamics, p. 79, Cambridge,

1967.

8. Marble, F.E.: "Recent Advances in the Aerospace Sciences" (C. Casci, Ed.),

p. 395, Plenum, 1985.

9. Karagozlan, A.R., and Marble, F.E., Combustion Science and Technology 45,

p. 65, 1985.

10. Baum, H.R., Ciment, M. , Davis, R.W. , and Moore, E.F.; "Seventh

International Congress on Numerical Methods in Fluid Dynamics" (W.C. Reynolds

and R.W. MacCormack, Eds.), p. 74, Springer-Verlag, 1981.

11. Rott, N.: ZAMP 9, p. 543 (1958).

12. Lundgren, T.S.: Phys. Fluids 25, p. 2193 (1982).

13. Landau, L.D. and Llfshitz, E.M. : Fluid Mechanics, p. 122, Pergamon,

1959.

14. Carrier, G.F., Fendell, F.E., and Marble, F.E.; SIAM J. App. Math 28, p. 463 (1975).

15. Driscoll, R.A., and Kennedy, L.A. : "Nineteenth Symposium (International)

on Combustion," p. 387, The Combustion Institute, 1982.

16. Dahm, W.J.A. and Dimotakis, P.E.: "Measurements of Entrainment and Mixing

in Turbulent Jets," AIAA 23rd Aerospace Sciences Meeting, AIAA Paper 85-0056,

Jan. 1985.

17
FIGURE CAPTIONS

1. Configuration of an axial vortex in an axially symmetric strain field.

The streamlines of the strain field are hyperbolic in each cylindrical plane.

The azimuthal velocity is in the direction of the curved arrows.

A 2. Swirl velocity profiles of various times t for the collapsing strained

vortex. The local Reynolds number (e)”l = 500.

A 3. Mixing configuration at various times t for an initially planar Interface

displaced one half vortex radius from the flow symmetry axis. The local

Reynolds number (e)~^ = 500. The vortlclty to strain rate ratio = S2/a = 2.

A 4. Mixing configuration at various times t for initially circular reactant

displaced one half radius from vortex center. Same flow parameters as Fig. 3.

A A 5. Lagranglan angular coordinate at time intervals t of 0.5 from t =0.5 to

At=5.0. Same flow parameters as Fig. 3.

6. Sketch of mixing and diffusing reactant configuration in Lagranglan

coordinates. The shaded annulus is the region of Intense vortlclty, the Inner

region has no swirl velocity, and the outer region is irrotatlonal

.

18
X

Figure 1

19

£=.002 [}= 2. T1ME=3.0

Figure 3

o

X) * 3/rh * Q

23

Figure d
4

psincp

eV)

o

a

o

I

9

- o

a

o
SECTION III

A SOLUTION FOR A TWO-DIMENSIONAL DIFFUSION FLAME IN A VORTEX FIELD

The theoretical study of chemical reactions in complex flow fields, for

example turbulent reacting flows, has received increased attention recently.

This attention is warranted because both numerical and analytical progress is

being made in addressing the problem; discussion of some of this progress is

presented in Reference 1. Also presented in Reference 1 is an outline of a rather general approach, using both analytical and numerical methods, for

attacking turbulent reacting flows and an analysis of a model of small-scale

mixing and reaction.

In Reference 2, a similar but more specialized model problem of small-scale

mixing and reaction has been posed and analyzed locally along the flame front

to provide dependencies of the solution upon dimensionless parameters. This

problem has several important features; it is a diffusion-controlled reaction

in a viscously spreading vorticity field which stretches the flame sheet. The

problem, named the Marble problem here, allows rather detailed treatment, as seen in Reference 2 which has slight variations in detail and notation from

the present write-up.

The novel features of this work compared with the analysis in Reference 2 are

(i ) it is observed that the convection-diffusion equation for the

Shvab-Zel dovi ch variable permits a similarity solution, reducing the number of

independent variables from three (radius, angle and time) to two (angle and

similarity variable); (ii) as in Reference 1, a Lagrangian coordinate system

is used to eliminate flame-sheet resolution problems induced by vortex

winding; (iii) Fourier analysis in angle and a numerical treatment in the

similarity variable allow one to solve the global problem. The analysis

outlined below has not yet been completed. The solution to this problem

should provide a special case on which to test methods for solution to the more general problems described in Referernce 1.

Initially we have fuel in the left half-plane and oxidizer in the right

half-plane in arbitrary proportions (see Figure 1). At t=0, the two half

spaces are brought into contact and simultaneously the line vortex with

tangential velocity ve is imposed:

d 0 r V 0 (r,t) = r -- = + -— [l-exp(-n)] ( 1 ) dt 2 Trr

where r is the circulation of the vortex, v is the kinematic viscosity and

n = r2/4vt. (The function ve(r,t) is a solution of the Navi er-Stokes

equati ons )

.

If the mass fraction of fuel in the left half-space initially is Yf q while

the mass fraction of oxidizer in the right half-space initially is q, and

if the single-step reaction occurs

vf[Yf] + vo[Yq] -» vp[Yp] ,
then with Mf, Mq = the molecular weight of fuel and oxidizer respectively,

Z E

{ ----- vfMf

^0 Yo,o . Yf^o Yq,o . +

I / I +

j \^o<^o '^oMo VfMf vqMq

is the Zhvab-Zel dovich variable for which

since

Z = 1 for X < 0

Z = 0 for X > 0 @ t = 0

Yf = Yf^o >

Yf = 0

Yq = 0 for X < 0

Yo = Yq,o for X > 0 @ t = 0 .

Z satisfies the convection-diffusion equation:

9Z

-- + V* (uZ) = DV^Z

9t

or in cylindrical coordinates

9Z V 0 (r,t) 9Z 92Z 1 9Z 1 92Z

9t r 90 9 r

2 r 9r r2 302

For the flame sheet approximation, Yf and Yq cannot coexist, so that

Yq,o Yq,o Yf^o

Zf =

/ { +

} VqI^O VqMq VfMf

determines the flame surface.

Integrating the tangenti al -vel ocity equation gives the angle 0(r,t,0o)at

any fluid element initially located at r,0Q:

( 2 )

(3)

(^)

(5)

( 6 )

t for
. r t

- 00 = — [^ ~ ! (-r'2/4vt" )dt"

}

ZttT^ 0

or

r 1 e(r,t) - 00 = - [1 - E2(n)] (7)

2tt»4v n

where

00 dt

E 2 (z) = / -- exp (-zt)

1 t2

and

Change independent variables,

r = p

n = r2/4vt

to Lagrangian coordinates.

r 1 9 = 0Q + - [1-E2(n)]

2-rr»4v n

t = T

Then Eq. (5) becomes, in Lagrangian coordinates.

3Z 32 z 1 3Z 2r 1 1 a2z

.. = of— + - - + f[— - (1 - e-0)]2 t 1} - —

9t 3p2 p 3p 8tt 2 n p^ 36 q

2r 2 8Z 4r 1 1 92 z + [e-^-(l-e-n)/n] — + (1-e-n) ---

} 8irv p2 39 q

8ttv p n 9p90o

( 8 )

( 9 )

This equation admits a similarity solution (which, in fact, exactly solves the

Marble problem since neither the velocity field generated by the line vortex,

nor the diffusion field produced by the two adjacent half-spaces of fuel and

oxidizer have a length scale associated with them.)
Let fi = p2/4vT as above, and assume Z(p,t,9o) = Z(n,0o).

Then Eq. (9) becomes

32 Z 3Z 2r 1 1 3^Z —- + (Sc + 1/n) — + {[ (1 - e-n)]^ + 1} --- —

3n2 3n 8irv n 4n^ 39q

r 1 1 3Z 2r 1 3 ^Z

+ [e-9 - - (1 - e-n)] —- + —- — (1 - e-n) = 0 (10)

Sttv n 39 q

8itv 3h39q

where Sc = v/D is the Schmidt number.

Fourier decomposition of Eq. (10) yields

d^Zn 2r 1 dZn ---- + [(Sc + 1/n) + in —- -- (1 - e-n)] --- dn^ Sttv dn

2r 1 r 1 + {-{[ - (1 - e-n)]^ + 1} --- + i n [e-n - - (1 - e-n)]}Zn = 0

8itv 4n‘ 8tt

V

n‘

( 11 )

The fact that a similarity solution exists has reduced the number of

independent variables upon which the solution depends by one. When Fourier

analyzed, the mathematical problem is reduced from a one-dimensional,

time-dependent partial differential equation, an i ni ti al -val ue problem, to an ordinary differential equation in the similarity variable, a two-point

boundary-value problem. For each Zn(n) we have a linear ordinary differential

equation for which boundary conditions are needed as n-*-” and as n>0. For

fixed p, as t-^0, n=P^/4vt-H» and Zn(n) must satisfy initial conditions for the

^ Marble problem:

Z(n,eo) Zn(n) exp(in0o)

As Z(r|x”,9o) = H(9o“ir/2) - H(9o-3tt/2)

for 0 < 9 g< 2 tt

( 12 )

= I Anexp(in9o)

n=-oo

( 13 '
Hence Ao = 1/2

A2j = 0,

(-l)j

A2i-1 =

j = l,2,-3,4,...

Tr(2j-1)

A-2j = 0

A-(2j-l)

(-l)j

»(2j-l) (14)

As n>0, Zn(n) must remain finite. Examination of the indicial equation for

Zn(n) as n>0 shows that the two solutions behave as where

u = -inr/STTv + n/2. Hence a change of independent variable to

Yn(n)

nm/2

l+nf^/2

Zn(n) (15)

where m = ] n |

,

and requiring Yn(n)>0 as n>0 implies that Zn(n) remains finite at n=0.

The equation for Yn(n) is

d^Yp m nr 1 dYp —r- + [(Sc + 1/n) + i —- -- (1-e-n)] —

dn“^ n(l+n'^/^) 4 ttv dn

m/2(m/2+l) m/2(Sc+l/n)

n2(i+^m/2) n(l+n'^/2)

r 1 n^

j[ ... . (i.e-'i)]2 + 1 i --

4ttv n

r 1 1 + i n — -- |[e"^ - - (l-e~'^)]

8itv n

m

l+n'"/2

- (l-e-i)ll Yn 0

( 16 )
It is convenient to map the interval 0 < n < <» into 0 < x < 1 using the

transformation

X = 1-e"^ ( 17 :

Then the equation for Yn(x) can be solved using boundary conditions that Yq(x)

= 0 at X = 0 and Yp(x) = Ap at x = 1. Equation (16) can be discretized

strai ghtforwardly using a second-order, central difference scheme, and the

resulting linear algebraic equations can be solved using a tridiagonal solver.

The Fourier synthesis to obtain the complete solution can be performed using

an FFT package. Location of the flame sheet and of the rate of fuel

consumption must then be computed from the complete solution. Work is in

progress on these tasks.

Plots of the interface, 0 = 0 and 9 = tt, when the diffusion 0 = 0 for four

values of the Reynold's number r/8irv are shown in Figures 2-5. When the

diffusion is zero, the solution depends only on this single parameter. It is

important to note that this sequence of plots does not represent a time

sequence; rather each plot shows the interface for that Reynold's number, the

similarity variable n as a function of the angle 0. This interface is

calculated from Eq. (7).
REFERENCES

1. Baum, H. R., Corley, D. M, and Rehm, R. C., "Time-Dependent Simulation of

Small-Scale Turbulent Mixing and Reaction," Paper submitted to the

Twenty-First International Symposium on Combustion.

2. Marble, F.E., "Growth of a Diffusion Flame in the Field of a Vortex",

Recent Advances in Aerospace Sciences (C. Casci , Ed.), p. 395, Plenum, 1985.
Figure 1
INTERFRCE SHAPE-MfiRBLE PROBLEM

GFIM/(8xPIxNU)-1.0

in

Fjgure 2
INTERFACE SHAPE-MARBLE PROBLEM

GflH/(8MPI><NU)-2.0

F igure ,5

o
INTERFRCE SHRPE-MRRBLE PROBLEM

SflM/(8«PI><NU)-3.0

in

80 "

CD

ci "

go

in

lO .

h jgure 4

i

I

I

I

I
INTCRFflCE SHflPE-MRRBLE PROBLEM

GRM/(8xPImNU)-1.0

inro"

CO

f>5“

LJO

CE

cJ*

in Figure 5
HBS-l 14A 1 BE V . 2 -ec . U.S. OEPT. OP COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions!

1. PUBLICATION OR

REPORT NO.

NBSIR 86-3334

2. Performing Organ. Report No. 3. Publication Date

DECEMBER 1986

4. TITLE and subtitle

Time-Dependent Simulation of Small-Scale Turbulent Mixing and Reaction

5. AUTHOR(S)

H, R, Baum, R, G, Rehm, D, M, Corley, and D, W, Lozier

6. PERFORMING ORGANIZATION (If yoint or other than NBS. see in struction s)

national bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Gram No. AFOSR-ISSA-85-0026

t. Type of Report & Period Covered

Annual Report

11/84 - 11/85

9.

SPONSORING organization NAME AND COMPLETE ADDRESS (Street. City. Stote, ZIP)

Air Force Office of Scientific Research

Bolling Air Force Base

Washington, DC 20332

10.

supplementary notes

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most si gnificant information. If document includes a significant

bi bl iography or literature survey, mention it here)

A mathematical model of the local transient diffusion-controlled reaction between

initially unmixed species is presented. It is intended ultimately as a computational

"molecule" to be imbedded in direct simulations of larger scale reacting flows. The

model consists of an interacting three-dimensional strain vortex field which exactly

satisfies the Navier-Stokes equation, an analytically determined Lagrangian representation of the mixing process and convection-diffusion equations for the reacting species

in Lagrangian coordinates. The length scale established by the stretching of the

vorticity field is shown to be directly relatable to the Kolmogoroff scale if the local

strain rate has a scale consistent with laminar boundary layer mixing. An analytical

solution to the convection-diffusion equation governing the diffusion-controlled

reaction is derived. The solution is valid for large Schmidt number and describes the

evolution of any initially two-dimensional configuration of reactants. A special twodimensional case of this model, in which vortex strain is excluded and fuel and

oxidizer initially occupy adjacent half-spaces, is also analyzed. This problem was originally formulated by F, Marble, who treated it in a very different manner from

that described here.

12.

KEY WORDS (Six to twelve entries; a Iph a bed cal order; capital! ze only proper nomes; and separate key words by semicolon s) combustion; convection; diffusion-controlled; flame-sheet; reaction; turbulent mixing.

13. AVAILABILITY

[~Xl Unlimited

I I For Official Distribution. Do Not Release to NTIS

I I Order From Suoermtendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

41

15. Price

$9.95

USCOmm-OC ©O43-P80
®

6

1 mmsttm ^

‘

- 1"

. ^ ‘'f

.Va 'nt lO './.'-* 'J/ws*

"<C .'i '"*T r-»v<*t'

•• I '»,•(/,'(, # '¥ ! f av&i'f ' i 8 uwA j -fe'\ff m; ;; . '

,

' S' isy5iiuf'|i''s

r • 0-5 , > .;>H

, 4 c > A'. >+.•»»> u , <«Ot

' ^ :-w-» &

- 'V f. i'"

^Sj r nribit

•^^^;-v ,r 'M-; r.. ; /; ‘vj ? fjt <(U

-'.' Pm .>^;,t:

' < r '- <- f r ’ A . < sm» tiSyA " t,

.

' Imm. 1^' tofi:

M, V

••

I '-Mr :^> ' /- *1 W*<«t r vv/.!-(.i :^'T^fu'Ptuj ^'^'?^:t^^T

''"h -': ".M,i'-,« „ ' ’‘!.rk,,

1*1 A |i»iiijwi* <1 HkS

( ; ^bi'\.-,t

A

" ..,;

;,

"

'.

. :

i

‘

'

1 i » .•* J t

: . ' .' ('• V‘-'X .

j

. ,

' ' /

! •

'' - ' . - 'f^'.'\

>

.

' „'. v|m

' 4r !

'

, ,

''.

,; '-

, Ait'...