Laminar boundary layers behind blast and detonation waves

, ".'~ "

LAMINAR BOUNDARY LAYERS

,.

, .~. ':.~ ~ ~ ~ ~~~ , I

BEHIND BLAST AND

'

DÈrONATION WAVES

TECHNISCHE HOGESCHOOL DELFT

LUCHTVAART-

EN RUIMTEVAARTTECHNIEK

BIBLIOTHEEK

Kluyverweg 1 - DELFT

by

_{8 FEB.}

₁₉₈₃

Xixing Du, W. S. Liu and I. I. Glass

~AlolgU,S.t,

1982

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..

UTIAS Report No. 259

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...

~

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~.'.

eN ISSN 0082-5255

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•

LAMINAR BOUNDARY LAYERS

BEHIND BLAST AND DETONATIOH WAVES

by

Xixing Du, W. S. Liu and I. I. G1ass

Submitted February, 1982

August, 1982

UTIAS Report No. 259

eN ISSN 0082-5255

Acknowledgements

One of us (X. X. Du) would like to express his gratitude to The Aerodynamics Research and Development Centre, Beijing, People's Republic of China, for providing this opportunity to work at UTIAS. The assistance he received from his fellows at UTIAS is very much appre-ciated.

We thank Dr. T. Saito for providing the random-choice computer program for solving the flow profiles behind a spherical Chapman-Jouguet detonation wave.

The financial support received from AFOSR (82-0092), the U.S. ARO and the Natural Sciences and Engineering Research Council of Canada is acknowledged with thanks.

Summary

The transformed coordinates devised by Mirels and Hamman have been modified in such a way th at the transformed nonstationary-boundary-Iayer equations become applicable to boundary-Iayer flows induced by both blast and detonation waves moving with a power-Iaw trajeetory in planar, cylindrical and spherical geometries. Investigations were made of boundary-Iayer flows in air behind nonuniform strong blast waves and in the burned gas óf a stoichiometrie mixture of hydrogen and oxygen behind uniform Chapman-Jouguet detonation waves. The results show that thePrandtl number has a profound influence on boundary-Iayer flow. For a blast wave and Pr less than unity it controls a boundary-layer velocity-overshoot as one moves away from the wave. The overshoot decreases with increasing Prandtl number. For a Chapman-Jouguet detonation wave similar results are obtained for a Pr = 0.72. However, for an actual Pr = 2.26, a flow reversal occurs away from the wave where the inviscid flow velocity approaches a small value. The results also show that the viscous exponent has a significant effect on the wall shear stresses and heat transfer, and that the effect of the wall temperature is small.

In order to show some of the physical features of the various boundary layers, actual velocity profiles we re computed for spherical and planar detonation waves in stoichiometrie hydrogen-oxygen and for blast waves in air. It is shown that owing to the rapid decrease in density behind a blast wave the boundary layer thickness becomes very much larger than their detonation-wave counterparts at the same wave velocity (but different physical conditions). In addition, the velocity-boundary-Iayer thickness in air behind a quasistationary planar shock wave is somewhat more than for aplanar det-onation wave at the same wave velocity (but in different gases).

In order to test the validity of the analysis, the heat transfer to the wall behind aplanar detonation wave was calculated. The profile of the variation of the heat trans-fer with time at any given position behind a C-J detonation wave is in good agreement with the experimental data, and adds confidence to the present analyses for cylindrical and spherical flows as weIl.

1. 2. 3. 4. 5. Acknowledgements Summary Notation I NTRODUCTI ON INVISCID FLOW FIELD 2.1 Basic Equations

2.2 Boundary Conditions

2.3 Similarity Solutions VISCOUS FLOW FIELD

CONTENTS

3.1 Governing Equations for Laminar Boundary-Layer Flows

3.2 Transformed Boundary-Layer Equations

3.3 Numerical Method and Procedure 3.4 Boundary-Layer Characteristics

3.5 Properties of Hydrogen-Oxygen Stoichiometrie Mixtures

3.6 Discussiorr of Boundary-Layer Equations BOUNDARY-LAYER DEVELOPMENT BEHIND EXPLOSIVE WAVES

4.1 Boundary Layer Behind Strong Blast Waves

4.2 Boundary Layer Behind C-J Detonation Waves 4.3 Some Discussions on Boundary-Layer Thickness 4.4 Heat Transfer to the Wall Behind a Detonation Wave

in 2Hz+Oz Mixture CONCLUSIONS ii iii v 1 2 2 3 4 5 5 5 6 7 8 8 9 9 9 10 10 11 REFERENCES 12 TABLES 13 FIGURES 18

APPENDIX A: DERIVATION OF TRANSFORMED BOUNDARY-LAYER EQUATIONS APPENDIX B: THERMAL CONDUCTIVITY AND VISCOSITY FOR MULTICOMPONENT

GAS MIXTURES

APPENDIX C: BOUNDARY-LAYER THICKNESS CLOSE TO THE ORIGIN APPENDIX D: DERIVATION OF HEAT TRANSFER EXPRESSION

a 00 A b B C Cf Cp D E f F g h Hl' H2 1 K Kl K 2 m M s M CJ n N p Q q qw R S S 0 Notation

speed of sound ahead of the wave front

constant used in the transformed coordinate

n

molecular-weight ratio Y(Yoo-l)Cpoo/Yoo(Y-l)Cp ratio _PlJ/PelJe

constant used in x

=

Ct m s

skin friction coefficient defined as Cf specific heat at constant pressure heat diffusivity defined as D = K/pC

p internal energy

nondimensional stream function

similarity parameter, F

=

pip u 2, for inviscid flow

00 s

enthalpy ratio, g = h/h

e (= T/Te) specific enthalpy

boundary-layer-thickness coefficients integral quantity defined by Eq. (14) thermal conductivity

energy coefficient defined by Eq. (22) invariant defined by Eq. (31)

exponent used in x _s

=

Ctm wave Mach number

C-J wave Mach number number of chemical species mole fraction of species pressure

chemica 1 energy released during combustion

2

nondimensional chemical energy, = _Q/aoo heat flux on the wal)

gas density ratio behind and ahead of the wave front, R = p/poo

quantity defined by Eq. (21)

w x y 8 y Ó À ~ p

n

S

~

x

cr cr W T T W Subseripts o r w e s

veloeity ratio aeross the boundary layer, w direetion along the wall surfaee

direetion normal to the wall surfaee

=

(m-l)/m

nondimensional speed of sound

speeifie-heat ratio or isentropie exponent boundary-Iayer thiekness

geometrie step-size ratio in n-direetion viseosity

density

transformed eoordinate transformed eoordinate veloeity ratio, ~

=

u/us sealar stream funetion

geometrie parameter for inviseid flow geometrie parameter for viseous flow viseous exponent

time-transformed variabie wall shear stress

value just behind wave front, or at

S

eonditions ahead of wave front referenee eondition

wall eondition

edge of boundary layer wave front

vi

o

1. INTRODUCTION

This study was motivated by the existence of a unique explosive-driven-implosion facility at the

Institute. Stabie focussed implosions can be pro-duced in this reliable facility yielding extremely high-pressure and temperature conditions at the focus. This facility has been used successfully for the generation of hypervelocity projectile flight and intense shock waves as weIl as the production of diamonds from graphite and neutrons and y rays from deuterium fusion [Ref. 1]. The performance of the implosion chamber was investigat-ed extensively [Refs. 2-8]. In all these investi-gations it was assumed th at effects of viscosity and heat conductivity were negligible. However, it is important for a complete assessment of per-formance for the imp los ion chamber to evaluate the effects of the viscous boundary layer on the im-plosion focus. It is necessary to consider these aspects of flow in several stages. The first stage concerns the induced boundary layer behind the gaseous detonation wave in a hydrogen-oxygen mix-ture and is the subject of the present report.

In general, all detonations or explosions fall into two categories: structure-independent and structure-dependent. For the former, the chemical-reaction zone is very narrow immediately behind the wave front and its structure has no effect on the characteristics of the flow field behind it. For the latter the relatively long chemical-reaction zone and its structure dominate the properties of the flow field. Both types of detonation were studied extensively [Refs. 9-17]. In this study of 2H₂+0₂ detonations only the structure-independent wave applies.

Mirels and Hamman [Ref. 18] made the first attempt to solve the laminar boundary layer behind a strong blast wave moving in a power-law path. They used a series-expansion method which was also employed later by Chen and Chung [Ref. 19]. This method is only applicable in the region near the wave front. Sichel and David [Ref. 29] made a modification of the shock-tube boundary-layer solu-tion by Mirels [Ref. 21] and Hartunian et al [Ref.

~2] to calculate the heat transfer to the wall behind a detonation wave with the assumption that the pressure, temperature and velocity behind the detonation remain constant and are equal to the C-J values. These assumptions are restrictive in describing the actual physics of the problem. Later on, numerical solutions we re obtained by Liu and Mirels [Ref. 23] for different one-dimensional flows (planar, cylindrical and spherical) over the entire flow region.

In previous analyses [Refs. 18, 19, 23] for boundary-layer flows behind strong blast waves, four major assumptions were made in order to sim-plify the analysis: (1) The explosion wave moves with a power-law pa th in the time-distance plane. Consequently, similarity solutions exist for the

boundary layers behind detonation waves. (3) The dynamic viscosity coefficient of the gas behind the wave front was assumed proportional to tem-perature. This assumption affects the wall skin-friction and heat-transfer coefficient but not the boundary-layer structure. (4) The Prandtl number is assumed constant across the boundary layer. This assumption is reasonable since the temperature dependence of the gas thermal-conduct-ivity is similar to the dynamic-viscosity coeffi-cient. Some of these assumptions must be removed and the characteristics of the flow field must be taken into account for a more realistic analysis of nonstationary boundary-layer flows behind deton-ation waves. It will be shown subsequently that the available experimental heat-transfer data agree very weIl with the present analysis for planar flow. Consequently this agreement also adds confidence to the analyses of cylindrical and spherical detonation boundary-layer flows.

In this work, the last three assumptions were removed. The transformed coordinates of Mirels and Hamman [Ref. 18] were modified in such a way th at their equations become applicable to flows behind blast waves as weIl as detonation waves.

It has been shown th at direct initiation of a detonation wave requires the instantaneous release of a finite amount of ignition energy to guarantee a sufficiently strong shock wave for auto-ignition in the shocked medium. When it is ignited by an exploding wire, the detonation wave is, at least initially, overdriven, then it decays rapidly to a Chapman-Jouguet (C-J) detonation wave with a con-stant velocity. When such a hemispherical C-J detonation wave propagates outward along the major diameter, as shown in Fig. 1, a boundary layer forms behind it.

For C-J detonation waves, similarity solutions exist. A complete description of the method for solving C-J detonation waves is given in Ref. 13. The detailed calculations are given by Saito [Ref. 8] for detonation waves in 2H₂+0₂ mixtures for various initial conditions. For the present application, it was necessary to recalculate the inviscid flow functions and their derivatives.

In doing so, the compatibility of the solution with the boundary conditions at the outer edge of the boundary layer was ensured.

As pointed out in Refs. 8 and 13 the detonation wave is followed by a rarefaction wave and therefore all the properties of the inviscid flow behind the wave front decrease until the velocity approaches a very small value very near the half-distance that the detonation wave has travelled. Beyond th at point the pressure and density remain constant and a uniform stationary state results. However, in the viscous flow behind a detonation wave the velocity boundary layer and the thermal boundary layer are both stabie and grow with time in the entire region, except at the origin, where the boundary-layer

geometries.

The effects of Prandtl number, the viscous ex-ponent and the wall surface temperature on the flow structure are considered and discussed in detail. Comparisons have also been made between planar and spherical boundary-layer growths behind blast waves.

It is shown that owing to the rarefaction wave profile behind the shock fronts the densities drop very rapidly and the boundary layers become very thick. The spherical boundary layer is thicker owing to the more rapid expansion. This phenomenon does not exist behind detonation waves (or constant speed shock waves) as the densities have finite values.

This is only a first step in trying to under-stand the viscous effects which exist in the UTIAS implosion facility at the focus of an implosion. As noted previously, the effects of viscosity, heat conduction and radiation are important for obtaining fusion in deuterium. However, since neutrons and y rays have al ready been obtained in this facility the foregoing effects cannot be too limiting.

Although the motivation was primarily for this purpose, the present analytical-numerical results are general and can be applied to many boundary-layer flow problems behind blast and detonation waves.

2. INVISCID FLOW FIELD 2.1 Basic Equations

In a mixture of chemical species of a reacting gas, a great simplification can be made in the analysis of the inviscid flow by assuming momentum and thermal equilibria in the system. The governing equations are then reduced to three-flow equations for the mixture. The only difference in the govern-ing equations between the unburned and burned gases is the equation of state. The basic equations des-cribing the one-dimensional flow for the mixture are, Mass: Momentum: Energy: (lp - +

at

au

+ u

a

u

+

l.

~

at

a

x

p

a

x

o

o

p

~~

+ u

~~

) - (

*

+ u

~

)

=

0 (1) (2) (3)

where cr is a numerical constant with values of 0, 1, 2 for planar, cylindrical and spherical symmetric inviscid flows, respectively, p is the mixture den-sity, u is the flow velocity, p is the mixture pres-sure, h is the mixture specific enthalpy, t is the time and x is the flow direction. The wall shear stress, radiation-energy transfer and the wall heat transfer are neglected for inviscid flows.

2

. !f the chemica 1 reaction is assumed to occur w1th1n a thin layer near the wave front, the

un-burne~ and burned gase~ ~an be treated as a perfect gas w1~h constant spec1f1c-heat ratios y and con-stant 1sentropic exponent y, then 00

State: _P =

_y

y-l

_p

h

(4 )

The isentropic exponent y of the burnt 2H

2+02 is

about 1.3 at a temperature of 2,000 K and 1.04 at a

~emperature of 20,000 K. The pressure effect on y 15 small and can be neglected when the pressure is below 104 atmosphere [Ref. 24] .

From similarity considerations, it was shown [Refs.~1,13] that for a self-similar motion the

expa~dlng wave front must either be a power law of t1me or an exponential function of time. For ~ost explosion problems, the power-law form is of 1nterest. Assume that the density ahead of the wav~ ~ront, Poo' is uniform and the wave front pos1t10n Xs satisfies

(5)

where C and mare constants and C is determined from the energy inte~ral equation. The following values of mare app11ed for explosive waves:

m

_---,

2 cr + 3

for a strong blast wave

)

m

=

1, for a Chapman-Jouguet wave

Define the following dimensionless coordinate,

1 x x 5

and the dimensionless variables,

F

_~

Poous R

..E...

Poo q> U U 5 (6) (7) (8 )

where.us is t~e wave front velocity which is a funct~on of t1me. The following self-similarity

equat10ns result: [q>-l+~]R~ + Rq>~ ÖCi>R

°

l-~ (9) [qH+~]q>~ + F~ _oq>

₌

°

R-

(10) [q>-l +~] F ~ + YFq>~ - 20F

-

yö

~ _l-~

=

0 (11)

where ~

=

0 corresponds to the wave front and ~

=

1

corresponds to the origin and y is the isentropic

exponent of the gas behind the wave front. The

sub-script ~ denotes the derivative with respect to ~.

The parameter a is defined by

m-l ₍₁₂₎

a =

-m

where a = 0 for a uniform wave front and a < 0 for

a decaying wave front.

Initially, we assume th at a finite amount of

energy Eo is released at time zero in a finite

volume of dimension Xo in a medium at rest. From

the energy integral, the constant C in Eq. (5) is

given by [Ref.ll]

C (13)

where Kj is 1, 2n and 4n for planar, cylindrical

and spherical symmetry, respectively, and I is

defined by I

=

1

f [

_F_ + R<p2 )

(l-Oäd~

y-l 2 (14) o 2.2 Boundary Conditions

To solve Eqs. (9)-(11), the boundary con'ditions

immediately behind a shock or detonation wave front

have to be determined. Consider the case where the

shock wave moves to the right with the velocity us· The gas on the left side of the shock, which was stationary before being traversed by the shock, is given a velocity Uo to the right. The situation illustrated above may be transformed to the case where the shock wave is stationary. In the new frame of reference, the shock wave appears to be stat ionary , while the gas on the left side of the shock appears to be flowing to the left with the

velocity (uo - us). In this frame of reference,

the conservation equations across the shock front are given by:

Mass: (15) Momentum: (16) Energy: h _co + "2 1 Us 2 + Q (17) (18)

y

[y

+

_1 _ S)

co M 2 s <Po (19) Yco +

~

+ yS Po M F s 0 - - 2 = yco(y+l) Pcous (20) where S { [ ; - M \ r 1

r

/2 K l - 2 _M (21) s s Yco(y-Yco) (y+l) 2 2 2 { Yco (y -1) q} Kl ₂ + ₂ Y (yco- l ) Y (22) q

l

₂ (23) aco 2 Pco a co Yco

P

_co (24)

For astrong blast wave (Ms .... co), the boundary

conditions at the wave front (~ 0) are

F 2

)

<Po 0

=

y+l (25) R y+l 0

=

y-l

For a C-J detonation wave, we have Ms

=

MeJ'

which

is defined by

and s

=

O. The boundary conditions at the wave

front are, Yco - YM_CJ <Po yco Cy +l ) -2 F Yco + MCJ yco(y+l) (26) (27)

2.3 Simi1arity Solutions

For a strong blast wave the shock speed is time-dependent, hence the entropy change across the shock front decreases with time. The simi1ar-ity conditions require an infinite strength wave.

However, in practice the strong-shock approxima-tion is adequat'e for shock strengths above three.

The solutions for a strong blast wave can be obtained by integrating Eqs. (9)-(11) with the boundary conditions at the wave front, Eq. (25). Tab1e 1 and Fig. 2 show the dimension1ess flow profiles for astrong spherica1 wave with y = Yoo = 1.4. Their corresponding derivatives are shown in Tab1e 2 and Fig. 3. The flow is stationary on1y at the origin (s = 1), which is a singu1ar point. These simi1arity solutions are va1id because the pressure ahead of the blast wave can be neg1ected compared with that behind an intense exp10sion.wave. The energy re1eased by the explo-sion and the undisturbed gas density p1ay major ro1es in a strong blast wave. It is noted that the pressure immediate1y behind the wave front decreased inverse1y as its radius cubed. The total energy contained in the region behind the blast wave is constant on1y if p ~ x-3 . For powers 1ess than or greater than 3, energy is being added or removed continuous1y with time. As seen in Fig. 2, the pressure is highest imme-diate1y behind the wave front. It then decays as s increases, levels off, and remains approxi-mate1y constant for s > 0.5. The particle ve10-city varies in an a1most linear fashion fr om the wave front to the origin and vanishes at the origin. The density approaches zero for s > 0.5.

For the C-J detonation wave, it is more con-venient to introduce a new variabie 6 defined as

_ [ yF )1/2

6-

R

(28)

Then the three similarity equations (9)-(11) can be reduced to two equations in terms of ~ and 6 as fol10ws [Ref.13]: CPs ;1JI32 (29) 2 2 [(~-l+s) -6 ](l-s) 6 s = y-l Öq>6 (~-1 +s) ₍₃₀₎ 2 2 2 [(~-l+s) -6 ](l-s)

In the C-J detonation wave, the shock front propa-gates at a constant velocity and the entropy increase is identical for all gas particles across the wave front. The invariant then reduces to the following form:

RY

f ' = K

2 = constant (31) K2 can be evaluated using the boundary conditions at the front, s = O.

4

For the planar motion

ä

equals zero. We can see that two solutions exist: (1) (cp-1+s)2_62 F 0, hence both ~ and 6

_s

are zero, which corresponds to a planar piston with constant speed driving a constant velocity shock or detonation wave.

(2) (cp-l+s)2_62 = 0, then both CPs and 6s are finite.

Integrating Eqs. (29) and (30) we obtain the solution for CP(s) and 6(s) with the proper signs as follows:

cp(s) __ _y+l 2_ s +

_%

(32) 6(s) - -_y+l y-l s + ₆₀ (33) where (for the C-J condition)

CJ>o + 60 = 1 (34) The subscript 0 denotes the value at

S

O. Using

Eqs. (28) and (31), we get

2 R

[6

_y K₂

r-

l (35) 62 F =

Y

R (36)

If CPo and 60 are known, using Eqs. (32), (33), (35) and (36), cp, R, F can be calculatedj ~ _{and 60 are}

determined from Eqs. (27) and (28).

Fo~ the cylindrical and spherical detonation wave, cr

F

_{0, CPo + 60} = 1. This'results in CPs ~ _00 and _6s ~ _00 at the front, s = O. Therefore, we cannot integrate Eqs. (29) and (30) directly fr om the boundary of the C-J detonation wave front. To overcome this difficulty, the perturbation method is used [Ref. 13] . The sOlution, in the form of a perturbation series, in the immediate neighbourhood of the C-J front is given by the following expres-sions (with proper signs):

[

2ä~ 6 ) 1/2

cp(S) =CPo-

y+~o

sl/2+ ... (37) 6(") _., = 6 ₀ _ (y-l) ₂ [ 2äCP060 _y+l ) 1/2 1/2 _{s + ...} (38) With Eqs. (37) and (38) the values cp(o+ös) and 6(0+ös) can be calculated from the values CPo and

6

0,

where Ös is very small. From these values, cP (o+ös) and 6(0+ös), Eqs. (29) and (30) can then be inte-grated to obtain CP(s) and 6(s). Using Eqs. (35) and (36), R(s) and F(s) are also obtained.

A stoichiometric mixture of hydrogen and oxygen was used. The specific heats, specific-heat ratios, isentropic exponent and equilibrium sound speed for the unburned and burned gases are given by Benoit

[Ref. 24].

In the Chapman-Jouguet detonation wave the shock front propagates at a constant velocity and the entropy increase for all gas particles across the

wave front are identical. Table 3 and Figs. 4-6 show the dimensionless pressure, velocity and density profiles behind the planar, cylindrica1 and spherical C-J detonation waves with initial conditions of Poo

= 13.6 atm,

_Yoo

= 1.4 and T

= 300 K.

The corresponding derivatives which are needed in the boundary-Iayer analysis are shown in Table 4 and Figs. 7-9. It is noted that there are three regions .. From the wave front CE; = 0) to I'; = 0.48 is the expansion region and from I';

= 0.48 to the

origin is the stationary zone. The particle velocity decreases sharply from its maximum value at I';

=

0 to zero at I';

=

0.48. Similar to the blast wave, the pressure behind the wave front decreases

as I'; increases and th en remains constant for

I'; > 0.48. However, the gas density resembles the pressure profile and does not approach zero close to the wave midpoint. It can be seen from Figs. 7-9 that a discontinuity exists in the flow deriva-tives at the position where the particle velocity is zero. The flow derivatives also approach an infinite value at the detonation-wave front. As mentioned previously, at the origin the basic equations are singular.

3. VISCOUS FLOW FIELD

3.1 Governing Equations for Laminar Boundary-Layer Flows

The laminar boundary layer behind astrong blast wave moving with nonuniform velocity was investigated by Mirels et al [Refs.18, 23) and Chen and Chang [Ref. 19). The laminar boundary layer behind a detonation wave has not been solved as yet. As mentioned in the Introduction, the last three assumptions made by Mirels et al [Ref. lSJ are not applicable to a detonation-wave induced boundary layer. In order to generalize the analysis, these assumptions are modified: (1) The specific heats and the specific-heat ratios behind and ahead of the wave front are different, but constant in each region. (2) The Prandtl number may be variabie across the boundary layer. (3) The temperature dependence of the dynamic-viscosity coefficient of the gas behind the wave front has a power law variation with temperature. Other assumptions are made that the chemical reaction takes place only at the C-J detonation-wave front and that the gases behind and ahead of the detonation wave are perfect. Therefore, the equations governing the laminar boundary-layer flow induced by a wave front are as

follows: élp +

..!...1....

0 él élt 0 élx (pux ) + ély (pv) x Ou

~

él [ élU) P Dt + élx

=

ély 11 ély Dh _ QE. _

1....

P Dt Dt - ély [ _{Pr ély} 11 élh) where

o

(39) (40) (41)

the x and y-directions, respectively, and Pr is the Prandtl number. The temperature dependence of the dynamic-viscosity coefficient 11 is assumed to be of the form:

(42) where the subscript r denotes a reference value and

w is the viscous exponent.

3.2 Transformed Boundary-Layer Equations

The following transformed coordinates were used by Mirels et al. 1 - x x 5 (43) (44) [At 2m (0+1)-1I';)1/2 2(0+1)

where A = _2mFoC v p

lp ,

v = lloo/Poo. As men-tioned previously thisOOs;t ~f t~ansformed coordi-nates cannot be used in chemical-reacting flows. By defining the transformed coordinate n as

n

[At2mo+2w(m-l)+II';) 1/2 (45)

where A = 2C2(0+W)m2w- l ll (bFop

lp

)W/p , C is given

00

oor

00

by Eq. (13), Fo is given by Eqs. 25) or (27) and b

=

[Y(Yoo-l)/Yoo(Y-l))Cpoo/Cp is the molecular weight ratio of the gases behlnd and ahead of the wave front, then n is applicab1e to chemical-reacting flows as weIl as non-reacting flows. It can be shown that when W

= 1, b

=

1 and y

= Yoo then A and

n become identical to those given by Mirels and Hamman [Ref .1\8t) .

Define the following dimensionless variables:

n f(E;, n)

_f

~

dn e 0 (46) g(l';, n) h he

where the subscript e denotes the edge of the boun-dary 1ayer. Since the pressure is constant in the y-direction, the density ratio is given by

Pe

-

=

g(l';, n) P

If a scalar stream function ~ is now introduced (47)

The continuity equation is th en satisfied

automati-cally. It is not difficult to show th at the

rela-tionship between the scalar stream function ~ and

the dimensionless variabIe f(~, n) is given by

Applying the transformed coordinates,

dimension-less variables and the relationship between ~ and

f to Eqs. (39) to (41), the transformed

boundary-layer equations are th en obtained,

(l-~)

20 Rl -w [ FF ] w (Bf ) + (n-epf) f

o nn n nn

=

2~

{[

fep~

+

epf~

-

Ï

[20 + a(2W-l)]] _{f nn} +

r.

~

+

(l-~-<pfn) [:~

+

~

]1

f_n -

:~g

} (50)

(l_

~

)20

{Rl-W [

:0

r [

:r gn ]n + y-l

R~2

Rl-w [

~

]W Bf2 } + (Tj-epf)g y f Fo nn Tj

2~

{[

fep

~

+

cpf~

-

Ï

[20 + a(2w-l)] ] gn + [2a +

(l-~-<pf)

[g~

+

F~

_

R

~

]J

g }

(SI) Y n g yF R

where a is defined by Eq. (12), the va lues F, R,

ep and their derivatives F~, R~ and ~ are obtained

from Eqs. (9) to (11), the subscripts n and ~

denote the derivatives with respect to n and S,

and B is defined by

(52)

It can be shown th at if w = 1 (i.e., B = 1) and

Pr = constant, Eqs. (50) and (51) reduce to those

derived by Mirels and Hamman [Ref.IS] . (The

details can be found in Appendix A.)

The required boundary conditions are given by

y

=

0: u(x, 0, t) 0

v(x, 0, t) 0

h(x, 0, t) hw(x, t) (53)

y :: 00: u(x,

_{'" ,}

t) ue(x, t)

h(x,

_'"

,

t) he(x, t) (54)

Af ter the trans format ion , they become

n

=

0: f(s, 0)

=

fn(s, 0)

=

0

n "': 1

(56)

where the subscript w denotes the wall surface. It

should be noted that from the definition of gw

the value T w T", or T w T '" (57) 1

;Z

5

should be a function of ~ only, for a self-similar

boundary-layer flow. Therefore, two cases exist:

(1) For a strong blast wave, according to its definition, we have 1 0 ;z~ 5 50 th at gw

=

O. 2

(2) For the C-J detonation wave, we have Ms2

=

McJ

=

constant. Consequently, gw is a iunction of

~ only.

It should be noted th at for the strong blast wave

or for the so-called cold-wall model (i.e., Tw

=

0)

the viscous exponent w has to be unity. Otherwise,

Eqs. (50) and (51) will be singular at the wall. 3.3 Numerical Method and Procedure

The nonlinear equations, (50) and (51), with the boundary conditions given by Eqs. (55) and (56), were solved numerically by an implicit six-point

finite-difference scheme developed in Ref. 25. The

initial profiles are required for a

finite-differ-ence methad. Consequently, the solution at the

start of the boundary layer has to be obtained

first. At ~

=

0, the partial-differential

equa-tions, Eqs. (50) and (51), become a set of ordinary-differential equations as follows:

R(l-w) (F/F )w(Bf ) + (n~f)fnn

=

0 (58)

o nn n

R(l-W) (F/F )w [[

~

g] + y-l R<p2 Bf2 ]

o Pr n n y F nn

+ (n-<Pf)gn

=

0

Equations (58) and (59) were solved using the Newton-Raphson method, or a shooting method.

(59)

Equations (50) and (51) are linearized in a farm suitable for an iteration scheme by introducing the following function

(60)

g(~, 0)

₌

gw

m

_{(55) It is really only required for the momentum}

tion and not for the energy equation. The momentum equation in terms of W can be written as

{

2a l-w W

+ .(l-E;) R (F /F 0) Bn + (n-qJf)

- 2E; [fq>E;+qJfE; -

~

[2a+a(2W-l)]]}

W~p)

- 2E; [a+(l-E;-<PW) qJ; ] W(p)

( ) FE;g

ZE;(l-E;-<pW)W P - ZE;

-E; RqJ (61)

and the energy equation in the linearized form is

+ (l_E;)zcrRl-W(F/Fo)W ( : Jn + (n-qJf) r

- ZE; [fCPE;+CPfE; -

~

[Zcr+a(Zw-l)] ]

g~~)

[ Za ( FE; RE; JJ (p)

- ZE;

y

+ (l-E;-cpI'I) yF - R g

Z

ZE; (l-E;-cpW)g(p) - (l_E;)Zcr Y-l ~ Rl-w(F/F )WBWZ

E; y F 0 n

(6Z)

where superscript p denotes the order of the itera-tion process and the quantities without the super-script denote those evaluated at the (P-l) itera-tion order.

A computer program was written based on the numerical scheme outlined in Ref. 25 and Eqs. (61) and (6Z) were solved for four cases of practical interest. The results are presented in Section 4. In this method, either equal intervals or nonequal intervals can be used in the n-direction. The interval in the n-direction is increased in a geo-metric progression as

where À is a constant set with a value slightly greater than unity and i is the index of the n-coordinate for the difference net. There are two differences in the numerical procedure between the present work and Liu and Mirels [Ref.

Z3]

.

First, in this work the momentum equation (50) is transformed into a second-order differential equa-tion through a funcequa-tion defined by Eq. (60). The

layer are not satisfied. In order to avoid a dis-continuity in the gradients of the dependent variables at the edge of the boundary layer, they were forced to select a curve for ne in their num-erical procedure. The boundary-layer profiles we re then recalculated using the selected ne distribution. However, in the present work the inviscid flow equations, Eqs. (9)-(11), and the boundary-layer equations, Eqs. (50) and (51), are solved simul-taneously with the same step size 6E;. No inter-polation errors are introduced and the compatibility conditions are satisfied. The selection of an ne distribution is not necessary in this work. The advantage of the present numerical procedure is the reduction of the number of convergent iterations for the nonlinear differential equations and the consequent computation costs.

In the calculations, 80 mesh points were used in the n-direction with À = 1.02 and an initial step size 6n

=

0.026. The step size 6E;

=

0.002 was applied. Normally two or three iterations were required to converge the solution within an error of 10-5 .

3.4 Boundary-Layer Characteristics

The relations between the transformed coor-dinates (E;, n) and the physical coordinates (x,y,t) are given by Eqs. (43) and (45),

y = Hl _{(t)H2(E;)So(E;,n)} where _2w-l [U x (b : : Fo

rf/

2 Hl s _Poo s \loo H 2 (2E;)1/2 (l-E;) aR n S

₌

_f

_gdn 0 0

The boundary-layer thickness oe, displacement thickness 0* and the momentum thickness

e

are expressed by ne 0 _{e = HlH2}

_f

gdn 0 0* _{r ( l} pu J dy _HlHZSl Peue 0

e

r

pu [1 - uU J dy HlH2SZ (63) (64) (65) (66) (67) (68)

and ne is the value of n at the edge of the boun-dary layer.

From the definition of the shear stress

'

00

and

the skin-friction coefficient cf:

,

_llw

_[

_~

_~

₎

w w

,

Cf ₁ w ₂

2

Peue

the dimensionless skin friction coefficient is expressed by (71) (72) 1 W 1 Cf lRë =

[~)2

[ : )2

R

2

(1-W) (H;)o B f U;,O) . 0 w nn

where the Reynolds nurnber Re is given by

Re = [_P_oo_R---,._) W Poo Ru x f;,

bpoou

s 2F lloo e s

The heat transfer at the wall is given by

where K is the thermal conductivity of gas. the definition of Stanton nurnber St:

St = PeueHe (73) (74) With (76)

where He = he + (1/2)ue 2. The normalized expression

for wall heat transfer is

o

(l-f;,) B g (f;"O) y-l Rq>2) w n

+ ~ ~ (77)

3.5 Properties of Hydrogen-Oxygen Stoichiometric Mixtures

A stoichiometric mixture of hydrogen and oxygen

is taken as the working fluid. The properties of

the unburned and burned gases, i.e., the gases behind and ahead of the detonation wave, are taken

from Ref. 24. The chemical-reaction equation is

as follows:

where NI, N2, ... , denote molar concentrations of

H20, OH, ... , in the burned gases.'

The initial conditions of the stoichiometric

mixture of hydrogen and oxygen are Poo = 13.6 atm, Too = 298.l5°K, yoo = 1.4 and Us = 2.98l5xl05 cm/sec.

The thermal conductivity and kinetic viscosity for the unburned and burned gases are calculated

using the semi-empirica 1 re1ations outlined in

8 Ref. 30: n N.K. K =

I

_n 1 1 (78) i=l

I

Nj<l>ij j=l

for the therma1 conductivity of the burned gas, and n Nill i II

L

_n i=l

L

Nj<l>ij j=l

for the viscosity of burned gas, where

[1

+ [

~~

f/2

r

l

:~

f/4

r

J 1 ( M. )1/2 /8 l+~ M. J (79) (80)

and N, ll, K, Mare the mole fraction, viscosity,

conductivity and molecular weight of the species,

respectively. Subscripts i and j denote the ith

and jth species. The details of the calculation

are given in Appendix B.

3.6 Discussion of Boundary-Layer Equations

As stated before, three regions are considered in treating the boundary layer behind a C-J

detona-tion wave. For the expansion region from f;, = 0 to

f;, ~ 0.5, the inviscid flow plays a significant role

in the deve10pment of thc boundary layer. Equa

-tions (50) and (51) are applied directly. For

the stagnation region from f;, ~ 0.5 to f;, ~ 0.95, because the velocity approaches a very small value,

Eqs. (50) and (51) become

(1_f;,)20R1-W [

~

)W(Bf ) Fo nn n + nfnn 2f;, {-

~

[2o+a(2w-l)]f nn + afn + (l-f;,)fnf;,}'" (81) (1_f;,)20Rl-W [

:0

)W[

:r gn)n + ngn 2f;, {-

~

[20+a(2w-l)]gn +

~a

g + (l-f;,)gf;,} ... (82) In the reg ion from f;, 0.95 to f;,

is given by

I, the solution

T(y,t) = (T _e -T )erf(y/l4Dt) + T

w w (83)

where T is temperature, erf (y/l4Dt) is the error function and

D K

= PC p

where C is the specific heat of the gas. The

tem-peratur~

gradient at the wall decreases in propor-tion to t- l / 2. In the initial phase, the tempera-ture gradient is 50 large that a temperature jump

exists at the wall. Details of the derivation of Eq. (83) are given in Appendix C.

Three interesting observations may be made regarding Eqs. (50) and (SI). First, as mentioned above, the equations are singular at ~ = 1. The solution near ~

=

1 is more of analytical interest, as the boundary-layer concept is not valid. Second, the effect of the viscous exponent w on the boun-dary layer depends on the inviscid solutions R and (F/Fo). When w < 1, the gas viscosity and thermal-conduction effects increase as w increases. Third, the coupling between the momentum and energy equations is through Band the last term on the right-hand side of Eq. (50) and the second term on the right-hand side of Eq. (SI).

It is convenient to rewrite Eqs. (50) and (SI) in the general form

(85)

where the ai are the coefficients and W stand for f

_n

and g. Equation (85) has the same form as the heat conduction equation. The coefficient

a

s,

which plays the same role as the thermal diffus-ivity in the heat-conduction equation, is negative in regions of reverse flow. For axisymmetric boundary-layer flows (i.e., cr

=

1), al becomes very small when ~ = 1. It means that the viscosity and thermal conduct ion effects become very small when al ~ O. When al

=

0, Eq. (85) is no longer parabolic, but hyperbolic. Consequently, all numerical methods for sol ving the boundary-layer equations fail in this case. The simple heat-conduct ion equation can then be applied.

The unsteady effect on the boundary-layer development can be estimated as follows. Consider the boundary layer induced by a uniform wave (i.e., wave velocity is constant, or m = 1). The steady-state boundary-layer equations are:

(86)

(87)

(2) l-~ « epf

n

The nonuniformity of the wave front increases the unsteady effect on the boundary-Iayer structure. The order of magnitude for the unsteady terms depends on the inviscid flow characteristics and it is dif-ficult to estimate.

4. BOUNDARY-LAYER DEVELOPMENT BEHIND EXPLOSIVE WAVES

4.1 Boundary Layer Behind Strong Blast Waves Three sets of solutions were obtained. The first set was for the boundary layer induced by a spherical strong blast wave. In order to check the numerical procedure applied here, the case with Y

=

Yoo

=

1.4, w

=

1, b

=

1, gw

=

0 and Pr

= 0.72

behind a spherical blast wave was examined. Re-sults were obtained beyond ~ = 0.95. Figures 10 and 11 show the normalized velocity and temperature profiles. For comparison, the corresponding results obtained by Liu and Mirels [Ref. 23] are shown in Fig. 12. It can be seen that agreement between the two sets of results is excellent. Further tabular comparisons are given in Tables 5 and 6. Agreement of the present results with the series expansion results [Ref.18] for the wall derivatives at ~

=

0 is bet ter than those obtained by Liu and Mirels [Ref. 23].

The thermal boundary-layer thickness is larger than the velocity boundary-layer thickness since Pr < 1 was applied. As ~ increases, the velocity gradient increases at a rate which is larger than that for the temperature gradient. Consequently, a velocity overshoot occurs. Results of other cases with different Pr indicate th at (1) the velocity overshoot decreases as Pr increases, (2) there is no flow reversal when the Pr is varied from 0.72 to more than 3, because the density decreases rapidly with ~. It can be seen from Figs. 10 and 11 that the velocity and thermal boundary-layer thicknesses decrease with ne as ~ increases (the actual thickness increases with ~). They become zero at the origin (~

=

_{1). Therefore, f n}

=

1 for all n at ~

=

1.

4.2 Boundary Layer Behind C-J Detonation Waves A second set of solutions was determined for the C-J detonation wave in a stoichiometric mixture of hydrogen-oxygen with w

=

0.75, Pr

=

2.26, Tw

=

T_oo

=

300o_{K, Y}

₌

1.14, Y_oo

=

1.4 and p

=

13.6 atm for different flow geometries. The ~istance used from the origin to the wave front is 10 cm (corre-sponding to the implosion chamber wall) and the wave velocity is 2982 mIs. Figures 13 to 20 show the normalized velocity and temperature distribu-tions across the boundary layer for the planar, cylindrical and spherical inviscid flows. The wall derivatives are shown in Table 7 and in Figs. 21 to 24. From the results presented above, the following observations are made: (1) Flow reversal occurs for

greater than for cr

=

0, as ~ increases. (4) The thickness of the velocity boundary layer is greater than that of the thermal boundary layer.

A third set of solutions was obtained for a spherical C-J detonation wave in a s~oichiometric

mixture of hydrogen-oxygen (T_ro

=

300 K, Y

=

1.14, Y

=

1.4 and p

=

13.6 atm) with different thermal p~operties: ro

As mentioned before, for Pr > 1, flow reversal exists when ~ > 0.3. In order to examine the effect of the Prandtl number on the flow structure, other cases with Pr

=

1.5, Pr

=

1.0 and Pr

=

0.72 were also studied. Figures 25 to 30 show the normalized velocity and temperature for different Prandtl numbers. From these figures, including Fig. 19, it can be seen that for Pr > 1, the mag-nitude of the flow revers al decreases with Pr decreasing, whereas for Pr < 1, the flow reversal disappears and a velocity overshoot occurs. The magnitude of velocity overshoot increases as the Pr decreases. It is clearly shown in Fig. 27 that for Pr

=

1 neither a flow revers al nor a velocity overshoot exists. Figures 31 and 32 show the wall derivatives of velocity and temperature, respec-tively. It can be seen from Fig. 31 th at when Pr < 1, fnn(~'O) increases monotonically with ~

increasing. However, when Pr > 1, fnn(~'O)

increases at first and reaches a maximum value at about ~

=

0.23, then decreases to a negative value. Separation is said to occur when the derivative fnn(~'O) changes from a pos~tive to a negative value. The boundary-layer th1cknesses of velocity and temperature are shown in Fig. 33. It is shown that when Pr > 1, the velocity boundary layer thickness increases slightly as Pr increases, whereas the thermal boundary-layer thickness de-creases considerably. It implies th at when Pr > 1, more kinetic energy of flow is lost from the main-stream than when Pr < 1. Due to imbalance between the velocity and thermal boundary-layer thicknesses, the flow is forced to separate at the wallor to have a velocity overshoot at the edge of the boun-dary layer. This separation is not observed in the blast-wave case.

Table 8 shows the variation of the thicknesses of velocity and temperature with Pro It was found th at the ratio of the velocity boundary-layer thick-ness to the thermal boundary-layer thickthick-ness is proportional to the square root of the Prandtl number as shown for a steady plane boundary-layer flow [Ref. 27]. For the spherical C-J detonation wave it can be seen from Table 8 th at when Pr > 1, the ~elocity boundary-layer thickness is greater than for the temperature layer. When Pr < 1, the converse statement is true and when Pr = 1, the two thicknesses are identical.

The effect of the viscous exponent and the surface temperature on the boundary-layer flow was also investigated. The normalized velocity and ternperature profiles are shown in Figs. 34 and 35 for the case of w = 1, and in Figs. 36 and 37 for the case of Tw = 0, respectively. The effects of wand Tw on the velocity and temperature gradients are shown in Table 9 and in Figs. 38 to 41. It can be seen from Table 9 and Figs. 38 to 41 that:

(1) when w changes from 0.75 to 1, the velocity and ternperature gradients on the wall increase by a factor of nearly 1.8. (2) The wall-temperature effect on the wall gradients is not significant.

The thermal boundary-layer thickness growing on the wallof the major diameter of the UTIAS hemispherical imp los ion chamber was obtained numer-ically from Eqs. (50) and (51), and analytically from Eq. (83). Figure 42 shows the variation of the thermal boundary-layer thickness with ~ just when the detonation wave reaches the hemispherical wallof the implosion chamber at 34 ~sec from init-iation, and Fig. 43 shows the variation with time at the origin for the case of Tro

=

300oK, b

=

1.26, F

=

0.1511, R

=

0.67, Us

=

2982 m/s and xs

=

10 cm. The predicted thermal boundary-layer thickness using the analytical solution is about 0.054 rnrn at the origin, which is very close to 0.058 rnrn predicted nurilerically.

4.3 Some Discussions on Boundary-Layer Thickness Figure 44 shows the variation of boundary-layer thicknesses for C-J spherical and planar detonation waves in 2H2+02, and strong spherical and planar blast waves in air under the same initial conditions and wave velocity Us and wave position xs. For a comparison, the constant speed nonstationary planar shock-wave case in air is also shown in Fig. 44. The initial conditions are: Pro

=

13.6 atm, Tro

300oK, wave speed Us

=

2981.7 m/sec, and wave position xs

=

10 cm. It can be seen from Fig. 44 th at the variation of the boundary-layer thickness with distance behind the wave behaves quite differ-ently for a C-J detonation wave, a strong blast wave and a constant speed planar shock wave. It is help-ful to recall the relation for the boundary-layer thickness, Eq. (67),

u2w-l x

[

P

)

W

]

1/2 1/2 fne

Ö

=

l

5 5

~

ro

b

~

F

(2~)

gdn (67)

Pro Pro 0 (l_~)crR o

in order to understand these variations. We can see from Eq. (67) that the density ratio plays a very important role in the development of the boundary layer behind.a given wave front. For a C-J detona-tion wave when ~ > 0.5, the velocity of the inviscid flow becomes very small, but it does not matter because the density and pressure remain at reason-able levels. For a constant speed planar shock wave, cr

=

0, R

=

constant and J~e gdn is also con-stant (that is, similarity exists and the boundary-layer profiles are identical anywhere), therefore the boundary-layer thickness ö can be expressed as

ö

=

c~

where c is a constant. For the strong blast wave, however, when ~ > 0.5, the density ratio becomes very small (as shown in Table 1 and Fig. 2). It means that when ~ > 0.5, the inviscid flow is so rarefied that the boundary layer gets comparatively thick (as shown in Fig. 44). Consequently, the assurnptions made in derivation of the boundary-layer equations would cease to apply and the plots in Fig. 44 for these cases for larger distances from the wave are no longer of physical interest.

4.4 Heat Transfer to the Wall behind a Detonation Wave in 2Hz+Oz Mixture

The relation for calculating wall heat transfer

behind a C-J detonation wave moving into a station-ary mixture of 2H_{2+02 is given by (see Appendix D)}

where TT

FR

_o ) W}1/2 (68) (l_~)cr ₍₆₉₎ (70) It can be seen th at BI is a function of ~ only. It decreases as ~ increases from 0 to 1. For planar flow with cr

=

0 it reduces to the following form

._ y

g~-l(gn)w

{ 2+w

[TT

)W}1/2

B - - F - (71)

1 y-l /2 Pr RoR

while B2 represents the effects of the initial conditions. For the structure-independent detona-tion uCJ is a funcdetona-tion of the initial condidetona-tions only. From the expression for BI, it can be seen that qw is also a function of position. For a given position x, ~ can be expressed as

(72)

where x is the distance of the point considered fr om the initiation position. Figure 45 shows the variation of heat transfer to the wall with time t, at different positions for plane flow. Also shown in Fig. 45 are the experimental results from Ref. 28. The agreement between the analytical-numerical results and the experiment al data is satisfactory. A bet ter relation for the temperature-viscosity might even improve the agreement.

To check the influence of the initial conditions on heat transfer, calculations we re also done for the variation of the average heat transfer over given periods of time for different pressures p , and is shown in Fig. 46. The crosses, closed a~d open circles shown in Fig. 46 represent similar experimental results from Ref. 29. It can be seen from Fig. 46 that the relation between heat transfer and initial pressure is not linear and does not justify the conclusion in Ref. 29 that it is linear.

As shown in Ref. 8, the structure-independent detonation inviscid flow properties in a 2H2+02 mixture are very insensitive to the initial condi-tions. For example, the detonation speed changes from 2.59 km/sec to 3.08 km/sec, the temperature from 2934 K to 4645 K, and the pressure ratio from 16 to 21.8 even when the initial pressure changes from 0.01 atm to 100 atm at an initial temperature

of Ref. 27 over the entire 200 ~sec. However, it only agrees weIl with the data of Ref. 15 from about 150 to 200 ~sec. For structure-independent detonations, as shown in Fig. 49, velocity, pressure and density behind the wave front decrease rapidly, then the values of pand p level off to a finite value at about ~

=

0.5. However, the velocity af ter ~ = 0.5 becomes 505mall that only conduction heat transfer becomes dominant. That may be the main reason why the present laminar boundary-layer analysis gives a good representation of the varia-tion of heat transfer to the wall with time behind C-J detonation waves.

5. CONCLUSIONS

A modification to the transformed coordinates of Mirels and Hamman [Ref. 18] makes it possible to apply the transformed equations to flows behind either blast or Chapman-Jouguet detonation waves. It was shown that the Prandtl number controls a reversed-flow phenomenon at the wall and a velocity overshoot near the edge of the boundary layer. No flow reversal occurs in the blast-wave case. The effect of Prandtl number on flow reversal in the blast-wave case is not as significant as in the detonation-wave case. The magnitude of the velocity overshoot for the Chapman-Jouguet detonation-wave case is larger than that for the blast-wave case. The viscous exponent W has a significant effect on the wall skin friction and heat transfer. For W < 1, the value of wall derivatives reduces by a factor of (F/RFo)l-w as w decreases. The effect of wall temperature Tw is found to be small. At lower wall temperatures the flow reversal is delayed to some extent, compared to the higher wall tempera-ture.

To test the analysis, the results of heat transfer to the wall behind a planar detonation wave moving into a stationary mixture of 2H2+02 were computed and the variation of heat transfer with time was compared with experimental data at different positions from the initiation point. Very good agreement was obtained between the present numerical results and the experimental data. This lends confidence to the present ana-lytical-numerical work which is less restrictive in its assumptions and models the actual flow more realistically than has been done previously.

'The boundary layer behind a detonation wave may undergo transition to a turbulent one. Conse-quently, a complete boundary-layer solution through the transition and turbulent flow is very difficult and must be done in stages, as noted in the Intro-duction. For the strong blast wave, the density behind the wave front decreases so rapidly that the inviscid flow becomes very rarefied and the boundary-layer thickness increases very quickly in a short distance (say, 0 < ~ < 0.3) away from the wave front. Consequently, the boundary-layer concept may no longer be valid in the region beyond ~ > 0.3. To

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24. Benoi t, A., "Properties of Chapman-Jouguet Detonations in Stoichiometric Hydrogen-Oxygen Mixtures Diluted with Helium and Hydrogen", UTIAS Tech. Note No. 104, 1966.

25. Glass, I. I. and Liu, ),/. S., "Effects of Hydrogen Impurities on Shock Structure and Stability in lonizing Monatomic Gases. Part 1, Argon", J. Fluid Mech. Vol. 84, Part 1, 1978, pp. 55-77.

26. Blottner, F. G., "Finite Difference Methods of Solution of the Boundary Layer Equations", AlM J. 8, 1970, pp. 193-205.

27. Schlichting, H., "Boundary Layer Theory", McGraw HilI Book Co. Inc., New Vork, 1968, pp. 99-102.

28. Laderman, A. J., Hecht, G. J. and Oppenheim, A. K., "Th in Film Thermometry in Detonation Research", Temperature - lts Measurement and Control, in Science and Industry, Vol. 3, Part 2, 1962.

29. Edwards, D. H., Brown, D. R., Hooper, G. and Jones A. T., "The Influence of Wall Heat Transfer on the Expansion Following a C-J Detonation Wave", J. Phys. 0: Appl. Phys. 3, 1970, pp. 365-376.

30. Bird, R. B. et al, "Transport Phenomena", John Wiley and Son Inc., New Vork, 1963. 31. Cars law, H. S. and Jaeger, J. C., "Conduct ion

of Heat in Solids", Oxford University Press,

Tab1e 1

Inviscid Flow Solution Behind a St rong Spherica1 Blast Wave

-cr 2, m 2/5 E; qJ F R 0.00 0.8333 0.8333 6.0000 0.10 0.68486 0.4231 1. 2318 0.20 0.58374 0.33714 0.39251 0.30 0.50309 0.31324 0.13251 0.40 0.42927 0.30656 0.40661x10-1 0.50 0.35738 0.30495 0.10284x10- 1 0.60 0.28596 0.30464 0.19214x10-2 0.70 0.21472 0.30460 0.22055x10-3 0.80 0.14384 0.30460

_I

0.10238x10-4 0.90 0.075392 0.30460

!

0.40588x10-7 i -~ Tab1e 2

Flow Derivatives Behind aStrong Spherica1 Blast Wave

._.

-cr

₌

2, m

=

2/5

E; _{qJE; FE; RE;}

0.00 1.8056 13.028 125.000 0.10 1.1956 2.2153 15.560 0.20 0.87500 0.58514 4.2368 0.30 0.75925 0.16773 1.4 782 0.40 0.72435

o

.43694x10-1 0.51333 0.50 0.71549 0.9187xlO -2 0.15468 0.60 0.71326 0.13726x10-2 O. 36112x10-1 0.70 0.71136 0.11818x10-3 0.55426x10-2 0.80 0.70435

I

0.36594xlO-5 0.39075x10-3 0.90 0.63478

I

0.73131x10-8 0.35366x10-5 I [

Tab1e 3

Solution for Inviscid Flow Field Behind C-J Detonation Wave

- -

-(]

=

0, m

=

1

(] =

1, m

=

(]

=

2, m

=

<: cp F R cp F R cp F R 0.00 0.4544 0.47752 1.8328 0.4544 0.47752 1.8328 0.45440 0.4 775 1.8328 0.05 0.40772 0.43301 1.6821 0.34050 0.36820 1.4595 0.30781 0.33779 1.3534 0.10 0.36105 0.39234 1.5430 0.28650 0.32261 1.3000 0.24717 0.28807 1.1773 I 0.15 0.31437 0.35527 1.4146 0.24088 0.28807 l..1773 0.19961 0.25350 1.0527 0.20 0.26769 0.32151 1.2962 0.19917 0.25966 1.0750 0.15871 0.22697 0.9556 0.25 0.22101 0.29077 1.1870 0.15984 0.23552 0.98700 0.12222 0.20576 0.8769 . 0.30 0.17434 0.26281 1.0865 0.12234 0.21474 O. n035 0.089251 0.18851 0.8122 , 0.35 0.12766 0.23738 0.99388 0.086556; 0.19684 0.84356 0.059572 0.17450 0.7591

I

0.40 0.08098 0.21428 0.90866 0.052765\ 0.18155 0.78591 0.033490 0.16146 0.7092 0.0343041 0.19329 0.83026 0.021685 I 0.73742 0.012032 0.15504 i 0.6845 I 0.45 0.16881 i I Tab1e 4

Derivatives of Inviscid Parameters Behind C-J Detonation Wave

-

-(] = 0, m

=

1

I

(] = 1, m

=

1

(]

=

2, m

=

1 0.00 0.93356 0.05 0.93356 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.93356 0.33356 0.93356 0.93356 0.93356 0.93356 0.93356 0.93356 0.93355 3.1361 0.85156 2.8958 , 0.77633 ! 2.6727 0.70735

i

_,

2.4655 0.64412 i 2.2732 0.58620 2.0949 0.53317 , 1.9295 I 0.48465 \ 1.7763 0.44028 1 1.6344 0.39972 1.5030 (53.840) (53.827) (180.92) (75.720) 1.2399 : 1.1030 3:8274 1.4515 0.97132 0.86560 0.80745 0.76743 : 0.73318 0.69719 i 0.65220 I 0.58542 0.77469 2.7328 ! 0.62075 2.2208 1.8902 : 0.52153 0.44701

I

1.6399 0.38556 1.4309 I 0.33137 0.28050 0.22807 1. 2432

:::::6

1

1.0467 0.87356 0.76941 0.69293 0.62663 0.55954 0.48058 0.36919

i

The derivatives are all negative.

14 (75.702) (254.51) 1.2615 4.4247 0.8045 0.59808 0.47150 0.38147 0.31085 0.25079 0.19507 0.13595 2.8783 2.1741 1.7377 1.4232 1.1725 0.95509

I

0.74903 0.52541

Tah1e 5

Comparison of Wa11 Derivatives Ohtained hy Three Different Numerical Methods

fnn(E,;,O) Gn(E,;,O)

Liu Liu

Series and Series and Expansion Mire1s , Expansion Mire1s

E,; Method Sol ut ion Present Method : Solution Present

0.0 0.66141 0.66198 0.66147 0.89693 0.89864 0.89735

Tah1e 6

Comparison of Wa11 Derivatives of Liu and Mire1s (1980) and Present Work for a Boundary Layer Behind a Strong Blast Wave

I

fnn(E,;,O)

I

gn(E,;,O)

I

Liu and Liu and

E,; Mire1s Present Mire1s Present 0.00 0.66198 0.66147 0.89864 0.89735 0.10 0.99598 0.99520 0.71326 0.70950 0.20 1.4215 1.4210 0.6506 0.6486

I

0.30 2.0170 2.0170 0.67915 0.6789 0.40 2.8813 2.8800 0.8664 0.8692 ! ! 0.50 4.03096 4.0240 1.2942 1.2980 ! 0.60 5.52177 5.5090 1.8290 1. 8400 0.70 7.94119 7.9040 2.61'496 2.6490 0.80

I

12.7369 12.570 4.0957 4.3220 0.90 27.2309 i _, 25.26 9.0235 9.770

~ 0.0 0.05 0.10 0.20 0.30 0.40 0.45 0.50 0.60 0.70 0.80 0.90 0.95 Tab1e 7

ComEarison of Wa11 Derivatives for Different Cases Behind a ChaEman-Jouguet Wave

J.T

w = 300 K, U)= 0.75, Pr = 2.26)

cr = 0 cr = 0 cr = 1 cr = 0 cr = 1 cr = 1

_I

cr = 2 fnn(~'O)

!

gn (~,O) f

nn (~,O) gn(~'O) fn'1(~'O) gn (~,O)

!

fnn(~'O) 0.4023 0.6354 0.4023 0.6354 0.4023 0.6354 0.4023 0.4149 0.6506 0.4421 0.6883 0.4673 0.7269 0.4800 0.4292 0.6699 0.4586 0.7191 0.5145 0.8043 0.5275 0.4467 0.7101 0.4656 0.7742 0.5943 0.9806 0.5882 0.4293 0.7587 0.4083 0.8314 0.6069 1.2120 0.5190 0.2145 0.8194 0.05685 0.8947 0.1353 1. 5320 -0.3740 -0.5084 0.8537 -0.9738 0.9248 -1. 7140 1. 7410 -3.6520 -0.1279 0.8895 -0.4341 0.9596 -0.8039 1.9860 -2.3680 0.2235 0.9310 0.04352 0.9994 0.2114 2.6110 -0.9636 0.4271 0.9585 0.3287 1. 0270 1. 2560 3.6320 0.3380 0.5605 0.9783 0.5208 1. 0470 2.9270 , 5.7040 2.2290 0.6513 0.9933 0.6577 1.0620 7.6860 _112.1500 7.3090 0.6836 0.9996 0.7111 1.0690 17.360 24.350 17.430 Tab1e 8

Ratios of Boundary Layer Thicknesses for Different Prandt1 Number, Pr

I

I

I

~ 0.00 0.10 0.20 0.30 0.40 0.50 0.60 Pr=O.72 °V/oT 0.8644 0.7695 0.6486 0.3456 0.2129 0.2676 0.70 0.2677 0.80 0.4186 0.90 0.6416 0.95 0.7407 Pr=1.0 °V/oT

I

1.000 1.000 J , 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 16 Pr=1. 5 Pr=2.256 °V/oT °V/oT

I

, 1.377 I 1. 791 i

I

I

1.303

!

1. 749 I , 1.364 1.702 j 1. 454 1.833

I

!

1.541 1.862 1.490 1.858 1.522 1.812 1.514 1.941 1.278 2.263 1.5120 2.158 cr = 1 gn(~'O) 0.6354 0.7482 0.8369 1. 0330 1. 2830 1. 6190 1.8320 2.0820 2.7320 3.7980 5.9640 12.690 25.250