COMBUSTION IN SUPERSONIC FLOW

V. K. Baev, V. I. Golovichev, and P. K. Tret'yakov

Interest in the problem of combustion in supersonic flow has arisen more than a quarter century behind the idea of creating a primary air-reactive engine for flight at hypersonic speeds, in which there occurs a braking of the air before entry into the combustion chamber at moderate supersonic speeds.

The most significant effect in combustion in supersonic flow is a pressure increase in the heat-release zone, which has served as the basis for investigation of various organiza- tion schemes of so-called external combustion, wherein the combustion is organized over the surface of the aircraft with the purpose of creating an uplift or side force, lowering of re- sistance, or providing thrust, i.e., in the general case in order to control the thrust vector.

Somewhat later, supersonic combustion processes were the object of interest in connec- tion with the development of work in chemical and chemical-gasdynamic cw lasers. It may be assumed that the sphere of application of reacting supersonic flows will further broaden in the future. It is possible to point to a number of monographs in which these questions have been reflected to greater or lesser degree [1-5]. In addition, there is a significant number of both foreign and domestic journal publications on details of investigations into various aspects of supersonic combustion [6-9]. In Baev et al. [10], entirely devoted to combustion in supersonic flow, results of theoretical and experimental investigations carried out at ITPM SO of the Academy of Sciences of the USSR between the years 1970 and 1980 have been gen- eralized.

The essence of the problem of the creation or control of the thrust vector of aircraft can be illustrated by means of an analysis of the simplest of relationships, following from the conservation equation for one-dimensional flow in a channel (stream tube) of variable cross section with liberation of heat.

For the variation of the total and the static pressure we have dp.._.s = • 2 dQ

Po 2 ( I + - ~ M ~) ~' (1)

~ dF p 1 _ ~ ~ F ' ( 2 )

where P0 is the total pressure, • is the adiabatic index, M is the Mach number, Q is the heat release, i is the enthalpy, p is the static pressure, and F is the cross sectional area. Total pressure losses due to heat uptake, as follows from Eq. (i), grow with increasing M in the heat release zone independent of flow regime~ Since they characterize the loss of ca- pacity for work of the gas, it is obvious that the higher the velocity in the combustion chamber, the lower the thermodynamic efficiency of the heat engine overall,

From Eq. (i) it is not hard to convince oneself that at significant sonic velocities upon increase of the temperature in the combustion chamber by a fewfold the total pressure losses stand at a few percent or fractions of a percent, whereas at supersonic velocities the pressure must fall a fewfold. Nevertheless, these losses are permissible since the degree of compression in the intake unit of the primary engine at supersonic speeds can stand at a few hundred units and incomplete braking of the stream allows it to have a low static pres- sure (and temperature) in the combustion chamber, which testifies in support of the appplica- tion of supersonic combustion in engines.

It is interesting to note that in Eq. (i) the relative variation of area does not enter directly, on the basis of which we may assume only a weak influence of the form of the com-

Novosibirsk. Translated from Fizika Goreniya i Vzryva, Vol. 23, No. 5, pp. 5-15, September-October, 1987.

0010-5082/87/2305-0511512.50 �9 1988 Plenum Publishing Corporation 511

p l o ~ ~ c

50 .

20L

Fig. Calculated values of the unit impulse for combustion of hydrogen.

bustion chamber on the characteristics of the engine, especially for large M. This is con- firmed by more detailed analysis [Ii]. At the same time, the quantity p can significantly grow or fall as a function of variation of F, which follows quite obviously from Eq. (2), thereby converting the combustion chamber in such an engine into a propulsion element. In this respect, its form can have great significance. The form of the combustion chamber is important also from the point of view of organization of the combustion process.

Understandably, estimates based on one-dimensional equations are rather approximate since the actual structure of the stream in supersonic combustion chambers is far from one- dimensional, and the combustion proceeds in the diffusion regime. The emergence of layered flows, wave structures, significant gradients of parameters leading to freezing out of chemi- cal reactions, to thermal choking or to flow instability is possible.

Organization schemes of external combustion have been based on creating organized regions with substantial elevation of pressure by taking into account heat release in super- sonic flow. (In sonic flow such a process is not considered because of its well known low efficiency [5].) In order to evaluate the potential of such schemes with external combus- tion, we may again make use of an analysis of a system of one-dimensional conservation equa- tions; however, for their closure in this case it is necessary to bring in an additional hypothesis (the equation of the process or law of 9ariation o~ the cross section of the dis- turbed region and so on).

Baev et el. [10] have shown that the relation between the relative increases of tempera- ture and pressure can be estimated by a power relationship of the form p = 8 k, where k = 0.5-1 as a function of the character of the combustion process. In particular, for a high reaction rate it may be assumed that the combustion takes place at constant volume and k = i. Having assumed further that the combustion products expand isentropically until they are at the pressure of the undisturbed stream, it is possible to obtain a simple analytical de- pendence for the unit impulse of fuel, which in the limiting case v = const has the form

I= (~n o+ I) ~cPT~ I+(~L o+i) cvTo I--~ ,

where ~ is the coefficient of surplus of air taking part in the process, L0 is the stoichiom- etric coefficient, g is the acceleration due to gravity, A is the mechanical equivalent of heat, H u is the heat-producing capacity of the fuel, ~ is the degree of completeness of com- bustion, c V is the heat capacity, To is the stagnation temperature, and % is the relative temperature rise.

In Fig. 1 the results of calculation of the unit impulse of hydrogen from Eq. (3) are represented by the solid line. The limits corresponding to pressure increase along the forward discontinuity are represented by the dashed lines~ In the steady process values of the impulses greater than the ordinates of these lines cannot be obtained since this would signify a greater pressure increase than in the forward discontinuity at the corresponding Mach number of the leading stream, which is possible only for movement of the discontinuity towards the stream. If hereby the discontinuity moves away from the fuel intake point, then the disturbed region broadens, and ~ grows. The region of experimental impulse values for hydrogen combustion obtained by Krause, Mhaurer, and Pfeifer [12] are shown in Fig. 1 by the dashed lines. From this comparison it is evident that estimates based on simple hypotheses give reasonable results and can, apparently, be used in an approximate consideration of the

512

efficacy of implementation of schemes of external supersonic combustion. For a more accurate analysis a detailed investigation of the structure of reacting flows in various schemes of organization of the process and the implementation of current methods of physical and mathe- matical modeling are required.

In contrast with sonic combustion, problems arising in the study of the gasdynamics of supersonic reacting flows are significantly more complicated as to the possibility of the generalization of results as well as from the point of view of experimental technique. It is possible in the case of turbulent sonic flames to apply similarity theory to the de- scription of the basic geometric characteristics and limits of flame stabilzation [13-16], implementing a small number of criteria, whereas in the case of supersonic combustion the number of criteria increases, and, as a rule, complete similarity does not obtain [ii]. Of course, it is still possible to apply simple criterial dependences with sufficient accuracy in partial cases to the calculation of the position of the flame in a supersonic stream of a homogeneous mixture [15] or to realize an approximate model [ii].

The fundamental character of supersonic combustion consists of a substantial nonequi- librium of multiparametric flow since the characteristic ranges of variation are determined by conditions of self-ignition of the fuel in mixing with the oxidant~ This predetermines the combination of processes of mixing and combustion (diffusive combustion), and, in addi- tion, because of the high level of flow velocities there takes place a coming together of characteristic times of chemical reaction, convection, and mixing. Hence, the rate of com- bustion cannot be characterized by one characteristic time. The dependence of the rate of combustion on composition and other parameters is significant. Also, in the case of high temperatures, when the condition of self-ignition is assured and the composition of the mix- ture exhibits only weak influence on the characteristic combustion time, in order to char- acterize the total reaction rate it is necessary to make use of the induction time and the energy release time. The dependence of these times on temperature and pressure varies ac- cording to the influence of admixtures. As an example, we can cite a comparison of results in the determination of the distance from the hydrogen injector to the flame in an accompany- ing supersonic flow in the experiments of Ao Ferri in the case in which an electrical heater was used and in experiments at ITPM SO, Academy of Sciences of the USSR, where a flame heater was used, i.e., the air contained the products of combustion and incomplete combustion H20, HO 2, H, and OH [16]. Coincidence of the dimensionless distances, expressed in stream units, in the case under consideration, allows us to obtain a close gasdynamic picture within settings of various dimensionality, which is a simple consequence of the identical ratio of linear scale to induction time.

Data presented in Baev, Tret'yakov, and Yasakov [16] and Suttrop [17] provide a basis for the assumption that the introduction into the stream of intermediate reaction products, for example, as a result of diffusion from the recirculation zones or from outflow from spe- cial gas generators, even for flow parameters which insure self-ignition, brings about a shortening of the induction period and makes the process dependent to a lesser degree on chemi- cal kinetics and to a relatively greater degree on mixing processes. This was realized in a planar channel with sudden expansion at supersonic speed at entry [18], where the fuel (hydrogen) moves into the recirculation zone. In connection with this, one more important circumstance related to problems of modeling combustion in supersonic flows should be men- tioned.

If the process is organized so that chemical nonequilibrium is insignificant, then the fundamental similarity parameters are the Mach number M and the relative preheating 0 = RT0/ RT i. The latter is determined by the composition of the mixture (coefficient of surplus of air), the stoichiometric reaction coefficients, the heat generating capacity of the fuel, and the initial temperature. Therefore, if we wish to model the flow in the combustion chamber of an aircraft flying at some speed corresponding to a definite~chnumber M, then in order to fulfill the modeling conditions of relative preheating it is necessary under laboratory conditions to have the same stagnation temperature or, if that is impossible, to use fuel with the same properties, but different heat generating capacity. In this case the principle of reversal of flow, frequently utilized in aerodynamical investigations (in aerodynamic tubes), runs up against serious limitations, all the more serious because the absolute value of the tempera- ture determines the rate of chemical reaction and the limits of self-ignition.

The organization of two-stage combustion (introduction of the fuel in the (sonic) recir- culation zone, where the first stage of its thermal decomposition and chemical reaction oc-

513

y, mm a b I ,~3 /

1,/n , '

o 400 ~oo o 5 ~b

y, mrn c

6~ Jo 1

0 400 800 ~.mm 7u, k g / ( m 2 - s ~ ) ~U, ,kg / (m~.sec)

~Lo+l Fig. 2. Distribution of consumption (a) of a mixture and (b) of hydrogen perpendicular to the channel, and (c) location of stream- lines (4~ = 0.I); x, mm: i) 182, 2) 362, 3) 542, and 4) 722. Dashed line corresponds to M = i, dash-double point line corre- sponds to isotherms. The region A: T = 1800~ ~ = 0.4.

!. ~'q/Po Me=2 0 [

,~ 19 1

! ~ f ', i

.. o..~,'~! ..,q~ I

' ~'"'"-" " ' " ' ~ ~i ~.~ '

, ' I i % ' ' P ' ~ "4t t o! t -- 6 0 400 80 .0 ::.-ram 4o 1,5 2,0 8

Fig. 3 Fig. 4

Fig. 3. Distribution of static pressure over channel length i) without preheating, 2) with preheating by a flame, and 3) with an electric preheater: i, 2) @ = 1.35, and 3, 4) 3.15. The parameter ~ is equal to i) 33, 2) 13.8, 3) 3.8, and 4) 3.7; To, K is equal to i) 290, 2) 528, 3) 473, and 4) 430~

Fig. 4. Variation of ground pressure at the offset of the chan- nel: To, C: i) 27, 2) i00, 3) 130, 4) 240, 5) 400, and 6) 500.

I";D'~ Ao' sec I \

IC L,

0 ~25 o.sg 0,75 ~%

Fig. 5. Variation of impulse 2) of intake and i) of combustion of hydro- gen in the bottom region of an axially symmetric body (M = 2).

curs) helps to a significant degree to soften the modeling conditions since the process now is characterized to a greater degree by a mixing of two streams (air and a rich mixture of reaction products), which differ only slightly in molecular weight, and heat liberation is limited by the low fuel concentration.

Baev, Tret'yakov, and Konstantinovskii [18] present a detailed investigation of the structure of such a flow in one of the regimes in which a number of interesting properties appear and it is possible to apply optical diagnostics to measure the intensity of local heat liberation and to identify the burnup curve. To the peculiar features of the flow which have been discovered experimentally should be added the quite distinctly expressed two-layered character of the flow (a burning sonic layer and a cold supersonic layer), in this connection it may be noted that the absolute values of the velocities in the sonic layer may be higher

514

than in the supersonic part of the stream. It is also interesting to note that the fraction of fuel which is burning in the supersonic part does not exceed 30% of the total. In Fig. 2 is shown the structure of the flow for combustion in a channel with sudden expansion (height of channel 60 mm, height of the offset 40 mm).

In subsequent works the influence of the configuration of the channel and the means of fuel injection into the main stream on the character of burnup of hydrogen in axisymmetric channels of varying shape under conditions of self-ignitionhas beeninvestigated [6, 19]. Experi- mental investigations into the influence of the dependence of the character of burnup on the wave structure of the flow(density discontinuities and rarefactionwaves) have been carried out [20,21]. On the whole, the accumulated experimental results allowa qualitative evaluation to be made of the influence of these and other factors on theevolution of combustion processes in channels of various configuration, in companion flows with free boundary, in detached zones behind bodies. They can be used for careful generalization on the basis of similarity theory or as test material for checking of unavoidably complicated mathematical models which may be applied in the cal- culation of reacting flows.

An example of some partial generalizations is the approximate similarity of static pres- sure distribution in the channel with sudden expansion for different modes of preheating (electrical or flame) of air (Fig. 3). The second example is the generalization of the ground pressure data Pg at the offset in the channel (Fig. 4). The characteristic salient point at which Pg begins to exceed P0 corresponds to the so-called secondary growth of pg.

A similar dependence is observed for combustion behind the bottom section of a body lo- cated in an aerodynamic tube. Analysis of results indicated secondary growth of pg when per- turbations from the walls of the tube or from specially located bodies reach the sonic wake [22]. If the density discontinuity falls back to the recirculation zone, the pressure in- crease can be so strong that the break zone propagates to the side surface [23]. The or- ganization of combustion in the bottom region as one of the cases of external combustion is of practical interest, in connection with which it is expedient to point to the decrease in unit efficacy upon increase of intake of reacting gas and of the corresponding total effect. This is illustrated by the example of the dependence of the impulse of hydrogen on the magni- tude of relative intake (Fig. 5).

In the literature (see, e.g., [24]) it has been noted fhat an improvement of the unit characteristics for external combustion in the bottom region can be achieved by additional supply of heat to the nonviscous part of the stream. However, an experimental check of such schemes is hindered by the substantial influence of the walls bounding the flow and by the impossibility of creating combustion regions of very small dimensions as a consequence of finite chemcial reaction rates at moderately high temperatures.

Definite prospects in the development of methods of experimental investigation of proc- esses of supersonic combustion are linked to the application of high-enthalpy set-ups of short-time action type impulse aerodynamic tubes [25.]. Here questions of thermal stability of objects are quite simple and the possibility of applying optical diagnostics is assured. At the same time, however, measurement technique becomes much more complicated than in the case of stationary flames.

The steady tendency towards increasing complication of the technical aspects of experi- ment and diagnostic methods is a result of the character of phenomena in which the interac- tion and interweaving of physical and chemical processes is important. In addition to this, the increase in the informativeness of an experiment which enables us to come to a deeper understanding of these processes does not, as a rule, make it possible to simplify the de- scription of the integral characteristics, which is important for engineering practice. Therefore, it is obvious that a unifying basis for such complicated processes must start with a sufficiently ideal mathematical model, including not only a properly complete mathematical description, but also the developed technology of calculational representation - a numerical program capable of realization on present-day computers.

This situation in regard to supersonic combustion was quite obvious from the onset of the problem, and mathematical models were developed simultaneously with the carrying out of experiments practically in all countries. The first attempt to put a mathematical model into final form with the appropriate programming provisions for the problems under consideration in our country was made by Baev, Golovichev, and Yasakov [26]. Further development of the model is reflected in Baev, Golovichev, and Tret'yakov [i0].

515

In order to illustrate the possibilities and perspectives of mathematical modeling meth- ods in the analysis of gasdynamic structures which are realized in various modes of heat and mass uptake into the supersonic stream, it is possible to take as an example the two-dimen- sional model of the full, nonstationary Navier-Stokes equations for either laminar or tur- bulent reacting flow in a planar, cylindrical, or radial channel with homogeneous supersonic airflow at the intake for various modes of injection of gaseous fuel (hydrogen). The system of fundamental conservation laws of this model is set forth in many works, while methods for its numerical solution are considered in a review article by Lapin, Nekhamkina, Stelets, et al. [27]. The method which is applied in the present work is expounded in Baev et al. [I0] and Golovichev [28]. In order to describe the kinetics of the chemical reactions, a suffi- ciently complete model of the oxidation of hydrogen in air [i0] was implemented.

Not all of the reactions taken into account are of equal significance: for the selec- tion of the most important ones at the progranmling step of soltuion of the direct kinetic problem, formal approaches can be used. In the case under consideration, following Pratt and Wormeck [29], we made the assumption that some of the reactions proceed in equilibrium while the rest proceed in the kinetic regime. In order to increase the efficiency of the numerical calculation of the right-hand side of the mass balance equations by eliminating arithmetic operations with zero matrix elements of the matrices of stoichiometric coeffi- cients of both forward and reverse reactions, we implemented a special technqiue for working with sparse matrices. A similar technique is often used in the solution of systems of linear algebraic equations or systems of ODE's, the Jacobian matrix of which has sparse structure [30]. Such an approach is a more universal means of increasing the efficiency of solution of the direct kinetic problem and economization of computer memory in the analysis of multi- step mechanisms of chemical reactions than, for example, classification of reactions by type, proposed by Pratt and Wormeck [29].

For the calculation of combustion processes in turbulent supersonic flow we made use of a semiempirical model [31] incorporating basic ideas of the two-parameter k - e model of turbulence and the assumption of the identity of processes of grid and viscous dissipation in a homogeneous turbulent flow. Here, the size of the finite-difference grid is taken to be equal to the Komolgorov microscale.

Finite-difference analogs of the fundamenta~ conservation laws were solved with the help of the modified semiimplicit method RICE, constructed on the. basis of implementation of the principle of "decoupling" of the effects of the component physical processes being modeled. In the solution of the initial equations we imposed "slipping" boundary conditions around the perimeter of the calculational region, special conditions for modeling the emergence of the sonic stream, "soft" conditions for the emergence of the supersonic stream beyond the limits of the boundary of the calculational region, and "adhesion" or "slipping" along the side and end surfaces of the streamlined pylon~ The energetic boundary conditons corresponded to the case of isothermal or thermally insulating walls. The basic features of the method in- clude the following:

a) The various components of the total physical process of combustion are summed to- gether in the source terms of the mass conservation laws. In this way the modular organiza- tion principle of the program RICE is maintained, and the "stiffness" of the total process can be lower than the "stiffness" of its individual components, which increases the effi- ciency of the calculations.

b) The contribution of the chemical process is also not "broken down" into the contri- butions of the elementary steps. In order to achieve automatic conservation of total mass balance a differentially transformed difference scheme was implemented, and by the method of families [30] at each point in the calculational region a system of ODE's of the chemical kinetics with additional terms reflecting the influence of convection and diffusion is solved, "freezing" at this calculational step the variations of chemical composition resulting from the evolution of the reactions. Such an approach in principle coincides with the well-known method of forward differences, with the difference that the differential approximation of spatial operators coincides with that used in RICE.

In such an approach to the solution of the initial value problem, the necessity of going through the special procedure at the start of the calculations which was needed earlier in the numerical integration of the equations of mass balance in the decoupled variant [32] practically never arises. After the determination of composition, the integration step chosen in the program is implemented in the hydrodynamic step, thereby determiming its tem- poral duration.

516

l l l l [ l [ I l l I l l l l l l [ l l l l l l l l ' l t l l l l l l l l [ I I l l i l

Fig. 6. Velocity vector field for organization of combustion in a channel with sudden expansion; t (dimensionless time) = 0.9; Re = i000.

[ I I I I 1 I I I " t 1 1 ! 1 1 I I I I I I I 1 1 1 ! I I I I I I I 1 1 I I I I I I 1 I [ ] ,

-

I

Fig. 7. Isoiines of static pressure in a channel with sudden expansion.

I I I I I I I I I I I I I I I I I I, I I ~I_I I I I I I I I I I I I I I I I I I I

XXXXX211XZIIIII.~-~..:----- . . . . . . . . . . . . . . . . . . . . . .

Fig. 8. Velocity vector field for normal injection of hydrogen into the supersonic stream; t = 0.5.

l l l l l i , , ! i t l l E , , \ ,.

\

~ . 17,~ -4~---- ....

.,~-00 /--.8~,-S, ; \ \ x x I I I \ .~%,.~a17,,.00\ .8i~-01

Yig. 9. Isolines of static pressure for normal injection.

517

| I I 1 1 1 ] 1 1 1 T I l i l l t l l l t , l l l I l l l l l l l I l i l t I l i t I T

y lw'.ll.t

Fig. 10. Isolines of static pressure for axial injection of hydrogen from an axisyn~netric pylon; t = 0.7.

l l l T l l t l l l T f l l l l l l l t l f l l t l l I i l l l l l l l l t I I t + 1

I I - I I 41 II - I I I Im I I + II+ I ~ II+ 41, ID I I 411.11 41b 41 4 1 . 4 p ~ + + ~ ~ ~ + ~ ~ + + ~ ~ + + + ~ + + +

. . . . . . . . . . . " ~ i ~ ' ~ . . . . . . . . . ~ 1 7 6 s I

Fig. ii. Velocity vector field for axial fuel injection.

10"5s

~56 ~,O. 56 \o~ [ Fig. 12. Isolines of static pressure with infiltration of sonic

jet of hydrogen on the wave structure; t = 0,3.

j?

-,kl

O,~ 0.62 0.62

/ Z Fig. 13. Isolines of static pressure; t = 0.7.

518

I l l l l l l l l l l l l I I I I I I I I I P I I I 1 1 1 t l l l l l l l l l I I I I - . _ I/vf_~ ..............

~ ~ ~ ~--Ib-b.~p~.~L~'L~'!~'~k ..~ ~ ~ ~ ~ ~

�9 ~ ~

�9 --'" Tllll ' '1"lJllJ'lJlJlZllJ ll l I I 1 I I I I I I I

Fig. 14. Formation of a "suspended" zone of recirculation; t= 0.5.

c) Realistic thermodynamic transport and thermochemical properties of the mixture are included in the model. The latter are shown on the basis of calculated data in the auto- matized regime at the stage of formation of the task in the preprocessor regime, which~ cilitates the implementation of the program for practical calculations. A generalization of the formulation of Fick's law for multicompnent mixtures [33] allows the elimination of the correction step for diffusive flow from the calculational process.

d) The calculational scheme allows the possibility of a natural parabolization of the conservation laws based on a realization of the idea of a sliding calculational region [34].

After an almost 10-yr period of application of the stated mathematical model to the study of the fundamental problem of the interaction of the component process in combustion in supersonic flows, diverse results have been obtained. Especially interesting is the nu- merical modeling of situations in which the kinetics of the exothermal reactions are subject to limitations imposed by the hydrodynamic conditions, and even the development of the motion of a continuous medium is part of the combustion process. We may note only that these are peculiarities of the structure of flows which, in our opinion, have still not been studied in detail. In Figs. 6 and 7 are displayed results of numerical analysis of the uptake of mass and energy into a supersonic (M = 4.2) axially symmetric airstream (T = 650~ means of axial injections of a cold stream (T = 300~ of hydrogen from the upper wall of a channel with sudden expansfon. The structure of the inert flow without injection of a secondary stream has been well studied and its elements can be well seen. In Fig. 6 the velocity vector field is shown, in which the points of the sonic region have been specially marked by little squares. The fact that small supersonic velocities are present in the reverse flow in the recirculation zone is of interest. The static pressure distributions shown in Fig. 7 also testify to this. This result has already been published by Golovichev and Yanenko [35].

Figures 8 and 9 illustrate the calculated flow parameter fields for normal injection of an unexpanded (n = 2) sonic stream of hydrogen into the stream of oxidant for which M = 3 in the entrance to the channel. In Fig. 9 the arising of an oblique density discontinuity, reflecting from the lower wall of the chamber with the formation of a sonic (for large in- tensity of the discontinuity - detached) region of flow is noteworthy. We may note that a similar mode of organization of mass uptake leads to choking of the stream [36]for M < 2 after taking into account thermal and mechanical effects.

In Figs. I0 and ii the mode of axial delivery of fuel from the surface of the axisym- metric injector located in the supersonic stream with M = 3 is analyzed. This mode of or- ganization of mass and heat uptake is characterized by somewhat smaller gasdynamic disturb- ances, in particular the sonic flow zones have smaller dimensions. Flows with smaller mag- nitude of M show themselves to be more sensitive to gasdynamic perturbations. Regimes of complete thermal stagnation of the flow for the given flow parameters were not realized in the numerical experiment. It is interesting to note that the bottom pressure for combustion in the bottom region is much higher in the case of injection of an inert gas into the founda- tion of the near wake.

The evolution of the supersonic combustion process in the unstationary regime is il- lustrated in Figs. 12-14. In a planar chamber with two wedge-shaped pylons lying near the upper and lower walls of the chamber, the supersonic airstream (M = 2.5) gives way and there are one or two sonic hot (to 600~ jets of hydrogen. The latter flow out through slits lo- cated near the walls of the channel. The results, which are displayed in Figs. 12 and 13,

519

show the variation in the structure of the static pressure field over the time course of the development of the combustion. Attendant to infiltration of the supersonic stream past the wedge (see Fig. 12), an oblique density discontinuity forms, which interacts with the jet of hot hydrogen, the sound velocity a 0 in which substantially exceeds a 0 in air. The up- take of heat leads to an increase in M and an elevation of the static pressure in the region of flow in front of the wedges. In this case the growth of pressure is sufficient for the formation of suspended recirculation zones by a process described in Ferri [6]. The density discontinuity becomes normal (see Fig. 13) and leads to movement upstream, reaching the en- trance section of the chamber. Combustion with relatively well mixed sonic flo w in the region of the chamber preceding the wedges with subsequent afterburning in the supersonic stream, subject to expansion as a consequence of the presence of bottom regions behind the wedges, occurs, Thus, an unstationary regime of thermal choking of the flow is realized with a "dislodged" shock wave and subsequent expansion of combustion products at supersonic velocities.

Imposition of boundary conditions on the exit of the stream beyond the bounds of the calculational region in all cases met the requirements of "soft" boundary conditions, which is equivalent to a calculational regime of outflow of the mixture from the chamber. An ex- ample of formation of a suspended recirculation zone is shown in Fig. 14-f~9~fuel delivery from a pylon located in the central part of the chamber. The above-noted qualitative regularity of supersonic combustion for such a mode of fuel delivery is essentially preserved.

The examples of numerical modeling of complex combustion processes in supersonic flows which have been adduced illustrate the possibility of obtaining useful information which may reveal their basic regularities, determined by the parameters of the entrance stream, mode of injection and type of fuel, and also the interaction of various factors pertaining to combustion.

LITERATURE CITED

i. E. S. Shchetinkov, Physics of Gaseous Combustion [in Russian], Nauka, Moscow (1965). 2. A. G. Prudnikov, M. S. Volynskii, and V. N. Sagalovich, Mixing and Combustion Processes

in Air-Reactive Engines [in Russian], Mashinostroenie, Moscow (1971). 3. V. S. Zuev and V. S. Makaron, Theory of Primary and Rocket-Primary Engines [in Russian],

Mashinostroenie, Moscow (1971). 4. R. I. Kurziner, Reactive Engines for High-Speed Supersonic Flight [in Russian], Mash-

inostroenie, Moscow (1971). 5. F. Bartelme, Gasdynamics of Combustion [Russian translation], Energoizdat, Moscow (1981). 6. A. Ferri, An. Rev. Fluid Mech., ~, 301 (1973). 7, V. L. Zimont, V. I. Ivanov, et al., in: Combustion and Explosion [in Russian], Nauka,

Moscow (1977). 8. V. N. Strokin and L. A. Klyachko, Inzh.-Fiz. Zh., 17, No. 3 (1969). 9. E. A. Meshcheryakov, V. M. Levin, and V. A. Sabel'nikov, Tr. TsAGI, Issue 2193 (1983).

10. V. K. Baev, V. I. Golovichev, P. K. Tret'yakov, et el., Combustion in Supersonic Flow [in Russian], Nauka, Novosibirsk (1984).

ii. V. K. Baev, V. A. Konstantinovskii, and P. K. Tret'yakov, in: Gasdynamics of Combustion in Supersonic Flow [in Russian], Novosibirsk (1979).

12. E. Krause, F. Mhaurer, and H. Pfeifer, ICAS Paper, No. 72-21 (1972). 13. Vo K. Baev, Zh. Prikl. Mekh. Tekh. Fiz., No. 4 (1966). 14. Investigations of the Combustion of Gaseous Fuels [in Russian], Novosibirsk (1979). 15. V. K. Baev and P. K. Tret'yakov, Izv. Sib. Otd. Akad. Nauk SSSR, Tekhnika, l, No. 3

(1969). 16. V. K. Baev, P. K. Tret'yakov, and V- A. Yasakov, in: Combustion and Explosion [in Rus-

sian], Nauka, Moscow (1972). 17. E. Suttrop, Catalytic Induction of Hydrogen Combustion in Hypersonic Jet Propulsion [in

German], Jahrburg, Braunschweig (1973). 18. V. K. Baev, P. K. Tret'yakov, V. A. Konstantinovskii, et el., Fiz. Goreniya Vzryva,

ii, No. 3 (1975). 19. V. A..Zabaikin, A. M. Lazarev, E. A. Solovova, et al., Vests. Akad. Navuk BSSR, Set.

Fiz.-Energ. Nauk, No. 3 (1986). 20. V. A. Zabaikin and A. M. Lazarev, Izv. Sib. Otd. Akad. Nauk SSSR, Tekhnika, No. 4 (1986). 21. A. F. Garanin, V. L. Krainev, and P. K. Tret'yakov, Fiz. Goreniya Vzryva, 20, No. 2

(1984). 22. V. K. Baev, S. A. Vuititskii, A. F. Garanin, et al., Fiz. Goreniya Vzryva, i__3, No. I

(1977).

520

23. V. K. Baev, A. V. Lokotko, and P. K. Tret'yakov, Fiz. Goreniya Vzryva, 9, No. 5 (1973). 24. W. C. Strahle, Twelfth Symp. (Intl.) on Combustion, The Combustion Institute, Pittsburgh

(1969). 25. V. K. Baev, ~. V. Shumskii, and M. I. Yaroslavtsev, Izv. Sib. Otd. Akad. Nauk SSSR,

Tekhn., ~, No. 4 (1984). 26. V. K. Baev, V. I. Golovichev, and V. A. Yasakov, Two-Dimensional Turbulent Flows of Re-

acting Gases [in Russian], Nauka, Novosibirsk (1976). 27. Yu. V. Lapin, O. A. Nekhannkina, M. Kh. Strelets, et al., in: Itogi Nauki i Tekhniki,

Vol. 19, VINITI, Moscow (1985). 28. V. I. Golovichev, in: Processes of Turbulent Transport in Reacting Systems [in Russian],

Minsk (1985). 29. D. T. Pratt and J. Wormeck, CREK: A Computer Program for Calculation of Chemical Reac-

tion Equilibrium and Kinetics in Laminar and Turbulent Flows, Report No. WSU-ME-TEL (76)1 (1976).

30. A. H. Sherman and A. C. Hindmarch, GEARS: Solution of ODE's Having a Sparse Jacobian Matrix, LLNL Report UCID-30116 (1975).

31. V. K. Baev and V. I. Golovichev, Tez. Dokl. Shkoly-Seminara Sotsialisticheskikh Stran "Vychislitel'naya Aerogidromekhanika," Moscow-Bukhara (1985).

32. S. Kojima, Combust. Flame, 22, No. 3 (1986). 33. V. I. Golovichev, Fiz. Goreniya Vzryva, 22, No. 3 (1986). 34. O. A. Farmer and J. D. Ramshaw, J. Comp. Phys., 24, No. 23 (1977). 35. V. I. Golovichev and N. N. Yanenko, Dokl. Akad. Nauk SSSR, 272, No. 3 (1983). 36. V. I. Golovichev, Fig. Goreniya Vzryva, 19, No. i (1983).

NONSTATIONARY COMBUSTION OF CONDENSED SUBSTANCES

SUBJECTED TO RADIATION

V. E. Zarko, V. N. Simonenko, and A. B. Kiskin

The study of the nonstationary combustion of condensed substances subjected to radia- tion is of interest from both the practical and the scientific viewpoints. Indeed, combus- tion occurs in the presence of notable radiation fluxes from an intrinsic flame in many cases of practical importance, where the effect of their infuence can increase substantially in the nonstationary combustion regimes. Meanwhile, thermal radiation is an effective means for physical modeling of nonstationary combustion processes. The radiation flux can be mea- sured reliably and comparatively easily by varying the amplitude and mode of its change in time. Periodic and step energetic actions on a burning system can here be realized success- fully and experimental data can be obtained on the stability of the combustion process as can information about the mechanism of nonstationary combustion under limit conditions. It is important to note that existing theoretical representations are based primarily on a priori assumptions about the nature of the progress of chemical reactions and the heat transfer be- tween zones. In this connection, the detailed experimental information about the combustion processes of condensed substances acquires great value.

Results of an experimental and theoretical investigation of the nonstationary combustion of condensed substances are represented in this paper, where the double-based propellants H, H + 1% C, H + 1% PbO as well as miscible compositions on an ammonium perchlorate base are the substances. Attention is paid to the specifics of physical and mathematical modeling of the processes under irradiation and interrelation of the responses of the burning system to pressure and radiation flux perturbations, and the possibilities of quantitative predic- tions of the combustion rate under irradiation are analyzed:

Novosibirsk. Translated from Fizika Goreniya i Vzryva, Vol. 23, No. 5, pp. 16-26, September-October, 1987.

0010-5082/87/2305-0521512.50 �9 1988 Plenum Publishing Corporation 521