ANALYSIS OF FLUID DYNAMICS OF SUPERSONIC COMBUSTION

PROCESS CONTROLLED BY MIXING

ANTONIO FERRI AND HEP~BERT FOX

New York University, University Heights, New York, New York

The fluid dynamics of supersonic combustion is discussed. The interference between the combustion process and the supersonic flow secondary to the combustion region is described. Multiple injector flow fields are described from the point of view of mixing and of interaction with the external flow. It is shown that the selection of i~,jector location and combustion proc- ess can be utilized to produce compression waves of controlled strength. Such waves can be utilized to reduce the flow Mach number in front of subsequent injectors. This effect is called thermal compression. Engineering criteria for utilization of thermal compression are presented An example of such utilization is described. The example indicates that the iateraction between combustion and geometry is of primary importance for the fluid-dynamic process. The effect of this interaction cammt be accounted for by a simple one-dimensional analysis. Only a judicious combhmtion of mkxing analyses and more complex analyses, that takes into account the for- mation a~d propagation of the waves due to combustion, can give detailed qualitative itfforma- tion on the fluid dynamics of supersonic combustion.

1. In t roduc t ion

The processes associated with supersonic com- bustion have been of some continuing interest. The purpose of this paper is to discuss some fluid- dynamic aspects of the process. The main problem in the burning process at supersonic speeds is related to the fact that the heat addition at supersonic speed tends to produce either large variations of streamtube area or large pressure rises, or both. Such disturbances propagate in the flow through compression waves and shock waves. The analysis of such phenomena is extremely complex in the general case where boundary conditions impose, simultaneously, variation of steamtube area and pressure, because a detailed analysis of the entire flow field is required.

From a practical point of view, one of the requirements of efficient engine design is that the entropy rise in the flow, due to formation of shocks, be minimized; therefore, design criteria must be developed to insure that such compres- sion waves produced by the combustion process are cancelled before strong focusing, and to prevent the shocks from propagating downstream into the nozzle. I t must be noted that the cancel- lation of compression waves or shock waves does not occur automatically during the expansion process unless the process produces expansion

waves of the same family as the shock waves that are in the region very close to the shock; there- fore, a detailed analysis of the mechanism of formation and cancellation of such waves is required for a satisfactory engine design.

The purposes of this paper are to discuss in detail the wave mechanism produced by super- sonic combustion, to outline engineering methods of analysis, and to illustrate such a method by applying it to a two-dimensional burner. The selection of a two-dimensional design, while not of direct practical interest, is due to the fact that it is easy to analyze and to present. I t may be noted, however, that the physical phenomena described and the basic criteria developed will not be altered when three-dimensional flow is analyzed. An important result of the analysis will be that the process of combustion, if correctly utilized, permits the design of burners which perform as if the inlet has variable aerodynamic contraction ratio without requiring actual varia- tion of the physical geometry of the shape of the channel.

The combustion process selected for the examples presented is one of constant-pressure burning that appears to be of direct practical interest (see Ref. 1); however, all of the con- siderations presented here are general and may be applied directly to any other process selected in the design provided that the flow external to the burning is supersonic.

1105

1106 SUPERSONIC COMBUSTION

Yj =0.05O ft.

y //"r ~ M=2Zl/'v ~ . / / " .. EDGE OF .... / / ~ ^ ~ ~ ~ / //-MIXING REGION / " /

. . . . X___':V, ,,NE

. . . .

o , I , \ . . . .

, - x ,, \ �9 .. ( , ) ' % . - - - - _ _ _ ~ , / STREAMLINES)

�9 -._ " ~ . . . - t . _ . _ _ , ~ - /

Fro. 1. Constant pressure supersonic combustion, M~ = 6.0, h = 90,000 ft.

2. Mix ing with H e a t Addit ion

The classical analysis of mixing with or without chemical reaction is performed by using bound- ary-layer type of equations. 2'~ In this approxima- tion, the pressure variation normal to the streamline is neglected, and an average value of pressure along the streamlines is used which is a function only of the streamwise coordinate. Such a value in the analysis must be given as a bound- ary condition. In actual problems, this boundary condition is determined by specifying the shape of a stream surface; therefore, the pressure distribution is a function of the mixing and com- bustion process and is not known a priori . A more advanced analysis then needs to be de- veloped, 3,4 where the pressure field is obtained from the running solution and from the available boundary conditions and is not assumed arbi- trarily a priori . This second method is more satisfactory; however, it is more complex and needs to be used only for the analysis at off- design conditions. When the design of the burner is given, the simple mixing analysis permits the selection of the physical boundary conditions for the burner design corresponding to a selected thermodynamic cycle for the design conditions. The results of both analyses give some important information on the interaction between the flow field and the combustion controlled by mixing, and indicate the possibility of obtaining simple

models for preliminary analysis of such an inter- action. Both techniques for solution have been programmed for a high-speed computer; the results presented here are a combination of the two.

Consider a combustion process controlled by mixing as illustrated in Fig. 1. A two-dimensional jet of gaseous fuel is injected parallel to the flow in a supersonic air stream. In the example presented in Fig. 1, the stagnation temperature of the air corresponds to a free-stream Mach number of 6, and the local Mach nmnber is 2, while the fuel is hydrogen at 2000OR, and the injection Mach number is equal to 1. Because of the high temperature, the combustion takes place immediately; therefore, the process is mixing controlled and in chemical equilibrium. The corresponding results have been calculated with the requirement of constant pressure along the edge of the mixing region. In Fig. 1, in the region of positive y, the streamline for constant pressure along the edge of the mixing region and the characteristic net that is consistent with this constant-pressure assumption are presented; in the region of negative y, the streamlines obtained by a simple mixing calculation are given.

As is shown in Fig. 1, the combustion process produces compression waves followed by expan- sion waves that propagate in the flow, and the process can be at constant pressure only if the waves of the opposite family reach the region of

FLUID DYNAMICS 1107

Q8

06

Y (ft) 04

0.2

0

1.0

P 0.9

O,e

/ / ,.,,~EDGE OF MIXING

~ . ~ /'-REGION . j

0 0.5 40 1.5 2.0 2.5 x l f t )

<)

05 1.0 1.5 20 2.5 X (ft.)

FIG. 2. Supersonic combustion in ~. uniform stream.

the combustion. The analysis based on boundary- layer theory assumes that the pressure is constant at each value of x. Therefore, the curvature of the streamlines obtained by this technique is not satisfactory, especially near the beginning of the mixing region. The actual pressure distribution is given by the external boundary conditions and a change in these conditions affects the process of mixing; the dependence of the mixing on gradual and small pressure gradients is, however, small. Therefore, the effects of a small variation of pressure can be analyzed in a first approximation with an hlviscid flow-field analysis, and then the calculation of the mixing process can be repeated. In Fig. 2, the pressure distribution along the edge of the mixing region consistent with the assump- tion of uniform flow upstream of the injector, the resultant streamlines, and the characteristic net for the same initial conditions of Fig. 1 are shown. The streamlines at the beginning of the mixing have a rapid and discontinuous deviation; and then the inclination decreases gradually as shown. The presence of the initial inclination is not consisteut with the assumption of constant pres- sure unless the required shock is produced by the injector as shown in Fig. 1 (for y > 0).

The variation of curvature of the streamline and the shape of the streamline depends on the rate of mixing and on the amount of gaseous-fuel mass flow injected; therefore, it can be controlled

by several different mechanisms. In general, the angle of the shock and the deviation of streamlines depend upon the amount of heat released locally; therefore, it depends upon the rate of diffusion of fuel in the oxidizer, which in turn depends on the eddy diffusivity of the mixing on the heating characteristics of the fuel. In Fig. 3, the variation of inclination 0w of the streamline at the edge of the mixing as a function of fuel jet conditions is shown at the edge of the mixing. If the chemical reaction is in equilibrium, a given change of the v~lue of the turbulent diffusivity changes the scale of the x axis proportionately for the mixb~g process. In addition, the mixing depends on the shape of the jet and on the contact suface betweeu fuel and oxidizer.

The results of Figs. 1 and 3 are of interest because they indicate that the combustion pro- cess is equivalent, from the wave point of view, to the presence of a physical body. Schematically the combustion process produces first compres- sions, then expansions as a wedge-slab for two- dimensional flow, or as a cone cylinder for the axially symmetric case.

3. T h e r m a l Compress ion

These considerations can be generalized to three-dimensional mixing and tend to indicate

1108 SUPERSONIC COMBUSTION

X-30 --=

4

Peue

PJ uj

3

x

I I ] I [ 1 o 6 8 IO 12 14 16 18

Ow

~ e

2 4

Fro. 3. Variation of the inclination of the stream- line at the edge of the mixing region for two-dimen- sional flow as function of the jet conditions at three distances from the injection point. Mo~ = 6.0, h = 90,000 ft, U e = 4,200 ft/sec, p~ = 1.18 X 10 -3 slugs/fta.

that the combustion process produces compression waves in the flow and is consistent with the con- cept of equivalence between heat sources and volume sources. The compression waves and the following expansion waves affect the flow field

outside of the combustion region. The propaga- tion of these thermally generated waves changes the local flow field outside of the combustion region and interferes with the waves produced by the walls; therefore, account must. be taken of these waves in the analysis of the combustion process. In addition, such waves can be utilized to decelerate the flow surrounding any given combustion region which is producing the waves. This phenomenon recognized first by the first �9 rather has been called "Thermal Compression."

In order to use such thermal compression to decelerate the uncombusted air outside a given combustion region, some basic features are required. The combustion must start locally in given region of the flow where the Maeh number is sufficiently ]ow. Then the combustion produces compression waves that propagate downstream. Any subsequent combustion region must be located downstream of the waves produced by the first combustion region if the waves are to be utilized to compress the air in front of a second combustor. Therefore, the injector must be located downstream of the first injector. In addi- tion, if uniform thermodynamic properties are required in all of the streamtubes of the flow, the inlet must be designed to produce nonuniform flow in the cross section of the region in front of the injectors. Therefore, the utilization of thermal compression as a device to decrease the compres- sion produced by the inlet requires nonuniform flowentering the combustion region in the absence of combustion, and a sweptback injector distribu- tion.

These requirements can be best obtained with

LEGEND - - STREAMLINE , / . .~ '~ -ED OF COMBUST,ON ZONE . ,

- - SHOCK / / ~ I "~ .../'-" " / / f . . --EXPANSION M.=C25349t m ,/.,~/~v . . ~ S'~ ' -

AT POINTS "P" M.= 60 \ - m / ~ , " - o / . ~ / / / M=2.O h = 60 kft. \ . . , / / , ~ / / - / /~/~ ' " P = . .

?> . . j -

- - , . o /

- - / ~

. . . . -----~. ~.".2--.-----

{

Fro. 4, Inlel, hiLt'tier' desigu which utilizes thermal compression.

FLUID DYNAMICS 1109

COMBUSTION INDUCED MACH NUMBER ~ -DISTRIBUTION

/// //" //

, / / / / /

. - - -> --- '7 \ MACH NUMBER / / / / / ~DISTRIBUTION BEFORE

/ /" / / COMBUSTION /_~ LMy

/ /

/ r

i i i ,

M b M l M2 M a M 4 ~M

--COMBUSTION INDUCED COMPRESSION WAVE

FLAME M O D E L M 4 ~ Mb

// M b DESIGN BURNER MACH NUMBER

M2MaM 4 iNLET MACH M._j.,~ NUMBER DISTRIBUTION

x

FIG. 5. Schematic wave distribution and Mach number distribution required in order to utilize the thermal compression effects.

three-dimensional flow, because it eliminates the necessity of compressing and expanding the gas; however, it can be easily illustrated with two- dimensional examples. Consider, then, the flow in Fig. 4. Assume that the Mach number distribu- tion produced by the inlet alone in the absence of combustion is such that the local Mach number

--- DOUBLE INJECTOR ..... SINGL F INJECTOR

+ / ! IM= 3 /

o 0.4 0,8 L2 t6 20 2.'~

X(ft) (a) S~reamline ~nd Wave Pattern

aH 2 0 0.1 0,2 03 04 05

3

o.s[ ' =3o.0 f -TEMPE ATURE

(14

I000 1400 1800 2200 2600 3000

T(~ (b) Hydrogem and Ten~,erature Profile

F[(~. 7. Comparison of colinear single and double injector configuration.

is equal to 2 near the first injector, that it becomes equal to 2.19 in front of the second injector, and equal to 2.39 in front of the third injector. The combustion process produces additional shock waves that cancel the gradient of Mach number produced by the expansion waves generated by the inlet and makes the Mach number of the flow in front of all injectors the same. I t must be noted that the local Mach number at which combustion

HYDROGEN INJECTED AT M = 3 . 0

P.ODUCEO ,F CONTINUES

- - S H O C K WAVE

Fro. 6. Flow field produced by parallel injectors. Air Mach number equal to 2. Hydrogen Mach number equal to 0.85; To air = 2000~ ToH2 = 2000~

l 110 SUPERSONIC COMBUSTION

t 2 - ~ \ . / . \ ~ 0

Y i / _. .J4rz-------r~/ 0 4 - / ~ . , . ~ ' ~ / ~ / ~ STREAMLINES

NEG OF ~ S I N LE INJ CTOR

0 0,4 0.8 12 16 2.0 2fl"

XCft.)

(a) Streamline aad Wave Pattern

(311-I 2 0 0 0 5 010 0.15 0.20

i - - 1 , 1

~.2~ X=so.o,. Y

0 8 . ~ " ~ TEMP~ATURE~

IO00 1400 1800 2200 2 6 0 0 3 0 0 0

TC*K) (I,) ff~ogea aad Temperature Profile

Fro. 8. Comparison of staggered injector configuration.

takes place is constant and is 1.82; however, the minimum geometrical area required by the channel is nmch larger than the area required for such conditions when the local Math number at

the injector station is obtained only by geometri- cal consideration.

In the design of Fig. 4, (tie first set of injectors is utilized to produce compression waves required by the condition that the local Mach number in front of the injector is constant. The second group of injectors are utilized to cancel the expansion waves that follow the compression. In addition, their location is selected in such a way that the mixing is accelerated; therefore, the fuel becomes fairly uniform in a short length.

The basic features required for utilization of thermal compressions are illustrated qualitatively in Fig. 5. The dotted line in the figure gives the Math number distribution generated by the inlet alone; the continuous line is the Math number at which the combustion takes place.

4. Multiple Injectors

In supersonic combustion controlled by mixing, the combustion process and mixing are simul- taneous; therefore, the injector pattern must be selected on the basis of two requirements: (1) wave pattern; and (2) fuel distribution. This requires that mixing from several injectors must be analyzed; numerical programs have been developed that permit the study of flows having several injection stations. Two quantities enter into the criteria for location of the injectors: The spacing in the direction normal to the streamlines

I .EGEYO

. . . . . . . . . STREAMLINE

. . . . . . SHOCK

. . . . . . . . . EXPANSION

DETAIL ~

COWL / T iP

~=54.1�88 ~'9: i 6 ~ - . . . . ~ . . . . / / / 7 ~ " . . . . . . M=a03 7 ~=0.74~ \P%--0~3/~ / _ . ~-3oo h ,7/fZ/_.z..z--~'--"~ ~---~L '~ . ~ ".,

. 1 . 7 5 . _ _ _ .

DETAIL

Fro. 9. Example of burner design using thermal compression in order to obtain uniform burner conditions.

FLUID DYNAMICS 1111

that affects the mixing and the strength of the waves, and the spacing along the streamlines that affects the position of the waves in the flow field. The spacing in the direction normal to the streamline depends on the fuel mass flow of each injector. If the spacing along the streamlines is zero, as is the ease of the unsweptbaek burner, then the length required to obtain a given uni- formity for a given number of injectors depends only upon the mixing characteristics of the flow. However, all the compression waves generated by eoInbustion superimpose and produce strong positive pressure gradients in the eoinbustion region, as shown schematically in Fig. 6. Because the combustion process increases the local temperature and, therefore, the speed of sound, the flow tends to become locally subsonic. Then, small subsequent pressure rises due to combustion will produce locally reverse flow as shown experi- mentally in Ref. 1. In the ease of Fig. 6, reverse flow would take place before the mixing regions of two adjacent jets meet, unless expansion waves are generated by the walls.

Figures 7(a), 7(b) and 8(a), 8(b) give the flow field and mixing distribution for two limiting eases of possible nmltiple injector configurations.

Figure 7(a) corresponds to two injectors in line. The second injector increases the concentra- tion of hydrogen at the axis and produees addi- tional combustion. In addition, it ehanges the curvature of the streamlines in the region outside the eoinbustion zone. In the example analyzed, the injector is placed in a region of the flow where, locally, surplus hydrogen is present and all of the oxygen present is completely burned. The second injector then changes only the hydrogen eoncen- tration and, therefore, the curvature of the streamlines. The change of concentration at the axis in this ease changes the streamline shape slightly downstream of the injector region, as shown in Fig. 7 (a). If the position of the second injector remains at the axis, but moves down- stream, this effect becomes more pronounced. Downstream of station x~ Y = 30, where Y is the height of the injector, exeess oxygen starts to be present at the axis and combustion can take plaee immediately if a second injector is introduced there. The heat release at the axis changes the sign of the curvature of the streamline because it produces local compression waves. Similar effects can be obtained by moving the second injector off the axis and closer to the first injector. Typical results are shown in Fig. 8 (a). In this example a strong discontinuity of curvature is obtained because the injector is placed in a region where a large amount of free oxygen is available. All the calculations have been made for the condition of constant pressure in the combustion region. Figures 7 (b) and 8 (b) also give the concentration

of unburned hydrogen and temperature distribu- tion at a station x /Y = 30.

5. Illustration of the Use of Thermal Compression in Burner Design

In order to illustrate some of the interesting features of the utilization of thermal compression in the design of the inlet burner combustion region, a two-dimensional configuration has been analyzed. The wave diagram of such a design is shown schematically in Fig. 9, where the free stream Math number is 6, and where the inlet produces a quite nonuniform flow at the entrance of the burner. The flow is expanded along the lower surface and the Maeh number at the en- trance of the eombustor varies from M = 2.65 at the upper surface to a value of 4.97 at the lower surface. The compression waves produced by combustion interact with the expansions pro- duced by the walls and turn the flow without changing the pressure. The second line of injectors carry much less mass flow than the first line (about 15%) and are located in such a way that the hydrogen mixing differs locally by only a small percentage from the average value. The combustion occurs practically at constant pres- sure and at an average Math number of 2.3 with maximum variation contained between 2.27 and 2.32. An alternate design could be conceived where all the expansion waves are of the same family as the compression waves produced by the combustion, as shown in Fig. 10. In this case, the inlet is designed to have uniform flow in front of the burner region. The static pressure in the combustion region is roughly constant in the two cases, and the initial Math number for the com- bustion is the same. However, the geometrical contraction ratio for the two inlets is quite different. The first design has a minimum area that is 1.95 times the minimum area of the second design, in spite of the fact that the combustion Mach number is the same, and, therefore, the efficiency of the two systems is the same.

The first design has much better performance at Mach numbers below the design Mach number because it can handle much higher flow before choking. The differences between the two designs give some indication of all the possibilities of utilizing thermal compression for the design of engines capable of performing in a wide range of Mach numbers.

6. Conclusions

The fluid dynamics of supersonic combustor design requires tile analysis of the interaction of the mixing phenomena, combustion, and inviseid flow regions. Such an analysis can be performed

11 [2 SUPERSONIC COMBUSTION

DETAIL

LEGEND

. . . . . STREAMLINE M ~ 6.0 - - SHOCK . . . . EXPANSION

r - ~ M = 2 .30 - - ]

M=2.64 [ P=91 t ~"-~ . . . , ~ - ~

COWL

F ~ INTERNAL CONTRACTION iS=o.zs7 L , . r ~ o ' ,, A 2 0 9

2 4 ~ - i = ~ , ~ =0 .454

DETAIL

FIG. 10. Inlet burner design having same conditions as inlet in Fig. 9 but not utilizing thermal compression.

for engineering purposes by a judicious combina- tion of mixing analyses based on boundary-layer approximations and more complex analyses that take into account the wave mechanism for prop- agation of pressure disturbances. Multiple injector analysis is required for such designs. The concept of "thermal compression" and its use are dis- cussed. I t is shown that large improvements in maximum contraction requirements can be obtained by suitable choice of the thermal compression phenomena.

ACKNOWLEDGMENTS

This work was carried out at New York University under the sponsorship of Contract No. F33615-68- O-1184, monitored by the Aerospace Research Laboratories of Wright-Patterson Air Force Base.

The authors are sincerely grateful to Gary Bleich for his effort in performing the requisite computer runs, and to Anthony Agnone for his work in the preparation of the characteristics program.

REFERENCES

1. FERRI, A.: J. Aircraft, 5, 3 (1968); also presented as Paper 66-826 at AIAA Third Annual Meeting, Boston, Mass., Nov. 29-Dec. 2, 1966.

2. FERRI, A., LIBBY, P. A., AND ZAKKAY, V.: Teoreoical and Experimental Investigation of Supersonic Combustion, ARL 62-467, Sept. 1962, Aeronautical Research Laboratories;also Third ICAS Congress, Aug. 27-31, 1962~ Stockholm, Sweden; also PIBAL Rept. 713, ARL 62-467, AD 291712, Sept. 1962, Polytechnic Inst. of

Brooklyn. 3. EDELMAN, R.: Diffusion Controlled Combustion

for SCRAM JET Application, Part I--Analysis and Results of Calculations, GASL TR-569, Dec. 1965, General Applied Science Labs., Inc.

4. FERRI, A., MORETTI, C., AND SLUTSKY, S.: J. Soc. Ind. Appl. Math. 13, 229 (1965).

5. MOaETTI, G.: Analysis of Two-Dimensional Problems of Supersonic Combustion Controlled by Mixiag, Paper 64-96, 1964, AIAA.

COMMENTS

F. A. Williams, University of California-- San Diego, La Jolla, California. What is the basis of your statement that, for supersonic com- bustion, liquid-fuel injection is not competitive

with gaseous-fuel injection? Liquid fuels atomize and burn readily when injected into supersonic air streams. They are likely to lead to mixing and wall-friction losses comparable to those of

FLUID DYNAMICS 1113

gases, and, for equal mass-injection rates, shock losses associated with the injection process are likely to be smaller for liquids because of the smaller volume injection rate.

pecially at Mach numbers where the stagnation air temperature is low.

A. Ferri. The main reason for the injection of gaseous fuel in place of liquid fuel is not related to the efficiency of mixing and combustion pro- cesses, but is related to the practical problems of cooling of the structures. The main source of coolant for a hypersonic vehicle is the fuel. The cooling capacity of the fuel is strongly increased if the latent heat of vaporization is utilized; therefore, it is very probable that, in any practi- cal application, the fuel will be injected as a gas at fairly high temperature and not as a liquid. In addition, the mixing and combustion length for gas is surely shorter than for liquid, es-

F. Suttrop, DVL Inst., Portz-Wahm, Germany. How is the supersonic combustion system, that you propose, adapted to a change of flight Mach number?

A. Ferri. The control of contraction ratio of the inlet depends in a large part upon the com- bustion process. When the flight Mach number decreases, the amount of fuel injected in the first line of injectors decreases and, therefore, tile shock produced by such injectors becomes weaker. The additional fuel is added in the downstream injector.