Crossed beam correlation measurements and model predictions in a rocket exhaust plume

COMBUSTIONAND

51

FLAME 43: 51-67 (1981)

Crossed Beam Correlation Measurements and Model Predictions in a Rocket Exhaust Plume G. T. KALGHATGP Department

of Aeronautics

and Astronautics,

University

of Southampton,

Highfield.

Southampton

SO9 5NH,

U.K.

J. M. COUSINS Procuremenr

Executive,

Ministry

ofDefence.

PERME,

Westcott,

U.K.

K. N. C. BRAY Department

of Aeronautics

and Astronautics,

University

of Southampton,

Highfield,

Southampton

SO9 5 NH, U.K.

The crossed beam correlation technique in its schlieren mode has been used to measure the rms value of the density gradient fluctuation and a weighted rms value of the density fluctuation ~LXBS the exhaust plume of a static, liquid bipropellant rocket engine at four different axial stations. A transverse.integral length scale, a form of time and length scales in the flow direction, and convection velocity have also been measmed. Some of these experimental results are compared with predictions from a computer model. where diit comparisons are possible, agreement is satisfactory. Where it is not, expected trends are observed. The results bring into question the validity of a key modeling assumption, that turbulent density fluctuations are negligible.

1. INTRODUCI’ION

Reliable prediction techniques are required to provide information on rocket exhaust plume properties under conditions where experimental measurements would be impracticable. ‘However, computer models can only be used with confidence if they have been tested against a wide variety of experimental data. Theoretical calculations of global features of exhaust jets such as infrared radiation emission intensity and microwave attenuation have been compared with measurements and agreement has been satisfactory [l, 21. However, very few measured data have been available for verifying predictions of the more * Present address: Shell Research Ltd., Thornton Centre, Chester CHl 3SH. Copyright Q 1981 by The Combustion Institute Published by Elsevier North Holland, Inc. 52 Vanderbilt Avenue, New York, NY 10017

detailed statistical structure of rocket exhausts such as profiles of time averaged velocity or turbulence characteristics. The purpose of the ‘work described in this paper is to provide experimental data concerning the detailed statistical structure of the rocket exhaust and to compare the results with theoretical predictions. This comparison tests the accuracy of the predictions and permits the validity of certain of the model assumptions to be assessed. In typical rocket exhaust flames, fuel rich gases from a single nozzle or a series of nozzles at temperatures between 700K and 25OOK mix turbulently with a moving stream of cooler air (Fig. 1). Shock waves may occur in the flow and external combustion can take place as the excess carbon monoxide and hydrogen mix with the ambient air. A recirculating flow region may exist behind the base wall of the missile, influencing the overall OOlO-2180/81/10051+17$02.50

52

G. T. KALGHATGI ET AL. Recirculation Region

I

I I

A

/ Mixing

Core

Region

A

Region

Exhaust

Axis

-Recompression Wave/Shock

Fig. 1. Diagrammatic view of exhaust flame geometry.

flame properties such as the onset of secondary combustion. The flow field may also be asymmetric. Theoretical techniques have been developed over a number of years by workers at Imperial College, CHAM Ltd., and PERME (Westcott) to take account of all these cases,although not always all at the same time (see Refs. [l-3]). These techniques have been demonstrated elsewhere [ 1,2] to provide satisfactory predictions of a variety of observed flame properties under a range of different conditions. The computer program used to model the flow in the present study treats only some of the features outlined above. The motor chosen for the experimental work had a single, axisymrnetric exhaust nozzle and was fired under static conditions. Thus, the flow could be assumed to be axisymmetric with no recirculation. In this case, the governing partial differentail equations, which include a twoequation model for turbulence closure [4], may be solved by the parabolic marching procedure of Patankar and Spalding [S]. Calculated exhaust properties include the time mean chemical composition, temperature, pressure, velocity, turbulence length scale, and the root mean square (rms)

fluctuation values of selected scalars such as species concentration, density, and temperature. A brief description of the model is given in Section 3. The experimental technique used is the crossed beam correlation method. This is an optical technique which relies on the detection of deflection and/or attenuation of narrow light beams transmitted through the flow field. Signal acquisition is more straightforward than in laser Doppler anemometry. The measured signal represents an integration along the beam path, but Fisher and Krause [6] have showed that it is possible to extract local&d information by statistically cross correlating such fluctuating signals from two mutually perpendicular light beams which intersect inside the flow. The signal that Fisher and Krause considered was the attenuation of the intensity of the light beams, brought about by seeding the flow. Wilson and Damkevala [7-j extended the crossed beam correlation technique to study the density field of a turbulent axisymmetric jet by measuring the deflections of two light beams crossing the flow. Since then, crossed beam correlation measurements have been conducted in axisymmetric jets [8-lo], premixed flames [ll], and diffusion flames

53

CROSSED BEAM CORRELATION MEASUREMENTS [12]. A general description and analysis of the method has been presented by Kalghatgi [13]. These methods are different from laser Doppler anemometry in another important sense as they give information about the scalar field of turbulence. This is of special interest when the flow field being studied involves combustion. In this work, the crossed beam correlation technique has been used in its deflection mode. The measurements are restricted to the outer part of the plume, where the beam deflections are found to be relatively highly correlated. The properties measured include the rms value of the fluctuation of the density gradient in the flow direction, the integral length scale of the density gradient tield in the direction transverse to the flow direction, certain time and length scalesin the flow direction, and the convection velocity for the density gradient field. The rms value of the density fluctuation, multiplied by a ratio of length scales, is also deduced from the measurements. An outline of the crossed beam correlation technique and the experimental setup are given in the next section of this paper. A brief description of the theoretical model is given in section 3 and the experimental results and a comparison with computer predictions are presented in Section 4.

2 THE CROSSED METHOD

BEAM

CORRELATION

(2.2)

where A, the Gladstone-Dale constant, is assumed to be constant across the flame, p is the density, and y and z are the coordinates as shown in Fig. 2. In our calculations, the value of 2.37 x 10e4 m3/kg, the value for air, was used for A. We will represent +/ax by G in all the discussion below. In general, the deflection 8 is related to the detector output u by v=a1s

(23)

where 1 is the optical lever, and s, the detector sensitivity. From Equations (2.1) and (2.2) we can write, in general, using the 5, q, i frame of reference.

eHxeVx - A2 =Qsb+S,~,z,d =-- 1 T w+5, T ss0 flow s

y+rl, z, 0

x WC, y, z+L t+d dt dv 4,

(2.4)

where 5 is the beam separation in the x direction, z is the time delay, the overbar denotes time averaging, and Tis the time over which averaging is done. The covariance Q, is determined experimentally. If the beams intersect (5 = 0), we can write its value at zero time delay

2.1. Outline of Technique The analysis that follows is based on the work of Wilson and Damkevala [7] and Davis [9]. Only the fluctuating components of the various parameters are considered during the analysis. The optical system consists of two mutually perpendicular beams which may either cross at a position ,4(x, y, z) or may be separated in the x direction by a distance 5 (Fig. 2). Each beam is in turn perpendicular to the x direction, which is the axial flow direction. The instantaneous x detlection 8, and 19, of the horizontal (z) and vertical Q beams, respectively, may be written as B,,=A

2 s flow 0 ax

dz

3

(2.1)

QsbY, z, 0) = (G(x, Y, z, O’L,L,

(2.5)

making use of the integral length scalesL,, and L, in the y and z directions, respectively. To proceed further, we must make assumption concerning the variations in gas density. 2.2 Expression for rms De&y

Fluctuation

If the turbulence could be assumed to be homogeneous, it is simple to show [7, 131 that we would have

G(x,Y,z, WJO+ 5, Y,z>t) = - $ QG ~30, (2.6)

54

G. T. KALGHATGI ET AL.

Ah, Y,

4

Laser

beam -

%

Detector

Fig. 2. Sketch of experimental setup.

where ~$7 and [ are separations between two points in the x, y, and z directions, respectively, and Q(5, q r) is the two-point density covariance given by

(2.4), (2.6) and (2.9)

Q((, q, r)=p(x, y, z, t)dx+t,

after formally extending the integration limits to & 00. Now, since l=O, s~=(~I~/A~)+(~~/AZ) and Eq (2.10) will reduce to

Y+V,

z+C, 0. (2.7)

We shall assume that Eq. (2.6) can be adapted to inhomogeneous turbulence as follows. Suppose that Q is a function of a separation parameter s, where

$=

-r2 + rt2 + -c2 A:

4’

AZ

(2.8)

and where A* A,, and AZare suitably defined length scalesof the turbulent density field in the x, y, and z directions, respectively. We can write

J2Q+!Q+-&~!-5g].(2.9) 85' Therefore, if the beams are intersectin& from Eqs.

+CC +,dQ 1 ds ; 4 Y,4=- &x s-ms-m

Qsb,

Qs(x, y, z)= -2n

y

[Q(CQ)--Q(o)l,

dt (2.10)

(2.11)

where Q(a) will be zero and Q(0) is the mean square density fluctuation at A&, y, z). Hence we may write

Y,4F KPrm=J Q&, ZR

(2.12)

where prmsis the root mean square density fluctuation and K is given by

K=- I A$. 7. V +

(2.13)

55

CROSSED BEAM CORRELATION MEASUREMENTS If the turbulence were isotropic in the region from which the contributions to the correlation are made, A, A,., and A, would be equal and K would be unity. 2.3. Turbulence Length and Time Scales

against the time delay r, we get a correlation curve such as those shown in Fig 3. If the beams are intersecting, the correlation curve will have the maximum value of unity at r = 0. If the beams are separated by (5,in the flow direction, the curve will show a positive peak, lessthan unity, at a time delay rp given by

From Eq. (2.1) we can write ~*=UV,; &;=A2

qx, y, z, Wo, Y, z + 5, t) 4 4

= A2

$L,

(2.14)

dz.

s In the present case, since the flow is an axisymmetric jet, we can assume the transverse integral length scales to be purely functions of the radius r. We also assume that L,=L,=L.

(2.15)

We then define a function&), only on the radius r, as

which is dependent

f(r) = G'(r),5 Substituting this in Eq. (2.14) and transforming into radial coordinates, we can write 8,.,3y)=2A2 sY

OD f(r) r p-3

.

(2.17)

The radial distribution ofAr) is derived from the measured distribution gti2(y) by a numerical technique known as “onion peeling” by writing Eq. (2.17) as (2.18) f(r) in the outermost layer, j = 1, is given by a direct measurement. Oncef(r) is known, L and -dz can be found from Eqs. (25) and (2.16). If we plot the spa-time correlation coefficient (2.19)

(2.20)

V o the convection velocity, can be estimated from Eq. (2.20). Finally, if we characterize the fluctuations by a time r0 defined as the area under the y(0, r) curve up to the first zero crossing, we can determine a length scale L, in the flow direction from Lx=qJJc

(2.21)

2.4 Experimental Setup The experimental measurements were made at PERME (Westcott) on a tethered liquid bipropellant rocket engine. Two mutually perpendicular 4mw He-Ne lasers (Coherent Radiation, Model 80) were made to intersect inside the exhaust plume, as sketched in Fig. 2. Each beam in turn was perpendicular to the flow direction x. The deflection of each beam in the x direction was measured using a position sensitive detector (United Detector Technologies, PIN K-25). These signals were amplified using specially designed amplifiers which gave voltages directly proportional to deflections of the beam spot on the face of the detector, irrespecive of any changes in the intensity of the beam.The amplifier bandwidth was O-20 kHz. Since the tests were done outdoors, it was also necessary to use a narrow band optical filter (Barr and Stroud DOSS) in front of each detector to keep out the sunlight. The tests were conducted under very hostile conditions. For instance, the jet generates an intense acoustic field, which was discovered to be strong enough to destabilize the lasers. Then, at the start and end of each ftig, large amounts of nitric acid fumes are expelled from the nozzle. To protect the equipment from these hazards, as well as from the weather, it was found n ecessary to mount the lasers

G.T.KALGHATGIETAL.

56

x = 0.51m Y = 0.155m

Fig. 3. Typical cross correlation curves.

and the detectors on shock absorber mounts which, in turn, were mounted inside wooden boxes lined with thick felt. These boxes were mounted on a heavy steel frame which could be moved vertically so that, at a given axial station, the beam intersection point could be moved across the plume in they direction. The amplified signals were carried to a room about 100 m away from the test bay by suitably screened cables and recorded on a FR1300 Ampex tape recorder running at 60 in&c. Each detector also gives a signal proportional to the beam spot movement across its face in a direction perpendicular to the plane containing the beam and the x direction, as well as a signal proportional to the intensity of light falling on it. These signals were also recorded during the tests, although the x deflection signals were the ones used in all the analyses reported here. The data was played back at 3.75 in./sec during analysis. Most of the analysis involved auto and cross correlation of the data. For this purpose, a Hewlett-Packard HP 3721 A correlator was used. Typically 128,000 samples were used during the correlation analyses at a sampling rate of 48 kHz. Spectral analysis was carried out at the Data Analysis Centre of the I.S.V.R. at Southampton University. The results are presented in Section 4.

lent density fluctuations may be neglected, the turbulent or Reynolds stresses are expressible in terms of mean velocity gradients by introduction of the eddy viscosity; equations for the conservation of energy and species concentration are closed by appropriate gradient approximations; no detailed explicit account need be taken of molecular transport as a rate determining process; the flow is adiabatic; laminar viscosity is small compared with turbulent viscosity; the later is expressible in terms of two parameters-implying that the turbulence is nearly isotropic and can be characterized by eddies of a single size-for which two additional modeled equations are solved, and fluid dynamics and chemical kinetics are coupled via time averaged values-that is, in evaluating rates of chemical reaction, turbulent fluctuations of temperature, density, and chemical species are neglected. Additionally, in the present case, it is assumed that the flow is axisymmetric and that axial diffusion is negligible. Some of these assumptions may readily be justified as acceptable in the context of a rocket exhaust. Others are reasonable only under certain conditions. Still others may be sources of significant error and exemplify the need for an improved theoretical technique. These points are discussed in Refs. [17] and [27] and in Section 4 of this paper.

3. THEORETICAL

32 GovemIng Equations

MODEL

3.1, BadI Assumptions The assumptions made in deriving the model equations are that the flow is quasi-steady and the governing equations may be time averaged; turbu-

Continuity (3.1)

57

CROSSED BEAM CORRELATION MEASUREMENTS Axial Momentum

(3.2) Radial Momentum

au puz=-$

ap

(3.3)

the local pressure, while p is a form of space averaged pressure over a cross section, introduced to render Eq. (3.2) of parabolic or boundary layer type, as will be explained later. Closure of the system of equations is obtained by incorporating the two-equation turbulence model described by Lauder and Spalding [4], which expresses the turbulence eddy viscosity in terms of two parameters, for which two additional differential equations are solved. The expression assumed for IL is

Conservation of Energy

(3.7)

P=CCDpk2e-l,

(3.4)

where C,, is an empirical constant, k is the turbulence kinetic energy per unit mass, and E is the eddy energy dissipation rate, taken to be E = CDkl.‘Ledl, where L, is the length scale of the energy containing eddies. For the axisymmetric, parabolic flow situation, the modeled transport equations for k and E are

Conservation of Chemical Species

aFi

PU,,

aFi

0

au 2

fPV% +p

z

(3.8)

-PC,

(3.5) Also required is the equation of state, which, after neglecting fluctuation correlations, is p = pRT/M.

(3.6)

In these equations the fluid properties are all time averaged; overbars have been omitted for clarity. The quantities u and v are the axial and radial components of velocity, respectively, p is the local density (turbulent density fluctuations being neglected), His the total (stagnation) enthalpy, Fi is the mass fraction of species i, li, i is the mass rate of production of this species, p the turbulence eddy viscosity coefficient, (TV and a,,, are the exchange coefficient ratios for enthalpy and mass (turbulent Prandtl and Schmidt numbers), respectively, M is the mixture molecular weight, R is the gas constant, and T is the temperature, which is calculated from local enthalpy, velocity, turbulence kinetic energy, and mixture composition. The symbol p stands for

++ 0g

2

-C,P$,

where C, and C, are empirical constants and ok and Q Eare the exchange coefficient ratios for k and E, respectively. A further equation may be written to describe the transport of the time mean value of the square of the fluctuating component of a scalarf

Cl, 161,

87 2- C,,P +CglP jy 0

&- ‘g,

(3.10)

where f is the instantanious value of this scalar$s the local mean value, g is the local mean value of (f-j’, and C,, andC, are empirical constants.

58 Effects of chemical reactions are incorporated via the source term tL i in Eq. (3.5), as described in Ref. [2]. However, it should be noted that scalar fluctuation effects (Bray [17]) are not included.; 3.3 Solution Procedure

The above equations form a set of simultaneous partial differential equations of parabolic or boundary layer type which are solved by the finite difference marching procedure of Patankar and Spalding [S]. The term parabolic implies that there exists a predominant direction of flow, that d&rsion has been neglected in that direction, and that the downstream pressure field has little influence on the upstream flow conditions. This last condition introduces an inconsistency into the treatment if the flow is subsonic when the downstream pressure does affect the upstream flow via the axial pressure gradient in Eq. (3.2). Thus, in subsonic flow, a form of space averaged pressure @ is used to calculate ap/ax; for a free jet this is simply taken to be the same as the longitudinal pressure gradient prevailing in the free stream adjacent to the boundary layer. The disadvantage of this technique is that, although calculation of pressure waves is permitted by inclusion of the radial pressure gradient in Eq. (3.3), shock waves cannot be treated properly since embedded subsonic flow regions, such as the Mach disk which occurs behind a normal shock wave, can be treated only approximately. However, work currently in progress at Imperial College [18] is aimed at removing this limitation. The advantage of the method is that it permits the freedom to use marching integration. This enables flow properties at a downstream axial grid station to be calculated from known upstream conditions, a process which is repeated until the domain of interest has been covered, and which, thus, only requires storage of the variables at two axial stations. The governing partial differential equations are transformed first to normalized streamline coordinates and then to finite difference form, by integration over a small control volume surrounding each grid point. The finite difference equations are solved by a line-by-line method in which the values of a variable at all the points along

G. T. KALGHATGI ET AL. a grid line are computed simultaneously by the use of a tridiagonal matrix algorithm (TDMA) [S]. 3.4. Rolmdary colMIitions

Properties at some upstream crosssection of the flow, usually the nozzle exit plane, must be specilied. In the present case, axisymetric conditions at the nozzle exit were obtained via a set of computer programs described in Ref. [19]. This provided radial profiles of velocity, pressure, temperature, and nonequilibrium chemical composition, but not turbulence initial conditions. These had to be estimated. The three temporally fluctuating components of velocity were initially assumed to be 10% of the nozzle exit velocity l’, and the length scale of energy containing eddies was taken as 10% of the nozzle exit radius rF These estimates are broadly consistent with experimental observations [4]. They give initial values of k and E of 0.015VE2 and lOC&%,‘, respectively. The resulting turbulence viscosity then becomes, from Eq. (3.7),

Scalar fluctuations were assumed to be very small at the nozzle exit, and initial values of g were set to lo- 30. The other computational boundaries are formed by the jet center line and a free boundary. At the jet axis, symmetry exists and the radial velocity and the radial gradients of all other quantities are zero. At the free boundary where the flow is unconfined and expands as it moves downstream, the value of all variables are set equal to their free stream values. 35. Computer Program

The above analysis forms the basis of the computer program REP3. This was developed by Concentration, Heat and Momentum (CHAM) Ltd for PERME (Westcott), although it has been much modified since. 4 RESULTS 41 Experimental Details

The experiments were conducted on a static, liquid bipropellant engine. Engine parameters are given

CROSSED BEAM CORRELATION MEASUREMENTS in Table 1. Measurements were made at four axial stations: x=0.51 m, 0.75 rn, 1.0 rn, and 1.52 m, that is, between about 7 and 21 nozzle exit diameters downstream. At each axial station many different firings were conducted, each time with the beam intersection point at a different vertical position. Such traverses of the plume were also made with the beams separated in order to estimate the convection velocity. Usually, only the upper half of the plume was studied. However, at x = 1.0 m, the traverse was also conducted across the lower half of the plume. In all the tests, the vertical beam passed through the jet axis. Statistical and instrumental errors associated with the data have been assessed elsewhere [14]. The error bars on some of the graphs presented below result from this assessment and do not include errors arising from the assumptions described in Section 2. 42 Computational Details Predictions were made with the computer program REP3. Calculated properties included the mean flow velocity, the length scale L, of the energy containing eddies (= C&‘.5 E- ‘), the density, and the r.m.s. density fluctuations. Although the equations in Section 3 were derived by assuming that density fluctuations are negligible, the intensity of the turbulent fluctuations was estimated by putting f=p in Eq. (3.10), and then p,,,=& The empirical constants used in the analysis are given in Table 2. It is generally accepted that there is no universal set of constants-see, for example, Lockwood and Syed [20] and Morse [21]-but TABLE 1 Liquid Bipropellant

Engine Parameters

Fuel: Mixed amine fuel (MAF-1) Oxidizer: Nitric acid (IRFNA) Oxidizer to fuel ratio (Stoichiometry at 3.3 : 1) Chamber pressure Nozzle throat diameter Nozzle exit diameter Exit temperature Exit velocity Exit pressure

3:l 6.1 M Pa (60 atm.) 40 mm 72.65 mm 2300 K 2200 ms-1 0.41 M Pa (4 atm.)

59 TABLE 2 Turbulence Model Constants CD

0.09

Cl

c2

1.64

1.92

cgl

Cgz

2.8

1.92

OH urn Ok ‘Je Qg 1.0 1.0 1.0 1.3 1.0

Table 2 contains the current best single set of values applicable to the whole rocket exhaust. The chemical reaction mechanism is given in Table 3. No error bounds are shown on the predicted theoretical curves in the following figures as these are difficult to quantify. However, because of the modeling assumptions outlined in Section 3 and the uncertainty in turbulence model constants and chemical reaction rates, the predictions are subject to considerable uncertainties [273. 4.3. Experimental Results and Compadson with Theoretical Pdictions Density Fluctuatiolrs The measured mean square value of the x deflection of the horizontal beam 8uX2 has been plotted against y, the distance of the horizontal beam from the jet axis, in Fig. 4 for the four axial stations considered. It can be seen that in all these casesas one starts from outside the plume and moves towards the center, the value of g&2 increases and, after reaching a maximum, decreases. This maximum value increaseswith increasing axial distance x. This is to be expected since Bk2 is integrated across the plume and the plume size increases as x increases. Figure 4 also serves to illustrate that the noise signal due to vibration of the optics was negligible. For instance, at x = 0.51 m, with the beam well outside the plume, say at y =0.29 m, oh2 is negligibly smaIl. The mean square value of the transverse detlection of the horizontal beam, 8nY2, was in general found to be slightly larger than the corresponding value of Bh2 [14]. A maximum plume radius rmaxat which ok2 was judged to have reached the noise level was found from Fig. 4 at each axial station. In Fig 5, rmar has been plotted against x. Also shown in Fig. 5 is the calculated plume boundary, that is, the value of r beyond which properties are predicted to be equal to the ambient values. The agreement between

60

G. T. KALGHATGI ET AL. TABLE

3

Chemical Reaction Mechanism Reaction O+O+M O+H+M H+H+M H+OH+M CO+O+M OH+H2 O+H, H+02 CO+OH OH+OH H+O,+M H+H02 H+H02 H2 + HO, CO + HO, O+H02 OH + HO2

Forward rate coefficient4 -+02+M +OH+M +H2+M +H20+M +C02+M +H20+H +OH+H +OH+O -CO,+H -H20+0 -tH02+M -*OH+OH +H2+02 +H20+OH *CO2 +OH +OH+02 -02+H20

0 In ml-molecule-second

3 X lo-34 exp (900/T) I x lo-29 ~-1 3 x 10-30 T-1 1 x 10-25 T-2 7 X lo-33 exp (-2200/T) 1.9 x lo-15 Tr.3 exp (-1825/T) 3 X lo-14 T exp (-4480/T) 2.4 X lo-10 exp (-8250/T) 2.8 X lo-17 T1.3 exp (330/T) 1 X lo-11 exp (SSO/TI 2 x lo-32 exp (500/n 4 X lo-10 exp (-950/T) 4 X lo-11 exp (-350/T) 1 X lo-12 exp (-9400/T) 2.5 X 10-10 exp (-11,900/n 8 X lo-11 exp (-500/T) 5 x 10-11

10 30 30 10 30 2 1.5 1.5 3 3 10 5 5 10 10 30 30

units. Rate coefficients and uncertainty factors are taken from Ref. [ 261.

theory and experiment is good. We now define a nondimensional distance y= all,,,,

Uncertainty factor

(4-l)

which will be used in the subsequent presentation of results. The measured values of the weighted rms density fluctuations Kprm scalculated from Eq. (2.13) are plotted in Fig. 6. The plot for x=0.51 m shows a smooth variation and a definite peak at Y=O.71. The points from the other downstream stations are more scattered. This is not very surprising considering that each point is the result of a different firing. However, taken together, they fall on a single curve which has a wider base and a slightly higher peak than the plot at x=0.51 m. The peak has also moved slightly inboard to around Y=O.63. As we approach the jet center line, the temperature increases and hence the density decreases rapidly. Probably for this reason, the covariance Q, was found to be extremely small in the central region of the jet and it became increasingly difficult to evaluate these small values with sullicient accuracy [14]. This is the reason why the plots in Fig. 6 are terminated at Y=O.3.

At this stage it should be pointed out that the shock structure in the exhaust plume does not appear to affect either the single beam signals or the cross correlation results. This is in line with the results of laser schlieren experiments conducted by Harper Boume [28] on a 76mmdiameter, choked air jet, preheated to 850°C. In contrast to the tests conducted on cold jets, these experiments failed to detect the presence of shock waves which were known to exist in that flow. The reason advanced by Harper Boume for this is that the density fluctuations caused by the static temperature differences in the shear layer are su&iently strong to mask the shock waves, especially since the jet core density is low. Such an argument is even more valid in the present experiments since the traverses were conducted at axial distances greater than 70, where D is the nozzle exit diameter. Harper Bourne’s experiments were conducted at 1D and 2.50 and hence in the region where shocks can be expected to be more sharply defined. The jet core temperature is also much higher than 850°C in the present experiments. Also shown in Fig. 6 are predicted radial profiles of Prms The peak values occur between Y=O.64 at x= 1.5 m and Y=O.68 at x=0.5 m, which is in

CROSSEDBEAMCORRELATIONMEASUREMENTS

61

4

Fig. 4. Distribution

0.28

6.32

0.36

of the mean square value of the x deflection of the horizontal beam.

0

-

Expt. Theory 0

/ 0.5

1 1.0 Xm

Fig. 5. Variation of maximum plume radius with axial distance.

I 1.5

0.40

-

62

G. T. KALGHATGI ET AL.

0.16

1.5 1-o 0.7 0.L

I

m

4

Theory) EXPt-

/

O-15 ,

erms

-

, K'erms

0

O.Slm

8

O-75 m

q

1.0

A

1.0 m,lower

0

1.52 m

m half

1.0

('r/rmax

Fig. 6. Distribution of weighted rms density fluctuation fluctuation (theory).

remarkably good agreement with the experimental measurements. Unfortunately, direct comparison of the magnitude of the measured and predicted curves is not possible because values of K, the ratio of length scalesdefined in Eq. (2.13), are not known, However, experimental results from previous work suggest that K will be less than unity. For the velocity field, Wygnanski and Fiedler [22] found a value of 0.43 for the ratio of transverse to axial integral length scalesin the self-preserving region of a cold air jet. Laurence [23] found values ranging between 0.4 and 0.5 whereas Davies and Fisher [24] found values as low as 0.33 for such a ratio in

(experiment) and rms density

cold air jets. From recent measurements of length scales in the temperature field of a turbulent diffusion flame, Kalghatgi and Roberts Cl23 found values ranging between 0.45 and 0.69 for K. The range of values for K is thus sufficient to reconcile the experimental data with the model predictions. However, the values of plrns from the computer prediction can also be modified by changing the values of C,, and C,, in Eq. (3.10) without changing the radial position at which the peak rms density fluctuation occurs. The value of K needs to be established accurately in the rocket exhaust before deciding upon any quantitative changes in the

CROSSED BEAM CORRELATION MEASUREMENTS

63 correlations between density and velocity cornponents. However, Fig. 7 shows that the density fluctuations are far from negligible. Even if the theoretical model is overpredicting prrns and K in Fig. 6 is given a value as large as unity, the maximum value of p,,,/p will still be greater than 0.2. We conclude, therefore, that the assumption in the theoretical model that density fluctuations are negligible is of doubtful validity.

44 Experimental results and Comparison with . . Pre!hmm Other Variables

Y = r/ rmox

Fig. 7. Theoretical predictions of ratio of nns density

fluctuation to local mean density.

model constants. The feasibility of a method for eliminating K from the measurements, involving the use of three orthogonal laser beams in a modified crossed beam correlation technique, has already been demonstrated in the laboratory [25], and it is hoped to extend this technique to the study of the rocket flame. Finally, the predicted rms density fluctuations, normalized by the local mean density, have been plotted in Fig. 7. In our opinion, one of the key assumptions contained in the present theoretical model is that turbulent fluctuations in density are negligible compared with the local mean density. If this assumption is not valid, the chemical source terms G i in Eq. (3.5) must be expanded to include terms proportional to prmr and many other scaler fluctuation variables and covariances will enter the source term expressions at the same time. Again, with large density fluctuations, turbulent transport terms must be mod&d due to the inlluence of

In Figure 8, measured values of & are plotted. The peaks appear to occur outboard of the corresponding peaks in Fig. 6. The peak value at x = 0.51 m is distinctly larger than for other axial stations. Once again, beyond x=0.75, the plots seem to scale with I,,,. The measured transverse integral length scale L is plotted in Fig. 9. Each of these plots shows a minimum whose position appears to correspond to the position of the maximum in the fl curves in Fig. 8. In Fig. 10, the time scale ?0 which is defined in Section 2, is plotted for each axial station. It can be seen that as one moves towards the center of the jet, this time scale decreases as expected because the high frequency content of the turbulent fluctuations increases. The measured convection velocity U, of the correlated optical disturbances is plotted in Fig. 11. This was estimated from Eq. (2.20). It becomes progressively more dinicult to establish the peak in the correlation curve and measure rP accurately as one moves further into the plume [14]. This uncertainty is reflected in the large error bars on the values for YcO.6. The predicted mean velocity is also plotted in Fig. 11. In all cases, for Y ~0.7 the measured convection velocity is smaller than the predicted mean velocity. For Y> 0.7 the convection velocity is measured to be broadly similar, or slightly larger than the theoretical mean velocity. This is consistent, qualitatively, with the results for cold subsonic jets where the convection velocity is found to differ from the local mean velocity [22,24] in this way.

G. T. KALGHATGI ET AL.

Expt., l

0 Cl A

0

I I

04 Fig. 8. Distribution

X= 031 0.75 1.0 1-O 1.52

m m m m,lower m

half

14

Y=r/r,,

of ms density gradient fluctuation.

An axial length scale& deflmed as the product of the measured convection velocity and the time scale z,, is plotted in Fig. 12. The distribution of& seems to be quite different from that of the transverse length scale plotted in Fig. 9, suggesting

that the turbulence is highly anisotropic. In general, the length scales at the downstream stations seem to be larger than at x =OSl m. Also plotted in Fig. 12 is the predicted length scale L, which is the length scale of the energy containing eddies. X 0 061 m @ 0*7S m 0 l.Om A 1-O m,lower 0 1.52 m

0 do '0 n

0.5 Fig. 9. Distribution

Y=r/rnx

1.0

of transverse length scale.

half

65

CROSSEDBEAMCORRELATIONMEASUREMENTS Xz

0 O-Slm e 0.75m A

0 140 m Al-00

Q

; 9 oa0

o(.p

.

u

m ,lower

half

01~52 m

:’ 00

l

O:s" 1.b Y sr/rmax Fig. 10. Distriiution of the time scale.

600

m/s Theory,U 4oc

Ill\\ I

-

Expt.,U,

X= 0 a 0

0.51 m 140 m 1.52 m

200

Fig. 11. Distribution

of convection velocity (experiment)

and mean velocity (theory).

66

G. T. KALGHATGI Expt.,Ls,

0.5 Y =r/r,,, Fig. 12. Distribution

ET AL.

X= 0 0.51 m

I.-o

of axial length scale and model length scale.

Implicit in the k- E model is an assumption that the turbulence is sulliciently close to isotropy to be characterized by the two scalar variables k and E. This assumption may perhaps be brought into question by the present experimental results. However, if the measured results from Fig. 9 and 12 are simply averaged, the resulting plots of length scalesare similar in shape to those predicted. In the mixing region(Y>0.7) they are also of roughly the same magnitude. The spectral content of the single beam signals is illustrated in Fig. 13, where the spectral density curves of the eKx signal for Y= 17.6 cm, 16.6 cm, 15.6 cm, and 14.6 cm at x=0.51 m have been plotted. It can be seen that the signal at Y= 17.6 cm, the outermost position, has a generally larger lowfrequency content than the other signals. However, the spectral density curves for the other positions are almost identical to each other. At the highfrequency end, the - 5/3 Kolmogorov power law [ 151 would lead to a - 8/3 power law variation for a line integrated signal. In Fig. 13, a line with a slope of -8/3 has been drawn. The spectra seem to decrease according to this power law in the approximate frequency range between 4 and 15 KHz.

CONCLUSIONS The crossed beam correlation technique in its schlieren mode has been used in very hostile

16,000 1600 Frequency. HZ. Fig. 13. Sample spectral density curves of the 6Hx signal at x = 0.51 m. 160

67

CROSSED BEAM CORRELATION MEASUREMENTS conditions to measure turbulence parameters in the exhaust plume of a stationary liquid bipropellant rocket engine. Measurements were made at four axial stations between 7 and 21 nozzle exit diameters downstream. Measured properties were the rms density gradient fluctuations, integral length scalesin the transverse direction, a form of time and length scalesin the flow direction, and the convection velocity. Also deduced from the measurements were weighted rms density fluctuations. When the measured maximum plume radius is used as a scaling factor, plots of most properties collapse onto a single curve for x> 10 exit diameters. Some of the experimental results are compared with predictions from a computer model. Most of the results could not be compared directly as different quantities were measured from those predicted. Where direct comparison was possible, agreement was satisfactory. Where it was not, expected trends were observed. Further experimental work is proposed that will reduce the uncertainty in rms density fluctuation measurements and thus permit direct comparison with theory. Allowing for this uncertainty, the results, nevertheless, show that the assumption contained in the computer model that density fluctuations are negligible is of doubtful validity. First and foremost thanks are due to Mr. R. E. Lawrence and his team at P.E.R.M.E., Westcott-in particular, Peter Musson-without whose help it would not have been possible to conduct the experiments. The amplijers and the associated electronics were designed and built by Mr. M. W. Stent, lately of the Department of Aeronautics and Astronautics, Southampton University. The experimental work wasfinanced by a contractfiom the U.K. Ministry of Defence, No. AT/2040/084.

REFERENCES 1. Jensen, D. E., andwilson, A. S., Cornbust. Flame 25: 43-55 (1975). 2. Jensen,D. E., Spalding, D. B.. Tatchell, D. G., and Wilson, A. S., Combust. Fkzme 34:309-326 (1979). 3. COUS~IS, J. M., Unpublished Ministry of Defense Report (1979).

4. Launder, B. E., and Spalding, D. B., Mathematical Models of Turbulence, Academis Press, New York, 1972. 5. Patankar, S. V., andSpalding, D. B., Znf. J. Heat Mass 7?uns. 15:1787-1806 (1972). 6. Fisher, M. J., and Krause, F. R., J. Fluid Mech. 28: 705-717 (1967). 7. Wilson, L. N., and Damkevala, R. J., Z. Fluid Mech. 43:291-303 (1970). 8. Fisher, M. J., and Johnston, K. D., NASA TN D5206 (1970). 9. Davis, M. R., J. FluidMech. 7Oz463-479 (1975). 10. de Belleval, J. F., and Maulard, J., ONERA TP No 1975-81 (1975). 11. Kalghatgi, G. T., and Moss, J. B., Proceedings, Second Symposium on Turbulent Shear Flows, Imperial College, London, 5.9-5.14 (1979). 12. Kalghatgi, G. T., and Roberts, P. T., ASSU Rep. No. 342, Department of Aeronautics and Astronautics, Southampton Univ. (1979). 13. Kalghatgi, G. T., AASU Rep. No. 339, Dept. of Aeronautics and Astronautics, Southampton University (1977). 14. Kalghatgi, G. T., AASU Rep. No. 340 Department of Aeronautics and Astronautics, Southampton Univ. (1979). 15. Davis, M. R.,J. FZuidMech. 46:631656 (1971). 16. Spalding, D. B., Chem. Eng. Sci. 26:95-107 (1971). 17. Bray, K. N. C., Seventeenth Symposium (Znternational on Combustion, The Combustion Institute, Pittsburgh, 1979, pp. 223-233. 18. Jennions, I. K., Ma, A. S. C., and Spaldmg, D. B., Private Communication. 19. Cousisn, J. M., Unpublished Ministry of Defense Report (1978). 20. Lockwood, F. C., and Syed, S. A., Cornbust. Sci. Technol. 19:129-140 (1979). 21. Morse, A. P., Axisymmetric flow with and without Swirl, Ph.D. Thesis, University of London to be submitted. 22. Wygnanski, I., and Fiedler, H., J. Fluid Mech. 38: 577-612 (1969). 23. Laurence, J. C., NACA Rept. 1292, (1956). 24. Davies, P. 0. A. L., and Fisher, M. J., AASU Rep. No. 233, Dept. of Aeronautics and Astronautics, Southampton Univ. (1963). 25. Kalghatgi, G. T., AASU Tech. Memo 80/3, Department of Aeronautics and Astronautics, Univ. of Southampton (1980). 26. Jensen, D. E., and Jones, G. A., Cornbust. Flame 32: I (1978). 27. Jensen, D. E., and Jones, G. A., Combustion and Flame, 41:71-85 (1981). 28. Harper Bourne, M., Laser Schlieren Measurements on the RJl Hot Jet Facility at Rolls Royce (Bristol), Private communication (1972). Received 4 June 1980; revised 1 October 1980