mixing in supersonic expansion ramp comb

Flow Turbulence Combust (2008) 80:489–506DOI 10.1007/s10494-008-9133-7

Mixing Measurements in a SupersonicExpansion-Ramp Combustor

Aristides M. Bonanos · Jeffrey M. Bergthorson ·Paul E. Dimotakis

Received: 17 July 2007 / Accepted: 3 January 2008 /Published online: 25 January 2008© Springer Science + Business Media B.V. 2008

Abstract This paper reports results on molecular mixing for injection via anexpansion-ramp into a supersonic freestream with M1 = 1.5. This geometry producesa compressible turbulent shear layer between an upper, high-speed “air” stream anda lower, low-speed “fuel” stream, injected through an expansion-ramp at α = 30◦ tothe high-speed freestream. Mass injection is chosen to force the shear layer to attachto the lower guide wall. This results in part of the flow being directed upstream,forming a recirculation zone. Employing the hypergolic hydrogen-fluorine chemicalreaction and pairs of “flip” experiments, molecular mixing is quantified by measuringthe resulting temperature rise. Initial experiments established the fast-chemistry limitfor this flow in terms of a Damköhler number (Da). For Da ≥ 1.4, molecularly mixedfluid effectively reacts to completion. Parameters varied in these experiments werethe measurement station location, the injection velocity of the (lower) “fuel” stream,the stoichiometry for the flip experiments, and the density ratio of the fuel and airstreams. As expected, mixing increases with increasing distance from the injectionsurface. The mixed fluid fraction increases by 12% when changing the fuel-to-airstream density ratio from 1 to 0.2. Comparisons with measurements at subsonic(high-speed) “air” stream velocities show that the trend of decreasing mixing withincreasing speed documented in free-shear layer flows is also encountered in theseflows. The current geometry produces higher mixing levels than do free shear layers.

Keywords Molecular mixing · Compressible turbulent shear layer ·Supersonic combustion

A. M. Bonanos (B) · P. E. DimotakisGraduate Aeronautical Laboratories, California Institute of Technology,1200 E. California Blvd, MC 301-46, Pasadena, CA 91125, USAe-mail: [email protected]

J. M. BergthorsonDepartment of Mechanical Engineering, McGill University, 3480 University Street,Montreal Quebec, Canada H3A 2A7

490 Flow Turbulence Combust (2008) 80:489–506

1 Introduction

Hypersonic air-breathing vehicle engines require, from a cycle efficiency pointof view, that the internal flow should remain supersonic through the combustor.Further, to shorten the combustor length and extract the benefit for the unavoidablepenalty paid to internal friction losses, the rate of heat release should ideallybe limited by the mixing rate and not by chemical reaction rates [1]. The samerequirements also dictate that mixing should be rapid, since this ensures compactnessof the propulsive device, minimizing internal losses and engine weight.

These requirements have led to studies of various mixing enhancement methodsin compressible flows. Several techniques for fuel injection into supersonic cross-flow exist, such as transverse jets, compression and expansion ramps, cavities, steps,strut injectors, and shear layers, to name a few [2–5]. The success of these designsis evaluated based on performance characteristics, such as fuel penetration anddispersion rates, mixing efficiency, and the minimization of total pressure losses.

Such assessments require measurement methods to quantify the efficiency ofthe mixing mechanism. Non-intrusive laser-based techniques for measuring fueldispersion and flowfield velocities, such as laser Doppler velocimetry, laser inducedfluorescence, and diode laser absorption spectroscopy have been developed [6].Mixing measurements rely on the scattering of light off a passive tracer introducedinto one of the freestreams, with the signal related to tracer concentrations. Otherstudies have employed aspirating probes to sample the flow, typically limited tothat of a binary gas, to deduce mixture concentrations [7, 8]. Such techniques,however, lack the spatio-temporal resolution required to measure molecular mixing,especially in the high Reynolds number and Mach number environments of interest.To address some of these limitations “cold-chemistry” techniques that employfluorescence quenching to provide a direct measurement of fluid unmixedness havebeen developed [9].

Exploiting that chemical product formation in the fast-kinetic regime is limitedby mixing, a technique called the “flip” experiment was developed to measuremolecular mixing [10–12]. In this technique, the temperature rise from an exothermicfast-kinetic reaction is used as a tracer for molecular mixing and statistics arecomputed from a pair of chemically reacting experiments with “flipped” stoichiome-tries. Although heat release itself affects mixing, its effects have been studied andunderstood [12–14]. In an extension of the original implementation, the heat releaseis tailored in each case such that the two flip experiments are matched, as indicatedby the total pressure profiles of the runs. The details are further discussed below.

This paper presents measurements of molecular mixing in an expansion-rampcombustor. A supersonic “air” stream is expanded over a ramp inclined at 30◦ to thehigh-speed freestream flow, with a “fuel” stream injected through perforations in theramp. The two streams are seeded with hydrogen and fluorine reactants, respectively.The consequent temperature rise provides the tracer of mixing. The terms “air” and“fuel” streams are used here in a non-conventional sense. In these experiments,hydrogen is actually carried in the high-speed stream that would typically be theoxidizer, while fluorine is carried in the low-speed stream that would typically carrythe fuel. This geometry entails many attributes of a scramjet’s design. However, it isdesigned to provide an archival environment for design studies and code validationand is not intended to represent any sense of scramjet design optimization. In

Flow Turbulence Combust (2008) 80:489–506 491

particular, a goal of the greater effort, of which the experiments reported hereare a part, is capturing the results of such experiments in numerical simulation offlows characterized by attributes relevant to scramjet propulsion. The attempt toprovide a well-characterized flow environment has led to many of the choices in theexperiments reported here.

2 Facility Description

Experiments were conducted in the supersonic shear layer (S3L) laboratory atCaltech, whose test section was developed for these experiments and is schematicallyshown in Fig. 1. The facility is a two-stream blow-down wind tunnel capable ofdelivering mass fluxes up to m1 = 10 kg/s in the upper stream and m2 = 2.5 kg/s in thelower stream, with a nominal run time of 2–6 s. The facility is designed to handlefast-kinetic reactants. Specifically, the top stream can be seeded with a mixture ofhydrogen (H2) and nitric oxide (NO), and the bottom stream with fluorine (F2). Theremainder of the gas in both streams is comprised of helium, argon and nitrogeninert diluents, such that the desired ratio of density (molar mass) and specific (molar)heat capacities of the two streams can be set. Nitric oxide is added to the hydrogenstream to generate radicals on contact with fluorine to initiate the hydrogen-fluorinereaction [10, 15]. The reaction then proceeds hypergolically, obviating externalignition sources.

Briefly, the top stream gas is supplied from a 1.2 m3 pressure vessel, typicallyfilled to 3.8 MPa (550 psi). Gas mixtures in this vessel are created using the partial-pressure technique. The flow to the top stream is controlled with a proportional-integral-derivative (PID) control algorithm applied to a servo-motor-actuated sonicvalve, set to deliver a constant pressure to the top stream plenum for the experimentsdescribed here. The flow to the lower stream is supplied from a Teflon bag within asmaller pressure vessel (0.57 m3), to minimize the amount of hazardous gas handled.The Teflon bag is also filled with the desired gas mixture using the partial-pressuretechnique. During a run, the outside of the bag is pressurized with nitrogen from

Fig. 1 Schematic of the expansion-ramp geometry


a 12.7 m3 surge tank, creating a very nearly constant supply pressure for the lowerstream for the duration of a run without the need for active control. The bottomstream flow is metered using a calibrated sonic valve.

The top stream is accelerated through a converging-diverging nozzle to the desiredMach number (M1 = 1.5 in the present study). The lower stream is acceleratedthrough a converging nozzle and injected through a perforated expansion rampangled at α = 30◦ with respect to the horizontal, with an open area fraction near65%, chosen to avoid the jet-coalescence instability [16].

The test section height is h = 8.38 cm (3.30 in.), with a top-stream height ofh1 = 3.30 cm (1.30 in.). The lower-stream mass flux, which is set by a choked meteringvalve, is calibrated without the ramp installed at the splitter plate tip, where theheight is 5.97 cm (2.35 in.), and the lower-stream velocity, U2, is computed from thismeasurement. A measurement rake is located downstream of the start of the expan-sion ramp, at either L = 38.1 or 46.7 cm (15.0 or 18.4 in.). The measurement rakeconsists of K-type thermocouples, static pressure probes, and total pressure probes.Pressure taps are included along the top and bottom guide walls, as well as the topand bottom stream plena. During a run, temperature and pressure data are recordedat 1 kHz/channel through a National Instruments LabView hardware/software dataacquisition and control system.

Schlieren flow visualization is also utilized. The schlieren setup employs a few-nanosecond duration light source (MiniStrobokin power supply with KL-L lamp)and a high-speed CMOS camera (Phantom v7.1). The schlieren setup is mountedon rails to allow imaging of both the forward and rearward portions of the testsection. Composite schlieren images are assembled by performing two experimentsat identical conditions, imaging the upstream and downstream portions of the testsection in separate runs. The images are then aligned using test section featurescommon to both images as fiducials.

The two gas streams form a shear layer that reattaches to the lower guide-wallsome distance from the ramp. The reattachment generates a re-entrant jet, similar tothat formed by a backward facing step [17, 18], which brings reacted fluid upstreaminto a recirculation flow region where it mixes with the bottom stream and is re-entrained into the shear layer (see Fig. 1). Beyond the measurement rake, the flowexpands into a large duct and is diverted into a chemically resistant catch bag. Theproducts are scrubbed down and neutralized prior to release in the atmosphere.

3 Data Reduction Techniques

To measure mixing, one would integrate the probability density function (PDF)of mixed fluid over the domain of interest. Since the PDF is unknown a priori, amethod of obtaining statistics about its form is required. In these experiments, thisis implemented using the hydrogen-fluorine reaction and the “flip” experimentaltechnique that relies on two experiments: one performed with the upper-streamreactant rich and the other with stoichiometries “flipped” such that the lower streamis reactant rich. The overall reaction is H2 + F2 → 2 HF. NO is premixed with thehydrogen stream to initiate the overall reaction, via the release of atomic fluorine,by the reaction, NO + F2 → NOF + F. The stoichiometric mixture ratio, φ, defined


as the number of moles of upper-stream fluid required to fully consume one mole oflower-stream fluid, is given for the reactants used here by,

φ = [F2][H2] + 1

2 [NO] , (1)

where brackets denote freestream (molar) concentrations. If Z is the mole fractionof high-speed fluid in a mixture, a stoichiometric mixture ratio of φ = 4 implies thatfour parts (moles) of upper-stream fluid per part of lower-stream fluid must mix forcomplete consumption of all reactants, or Zφ = 0.8.

In computing the probability of mixed fluid, the flip experiment technique ex-cludes contributions from the edges of the PDF. These are dominated by unmixedfluid and as such do not contribute to the probability of mixed fluid calculation. Thisis expressed as

Pm(y) =∫ 1−ε

ε

p(Z , y) dZ , (2)

where p(Z , y) is the PDF of mixed fluid and ε � Zφ /2 [11, 12].The flip experiment is used to measure the total amount of molecular mixing in

the flow. This can be expressed as a fraction of the mixed fluid in the duct, δm/h,or as a fraction of the mixed fluid in the mixing layer, δm/δt. The details of the flipexperiment have been previously documented [10–13]. A brief description of thedata reduction steps is included in the Appendix for completeness.

A key assumption in this technique is that chemical reactions are kinetically fast.For a chemical reaction to be mixing limited, the reaction time scale, τχ , must beshorter than the fluid dynamic mixing time scale, τm. The later is proportional to theconvection time, τc = L/Uc, where Uc is the convective velocity, given by,

Uc

U1= 1 + r s1/2

1 + s1/2, (3)

where r ≡ U2/U1 is the freestream speed ratio, s ≡ ρ2/ρ1 is the freestream densityratio [12] and the subscripts 1 and 2 refer to upper- and lower-stream quantities,respectively. This is expressed in terms of the Damköhler number defined herein terms of the outer (convective) time scale, Da = τχ/τc. Other characteristicmixing times that could be used are proportional to this choice, with a constant ofproportionality that depends on known ratios of large-scale variables.

The chemical time scale, τχ , is estimated using the “balloon reactor” model [19].In this model, the chemical and thermodynamic state of a fluid parcel is solved as afunction of time at constant pressure. Fluid from the two streams is entrained intothe shear layer mixed-fluid core volume, at rates governed by shear layer growthand entrainment, where reactions proceed, with a rapid increase in temperature. Themaximum slope of this temperature rise vs. time is extrapolated to the asymptotictemperature to determine the chemical time scale, τχ . Calculations are performedusing the Cantera software package [20]. Cantera is also used to calculate the adi-abatic flame temperature rise, Tf, computed as the difference between the temper-ature of the stoichiometric mixture at constant enthalpy and pressure and the initialambient temperature.


An assumption of the flip technique is that the flow remains unchanged betweenthe high- and low-φ experiments. This assumption has been relaxed over that used byMungal and Dimotakis [10], who required that the heat release be minimal in orderfor its effect on the flow to be negligible, i.e., the reacting and a corresponding non-reacting runs should have identical total-pressure profiles at the rake location. In thepresent study, the magnitude of the heat released was allowed to be relatively high,so long as the flow in the two reacting experiments was matched, as indicated by thetotal-pressure profiles.

4 Results and Discussion

Figure 2 presents two composite schlieren visualizations of the flowfield: (a) for non-reacting flow and (b) for reacting flow. In both cases, M1 = 1.5 and U2 = 45 m/s. In thenon-reacting case, the mismatch in the pressure of the two streams causes the shearlayer to curve downwards and a shock train beginning with an expansion fan is set upat the tip of the splitter plate. However, with heat release, the shear layer growth rateis reduced [13, 14], and, as a result of the change in angle of the dividing streamline,the shock train begins with a compression wave. A change in the gradient of the heatrelease, registered as the transition from dark to light in the schlieren image, is seen inthe lower portion of the reacting shear layer, and indicates the location of maximumheat release. It is somewhat shifted toward the lean reactant, as expected [10, 21].The recirculation zone captures unmixed high-speed upper-stream fluid as well aspartially reacted lower-stream fluid and pumps it upstream towards the ramp (see

Fig. 2 Schlieren visualization of a M1 = 1.5 and U2 = 45 m/s shear layer with the rake at L = 38.1 cm,visible at the right edge of the figures. Each image is a composite from two separate experiments.a Non-reacting, top and bottom streams are both are 100% [N2]; b reacting with φ = 1/6, reactantcompositions for top stream are 17.8% [H2], 0.4% [NO] and for bottom stream is 3% [F2]


Table 1 Stoichiometric run(φ = 1) conditions forU1 ≈ 430 m/s (M1 = 1.5)and U2 = 25 m/s

Upper-stream Lower-stream Tf φ Da

[H2] (%) [NO] (%) [F2] (%) (K) – –

0.96 0.10 0.98 98 1.03 0.091.95 0.10 1.97 197 1.01 0.592.95 0.10 2.97 294 0.99 1.443.96 0.10 3.93 389 1.02 2.525.95 0.10 5.88 573 1.02 4.65

Fig. 1). A secondary, low strain-rate shear layer is established at the intersectionof the ramp with the lower guide wall that can serve as a flameholding region in ascramjet combustor. The instrumentation rake, located at L = 38.1 cm, is visible atthe far right edge of the schlieren images.

Flows examined to validate that the runs used to determine mixing were in thefast kinetic regime are summarized in Table 1. Subsequently, flip experiments arecompared, measuring mixing and highlighting differences between various pointsin the test matrix. A list of experimental conditions, namely, lower-stream speed,stoichiometry, reactant composition, adiabatic flame temperature rise, Damköhlernumber, and mixed fluid fractions are listed in Table 2.

Figure 3 presents the normalized temperature-rise profiles, T/Tf, fromeach case of Table 1. T is computed as the difference between the measuredtemperatures of a reacting and a corresponding non-reacting run. Insight can begained as to the minimum Da, above which reactions are mixing limited, bycomparing these curves. Normalization by the adiabatic flame temperature risefurther removes any bias of different molar concentrations on overall temperaturerise and allows a direct comparison of relative effective kinetic rates [15]. Beyondconvergence of the normalized temperature rise profiles, further increases in kineticrate do not increase overall chemical product formation and mixing-limited reactingflow is established. A maximum of 80% of Tf is seen to be attained, as limited bythe entrainment of unreacted fluid into the mixing layer.

Table 2 Summary of run conditions

Case U2 φ [H2] [NO] [F2] L Tf Da δm/δt δm/h

a 45 6 1.30 0.1 8.0 0.38 223 6.74 0.54 0.441/6 17.80 0.4 3.0 0.38 495 2.48

b 45 6 1.30 0.1 8.0 0.47 223 8.16 0.66 0.581/6 17.80 0.4 3.0 0.47 495 3.01

c 15 4 1.45 0.1 6.0 0.47 234 6.23 0.61 0.451/4 11.95 0.1 3.0 0.47 465 2.05

d 45 4 1.45 0.1 6.0 0.47 234 5.97 0.66 0.541/4 11.90 0.2 3.0 0.47 464 2.49

e 45 4 1.40 0.2 6.0 0.47 291 9.67 0.74 0.541/4 7.90 0.2 2.0 0.47 397 1.89

Note that M1 = 1.5 (U1 ≈ 430 m/s), U2 in m/s, L in m and Tf in K


Fig. 3 Normalizedtemperature rise profilesfor stoichiometric runs withincreasing heat release.M1 = 1.5, U2 = 25 m/s

ΔT / ΔTf

y/h

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.51% [H2; F2]2% [H2; F2]3% [H2; F2]4% [H2; F2]6% [H2; F2]

Transition to mixing-limited kinetics in this flow occurs at reactant concentrationsabove 3%, corresponding to Da ≥ 1.4. This is a conservative estimate, as the“balloon reactor” model was formulated for a free shear layer and does not accountfor the benefits of the recirculation zone that makes hot products and radicalsavailable for entrainment, increasing the overall kinetic rate. Note that, for the 3%reactant concentration case, a higher fraction of the adiabatic flame temperaturerise is reached near the lower guide wall than in the other “fast” cases. Thisis attributed to the beneficial secondary effect of the recirculation zone. In the3% case, the recirculation bubble re-entrains partially mixed fluid into the lowerportion of the shear layer, allowing more time for reactions to occur. Increasingreactant concentrations, and hence heat release, leads to a reduction in shear layerentrainment and growth rates [13]. Consequently, the attachment location is pushedfurther downstream and so the probability of finding mixed fluid at the measurementlocation near the lower guide wall decreases, more-closely resembling the behaviorof a free shear layer.

One way of quantifying the effect of the heat release on the flowfield is throughthe pressure coefficient, defined as:

Cp = pe − pi12ρ1U2

1

= 2

γ M21

(pe

pi− 1

), (4)

where pe and pi are the upper guide wall exit and inlet (static) pressures, respectively,ρ1 is the upper-stream density, and γ the ratio of specific heats (matched in thetwo streams). Figure 4 shows the overall pressure coefficient of the device for inertand reacting mass injection through the lower stream. Increasing the (inert) massflux through the lower stream deflects the upper stream upwards and increases thepressure coefficient within the test section, eventually recovering the (near) zero


Fig. 4 Overall pressurecoefficient with M1 = 1.5 forreacting (open symbols -U2 = 25 m/s) and non-reacting(closed symbols) flow

r = U2 / U1

Cp

0 0.05 0.1 0.15 0.2 0.25-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

Inert (M1 = 1.5)1% [H

2; F

2]

2% [H2; F

2]

3% [H2; F

2]

4% [H2; F

2]

6% [H2; F

2]

value typical of a free shear layer. As mentioned previously, heat release reduces theshear layer’s growth rate and entrainment requirements, and this dilatation effectsthe flow by increasing the effective pressure coefficient. Therefore, increasing heatrelease in the flow acts in a similar fashion to increasing mass injection, albeit fordifferent reasons.

4.1 Variable measurement location

Table 2 summarizes the velocity, composition, adiabatic flame temperature rise, andmixing results for the flip experiments. Figure 5 presents results from the series offlip experiments performed, showing the normalized temperature rise for the twocases as well as the probability of mixed fluid. For each composition, the location ofmaximum heat release is shifted toward the stream that is lean in reactants, howeverthe peak values are almost equal, indicating a near unity entrainment ratio forthis flow.

Normalized temperature profiles from a pair of “flip” experiments with M1 = 1.5,U2 = 45 m/s, and φ = 6 and 1/6 are presented in Fig. 5a. The measurement stationis at L = 38.1 cm. The stoichiometric adiabatic flame temperature rise, Tf, forthe low-φ experiment was approximately twice that of the high-φ experiment, tomatch the two flows. The flows are considered matched when the total pressureprofiles for the two cases are very nearly matched, as discussed in the Appendix and

�Fig. 5 Normalized temperature rise and probability of mixed fluid profiles for M1 = 1.5. For Casesa–d, s = 1, whereas for Case e, s = 0.2. a U2 = 45 m/s, φ = 6 and 1/6, L = 38.1 cm; b U2 = 45 m/s,φ = 6 and 1/6, L = 46.7 cm; c U2 = 15 m/s, φ = 4 and 1/4, L = 46.7 cm; d U2 = 45 m/s, φ = 4 and 1/4,L = 46.7 cm; e U2 = 45 m/s, φ = 4 and 1/4, L = 46.7 cm


ΔT/ΔTf ; Pm

y/h

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5φ = 1/6; ΔTf = 495 Kφ = 1/6 fitφ = 6; ΔTf = 223 Kφ = 6 fitPm (from raw data)Pm (from fits)

ΔT/ΔTf ; Pmy

/h

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4


ΔT/ΔTf ; Pm

y/h

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4


ΔT/ΔTf ; Pm

y/h

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4


ΔT/ΔTf ; Pm

y/h

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4


a b

c d

e


illustrated in Fig. 6. Figure 5a also plots the probability of mixed fluid at any locationwithin the mixing region [see (2)]. Fits are computed using a matched-exponentialfunction, characterized by a single peak but different widths for the two branches(see Appendix for more details). The fit to the probability of mixed fluid is computedas a sum of the fits of the normalized temperature rise profiles and not as a fit to theprobability of mixed fluid data.

Figure 5b presents data from a set of flip experiments at identical conditions,with the measurement location shifted downstream to L = 46.7 cm. The mixed fluidfraction in the duct, δm/h, defined in the Appendix, has increased by 32%, with theshear layer continuing to mix previously and freshly entrained fluid from the freestreams. The upper portion of the normalized temperature profiles, up to the locationof maximum heat release, is very similar, indicating that little or no additional heatrelease occurs in the high-speed section of the flow. However, there is increasedmixing in the lower portion of the profiles, with Case b yielding much higher mixing.This is attributed to the resulting re-entrant jet into the recirculation zone. The re-entrant jet is rich in upper-stream fluid and also contains unreacted lower-streamfluid. These have ample time to react in the relatively slow moving recirculationzone. This causes an increase in the mixed fluid fraction within the shear layer, δm/δt,of 22%.

In the core of the mixing layer, a probability of mixed fluid of unity is achieved, in-dicating that all fluid in that region has mixed on a molecular level by the downstreammeasurement station. In some locations, a probability larger than unity is estimatedas a result of compounded measurement errors from the pair of experiments, asdiscussed in the Appendix.

Fig. 6 Flow matching betweenlow- and high-φ runs of Fig. 5d


In gas-phase free shear layers, both the shear layer thickness, δt, and the mixed-fluid thickness, δm, are linear functions of the distance from the splitter plate, andtheir ratio has been found to have the constant value of δm/δt ≈ 0.49 [12]. Therefore,the increase in mixed fluid fraction with increasing distance of the measurementstation indicates a higher mixing efficiency of the expansion-ramp geometry overthat in a free shear layer.

4.2 Variable injection

Figure 5c and d presents the normalized temperature rise profiles from a pair of flipexperiments and the calculated probability of mixed fluid in a flow with M1 = 1.5,φ = 4 and 1/4, and L = 46.7 cm. The difference between the two cases is thelower stream injection velocity, namely U2 = 15 and 45 m/s, respectively. Case c isinteresting. In the low-φ case (φ = 1/4), the low injection speed of the lower stream,U2, and the lower mass flux, limit the available reactants. The reaction is completedshortly after the two streams meet, fully consuming the lower-stream reactants. Oncethe lower stream has been entrained (and consumed), the shear layer attaches tothe lower guide wall and the recirculation zone is established. The hot products arefurther diluted by the entrained upper-stream fluid leading to a lower maximumtemperature rise. However, the temperature rise profile is wider, indicating thatgood mixing extends throughout the reaction zone. For the high-φ case (φ = 4), therecirculation zone causes the location of the maximum heat release to shift downward(towards the lower stream).

Comparing these flip experiments, the maximum value of the probability of mixedfluid, Pm, increased from 0.9 for Case c, to 1 for Case d, indicating that the shear layerinterior is fully mixed on a molecular scale in this region. Increasing the lower-streamspeed by a factor of 3 produces a 20% increase in the mixed fluid fraction, δm/h. Theincrease in mixing with increased injection can be related to the increased availabilityof lower-stream reactants and the enhancement of the recirculating flow associatedwith the chosen geometry. Further, the location of the maximum is shifted towardsthe high-speed fluid. In the context of a scramjet engine, for example, this impliesthat more high-speed fluid has mixed and reacted, and so a higher flux of reacted,thrust-producing fluid exits the combustor.

4.3 Variable stoichiometry

To assess the error introduced by neglecting the edge contributions of the PDF [see(2)], flip experiments were conducted at the same flow conditions at two different sto-ichiometries. Figure 5b and d presents the normalized temperature rise profiles andthe probability of mixed fluid for M1 = 1.5, U2 = 45 m/s, L = 46.7 cm, with φ = 1/6 and6, and φ = 1/4 and 4, respectively. The high- and low-φ normalized temperature riseprofiles are quite similar, both in the maximum T/Tf achieved and in their overallshape. Slight differences may be attributable to the different levels of heat releasein the two cases and are localized in the low-speed region of the flow (y/h <-0.3).The mixed-fluid fraction within the mixing layer, δm/δt, is identical for the two casesand equal to 0.66. A 7% difference in the mixed fluid fraction within the duct, δm/h,in the two cases can be attributed to the differences mentioned above and in thedifferent resolution in the temperature rise data. Good agreement in mixing between


the two cases validates the assumptions in the flip experiment, despite the relativelylarge value of ε corresponding to the selected equivalence ratios (for φ = 1/4, ε ≈ 0.1and for φ = 1/6, ε ≈ 0.07).

4.4 Variable density ratio

Previous research has focused on freestream density ratio effects on mixing of a freeshear layer [7, 22]. These studies showed that “the mixed fluid fraction is...essentiallyindependent of the density ratio” [12]. The mean mixed fluid composition, Z m, isdefined as

Z m = δp(1 − Zφ)

δp(Zφ) + δp(1 − Zφ)(5)

where δp is the chemical product thickness and Zφ is the stoichiometric mixturefraction, as defined in the Appendix.

The mixing region, formed between the upper- and lower-stream fluids, entrainsat a certain ratio, which can be solved for in terms of the mean mixed fluidcomposition [23],

En = Z m

1 − Z m= δp(1 − Zφ)

δp(Zφ)(6)

In contrast to the mixed-fluid fraction, both the mean mixture composition and theentrainment ratio depend strongly on the density ratio [12].

Flip experiments conducted at M1 = 1.5, U2 = 45 m/s, φ = 4 and 1/4, and s = 1 and0.2, are presented in Fig. 5d and e, respectively. Considering the difference in densityof the two streams, care was taken in the mixture preparation to match molar specificheats. In both cases, the mixed fluid fraction within the duct, δm/h, remains constant,and the probability of mixed fluid profiles are quite similar, both in the locationand width of the curve, in good agreement with the results for a free shear layer.However, the mixed fluid fraction within the mixing layer, δm/δt, increases by 12% forthe flow with s = 0.2 (Case e). Both cases exhibit significantly higher levels of mixingover the value of δm/δt = 0.49 quoted for a free shear layer [12], indicating a highermixing efficiency for the expansion-ramp injection geometry. The entrainment ratio,which is near unity for Case d, increases to 1.4 for Case e, indicating increased upper-stream fluid entrainment in the latter case.

5 Summary and Conclusions

Molecular mixing was measured using the flip experiment technique in a two-streamblow-down wind tunnel. The “air” stream was supersonic (M1 = 1.5) and expandedover a 30◦ ramp. “Fuel” was injected through perforations in the ramp forming amixing layer between the two streams. Once the lower stream is fully entrained, themixing layer attaches to the lower guide wall, establishing a recirculation zone. Thisgeometry is characterized by many desirable features of a scramjet combustor, suchas high mixing rates, low pressure losses, and low strain rate regions for flameholding.


The fast chemistry limit for this flow is established by increasing reactant concen-tration in a stoichiometric (φ = 1) reaction. The reacting flow is found to be mixing-limited for Da ≥ 1.4, a value accepted as the minimum Da required for the flipexperiments.

A significant increase in mixing is observed when the measuring station is movedfrom L = 38.1 to 47.6 cm, indicating the effectiveness of the recirculation zone as amechanism to enhance mixing in compressible high-speed flows. Such an increase inmixing is not seen in free shear layers, which maintain a constant amount of mixedfluid, independent of the measurement station.

Mixing is found to depend on the velocity ratio, r, for a fixed top stream velocity.Low injection levels lead to full entrainment and consumption of lower stream fluid.The mixed fluid is then further diluted until the measurement station, yielding lowtotal mixing due to limited available reactants. For large lower stream velocities,a self-similar free shear layer flow is recovered, where the mixing level becomesindependent of the measurement position. However, an intermediate range ofvelocity ratios exists that results in higher levels of mixing than for free shear layers.This mixing enhancement is attributed to the action of the recirculating fluid.

Molecular-mixing measurements were insensitive to the selected flip experimentstoichiometric ratios chosen. This indicates that edge-effects of the PDF do notsignificantly contribute to the mixed fluid fraction measurements and that there existsa region in Z -space where mixing is very nearly independent of the particular choiceof stoichiometry selected for the pair of experiments.

For reduced density ratios, s, more high-speed fluid is entrained into the mixinglayer. The overall mixing within the duct (δm/h) achieved is independent of thedensity ratio, however the particulars of the mixing region change, and the mixingwithin the shear layer (δm/δt) is found to increase over the density-matched case.

Overall, the expansion ramp geometry produces higher levels of mixing thanthose exhibited by free shear layers and appears to be a promising combustor flowgeometry element for high-speed propulsion systems.

Acknowledgements The authors would like to acknowledge constructive discussions with C. Bondand G. Matheou. D. Lang assisted with the computers and electronics associated with the facilitycontrol and data acquisition. The experiments were conducted with the expert assistance of E. Dahl.This work was funded by the AFOSR under Grants FA9550-04-1-0020 and FA9550-04-1-0389, whosesupport is gratefully acknowledged.

Appendix: Flip-Experiments, Flow Matching and Error Estimation

Let the mixture mole fraction be defined as Z = n1/(n1 + n2), where n1 and n2

represent the number of moles of upper- and lower-stream fluid. Accordingly, thecomplete consumption of all reactants will occur at a stoichiometric mixture molefraction [12],

Zφ = φ

φ + 1(7)

and a maximum amount of chemical product, θ(Z ; Zφ), will be formed. At off-stoichiometric conditions and with complete consumption of the lean reactant, theamount of chemical product decreases to zero at Z = 0 and Z = 1, yielding a


triangular normalized product function with a peak of 1 at Z = Zφ . Exploiting thetriangular shape of the chemical product function for off-stoichiometric mixtures,the integral of the PDF of mixed fluid can be estimated by adding the normalizedtemperature rise values from a low-φ, φ1, and a high-φ, φ2, experiment at any locationin the flow (see discussion in [12] and references therein).

For a chemically fast (high-Da) system accompanied by heat release, the meannormalized temperature rise, T/Tf, at any location in the flow, equals the integralof the product of the probability density function (PDF) of mixed fluid, p(Z , y), andthe normalized chemical product function, θ(Z ; Zφ) [12]. Therefore, the probabilityof mixed fluid at a location in the flow is given by [13],

Pm(y) =∫ 1

0

θ(Z ;φ = φ1) + θ(Z ;φ = φ2)

1/(1 − Zφ1)p(Z , y) dZ , (8a)

or

Pm(y) = [1 − Z (φ1)][

T(y)

Tf

∣∣∣∣φ=φ1

+ T(y)

Tf

∣∣∣∣φ=φ2

]. (8b)

A measure of the total amount of mixing in the flow is the mixed fluid fraction,δm/h, given by

δm

h=

∫ 1/2

−1/2Pm

( yh

)d

( yh

), (9)

yielding an estimate of the total (mole) fraction of fluid within the duct that has mixedon a molecular scale. It takes into account the effects of changing growth rate andamount of mixing in the layer with increasing upper stream velocity. The fraction ofmixed fluid within the mixing layer, δm/δt, is given by

δm

δt= (1 − Zφ=φ1)

[δp

δt

∣∣∣∣φ=φ1

+ δp

δt

∣∣∣∣φ=φ2

], (10)

where δp is the product thickness,

δp =∫ h/2

−h/2

T(y)

Tfdy (11)

is the integral of the normalized temperature rise profile, and δt is the distance fromthe lower guide wall at which point the normalized temperature profile reaches 1%of its maximum value [10, 12, 13]. A pair of flip experiments yields the probability ofmixed fluid, Pm, by measuring the resulting temperature rise. The amount of mixedfluid is then obtained through the mixed fluid fraction, δm/δt, or δm/h.

The T/Tf profiles were fit using the (asymmetric) exponential function

T(y)

Tf= T

Tf ∞,i+

(T

Tf,max− T

Tf ∞,i

)exp

⎛⎝ 4∑

j=2

λi, j(y − y0)j

⎞⎠ , (12)

where y = y/h and T/Tf,max is the single characteristic peak located at y0. The fiton either side of the peak has different slopes, fitted by the coefficients λi, j, where i = 1


for y < y0 and i = 2 for y > y0 and j = 2–4. Exploiting the fact that the temperatureprofiles asymptote to zero in the top stream, the parameter (T/Tf)∞,2 was set tozero and not included in the optimization in that portion of the flow.

An assumption of the flip experiment is that the flow remains unchanged with thechanging location of the temperature rise between the low- and the high-equivalenceratio run. The criterion for flow matching is taken to be the total pressure profile,p0(y). The flows in a pair of flip experiments are said to be matched when the totalpressure profiles are in good agreement. Example total pressure profiles from Fig. 5dare shown in Fig. 6. The total pressure profiles are in good agreement if the adiabaticflame temperature of the stoichiometric mixture is chosen to be twice as high forthe low-φ experiment as for the high-φ experiment. Even at moderate levels of heatrelease, changes in the amount of heat release do not significantly affect the flow [13].

In calculating the probability of mixed fluid utilizing the flip experiment technique,two sources of measurement errors affect the determination of Pm: the temperaturedata, and the mixture composition of each stream, which affects the adiabatic flametemperature rise, Tf. Temporal temperature measurements are averaged over aperiod of time in which the flow has achieved steady state. The averaging timeis chosen to allow sufficient flow-through times, based on the convective velocityof the shear layer. Statistical convergence in this flow typically requires averagingtimes of the order of 0.5-1 s, while typical flow through times are on the order of1-2 ms. The thermocouple response time was measured to be less than 3 ms. Prior toeach run, “tare” reference temperature values are obtained for each probe, whichare subtracted as offsets from measured values. For a typical run, uncertainties intemperature due to flow unsteadiness are typically less than ±3% F.S., with 95%confidence, which translate to a ±4.5% error in the mixed fluid measurement, δm/h.Errors are slightly higher in the recirculation zone, because lower speeds result in

Fig. 7 Figure 5b includingerror estimates. Temperatureprobes are sampled at 1 kHzwith data averaged over 1 s.Error bars increase in theregion near the lower guidewall due to longer flow timescales in the recirculation zone

ΔT/ΔTf ; Pm

y / h

0 0.2 0.4 0.6 0.8 1-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4



longer characteristic time scales for a given averaging period. Radiative losses werecalculated to be less than 1% for these conditions, and so no further correctionswhere applied to the measurements. Sample error estimates in the flame temperaturerise and the probability of mixed fluid for the case of Fig. 5b presented previously areshown in Fig. 7.

The mixtures for each stream are prepared with the partial pressure technique.Thus, the accuracy of the mixture is limited by the accuracy of the pressure gageused. The gages were Druck type PDCR-920 with a reported combined non-linearity,hysteresis and repeatability of ±0.1% full scale (best straight line). Errors whilepreparing the mixtures typically resulted in a deviation of less than 1% in the targetedvalues of the mixture’s density and specific heat, and thus are considered negligible.

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