43rd AIAA Aerospace Sciences Meeting and Exhibit, 10–13 Januray 2005, Reno, Nevada

Mixing in crossflowing jets: Turbulence quantities

L. K. Su∗

Applied Fluid Imaging Laboratory, Department of Mechanical Engineering

Johns Hopkins University, Baltimore, MD 21218, USA

M. G. Mungal†

Thermosciences Division, Mechanical Engineering Department

Stanford University, Stanford, CA 94305, USA

Simultaneous planar laser-induced fluorescence (PLIF) and particle image velocimetry(PIV) yield measurements of two-dimensional jet fluid concentration and velocity fields,both the mean and fluctuating terms, in turbulent crossflowing jets. The jet-to-crossflowvelocity ratio is r = 5.7 and the jet exit Reynolds number is approximately 5000. Themeasurements concern the developing region of the flow. Two flow configurations arestudied, one in which the jet nozzle is flush with the tunnel wall and the other where thenozzle protrudes into the uniform region of the tunnel flow. The jet nozzle in both casesis a simple pipe. Previous work focused on the averaged scalar and velocity fields. Here,we discuss the resolution issues bearing on the determination of small scale fluctuations,and present results for the averaged scalar variance, 〈C′2〉, the scalar flux components,〈u′

iC′〉, the turbulent normal stresses, 〈u′2〉 and 〈v′2〉, and the turbulent shear stress, 〈u′v′〉.

These results should prove particularly useful for assessment of crossflowing jet simulationefforts. The range of length scales spanned by the measurements also makes possible directassessment of models and model assumptions used for scalar transport and mixing in large-eddy simulation (LES). Sample results demonstrate the use of the present data in theseLES model assessments.

I. Introduction.

The crossflowing turbulent jet, in which a jet is injected into a perpendicular fluid stream, is of con-siderable practical significance in engineering systems, inclucing combustion applications such as aerospacepropulsion and gas-burning power generation. However, the body of research devoted to mixing in thecrossflowing jet is small. Among other issues, the complicated vortical structure of the crossflowing jet1

makes theoretical treatment difficult. Modeling efforts have been reported by various authors.2–5 A commonproblem is the difficulty in modeling the near field of the flow. Experimental evidence also suggests that theflow is very sensitive to the ratio of jet velocity to crossflow velocity, which constrains efforts to make generalconclusions about the flow configuration.

A schematic of the characteristic vortical structures of the crossflowing jet is given in Fig. 1. Shown arethe horseshoe vortices that form at the upstream side of the jet exit, the jet shear layer instability on thejet windward surface, the wake vortices, and the counter-rotating vortex pair (CVP). The CVP becomes thedominant structural feature of the crossflowing jet as the flow develops, and is responsible for much of thedifficulty in modeling the flow.

The present measurements are intended to provide a comprehensive view of the velocity and conservedscalar fields in the developing region of the flow. A previous paper6 dealt primarily with the flow developmentin terms of the mean scalar and velocity fields. In this paper, we will focus on the fields of fluctuatingquantities, including the scalar variance, C

′2, the scalar flux components, u′iC

′, the turbulent normal stresses,∗Assistant Professor, lsu@jhu.edu, Member AIAA.†Professor, mungal@stanford.edu, Associate Fellow AIAA.Copyright c© 2005 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with

permission.

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x, u0

y

v�

z

Figure 1. The vortical structure of the crossflowing jet.

u′2 and v′2, and the turbulent shear stress, u′v′. All measurements are made at a single jet-to-crossflowvelocity ratio (r = 5.7). Velocity ratios of around 5 are of particular interest in aerospace propulsionapplications. The effect of the crossflow velocity profile is considered by placing the jet exit nozzle bothflush with the wind tunnel wall, and also outside the crossflow boundary layer. By performing the planarmeasurements in the center plane (i.e. the jet symmetry plane) and at various positions off the center plane,we are able to evaluate the three-dimensionality of the flow. These results are also of direct interest toongoing efforts to compute mixing and combustion in the crossflowing jet.7, 8

These measurements span, instantaneously, from the small scales of the mixing and velocity fields nearlyto the full jet width. This makes the data ideally suited for assessment of subgrid-scale (SGS) modelsfor large-eddy simulation (LES), since the relationships between flow scales can be evaluated for a widerange of such scales. This paper will demonstrate two methods whereby the data are used to assess SGSscalar transport and mixing models. In the first method, known as a priori testing, the spatially resolvedexperimental measurements are first filtered to simulate LES data, then SGS models for quantities such asscalar flux or dissipation are applied; the results from the SGS models are then compared with the actualsmall-scale flux or dissipation values detemined directly from the measurements. The second method entailsthe direct assessment of the assumptions used in constructing the SGS models. Here we present sampleresults for both of these methods (Sun & Su9 provide a more complete discussion of LES model assessmentsusing these data).

II. Experimental conditions.

These experiments are performed in an updraft wind tunnel with air as both jet fluid and crossflow fluid.The maximum crossflow velocity in the tunnel is v∞ = 2.95 m/s. The jet nozzle is a simple pipe with 6.35 mmouter diameter, inner diameter d = 4.53 mm, and length 320 mm. The average (bulk) jet velocity based onvolumetric flow rate is 16.9 m/s, giving a velocity ratio of 5.7 and a jet exit Reynolds number of 5000.

Particle image velocimetry (PIV) is used to measure the velocity fields while planar laser-induced fluores-cence (PLIF) is used for the scalar field measurements. To provide the Mie scattering signal for PIV, the jetflow is seeded with submicron aluminum oxide particles, while the crossflow is seeded with a glycerol-waterfog. A single, dual cavity Nd:YAG laser (Spectra-Physics PIV-400) is used to produce two 532 nm laser sheetpulses in quick succession. The time delay between pulses is as short as 8 µs. The resulting scattering signal iscollected by an interline transfer CCD camera (Kodak Megaplus ES 1.0, 1k×1k pixel resolution). The frametransfer capability of this camera allows each of the two closely spaced laser sheet pulses to be captured ina separate image. This permits the use of a cross-correlation PIV algorithm,10 which eliminates directionalambiguity and yields improved resolution over single-image autocorrelation techniques. The algorithm usedhere also incorporates iterative interrogation window offset to increase vector yield.

For PLIF, acetone is seeded into the jet fluid stream, to approximately 10% by volume. To excite thefluorescence signal, a 308 nm XeCl excimer laser (Lambda-Physik EMG 203MSC) is used. The PLIF signalis captured by a thermoelectrically-cooled, slow-scan CCD camera (Photometrics AT200), with 512 × 512pixel resolution. BG-25 bandpass filters isolate the PLIF signal (which peaks in the range 400 − 500 nm)from the much brighter Mie scattering signal at 532 nm.

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log

10 (

C/C

0 )

0.

1.5 20.5

1

1.5

-1.91 2 3

4 5 6

7 8

y/rd

0.

1.

2.

(a)

x/rd

0. 1. 2. 3.

(b)

Figure 2. A sample image pair, from the center plane of the protruding nozzle case. (a) The scalarfield, showing the 8 imaging subwindows used for PIV. (b) The PIV vector field corresponding to thesolid box (subwindow number 5) in (a).

Prior to entering the measurement area, the laser beams are passed through a focusing spherical lens, adiverging cylindrical lens to form the sheets, and a converging cylindrical lens to control the sheet spreadingangle. The centers of the two sheets are separated by no more than 50 µm throughout the measurementarea, where the thickness of the 532 nm sheet varies from approximately 300 to 800 µm, and the thicknessof the 308 nm sheet ranges from 500 to 1000 µm.

In analyzing the scaling properties of the crossflowing jet, the flow may be divided into three regimes.In the near field, for sufficiently large velocity ratio r, the jet momentum dominates and the flow can beexpected to approximate a pure jet. The flow then bends through an intermediate region until, in the farfield, memory of the initial conditions will be lost and the flow is expected to resemble a wake. The planarmeasurement area in these experiments extends from the jet exit; prior work6 indicates that the measurementarea encompasses the onset of the region in which the velocity field exhibits wake-like scaling properties.To permit assesment of the effect of the crossflow boundary layer on the flow development, two jet nozzlepositions are considered, the first in which the nozzle exit is flush with the wind tunnel wall, and the secondin which the nozzle protrudes 100 mm into the crossflow. The 80% point of the boundary layer profile lies6 mm from the wind tunnel wall, so in this protruding nozzle case the jet is well outside of the crossflowboundary layer. To evaluate the three-dimensionality of the flow, the planar measurements are taken in thejet center plane, and in planes located at 0.22, 0.45, 0.67, 0.89, and 1.11 rd off of the center plane. It has beenshown11 that flow trajectories, jet widths etc. for different r values are in good agreement when distancesare normalized by rd.

A sample PLIF image, taken in the center plane with the protruding nozzle, is shown in Fig. 2a. Asshown in Fig. 1, x is the initial jet direction, y is the crossflow direction, and z is the out-of-plane direction.The PLIF imaging window spans roughly 3.5 rd per side, and is identical for all of the measurements. ThePIV processing inherently compromises spatial resolution (yielding here a final vector resolution of 100 x 100pixels from the original 1k × 1k Mie scattering images), so in order that the processed PIV results resolvefluctuations in the velocity field, the full PLIF imaging region is tiled by eight smaller PIV imaging windows,indicated by the dashed boxes in Fig. 2a. For the particular scalar field shown in the figure, the simultaneousPIV field is given in Fig. 2b, and corresponds to the solid box (subwindow number 5) in 2a.

III. Results.

Turbulence quantities involving the fluctuating velocity and scalar field terms are available due to thehigh resolution of the present data. In particular, the simultaneous nature of the measurements permitsthe scalar flux components, u′C′ and v′C′, to be determined. These represent unresolved terms in theReynolds-averaged scalar transport equation, and correspond to subgrid production terms in LES-filteredscalar transport. From the evidence of previous computations, these scalar flux terms are very difficult tosimulate accurately.12 This section also presents results for the scalar variance, C′2, and the u′2, v′2 andu′v′ components of the turbulent stress tensor.

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s/rd U (m/s) δ (m) Reδ λν (µm) λD (µm) ∆xC (µm) ∆xu (µm)1 11.3 0.019 14100 220 180 175 2302 4.0 0.041 10800 580 480 175 3103 3.0 0.046 9100 740 610 175 420

3.5 2.6 0.052 8900 850 700 175 420

Table 1. Outer scale parameters and resolution estimates. The estimated finest length scales in thevelocity and scalar fields are given by λν and λD, respectively. The grid resolution of the scalar fieldimages is ∆xC, and the grid resolution of the velocity fields is given by ∆xu.

A. Resolution estimates.

The accuracy of the measured fluctuation quantities is dependent upon the resolution of the data. FollowingSu & Clemens,13 we define λν , the characteristic full width of a structure in the kinetic energy dissipationrate field, as

λν = Λ · δRe−3/4δ . (1)

Here δ is a measure of the local mean flow width, and the local outer scale Reynolds number, Reδ, is definedin terms of δ and a velocity U quantifying the mean shear. Defining the flow width δ as the distancebetween the 20% points of the velocity magnitude profile, a suggested value for the proportionality constantis Λ ≈ 15.13 For the scalar mixing, the characteristic full width of a structure in the scalar energy dissipationrate field is given by

λD = λν · Sc−1/2, (2)

where the Schmidt number, Sc, for this flow system is 1.49.To determine λν and λD for various downstream positions in the present data, we estimate δ and U using

prior results for the mean flow field evolution.6, 14 These outer scale parameters, and the resulting outerscale Reynolds number Reδ, are given in Table 1 for s/rd = 1, 2, 3 and 3.5. (The quantity s is a downstreamcoordinate defined by local maxima in the averaged velocity magnitude field.) Table 1 also presents theresolution estimates determined from (1) and (2).

For s/rd = 1, the flow still follows the initial trajectory, so the relevant U is the maximum mean velocitymagnitude. For s/rd = 2, 3 and 3.5, the flow begins to exhibit wake-like properties, so the proper U is themaximum mean crossflow-subtracted velocity magnitude. The resulting length scale estimates λν and λD

are found from (1) and (2) using Λ = 15. The grid spacing in the scalar field data is ∆xC = 175.1 µm,which resolves λD at all positions shown, with Nyquist resolution except at s/rd = 1. The grid spacing inthe velocity fields, ∆xu, slightly exceeds λν at s/rd = 1, but easily resolves λν for the remaining positions.(∆xu varies because of the different sizes of the PIV imaging windows, shown in Fig. 2.)

B. Scalar variance.

Figure 3 shows the averaged scalar variance fields, 〈C′2〉, for the flush nozzle case, in the center plane and thez = 0.22 rd plane. The fields show local maxima both on the jet outer boundary and in the jet wake region.This bifurcated structure is familiar from mixing in canonical shear flows, such as mixing layers or jets, withthe peak variances being associated with steep gradients in the mean scalar fields. A more quantitative viewis available by considering profiles of 〈C′2〉. Figure 4 presents x- and y-profiles of 〈C′2〉 in the z = 0 plane forboth the flush and protruding nozzle cases. In the jet near-field, at x = 0.1 rd and x = 0.5 rd, the profiles areapproximately symmetric about the centerline, and the discrepancy between the two nozzle configurationsis small. Just outside of the potential core, at x = 1.0 rd, the bifurcated structure of the variance profile isdistinctly asymmetric, with the windward peak being much stronger than the wake-side peak; additionally,the variances are noticeably higher for the flush nozzle than for the protruding nozzle configuration. Thesetrends persist as the jet turns in the crossflow direction, as seen in the x = 1.5 rd, y = 0.5 rd and y = 1.0 rdprofiles. Both the profile asymmetry and the discrepancy between nozzle configurations are most significantat y = 0.5 rd. By y = 1.5 rd, the two nozzle configurations show similar profile magnitudes, while at

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log

10 (⟨C′2

⟩/C

0

2

)

-1.4

-3.4

x/rd

y/rd

0. 1. 2. 3.

0.

1.

2.

3.

(a) (b)

x/rd

0. 1. 2. 3.

0.

1.

2.

3.

Figure 3. Averaged scalar variances, 〈C′2〉, for the flush nozzle case, in the planes (a) z = 0, and (b)z = 0.22 rd.

y/rd

·C¢2Ò/C0

2

x = 0.1 rd x = 0.5 rd x = 1.0 rd x = 1.5 rd (a)

0 1 2 3

center streamlines

0

0.004

0.008

0.012

0

0.004

0.008

0

0.004

0.008

0

0.004

0.008

x/rd

y = 2.5 rd

y = 1.5 rd

y = 1.0 rd

y = 0.5 rd

flush nozzle

100 mm nozzle

(b)

⟨C′2⟩/

C02

Figure 4. Averaged scalar variance profiles in the jet center plane, for both the flush and protrudingnozzle cases. (a) The profiles for x = 0.1 rd, 0.5 rd, 1.0 rd and 1.5 rd, and (b) the profiles for y = 0.5 rd,1.0 rd, 1.5 rd and 2.5 rd.

y = 2.5 rd, the variance profiles are again nearly symmetric, with symmetry axes to the wake side of thecenter streamline.

The variance profiles in the off-center planes suggest, however, that the asymptotic downstream stateof the scalar variance is not characterized by symmetry between the windward and wake sides of the flow.Figure 5 shows the profiles of 〈C′2〉 in the z = 0.22 rd plane. In this plane, the variance profiles observesimilar trends to those seen in the z = 0 plane, except that the wake-side variance peak is larger thanthe windward-side peak at large y values. This can be seen in the y = 2.5 rd profile in the z = 0.22 rdplane. This implies that as the flow moves downstream, there is more pronounced mixing between jet andambient fluid on the wake side of the flow than on the windward side, possibly due to the persistence of thecoherent vortical structures in the wake region.15 This asymmetry in the variance profiles is inconsistentwith the assumption of wake-like asymptotic similarity in the mixing field; prior analysis of the data6, 14 hasshown that the scalar field, unlike the velocity field, fails to observe wake scaling with increasing downstreamdistance.

C. Turbulent scalar flux profiles.

The turbulent scalar flux terms, u′iC

′, appear in the Reynolds-averaged scalar transport equation (here innon-dimensional form):

〈ui〉∂〈C〉∂xi

− 1ReSc

∂2

∂xi∂xi〈C〉 = − ∂

∂xi〈u′

iC′〉, (3)

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American Institute of Aeronautics and Astronautics Paper 2005-0305

y/rd

x = 0.5 rd x = 1.0 rd x = 1.5 rd (a)

·C¢2Ò/C02

0 1 2 3

center streamlines

0

0.004

0.008

0

0.004

0.008

0

0.004

0.008

0

0.004

0.008

x/rd

y = 2.5 rd

y = 1.5 rd

y = 1.0 rd

y = 0.5 rd

flush nozzle

100 mm nozzle

(b)

⟨C′2⟩/

C02

Figure 5. Averaged scalar variance profiles in the z = 0.22 rd plane, for both the flush and protrudingnozzle cases. (a) The profiles for x = 0.5 rd, 1.0 rd and 1.5 rd, and (b) the profiles for y = 0.5 rd, 1.0 rd,1.5 rd and 2.5 rd.

y/rd

·ui¢C¢Ò/v

•C

0

x = 0.1 rd x = 0.5 rd x = 1.0 rd x = 1.5 rd (a)

-0.02

0

0.02

0.04

0 1 2 30 1 2 3

-0.02

0

0.02

0.04

-0.02

0

0.02

0.04

-0.02

0

0.02

0.04

0.06

center streamline

x/rd

⟨ui′C

′⟩/v

∞C

0

y = 2.5 rd

y = 1.5 rd

y = 1.0 rd

y = 0.5 rd

⟨u′C′⟩⟨v′C′⟩

(b)

Figure 6. Scalar flux profiles in the jet center plane, for the flush nozzle case. (a) The profiles of 〈u′C′〉and 〈v′C′〉 for x = 0.1 rd, 0.5 rd, 1.0 rd and 1.5 rd, and (b) the profiles for y = 0.5 rd, 1.0 rd, 1.5 rd and 2.5 rd.

where Re is the Reynolds number formed from the length and velocity scales used for normalization, and Scis the Schmidt number. The 〈u′

iC′〉 terms are unresolved in a RANS simulation, and must be modeled to

close (3). (In the context of large-eddy simulation, where quantities are decomposed into filtered (denoted by( )) and subgrid (denoted by ( )′′) parts, the divergence of the corresponding filtered LES scalar flux vector,u′′

i C′′, appears in the production term for the filtered scalar field, C. In LES of mixing, these scalar fluxesare also unresolved and are thus entrusted to subgrid models.)

Figure 6 shows profiles of the scalar flux components 〈u′C′〉 and 〈v′C′〉, in the center plane for the flushnozzle case. The profiles for the protruding nozzle configuration are qualitatively similar and are not shown.The profiles indicate that 〈u′C′〉 and 〈v′C′〉 are negatively correlated throughout the measurement area.Beyond the potential core, on the windward side of the jet, 〈u′C′〉 is positive, with the peak positive valueoccuring slightly to the outside of the jet center streamline, and 〈v′C′〉 is negative. On the lee side, 〈u′C′〉 < 0and 〈v′C′〉 > 0. The location at which the signs of these terms change lies slightly to the wake side of thecenter streamline. The profiles for the flush and protruding nozzle cases differ in that the peak values of〈u′

iC′〉 are slightly higher for the flush case, in the near field. Toward the downstream side of the measurement

area (specifically, y ≥ 1.0 rd), the magnitudes of 〈u′iC

′〉 are similar for both nozzle configurations.The form of the 〈v′C′〉 profile, in particular, seems counterintuitive. A simple interpretation of this

term might hold that since the crossflowing jet imposes a deficit velocity in the y-direction, fluid elements

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American Institute of Aeronautics and Astronautics Paper 2005-0305

-0.5

0

0.5

1

1.5

2

-0.02 0 -0.02 00.02 0.02 0.04

y/rd

⟨ui′C′⟩/v∞C

0

x = 1.0 rd x = 1.5 rd (a)

-0.02

0

0.02

-0.02

0

0.02

0 1 2 3

-0.02

0

0.02

-0.02

0

0.02

0.04

center streamline

x/rd

⟨ui′C

′⟩/v

∞C

0

y = 2.5 rd

y = 1.5 rd

y = 1.0 rd

y = 0.5 rd

⟨u′C′⟩⟨v′C′⟩

(b)

Figure 7. Scalar flux profiles in the z = 0.22 rd plane, for the flush nozzle case. (a) The profiles of 〈u′C′〉and 〈v′C′〉 for x = 1.0 rd and 1.5 rd, and (b) the profiles for y = 0.5 rd, 1.0 rd, 1.5 rd and 2.5 rd.

originating in the crossflow will have v-component of velocity in excess of the mean (v′ > 0), while havingscalar concentration less than the mean (C′ < 0). Then 〈v′C′〉 would be uniformly negative, which is notthe form of 〈v′C′〉 measured here. Instead, both the 〈v′C′〉 and 〈u′C′〉 profiles can be readily interpreted interms of the spread and centerline concentration decay of the jet. To interpret the 〈v′C′〉 term, we observethat turbulent advection into the jet is marked by v′ > 0 on the jet outer boundary, and v′ < 0 on the innerboundary. As a result, negative 〈v′C′〉 values to the outside of the jet, and positive values to the inside,show that fluid elements advected in the y-direction towards the center of the jet are characterized by excessambient fluid concentration (C′ < 0). Positive values of 〈u′C′〉 to the outside (increasing x) of the jet, andnegative 〈u′C′〉 to the inside, indicate that fluid elements undergoing turbulent advection in the x-directiontowards the boundaries of the jet (i.e. on the outer part of the jet, u′ > 0, and on the inner part, u′ < 0) arecharacterized by excess jet fluid concentration (C′ > 0). The measured 〈u′

iC′〉 thus reflect the spread of the

jet boundaries, and the decay of maximum scalar concentration, with increasing downstream distance.The present data are consistent with the heat flux profiles reported by Andreopoulos,16 in showing a

negative correlation between 〈u′C′〉 and 〈v′C′〉, although that earlier study does not show sign changes inthe profiles. Instead, the 〈u′

iC′〉 profiles of Andreopoulos resemble the portion of the profiles of Fig. 6 for

the outer part of the jet, with 〈u′C′〉 > 0 and 〈v′C′〉 < 0. The data of Andreopoulos are for a jet with nozzleexit flush with a wall, and with very low velocity ratio, r = 0.5; with that configuration, the lee side of thejet impinges on the wall. The r = 0.5 jet thus has no free inner (lee-side) boundary, which explains why the〈u′

iC′〉 profiles for the r = 0.5 flow agree only with the windward portion of the profiles for the r = 5.7 flow.

In the large-eddy simulations by Yuan17 of r = 3.3 scalar mixing, in which the jet nozzle exit is alsoflush with a wall, the profiles of mean scalar concentration show that the jet stands clear of the wall, witha distinct inner boundary in the mixing field. Accordingly, the 〈u′C′〉 profiles show sign changes similar tothose for the present r = 5.7 jet. The work of both Andreopoulos16 and Yuan17 suggests that the transitionalvalue of r, above which the jet inner boundary is free of the wall and the 〈u′

iC′〉 profiles resemble those of

Fig. 6, is between 1.0 and 2.0.It is also instructive to compare the magnitudes of the 〈u′C′〉 and 〈v′C′〉 profiles in Fig. 6. The 〈u′C′〉

component consistently has a higher magnitude than the 〈v′C′〉 component, suggesting that in the centerplane, turbulent scalar transport in the cross-stream direction is more significant than transport in thestreamwise direction. Also, the highest magnitudes of 〈u′C′〉 are on the outer portion of the jet, indicativeof stronger turbulent scalar transport on the windward side, though the discrepancy between |〈u′C′〉| onthe windward and wake sides of the jet weakens with increasing downstream distance. This is consistentwith the 〈C′2〉 results of §B, which showed higher scalar variance values on the windward side in the centerplane, with the values on the windward and wake sides approaching equality at the downstream limit of theimaging region.

Figure 7 shows the profiles of 〈u′iC

′〉 for the flush nozzle case in the z = 0.22 rd plane. In the z = 0.22 rdplane, the primary difference with the results for the center plane is that 〈u′C′〉 > 0 on the wake side of thejet for y = 0.5 rd (this is also weakly evident for y = 1.0 rd). It has been proposed17 that positive values of

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American Institute of Aeronautics and Astronautics Paper 2005-0305

y/rd

·ui¢u

j¢Ò/v

•2

x = 0.1 rd x = 0.5 rd x = 1.0 rd x = 1.5 rd (a)

0 1 2 3

center streamline

-0.4

-0.2

0

0.2

0.4

0.6

-0.4

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4

x/rd

⟨ui′u

j′⟩/v

∞2

y = 2.5 rd

y = 1.5 rd

y = 1.0 rd

y = 0.5 rd

(b)

⟨u′2⟩⟨v′2⟩⟨u′v′⟩⋅3

Figure 8. Stress profiles in the jet center plane, for the flush nozzle case. (a) The profiles of 〈u′2〉, 〈v′2〉,and 〈u′v′〉 for x = 0.1 rd, 0.5 rd, 1.0 rd and 1.5 rd, and (b) the profiles for y = 0.5 rd, 1.0 rd, 1.5 rd and 2.5 rd.The fixed-y profiles of 〈u′v′〉 are stretched by a factor of 3.

〈u′C′〉 in the off-center planes reflected the probability that fluid parcels with excess velocity in the initialjet momentum direction would also have excess jet fluid concentration.

This interpretation can be cast in terms of the large-scale organization of the velocity field. At the edgesof jet in the z-direction, the x-component of the crossflow fluid velocity relative to the jet fluid velocity isnegative, as the crossflow fluid is diverted around the jet into the wake region (this also agrees with the senseof rotation of the counter-rotating vortex pair). Then u′ < 0 correlates with C′ < 0 (crossflow fluid), so〈u′C′〉 > 0. This picture is also consistent with the explanation given above of the negative values of 〈u′C′〉observed on the wake side of the flow in the center plane, and in the z = 0.22 rd plane for y ≥ 1.5 rd. TheCVP is associated with positive x-component of velocity on the wake side in the center plane, but those fluidparcels with u′ > 0 carry crossflow fluid (C′ < 0) into the jet, giving 〈u′C′〉 < 0.

D. Turbulent stresses.

Figure 8 shows profiles of the in-plane turbulent normal stress components (equivalently, components of theturbulent kinetic energy), 〈u′2〉 and 〈v′2〉, and the turbulent shear stress, 〈u′v′〉, for the center plane for theflush nozzle case. (Again, the profiles for the protruding nozzle case are qualitatively similar and are notshown.) The results for the flush and protruding nozzle positions are very similar, the primary differencebeing the slightly smaller magnitudes for the protruding nozzle case, so the conclusions drawn here aregeneral to both nozzle configurations.

Near the jet exit (at x = 0.1 rd), the averaged normal stress components 〈u′2〉 and 〈v′2〉 show peaks onboth the windward and wake sides of the jet. These reflect the dominance of the jet shear layer instabilityin the near field. The stress component in the jet initial momentum direction, 〈u′2〉, has higher magnitudethan 〈v′2〉 in the near field, which was also noted in the LES of Yuan et al.;18 those authors attributed thereduced 〈v′2〉 to suppression by strong pressure gradients in the y-direction. At x = 1.0 rd, approximatelythe end of the potential core, the roll-up structures in the shear layer have met, giving single peaks in the〈u′2〉 and 〈v′2〉 profiles. The peak values of 〈u′2〉 and 〈v′2〉 are then comparable, though the maximum 〈u′2〉occurs just to the windward side of the center streamline, and the maximum 〈v′2〉 lies to the wake side.This slight misalignment of the maximum 〈u′2〉 and 〈v′2〉 persists througout the measurement region. Yuanet al.18 found that sufficiently far downstream, the 〈u′2〉 profile once again takes on a bimodal shape, asturbulence generated by shear between the jet and crossflow fluid produces a local peak in the wake region.This was not observed in the present measurements, which is likely due to insufficient downstream extent ofthe measurement area.

The turbulent shear stress, 〈u′v′〉, in the near field of the jet takes on high negative values on the windwardside, and high positive values on the wake side. These stresses also result from the jet shear layer instability.As the flow evolves, the dominant mechanism for generation of 〈u′v′〉 becomes shear between the jet andcrossflow fluid. At y = 1.0 rd, regions of negative 〈u′v′〉 begin to dominate, slightly to the wake side of the

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0

-0.5

0

0.5

1

1.5

2

-0.40.4 00.8 0.8-0.4 0.4 1.2

y/rd

⟨ui′u

j′⟩/v∞

2

x = 1.0 rd x = 1.5 rd (a)

0 1 2 3

center streamline

-0.2

0

0.2

0.4

-0.2

0

0.2

-0.2

0

0.2

-0.2

0

0.2

x/rd

⟨ui′u

j′⟩/v

∞2

y = 2.5 rd

y = 1.5 rd

y = 1.0 rd

y = 0.5 rd

(b)

⟨u′2⟩⟨v′2⟩⟨u′v′⟩⋅3

Figure 9. Stress profiles in the z = 0.22 rd plane, for the flush nozzle case. (a) The profiles of 〈u′2〉,〈v′2〉, and 〈u′v′〉 for x = 1.0 rd and 1.5 rd, and (b) the profiles for y = 0.5 rd, 1.0 rd, 1.5 rd and 2.5 rd. Thefixed-y profiles of 〈u′v′〉 are stretched by a factor of 3.

center streamline, with positive values to either side. The regions of positive 〈u′v′〉, particularly that on thewindward side, become progressively weaker relative to the regions of negative values for increasing y. Thesetrends are consistent with those found by19 in an r = 2 jet. As the jet bends in the crossflow direction,the sign of 〈u′v′〉 generally correlates with the sign of −∂〈v〉/∂x. The deficit velocity due to the wake-likecharacter of the jet is manifested in the positive 〈u′v′〉 on the wake side of the jet and negative 〈u′v′〉 onthe windward side. The small region of positive 〈u′v′〉 on the outer boundary of the jet for small y arisesbecause as the jet turns, the streamwise velocity on the outer edge exceeds slightly the crossflow velocity.

The turbulent stress profiles in the z = 0.22 rd plane (Fig. 9) largely reproduce the trends from the centerplane, outside of the near field. A significant difference between the profiles in the center and off-center planesis that the profile peaks for z = 0.22 rd are noticeably shifted toward the wake region. The structure of theturbulent stresses thus shows evidence of the characteristic kidney-like structure, depicted in Fig. 1; themean scalar fields also exhibited this kidney-like structure, but the velocity magnitude fields did not.6, 14

E. LES model assessments.

From Fig. 2a, it is clear that individual PIV measurement windows span nearly the full flow width, whileTable 1 demonstrates that, on the downstream side of the imaging region, both the scalar and velocity fieldmeasurements resolve the fine scales of the flow. We can exploit these properties of the data in assessmentsof LES scalar mixing and transport models.

A quantity that is significant in LES of non-premixed combustion is the subgrid scalar dissipation rate,defined as

χ ≡ 2D∂C

∂xi

∂C

∂xi, (4)

where the overbar ( ) denotes LES filtering, D is the scalar diffusivity, and C is a conserved scalar quantity.In an LES, only C and its derivatives can be determined directly, so χ must be modeled. Among the SGSmodels proposed for χ in the literature are a gradient-based model by Pierce and Moin:20

χ ≈ 2αs∆2(2SijSij)1/2 ∂C

∂xi

∂C

∂xi, (5)

where αs is a proportionality constant determined using the dynamic procedure,21 ∆ is the LES filter width,and Sij is the strain rate tensor. This model derives from the assumption that the production and dissipationof scalar variance are in balance, together with eddy diffusivity ideas. A second model is proposed by Jimenezet al.:22

χ = αχ2νSijSij

12 (ujuj − uj uj)

(CC − C C) = αχε

k(CC − C C), (6)

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Figure 10. Scatterplots of χ, as computed by (a) the gradient-based model (5), (b) the Jimenez model (6),and (c) the dynamic structure model (8), vs. χ as determined directly from the resolved experimental data.

The LES filter size is ∆ = 8∆xu and the test filter size is ∆ = 2 ∆. (The term χ has units 1/s.)

where αχ is a proportionality constant taken to be 1/Sc. Neither the SGS kinetic energy, k, nor the filteredkinetic energy dissipation rate, ε, can be computed explicitly, so Jimenez et al.22 recommend that theseterms be modeled as

ε ≈ 2(ν + ατ∆2|S|)SijSij , k ≈ 2αI∆

2SijSij (7)

with ατ and αI both being determined using the dynamic procedure. This model assumes that characteristictime scales in the velocity and scalar fields are proportional. Finally, Chumakov et al.23 propose a so-calleddynamic structure model for the dissipation:

χ ≈ 2DCC − C C

C C − C C

( ∂C

∂xi

∂C

∂xi− ∂C

∂xi

∂C

∂xi

)− 2D

∂C

∂xi

∂C

∂xi, (8)

where ( ) denotes test filtering, i.e. an additional LES filter applied to the original LES data. This dynamicstructure model assumes that the dissipation is scale similar, and that the structure of the dissipation fieldis well-represented by the Leonard-type parenthetic term in (8).

Sample results from a priori testing of these χ models are shown in Fig. 10, which presents scatterplotsof the modeled χ values versus the χ values determined directly from the experimental data. The modeledvalues are found from the mock LES fields resulting from LES filtering of the experimental data. Theexperimental data are from PIV window 8 in Fig. 2; the scalar field data is mapped onto the PIV vectorgrid, which has spacing ∆xu = 420 µm. For the results in the figure, the LES filter size is ∆ = 8 ∆xu and thetest filter size is ∆ = 2 ∆. The Jimenez model evidently gives a superior correlation of modeled and measuredχ values for these filtering parameters. More extensive analyses9 show that the superior performance of theJimenez model holds both for measures of correlation and of absolute error, and is maintained for a rangeof LES filter sizes.

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Figure 11. The mechanical-to-scalar time scale ratio, t/tC , for different filter sizes.

It is possible to apply the present measurements to test not only the performance of the actual SGSmodels, but also to test their fundamental assumptions. For example, the Jimenez model (6) assumes thatthe mechanical and scalar time scales, defined as

t = k/ε and tC = (CC − C C)/χ, (9)

respectively, are proportional. Jimenez et al.22 recommend a value t/tC = 1/Sc. Figure 11 presents meanprofiles of t/tC along the x-direction for different LES filter sizes. It is evident that the time scale ratiochanges in magnitude with different filter sizes, but that its profile retains the same shape. It can alsobe seen that the time scale ratio is not constant, and deviates from the recommended value 1/Sc by asmuch as 50%. There is, however, a region of the flow where t/tC is within 25% of 1/Sc. This region,between x ≈ 0.05 m and 0.075 m, accounts for approximately 60% of the data domain. The approximationt/tC ≈ 1/Sc thus appears to be acceptable for a majority of the spatial region.

The sample results in this section give an indication of the flexibility of the present experimental data inassessments of SGS scalar models. In light of the presumed generality of small turbulent scales, the resultsare applicable not only to the crossflowing jet configuration, but to general turbulent shear flows as well.In combustion applications, chemical reactions depend on small-scale molecular mixing, which in LES isrepresented almost entirely by SGS models. It is therefore essential that those SGS scalar models have higha priori accuracy. Experimental data that simultaneously resolve wide length scale ranges in the velocityand scalar fields are especially well suited for SGS scalar model assessments.

IV. Conclusions.

Simultaneous, planar measurements of scalar mixing and two-dimensional velocity fields with high spatialresolution have permitted the detailed study of turbulence quantities in the developing region of turbulentcrossflowing jets with velocity ratio r = 5.7. Measurements in off-center planes, in addition to the symmetryplane of the flow, give a direct view of the complex three-dimensional nature of the flow.

Profiles of the scalar variance, the x- and y-components of the turbulent scalar flux, and the in-planeturbulent stresses were presented in the jet center plane and the z = 0.22 rd plane. The results show that theintensity of the mixing, as quantified by the scalar variance and the magnitude of the turbulent scalar fluxes,is initially higher on the jet windward side, but eventually becomes higher on the wake side. The scalarflux results also indicate that cross-stream turbulent transport is more significant the streamwise transport,particularly in the near field of the flow. Meanwhile, no qualitative differences were seen between the twonozzle configurations studied, suggesting that the presence of the wall in the flush nozzle configuration doesnot have a significant effect on the flow development, for the present velocity ratio r = 5.7.

The results of this paper should find particular application in the assessment of simulations of crossflowingjet mixing. The disparate scaling properties of scalar and velocity fields seen in prior work6, 14 illustrate theinadequacy of treating the mixing problem as a direct analogue of the flow field. The profiles presented here

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of the turbulent scalar fluxes, for example, will be of direct use in evaluating the accuracy of scalar mixingmodels used in simulations. Sample results from assessments of SGS scalar models also indicate the valueof this type of experimental data to progress in LES of mixing and combustion.

Acknowledgments.

Much of this work appears in more complete form in Su & Mungal.14 The support of the Center forTurbulence Research, the National Science Foundation, and the former Gas Research Institute in this workis greatly appreciated. Dr. D. Han and Mr. R.M. Miraflor provided valued assistance with the experiments,while Ms. O.S. Sun performed the computations involved in the LES model testing.

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