the mean and turbulent ﬂow structure of a weak hydraulic jump et al 2008 pof weak hydraulic... ·...

The mean and turbulent flow structure of a weak hydraulic jumpS. K. Misra,1,a� J. T. Kirby,1 M. Brocchini,2 F. Veron,3 M. Thomas,4 and C. Kambhamettu4

1Center for Applied Coastal Research, University of Delaware, Newark, Delaware 19716, USA2Istituto di Idraulica e Infrastrutture, Universit Politecnica delle Marche, Via Brecce Bianche,60131 Ancona, Italy3Graduate College of Marine Studies, University of Delaware, Newark, Delaware 19716, USA4Department of Computer and Information Science, University of Delaware, Newark, Delaware 19716, USA

�Received 1 June 2007; accepted 16 January 2008; published online 12 March 2008�

The turbulent air–water interface and flow structure of a weak, turbulent hydraulic jump areanalyzed in detail using particle image velocimetry measurements. The study is motivated by theneed to understand the detailed dynamics of turbulence generated in steady spilling breakers and therelative importance of the reverse-flow and breaker shear layer regions with attention to theirtopology, mean flow, and turbulence structure. The intermittency factor derived from turbulentfluctuations of the air–water interface in the breaker region is found to fit theoretical distributions ofturbulent interfaces well. A conditional averaging technique is used to calculate ensemble-averagedproperties of the flow. The computed mean velocity field accurately satisfies mass conservation. Athin, curved shear layer oriented parallel to the surface is responsible for most of the turbulenceproduction with the turbulence intensity decaying rapidly away from the toe of the breaker �locationof largest surface curvature� with both increasing depth and downstream distance. The reverse-flowregion, localized about the ensemble-averaged free surface, is characterized by a weak downslopemean flow and entrainment of water from below. The Reynolds shear stress is negative in thebreaker shear layer, which shows that momentum diffuses upward into the shear layer from the flowunderneath, and it is positive just below the mean surface indicating a downward flux of momentumfrom the reverse-flow region into the shear layer. The turbulence structure of the breaker shear layerresembles that of a mixing layer originating from the toe of the breaker, and the streamwisevariations of the length scale and growth rate are found to be in good agreement with observedvalues in typical mixing layers. All evidence suggests that breaking is driven by a surface-paralleladverse pressure gradient and a streamwise flow deceleration at the toe of the breaker. Both effectsforce the shear layer to thicken rapidly, thereby inducing a sharp free surface curvature change at thetoe. © 2008 American Institute of Physics. �DOI: 10.1063/1.2856269�

I. INTRODUCTION

Turbulent hydraulic jumps have long been of interest toengineers as efficient energy dissipaters in the design of con-trol structures. In attempts to clarify the details of the turbu-lence structure, many laboratory experiments have been per-formed with various techniques like hot-wire anemometry�e.g., Rouse et al.1 and Resch and Leutheusser2� and laserDoppler anemometry �Long et al.3�. Rouse et al.1 showedthat the air entrainment process, the momentum and energytransfer, and the energy dissipation are strongly dependent oninflow conditions. Cheaper and easier to use point measure-ment probes such as micro acoustic Doppler velocimetershave also been used by Liu et al.4 However, the authorsfound that the signal quality deteriorated with increasednoise levels near the free surface due to air entrainment andthe turbulent fluctuations of the interface. Consequently, theywere unable to obtain any information about the flow in the“roller” region or near the free surface. It is also doubtfulwhether an intrusive method is appropriate to measure thekinematics of a flow which is very sensitive to perturbations.Being single point measurement techniques, these at best

provide a sparse picture of the flow field, and reconstructinga spatial map can only be done in an ensemble-averagedsense. Particle image velocimetry �PIV�, on the other hand,is an established nonintrusive technique which provides ac-curate and detailed instantaneous spatial maps of a flow fieldcomparable in accuracy to those obtained from laser Dopplervelocimetry �LDV� measurements �Hyun et al.5�. Hornunget al.6 performed PIV experiments on traveling hydraulicjumps/bores with Froude number range of 2–6. Lennon andHill7 did PIV experiments analyzing undular jumps in theFroude number �Fr� range 1.4–3. Fully turbulent breakingjumps could not be generated at these low Froude numbers.The complete flow field could not be investigated even usinga mosaic of ensemble-averaged spatial maps because the la-ser sheet access was blocked by optically opaque metal sup-ports on the channel bottom. The free surface for the lowerFroude number cases could not be measured by a wave gagesince it disturbed the flow and caused the jump to move.Standard point gage and image processing techniques wereused but with limited success. All the above studies, andnumerous others that the authors are aware of, have eitherfocused on undular jumps where some of the energy deficitacross the jump is accounted for by the mean flux of wavea�Presently at: Halcrow, 22 Cortlandt Street, New York, New York 10007.

PHYSICS OF FLUIDS 20, 035106 �2008�

1070-6631/2008/20�3�/035106/21/$23.00 © 2008 American Institute of Physics20, 035106-1

Downloaded 06 Nov 2009 to 128.175.13.10. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

http://dx.doi.org/10.1063/1.2856269

http://dx.doi.org/10.1063/1.2856269

energy being swept downstream, or fully turbulent jumpswith Fr�1.5.

Weak hydraulic jumps, on the other hand, are believed toprovide a simplified description of the flow in spilling break-ers in the inner surf-zone in a frame of reference movingwith the broken wave. The similarity with weak hydraulicjumps is particularly true because of the low Froude numberwhere the downstream-to-upstream depth ratio �� h1 / h0 ;h0, and h1 are the upstream and downstream depths, respec-tively� is similar to that of typical inner surf-zone breakers.These jumps fall under the general category of steady spill-ing breakers such as those generated by submerged hydro-foils �Duncan8 and Battjes and Sakai9� which are strictly“quasi-steady” because of experimentally observed oscilla-tions of the breaking zone �Duncan10 and Banner11�. In thesaturated breaking zone for irregular waves, both field�Thornton and Guza12� and laboratory �Ting13� measure-ments have shown that the wave height-to-water depth ratio,known as the breaker index, varies from 0.4 to about 0.7. Fora bore or, equivalently, a hydraulic jump, this gives � varyingfrom 1.2 to 1.35. Based on Belanger’s equation �see Eq. �1��,the corresponding upstream Froude number �Fr� varies from1.15 to 1.26. For such low Froude number turbulent jumps,the energy dissipation in the transition region from super-critical to subcritical flow comes primarily from turbulencegenerated by the shearing action of the flow, similar to spill-ing breakers and bores �Battjes14�. The resulting flow in thisbreaker region is highly intermittent both in space and timeand encompasses a wide range of temporal and spatial scales.This makes it a challenging flow to be investigated experi-mentally.

The most comprehensive study to date of low Froudenumber, weakly turbulent jumps with breaking is that of Sv-endsen et al.15 �SVBK00 hereafter� based on LDV data col-lected by Bakunin16 at several streamwise and vertical loca-tions. The Froude numbers for the jumps analyzed are in therange 1.37–1.62. They found that, contrary to the tradition-ally simplified formulations for a hydraulic jump, the effectsof vertically nonuniform velocities and nonhydrostatic pres-sure are important for mass and momentum conservation.Although a significant amount of the turbulent structure wasmapped by vertical profiles of LDV data at several stream-wise locations, simple analytical approximations and as-sumptions had to be made to calculate streamwise gradientsof flow properties, or the gradients had to be neglected alto-gether �for example, in the calculation of vorticity, thestreamwise gradient of the vertical velocity had to be ne-glected since it could not be estimated accurately�. Littleattention was focused on the intermittency of the surface andits influence on the flow structure. A single recirculating flowregion was found, bounded from below by a dividing stream-line �above which the net volume flux was zero� and fromabove by the mean surface. The peak values of the turbu-lence intensities and Reynolds shear stress were found to beat or below the bottom of the dividing streamline.

Peregrine and Svendsen17 were perhaps the first to sug-gest a mixing-layer model describing the similarities in theflow structure between breakers, bores, and weak hydraulicjumps based on qualitative photographic visualizations of

bubble tracers. They suggested that “the surface roller doesnot play a dominant role in the dynamics of the wave.” Theanalogy with mixing layers is further supported qualitativelyby the photographic evidence in Hoyt and Sellin18 whichwas, however, based only on the structure of large scale ed-dies since the high polymer solution used to visualize theflow suppressed the smaller scale motions. The LDV mea-surements of Battjes and Sakai9 for short wavelength break-ers have shown that a self-similar wake is generated behindthe breaker. An analogy between hydraulic jumps and planeturbulent wall jets has also been suggested by several authors�e.g., Rajaratnam19 and Narayanan20�. However, recently,Chanson and Brattberg21 have shown that a quantitativecomparison between the parameters of the two flows is sig-nificantly affected by the air entrainment. This short sum-mary reveals that many questions are still open regarding thedetailed turbulence structure and dissipative mechanisms ofsteady breakers, in particular, weak hydraulic jumps. What isthe appropriate topological description for the reverse-flowregion? How important are the effects of surface turbulencecompared to the turbulence generated in the breaker shearlayer? Can the breaker shear layer be classified as a mixinglayer as is often assumed in theoretical models for breakers�Svendsen and Madsen,22 Cointe and Tulin,23 and Rhee andStern24�?

In this paper, a PIV laboratory experiment of a turbulenthydraulic jump of Fr�1.2 and a depth ratio ��1.3 is dis-cussed in detail. In the next section, we describe the experi-mental setup and the flow parameters. Section III describesthe characteristics of the turbulent air–water interface. Inter-mittency characteristics and toe oscillations are analyzed.The mean and turbulent flow structure are described in detailin Sec. IV with specific emphasis on the breaker shear layerand the reverse-flow region. Further discussion and conclu-sions are presented at the end of the paper.

II. EXPERIMENTAL SETUP

The experiment was performed in a recirculating Arm-field S6 tilting flume that is 4.8 m long and 30 cm wide, withglass side walls �9 mm thick� and a solid opaque bottom, thesame facility used to collect the data analyzed by SVBK00,even though no extensive attempts were made to replicatethe exact flow geometry and conditions. Indeed, with theknown sensitivity of the geometry and flow in hydraulicjumps to external perturbations such as bottom roughness,inflow velocity, prejump entrainment �Henderson25�, it wasextremely difficult to replicate the exact flow conditions. Wa-ter was pumped into the upstream end of the channel througha number of screens and flow straightening devices, afterwhich it flowed past an undershot weir. The undershot weirwas constructed from a plexiglass plate with a thickness of1 cm, and was wedged in between the flume walls. A metalsupport fixed the plate to the side rails of the flume. Thedistance from the bottom of the flume to the bottom edge ofthe weir was controlled by a screw attached to the metalsupport. The lower end of the plate was machined in order toyield a sharp uniform edge on the upstream side to minimizelateral variations of the inflow. The gaps between the weir

035106-2 Misra et al. Phys. Fluids 20, 035106 �2008�


and the side walls of the flume were made water tight byinserting flexible plastic tubes through grooves cut into thesides of the plate. The flow rate and the height of the weirwere used to control the upstream flow velocity and waterdepth, thereby determining the upstream Froude number.Further downstream, the water flowed over the downstreamgate �which was lowered or raised to fix the location of thejump� and into the reservoir. Just before the weir, the freesurface was capped by a foam plate to damp out surfacefluctuations on the upstream side. The flume was kept hori-zontal throughout the experiments. Figure 1 shows a sche-matic diagram of the flume and the PIV system configurationand Fig. 2 is a photograph of the jump. Figure 2 shows thatcross-channel disturbances were negligible in the upstreampart of the flow and reduced with increasing downstreamdistance from the breaker region. The jump did not have anyvisible undular characteristics. The jump characteristics aretabulated in Table I, where Qm is the mean, vertically inte-grated volume flux.

The PIV setup consisted of a 120 mJ /pulse Nd-Yag newwave solo laser source with a pulse duration of 3–5 ns. Thiswas mounted onto a custom-built submersible waterproofperiscope which was lowered into the water. The optics werearranged in such a way that the laser beam emerged as aplanar light sheet parallel to the flume wall. The laser sheetwas aligned with the center plane of the flume away from theside wall boundary layers. Although the submersible wassuitably streamlined, it changed the location of the jump andits position had to be adjusted so as to keep the jump station-

ary within the camera target area. It was kept far enoughdownstream of the jump to minimize flow interference anddid not cause any visible changes to the flow. The flumebottom and the undershot weir were painted with water re-sistant black marine paint to minimize reflections from thelaser sheet. As tabulated by Chow,26 for painted, smooth steelsurfaces, the Manning roughness coefficient is estimated tobe 0.013. The water was seeded with 14 �m diameter silvercoated hollow glass spheres with a specific gravity of 1.8obtained from Potters Industries. From Maxey27 and Wang

Laser source

Camera target area

Plate

Undershot weir

Flume bottom

Floating board

Screens

Pump

Water reservoir

Submersible periscopelaser light sheet

Honeycomb flow straightener

Downstream gate

Water surface

FIG. 1. Experimental setup.

FIG. 2. Photograph of the hydraulic jump.

035106-3 The mean and turbulent flow structure Phys. Fluids 20, 035106 �2008�


et al.,28 the response time Tp of the particle is calculated tobe Tp=1.96�10−5 s. Flow time scales comparable to Tp, orlarger, should be faithfully represented by the particles. AKodak Megaplus 1.0 camera with a 1016 �vertical� �1008�horizontal� pixel CCD array with its image plane parallel tothe flume wall, was used to visualize the flow. The target areadimensions were 23.3 cm �vertical� by 23.1 cm �horizontal�.The distance of the CCD array from the laser sheet was70.2 cm. The active area of the CCD sensor was 9.1 mm�horizontal� by 9.2 mm �vertical� and the fill factor was 55%.A 532 nm filter was used to reduce noise due to ambientlight. A Dantec acquisition system was used to acquire the8-bit gray scale images and store them onto a hard drive. Thelaser pulses were synchronized with the 30 Hz camera framerate which ultimately led to a 15 Hz sampling rate for theinstantaneous velocity fields. The time interval between twopulses in each image pair was 300 �s and each experimentalrun consisted of an ensemble of 1020 image pairs equivalentto 1020 instantaneous velocity maps.

III. THE AIR–WATER INTERFACE

There have been recent attempts to estimate air–waterinterfaces from PIV images with varying degrees of success�Hassan et al.,29 Peirson,30 Law et al.,31 Lin and Perlin,32 andTsuei and Savaş,33 among others�. We have developed a ro-bust and completely automated image processing algorithmto estimate the spatio-temporal evolution of the air–waterinterface for the present experiment. Only a brief descriptionis given below. More details of the algorithm can be found in

Misra et al.34 and a recent application in Veron et al.35 A rawPIV image, shown in Fig. 3, has four distinct regions. Thecoordinates are in pixels. The specularity seen in this particu-lar raw image is from air bubble entrainment near the toe ofthe jump, due to breaking. Upon a careful visual analysis ofthe complete set of raw images, very few showed air bubblesresulting from prejump entrainment upstream of the sluicegate and minimal entrainment at the toe of the jump. Theintersection of the laser light sheet with the bottom dividesthe lower region in the image into the fluid region and thesolid flume bottom. This boundary is easily detected by stan-dard edge detection techniques such as Canny or Sobel edgemethods, and the result is shown as the dashed line. Theblack region at the top of the image is air and forms a sharpboundary from the rest of the image. This interface is alsocalculated by a Canny edge algorithm and is shown in thefigure as the solid line. The diffuse region between thisboundary and the actual surface �which should be the upperlimit of the seeding particles� is due to the viewing angle ofthe camera which causes the water surface between the lasersheet and the flume wall to be reflected onto the image.

Using image segmentation methods, the air–water inter-face is calculated and is shown as the dotted line in Fig. 3.No attempt was made to resolve the small-scale surface un-dulations or account for the breakup of the surface due tobreaking; a continuous, smooth contour was calculated torepresent the interface. For two arbitrary images, the averageabsolute error across the whole domain was less than 6 pix-els. The maximum error was found in the far upstream re-gion where the elevation of the real surface was at timesoverpredicted because of the degradation of image contentdue to laser attenuation. The error was approximately 10pixels or 2.3 mm �Misra et al.34�. The overprediction of theactual surface leads to contamination of the near-surface ve-locities in the far upstream region and this is discussed fur-ther in the following. Due to the undulations in the water

TABLE I. Jump parameters.

h0 �cm� h1 �cm� � Qm �l/s� Fr

8.62 10.85 1.26 21.78 1.19

100 200 300 400 500 600 700 800 900 1000

400

500

600

700

800

900

1000

FIG. 3. Typical raw PIV image show-ing surface reflections and the inter-faces: Flume bottom �dashed line�,boundary between surface reflectionregion and air �solid line�, and calcu-lated actual free surface �dotted line�.



surface between the laser light sheet and the sidewall, thecamera line of sight can sometimes be blocked, leading to anunderprediction of the interface. A quantification of such er-rors is not possible because of a lack of cross-channel mea-surements.

The vertical location of the Cartesian coordinate system�x ,y� origin is defined at the flume bottom and the horizontalposition at an arbitrary distance upstream of the mean toelocation, similar to the coordinate system adopted bySVBK00. In SVBK00, the origin of the horizontal coordi-nate is placed at the mean position of the toe of the turbulentfront and the mean position of the toe is, a posteriori, fixedat the beginning of the calculated roller geometry �see theirFig. 1�. We believe it is more appropriate to define the toelocation based on the surface curvature, and its estimation isdiscussed in detail in the following section. The supercriticaland subcritical depths, taken here as the distances from theflume bottom to the end points of the ensemble-averaged freesurface where the horizontal gradient of the surface wasnearly zero, are found to be h0=8.62 cm and h1=10.85 cm,respectively, giving an upstream-to-downstream depth ratio�=1.26. The calculated values agree well with visually ob-served upstream and downstream depths. We note the differ-ence in the definition of upstream depth with SVBK00 whereit was fairly arbitrary and ambiguous because of the contrac-tion immediately downstream of the sluice gate. Further, inSVBK00, the actual position where measurements weretaken in front of the jump was somewhat arbitrary. In thepresent case, the toe of the jump was located further down-stream of the sluice gate �approximately 20 cm; �2h0� andmultiple streamwise measurements were available upstreamof the toe location. This led to, as shown in Fig. 4 �see alsoFig. 2�, a fairly spatially uniform upstream flow, and conse-

quently, a more objective definition for the upstream depth.From Belanger’s equation,

Fr =�1

2

h1

h0�h1

h0+ 1 , �1�

the upstream Froude number is calculated to be Fr=1.1932.Due to the error in the surface estimation, the upstream depthis slightly overpredicted and this leads to an underpredictionof the Froude number of about 3%. The Froude number canalso be determined based on a depth-averaged flow velocityor volume flux and is explored in the next section. It is per-haps surprising that we were able to obtain a breaking jump,which showed no visible undular characteristics, at such alow Froude number. The nature of the jump is very sensitiveto the flow conditions and channel characteristics and itwould require carefully designed control experiments to de-termine the specific cause�s� of breaking. Drastic qualitativeand quantitative changes are known to occur in the nature ofthe jump through minor changes in the flow and channelcharacteristics, such as bottom roughness �Henderson25�, in-flow velocity profile �Resch and Leutheusser2�, and prejumpair entrainment �Chanson36�. Chanson and Montes37 havesuggested that “the transition between an undular jump and aweak jump may occur for upstream Froude numbers in therange 1.0–3.6, the transition being a function of the upstreamflow conditions.”

A. Interface fluctuations and toe oscillation

For laboratory-generated steady spilling breakers, domi-nant coherent motions of the surface fluctuations and thebreaker toe with peak frequencies in the range 0.47–3.75 Hzhave been reported by various authors �Duncan38�. In the

0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x / h0

y/h

0 FIG. 4. Ensemble-averaged free sur-face �dotted line�. The measurementgrid is also shown at every fifth verti-cal and horizontal location.



present study, even though the sampling rate �15 Hz� is nothigh enough for capturing the smaller turbulent time scales,it is instructive to look at the unsteady nature of the fluctua-tions. Figure 5 shows a time stack of the dimensionless �withrespect to h0� surface profiles of the entire jump. The en-semble spans a total time of 67.93 s. The transitional break-ing region shows coherent horizontal oscillations with a timescale of about 20 s or a frequency of 0.05 Hz. Since thelength of the toe time series is small for a reliable Fourierspectral analysis and there are possible nonlinear effects inthe signal, wavelet and Hilbert–Huang transforms �HHTs�were also used to compute the power spectrum of the timeseries in order to isolate the dominant frequency component.All three peaks were found to coincide at a frequency ofapproximately 0.05 Hz, which is the observed fluctuationfrequency in the surface elevations. In order to determine thesource of this slow oscillation, two Sontek Acoustic DopplerVelocimeters were placed near the inflow and outflow sec-tions of the channel, 200 cm upstream of the sluice gate and160 cm downstream of the toe. Time series of streamwisevelocities were collected at 25 Hz, and spectral analysesshowed a predominant peak at approximately 0.05 Hz atboth locations. We believe that the long-time scale oscillationis intrinsic to the reservoir and pump system and unrelated tothe turbulent dynamics at the toe of the breaker.

Tests with analytically constructed shear layer geom-etries showed that the contamination due to such a low fre-quency oscillation to the dynamics of the flow in the breakershear layer is negligible. However, for increased confidencein the analysis, the effects of this oscillation were removedfrom both the raw computed interfaces and flow fields. Fromthe ensemble of instantaneous interface realizations, we ex-tracted time series of the horizontal oscillations at ten fixedelevations in the breaker region. These were then averagedand a fifth-order Butterworth low-pass filter with a cut-offfrequency of 0.33 Hz was used to extract the low frequencyoscillation. Both the averaged oscillations and the filteredtime series are shown in Fig. 6. The maximum deviation

caused by the oscillation is approximately equal to two mea-surement grid points. This filtered time series was then usedto interpolate �using a bilinear interpolation� the interfaceand flow fields onto a common fixed grid at which theensemble-averaged properties were calculated.

The toe itself is often loosely defined in the literature asnoted by Duncan.38 For incipient breaking of surfacetension-dominated waves, the toe is defined as the leadingedge of the bulge at the crest where the slope of the surfaceundergoes an abrupt change �Qiao and Duncan39�. There is asharp curvature at this point, and at a larger scale, the flow ismarked by flow separation �Longuet-Higgins40 and Duncanet al.41�. For a fully evolved breaker, with a “roller” on thefront face of the wave, the underlying turbulence and thesignificant corrugations of the surface make it difficult toprecisely locate the toe. Banner11 and Qiao and Duncan39

have referred to the leading edge of the “roller” region as thetoe. In saturated breakers or bores where the entire front faceof the breaker is turbulent, Brocchini and Peregrine42 haverecently proposed a descriptive terminology of a “foot” asthe base of the breaker, when looking down from above, withthe fluctuations of the boundary between smooth and turbu-lent flow termed as “toes.” In the present case, the air–waterinterface is approximated by a smooth unbroken curve. Thetoe is defined as the location of the maximum positive cur-vature of the free surface. The curvature �� of the instanta-neous free surface �� can be calculated as

� =

d2�

dx2

1 +d�

dx2�3/2

. �2�

Therefore, the toe motion in time can be tracked by comput-ing the time series of the location of the peak curvature. Anensemble average of the locations of the maximum curvaturecalculated from the instantaneous surfaces is found to coin-

FIG. 5. Time stack of nondimensionalized surface pro-files �� / h0 �. The horizontal grayscale bar shows themagnitude of the nondimensional surface height. Thefive contours plotted, progressing from dark to light, arefor � / h0 =1.05, 1.10, 1.15, 1.20, and 1.25.



cide, within the measurement resolution, with the location ofthe maximum curvature of the ensemble averaged surface atx= x / h0 �0.47 as shown in Fig. 7. The negative peak for thecurvature is located at x�1.18. The oscillatory nature of thecurvature in the breaker region is linked to the surface esti-mation process with any small irregularities amplified by thesecond derivative in the curvature. These can easily be re-moved by manually drawing a smooth contour through the

breaking region for the instantaneous images, but we do notthink this is any more appropriate than that estimated by thealgorithm since an air–water interface in this region cannotbe defined unambiguously. Quantifying the numerical errorassociated with a second degree representation of the geo-metric curvature is extremely difficult, as, in addition to thenumerical discretization error, the resolution in the horizontaldirection ��8 pixels in our case� also comes into effect since

0 10 20 30 40 50 60

−0.1

−0.05

0

0.05

0.1

0.15

x/h

0

Time (secs)

FIG. 6. Averaged surface oscillationsin the breaker region �gray line� andlow-pass filtered oscillation �dashedblack line�.

1

1.1

1.2

1.3

1.4(a)

y/h 0

0

5

10

15

20

25(b)

θ(d

egre

es)

0.5 1 1.5 2 2.5−20

−10

0

10

20

x/h0

κ(1

/m)

(c)

FIG. 7. �a� Ensemble-averaged freesurface, �b� slope of the mean surface�degrees�, and �c� curvature of meansurface.



we are primarily concerned with the horizontal location ofthe curvature peak. In similar experimental investigations,other authors such as Dabiri and Gharib43 and Lin andRockwell44 have identified peaks in curvature by visual ob-servation of the corrugations of the instantaneous surface,which itself was not directly measured, but either visuallyinterpreted or estimated. In the present case, the slope of themean surface shown in Fig. 7�b� at the location of the maxi-mum curvature is found to be around 14°. This is close to thestable equilibrium value �10°–15°� typically used in the innersurf-zone for Boussinesq modeling of broken waves�Schäffer et al.45 and Briganti et al.46� and is also in reason-able agreement with incipient breaking angles found bySalvesen and von Kerczek47 and Duncan.10

Longuet-Higgins48 showed in a theoretical model of flowseparation at a free surface that, immediately upstream of thetoe, the free surface should be inclined to the horizontal at anangle lying between 10° and 20°.

B. Intermittency factor

It is essential to implement a conditional averaging tech-nique for defining ensemble averages of the turbulent flowfield for the intermittent reverse-flow region near the surface.Here, since the interface is assumed to be continuous andunbroken, separating the air and water phases, the intermit-tency arises from the fluctuation of the instantaneous surface.To deal with the interface intermittency, we apply a zone-averaging technique �Antonia49� obtained as the ensemble ofvalues during times when a fixed measurement point is eitherin water or air. We define an intermittency function I�x ,y , i�at a spatial location �x ,y� for the ith realization, as

I�x,y,i� = �1, if ��x,i� � y, i.e., if the point is in water

0, otherwise, i.e., if the point is in air,

where ��x , i� is the instantaneous water surface. Theensemble-averaged velocity thus becomes

v� =� i=1

N Iv

� i=1N I

, �3�

where N is the total number of realizations; N=1020. Thezone-averaged intermittency factor ��x ,y� is given by

� =� i=1

N I

N. �4�

Corrsin and Kistler50 have shown that the inferred distri-butions of turbulent interface fluctuations closely followGaussian error law distributions. A similar result was alsoobserved at an air–water interface, on the basis of data oflower quality than that used here, by Killen and Anderson51

and by Ervine and Falvey.52 This illustrates that such similarbehavior is a direct response of the interfaces �turbulent/laminar or air/water� to a random forcing �i.e., turbulence� asopposed to a deterministic one �i.e., waves�. An illustrationof this is also shown in Fig. 3 of Brocchini and Peregrine.42

The present raw measurements have been shown to comparewell to the error function profile at various locations insidethe breaker region �see Fig. 10 in Misra et al.34�. With theremoval of the low-frequency oscillation, the comparison issignificantly improved, as shown in Fig. 8. The Gaussiandistribution of the surface fluctuations was further verified

1.02 1.04 1.06 1.08 1.10

0.2

0.4

0.6

0.8

1

x/h0

= 0.5098γ

1.06 1.08 1.1 1.12

x/h0

= 0.5947

1.1 1.12 1.14 1.160

0.2

0.4

0.6

0.8

1

x/h0

= 0.7009

γ

y / h0

1.14 1.16 1.18 1.2 1.22

x/h0

= 0.8071

y / h0

FIG. 8. Intermittency factor � in thebreaker region. Measured data �� andtheoretical error function �solid line�.



from the observation that the ensemble-averaged free surfacecoincided with the contour of �=0.5. Defining the locationof the mean interface at �=0.5 is an arbitrary choice whichreflects a specific bias towards mass conservation arguments��=0.5 reflects equivalent phase volumes at that level�.However, as shown by Brocchini and Peregrine,53 for a sim-pler deterministic boundary like the shoreline, different meaninterfaces can be found which match other conservation laws�e.g., flow-parallel momentum� and distinguishing which isthe more dynamically important is rather subjective. Here,we simply note that measurements show that the ensemble-averaged air-water interface coincides with the location�=0.5.

IV. FLOW STRUCTURE

A. The PIV algorithm

The PIV algorithm used here is described in detail inThomas et al.54 We have developed a phase-correlationbased algorithm that first estimates the global motion fieldfrom a pair of PIV images and then uses this as an initializa-tion for the local motion calculation. A pyramid hierarchy isused to percolate information from a coarse to fine resolutionin a computationally efficient fashion. This helps achieve avery high spatial resolution without loss in accuracy�Nogueira et al.55�. The velocity convergence at multiple spa-tial locations is found to be good and provides confidence inthe final ensemble-averaged quantities. The motion estima-tion algorithm has been extensively tested for transient andnontransient strongly sheared flows such as the formation ofa wake vortex, sensitivity to PIV image parameters, and re-produces accurate results when compared with ground-truthdata with mean errors of 1%.

Once the real surface and bottom have been calculated,the raw image is cropped off at the location of the flumebottom and above the envelope of the highest surface fluc-tuations. The cropped image is then subdivided by finite win-dows of size 16�16 pixels with a 50% �8 pixels� overlap inthe horizontal and vertical directions. The horizontal and ver-tical resolutions are, respectively, x=y=0.184 cm. In thevertical direction, the first calculation point is 8 pixels belowthe instantaneous surface, whereas the last calculation pointis 16 pixels �y1=3.66 mm� above the calculated bottom.These limitations were imposed by the size of the windowused for computing the global motion �32�32 pixels�. Asnoted before, all instantaneous raw velocity fields weremapped to a fixed grid using the extracted time series of thelow frequency oscillation before ensemble-averaged quanti-ties were calculated.

B. Mass flux conservation

The conditionally averaged mean velocities are calcu-lated according to Eq. �3� and are represented hereafter as Uand V in the x and y directions. Although there were no insitu measurements of the flow velocities, the accuracy of thecomputed flow field can be checked by examining an integralconstraint based on mass flux conservation. The mean vol-ume flux across a vertical section can be computed as

Q�x� = b�y1

h�x�

U�x,y�dy , �5�

where b is the width of the channel, and the integration isdone numerically. Note that the lower limit of integration isat y1 and not the flume bottom. The contribution to the fluxnear the bottom, including the boundary layer, cannot beestimated because of lack of data. A comparison of thestreamwise variation of the vertically integrated �to the meansurface� nondimensionalized volume fluxes between thepresent measurements, from x�0.47, the toe location, to x�2.48, and the Fr=1.46 case considered by SVBK00 isshown in Fig. 9. The standard deviation �Q�� and mean �Qm�for the present measurements, from x�0.47 to x�2.48, are0.5198 l /s and 21.784 l /s, respectively, which givesQ� /Qm=0.0239. For the data analyzed in SVBK00, from thetoe location to x�4.0, Q� /Qm=0.0221, which shows thatvolume fluxes calculated from the raw data for the two stud-ies are comparably accurate. The effects of the bottom andsidewall boundary layers are not included in the volume fluxestimate. By fitting analytic velocity profiles to the raw dataand incorporating the non-negligible reductions in the totalvolume flux due to sidewall and bottom boundary layers,SVBK00 found that the streamwise deviations of the volumeflux could be reduced further.

The Froude number can be calculated from the volumeflux as

Fr� =Qm

�bh0��gh0�, �6�

where h0�=h0−y1. The upstream Froude number is thus cal-culated to be Fr�=1.023 which is smaller than Fr calculatedfrom Eq. �1�. This is in agreement with SVBK00 who foundthat the Froude number calculated from Belanger’s equationconsistently overpredicted a depth-averaged velocity or vol-ume flux based estimate.

C. Mean flow structure

Figure 10 shows the vertical profiles of the mean hori-zontal velocity at x=0.15 and x=2.50. The dashed line is theelevation of the mean free surface at this location. As notedbefore, due to the overprediction of the height of the instan-taneous surface in the far upstream region, the errors in thevelocity at the measurement location just below the surfaceresults in contamination of the shear layer near the surfaceshown in Fig. 10�a�. The location of the velocity profileshown in Fig. 10�a� is further upstream of the toe than mea-surements reported in SVBK00 and a direct comparison can-not be made. As noted earlier, the jump location in thepresent case was further downstream of the sluice gate thanin SVBK00. It is well established that the hydraulic jumplocation and the boundary layer structure are interdependent�Wilson and Turner56� and that the flow characteristics, par-ticularly near the toe and the free surface, depend signifi-cantly on the inflow characteristics. The present measure-ments show that the bottom boundary layer at the toelocation was more developed than was the case in SVBK00.At the downstream location shown in the right panel in Fig.



10�b�, the thickness of the shear layer is approximately 3 cm,showing that the flow has not relaxed to its depth-uniformopen channel configuration. The velocity estimates at loca-tions downstream of the toe are accurate up to the meansurface. For downstream flow conditions, Fig. 10�b� is verysimilar to that reported by SVBK00 �see their Fig. 2�.

Next, we examine contour plots which present the wholespatial map of the flow field. In Fig. 11, the dotted line is theensemble-averaged free surface. The flow fields are shownonly up to the mean surface. The flow structure in the inter-mittent region is analyzed later. Although the mean horizon-tal velocities are nearly uniform around mid-depth, there are

0 1 2 3 4 5 6 7 80.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

x/h0

Q/Q

m

FIG. 9. Nondimensionalized verticallyintegrated volume flux. Raw data ana-lyzed in SVBK00 �� and presentmeasurements ��.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8(a)

y/h 0

U (m/s)0 0.2 0.4 0.6 0.8 1

U (m/s)

(b)

FIG. 10. Vertical profiles of theensemble-averaged horizontal velocityU �m/s�. �a� x=0.15, �b� x=2.50. Thedashed lines show the location of themean free surface.



significant deviations from the assumption of depth-uniformflow, particularly in the breaker shear layer. The bottomboundary layer with positive vorticity also grows in thedownstream direction. The vertical velocity is almost an or-

der of magnitude smaller than the horizontal velocity andreaches its peak value, 0.2 m /s, downstream of the toe atx�0.68, y�0.99.

There is a thin, concentrated region of negative vorticity

0.20.40.60.8

11.21.4

y/h 0

(a)

0.4

0.6

0.8

1

1.2

1.4

y/h 0

(b)

0.2

0.4

0.6

0.8

0

0.05

0.1

0.15

0.5 1 1.5 2 2.5

0.20.40.60.8

11.21.4

x/h0

y/h 0

(c)

−200

−150

−100

−50

0

FIG. 11. Mean flow structure. �a�Ensemble-averaged horizontal veloc-ity U �m/s�, �b� ensemble-averagedvertical velocity V �m/s�, and �c�ensemble-averaged vorticity �1/s��detail shown in Fig. 12�. The dottedline is the ensemble-averaged free sur-face. The toe of the breaker is at x�0.47.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

y/h 0

x/h0

−235 −200 −150 −100 −50 0

FIG. 12. �Color online� Ensemble-averaged vorticity �1/s�. The dotted line is the mean free surface. The toe of the breaker is at x /h0�0.47.



in the breaker shear layer �a special mixing layer whose fea-tures are discussed in detail in Sec. IV E 2�, shown in moredetail in Fig. 12. Near the toe, the negative vorticity decaysrapidly away from the mean surface reaching 1% of its peakvalue at a distance of approximately 2.2 cm. Therefore, thebulk of the flow can be modeled as irrotational. In SVBK00,the vanishing vorticity at the mean surface was a conse-quence of the imposed zero mean shear stress condition forthe analytic fit to the horizontal velocity. In the present study,we find a nonzero vorticity at the mean surface. As shown inFig. 12, the mean vorticity has its peak, negative value up-stream of the toe, at x�0.42, y�0.99, and decreases in thedownstream direction. With the extent of available data, it isalso clear that the vorticity decays to zero in the upstreamstreamwise direction as there cannot be any physical vortic-ity in the far upstream region away from the jump. The in-stantaneous vorticity is shown in Fig. 13. The streamwisevariations of the instantaneous vorticity show well definedpeaks, indicating coherent vortical motions in the shear layer.This is consistent with the observations made by Hornunget al.6 during the initial stages of bore formation �see theirFigs. 10 and 11�. For detailed experimental observations onthe vorticity generation mechanisms in steady breakers withrespect to the free surface geometry, see Lin and Rockwell44

and Dabiri and Gharib.43

The flow deceleration due to the convective term,

Us�Us / �s , is shown in Fig. 14. This is computed on the basisof locally orthogonal, curvilinear coordinates �s ,n� fixed onthe mean surface �Brocchini and Peregrine42�. s and n are

measured along the surface parallel and normal directions,respectively. The mean surface-parallel and surface-normalvelocities are, respectively, calculated as

Us = U cos � + V sin � �7�

and

Vn = − U sin � + V cos � . �8�

��x�=tan−1�� x�� / �x� is the angle that the tangent to themean surface makes with the x-axis, shown in Fig. 7�b�.Since � is determined from the local slope of the mean sur-face, the transformation is accurate only near the mean sur-face where the streamlines closely follow the surface. Thesurface-normal and surface-parallel gradients are thereforeonly shown until five measurement points below the meansurface. Although the peak negative value of the surface-parallel convective acceleration occurs at x�0.49, y�0.98which is coincident within the measurement resolution withthe toe location, intense flow deceleration is seen to occurupstream of the peak vorticity location. The localization ofthe deceleration, indicative of the strong reduction in veloc-ity over a small streamwise distance, is in good agreementwith the observations of Lin and Rockwell.44

Next, the flow in the intermittent region is analyzed. Fig-

ure 15 shows the mean surface-parallel �Us� and surface-

normal �Vn� velocities near the free surface, which, due tothe curvature of the streamlines, are more appropriate in un-derstanding the dynamics of the flow in this region, com-

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

−350 −300 −250 −200 −150 −100 −50 0 50 100

FIG. 13. Instantaneous vorticity �1/s�. The dotted line is the mean free surface. The toe of the breaker is at x /h0�0.47.



pared to the Cartesian forms presented in SVBK00. The dot-ted line is the mean surface. A weak reverse flow with thelargest negative velocity found to be 2.38 cm /s is clearlyseen and is localized entirely above the mean surface.SVBK00 determines the location of the lower boundary of arecirculating region by calculating the elevation below whichthe depth-integrated volume flux �obtained from an analyticfit to the measured velocities� goes to zero. This is in no waya trivial calculation. In the present case, we do not have

enough vertical resolution in the very thin reverse-flow re-gion to accurately estimate the volume flux. However, toconceptualize a recirculating region, the bottom boundarycan be estimated by assuming that the positive velocities atthe bottom boundary should be approximately equal to themaximum negative velocity in the reverse-flow region.Therefore, the bottom boundary should be just below themean surface. This is a different scenario than SVBK00where the entire recirculating region was confined below the

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

x / h0

y/h

0

−35

−30

−25

−20

−15

−10

−5

FIG. 14. Ensemble-averaged surface-parallel convective acceleration

Us�Us / �s �m /s2�. The toe of thebreaker is at x /h0�0.47.

1

1.1

1.2

1.3

(a)

y/h

0

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1

1

1.05

1.1

1.15

1.2

1.25

x / h0

y/h

0

0

0.02

0.04

0.06

0.08

0.1

(b)

FIG. 15. �a� Mean surface-parallel ve-

locity Us �m/s� and �b� mean surface-

normal velocity Vn �m/s�. Dotted lineis the ensemble-averaged free surface.



mean surface. Since no velocity measurements were avail-able above the mean surface in SVBK00, it is clear that thedifference arises because of the treatment of intermittency inthe present case.

The surface-normal velocity is nearly zero in thereverse-flow region. However, there are large, positive values

of Vn below the mean surface and are an indication of theentrainment of fluid from below. This is in agreement withthe conceptual “starting plume” model of Longuet-Higginsand Turner57 �see their Fig. 7� where the spilling region isregarded as a turbulent gravity current riding down the frontface of a wave, with entrainment of fluid from below. It isalso clear from the present data that, though most of theentrainment occurs near the toe, inflow from below occursall over the reverse-flow region. Note that the small-scalestructure seen in the vertical velocity is resolved by the mea-surement location grid �x=y=0.0213; see Fig. 4 for mea-surement grid� and is not merely an artifact of the contourplotting routine. As opposed to the assumption made bySVBK00, the mean shear is finite and negative at the meansurface. Lastly, regarding the geometry of the reverse-flowregion, the mean inclination is calculated to be 18°, which isclose to the value ��17°� that have been found for steadybreakers generated by submerged hydrofoils �Duncan10 andWalker et al.58�.

D. Convergence of turbulent quantities

The horizontal root mean square turbulent velocity iscomputed as

u� =��i=1N �I�u − U��2

N��9�

and similarly for v�. u and v are the instantaneous horizontaland vertical velocities, respectively. The turbulent kinetic en-ergy �k� and Reynolds shear stress �� are computed as

k =

�i=1N I

1

2��u − U�2 + �v − V�2�

�N�10�

and

�� = −�i=1

N I�u − U��v − V��N

. �11�

The convergence of turbulent quantities of interest is exam-ined at a number of spatial locations prior to analyzing theturbulence structure of the flow field. This is performed bycalculating running averages of the turbulence statistics overthe entire ensemble. The convergence rates for the turbulentstatistics are shown in Figs. 16 and 17. The horizontal coor-dinate is shown in terms of the ensemble size, since thisillustrates the size of the ensemble needed to achieve satis-factory convergence, which is often an issue in PIV investi-gations of root mean square turbulent quantities �Ullumet al.59�. In Fig. 16, convergence rates at a location in thebreaker shear layer near the toe of the breaker at x=0.53,y=0.96 are shown. Figure 17 shows similar convergencestatistics at x�1.0 and y�1.1, further downstream of the toelocation, and as can be seen, satisfactory convergence is

0.06

0.08

0.1

0.12(a)

u’(m

/s)

0.06

0.08

0.1

0.12(b)

v’(m

/s)

6

8

10x 10

−3

k(m

2 /s2 )

(c)

100 200 300 400 500 600 700 800 900 1000−10

−5

0

5x 10

−3 (d)

τ’(m

2 /s2 )

Ensemble size

FIG. 16. Convergence of turbulencestatistics at x�0.53, y�0.96. �a�Horizontal turbulent velocity u� �m/s�,�b� vertical turbulent velocity v� �m/s�,�c� turbulent kinetic energy k �m2 /s2�,and �d� Reynolds shear stress ��m2 /s2�.



attained faster. The convergence of the mean quantities are,everywhere, much faster than that of the turbulent quantities.

E. Turbulence structure

Figure 18 shows the spatial structure of the stronglynonisotropic turbulence of the breaker shear layer. Only asubsection of the entire domain is shown here for more detail

and clarity. As expected, the turbulence intensity decays rap-idly over depth outside the breaker shear layer. The horizon-tal and vertical root mean square turbulent velocities in theshear layer are of the same order as the mean velocities, withu� being slightly larger than v�. The peak value for the hori-zontal turbulence intensity is found at the toe. This is differ-ent from the measurements of Lennon and Hill7 who had

0.12

0.14

0.16

0.18

(a)u’

(m/s

)

0.12

0.14

0.16

0.18

0.2

v’(m

/s)

(b)

0.01

0.02

0.03

0.04

k(m

2 /s2 )

(c)

100 200 300 400 500 600 700 800 900 1000−0.02

0

0.02(d)

τ’(m

2 /s2 )

Ensemble size

FIG. 17. Convergence of turbulencestatistics at x�1.0, y�1.1. �a� Hori-zontal turbulent velocity u� �m/s�, �b�vertical turbulent velocity v� �m/s�, �c�turbulent kinetic energy k �m2 /s2�, and�d� Reynolds shear stress �� m2 /s2�.

1

1.2

1.4

y/h

0

(a)

0.05

0.1

0.15

0.2

0.25

1

1.2

1.4

y/h

0

(b)

0.05

0.1

1

1.2

1.4

y/h

0

(c)

0

0.02

0.04

0.4 0.6 0.8 1 1.2 1.4 1.6

1

1.2

1.4

x / h0

y/h

0

(d)

−6

−4

−2

x 10−3

FIG. 18. Turbulence structure. �a�Horizontal root mean square turbulentvelocity u� �m/s�, �b� vertical rootmean square turbulent velocity v��m/s�, �c� turbulent kinetic energyk �m2 / s2 �, and �d� Reynolds shearstress �� m2 / s2 �. The dotted line isthe ensemble-averaged free surfaceand the toe of the breaker is at x /h0

�0.47.



noted the general similarity of the structure of the horizontalturbulence intensity �for their strongest jump, at Fr=3� withthe present measurements. However, due to the undular char-acteristics, the location of the peak in u� in their case wasdownstream of the toe. In the present study, for the verticalturbulence velocity, the largest values occur at x�0.81,y�1.04 downstream of the toe of the breaker. The turbulentkinetic energy has its peak value at the toe of the breaker.This arises from the dominant contribution to k from thehorizontal turbulence intensity. The Reynolds shear stress isnegative throughout the shear layer, implying an upward dif-fusion of momentum. The Reynolds shear stress has its larg-est value at x�0.53, y�1.02, downstream of the toe and thelocation of the maximum negative mean simple shear �U / �ywhich occurs at x�0.40, y�0.98, indicating that in additionto the simple shear, there are extra strain rates such asstreamline curvature which affect the Reynolds stress struc-ture �Bradshaw60 and Misra et al.61�. In the following sec-tion, we take a more detailed look at the Reynolds shearstress in the reverse-flow and breaker shear layer regions.

1. The reverse-flow region

Figure 19 shows that the Reynolds shear stress is posi-tive about the mean surface indicating a downward flux ofmomentum from the reverse-flow region into the breakershear layer. The magnitudes of the Reynolds stress are, onaverage, smaller than the breaker shear layer values. Further,the turbulent kinetic energy in the reverse-flow region is anorder of magnitude smaller than in the shear layer suggestingthat, even though in the intermittent region turbulent fluctua-tions are as important as the mean velocities, the turbulenceintensity is much smaller than in the breaker shear layer assuggested by Peregrine and Svendsen.17 The distribution of

both horizontal and vertical turbulence intensities are quali-tatively very similar to the turbulent kinetic energy distribu-tion and decay monotonically away from the mean surfacetoward �=0.

2. The breaker shear layer region

It is fairly well-established that there is an intense shearlayer formed below a fully formed, spilling breaker whichspreads downstream from the toe of the breaker. The dynam-ics of the breaker shear layer are different from those of atypical shear layer in several respects. The outer boundary ofthe breaker shear layer is marked by a turbulent interfacewith a strong density difference between air on one side andwater on the other. The localized effects of an adverse pres-sure gradient and streamline curvature along with the un-steadiness of the external flow also modify the flow struc-ture. There is still so little knowledge of the detailedturbulence structure of spilling breakers that it is a meaning-ful pursuit to search for similarities with well known turbu-lent flows and use established results as a starting point for amore detailed understanding and modeling of the compli-cated processes accompanying breaking. Although the con-cept of a mixing layer has been assumed in theoretical mod-els for breakers �Svendsen and Madsen,22 Cointe and Tulin,23

and Rhee and Stern24� there is no experimental evidence forthe suggested classification of the shear layer in a fullyformed breaker as a mixing layer.

In Fig. 19, we show the detailed structure of the Rey-nolds shear stress in the breaker shear layer. A robust classi-fication of the comparative structure of two-dimensionalplane wakes, jets, and mixing layers has been suggested byTennekes and Lumley62 by describing the typical down-stream variation of Us and l. Us is defined as the scale for the

0.6 0.7 0.8 0.9 1 1.1 1.2

1

1.05

1.1

1.15

1.2

1.25

x / h0

y/h

0

−7 −6 −5 −4 −3 −2 −1 0 1 2

x 10−3

FIG. 19. Reynolds shear stress, ��m2 /s2�, in the reverse flow andbreaker shear layer regions. The dottedline is the mean free surface.



cross-stream variation of the streamwise mean velocity com-ponent and for shear layers is equal to the maximum free-stream velocity. l is the cross-stream length scale defined asthe distance from the center line to the location at which�U−U0� is equal to 1 / 2Us, where U0 is the scale for thevelocity of the mean flow in the streamwise direction. It isexpected that for mixing layers, Us is constant and l in-creases linearly with downstream distance. In turbulent mix-ing layers, the turbulent region is separated from the irrota-tional region by irregularly distorted bounding curves. In apractical sense, a finite percentage value of the maximum�magnitude� Reynolds shear stress in the region of interesthas to be chosen to define the boundaries of the breaker shearlayer. Due to the complicated turbulent structure seen in Fig.19, we chose to define the boundaries at the location wherethe Reynolds shear stress decayed to 90% and 10% of itsmaximum value ��max� �−0.0071 m2 /s2�. These contours areshown as solid lines in Fig. 20. It was found that the meanstreamwise velocity contours of 0.12 m /s and 0.73 m /s fitthe �� contours reasonably well, as shown in Fig. 20 by“cross” and “plus” signs, respectively. These values wereapproximately 10% of the maximum mean streamwise ve-locity in this region, and were chosen as U0 and Us, respec-tively. The constancy of Us with downstream distance is,thus, automatically satisfied. The centerline of the layer wasdefined as the location of maximum shear, and is shown asthe dashed line in Fig. 20. The velocity contour for Ul�U0

+ Us / 2 =0.485 m /s is shown as the dotted line. The lengthscale is calculated as the distance from this line to the cen-terline. The mean free surface is shown as circles.

In Brown and Roshko,63 the virtual origin for the down-stream coordinate is defined based on characteristic thick-nesses calculated from their density and velocity profiles.The intersection of a linear fit to these thicknesses with the

x-axis yielded the virtual origin. A similar approach was usedby Battjes and Sakai9 based on a linear fit to the calculatedvelocity defects. In the present case, a simpler visual ap-proach is taken. By drawing tangents to the 10% contours ofthe Reynolds stress, we find that the location of their inter-

section is at x0�0.4, upstream of the toe. This is a physicallyreasonable choice for the virtual origin, since the Reynoldsshear stress reaches its characteristic value further down-stream of this location.

The width of the shear layer, W, and the length scale arenondimensionalized by h0 and are shown in Fig. 21 alongwith the linear least squares regression fits. The regressionhas been calculated using a 95% confidence interval and theR2 statistical value �defined as the ratio of squares to totalsum of squares� is also shown along with the estimated linearfit. Both fits are statistically accurate in representing the lin-ear trend of the increase in the width and length scale of thebreaker shear layer, as is evident from the high values of theR2 estimate. This is in good qualitative agreement with thespreading rate of mixing layers where the “wedge” is knownto spread in a linear fashion, and indicates that the breakershear layer can be classified as a mixing layer with its origindownstream of the toe and extending downstream until itchanges to a wake-type flow. This transition occurs at ap-proximately the location of the peak negative curvature. Thedimensionless length scale in the breaker shear layer is alsoshown in Fig. 21 and shows a linear increase from x�0.6until x�1.2.

Another parameter of interest is the ratio W / x−x0 re-ferred to as the growth rate by Brown and Roshko.63 InBrown and Roshko,63 the linear fit to the data for a densityratio of unity is

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40.95

1

1.05

1.1

1.15

1.2

1.25

x/h0

y/h

0 FIG. 20. Contours defined in thebreaker shear layer region shown inthe legend plot. See text fordescription.



W

x − x0= 0.38Us − U0

Us + U0� . �12�

For Us−U0 / Us+U0 =0.72 �Us and U0, here, are takenequivalent to U1 and U2 in their case�, this gives W / x−x0

=0.27. This is in good agreement with the mean value �av-eraged over the values from x�0.6 to x�1.2� for the growthrate in the present case, equal to 0.26.

F. Index of entrainment

For free turbulent flows, the ratio of standard deviationto the mean value of the interface fluctuations�R= �� / �0� is a measure of relative depths of the corru-gations of the bounding surface and in conjunction with theintensity of the interface vortices gives a measure of the ac-tivity of the entrainment at the interface �Townsend64�. R isknown to vary considerably depending on the turbulent flowregime, and since the definition of �0 is dependent on thechoice of the datum, it is first necessary to choose an appro-priate length scale characterizing the surface fluctuations. Wetake the width of the intermittent region �where � varies

from 0 to 1� as �0. The calculated entrainment ratio index isshown in Fig. 22�b� and is seen to vary between 0.15 and0.27 in the breaking region. This is comparable to the valuesreported by Townsend64 �see Table 6.2� for boundary layers�0.17�, jets �0.22�, and wakes �0.38�. The entrainment indexis seen to increase into the breaking breaker region where itstays approximately constant and then decreases furtherdownstream. This gives a good qualitative insight into thevariation of the intensity of the surface turbulence of theflow. Previous results of the intensity of surface turbulencefor steady breaking waves found by Duncan and Dimas65

and Walker et al.58 agree with the trends observed above.An important factor influencing the entrainment index is

the intense shear generated at the toe of the breaker shearlayer. A qualitative, and admittedly simplified, argument canbe made for the trends observed in R by looking at the cur-vature of the ensemble-averaged surface shown in Fig. 7�c�.A positive curvature implies concavity of the surface. Thenormal pressure gradient �with the positive normal being di-rected away from the surface� is, therefore, negative. This, inturn, implies an increase in entrainment downstream since aparcel of entrained fluid encounters a favorable pressure gra-dient. The situation is reversed for convexity when the cur-vature is negative, leading to a positive pressure gradientwhich opposes entrainment. In the presence of an adversepressure gradient as in the present case, the influence of thecurvature on the entrainment is reduced compared to a zero-pressure-gradient case, since the outer region streamlines areless curved than the surface �Simpson66�. This might be thereason for the increase in entrainment taking place furtherupstream than the location of the peak positive curvature.Although the pressure was not directly measured, the meanpressure gradients can be calculated from the Cartesian mo-mentum equations where it is assumed that �U / �t�0, �V / �t �0. The horizontal and vertical pressure gradi-ents are thus calculated as

1

�

�P

�x� − U

�U

�x− V

�U

�y−

�

�x� u�2�� −

�

�y� u�v�� 13�

and

1

�

�P

�y� − U

�V

�x− V

�V

�y−

�

�x� v�2�� −

�

�x� u�v�� . �14�

P is the ensemble-averaged pressure. The streamwise andsurface-normal gradients are given by

�P

�n= − sin��

�P

�x+ cos��

�P

�y�15�

and

�P

�s= cos��

�P

�x+ sin��

�P

�y. �16�

As before, the results are shown only for five measure-ment locations below the mean surface. It can be seen from

0.6 0.7 0.8 0.9 1 1.1 1.20.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22W

/h0

x/h0

0.7 0.8 0.9 1 1.1 1.20

0.005

0.01

0.015

0.02

0.025

0.03

x/h0

l/h0

(a)

(b)

FIG. 21. �Top panel� Dimensionless width of the mixing layer W /h0, best fitline given by W /h0=0.297x /h0−0.137 with R2=0.993. �Bottom panel� Di-mensionless length scale l /h0, best fit line given by l /h0=0.0536x /h0

−0.0358 with R2=0.9953.



Fig. 23�b� that there is a strong adverse �positive� streamwisegradient of the specific pressure �per unit density� with itspeak value located at x�0.47, y�1.0, i.e., at the toe of thebreaker. This leads us to believe that the sharp curvature ofthe surface is induced by the flow separation which, in turn,is caused by the adverse pressure gradient. A short distanceupstream of the peak value, the magnitude of the streamwise

pressure gradient decreases drastically. This is typical offlows approaching separation and is attributed to the rapidthickening of the layer as a separation point is approached�Townsend64�. The normal gradient has a negative peak atx�0.47, y�0.98. The highly anisotropic turbulence down-stream of the toe of the breaker also contributes to a largerentrainment in the breaker shear layer.

1

1.1

1.2

1.3

1.4(a)

y/h

0

0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

(b)

R=

σ(η)

/η0

x / h0

FIG. 22. �a� Ensemble-averaged freesurface and �b� ratio of standard devia-tion to ensemble mean of the free sur-face, R= �� / �0 . The toe of thebreaker is at x /h0�0.47.

0.95

1

1.05

1.1

1.15

1.2

1.25

y/h

0

(a)

−30

−25

−20

−15

−10

−5

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

0.95

1

1.05

1.1

1.15

1.2

1.25

x / h0

y/h

0

(b)

−5

0

5

10

15

20

25

30

FIG. 23. �a� Surface-normal gradientof the specific pressure, �P / �n �m /s2�and �b� surface-parallel gradient of thespecific pressure �P / �s �m /s2�. Thetoe of the breaker is at x /h0�0.47.



V. CONCLUSIONS

PIV measurements of a weakly turbulent hydraulic jumpof Fr=1.2 are analyzed in detail. An interface calculationmethod based on image segmentation was used to calculatethe instantaneous free surface directly from the PIV images.This avoided the inclusion of spurious interface reflections.The turbulent surface fluctuations were seen to fit error func-tion profiles well and give an indication of the entrainmentindex. In regions of upward surface curvature, a negativesurface-normal pressure gradient leads to increased entrain-ment further downstream; the opposite occurs for regionswith downward curvature.

The calculated velocity field is found to accurately sat-isfy mass conservation. An intense and thin shear layer origi-nates near the toe of the breaker. The deceleration of thesurface-parallel velocity has its largest �negative� value at thetoe but large values are found at the peak vorticity location,which is slightly upstream of the toe location. The surface-parallel pressure gradient is highly localized and has its peak,positive value at the toe location. This leads us to believe thatbreaking for gravity-dominated waves is driven by a surface-parallel adverse pressure gradient and a streamwise flow de-celeration, both occurring at the toe location. Both effectsforce the shear layer to thicken rapidly, inducing a sharp freesurface curvature change at the toe. The mean shear is finiteand negative at the mean surface. A weak reverse flow isfound in the intermittent surface region. In this region, themean surface-normal velocity was positive indicating en-trainment of fluid from below. This is in good agreementwith the “entraining plume” model of Longuet-Higgins andTurner.57 The present topology of the reverse-flow region isdifferent from the concept of an isolated recirculating eddy/“roller” bounded on the top by a mean surface and on thebottom by a dividing streamline as indicated by SVBK00,since there clearly is reverse flow above the mean surface. Asnoted by Banner and Phillips,67 we believe it essential toreplace the idea of a mean surface in the reverse-flow regionby an intermittency zone.

The turbulent kinetic energy is maximum at the toe ofthe breaker and the Reynolds shear stress is positive justbelow the mean surface, signifying the downward transfer ofmomentum from the reverse-flow region, into the shearlayer. The turbulence intensity in the intermittent surface re-gion is an order of magnitude smaller than that in the breakershear layer. In the shear layer, the Reynolds shear stress isnegative, indicative of an upward diffusion of momentumfrom the nearly irrotational flow below. The structural simi-larity of the turbulence structure in the shear layer was com-pared to those of well-known plane shear flows. The breakershear layer is seen to resemble a plane mixing layer; thewidth of the layer, and the length scale grow linearly withdistance from the toe until they become constant around thelocation of the maximum downward curvature. The meanvalue of the nondimensional growth rate in the mixing layeris found to be in good agreement with values found earlierby Brown and Roshko.63 A more robust quantification of thelength scales can be obtained by an analysis of the coherentturbulent structures.

ACKNOWLEDGMENTS

S.K.M. and J.T.K. acknowledge the support of the Na-tional Oceanographic Partnership Program �NOPP� GrantNo. N00014-99-1-1051. S.K.M. acknowledges Dr. R. Brig-anti’s help with the spectra calculations, P. Teran Cobo’s helpwith the ADVs, and stimulating discussions with F. D. San-tos. This paper is dedicated to the memory of our esteemedcolleague, Dr. Ib Arne Svendsen �1937–2004�.

1H. Rouse, T. T. Siao, and S. Nagarathnam, “Turbulence characteristics ofthe hydraulic jumps,” J. Highw. Div. 84, 926 �1959�.

2F. J. Resch and H. J. Leutheusser, “Reynolds stress measurements in hy-draulic jumps,” J. Hydraul. Res. 10, 409 �1972�.

3D. Long, P. M. Steffler, and N. Rajaratnam, “LDA structure of flow struc-ture in submerged hydraulic jumps,” J. Hydraul. Res. 4, 437 �1990�.

4M. Liu, N. Rajaratnam, and D. Z. Zhu, “Turbulence structure of hydraulicjumps of low Froude numbers,” J. Hydraul. Eng. 130, 511 �2004�.

5B. S. Hyun, R. Balachander, K. Yu, and V. C. Patel, “Assessment of PIVto measure mean velocity and turbulence in open-channel flow,” Exp.Fluids 35, 262 �2003�.

6H. G. Hornung, C. Willert, and S. Turner, “The flow field of a hydraulicjump,” J. Fluid Mech. 287, 299 �1995�.

7J. M. Lennon and D. F. Hill, “Particle image velocimetry measurements ofhydraulic and undular jumps,” J. Hydraul. Eng. 132, 1283 �2005�.

8J. H. Duncan, “An experimental investigation of breaking waves producedby a towed hydrofoil,” Proc. R. Soc. London, Ser. A 377, 331 �1981�.

9J. A. Battjes and T. Sakai, “Velocity field in a steady breaker,” J. FluidMech. 111, 421 �1981�.

10J. H. Duncan, “The breaking and non-breaking wave resistance of a two-dimensional hydrofoil,” J. Fluid Mech. 126, 507 �1983�.

11M. L. Banner, “Surging characteristics of spilling zones of quasi-steadybreaking water waves,” in Nonlinear Water Waves, International Union ofTheoretical and Applied Mechanics Symposium, edited by K. Horikawaand H. Maruo �Springer-Verlag, Germany, 1988�, pp. 151–158.

12E. B. Thornton and R. T. Guza, “Transformation of wave height distribu-tion,” J. Geophys. Res. 88, 5925, DOI: 10.1029/JC088iC10p05925�1983�.

13F. C. K. Ting, “Wave and turbulence characteristics in narrow-bandedirregular breaking waves,” Coastal Eng. 46, 291 �2002�.

14J. A. Battjes, “Surf-zone dynamics,” Annu. Rev. Fluid Mech. 111, 257�1981�.

15I. A. Svendsen, J. Veeramony, J. Bakunin, and J. T. Kirby, “The flow inweak turbulent hydraulic jumps,” J. Fluid Mech. 418, 25 �2000�.

16J. Bakunin, “Experimental Study of Hydraulic Jumps in Low FroudeNumber Range,” M.S. thesis, Department of Civil and Environmental En-gineering, University of Delaware �1995�.

17D. H. Peregrine and I. A. Svendsen, “Spilling breakers, bores and hydrau-lic jumps,” in Proceedings of the 16th International Conference onCoastal Engineering �ASCE, Hamburg, 1978�, pp. 540–555.

18J. W. Hoyt and R. H. J. Sellin, “Hydraulic jump as ‘mixing layer’,” J.Hydraul. Eng. 115, 1607 �1989�.

19N. Rajaratnam, “The hydraulic jump as a wall jet,” J. Hydr. Div. 91, 107�1965�.

20R. Narayanan, “Wall jet analogy to hydaulic jump,” J. Hydr. Div. 101, 347�1975�.

21H. Chanson and T. Brattberg, “Experimental study of the air-water shearflow in a hydraulic jump,” Int. J. Multiphase Flow 26, 583 �2000�.

22I. A. Svendsen and P. A. Madsen, “A turbulent bore on a beach,” J. FluidMech. 148, 73 �1984�.

23R. Cointe and M. P. Tulin, “A theory of steady breakers,” J. Fluid Mech.276, 1 �1994�.

24S. H. Rhee and F. Stern, “RANS model for spilling breaking waves,” J.Fluids Eng. 124, 424 �2002�.

25F. M. Henderson, Open Channel Flow �MacMillan, New York, 1970�.26V. T. Chow, Open Channel Hydraulics �McGraw-Hill, New York, 1959�.27M. R. Maxey, “The gravitational settling of aerosol particles in homog-

enous turbulence and random flow fields,” J. Fluid Mech. 174, 441�1987�.

28Q. Wang, K. D. Squires, and L. Wang, “On the effect of nonuniformseeding on particle dispersion in two-dimensional mixing layers,” Phys.Fluids 10, 1700 �1998�.



http://dx.doi.org/10.1061/(ASCE)0733-9429(2004)130:6(511)

http://dx.doi.org/10.1007/s00348-003-0652-7

http://dx.doi.org/10.1007/s00348-003-0652-7

http://dx.doi.org/10.1017/S0022112095000966


http://dx.doi.org/10.1017/S0022112083000294

http://dx.doi.org/10.1029/JC088iC10p05925

http://dx.doi.org/10.1017/S0022112000008867

http://dx.doi.org/10.1016/S0301-9322(99)00016-6

http://dx.doi.org/10.1017/S0022112084002251

http://dx.doi.org/10.1017/S0022112084002251

http://dx.doi.org/10.1017/S0022112094002442

http://dx.doi.org/10.1115/1.1467078

http://dx.doi.org/10.1115/1.1467078

http://dx.doi.org/10.1017/S0022112087000193

http://dx.doi.org/10.1063/1.869687

http://dx.doi.org/10.1063/1.869687

29Y. A. Hassan, K. Okamoto, and O. G. Philip, “Investigation of the inter-action between a fluid flow and the fluid’s free surface using particleimage velocimetry,” in 9th International Symposium on Transport Phe-nomena in Thermal-Fluids Engineering edited by S. H. Winoto, Y. T.Chew, and N. E. Wijeysundera �1996�, pp. 566–574.

30W. L. Peirson, “Measurement of surface velocities and shear at a wavyair-water interface using particle image velocimetry,” Exp. Fluids 23, 427�1997�.

31C. N. S. Law, B. C. Khoo, and T. C. Chen, “Turbulence structure in theimmediate vicinity of the shear-free air-water interface induced by adeeply submerged jet,” Exp. Fluids 27, 321 �1999�.

32H. J. Lin and M. Perlin, “Improved methods for thin, surface boundarylayer investigations,” Exp. Fluids 25, 431 �1998�.

33L. Tsuei and T. Savaş, “Treatment of interfaces in particle image veloci-metry,” Exp. Fluids 29, 203 �2000�.

34S. K. Misra, M. Thomas, C. Kambhamettu, J. T. Kirby, F. Veron, and M.Brocchini, “Estimation of complex air-water interfaces from PIV images,”Exp. Fluids 40, 764 �2005�.

35F. Veron, G. Saxena, and S. K. Misra, “Measurements of the viscoustangential stress in the airflow above wind waves,” Geophys. Res. Lett.34, L05402, DOI: 10.1029/2006GL029147 �2007�.

36H. Chanson, Air Bubble Entrainment in Free-Surface Turbulent ShearFlows �Academic, New York, 1996�.

37H. Chanson and J. S. Montes, “Characteristics of undular hydraulic jumps.Experimental apparatus and flow patterns,” J. Hydraul. Eng. 121, 129�1995�.

38J. H. Duncan, “Spilling breakers,” Annu. Rev. Fluid Mech. 33, 519�2001�.

39H. Qiao and J. H. Duncan, “Gentle spilling breakers: Crest flow-fieldevolution,” J. Fluid Mech. 439, 57 �2001�.

40M. S. Longuet-Higgins, “Shear instability in spilling breakers,” Proc. R.Soc. London, Ser. A 446, 399 �1994�.

41J. H. Duncan, H. Qiao, V. Philomin, and A. Wenz, “Gentle spilling break-ers: Crest profile evolution,” J. Fluid Mech. 379, 191 �1999�.

42M. Brocchini and D. H. Peregrine, “The dynamics of strong turbulence atfree surfaces. Part 2. Free-surface boundary conditions,” J. Fluid Mech.449, 255 �2001�.

43D. Dabiri and M. Gharib, “Experimental investigation of the vorticitygeneration within a spilling water wave,” J. Fluid Mech. 330, 113 �1997�.

44J. C. Lin and D. Rockwell, “Evolution of a quasisteady breaking wave,” J.Fluid Mech. 302, 29 �1995�.

45H. A. Schaffer, P. A. Madsen, and R. Deigaard, “A Boussinesq model forwaves breaking in shallow water,” Coastal Eng. 20, 185 �1993�.

46R. Briganti, R. E. Musumeci, G. Bellotti, M. Brocchini, and E. Foti,“Boussinesq modeling of breaking waves: Description of turbulence,” J.Geophys. Res. 109, C07015, DOI: 10.1029/2003JC002065 �2004�.

47N. Salvesen and C. von Kerczek, “Nonlinear aspects of free-surface flowpast two-dimensional bodies,” in 14th International Union of Theoreticaland Applied Mechanics Delft �Springer-Verlag, Germany, 1976�.

48M. S. Longuet-Higgins, “A model of flow separation at a free surface,” J.

Fluid Mech. 57, 129 �1973�.49R. A. Antonia, “Conditional sampling in turbulent measurement,” Annu.

Rev. Fluid Mech. 13, 131 �1981�.50S. Corrsin and A. L. Kistler, “Free-stream boundaries of turbulent flows,”

National Advisory Committee for Aeronautics, Washington, Report No.1244 �1955�, p. 1244, available at htpp://naca.central.cranfield.ac.uk/reports/1955/naca-report-1244.pdf.

51J. M. Killen and A. G. Anderson, “A study of the air-water interface inair-entrained flow in open channels,” in Proceedings of the 13th Interna-tional Association of Hydrautic Engineering and Research Congress,Kyoto, Japan �1969�.

52D. A. Ervine and H. T. Falvey, “Behavior of turbulent water jets in theatmosphere and in plunge pools,” in Proceedings of the Institute of CivilEngineering �Thomas Telford, London, 1987�, Part 2, Vol. 83, pp. 295–314.

53M. Brocchini and D. H. Peregrine, “Integral flow properties of the swashzone and averaging,” J. Fluid Mech. 317, 241 �1996�.

54M. Thomas, S. K. Misra, C. Kambhamettu, and J. T. Kirby, “A robustmotion estimation algorithm for PIV,” Meas. Sci. Technol. 16, 865�2005�.

55J. Nogueira, A. Lecuona, and P. A. Rodriguez, “Limits on the resolution ofcorrelation PIV iterative methods. Fundamentals,” Exp. Fluids 39, 305�2005�.

56E. H. Wilson and A. A. Turner, “Boundary layer effects on hydraulic jumplocation,” J. Hydr. Div. 98, 1127 �1972�.

57M. S. Longuet-Higgins and J. S. Turner, “An ‘entraining plume’ model ofa spilling breaker,” J. Fluid Mech. 63, 1 �1974�.

58D. T. Walker, D. R. Lyzenga, E. A. Ericson, and D. E. Lund, “Radarbackscatter and surface roughness measurements for stationary breakingwaves,” Proc. R. Soc. London, Ser. A 452, 1953 �1996�.

59U. Ullum, J. J. Schmidt, P. S. Larsen, and D. R. McCluskey, “Statisticalanalysis and accuracy of PIV data,” J. Visualization 1, 205 �1998�.

60P. Bradshaw, “Effects of streamline curvature on turbulent flow,” AGAR-Dograph, Vol. 169 �1973�.

61S. K. Misra, J. T. Kirby, M. Brocchini, M. Thomas, F. Veron, and C.Kambhamettu, “Extra strain rates in spilling breaking waves,” in Proceed-ings of the 29th International Conference on Coastal Engineering, Lisbon�ASCE, Reston, Virginia, 2004�. pp. 370–378.

62H. Tennekes and J. L. Lumley, A First Course in Turbulence �MIT, Cam-bridge, 1972�.

63G. L. Brown and A. Roshko, “On density effects and large structures inturbulent mixing layers,” J. Fluid Mech. 64, 775 �1974�.

64A. A. Townsend, The Structure of Turbulent Shear Flow, 2nd ed. �Cam-bridge University Press, Cambridge, 1976�.

65J. H. Duncan and A. Dimas, “Surface ripples due to steady breakingwaves,” J. Fluid Mech. 329, 309 �1996�.

66R. L. Simpson, “Turbulent boundary layer separation,” Annu. Rev. FluidMech. 21, 205 �1989�.

67M. L. Banner and O. M. Phillips, “On the incipient breaking of small scalewaves,” J. Fluid Mech. 65, 647 �1974�.



http://dx.doi.org/10.1007/s003480050131

http://dx.doi.org/10.1007/s003480050356

http://dx.doi.org/10.1007/s003480050249

http://dx.doi.org/10.1007/s003489900068

http://dx.doi.org/10.1029/2006GL029147


http://dx.doi.org/10.1146/annurev.fluid.33.1.519

http://dx.doi.org/10.1017/S0022112001004207

http://dx.doi.org/10.1017/S0022112098003152

http://dx.doi.org/10.1017/S0022112096003692

http://dx.doi.org/10.1017/S0022112095003995

http://dx.doi.org/10.1017/S0022112095003995

http://dx.doi.org/10.1016/0378-3839(93)90001-O

http://dx.doi.org/10.1029/2003JC002065

http://dx.doi.org/10.1029/2003JC002065

http://dx.doi.org/10.1146/annurev.fl.13.010181.001023


http://dx.doi.org/10.1017/S0022112096000742

http://dx.doi.org/10.1017/S002211207400098X

http://dx.doi.org/10.1017/S002211207400190X

http://dx.doi.org/10.1017/S0022112096008932



http://dx.doi.org/10.1017/S0022112074001583

the mean and turbulent ﬂow structure of a weak hydraulic jump et al 2008 pof weak hydraulic... ·...

Documents