Water Resources Research

Spatial and temporal structure of shear layer turbulence at a stream confluence

Authors


Abstract

[1] The shear layer that develops at the interface between converging flows is a prominent but poorly understood hydrodynamic feature at stream confluences. To examine the spatial and temporal characteristics of turbulence and coherent turbulent structures within a shear layer, three-dimensional velocity measurements were obtained at a small stream confluence in Illinois using two acoustic Doppler velocimeters spaced at various separation distances. Results indicate that coherent structures can be readily identified through cross-correlation analysis of the velocity time series for different measurement locations. Maximum correlations decrease relatively slowly over short separation distances but decrease rapidly for separation distances greater than about 2 times the flow depth. Convective velocities of coherent structures derived from the correlation analysis differ from local mean velocities and from spatially averaged mean velocities, indicating that coherent structures evolve as they are transported along the shear layer, a finding consistent with dye-tracing experiments. Despite this evolution, correlation and spectral analyses indicate that the frozen turbulence hypothesis holds for frequencies corresponding to length scales of at least some shear-generated coherent vortices. A domain of quasi-two-dimensional turbulence energy at low frequencies appears to be associated with large-scale intrusions of fluid from one stream into the other along the mixing interface.

1. Introduction

[2] The shear layer that develops along the interface of two merging flows is a prominent hydrodynamic feature of stream confluences. This shear layer is characterized by distinct vortices that rotate about vertical axes [Biron et al., 1993; Sukhodolov and Rhoads, 2001]. Coherent turbulent vortices within confluence shear layers are similar to the predominately two-dimensional vortex structures that develop in plane mixing layers (i.e., mixing layers that develop between two parallel streams) and that emanate from the Kelvin-Helmholtz-like instability of such layers [Rogers and Moser, 1992]. Plane mixing layers generally increase in width downstream from the point of initial contact between the two fluids via vortex pairing [Winant and Browand, 1974]. The process of pairing occurs when differential rates of vortex advection and evolution between successive vortices along the mixing layer results in vortex encroachment, interaction and amalgamation. Vortex pairing is not necessarily limited to two vortices, but can be quite complex and involve multiple vortices [Ho and Huerre, 1984].

[3] Most confluence shear layers develop in depth-limited environments, i.e., the distance between the channel bed and water surface is small relative to the size of turbulent structures. This constraint limits fluid motion in the vertical direction [Sukhodolov et al., 1998], but has little effect on velocity components in the transverse and streamwise directions. Fluid motion within depth-limited plane mixing layers can be characterized as quasi-two-dimensional [Babarutsi and Chu, 1998], a characterization that acknowledges fluid motion is fully three-dimensional at high frequencies, but that also recognizes the importance of strong two-dimensional effects at low frequencies corresponding to the domain of large coherent vortices [Babarutsi and Chu, 1998; Uijttewaal and Tukker, 1998]. The enhanced effects of bottom friction in depth-limited environments constrains widening of plane mixing layers via vortex pairing; rates of mixing layer growth in shallow flows are much less than in deep flows [Chu and Babarutsi, 1988; Uijttewaal and Booij, 2000].

[4] Understanding the spatial and temporal structure of shear layers at stream confluences is important for verifying assumptions underlying numerical hydrodynamical models and for attempts to derive scalar properties of turbulence structure from point measurements of velocity time series. Some numerical models assume that a single length scale is adequate to characterize turbulence at confluences [Weerakoon et al., 1991; Bradbrook et al., 2000a]. On the other hand, Babarutsi and Chu [1991, 1998] developed a plane mixing layer model with two length scales: one to account for small-scale turbulence generated by bed friction and the other to account for large-scale turbulence generated by transverse shear. One-scale and two-scale models assume that the relation among turbulence kinetic energy (TKE), turbulence dissipation rate and turbulence length scale for large-scale turbulence can be described by the relation for small-scale turbulence. This assumption implies that frequency spectra are equivalent to wave number spectra (i.e., the frozen turbulence hypothesis holds for large and small length scales) and that Kolmogorov's −5/3 power law relation defines the cascade of turbulence energy at large and small scales. The extent to which Kolmogorov's law holds at large scales has not been evaluated for shear layers at stream confluences. Laboratory experiments indicate that spectra of transverse velocity fluctuations for depth-limited mixing layers exhibit a characteristic “hump” at low frequencies [Uijttewaal and Tukker, 1998; Uijttewaal and Booij, 2000]. The high-frequency side of this hump has a slope of −3 rather than −5/3, indicating that Kolmogorov's law may not hold for large-scale turbulence in depth-limited mixing layers. A spectral slope of −3 is indicative of an inverse cascade of turbulence energy of the type that typically occurs in strongly two-dimensional turbulence, providing support for the importance of two dimensional effects at low frequencies [Frisch, 1995].

[5] The large-scale complexity of turbulence at confluences may also undermine the assumptions of conventional methods for deriving spatial properties of turbulence from point measurements of velocity time series. These properties commonly are obtained from spectral analysis of time series under the assumption that Taylor's frozen turbulence hypothesis (FTH) holds for the flow under consideration. This hypothesis assumes that coherent turbulent structures remain unchanged as they pass by the measurement point and that the temporal lag of the structure can be scaled to its spatial extent using the local mean velocity. Complex patterns of vortex growth, interaction and amalgamation in a shear layer may produce hydrodynamic conditions that do not conform to the FTH. Field studies at confluences indicate that streamwise turbulence intensities (I) can greatly exceed 10% of the mean streamwise velocity [Sukhodolov and Rhoads, 2001], calling into question the assumption that I ≪ 1, an assumption underlying the FTH [Frisch, 1995].

[6] This study examines the spatial-temporal structure of turbulence within the shear layer at a stream confluence. The analysis of turbulence structure is based on simultaneous two-point, high-frequency velocity measurements obtained with two three-dimensional acoustic Doppler velocimeters (ADVs). The goals are to explore (1) spatial relations among time series of velocity data collected at different locations along the shear layer, (2) the degree to which the temporal characteristics of turbulence at a single measurement location conform to the spatial characteristics of turbulence upstream from this location, (3) the extent to which the frozen turbulence hypothesis can be applied to spectra derived from time series data, and (4) the low-frequency properties of turbulence at confluences and the relation of these properties to assumptions underlying extant models of shallow mixing layers.

2. Data Collection and Processing

[7] Data on shear layer turbulence were collected on 19 May 2000 at the confluence of the Kaskaskia River and the Copper Slough in East Central Illinois (Figure 1). The two streams forming the confluence are contained within trapezoidal drainage ditches that are about 4 m deep and have bankfull widths of about 30 m. The confluence has a junction angle of approximately 60° and bed material consists of mixed sand and fine gravel. Bed morphology is concordant in the sense that the channel beds upstream and within the confluence have the same elevations. A distinct scour hole existed in the Kaskaskia River in May 2000, but this feature was located downstream of the region of data collection.

Figure 1.

Kaskaskia River–Copper Slough confluence showing locations of sampling points (dots). Shaded region corresponds to stagnation zone.

[8] Prior to the velocity measurements, the surficial position of the shear layer within the confluence was determined by injecting large amounts of a highly visible dye tracer (potassium permanganate) into the flow about 10 cm below the water surface near the upstream junction corner. The dye-tracing experiments revealed that the shear layer developed downstream of a region of flow stagnation, a characteristic feature generated by superelevation of the water surface and the development of an adverse pressure gradient at the upstream junction corner [Best, 1987; Bridge, 1993; Rhoads and Sukhodolov, 2001]. The stagnation zone contained recirculating fluid and was bounded on each side by small shear layers. At the downstream end of the stagnation zone, the two small shear layers merged to form a single, large shear layer. The lateral margins of the shear layer downstream of the stagnation zone, as indicated by the lateral extent of coherent structures made visible by the dye, were marked at 2–3 m intervals within the confluence with pairs of wooden stakes inserted into the streambed. Dye was injected into the shear layer several times over a period of 30 minutes to ensure consistent delineation of the margins of the layer. The set of locations halfway between pairs of wooden stakes defined the center of the shear layer. The wooden stakes were removed prior to the velocity measurements to avoid disturbance of the flow. While shear layer mapping was being conducted, the discharges of the Kaskaskia River and Copper Slough were measured at two locations about 25 m upstream of the confluence using the six-tenths depth method [Buchanan and Somers, 1969] (Table 1).

Table 1. Hydraulic Characteristics at KRCS Confluence on 19 May 2000a
 Q, m3 s−1U, m s−1H, mW, mρQU, kg m s−1
  • a

    Q, discharge; U, mean velocity of the stream; H, mean flow depth; W, width of water surface; ρQU, momentum flux.

Kaskaskia River0.550.210.357.35115
Copper Slough0.680.280.298.25192

[9] Past work at the Kaskaskia River-Copper Slough confluence has involved measurements of time-averaged velocities at several cross sections within and immediately downstream of the confluence [Rhoads and Kenworthy, 1995, 1998; Rhoads, 1996; Rhoads and Sukhodolov, 2001]. The existing system of cross sections provided a downstream reference location for turbulence measurements within the shear layer. A stationary ADV was positioned within the center of the shear layer at cross section A, located at the entrance to the downstream channel, and the second ADV was moved progressively away from the stationary probe in the upstream direction along the center of the shear layer (Figure 1). This measurement strategy was based on results of previous research at the site, which have shown that the shear layer can be delineated clearly on the basis of turbulence data upstream of cross section A and that helical motion associated with curvature of flow from the Copper Slough does not have a disruptive effect on shear layer structure upstream of this cross section [Rhoads and Sukhodolov, 2001; Sukhodolov and Rhoads, 2001]. Downstream of cross section A, the shear layer becomes difficult to identify because turbulence generated by transverse shear begins to dissipate rapidly and the vertical structure of the mixing interface is distorted strongly by flow helicity [Rhoads and Kenworthy, 1995; Rhoads, 1996; Rhoads and Sukhodolov, 2001].

[10] Each ADV consists of a laboratory probe connected to fully submersible field electronics. The probe has a sampling volume of 0.125 cm3 located approximately 5 cm from the probe head. The stationary probe was mounted on a custom-built wading rod attached to a steel cable stretched tautly between iron pipes at the endpoints of cross section A. The wading rod is plumbed using a rod level. The movable probe was mounted on a custom-built tripod that rests on a 0.45 m × 0.60 m flat aluminum plate. This ADV is plumbed using adjustable screws on the three legs of the tripod. The plate sits on a 0.6 m × 3 m wooden cross piece that spans a 3 m × 3 m wooden frame. The frame, in turn, rests on four metal anchors screwed into the streambed (Figure 2). The alignments of the probes on the two mounting systems were checked with a total station to ensure that verticality of the ADVs obtained when the wading rod was plumbed (stationary probe) or the tripod leveled (movable probe).

Figure 2.

Measurement apparatus for field experiment of shear layer turbulence. Stationary ADV is mounted on wading rod attached to steel tagline at right side of wood frame, and mobile ADV is located on mount sitting on wood cross piece toward the left side of the wood frame. Direction of flow is from left to right.

[11] The wooden frame initially was set up immediately in front of the wading rod so that measurements could be obtained for small distances of separation between the two probes (Figure 2). The crosspiece containing the tripod mount was shifted along the wooden frame to increase the distance of the movable probe from the stationary ADV. When the crosspiece reached the upstream end of the frame, the frame was repositioned immediately upstream of its previous location. For each new position of the crosspiece, the alignment and position of the movable probe relative to the shear layer was determined using a total station.

[12] In total, a set of 20 simultaneous ADV measurements were obtained for probe spacings (Δx) of 0.10, 0.22, 0.30, 0.44, 0.53, 0.66, 0.78, 0.88, 1.00, 1.25, 1.49, 1.75, 2.00, 2.52, 3.00, 3.57, 4.04, 4.97, 6.04, and 7.04 m (Figure 1). Departures of probe spacings from regular intervals (e.g., 0.10, 0.25, 0.50, 1.0 m) are due to difficulties in repositioning the mobile probe precisely while maintaining proper probe alignments. Sampling volumes for all measurements were located between 10 and 12 cm below the water surface. Water depth at the measurement locations (h) was relatively constant (40 cm ± 2 cm). CollectADV software, which allows real-time monitoring of 3-D velocity series for multiple probes, was used to operate the two ADVs. For each spacing, simultaneous 3-D velocity measurements were obtained at a frequency (f) of 25 Hz over sampling intervals of 300–350 s. Sampling duration depended on the time required for the variance of the time series to achieve a stable value, an indication that the record length was sufficiently long to ensure stationarity of the turbulence. Velocities were measured in the streamwise (u), transverse (v), and vertical (w) directions in relation to the general path of the shear layer by aligning the two probes in the same orientation and maintaining this alignment as the mobile probe was moved progressively away from the stationary probe.

[13] Time series of velocity measurements were inspected visually during sampling to identify obvious errors, such as systematic acoustic noise caused by debris on the sensor or in the water. If necessary, the sampling duration was increased to obtain a clean record of data. The effect of Doppler noise was evaluated by placing the probe in a bucket of still stream water, obtaining velocity measurements over an appropriate sampling interval and generating power spectra from the velocity time series [Nikora and Goring, 1998]. These tests indicated that the variance of acoustic noise for each velocity component is at least 2 orders of magnitude less than the variance of each measured velocity component in the shear layer. Following the recommendation to perform as little conditioning of the data as is necessary based on its quality [Stull, 1988], no low-pass filtering of the data at high frequencies was conducted. Under low noise conditions such filtering can modify the data to a greater extent than the noise [Bath, 1974]. However, a few of the time series contained isolated “spikes” caused by interruption of the acoustic beam by debris. Such spikes typically span 3 to 6 individual velocity readings (at 25 Hz) and have a noticeable influence on high-frequency portions of power spectra [Sukhodolov and Rhoads, 2001]. A 3σ filter, where σ is the standard deviation of the unfiltered data, was used to remove spikes from the time series. In all cases, application of the filter removed the spikes without altering the rest of the time series.

[14] Possible effects of sensor proximity on velocity signals were evaluated by comparing power spectra for data collected at the stationary probe with and without placement of the mobile probe immediately upstream. Comparisons of these spectra indicated no obvious systematic influence of the upstream probe on characteristics of the spectrum for the downstream probe, even when the upstream probe was placed only 10 cm away from the downstream probe. The greatest concern is systematic shedding of eddies from the wake of the upstream probe into the sampling volume of the downstream probe. This effect, if present, should be detectable as a “hump,” or local increase in spectral energy over a narrow range of frequencies, in the spectral curve for the downstream probe, similar to that produced by probe vibration [e.g., Lapointe et al., 1996]. No evidence of this effect was apparent in spectral curves for the downstream probe. Apparently, any turbulent eddies shed from the upstream probe did not move downward into the sensing volume of the downstream probe, but remained at the level of the sensors.

3. Analysis and Results

[15] Analysis of the spatial and temporal structure of turbulence within the shear layer consists of four components. First, the spatially distributed velocity measurements are examined to define spatial-temporal correlations among time series of velocity fluctuations measured at different separation distances along the shear layer. Second, based on results of the spatial-temporal correlation analysis, the streamwise convection rate of coherent structures within the shear layer is determined and compared to the spatial pattern of local and spatially averaged mean velocities. The purpose of this analysis is to assess the extent to which the local mean velocity serves as an accurate index of the streamwise convection of turbulent eddies. Third, correlation functions for the spatial data are compared to estimated spatial correlation functions derived from the ensemble-averaged time series at the stationary probe to evaluate the extent to which temporal data at a single location capture the spatial structure of the turbulence. Fourth, power spectra are computed for temporal data at individual locations along the shear layer and for the ensemble-averaged data at the stationary probe to examine the spectral characteristics of turbulence, including potential differences in spectral curves at high and low frequencies. In addition, estimated spectral frequency curves are computed from wave number spectra for the spatial data to assess the extent to which these estimated spectra match ensemble-averaged frequency spectra for the stationary probe – a test of the frozen turbulence hypothesis.

3.1. Space-Time Structure of Turbulence Within the Shear Layer

[16] For the incoming flow conditions measured on 19 May 2000 (Table 1) the magnitude of standardized turbulence kinetic energy:

equation image

does not vary systematically over distance (Figure 3). In equation (1), u′, v′, and w′ are velocity fluctuations in the streamwise, transverse, and vertical directions, respectively, k is the turbulence kinetic energy, and Kd is the kinetic energy of the mean flow averaged over all velocity measurements at the stationary probe (1535 m2 s−2):

equation image

where U, V, and W are time-averaged velocities in the streamwise, transverse, and vertical directions. Despite the spatial uniformity of ks, the relative contribution of fluctuations in each velocity component to k varies along the shear layer (Figure 3). Near the stationary probe the contribution of u′ exceeds slightly that of v′, whereas farther upstream, near the beginning of the shear layer, the contribution of v′ exceeds that of u′. At all locations the contribution of w′ is less than that of u′ and v′. These findings are consistent with previous results showing that within the shear layer at KRCS the domains of percentage contributions of u′ and v′ to k strongly overlap and are greater than the percentage contribution of w′ to k [Sukhodolov and Rhoads, 2001]. Values of turbulence intensity, I = equation image/U, are fairly constant along the shear layer, equaling or slightly exceeding 0.10, a magnitude that does not severely challenge the FTH assumption that I ≪ 1.0 (Figure 3).

Figure 3.

Spatial patterns of time-averaged turbulence characteristics upstream from the stationary probe (x = 0) (see equations (1) and (2) for definitions of variables).

[17] Time-averaged properties of turbulence do not reveal information about the spatial properties of coherent turbulence structures within the shear layer. To examine these properties, analysis must focus on the degree of correspondence among temporal patterns of velocity fluctuations at different locations along the shear layer. Time series of streamwise velocity (u) indicate that closely spaced locations have similar patterns of velocity fluctuations (Figure 4). Time series for probes 1 and 2 at Δx = 0.22 m and Δx = 0.53 m are virtually identical with only a small temporal lag between patterns of velocity fluctuations for upstream and downstream probes. As expected, the lag between the times series for Δx = 0.22 m is less than the lag for Δx = 0.53 m (Figure 4). As distance increases (e.g., Δx = 1.00 m, Δx = 2.00 m), the visual similarity of the time series for the two probes decreases.

Figure 4.

Corresponding time series of streamwise velocity fluctuations for the stationary probe (left-hand axes, probe 1) and mobile probe (right-hand axes, probe 2) at four different separation distances (Δx).

[18] Detailed investigation of the spatial properties of turbulence within the shear layer is possible through space-time cross-correlation analysis. The cross-correlation function of velocity fluctuations for the stationary probe and any other point along the shear layer is defined as:

equation image

where subscripts 1, 2 refer to the separated measurement points, t is time, τ is time lag, σ is standard deviation, and T is a period of time. This function is asymmetrical for positive and negative lags and its value at τ = 0 equals the value of the spatial correlation function [Rξξ(Δx)] for any given displacement Δx. Cross-correlation analysis of the field data shows that the function for each Δx contains a pronounced peak and that the displacement of these peaks from τ = 0 increases with increasing distance between the two probes (Figure 5). The ideal case of frozen turbulence along the entire shear layer corresponds to Rξx, τ = τmax) = 1 for all Δx, where τmax is the time lag corresponding to the maximum of the cross-correlation function. For the field data, Rξx, τ = τmax) decreases systematically along the shear layer and the pattern of decay exhibits two distinct domains (Figure 6). At displacement lengths shorter than 0.53 m, values of Rξx, τ = τmax) decrease relatively slowly with increasing distance [Rξx, τ = τmax) ∼ Δx−0.1], whereas for displacements larger than 0.53 m, the decay is more pronounced [Rξx, τ = τmax) ∼ Δx−0.5]. The rate of distance decay in maximum correlation is approximately the same for all three components, but the magnitude of correlation is less, and degree of scatter greater, for the vertical component than for the streamwise and transverse velocity components, especially for displacements greater than 0.53 m (Figure 6).

Figure 5.

Cross-correlation functions at different separation distances for (a) streamwise velocity component, (b) transverse velocity component, and (c) vertical velocity components.

Figure 6.

Change in maximum correlation of cross-correlation analysis with increasing distance upstream of the stationary probe.

3.2. Comparison of Convective, Local, and Spatially Averaged Velocities

[19] Multiprobe velocity measurements allow detailed analysis of the relation between the advection of coherent turbulent structures within the shear layer and the mean velocity of flow in the shear layer. Assuming that the peaks of the cross-correlation function represent the passage of distinct eddies through the upstream and downstream probes, the average convective velocity of coherent structures UC within the shear layer can be computed as:

equation image

where Δτm = τ1,max − τ2,max [Cenedese et al., 1991; Romano, 1995]. Local mean velocity of the flow at position x (UX) along the shear layer is defined as:

equation image

where U, V and W are the local mean velocities in the streamwise, transverse and vertical directions, respectively. Since the local mean velocity changes substantially along the shear layer, the spatially averaged velocity 〈UX〉 serves as a representative velocity for advective transport processes between any upstream location and the downstream location along the shear layer:

equation image

where L is the total distance between the two probes.

[20] The spatial distributions of UC, UX and 〈UX〉 along the shear layer indicate that UX and 〈UX〉 deviate systematically from UC (Figure 7). The deviations have different patterns: UX deviates mostly at the upstream portion of the shear layer where the local velocity is relatively low and flow gradually accelerates. The effect of this convective acceleration on the advection of coherent structures seems to be captured fairly well by the spatial-averaging procedure (〈UX〉). The greatest discrepancy between UC and 〈UX〉 occurs about 3 m upstream of the downstream probe. Here UC is about 12% greater than both UX and 〈UX〉. Within 1.5 m of the stationary probe, the difference between UC and 〈UX〉 changes from positive to negative values, indicating that the relation between these two measures is complex over short separation distances.

Figure 7.

Spatial distributions of local mean velocity (Ux), spatially averaged mean velocity (〈Ux〉), and estimated mean convective velocity of turbulence structures (Uc).

3.3. Comparison of Spatial Correlation Functions

[21] The spatial correlation function can be computed from the multiprobe measurements using the values of two-point cross-correlation functions (3) at zero lag times Rξx) = R12ξx, τ = 0). The autocorrelation function for velocity time series for the stationary probe is:

equation image

Rξξ(τ) is a symmetrical function with regard to positive and negative lag times. Under the assumption of frozen turbulence, an estimate of the spatial correlation function can be derived from the temporal autocorrelation function as Rξx) = Rξξ(UC τ). An average autocorrelation function obtained from the ensemble of 20 individual time series was used to estimate Rξξ(UC τ).

[22] Overall the estimated spatial correlation Rξξ(UC τ) mirrors closely the actual spatial correlation over the length of the shear layer. Results indicate that R12ξx, τ = 0) ≈ Rξξ(UC τ) for all three velocity components over short distances (Figure 8). Both functions reveal a sharp decay in correlation as distance from the stationary probe initially increases. The decline is greatest for the vertical velocity fluctuations, intermediate for transverse velocity fluctuations, and most gradual for the streamwise velocity fluctuations (Figure 8). Correlations decline to near zero over a distance of 1.5 m for u′, 1.0 m for v′ and 0.5 m for w′. Over these ranges, values of R12ξx, τ = 0) and Rξξ(UC τ) generally differ by 10% or less. At distances greater than these thresholds, the small magnitudes and random patterns of Rξx) lead to large percentage differences between R12ξx, τ = 0) and Rξξ(UC τ).

Figure 8.

Spatial correlation functions derived from spatially distributed velocity measurements (symbols) and from ensemble-averaged time series at the downstream probe (solid lines) for (a) streamwise, (b) transverse, and (c) vertical velocity components.

3.4. Turbulence Spectra

[23] Power spectra were calculated by applying Fourier transformations to autocorrelation functions for individual times series at each measurement location:

equation image

Repetitive measurements at the stationary probe permit computation of ensemble-averaged frequency spectra

equation image

where N is number of time series in the ensemble (number of repetitions). Spatial spectra were calculated applying Fourier transformations to the spatial correlation functions derived from the multiprobe measurements

equation image

where λ is wave number. For comparison, spatial spectra can be converted to the frequency domain by assuming the validity of the frozen turbulence hypothesis Sξξ(f) ≈ Sξξ(Ucλ/2π).

[24] The most notable characteristics of spectra for individual time series and the ensemble-averaged data are (1) a high-frequency region (>0.75 Hz) of power law scaling between Sξξ(f) and f, (2) an intermediate frequency range (0.3–0.75 Hz) where Suu(f) ≈ Svv(f) > Sww(f), and (3) a low-frequency region (0.02–0.06 Hz) with a well-defined spectral peak in Svv(f) (Figures 9 and 10). The power law relations for individual velocity components all have exponents of approximately −5/3, a value characteristic of the systematic cascade of turbulence energy within the inertial subrange [Frisch, 1995]. Over this range of frequencies all three spectral curves are similar, indicating that turbulence is three-dimensional and isotropic [Sukhodolov and Rhoads, 2001]. The effects of electronic noise at the highest frequencies (>6 Hz) lead to flattening of the spectral curves and slight departures from the −5/3 relation.

Figure 9.

Power spectra of streamwise, transverse, and vertical velocity components at three locations along the shear layer.

Figure 10.

Ensemble-averaged power spectra of streamwise, transverse, and vertical velocity components for stationary probe.

[25] The validity of the energy cascade interpretation of the −5/3 power law relation depends on the extent to which Taylor's hypothesis holds for frequencies defining the inertial subrange. Only under the assumption that turbulent eddies can be treated as “frozen” are frequency and wave number equivalent. The comparison of frequency and wave number spectra confirms that these two spectra differ by only 10%–15% at frequencies between 0.75 and 6 Hz (Figure 11). At frequencies less than 0.75 Hz, discrepancies between frequency and wave number spectra increase, but relative relations among spectra for the three velocity components are maintained for both types of spectra. The total range of separation distances between the stationary and mobile probes is not great enough to determine whether the wave number spectra exhibit low wave number humps similar to those for the frequency spectra.

Figure 11.

Comparison of ensemble-averaged frequency functions for stationary probe with estimated frequency spectra derived from wave number spectra for spatially distributed velocity data.

[26] The low-frequency spectral peak is best defined for the transverse velocity spectra [Svv(f)], especially the ensemble-averaged spectrum (Figures 9 and 10). A corresponding, less prominent peak is apparent for the ensemble-averaged spectrum of Suu(f) and the ensemble-averaged spectrum of Sww(f) exhibits a slight hump with a peak at a slightly lower frequency than the peaks of the Suu(f) and Svv(f) humps (Figure 10). Not all transverse velocity spectra for individual locations exhibit low-frequency peaks, but those that do have peaks of roughly similar frequency (≈0.04 Hz) (Figure 9). Low-frequency peaks for the individual spectra of Suu(f) and Sww(f) both are less pronounced and more variable in frequency than peaks for the Svv(f) spectra (Figure 9).

4. Discussion

[27] The results of the various analyses provide the basis for interpreting the spatial and temporal structure of turbulence within the shear layer. Both the visual similarity of the time series for small probe separation distances (<0.6 m) (Figure 4) and the large maximum correlations (Figure 5) among these time series suggest that the structure of the coherent eddies responsible for velocity fluctuations is preserved over short distances. The time lag between the patterns of fluctuations for closely spaced time series presumably reflects differences in the time of travel of eddies past the probes. As expected, correlations are greatest for the streamwise and transverse velocity components and weakest for the vertical velocity component, a finding consistent with the type of velocity signal expected to be produced by shear-generated quasi-two-dimensional coherent structures rotating about a vertical axis [Sukhodolov and Rhoads, 2001].

[28] Both the decreasing visual similarity and increased rate of decline in values of maximum correlation among the u′ and v′ time series at separation distances greater than about 0.5–0.6 m indicate that the coherent structures are evolving to some extent over the time interval required for these structures to travel from one probe to the next. The −0.5 scaling exponent for the rate of decline in maximum correlation at separation distances greater than 0.5–0.6 m is suggestive of a diffusion-dispersion process (Figure 6); however the domain of this relation is located upstream of the stationary probe where rapid diffusion of turbulence should diminish the spatial organization of coherent eddies. This argument is difficult to reconcile with the fact that the rate of decline in maximum correlation decreases to a value of −0.1 near the stationary probe, implying that well-organized coherent structures exist at this location. A more likely explanation is that the abrupt change in the Rξx, τ = τmax) − Δx relation at a separation distance of about 0.5 m denotes the spatial scale of pronounced individual coherent eddies and that beyond this distance temporal evolution of the advecting eddies leads to a marked decline in their correlation structure. Support for this argument is provided by the comparison among convective, local and spatially averaged velocities, which reveals a maximum discrepancy between convective and spatially averaged velocities at a separation distance of about 3 m. A possible explanation for this discrepancy is that pronounced vortex pairing, stretching and growth may occur within the vicinity of this separation distance. If vortex pairing increases the size of coherent structures at a rate that exceeds the spatially averaged velocity, the result will be an apparent increase in the rate of advection of the structures. The occurrence of this pairing immediately upstream of the stationary probe could also account for the short separation distance of strong correlation among temporally lagged velocity fluctuations (Figure 6).

[29] Supporting evidence for growth of coherent structures at a distance about 3 m upstream of the stationary probe is provided by dye-tracing studies, which provide an important complement to detailed velocity measurements [Roy et al., 1999]. Dye tracing reveals the growth, elongation and individuation of discrete rotating eddies with vertical axes about 2 to 4 m upstream of cross section A in the vicinity of the location where the convection velocity locally increases (Figures 7 and 12). Many of the discrete eddies have sizes roughly on the order of 0.5 to 1.0 m, a size consistent with the change in the maximum cross-correlation relation at a separation distance of slightly greater than 0.5 m (Figures 6 and 12). The process of eddy development appears to be similar to traditional vortex pairing, whereby small individual vortices moving at different speeds on each side of the shear layer amalgamate in the downstream direction to form large, coherent structures [Winant and Browand, 1974].

Figure 12.

Dye-tracing experiment in the shear layer at KRCS (dark tones) showing downstream growth and isolation of turbulent structures. Coherent vortex marked by arrow appears to be embedded in the upstream portion of an intrusion of fluid from the Copper Slough (top) into the Kaskaskia River (left).

[30] The similarity of the estimated and actual spatial correlation functions suggests that the frozen turbulence hypothesis can be reasonably applied to explore spatial characteristics of turbulence structure over short distances using time series data for a single stationary probe. The threshold frequency in this case is about 0.75 Hz, which corresponds to a wavelength of about 1.5 m or Δx/h ≈ 4, where h is the flow depth. The threshold is defined best for the streamwise and transverse velocity components, emphasizing the quasi-two-dimensional structure of the turbulence.

[31] Previous research on spectral characteristics of turbulence within the KRCS shear layer has shown that over a certain range of frequencies between 1 and 0.1 Hz Suu(f) ≈ Svv(f) > Sww(f) [Sukhodolov and Rhoads, 2001]. Both the individual and ensemble-averaged spectral curves indicate that such a relation exists among the velocity spectra within a region of intermediate frequencies (0.3–0.75 Hz) (Figures 9 and 10). Within this frequency domain, turbulence becomes increasingly anisotropic and quasi-two-dimensional. Coherence between streamwise and transverse velocity fluctuations also reaches a maximum within this range of frequencies [Sukhodolov and Rhoads, 2001], suggesting that intermediate-scale Kelvin-Helmholtz-like vortices are responsible for the apparent quasi-two-dimensionality of turbulence.

[32] The occurrence of a prominent low-frequency peak in the transverse velocity spectra is consistent with the development of quasi-two-dimensional turbulence and an inverse cascade of turbulence energy [Kraichnan, 1967]. The hump in the power spectra of transverse velocity fluctuations also conforms to results of experimental work on shallow mixing layers and the −3 slope of the high-frequency side of this hump is consistent with the interpretation of the existence of an inverse cascade of energy within the shear layer [Uijttewaal and Tukker, 1998; Uijttewaal and Booij, 2000]. Although laboratory investigations suggest that the peaks tend to shift toward lower frequencies with increasing distance downstream along the mixing layer, such a trend is not evident in this study.

[33] Both the individual and ensemble-averaged spectra for S(f) exhibit marked “flattening” of the spectral curve at frequencies slightly higher than those of the spectral hump (Figures 9 and 10). This stepped spectral signature is consistent with a direct transfer of energy from low to high frequencies, a hallmark of quasi-two-dimensional turbulence, rather than a continuous cascade of energy through the intermediate range of frequencies [Babarutsi and Chu, 1991]. These findings indicate the need for revision of two-length-scale turbulence models of shear layers [e.g., Babarutsi and Chu, 1998] wherein the length scale of large-scale turbulence (Lt) is related to the turbulence kinetic energy (k) and dissipation rate (ɛ) using the same relation as for the direct cascade in the inertial subrange

equation image

As this study shows, relationship (11), which is based on the −5/3 power law relation, is not valid for the high-frequency side of the low-frequency hump in the spectral curve.

[34] Laboratory studies have attributed the spectral hump in transverse velocity fluctuations to two-dimensional turbulence with the downstream increase in the size of quasi-two-dimensional coherent structures explaining the progressive shift of spectral peaks toward low frequencies [Uijttewaal and Tukker, 1998; Uijttewaal and Booij, 2000]. The exact mechanism responsible for the low-frequency hump in Sv(f) observed in this study is uncertain. Although it was not possible to determine through correlation analysis whether the FTH holds for frequencies associated with the spectral hump, consideration of frequency-wave number characteristics under the FTH provides a rough guide for exploring possible scale relations. The frequency of the peak for the ensemble-averaged spectra (0.04–0.05 Hz) corresponds to a periodicity of about 20–25 s. Given the mean velocity of the flow (≈0.50 m s−1) (Figure 7), this periodicity implies a length scale of about 10 to 12 m – approximately 1.5 to 2 times the measured length of the shear layer. The results of dye tracing (Figure 12), cross-correlation analysis (Figures 5 and 6) and correlation analysis (Figure 8) suggest that individual coherent vortices at or upstream of the stationary probe typically are at most only 1–2 m in size. The largest of these structures should be advected past the probe in a few seconds, rather than over several tens of seconds. Support for this assertion is provided by Sukhodolov and Rhoads [2001], who found that maximum coherence of streamwise and transverse velocity fluctuations at KRCS occurred at a frequency of 0.5 Hz or periodicity of 2 sec, which, for the mean flow rate of 0.5 m s−1, corresponds to a length scale of about 1 m. Although approximate, these considerations suggest that coherent vortices captured by dye tracing and spatial correlation analysis are too small to be responsible for the low-frequency spectral hump.

[35] Visualization studies involving injection of tracer material into one of the upstream tributaries only reveal periodic lateral intrusions of fluid from one confluent stream into the other, resulting in large-scale changes in the location of the mixing interface within the confluence (Figure 13). Such intrusions of fluid appear to be related to local temporal and spatial variations in momentum relations between the flows as the two streams converge at an angle and collide along the shear layer. This phenomenon may be similar to low-frequency oscillatory movements of the shear layer described in other studies and attributed to periodic changes in local pressure gradients within the confluence [Biron et al., 1993; Roy et al., 1999; De Serres et al., 1999; Bradbrook et al., 2000b]. However, as is evident from flow visualization, the intrusions generally do not produce changes in the position of the shear layer along its entire length (Figure 13). Moreover, rotating, two-dimensional eddies are embedded within these intrusions (Figure 12). The intrusions appear to scale roughly with the widths of the two confluent streams (7–9 m) and may reflect periodic pulsations of incoming flow caused by mutual hydrological interaction (i.e., backwater effects) between the streams. Recent work has revealed pronounced intermittency in the pattern of water surface topography at confluences over a variety of spatial and temporal scales that could possibly account for such pulsating hydrological behavior [Biron et al., 2002].

Figure 13.

Flow visualization illustrating changes in mixing interface (solid lines) resulting from large-scale intrusion of fluid from one stream into the other within the confluence. Wooden stakes in the streambed within the confluence and submerged ADV cable (to the right of the mixing interface) provide points of reference.

[36] The study here involves only two-point measurements of turbulence characteristics at a relatively constant depth within the shear layer. Therefore it is difficult to generalize the results in relation to the entire vertical and lateral extent of the shear layer or to flow structure outside of the shear layer. Past work at the KRCS confluence suggests that the time-averaged turbulence structure of the shear layer upstream of the stationary probe, as defined on the basis of spatial patterns of turbulence kinetic energy throughout the flow, remains relatively uniform over depth [Sukhodolov and Rhoads, 2001]. Whether or not the uniformity of time-averaged properties can be extended to the spatial and temporal structure of coherent eddies cannot be ascertained without a more extensive set of simultaneous measurements throughout the shear layer than those conducted in this study. Even less certain is the relation of coherent turbulent structures within the shear layer to properties of the time-averaged flow, such as the helical motion documented at the KRCS confluence. Prior research indicates that helical motion largely is confined to the ambient flow outside of the shear layer and that this motion is only weakly developed upstream of cross section A, the position of the stationary probe in this study [Rhoads and Kenworthy, 1998; Rhoads and Sukhodolov, 2001]. Only as flow moves downstream of this location does helical motion produce substantial distortion of the vertical structure of the mixing interface. The interaction between large-scale helical motion and coherent turbulence structures within the shear layer is a problem in need of further investigation.

5. Conclusion

[37] The results of this study contribute to the understanding of the three-dimensional spatial and temporal structure of turbulence within the shear layer of river confluences. Cross-correlation analysis of spatially distributed 3-D velocity measurements reveals that the magnitude of maximum correlation decreases with separation distance between probe locations, whereas the temporal lag of maximum correlation increases with separation distance. The spatial-temporal pattern of maximum correlation reflects the extent to which discrete eddies maintain similarity of turbulence structure as these eddies travel through specific locations along the shear layer. The existence of two distinct domains of rates of decrease of maximum correlation suggests that an abrupt change in the similarity, or spatial organization, of coherent structures occurs at a separation distance of about 0.50–0.60 m. Upstream of this threshold, the rapid decline in maximum correlation with increasing separation distance suggests that turbulent structures at the stationary probe typically differ substantially from those greater than 0.5–0.6 m upstream. Dye-tracing studies confirm that coherent rotating vortices within the shear layer evolve substantially upstream of the downstream probe. Tracing experiments also confirm that individual vortices near the probe generally are about 0.5–1.0 m in size, a range that encompasses the separation distance over which strong correlation among velocity components occurs.

[38] Comparison of the correlation function for the spatially distributed velocity measurements and an estimated spatial correlation function derived from the ensemble-averaged time series for the stationary probe suggests that these two functions are reasonably similar for a separation distance less than 1.5 m. Similarly, the frequency spectra for the ensemble-averaged data and the estimated frequency spectra derived from wave number spectra for spatially distributed measurements correspond closely for frequencies greater than 0.3 Hz, which, for an average convection velocity of about 0.5 m s−1, corresponds to a separation distance of approximately 1.5 m or less. Together these findings indicate that the frozen turbulence hypothesis is valid for frequencies greater than 0.3 Hz and wavelengths of coherent structures less than 1.5 m. This domain includes the frequencies greater than 0.75 Hz where turbulence is isotropic and power law scaling obtains between turbulence energy and frequency. In other words, the range of frequencies greater than 0.75 Hz conforms to the inertial subrange characterized by a smooth cascade of turbulence energy. Between 0.3 and 0.75 Hz the power spectra of streamwise and transverse velocity fluctuations overlap and exceed the spectrum for vertical velocity. Thus the frozen turbulence hypothesis holds for a portion of the frequency range corresponding to quasi-two-dimensional turbulence characterized by coherent rotating structures generated via Kelvin-Helmholtz-like instability of the shear layer.

[39] Low-frequency peaks in the streamwise and transverse velocity spectra appear to be associated with periodic large-scale intrusions of fluid from one stream into the other along the shear layer, a phenomena documented by dye-tracing experiments. Further work is required to determine whether such intrusions are indeed the cause of the low-frequency humps and to establish how turbulence mechanisms responsible for low-frequency characteristics of spectra at stream confluences relate to those responsible for low-frequency characteristics of spectra for depth-limited shear layers between parallel laboratory streams. Also, previous work on confluence shear layers has indicated that turbulence characteristics can vary both among different types of concordant confluences [Sukhodolov and Rhoads, 2001] as well as between concordant and discordant (i.e., those with different bed elevations of upstream tributaries) confluences [Best and Roy, 1991; Biron et al., 1993, 1996a, 1996b; Bradbrook et al., 2000b, 2001; De Serres et al., 1999; McLelland et al., 1996]. Future research should seek to identify and determine the relative importance of various factors that influence the spatial-temporal structure of confluence shear layers.

Acknowledgments

[40] Funding for this research was provided by the Geography and Regional Science and Eastern European Programs of the National Science Foundation (SBR 9710068) and by the University of Illinois Research Board. Rebecca Wade, Melinda Daniels and Scott Rayburg assisted with data collection. Wim Uijttewaal provided helpful comments on an initial draft of this paper. Suggestions by two anonymous reviewers greatly improved the quality of the paper.

Ancillary