Fundamental study on mixing layer and horizontal circulation in open-channel flows with rectangular embayment zone＊

Michio Sanjou, Iehisa Nezu

Department of Civil and Earth Resources Engineering, Kyoto University, Kyoto, Japan, E-mail: michio.sanjou@water.kuciv.kyoto-u.ac.jp

(Received October 16, 2015, Revised March 12, 2016)

Fundamental study on mixing layer and horizontal circulation in open-channel flows with rectangular embayment zone＊

Michio Sanjou, Iehisa Nezu

Department of Civil and Earth Resources Engineering, Kyoto University, Kyoto, Japan, E-mail: michio.sanjou@water.kuciv.kyoto-u.ac.jp

(Received October 16, 2015, Revised March 12, 2016)

A part of mean kinetic energy in a main-channel is used for production of a large-scale horizontal circulation in the side cavity. However, the details of the mechanism such as energy transport are poorly understood. Therefore, we conducted PIV measurements in a laboratory flume and compared space distributions of mean velocity components and Reynolds stress by varying a cavity geometry. In particular, a practical calculation method of Reynolds stress was also developed and its accuracy was examined by comparison with the measured data. Furthermore, contributions of components in an energy transport equation were revealed quantitatively.

Embayment, open-channel turbulence, mixing layer, Reynolds stress, mean kinetic energy

Introduction

Dead water zones in natural rivers occur in groin fields and embayments. A mixing layer is typically formed in the cavity opening attached to the main channel, and it promotes mass and sediment transportation between the mainstream and the cavity zone. It is well known that nonuniformity of the bed shear stress induces local scour and sedimentation. A largescale horizontal circulation within the cavity generated by a lateral shear force in the mixing layer often traps pollutants and nutrient salts and results in a sudden drop in the water quality of the dead water zone. Though the development of the mixing layer has a close relation to the shedding vortex along the cavity opening, there are many unresolved issues in understanding the turbulence structure around such an embayment zone.

The mean velocity profile and development of the mixing layer have been studied theoretically by many researchers. Particularly, van Prooijen and Uijttewaal[1]focused on shallow mixing layers in which bed friction stabilizes the coherent structure. They proposed a theoretical model derived from depth-averaged shallowwater equations and examined the predicted mean flow field, particularly the streamwise development of the velocity difference between the high- and low-velocity sides, the development of the mixing layer width, and the horizontal distribution of the streamwise velocity component, by comparing these predictions with the measured data. Furthermore, linear stability analysis based on the mean flow field allowed them to determine the spatial evolution of the energy densities and the characteristic length scales.

Local dead water zones in rivers are observed behind isolated spur dikes, within groin fields, and in the embayment attached to the mainstream, and a similar mixing layer due to shear instability should be considered for practical problems concerning sedimentation, mass transport, and water preservation in these locations. Therefore, many previous works have studied these issues from various perspectives.

Ettema and Muste[2]have classified the horizontal flow field around a spur dike into characteristic zones, including flow separation and thalweg alignment, and these length scales and their related hydrodynamic characteristics were investigated in detail. The results imply that single spur dikes produce shedding vortices accompanied by surrounding complex three-dimensional behavior. Duan[3]conducted turbulence measurements around a spur dike model in a gravel bed flume using acoustic Doppler velocimetry(ADV). After the uncertainty analysis of ADV, threedimensional distributions of velocity components and Reynolds stress were obtained. The measured data revealed a secondary current structure and the relation between the local scoring and the shear stress. Sharma and Mohapatra[4]measured the velocity components around a spur dike in a meandering trapezoidal flume with a rigid bottom, in their study, the dependence of the separation zone and the streamwise velocity distribution on the dike position was examined. Particularly, characteristic lengths, such as the reattachment distance and the width of the separation layer, may be evaluated quantitatively.

Dead water zones have also been observed in consecutive spur dikes and groin fields. Previous studies have reported that horizontal vortices, related threedimensional structures, and sediment processes are significantly influenced by cavity geometry and groin rank number. Particle image velocimetry (PIV) measurements by Weitbrecht et al.[5]suggested that the gyre structure within the cavity also depends on the mounting angle on the sidewall of the channel. Some fundamental studies have provided detailed knowledge of mean flow fields, turbulence in the cavity opening, and related mass and sediment transport.

Uijttewaal et al.[6]examined the dependence of the mass transfer coefficient on groin geometry, water depth, groin rank number, and bottom elevation, and the results showed that transfer speed is significantly affected by aspect ratio. In a small gyre field where the aspect ratio is close to 1.0, i.e., a square horizontal shape, the mass transfer is dominated by the mixing layer formed along the cavity opening, whereas in a large groin field, coherent shedding vortices are generated from the tip of the higher ranking groin. The measurement results of a study by Weitbrecht et al.[7]provided the relation between the exchange rate and the bed configuration of the cavity, and they suggested that there is a linear relation between the mass transfer coefficient and the morphometric groin field parameter normalized by water depth, width, and groin length. Yossef and De Vriend[8]conducted experiments by analyzing the time variation of bedform migration using the PIV technique, and these experiments yielded important results. Notably, their results suggested that the mixing layer dynamics induce variation in bedform celerity, and the time series of the instantaneous celerity vector field revealed a sediment transport process toward the cavity zone. Yossef and De Vriend[9]indicated that there are significant differences in the developing processes of the mixing layer and turbulence structures under emerged and submerged conditions.

Some researchers have conducted studies on groin fields in natural rivers. For example, Sukhodolov et al.[10]conducted velocity measurements with ADV in the Elbe River in Germany, and their analysis of the collected data provided mean horizontal velocity vectors, vertical profiles of the mean velocity, turbulent kinetic energy, and spectral properties. It is noteworthy that mean velocity profiles are consistent with the conventional power law, except near the free surface where the effects of wind-induced waves could not be ignored and upwelling occurs. Furthermore, they indicated that the spectral density is consistent with Kolmogorov’s −5/3 power law over the whole depth. Numerical simulations by Ercan and Younis[11]predicted mean velocity components, turbulent kinetic energy, and bed shear stress for the beach of the Sacramento River and calculated the maximum erosion rate of the river bank.

The embayment zone adjacent to the mainstream produces the mixing layer and cavity gyre in the same manner as the groin field. Previous theoretical works have been conducted on two-dimensional simple rectangular cavities. For example, Mizumura and Yamasaka[12]introduced a streamwise function within square- and rectangular-shaped cavities under limited conditions, i.e., two-dimensional assumption, steady flow, ignoring inertia, and rigid free surface assumption. The predicted results agree with the measured data except in the wall region. They deduced that the gap near the wall was caused by neglecting viscidity and three-dimensional behavior. Hill[13]assumed the horizontal gyre in the cavity to be in rigid body circular motion and gave the linear velocity distribution. Considering the shear stress and torque on the cavity opening and three walls, they introduced a rotational velocity profile normalized by the mainstream velocity on the basis of balancing resistance torques acting on the horizontal gyre. Furthermore, they examined the relation between the cavity geometry and the coefficients of friction and moment.

Sedimentation in the side cavity is an irreversibly damaging problem that must be solved as soon as possible. It is necessary to reveal the three-dimensional flow pattern and the related coherent turbulence. Previous studies have noted that the sedimentation and scour processes within the cavity are relevant to not only the horizontal gyre but also the upward and downward currents and the secondary currents with a longitudinal axis. The numerical results proposed by Sanjou et al.[14]provided the time series of cross-sectional secondary currents and a phenomenological threedimensional model in which the transverse direction of momentum transfer depends on the elevation from the bottom. Measured data obtained by the threedimensional particle tracking velocimetry (PTV) technique suggested that the rotation axis of the horizontal gyre is tilted by the vertical velocity gradient, which induces outward flow from the embayment near the free surface and inward flow to the embayment near the bottom[15].

Gravity waves and oscillations induced by theexistence of the cavity are known to generally increase drag force or momentum transfer. Meile et al.[16]focused on transverse oscillations in an open channel with axisymmmetric cavities on both sides by performing experimental tests with an ultrasonic velocity profiler. They asserted that the momentum transfer from the mainstream to the embayment has an impact on the sloshing phenomenon, and the free-surface oscillation is influenced by the impingement of a shedding vortex at the downstream edge. They emphasized that the Strouhal number is the key parameter that promotes wave excitation, and their measured data proved the existence of critical values of the Strouhal number where peak excitations of oscillations are clearly observed. They suggested that the critical Strouhal number should be avoided for practical river management, e.g., reduction of water oscillations in cavity zones, such as the harbor, groin field, and embayment. Wolfinger et al.[17]investigated the relation between the inherent instability of the mixing layer in a cavity opening and streamwise-oriented gravity standing waves, which results in the occurrence of highly organized oscillations. They measured not only horizontal velocity components but also the pressure at the impingement wall of the cavity. Their results indicate that the frequency of the separated turbulent layer in the cavity opening is consistent with the frequency of the gravity wave and there exists a threshold inflow velocity where the amplitude of the spectral peak of the pressure fluctuation reaches a maximum value. Furthermore, they concluded that the coupling between the mixing layer and the gravity standing waves substantially enhances the increase in Reynolds stress, turbulence intensity, and transverse velocity through the mixing layer. Tuna et al.[18]conducted laboratory experiments with a PIV system to investigate the effects of standing waves on streamlines, turbulence structures, and mass exchange in shallow flows past a single cavity. In particular, the streamline near the bottom was found to be deflected by the standing wave. Their phase-averaged analysis yielded detailed information about the development of the undulating vortex layer over time. Furthermore, they suggested that the exchange velocity evaluated by the integration of the transverse velocity has a strong correlation with the Reynolds stress and has a peak value at the same position as does the Reynolds stress. The mass exchange velocity indicated a 40% enhancement due to the gravity standing wave.

The abovementioned studies evaluated not only simple conditions limited to theoretical analysis but also complex flow fields, including bed formation, free-surface oscillation, and three-dimensional behavior, for practical problems. Furthermore, the development of measurement techniques and calculation methods allow us to understand detailed turbulence structures and coherent events.

The present study considers formation characteristics of horizontal gyre inside the cavity from the viewpoint of transverse transfer of mean kinetic energy from the mainstream toward the cavity. We indicate that distribution of Reynolds stress related to the shedding vortices contributes significantly to the energy transfer. Especially, longitudinal variation of Reynolds stress along the centerline of the mixing layer is obtained theoretically and comparison of the theoretical value with measured data is conducted quantitatively. When such a physical model which is improved in credibility by laboratory experiments is applied to actual water field, scale dependency is an unavoidable problem. The scale similarity of hydraulic characteristic in the mixing layer phenomenon caused by the embayment is examined based on detailed measurement database containing results obtained in two laboratory flume with different width in order to raise practicability of the present results.

Therefore, the present study has the following objectives. (1) Turbulence measurements are generally difficult in field work. We propose a practical method to predict Reynolds stress along the cavity opening from the spanwise profile of the mean streamwise velocity component. (2) We consider the generation mechanism of horizontal circulation in the cavity, particularly how mean energy in the main channel is supplied toward the cavity to maintain the gyre formation. To achieve our goals, we conducted PIV measurements in a laboratory flume and compared the space distributions of the mean velocity components and the Reynolds stress for various cavity geometries. In particular, a practical calculation method for Reynolds stress was developed, and its accuracy was examined by comparison with the present measured data. Furthermore, the contributions of the components in an energy transport equation were quantitatively determined.

Fig.1 Experimental setup and coordinate system

1. Experimental procedure

We previously studied turbulence structure in theside-cavity zone using a laboratory flume of 0.4 m in width[19]. However, the present paper mainly focuses on the results obtained in a flume of 1.5 m in width to remove the influence of the opposite bank. The longitudinal length from the upstream honeycomb to the tailgate is 9 m, and a sidewall made of tempered glass allowed us to observe the flow pattern.

Figure 1 shows the experimental setup, in which the floodplain was modeled by placing acrylic boxes with dimensions of 0.2 m (width)×0.25 m (height)× 0.6 m (length). One box was removed to create the side-cavity zone 6 m from the honeycomb section.B, Bw,Lwand Hare the total flume width, cavity width, cavity length in the streamwise direction, and water depth, respectively.x,y and zare the streamwise, vertical, and spanwise axes, respectively, and their respective origins are situated on the upstream side of the cavity, at the flume bottom, and at the junction between the cavity and the mainstream. U,V and Ware the time-averaged velocity components in thex,yandzdirections, respectively, andu,vandware turbulence fluctuation components in the same respective directions.

The time series of the horizontal velocity components were measured using the PIV technique. A 2 mm-thick continuous laser light sheet (LLS) was projected to illuminate the side cavity. Under the experimental flow conditions, the momentum transfer is much larger in the horizontal direction than in the vertical direction. Our previous study examined measurement uncertainty for the same PIV system as used in the present study. A maximum repeatability standard deviation was 0.004 m/s and 0.12×10−4m2/s2for mean streamwise velocity and Reynolds stress respectively[19].

Jamieson and Gaskin[20]identified secondary current within the cavity by dye observation technique, in which radially outward (diverging) flow and radially inward (converging) flow occur in the free-surface and in the bottom, respectively. Examples of some particles paths showed existence of three-dimensional currents in the recirculation zone and the shear layer. Tuna and Rockwell[21]could attenuate free-surface oscillation by placing small submerged cylinder near the leading edge of the single cavity. They indicated that mean flow, turbulence structure and free-surface oscillation depend significantly on the relative height of the cylinder, despite the cylinder size being much smaller than the cavity. Flow instability could not be fully explained by a two-dimensional stability parameter defined by Chu and Babarutsi[22], and therefore, it was concluded that three-dimensional current patterns cannot be ignored. As there are various roughness elements such as above-mentioned submerged cylinder in natural river flow past side cavity, three-dimensional currents are expected to be induced within natural cavity zone. Although the flow exhibits threedimensionality induced by secondary currents in this way, our previous study proved that the spatial variation of horizontal circulation is comparatively small in the vertical direction[19]. Therefore, the present study discusses mainly two dimensional properties as practical use was given the first priority and the half depth yLLS/H=0.47was chosen as the representative measurement elevation yLLS.

It is difficult to illuminate the entire width of this flume, though the present setup is able to cover the whole area of the side cavity and a portion of the mainstream. An Ar-ion laser with a maximum of 5 W was used as the light source. The specific density and diameter of the tracer particles are 1.02 μm and 100 μm, respectively. The space and time distributions of the tracer pattern were taken by a high-speed CMOS camera situated over the free surface. The resolution is 0.5 mm/dot, and the frame rate, which corresponds to the time lag of the two images used for the correlation process to calculate the two-dimensional velocity components, is 100 Hz. The sample rate, which corresponds to the time interval of the obtained velocity data, is 30 Hz, as given by the function generator. The extent of photographed region is 0.53 m × 0.53 m, and that of the interrogation window is 15 mm × 15 mm. A Gaussian approximation was applied in the subpixel analysis.

2. Theoretical background

Van Proojen and Uijttewaal[1]have given theoretical profile of mean streamwise velocity in plane mixing layer through shallow momentum equation. They assumed following form by using a hyperbolic tangent curve.in whichUcis mean velocity in the mixing layer center. Information of three variables of the velocity differenceΔU(x), the mixing layer thicknessδ(x) and spanwise center position of mixing layeryc(x) are required to evaluate the velocity profile by using Eq.(1). They lead one-dimensional momentum equations of high-speed and low-speed layers assuming these layers are not influenced by the mixing layer, and obtained mathematical form of ΔU(x). A growth rate of the thickness in the steamwise direction was expected to be proportional to the velocity difference.

Therefore,δ(x)could be given by integration of Eq.(2).yc(x)could be calculated by the mass balance of high-speed layer between the inflow and the test sections. They verified that theoretical profile obtained in this manner agrees well with the measured data.

In contrast, it is harder to lead theoretically profiles of Reynolds stress. Reynolds stress appearing in the cavity opening is a statistically significant turbulence value, and it increases the flow resistance and promotes the formation of a mixing layer. However, a direct measurement of Reynolds stress is generally hard in natural rivers.

The present study focuses on transverse Reynolds stress −in the cavity opening, and proposes a practical model of calculating streamwise variation of Reynolds stress using the momentum equation and the mean streamwise velocity profile, which is comparably easy to obtain in natural environments.

Fig.2 Definition sketch for prediction model of Reynolds stress

Figure 2 defines momentum vectors related to a control volume ABCD situated in the horizontal boundary zone, referring to the derivation of the von Karman momentum integral. The sides AD and BC correspond to the open end of the cavity and the outer edge of the boundary layer, respectively.Umcis the mean velocity in the main channel.MABandMCDindicate the inflow and outflow momenta per unit time of sides AB and CD. They are given by

MsCDandMsABindicate the inflow and outflow masses per unit time of sides AB and CD. They are given by

When the pressure gradient ∂P/∂xand shear stress of side BC can be ignored, the primary forces acting on the control volume are the shear stress on side AD and gravity.

Thus, substitution of Eqs.(3), (4) and (5) in the momentum equation yields

As the present study targets developed turbulent flows, i.e.,τ/ρ≅−, the Reynolds stress profiles in the main channel and the cavity junction can be predicted given the spanwise distribution of the mean velocityU(z).

Fig.3 (Color online) Horizontal distributions of mean velocity vectors, in which contour means time-averaged streamwise velocity

3. Results and discussion

3.1 Mean circulation within side cavity

The horizontal distributions of mean velocity vectors for Um=0.12 m/sis shown in Fig.3, the contour in these corresponds to the value of the streamwise velocity. Large-scale circulations are observed in sizes comparable to the cavity area, irrespective of main channel streamwise velocity. Particularly, the circulation occupies the entire field of the cavity in the case ofLw/Bw=1. In contrast, a counter-rotating circulation is produced in the upstream side of the cavity whenLw/Bw=2, although the primary circulation occupies the majority of the dead water zone. A bulge inUsituated at the upstream inside corner also supports the generation of a small horizontal vortex. Uijttewaal et al.[6]previously noted the existence of this kind of secondary circulation, and they asserted it plays an essential role in local stagnation and sedimentation despite having a smaller length scale than that of the primary circulation. Both the primary gyre (PG) and the secondary gyre (SG) are formed in the case of Lw/Bw=3in the same fashion as with Lw/ Bw=2. The occupation area of the SG when Lw/ Bw=3is larger than that when Lw/Bw=2. The mainstream momentum flows in through downstream side of the junction and flows out through the midsection (x/ Bw=1.5)of the junction after making one circulation in the downstream area of the side cavity. This fact suggests that the primary circulation is induced by intensive inflow of the mainstream, and furthermore, the width of the side cavity has a significant impact on the mean flow structure, e.g., the production of the SG, the length scale of the horizontal gyre, and the position of the outflow on the cavity opening toward the mainstream.

Incidentally, there are many studies on 2D-driven cavity flow without free-surface in the mechanical engineering, aeronautical engineering and chemical sciences and so on. The experiment conducted by Haigermoser[23]which is similar to the present hydraulic case for Lw/Bw=3could be compared with. Reynolds number using the cavity length is 7 800 which is lower than that of the present study (36 000). The both of them have similar horizontal currents structure. In particular, the secondary gyre could be commonly observed in the leading side of the cavity. A streamwise position of the PG core isx/ Bw=2.4 and 2.3 for the Haigermoser’s and our results, respectively. That of SG core appears inx/ Bw=1.1 and 0.4 for the Haigermoser’s and our results, respectively. Thus, core position of PG in the 2D-driven cavity matches that in the present study, although there is a little gap for the SG core position.

Figure 4 shows the longitudinal profile of the mean spanwise velocityW normalized by Uconsidering its dependence on the aspect ratio. The distri-bution of the outflow (W＞0)is found to peak in the midsection for the cases ofLw/Bw=2 and 3. In contrast, a well-defined peak is not observed in the midsection in the case ofLw/Bw=1, where local variation is much smaller thanLw/Bw=2 and 3. The maximum position shifts downstream ofLw/Bw=3compared to that ofLw/Bw=2, corresponding to development of secondary gyre as shown in Fig.3. Anyway, the outflow is prominent around boundary between the primary gyre and the secondary one. Because the present theory ignores spanwise momentum across the control volume, whenW/Uis larger, the prediction accuracy significantly decreases. Our theory is applicable to hydraulic cases with a small ratio of the spanwise velocity to the mainstream. The results suggest that absolute value ofWfalls in the range smaller than 10% ofU.

Fig.4 Longitudinal profiles of ratio of spanwise mean velocity normalized by streamwise one along the cavity opening

Fig.5 Longitudinal profiles of product of spanwise mean velocity to streamwise one along the cavity opening

Figure 5 shows the momentum transfer that was contributed by the spanwise velocityUWin the cavity opening. All cases have a local negative peak at approximatelyx/Bw=0.8 to 0.9, which implies momentum transport within the mainstream toward the cavity. Furthermore, positive distributions observed in the rage ofx/Bw=0.4 to 0.6 for the cases ofLw/Bw=2 and 3 correspond to the reverse transport of the momentum toward the main channel.

Fig.6 (Color online) Horizontal distributions of Reynolds stress

3.2Distribution of Reynolds stress and comparison to predicted values

The horizontal distributions of the Reynolds stress −is drawn in Fig.6. These results suggest that large Reynolds stress values are locally observed in the boundary zone, i.e., the cavity opening. Although the mixing layer developed downstream for all cases, the peak value is the smallest whenLw/Bw=1. The spanwise scale of the cavityBwmay influence the formation of the mixing layer. The outflow toward the main channel increases as the aspect ratioLw/Bwincreases, as mentioned above, and this implies that such spanwise velocity is closely related to the development of the mixing layer and turbulence. The V-shaped distribution is also formed in 2D-driven cavity measured by Haigermoser[23]. The distance of maximum Reynolds stress in the boundary of mainstream/cavity isx/Bw=0.67 and 0.60 for the 2D-driven and the present free-surface cavities, respectively. Therefore, results of present study can be applicable to 2D-driven cavity phenomena. Haigermoser[23]varied the cavity length under the certain cavity width, and pointed out that maximum Reynolds stress along the mixing layer becomes larger because the shear instability has more time and space to develop in the case of larger aspect ratio corresponding to longer cavity. The present study also has the same feature, that is to say, maximum Reynolds stress increases with increase of the aspect ratio. It is well known that flow pasts a cavity has two significant frequency mode[24]. One is a shear layer mode induced by the K-H instability and another is a wake mode induced by interaction between the separated vortices at the leading edge and recirculating eddy within the cavity. Particularly, the wake-mode becomes remarkable in the larger aspect ratio and it results in the intensive production of the Reynolds stress along the junction between the main-channel and the cavity zone. Bian et al.[24]compared several power spectrum of velocity fluctuation in the cavity shear layer and suggested that the spectra peaks move to slightly lower frequencies with increasing downstream distance. This fact implies instable shedding vortices at leading edge grow up while convection downstream. Furthermore, increase of vortex scale induces intensively spanwise spread of mixing layer where significant momentum transfer could be observed between the mainstream and the cavity. It results in the V-shaped distribution of −. The impinging at the trading edge influences the initial condition of the mixing layer at the leading edge[25], and this unique feedback mechanism peculiar to the cavity flow is quite different from that observed in finite-length mixing layer.

Tuna and Rockwell[26]focused on properties of velocity field and free-surface oscillation in flows past two consecutive cavities. Their result suggested that the peaks of turbulence intensity and Reynolds stress in the mixing-layer appear in downstream-side within a leading cavity. Whereas, the peaks within a trailing cavity shift upstream-side and become large. They explained that strong unsteady motions are induced by the impinging of the mixing-layer in the downstreamedge of the leading cavity. The interaction of the impinging and the mixing-layer produces unsteady initial condition of the developing mixing-layer in the trailing cavity, and it results in the enhancement of the turbulence structure such as Reynolds stress of the trailing cavity.

The present study considers turbulence structure of the mixing-layer using statistics such as Reynolds stress. In contrast, a frequency characteristics is also one of significant factors to govern the mixing-layer development in flow past the cavity. Meile et al.[16]examined generating condition of free-surface oscillation, and indicated the transverse oscillation is controlled by not only the aspect ratio but also the Strouhal number. In particular, it was found that there exist a critical Strouhal number together with the maximum amplitude. Furthermore, the oscillation was found to be insignificant in larger aspect ratio cases where reattachment point appears within the cavity.

Fig.7 Examples of fitting curve for lateral profile of mean streamwise velocity (fitting-curve Eq.(9))

The present theory of Eq.(8) could be utilized together with the fitting curve obtained from the measured profile of the streamwise velocity and numerical integration. The equation used for this process is

whereδ(x)andUmc(x)are the mixing layer width and streamwise velocity of the mainstream, respectively. They are calculated such that Eq.(9) best fits the measured data. This operation is relatively reasonable, because the coefficients of determinationR2are 0.989, 0.990, and 0.978 atx/Lw=0.14, 0.70, and 0.98, respectively for the case ofLw/Bw=1. Figure 7 shows two fitting examples atx/Lw=0.17 and 0.87 corresponding to near the leading and trailing edges, respectively, whenLw/Bw=1, which suggests the validity of Eq.(9).

The longitudinal profile of the mixing layer widthδgiven by Eq.(9) is shown in Fig.8. It is found that the width is not dependent on the aspect ratio on the upstream side of the cavity; however, the value ofδincreases asLw/Bwbecomes larger on the downstream side. The formation of the mixing layer may be promoted by the outflow of low momentum from inside the cavity toward the mainstream observed in the midsection for the cases ofLw/Bw=2 and 3.

Fig.8 Longitudinal profiles of mixing layer width evaluated by fitting operation

Figure 9 indicates the Reynolds stress predicted by the proposed method, and the measured data by PIV is also included in this figure for comparison. The Reynolds stress tends to increase for larger aspect ratios over the whole region. The profile of the Reynolds stress corresponds with the development of the mixing layer. It should be noted that the width of the mixing layer continues developing within the whole region in the junction, and in contrast, the variation in the Reynolds stress is almost constant in the midsection. After then it decreases slightly downstream. In particular, such transition positions depend on the aspect ratio and arex/Lw=0.4, 0.3, and 0.5 in the cases ofLw/Bw=1, 2, and 3, respectively. This is because an increase in shear stress inducing turbulence production is not expected due to the small velocity gradient∂U/∂zin the midsection of the cavity opening.

Fig.9 Comparison of measured and calculated Reynolds stress along the cavity opening

The maximum position of transverse shear stress measured by Yossef and De Vriend[9]is included in the results ofLw/Bw=2. Their experiment was conducted under the condition ofLw/Bw=2.25, in which both of PG and SG were formed inside the embayment. The Reynolds number is 60 000 and measurement elevation isy=0.2H. The shear stress has maxima aroundx/L=0.35and this position corresponds_to the outflow from boundary between the PG and the SG. This tendency is almost consistent with our results forLw/Bw=2. They suggested that the shear stress diffuse in the lateral direction with increase of streamwise travel distance in the same manner asshown in Fig.6. Their flume size is 10 times larger than ours and thus, this implies these hydrodynamic properties are expected to have similarity in scale.

There is no tangible correlation between the profile ofUW, shown in Fig.5, and the gap between the predicted and measured data, as shown in Fig.9. It is therefore concluded that the momentum transport by the spanwise velocity does not have a large impact on the present prediction of Reynolds stress. The predicted and measured results have similar tendencies, irrespective of the aspect ratio, and relative errors between the measured and predicted results are 17.2 %, 16.0%, 13.7%, for case ofLw/Bw=1.0, 2.0 and 3.0, respectively.

Increasing discharge in the main-channels, the existence of spanwise flow cannot be ignored in the interface between the mainstream and the embayment zone, and it is an error source in evaluation of mass balance within a control volume. Furthermore, vertical component is also induced, and it results in formation of secondary current. McCoy et al.[27]suggested based on LES results that the shallow embayment flow has a strong two-dimensional property by the power-spectrum of the spanwise velocity component in which−3 subrange is observed related to a quasi two-dimensional flow. In contrast, predominant three-dimensional flows accompanied by secondary currents are confirmed in the cavity opening. Secondary currents cause depth profile of streamwise velocity and vertical shear stress. The present model presupposes a kind of shallow depth assumption, and hence, cannot consider the momentum consumption by vertical shear friction.

3.3Calculation of mean kinetic energy

The primary gyre inside the cavity is found to be induced by the momentum transfer from the mainstream. By changing perspective, we highlight contribution of mean kinetic energy to production of the cavity gyre. It is very important to understand how the mainstream supplies mean kinetic energy to the largescale horizontal circulation within the cavity. The mean kinetic energy of the circulation is defined as

whereAis the horizontal area of the dead water zone andEis the mean kinetic energy at any given position. Assuming that the vertical velocity component is negligible due to horizontal two-dimensional flow,Eis given by

Fig.10 (Color online) Horizontal distributions of mean kinetic energy in the cavity zone

3.4Energy budget of mean kinetic energy

The energy production of the horizontal circulation is influenced by not only the energy flux between the mainstream and the cavity but also diffusion and dissipation. The mean kinetic energy equation (MKE) is very useful for quantitatively evaluating the energy budget. The MKE in two-dimensional horizontal flows is given by

whereGr,Gt,Dandφare gravity, generation of turbulence, diffusion, and energy dissipation, respectively. They are defined as

in whichθis bed slope of the experimental flume.

The diffusion term is composed of three terms related to viscosity, turbulence, and pressure, as given on the right-hand side of Eq.(15). The viscous diffusion is often negligible in fully developed turbulent flows, as in the hydraulic conditions observed in this study. Furthermore, the pressure diffusion is also ignored due to the static pressure assumption. Consequently, the total diffusion termDis considered to be equal to the turbulence diffusionDt=−Ui, which is indicated by the second term on the right-hand side of Eq.(15). The dissipation termφis generally much smaller than other terms on the right-hand side of Eq.(12).

The convection term which is the first term of right-hand-side of Eq.(12) could be decomposed into streamwise and spanwise componentsCxandCzin the following way.

Figure 11 shows the lateral profile of the convection termsCxandCz, the gravity termGr,the turbulence generation termGt, and the turbulence diffusion termDtat the longitudinal sections ofx/Lw=0.2, 0.4, 0.6, and 0.8. In this figure, all results are normalized by the bulk mean velocityUm. All terms were found to have noticeable local variation in the junction between the mainstream and the side cavity. The streamwise convectionCxis negative at the junction (z=0). This means that the energy being transferred downstream is larger than that supplied from upstream.

In contrast,Czis negative in the region ofx/Lw≤0.6and is positive whenx/Lw=0.8. The turbulence generationGtis negative at the junction, and this tendency is better defined in the range ofx/Lw≥0.6. This finding implies that a part of the mean kinetic energy is consumed for turbulence production.Dtis found to be positive at the junction, which means that turbulence contributes significantly to energy transportation. The termsGtandDt, which are closely related to the turbulence structure, have the same order as the conventional terms, and therefore, turbulence and accompanying vortices are of essential importance when the transport and generation of the mean kinetic energy are investigated on a small scale.

3.5Examination of dependence on main channel width

The aspect ratio of the water depth to the main channel width used in most hydraulic experiments is not sufficiently large compared to the natural rivers owing to the space limitation of the laboratory room. The narrow width may give rise to some troublesome problems, e.g. the generation of secondary currents and the sidewall effect. To solve this problem, this study used a glass flume with a relatively large width of 1.5 m and a length of 9.5 m, while our research group has been continued to study turbulence dynamics primarily in side-cavity flows using 0.4 m-wide glass flumes[19]. The 40cm-wide flume has the advantage of obtaining accurate measurements with optical and laser instruments using methods, such as laser Doppler anemometry (LDA) and PIV.

Fig.12 (Color online) Horizontal distributions of mean velocity vectors for flumes with widths of 0.4 m, 1.5 m (Lw/Bw=3)

This section compares the horizontal distributions of the mean velocity and Reynolds stress measured in flumes with widths of 0.4 m and 1.5 m. The width ratios of the main channel to the cavityBm/Bware 7 and 12 for the 0.4 m and 1.5 m flumes, respectively, meaning the width ratio of the wider flume is 1.7 times larger than that of the narrower one.

Figures 12 to 14 show the comparisons of the mean velocity vectors(U,W), the spanwise velocityW, and the Reynolds stress −uw, respectively. The streamwise length of the cavity opening isLw= 0.15 m and 0.3 m. The hydraulic conditions are quite similar in the two types of flumes: the bulk mean velocityUm=0.12 m/s, the water depthH=0.053m, the aspect ratioLw/Bw=3, and the measured elevationy/H=0.47.

Fig.13 (Color online) Horizontal distributions of spanwise mean velocity for flumes with widths of 0.4 m, 1.5 m (Lw/Bw=3)

4. Conclusions

Turbulence measurements were conducted in an open channel flow with a side cavity zone to investigate the circulation mechanism and the related transport phenomenon. In addition, a practical method to predict the Reynolds stress distribution was reasonably developed.

The horizontal distributions of the velocity vecto- rs suggest that the aspect ratio of the cavity greatly influences circulation formation. A single large-scale circulation appears in the case ofLw/Bw=1, which is comparable to the length scale of the cavity, whereas there are twin vortices, i.e., the primary gyre genera- ted by the mainstream and the secondary gyre induced by the primary gyre, in the case of larger aspect ratios,Lw/Bw=2 and 3. This tendency agrees well with pre- vious data[6,7,9,19].

Larger Reynolds stress values in the junction between the mainstream and the cavity are distributed as the aspect ratio increases. The mixing layer width calculated using the spanwise profile of the streamwise velocity tends to increase over the whole length of the cavity opening, whereas the Reynolds stress remains constant in the mid-section. This is because the mixing of the momentum, which makes the spanwise gradient of the streamwise velocity smaller, causes the prevention of turbulence production.

This study developed and examined a method of predicting the horizontal Reynolds stress in the junction using the mean streamwise velocity. The longitudinal profiles of the present prediction and the measured data were in good agreement. The present examination is under the limited condition that the spanwise velocity is small compared with the streamwise velocity in the present hydraulic condition. It is inferred that the prediction accuracy may decrease when the effect of the spanwise momentum transfer cannot be ignored, and thus, further studies are necessary to determine the applicability of this method in the next research stage.

Horizontal PIV measurements were conducted in the same hydraulic conditions for laboratory flumes with different length scales to examine the influence of the mainstream width on the spanwise mixing phenomena between the main channel and the side cavity. Comparisons of the mean horizontal velocity components and the Reynolds stress show that the dependency of these characteristics on spanwise length of the mainstream could not be clearly observed.

Acknowledgements

The authors would like very much to thank editors of Journal of hydrodynamics, and anonymous reviewers for very valuable and encouraging comments. The present experiments were carried out under the support from the graduate school students, Mr. Mitsuyoshi Ikeuchi and Mr. Taikou-Paul Kaneko. The authors gratefully acknowledge this support.

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* Biography:Michio Sanjou (1975-), Male, Ph. D., Associate Professor