Water Resources Research

Vegetated mixing layer around a finite-size patch of submerged plants: Part 2. Turbulence statistics and structures

Authors

  • Alexander N. Sukhodolov,

    Corresponding author
    1. Department of Ecohydrology, Institute of Freshwater Ecology and Inland Fisheries,Berlin,Germany
      Corresponding author: A. N. Sukhodolov, Department of Ecohydrology, Institute of Freshwater Ecology and Inland Fisheries, Müggelseedamm 310, Berlin DE-12587, Germany. (alex@igb-berlin.de)
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  • Tatiana A. Sukhodolova

    1. Department of Ecohydrology, Institute of Freshwater Ecology and Inland Fisheries,Berlin,Germany
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Corresponding author: A. N. Sukhodolov, Department of Ecohydrology, Institute of Freshwater Ecology and Inland Fisheries, Müggelseedamm 310, Berlin DE-12587, Germany. (alex@igb-berlin.de)

Abstract

[1] Dynamics of vegetated mixing layer displays analogy with canonical and shallow mixing layers. Universality of turbulence and minor influence of flexible aquatic plants on the individual turbulent eddies are anticipated as the prerequisites for the analogy. This paper explores theoretically and experimentally some aspects of analogy by examining spatial patterns of turbulence characteristics around a finite-size patch of submerged flexible vegetation the population density of which was varied between the experimental runs [Sukhodolova and Sukhodolov, 2012]. It introduces analysis of turbulent shear stress, kinetic energy, dissipation rate, and periodicity of organized motions inside the vegetated mixing layer. The theory is compared with the field experiments. The results of the study indicate that the flow structure around the patch with dense population agrees well with the proposed theoretical scaling relations and reveals many aspects of the analogy to hydrodynamic mixing layers. Around the patch with sparse vegetation the flow structure is best represented by a perturbed boundary layer model. These conclusions on the dynamics of the flow are supported by the analysis of scaled turbulence characteristics. The patterns of turbulence profiles are approximated by scaling functions fairly well while the values of scaling parameters differ from the values reported for canonical and shallow mixing layers. This is an important result which agrees with the previous reports of laboratory research with model vegetation and is explained by a combined effect of internal shear and friction at the top of the canopy.

1. Introduction

[2] Aquatic vegetation is ubiquitous in shallow fluvial systems and it often forms patchy mosaics that mutually interact with turbulent flow at various scales ranging from an individual plant scale to a scale of a river reach [Carr et al., 1997; Marba and Duarte, 1998; Nikora, 2010; Sukhodolov and Sukhodolova, 2010]. The interactions at the scale of a finite-size patch of vegetation (intermediate scale), which are of importance for in-stream retention processes and flow resistance in vegetated channel, alter the vertical flow structure along and downstream of the patches. The models analogous to the models of canonical mixing layer [Monin and Yaglom, 1971; Tennekes and Lumley, 1972; Pope, 2000] were shown to describe well the vertical structure of turbulent flow with uniformly distributed submerged aquatic vegetation [Ghisalberti and Nepf, 2002]. The analogy between canonical hydrodynamic mixing layer and a vegetated mixing layer developing at the leading edge of a finite-size patch of vegetation was the focus of companion paper [Sukhodolova and Sukhodolov, 2012]. The companion paper examines the characteristics of the mean flow using a combination of theoretical and field experimental approaches which provided relationships between characteristics of vegetation and principal variables of the mixing layer. This paper introduces analysis of scaling relationships for turbulence characteristics and reports on the application of the analysis to the results of field experiments already presented in the companion paper.

[3] In turbulent flows with flexible submerged vegetation, turbulent eddies populating the mixing layer significantly affect the momentum transfer into the vegetation canopy [Ghisalberti and Nepf, 2002]. Propagation of turbulent eddies over the top of the canopy results in quasiperiodic fluctuations in the drag force causing swaying of the plants called monami motions [Okamoto and Nezu, 2009]. Laboratory studies show that turbulent eddies, that are also known as organized motions or coherent structures, are the reason for appearance of monami and that certain velocity thresholds have to be triggered to initiate the monami [Ghisalberti and Nepf, 2002]. The consequence of monami is the increased bending of individual plants in the patch and consequently an increase in the distance that fluid from the ambient layer is able to penetrate into the vegetation layer [Okamoto and Nezu, 2009]. Most probably that characteristics of monami can be accounted by considering the momentary balance between physical forces imposed by coherent structures and reaction forces from plants tissue. This complex task is composed of two parts: Assessment and characterization of physical forces imposed by turbulent flow, and quantitative description of reaction forces due to flexural rigidity and buoyancy of the plant's tissue. This paper examines the structure of turbulence in vegetated mixing layers and respectively contributes to the assessment of physical forces induced by turbulent flows.

[4] Although turbulence is habitually regarded as a random phenomenon, application of turbulence models in hydrodynamics implies a certain degree of determinedness in seemingly chaotic turbulent signals [Monin and Yaglom, 1971]. Records of fluctuating velocity and pressure are conventionally characterized by the statistical moments and correlations of their probability density functions. These moments are related in a deterministic way to the structural properties of the flow described by specific hydrodynamic models; for example, by mixing layer model [Monin and Yaglom, 1971; Tennekes and Lumley, 1972; Pope, 2000; Ghisalberti and Nepf, 2002; Sukhodolov et al., 2010; Lacy and Wyllie-Echeverria, 2011]. Hitherto the analogy between canonical mixing layers and vegetated mixing layers was examined mainly by exploring the similarity of shape in profiles of turbulence characteristics and we hypothesize that the analogy can be further understood by investigating relations between the principal parameters of the mixing layer model and characteristics of vegetation.

[5] The concept of coherent structures introduces spatially coherent, temporally evolving vortical motions possessing distinctive correspondence between components of the velocity vector and pressure that distinguishes the structure from the surrounding incoherent liquid [Cantwell, 1981; Lumley, 1981; Nezu and Nakagawa, 1993; Schoppa and Hussain, 2000; Jirka, 2001; Constantinescu et al., 2011]. Coherent vortices evolving in the canonical mixing layers are shown to develop through complex interactions periodically appearing along the mixing layers [Roshko, 1992; Rogers and Moser, 1992; Winant and Browand, 1974]. Coherent structures in canopy edge flow were recently examined for atmospheric flows with forest vegetation using large-eddy simulation (LES) technique [Dupoint and Brunet, 2009]. Their study revealed the strong similarities in the development of coherent structures between averaged characteristics of vegetated and canonical hydrodynamic mixing layers. Besides, as it was demonstrated by earlier laboratory investigations [Poggi et al., 2004] and confirmed by recent LES studies [Dupoint and Brunet, 2009; Huang et al. 2009], vegetation properties and population density are important factors controlling the dynamics of coherent structures. Therefore the search of relationship between statistical moments of turbulence, organized motions or coherent structures, and characteristics of vegetation is an appealing task which can further clarify the dynamics of complex vegetated flows.

[6] The goals of this paper are (1) to explore semitheoretical scaling relationships for the principal hydrodynamic variables controlling the dynamics of mixing layers; (2) to introduce turbulence characteristics obtained as a result of field experimental studies and to analyze the dynamics of vegetated mixing layers by comparing field experimental data with semitheoretical scaling relations; and (3) to examine the effect of population density on the dynamics of coherent structures in a vegetated mixing layer. This study explores the hypothesis on local equilibrium between production and dissipation of turbulent kinetic energy which stabilizes vegetated mixing layer and allows for deduction of scaling relations for coherent structures.

2. Theoretical Framework

[7] The theory of hydrodynamic mixing layers which is a core element of the theoretical framework in this study is introduced in details in the companion paper [Sukhodolova and Sukhodolov, 2012]. Therefore most of the theory is omitted in this paper and only minor repetitions are made to avoid cross referencing that would impede readability.

2.1. Theoretical Backgrounds

[8] Previous studies report that the profile of mean streamwise velocity u in canonical, shallow, and vegetated mixing layers are well described by a hyperbolic tangent function [Ghisalberti and Nepf, 2002; Sukhodolov and Sukhodolova, 2012]

display math

where u and inline image are the streamwise velocity and the convective velocity, respectively, z is the cross-stream coordinate, inline image is the coordinate of the mixing layer center, inline image is the velocity differential, and δis the mixing layer width. Although a hyperbolic tangent profile is evident on the time-averaged patterns, it never exists at any given instant in the instantaneous flow patterns. The instant interface between merging flows appears to be sharp and corrugated due to the presence of large-scale vortex structures [Ho and Huerre, 1984]. Initially, close to the origin, the downstream development of the mixing layer is dominated by a linear instability mechanism. Vorticity distribution in parallel flows of different velocities has a distinctive maximum and hence is inviscidly unstable to small perturbations via the Kelvin-Helmholtz instabilities which grow exponentially with distance and roll up into vortices [Ho and Huerre, 1984]. The characteristic period inline image of instabilities and vortices in the mixing layer is related to bulk flow characteristics by the Strouhal number

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where inline image is momentum thickness, and inline image is frequency. The linear stability analysis completed using (1) provided the critical value of the Strouhal number S = 0.032 which corresponds to a natural frequency of the unforced mixing layers between parallel flows and was validated by experimental studies [Ho and Huerre, 1984; Ghisalberti and Nepf, 2002].

[9] Recent laboratory studies of coherent structures in the shear layers developing over riverbeds with flexible submerged vegetation confirmed applicability of equation (2) with the nominal Strouhal number 0.032 for predicting the frequency of characteristic peaks in turbulence spectra [Ghisalberti and Nepf, 2002]. However, stability analysis [Ho and Huerre, 1984] has also revealed uncertainty in determination of the nominal frequency. Despite the fact that nominal frequency appeared to be independent of velocity differential in equation (2), in nonparallel flows and forced mixing layers its value is shown to deviate significantly (S = 0.076) from the nominal [Ho and Huerre, 1984]. In vegetated mixing layers characteristics of turbulence, and scales of structures (Strouhal number) depend on the characteristics of the approaching flow, plants, and population density of a patch [Dupoint and Brunet, 2009; Huang et al. 2009; Okamoto and Nezu, 2009]. This dependency is explored applying theoretical considerations in the following subsections.

2.2. Scaling Coherent Structures

[10] Vortex revolution period te and radius of a vortex rare related by the Richardson-Obukhov scaling law [Monin and Yaglom, 1974]

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where inline image is the dissipation rate. This phenomenological model can be specified for a case of a vegetated mixing layer assuming that the radius of vortex in the layer is proportional to the layer width inline image and further using a hypothesis of local equilibrium between mean turbulence production inline image and dissipation

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where inline image is the shear stress at the centerline of the mixing layer, and ρ is the density of water. The validity of this hypothesis is supported by the fact that vegetated mixing layers attain stabilization [Ghisalberti and Nepf, 2002; Sukhodolov and Sukhodolova, 2012]. Turbulent shear stress at the centerline of the layer is related to the convective velocity in the layer as

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where inline image is friction factor. Transforming (3) by substitution of (4) and (5) yields

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which together with frozen turbulence approximation inline image converts to

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Equation (7) explicitly relates characteristics of vegetation represented by the bulk friction factor (5) and integral characteristics of the flow around the vegetated mixing layer. Moreover, it can be shown that other variables in (7) can be expressed as the functions of characteristics of population density of vegetation and hydrodynamic characteristics of vegetated mixing layer [Sukhodolova and Sukhodolov, 2012].

2.3. Scaling Turbulence Statistics

[11] In this section theoretical analysis is focused on the most important characteristics of turbulence which are required for theoretical studies and numerical computations. These characteristics are turbulent shear stress, dissipation rate, and turbulent kinetic energy. Analytically these turbulence characteristics can be assessed using the turbulence viscosity model

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where inline image and inline image are turbulent fluctuations of streamwise and vertical velocity, respectively, inline image is shear stress, and inline image is turbulent viscosity. Applicability of the turbulence viscosity model with a constant inline image is already confirmed experimentally and theoretically for canonical and shallow mixing layers [Rodi, 1980; Pope, 2000; Sukhodolov et al., 2010]. Besides, past research has shown that the eddy viscosity can be presented as the multiplicative product of mixing layer width, as characteristic length scale, and velocity differential as characteristic velocity scale inline image, where γ is the empirical coefficient which value equals 0.01 [Rodi, 1980; Pope, 2000; Sukhodolov et al., 2010; Sukhodolov and Sukhodolova, 2010].

[12] A scaling law for turbulent shear stress is readily obtained from (8) by specifying the velocity gradient with a derivative of hyperbolic tangent function (1)

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Substituting (9) into (8) and performing simple transformations yields

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The scaling law (10) together with a fact that hypothesis of local equilibrium between turbulence production and dissipation holds for the stabilized mixing layers provides the way for deriving an analytical relation for the dissipation rate. Substituting (9) and (10) into inline image and performing transformations, one can obtain the scaling relation for the dissipation rate

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[13] The scaling law for turbulent kinetic energy can be obtained using the k-ε model of turbulence in which turbulent viscosity is related to turbulent kinetic energy and dissipation as [Rodi, 1980]

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where inline image is the empirical coefficient equal to 0.09. Substituting (11) into (12) and performing simple transformation finally yields

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It is noteworthy that relationship (13) is very similar to (10) because turbulent kinetic energy is an integral characteristic of the diagonal components of the turbulent stress tensor. The magnitude of kinetic energy appears to be about three times larger than for shear stress component inline image which is also anticipated as a characteristic ratio [Monin and Yaglom, 1971].

[14] Theoretical analysis of this section introduces analytical functions (7), (10), (11), and (13) which predict the shape of profiles of turbulence characteristics. Although the magnitudes of the functions are explicitly related to the characteristics of vegetation in the patch via velocity differential as a function of downstream distance from the leading edge of a patch [Sukhodolova and Sukhodolov, 2012], there is still a need for evaluation of parameters γ and inline image from the field experimental data.

3. Methods

3.1. Experimental Data Set

[15] Field experiments for this study were completed in a lowland river Spree [Sukhodolov and Uijttewaal, 2010; Sukhodolova and Sukhodolov, 2012]. In a straight reach of the river the aquatic plants of Sagittaria sagittifoliawith filamentous flexible blades were implanted in the central part of the river to form an experimental patch. A series of experimental runs was completed in which three-dimensional velocities were measured on vertical profiles distributed along the centerline of the patch. Between the experimental runs the population density of vegetation in the patch was reduced by pruning. Instantaneous velocity vectors were recorded by a probe located in a stationary position near the origin of the patch and by two probes traversed along the centerline of the patch. All three ADV probes were sampling synchronously at a sampling rate of 25 Hz. Details of the hydraulic and morphologic characteristics of the site and experimental runs, description of measurements procedures, data post-processing, and analysis of mean velocity fields are provided in detail bySukhodolova and Sukhodolov [2012].

[16] Flow velocities were recorded for 300–400 s at each sampling point providing the series sufficiently long for unambiguous characterization of organized motions. Examples of typical velocity records sampled in the ambient flow above the vegetation patch and inside the mixing layer are shown in Figure 1. Inside the vegetated mixing layer the velocity records exhibit quasiperiodic signals with a distinctive behavior of the velocity vector components—the streamwise velocity component always appeared in the opposite phase to the vertical one (Figure 1b). The sampling period was long enough to capture 15 to 20 structures.

Figure 1.

Flow velocity records measured in the ambient flow (a) above the plants, and (b) inside the vegetated mixing layer (run 1, 9.1 m, point 7 and 3, respectively).

[17] Application of underwater video-system RAY provided digital records of plants in the patch synchronized with the velocity measurements in the centerline of the shear layer. Digital post-processing of the obtained video records provided quantitative assessment of plant motions. Typical synchronized velocity record and a digitized sequence of trajectories of selected swaying plant are illustrated inFigure 2. The decelerated streamwise velocities inline image coincide with the positive vertical velocities ( inline image; Figure 2b), and cause uplifting of the plant blades (Figure 2c). These organized motions are analogous to ejection events in the classical horseshoe vortex model [Nezu and Nakagawa, 1993]. Another type of interactions with accelerated streamwise velocities inline image and decelerated vertical component inline image, analogous to sweep events, result in downward motion of the plant's blades and bending of the stems. The amplitude of swaying of the plants as a result of these dynamical interactions on average was about 20 cm that corresponds to 1/3 of the deflected patch height under these experimental conditions.

Figure 2.

Flow velocity record measured in the (a) centerline of the shear layer, (b) enlarged part of the record, and (c) the trajectories of an individual plant recorded by underwater video system synchronously with velocity measurements, run 1.

3.2. Auto- and Cross-Spectral Analysis

[18] Multipoint synchronous velocity measurements employed in this study allow examination of structural properties with the statistical methods of analysis of random processes. Turbulent signal is a result of certain motions that pass the point of observation as flow evolves in space and time. For a stationary process it is possible to define a multitime statistic called autocovariance inline image and, respectively, autocovariance function as [Jenkins and Watts, 1968]

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where T is the sampling period, and τ is the time lag. Normalizing autocovariance function with the variance of random variable yields an autocorrelation function of the process inline image with the following properties: inline image, inline image.

[19] Records of instantaneous flow velocities can be approximated by Fourier series, and the variance of random variable then is the sum of contributing Fourier modes. Respectively, the variance can be decomposed into spectrum of Fourier modes inline image by applying Fourier transformation to autocovariance function

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For isotropic turbulence inline image, turbulence spectrum is related by Kolmogorov's theoretical relationship [Kolmogorov, 1942; Monin and Yaglom, 1971]

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where inline image is the universal constant. Although according to previous studies of river turbulence [Sukhodolov et al., 1998], the condition inline image approximately holds for the range from 0.6 to 5 Hz and the spectral density of the streamwise component appears to be about 10 times larger than the spectral density of the vertical component in the range from 0.01 to 0.6 Hz. Therefore river turbulence is always referenced as locally isotropic [Sukhodolov and Uijttewaal, 2010]. Presence of quasiperiodic motions because of the passage of coherent structures in the shear layer is pronounced as the distinctive peak in the spectral densities [Ghisalberti and Nepf, 2002]. This peak is located at lower frequencies than those of the inertial subrange [Sukhodolov et al., 2010].

[20] Spatial distributions of flow variables represented by a random processes inline imageform the fields of random variables. Understanding of structural properties requires analysis of statistical properties for these fields and can be achieved by the analysis of multipoint cross-covariance functions [Jenkins and Watts, 1968]

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which can be normalized by a pair of standard deviations (root squares of variances) of the variable at each point inline image and inline imageto obtain a cross-correlation function. The examples of cross-correlation maps characteristic for the experimental site of this study (without vegetation) are reported bySukhodolov et al. [2011] while turbulence spectra with expanded low frequency range are presented by Sukhodolov and Uijttewaal [2010].

4. Results and Analysis

4.1. Turbulence Statistics

[21] Measured patterns of shear stress inline image are presented in the companion paper [Sukhodolova and Sukhodolov, 2012]. In this subsection other important turbulence statistics, dissipation rate and turbulent kinetic energy, are presented together with the comparative analysis of experimental data and scaling laws (10)(13) introduced in section 2.3.

[22] The spectral densities (periodograms) were computed using equations (14) and (15) and smoothed with a Tukey's spectral window [Jenkins and Watts, 1968]. The values of dissipation rate are determined by fitting equation (16) to spectra in the inertial subrange. The patterns of dissipation rate are shown in Figure 3. These patterns clearly illustrate the effect of population density of vegetation on the properties of turbulent flow. Near the origin of the vegetation patch the profiles of dissipation rates are similar to the profiles of dissipation rate measured in this river reach without vegetation [Sukhodolov et al., 1998]. They possess a characteristic shape which is described by a semitheoretical solution proposed by Nezu and Nakagawa [1993]. When the population density of vegetation is reduced, a larger region downstream the patch leading edge is exhibiting this characteristic shape of the profile (Figure 3b). The maxima of dissipation rate in the near field ( inline image m, run 1, and inline image m, run 2) are located near the lower boundary of the mixing layer. In the far field from the origin the maximum of dissipation rate as well as the pattern of the profile vary insignificantly and indicate a steady condition with respect to spatial evolution of the dissipation rate.

Figure 3.

Vertical profiles of measured values of dissipation rate as a function of position in the experimental patch of vegetation: (a) run 1, (b) run 2, (c) run 3, and (d) run 4, dashed lines indicate the boundaries of flow layers.

[23] Spatial distributions of turbulent kinetic energy are also indicative of the distinctions in spatial dynamics of mixing and boundary layers, as well as in the manifestation of population density effect on the structure of flow (Figure 4). Turbulent kinetic energy increases along the vegetation patch and is maximal in the far downstream. However, the maxima of kinetic energy are located closer to the riverbed near the origin of the vegetation patch, and, similarly to shear stress component inline image, their loci coincide with the upper boundary of the vegetation in the far downstream. The changes in magnitude of the energy reflect dependency on the population density of vegetation, while the pattern of the distribution attains similarity with the boundary layer without vegetation.

Figure 4.

Vertical profiles of measured turbulent kinetic energy as a function of position in the experimental patch of vegetation: (a) run 1, (b) run 2, (c) run 3, and (d) run 4, dashed lines indicate the boundaries of flow layers.

[24] Analytical scaling laws introduced in section 2.3 provide the theoretical framework in which the measured data can be analyzed in the context of anticipated model by applying the similarity principle. The models of vegetated mixing layer and boundary layer provide the two limiting cases of flow between which the actual flow can alternate depending on the population density of vegetation. Equation (10) provides the scaling law for generalization of measured vertical profiles of shear stress. The distance from the river bed for the measured data is scaled the same way as for mean velocities [Sukhodolova and Sukhodolov, 2012], and measured values of the shear stress are normalized by density of water and the square of velocity differential (Figure 5). In the vegetation canopy with high population density the normalized measured data agree well with the function (10). At locations with distance x > 3 the collapse of points around the same theoretical curve indicates that constant turbulence viscosity is a good approximation for data in run 1. However, the value of coefficient γ which provides the best fit is two times larger than the value 0.01 reported for canonical and shallow mixing layers [Rodi, 1980; Sukhodolov et al., 2010]. When the vegetation is sparse and its effect on flow structure is minor, the distribution of turbulent shear stress is expected to resemble a linear profile inline image, where g is the specific gravity, h is the flow depth, and S is the slope of the free surface. This linear profile is also scaled with velocity differential and represented by a dashed line in Figure 5. For runs 2 and 3 the near bed points were used as the velocities characteristic for the slow part of the mixing layer, while the velocity differential for run 4 was determined as the difference between outer flow in which velocity was nearly constant and the points closest to the riverbed. The linear profile of shear stress fits well the measured data in run 3, therefore revealing the fact that this run is exhibiting the dynamics characteristic for the perturbed boundary layer. The intermediate population density in run 2 results in an intermediate behavior. Partly, specifically in the upper part, the shear stress profile in run 2 is in agreement with the boundary layer model, while at the location of the top of canopy there is an excess of shear stress which is of the same magnitude as the values produced by internal gradient of the mixing layer. In the lower part of the profile the values of stress are significantly smaller than predicted by the model of the boundary layer, though they are somewhat larger than the values predicted by a mixing layer model. In run 4 the measurements indicate the presence of an internal boundary layer developing in the river reach with aquatic vegetation above the riverbed devoid of vegetation. This observation is similar to those reported in the previous field studies [Sukhodolov and Sukhodolova, 2010]. The height inline image of the internal boundary layer corresponds to the vertical distance from the riverbed at which the value of shear stress is nearly zero. The measured profiles of shear stress indicate a linear profile of stress inside the boundary shear layer and approximately zero stress profile in the outer layer. The value of velocity differential was chosen as the difference between the mean velocity in the outer layer and the velocity near the riverbed.

Figure 5.

Normalized vertical profiles of shear stress (arrows indicate position of the riverbed).

[25] The values of dissipation rate calculated from the measured turbulence spectra are normalized in accordance with equation (11) (Figure 6). The experimental data indicate a high degree of correspondence with (11) for larger population density of vegetation in run 1, though the value of coefficient γwhich provides the best approximation of the experimental data is 0.02; similar to the shear stress profile. The profile of dissipation rate in an open-channel flow without vegetation is described by a semitheoretical function inline image [Nezu and Nakagawa, 1993] which is also valid for the experimental river reach of this study [Sukhodolov et al., 1998]. This semitheoretical function scaled with the variables of the mixing layer model is represented by a dashed line in Figure 6. Experimental data agree well with the scaled function for run 3 in which influence of vegetation is minimal. In run 2 with intermediate population density of vegetation experimental data agree with the semitheoretical function in the upper part of the water column, while near the bed they are smaller. In run 4 the profiles of dissipation rate normalized with the height of internal boundary layer and velocity difference between the outer flow and velocity near the riverbed are in good agreement with the theoretical curve which was obtained by matching the theoretical curve of an open channel flow with the background value of dissipation in the outer layer.

Figure 6.

Normalized vertical profiles of dissipation rate (arrows indicate position of the riverbed).

[26] The scaled values of turbulent kinetic energy are shown in Figure 7. These profiles are compared to equation (12) and indicate that measured data agree with scaling relation (12) for the dense population of vegetation in run 1 (Figure 7). Similar to the profiles of turbulent shear stress and dissipation rate the value of coefficient γ, which provided the best approximation of measured data, is two times larger than the value reported for canonical and shallow mixing layers. For this experimental reach without vegetation a semitheoretical profile deduced from a standard inline image model [Nezy and Nakagawa, 1993] inline image was shown to fit the profile of turbulent kinetic energy [Sukhodolov et al., 1998]. The values of turbulent kinetic energy scaled with the square of velocity differential are represented by dashed lines in Figure 7. The normalized profiles of turbulent kinetic energy indicate the same trend in flow response to changes in population densities as it was already outlined for the profiles of shear stress and dissipation rate. While in dense canopy the profile of turbulent kinetic energy agrees with the model of mixing layer, in sparse vegetation the profile is similar to boundary layer profile. In run 4 the profiles of turbulent kinetic energy scaled with velocity difference between the outer flow and velocity near the riverbed are well approximated by the theoretical curve which was obtained by matching the theoretical curve of an open channel flow with the background value of turbulent kinetic energy in the outer layer.

Figure 7.

Normalized vertical profiles of turbulent kinetic energy (arrows indicate position of the riverbed).

4.2. Coherent Structures

[27] The method of two-point cross-correlation functions (section 3.2) is applied for mapping the spatial patterns of coherent structures. It is assumed that from these maps one can directly obtain characteristic spatial scales of the organized motions without applying frozen turbulence approximation [Sukhodolov et al., 2011]. Typical two-point cross-correlation functions computed according toequation (17) are shown in Figure 8. These functions display systematic decrease in the maxima of correlation at increasing lag distances—a pattern reported to be characteristic for many shear-driven turbulent flows. The ordinates of two-point cross-correlation functions atτ = 0 represent the value of correlation at a given displacement Δx, Δy, and can be mapped to obtain spatial patterns of coherent structures [Sukhodolov et al., 2011]. The correlation maps for streamwise velocity are shown in Figure 9, and for vertical velocity in Figure 10. The spatial correlation maps clearly depict the pattern representative for mixing layers—three distinctive zones of positive correlation values mark the positions of coherent structures along the mixing layer in run 1 (Figure 10a). The zones are elongated in the streamwise direction and their vertical extent corresponds to the width of the mixing layer, though the spatial arrangements of the zones indicate that they are centered along the upper boundary of the vegetative canopy at the near field of the mixing layer origin. In the far field ( inline image m) the center of coherent structures is aligned with the central position of the mixing layer. With decreasing population density of the vegetation and respectively decreasing resistance (roughness) of the flow, the spacing between centers for area of flow with positive correlations becomes larger (Figure 10b). At lowest densities the flow structure resembles the pattern of boundary layer structure with characteristic narrow band segmentation and evident uplift of the second positive maxima toward the free surface. This uplift is indicative for ejection events and is a phenomenon of surface boils that is reported by many studies [Nezu and Nakagawa, 1993; Sukhodolov et al., 2011]. Two-point correlation maps provide comprehensive source of qualitative information and also some valuable quantitative information about spatial scales related to organized motions.

Figure 8.

Examples of cross-correlation functions for (a) streamwise velocity and (b) vertical velocity components, experimental run 1.

Figure 9.

Spatial two-point correlation maps for streamwise velocity component: (a) run 1, (b) run 2, (c) run 3, and (d) run 4, the dashed lines mark the boundaries of the flow layers.

Figure 10.

Spatial two-point correlation maps for vertical velocity component: (a) run 1, (b) run 2, (c) run 3, and (d) run 4, the dashed lines mark the boundaries of the flow layers.

[28] Additional quantitative information can be provided by the analysis of turbulence spectra and comparison with the predictions of models (2) and (7). Examples of turbulence spectra measured in the different parts of the flow are shown in Figure 11. The spectral densities measured inside the area occupied by the mixing layers (Figure 11a) resemble those reported in laboratory studies [Ghisalberti and Nepf, 2002]. A distinctive peak is evidently present at low frequencies (f = 0.05 Hz) on both spectra measured in the free flow above and inside the mixing layer, though the spectral densities of vertical velocity in the free flow are an order of magnitude smaller than those inside the layer. The spectral densities reveal the presence of an inertial subrange −5/3 at higher frequencies ( inline image Hz inside, and inline image Hz in free flow), while at lower frequencies the spectral density display the presence of the distinctive peaks.

Figure 11.

Examples of turbulence spectra measured (a, c, e, and g) near the center of the mixing layer and (b, d, f, and h) in the free flow over the vegetation patch, vertical 7 for all runs.

[29] The data on mean flow and turbulence presented in the previous sections permit the analysis of performance for the scaling relations (2) and (7). The main characteristics of the mixing layer and the results of computations according to equations (2) and (7) are presented in Table 1. Comparison of measured and predicted frequencies for coherent structures indicates that equation (2) underestimates (about 30%–50%) the size of coherent structures at higher densities of vegetation in run 1, while equation (7) predicts the frequency with accuracy of about 15%. In run 2 predictions with equation (2) yielded overestimates (around 30%–50%), while predictions with relation (7) provided the same accuracy as for run 1 except the distance closer to the leading edge x = 4 m. In run 3 predictions with equations (2) are characterized by large overestimates while equation (7) at shorter distances provided reasonable accurate estimates and failed at the large (x = 9 m) indicating probably the fact that at low population density the flow arrangement is no more adequately presented by a model of a mixing layer. The spatial scale of coherent structures can be also obtained from the correlation map (Figure 10a) as the distance between the centers of positive correlation regions in the far field ( inline image m), and hence the accuracy of an estimate of the spatial scale is slightly larger than 20%. Until now the value of the coefficient of proportionality was always equal to 1 assuming perfect agreement with frozen turbulence hypothesis. The slight underestimate of the spatial scale can be most probably attributed to the fact that coherent structures propagate faster than the mean velocity in the mixing layer. This fact was confirmed by laboratory experiments [Ghisalberti and Nepf, 2002] as the precondition for development of monami motions in submerged vegetation. In fact the video records obtained synchronously with instrumental (ADV) measurements also revealed monami motions (magnitude of about 20–30 cm) directly related to the ejection and sweep phases of structures.

Table 1. Characteristics of the Mixing Layer, Turbulent Stress, Resistance Factors, Measured and Predicted Frequencies of Coherent Structures
x (m)δ(x) (m)Δu (m s−1)uc (m s−1)τ/ρ (cm2 s−2)cff (Hz)
MeasuredPredicted
(2)(7)
Run 1
5.40.850.220.167.00.0550.0900.0420.063
7.30.950.220.159.00.0800.0700.0350.062
9.21.050.240.1610.00.0780.0500.0340.058
 
Run 2
4.00.570.110.194.60.0260.0800.0750.064
7.10.630.110.192.30.0130.0400.0680.047
8.90.610.110.194.20.0230.0500.0700.059
 
Run 3
6.30.420.100.202.00.0100.0600.1070.065
7.60.460.110.203.20.0170.0620.0970.073
9.00.510.100.192.00.0110.0750.0830.054

5. Conclusions and Discussion

[30] This study examines the structure of turbulence in the flow evolving around a patch of submerged flexible vegetation in a river. Theoretical analysis is presented which explored scaling relationships for the turbulence statistics and coherent structures in the far field from the leading edge of the vegetation patch. Application of hypothesis of local equilibrium between production and dissipation of turbulence yielded scaling relationships for the profiles of turbulent shear stress, dissipation rate, and turbulent kinetic energy. The same hypothesis provided the scaling relationship for the nominal frequency of periodic fluctuations because of passage of coherent structures in the mixing layer. The obtained relations explicitly account for characteristics of the vegetation forming the patch because principal scaling variables (mixing layer width and velocity differential) are the functions of the drag coefficient and population density of vegetation [Sukhodolova and Sukhodolov, 2012]. The results of theoretical analysis are compared to the experimental data obtained in the field. A set of experimental measurements was completed in a lowland river using natural plants Sagittaria sagittifolia that are ubiquitous for the conditions examined in this study and are known to be aquatic macrophytes widely abound in the regions with temperate climates. In this study the plants with filamentous elongated blades were selected and implanted into the experimental patch thus providing direct analogy to the study design with the laboratory studies completed with model vegetation [Ghisalberti and Nepf, 2002, 2004].

[31] The analysis and comparison with experimental data presented in this paper indicate that scaled profiles of turbulence characteristics exhibit patterns which reveal the effects of population density of vegetation on the flow structure around a patch of submerged flexible plants. These patterns are comprised between the two extreme cases: In a patch with large population density the structure of flow is best represented by a mixing layer model, while in a patch with sparse vegetation the behavior of flow much better fits the model of perturbed boundary layer. This result is in agreement with conclusions drawn from the examination of mean velocity field [Sukhodolova and Sukhodolov, 2012].

[32] Although the profiles of turbulence characteristics qualitatively measured in densely vegetated flow agree well with the proposed theoretical relationships, the coefficient of eddy viscosity model γ appears to be larger than the nominal value reported for free and shallow mixing layers. Similar results (γ about 1.5 times larger than nominal) were shown by the scaled profiles in laboratory [Ghisalberti and Nepf, 2002, 2004] and field measurement studies [Sukhodolov and Sukhodolova, 2010]. This difference in the values of coefficients can be explained by the fact that in the vegetated flow besides the internal shear imposed by the mixing layer, the upper surface of vegetation interacts with the upper layer of the flow and generates an additional source of turbulence due to the friction over the plant blades. The estimates of the magnitudes of this additional shear completed for run 1 provided quite close values, thus supporting the suggested hypothesis. This hypothesis expands the concept of two-process spectra accounting for the shear production due to inflection-point instability at lower frequencies and the wake production at the scale of individual plants [Finnigan, 2000, Figure 14]. However, there is also an opinion that the larger nominal values of turbulent eddy viscosity might be attributed to stronger organization of coherent structures in vegetated shear layers compared to the canonical free mixing layers [Ghisalberti and Nepf, 2006].

[33] Analysis of structural properties of turbulence completed with multiprobe technique indicated that analogy with mixing layers most probably accounts for the universality of structural properties of turbulence. However, this study also clearly demonstrates the value of the perturbed boundary layer model that can account for either local variations in the population density of plants communities, or for their seasonal variations directly related to the phenological cycles of the plants.

[34] Particular results of the study are of remarkable interest for the development of the special theory of vegetated flows. Scaling relationships developed for turbulence statistics and coherent structures in which the properties of vegetation are implicitly related to characteristics of flow comprise the theoretical framework for analysis and interpretation of result of field investigations [Sukhodolov and Sukhodolova, 2010]. This framework can be effectively used for examining the relation between biomechanical properties of the plants and resultant interactions between flow and plants. The importance of unified approaches was a subject of recent debates [Sand-Jensen, 2003, 2005; Sukhodolov, 2005; Green, 2005; Statzner et al., 2006] and hence this study contributes to the clarification of theoretical backgrounds. Particularly the proposed phenomenological model for frequency of coherent structures is shown to predict frequencies accurately because it accounts for integral characteristics of vegetation additionally to the inflection-induced factor characteristic of canonical mixing layers. Although in this paper the biomechanical properties ofSagittaria sagittifolia plants were omitted due to space limitations, the relevant information can be found by Sukhodolova [2008].

[35] The design of this study represents a link between laboratory research under idealized conditions [Ghisalberti and Nepf, 2002, 2004] and the coarse observations in natural conditions [Sand-Jensen, 1998; Sans-Jensen and Pedersen, 1999; Sukhodolova and Sukhodolov, 2010]. It demonstrates that because of the control provided by selection of plants, uniformity of their morphological, physiological, and biomechanical properties, spatial homogeneity, the results obtained under field conditions remarkable agree with the theoretical approaches and the results of laboratory studies.

Acknowledgments

[36] Funding for this research was provided by the Deutsche Forschungsgemeinschaft (DFG grants SU 629/1 and SU 405/4). The authors acknowledge the value of comments by Heidi Nepf and late Gerhard Jirka on parts of this research.

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