Rigid Canopy Research Articles

An efficient numerical solver for highly compliant slender structures in waves: Application to marine vegetation

This paper presents a fully explicit coupled wave–vegetation interaction model capable of efficiently solving the coupled wave dynamics and flexible vegetation motion with large deflections. The flow model is formulated using the continuity equation and linearized momentum equations of an incompressible fluid, with additional terms within the canopy region accounting for the presence of vegetation. This linearized flow solver is unconditionally stable and second-order accurate. The flow model is validated and verified against experimental measurements and analytical solutions for waves over a rigid canopy, demonstrating its capability to accurately capture the wave dissipation and flow velocity profiles, even with a relatively coarse grid. A truss-spring model is proposed to capture vegetation motion with substantial deflections, and is proven to be mathematically consistent with the governing equation for the flexible vegetation motion. It allows for explicit time integration with large time steps when dealing with highly compliant vegetation. The truss-spring model is validated and verified by experimental and numerical results for large-amplitude motions of a single elastic blade subjected to waves and sinusoidal oscillatory flows. The coupled model, combining the linearized flow solver and the truss-spring model, is applied to investigate wave propagating over a heterogeneous, suspended, and flexible canopy, showing high efficiency and good agreement with the experiments concerning wave attenuation and the hydrodynamic loads on the vegetation.

Variation in Zero Plane Displacement and Roughness Length for Momentum Revisited

Zero plane displacement height (d0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_0$$\\end{document}) and momentum roughness length (z0m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z_{0m}$$\\end{document}), describe the aerodynamic characteristics of a vegetated surface. Usually, d0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_0$$\\end{document} and z0m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z_{0m}$$\\end{document} are assumed to be constant functions of the physical characteristics of the surface. Prior evidence collected from the literature and our examination of flux tower data show that d0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_0$$\\end{document} and z0m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z_{0m}$$\\end{document} vary in time at sites with tree and shrub canopies, but not grasslands. The conventional explanations of these variations are based on linear functions of wind velocity and friction velocity, with little theoretical basis. This study explains the variation in aerodynamic parameters by matching four analytical canopy velocity models to a logarithmic above-canopy velocity profile at canopy height. d0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_0$$\\end{document} and z0m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z_{0m}$$\\end{document} come out as functions of 2 non-dimensional terms, the canopy momentum absorption capacity (parameter) and a (measurable) Péclet number. To test the theories of variation, we analysed the velocity profiles from Ozflux and Ameriflux sites. None of the theories could recreate d0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_0$$\\end{document} and z0m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z_{0m}$$\\end{document} at half-hourly intervals. However, the canopy velocity models were able better to recreate the distribution of the variations in d0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_0$$\\end{document} and z0m\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z_{0m}$$\\end{document}. Additionally, the estimates of canopy momentum absorption capacity varied consistently with phenological changes in the canopies, whereas, the fitting parameters of the linear regression of using wind speed and friction velocity did not exhibit physically interpretable variations. The canopy velocity models may offer better predictions with an accurate estimation of the canopy height, a horizontally homogeneous and rigid canopy, and incorporation of the roughness sublayer.

Experimental investigation of the total flow resistance in emergent and submerged rigid canopy flows

In canopy flows, flow resistance mainly originates from vegetation drag and depends on vegetation characteristics and flow conditions. In the present study, a series of experiments were performed in various hydraulic scenarios with high stem Reynolds numbers (2641 ≤ Red≤ 17333) using relatively sparse rigid canopies, represented with four different dimensionless vegetation densities (0.0044, 0.0098, 0.0174 and 0.0392), on a smooth bed. A novel drag plate mechanism was developed to measure the total flow resistance due to the emergent and submerged vegetation arrays in a staggered pattern under subcritical flow conditions. Manning’s roughness coefficient and Darcy–Weisbach friction factor were adopted to represent the total flow resistance in the analyses. Simple empirical relationships based on roughness concentration and submergence ratio were derived to determine the total flow resistance parameters within a broad range of stem Reynolds numbers. Although relationships were proposed in a simple form to be used for direct practical applications, they show similar or better performance in the prediction of total flow resistance parameters than the existing equations in the literature, which require considerable computational effort. Additionally, analyses demonstrated that the results of the present study and those of similar studies regarding canopy flow resistance are in good agreement. Accordingly, the novel drag plate looks promising for measuring flow resistance due to vegetation and bed conditions similar to those in nature.

SPECTRAL WAVE ENERGY DISSIPATION BY A SEAGRASS MEADOW: INTEGRATING ANALYTICAL AND EMPIRICAL MODELS

Waves propagating across partially emerged and submerged vegetation (such as a seagrass meadow) gradually lose energy due dissipation generated by the drag forces exerted by the vegetation elements (e.g., stems, blades, etc.) (Mendez and Losada, 2004). How vegetation characteristics influence wave dissipation is important to quantify how a canopy affects the physical processes that ultimate lead to the attenuation of waves. However, field studies on wave dissipation in seagrass meadows have often not considered the impact of plant flexibility on both in-canopy flows and wave attenuation. In this study we aim to (1) understand how wave interactions with flexible seagrass canopies affect wave dissipation, (2) assess the role of flow modifications by canopies to explain frequency-dependent wave dissipation, and (3) evaluate the role of flexibility on wave attenuations relative to assuming rigid canopy flow theory.

Simulation-based study of turbulent aquatic canopy flows with flexible stems

Large-eddy simulation (LES) is performed to investigate the dynamics of flow and canopy motions and the energy transfer in turbulent canopy flows. Different from the traditional approach that models the canopy as a continuous medium with a drag coefficient prescribed a priori, an immersed boundary method together with a beam model is employed to explicitly capture the dynamics of individual stems and resolve monami. The simulation cases cover a broad range of stem flexibilities from rigid stems to oscillatory stems to stems yielding to the flow. For highly flexible canopies, the stem fluctuation is small such that the canopy behaves like a rigid canopy, which is used to explain the similarities of the flow features between rigid and highly flexible canopies. Analyses of the turbulent kinetic energy (TKE) budget show that, in the flexible canopy cases, the waving term associated with the canopy drag–flow velocity correlation can be as large as one-half of the shear production term near the canopy top. Spectral TKE budget analyses further reveal dominant effects at two characteristic scales: the monami scale associated with the coherent structures in the mixing layer and the wake scale associated with the interval between adjacent stems. For the TKE in flexible canopies, the waving term is found to play an important role in the interscale and wall-normal transport terms. Our LES data show that the spectral shortcut mechanism proposed by previous studies is caused by the waving term.

On the submerged low-Cauchy-number canopy dynamics under unidirectional flows

The unsteady dynamics and induced turbulence associated with flow development of a fully-submerged canopy with relatively low flexibility were explored experimentally under small Cauchy numbers Ca≲9 using particle image velocimetry (PIV) and particle tracking velocimetry (PTV). Inspection of the flow around a rigid canopy supported the identification of significant modulation of the flow dynamics due to minor blade reconfiguration and oscillations within the first rows. The canopies consisted of relatively short arrays of rectangular blades of height h=37.5mm placed in a staggered layout and extending Δx/h=5 in the streamwise direction. The results show a downstream shift in the onset of the internal boundary layer. Canopy top friction velocity was lower than its rigid counterpart, even where blades undergo negligible deflection, evidencing the impact of the leading structure dynamics on flow adjustment. A transition from vortex-induced blade motions to dominance by natural blade frequency occurs with distance from the canopy edge. A frequency synchronization occurs several rows into the canopy, resulting in peak displacement intensity of the blades. The relationship between flow-induced and blade natural frequencies is described through a developed transfer function, enabling the determination of the characteristic vortex frequency. Natural and flow-induced frequencies dominate the flow-structure interaction near the leading edge region, resulting in increased small-scale turbulence.

Impact of gaps on the flow statistics in an emergent rigid canopy

High-resolution large eddy simulations and complementary laboratory experiments using particle image velocimetry were performed to provide a detailed quantitative assessment of flow response to gaps in cylinder arrays. The base canopy consists of a dense array of emergent rigid cylinders placed in a regular staggered pattern. The gaps varied in length from Δg/d=4 to 24, in intervals of 4d, where d is the diameter of the cylinders. The analysis was performed under subcritical conditions with Froude numbers Fr∈[0.08,0.2] and bulk Reynolds numbers Re∈[0.8, 2]×104. Results show that the gaps affect the flow statistics at the upstream and downstream proximity of the canopy. The affected zone was Δx/d≈5 for the mean flow and Δx/d≈3 for the second-order statistics. Dimensionless time-averaged streamwise velocity within the gap exhibited minor variability with gap spacing; however, in-plane turbulent kinetic energy, k, showed a consistent decay rate when normalized with that at x/d≥1 from the beginning of the gap. The emergent canopy acts as a passive turbulence generator for the gap flow for practical purposes. The streamwise dependence of k follows an exponential trend within 1≤x/d≲2.5 and transitions to a power-law at x/d≥4. The substantially lower maximum values of k within the gap compared to k within the canopy evidence a limitation of gap measurements representative of canopy flow statistics. We present a base framework for estimating representative in-canopy statistics from measurements in the gap.

On the turbulence dynamics induced by a surrogate seagrass canopy

The distinct turbulence dynamics and transport modulated by a common seagrass species were investigated experimentally using a flexible surrogate canopy in a refractive-index-matching environment that enabled full optical access. The surrogate seagrass replicated the dynamic behaviour and morphological properties of its natural counterpart. The flows studied were subcritical with Froude numbers $Fr<0.26$ and concerned five Reynolds numbers $Re\in [3.4\times 10^{4}, 1.1\times 10^{5}]$ and Cauchy numbers $Ca\in [120, 1200]$ . Complementary rigid canopy experiments were also included to aid comparative insight. The flow was quantified in wall-normal planes in a developed region using high-frame-rate particle image velocimetry. Results show that the deflection and coordinated waving motion of the blades redistributed the Reynolds stresses above and below the canopy top. Critically, in-canopy turbulence associated with the seagrass lacked periodic stem wake vortex shedding present in the rigid canopy, yet the flexible canopy induced vortex shedding from the blade tips. Inspection of spatial and temporal characteristics of coherent flow structures using spectral proper orthogonal decomposition reveals that Kelvin–Helmholtz-type vortices are the dominant flow structures associated with the waving motion of the seagrass and that modulated the local flow exchange in both rigid and flexible canopies. A barrier-like effect produced by the blade deflections blocked large-scale turbulence transport, thereby reducing vortex penetration into the canopy. In addition, we uncovered a transition from sweep-dominated to ejection-dominated behaviour in the surrogate seagrass. We hypothesise that the vortices created during the upward blade motion period play a major role in the sweep-to-ejection-dominated transition. Conditionally averaged quadrant analysis on the downward and upward blade motion supports this contention.

New dynamic two-layer model for predicting depth-averaged velocity in open channel flows with rigid submerged canopies of different densities

The depth-averaged velocity is the commonly used engineering quantity in natural rivers, and it needs to be predicted in advance, especially in flood seasons. A model that can provide a unified physical foundation for open channel flows with different canopy densities remains lacking despite ongoing researches. Here, we use the concept of the auxiliary bed to describe the influence of momentum exchange on rigid canopy elements with varying density and submergence. The auxiliary bed divides the vegetated flow into a basal layer and a suspension layer to predict average velocity in each layer separately. In the basal layer, the velocity profile is assumed to be uniform. In the suspension layer, a parameter called “penetration depth” is applied to present the variations in velocity distribution. We also apply a data-driven method, called genetic programming (GP), to derive Chezy-like predictors for average velocity in the suspension layer. Compared to the hydraulic resistance equation for rough-wall flows, the new formulae calculated by the weighted combination method show sound physical meanings. In addition, comparison with other models shows that the new dynamic two-layer model achieves high accuracy in flow rate estimation, especially for vegetated flow with sparse canopies.

Large-eddy simulation of an open-channel flow bounded by a semi-dense rigid filamentous canopy: Scaling and flow structure

We have carried out a large-eddy simulation of a turbulent open-channel flow over a marginally dense, fully submerged, rigid canopy. The canopy is made of a set of rigid, slender cylinders normally mounted on a solid wall. The flow in the canopy is resolved stem-by-stem by means of an immersed boundary method. It is found that the flow behavior can be categorized according to the velocity distribution into two separate spatial regions: the canopy itself and the outer region above it. Within the region occupied by the canopy elements, the velocity magnitude is found to be related to the local shear stress. Outside the canopy, a logarithmic velocity profile matching the canonical turbulent open-channel flow over rough walls is recovered albeit the strong manipulation exerted by the canopy on the buffer layer. In the innermost layer, the presence of the stems is responsible for redistributing the local momentum fluctuations from a streamwise to a spanwise leading component, inhibiting the survival of the wall streamwise velocity streaks. On the other hand, the outer region presents a structure very similar to the well-known logarithmic boundary layer with the presence of large and energetic streamwise velocity streaks generated by a system of quasistreamwise vortices. These vortices strongly contribute to the intracanopy fluctuations through vigorous sweep and ejection events that affect all the region occupied by the stems. Consistent with the results of previous investigations [H. Nepf, “Flow and transport in regions with aquatic vegetation,” Annu. Rev. Fluid Mech. 44, 123–142 (2012)], it is found that the inflection point in the mean velocity profile, produced by the drag discontinuity at the canopy tip, promotes the appearance of another system of spanwise oriented vorticity structures. However, different from previous results reported in the literature [J. Finnigan, “Turbulence in plant canopies,” Annu. Rev. Fluid Mech. 32, 519–571 (2000)], in our simulations, the presence of alternating head up–head down hairpin vortices generated by a mutual induction of the counter-rotating spanwise vortices is not observed. Instead, we advocate that the modulation of the spanwise coherent vorticity is due to the action of the external logarithmic layer structures (i.e., the outer streamwise vortices that penetrate the canopy) rather than by upwash and downwash motions induced by the mutual interaction of the spanwise rollers.

Numerical investigation of effects of angle-of-attack on a parachute-like two-body system

The supersonic flows around a typical parachute-like two-body system are numerically studied at a freestream Mach number of 2.0. Both rigid and flexible models, composed of a capsule and a canopy, are considered. The objective is to analyze the effects of angle of attack (AoA) on the unsteady flow produced by the parachute-like two-body system and the difference in flow behaviors between the rigid and flexible systems. The following two model configurations are examined for the rigid canopy case: (1) the canopy is connected to the capsule with a rigid rod and (2) the canopy is separate from the capsule. It is found that at a moderate capsule AoA, the most unstable flow field is generated in the interior of the canopy and the connecting rod has a significant effect on the flow field. For the flexible case, the supersonic flows over the model (2) configuration and the canopy alone are investigated. An area oscillation phenomenon is observed in the flexible canopy. When the capsule AoA is increased, the flexible canopy has greater deformation, which results in a lower drag coefficient. The effect of the canopy AoA is also examined for the flexible canopy alone. A small deformation in the canopy shape is observed.

Experimental observations on sediment resuspension within submerged model canopies under oscillatory flow

A set of laboratory experiments were conducted to study the effect of submerged aquatic vegetation in sediment resuspension under progressive waves. Three vegetation models (rigid, flexible and real plants of Ruppia maritima), six wave frequencies (in the range F=0.6–1.6Hz) and four plant densities (Solid Plant Fractions, SPF in the range of 1–10%) were used. The sediment bed properties corresponded to a salt marsh wetland with a bimodal particle size distribution with two particle populations (population 1: particle diameters in the range of 2.5 to 6.0μm, and population 2: particle diameters in the range of 6.0 to 100μm).Within the canopy, wave velocities were attenuated for all the canopies studied and for all the frequencies analyzed. The change in the TKE (ΔTKE) compared with the case without plants was studied. For the rigid canopy model, in comparison to the unimpeded experiment, an increase in ΔTKE inside the canopy for smaller frequencies (F=0.6–1.2Hz) was observed together with stem Reynolds numbers Rep above 250. As a result, sediment resuspension for both sediment populations was higher than that of the unimpeded experiment. However, at higher frequencies (F=1.4 and 1.6Hz) and higher plant densities (SPF=5%, 7.5% and 10%), the ΔTKE inside the canopy decreased, coinciding with stem Reynolds number Rep below 250. As a result, sediment resuspension for larger canopy densities and larger frequencies was reduced.For the flexible vegetation model, in comparison with the unimpeded experiment, a reduction in the ΔTKE inside the canopy was nearly always found. Resuspended sediment concentrations were found to decrease as flexible canopy densities increased. For the flexible vegetation the stem Reynolds number was Rep<250 and no production of ΔTKE was observed. The real case of a canopy of R. maritima behaved similarly to the flexible model canopy.

Canopy-scale turbulence under oscillatory flow

The aim of this study is to understand the turbulent flow structure within diverse canopy models dominated by progressive waves. A set of experimental conditions were considered in a laboratory flume: three vegetation models (submerged rigid, submerged flexible and emergent rigid), three canopy densities (128, 640 and 1280 stems/m2) and three wave frequencies (f=0.8, 1 and 1.4Hz). The canopy morphology through both the plant flexibility and height and the canopy density were the characteristic parameters that exerted a control on wave induced turbulence within the canopy bed. In the flexible canopy model, sheltering at the bed was observed and was associated with the movement of the blades. In contrast, in the rigid canopy model larger TKE was found as compared with the case without canopy. The increase of TKE was associated with the production of the stem-wake turbulence. Sheltering in the submerged rigid canopy model was found at the lower layer for the largest canopy density and highest wave frequency because of a great loss of wave velocity, confined below the top of the canopy. Sweeps and ejections were the predominant events, enhancing the transfer of momentum at the top of the canopy. Therefore, below the top of the submerged rigid canopy was characterized by a vertical energy exchange zone. Unlike the submerged model, sheltering was always found for emergent rigid vegetation, and attributed to the inhibition of the wave energy at all depths.

Momentum Transfer and Turbulent Kinetic Energy Budgets within a Dense Model Canopy

Second-order closure models for the canopy sublayer (CSL) employ aset of closure schemes developed for `free-air' flow equations andthen add extra terms to account for canopy related processes. Muchof the current research thrust in CSL closure has focused on thesecanopy modifications. Instead of offering new closure formulationshere, we propose a new mixing length model that accounts for basicenergetic modes within the CSL. Detailed flume experiments withcylindrical rods in dense arrays to represent a rigid canopy areconducted to test the closure model. We show that when this lengthscale model is combined with standard second-order closureschemes, first and second moments, triple velocity correlations,the mean turbulent kinetic energy dissipation rate, and the wakeproduction are all well reproduced within the CSL provided thedrag coefficient (CD) is well parameterized. The maintheoretical novelty here is the analytical linkage betweengradient-diffusion closure schemes for the triple velocitycorrelation and non-local momentum transfer via cumulant expansionmethods. We showed that second-order closure models reproducereasonably well the relative importance of ejections and sweeps onmomentum transfer despite their local closure approximations.Hence, it is demonstrated that for simple canopy morphology (e.g.,cylindrical rods) with well-defined length scales, standard closureschemes can reproduce key flow statistics without much revision.When all these results are taken together, it appears that thepredictive skills of second-order closure models are not limitedby closure formulations; rather, they are limited by our abilityto independently connect the drag coefficient and the effectivemixing length to the canopy roughness density. With rapidadvancements in laser altimetry, the canopy roughness densitydistribution will become available for many terrestrialecosystems. Quantifying the sheltering effect, the homogeneity andisotropy of the drag coefficient, and more importantly, thecanonical mixing length, for such variable roughness density isstill lacking.

Computation of dynamic stresses in the retrimming of an axially symmetric parachute

We propose a method of computing the dynamic stresses that arise in the elements of a parachute system during retrimming. The method is based on a simplified mathematical model of a deformable elastic parachute. The results of numerical computations from the model developed are presented, and a comparison of these results with the computations using a rigid canopy model is given. It is established that the simplified mathematical model is a qualitatively correct description of the properties of parachute functioning during retrimming.

Computation of dynamic stresses in the retrimming of an axially symmetric parachute

We propose a method of computing the dynamic stresses that arise in the elements of a parachute system during retrimming. The method is based on a simplified mathematical model of a deformable elastic parachute. The results of numerical computations from the model developed are presented, and a comparison of these results with the computations using a rigid canopy model is given. It is established that the simplified mathematical model is a qualitatively correct description of the properties of parachute functioning during retrimming.