The derivation of flow and mass transfer models in canopy and porous media environments involves the spatial-averaging of the flow properties and their subscale equations. The averaging of the momentum equation generates the dispersive stress terms that represent the subscale spatial variations of the unresolved velocity field. While previous studies ignored the dispersive stresses in their flow models, recent evidence indicates that the dispersive stresses may be important. Here we focus our attention on the magnitude of the normal dispersive stresses in the entry region of a 'forest patch', where the in-canopy velocities are large and the longitudinal derivatives do not cancel out. Highly detailed particle image velocimetry measurements, at a temporal and spatial resolution of 5 Hz and 1.4 mm, are obtained inside and around a 1-m long model canopy which consists of transparent vertical cylinders 6 mm in diameter and 74.3 mm high (h). The cylinders are randomly distributed to form a relatively sparse forest patch with a leaf area density of 7.56 m −1 and a fluid volume fraction (porosity) of 0.965. We present results of the double averaged flow properties at three different regions of the forest patch; the upstream edge (x ≈ 0), the fully-developed interior region (x ≈ 10h) and the downstream edge (x ≈ 13h). We find that the normal dispersive stresses around the entry region of the forest patch are significantly larger than the normal Reynolds stresses. An order of magnitude analysis of the relevant terms in the momentum equation indicates that the longitudinal derivatives of the dispersive stresses are of the same order of magnitude as that of the drag force and similar to that of the horizontal convection term. The longitudinal derivatives of the Reynolds stresses are smaller, though cannot be ignored. Comparing these results with the characteristic profiles measured in the fully-developed region indicates that the dispersive stresses, which are generated at the forest patch entrance, decrease along an adjustment region while maintaining their profile shape. We find that the dispersive stresses influence the rate at which momentum penetrates into the 123 334 S. Moltchanov et al. canopy. These observations suggest that under certain flow conditions, dispersive stresses may dominate the momentum balance and therefore must be considered in future canopy and porous media flow models.
Canopy flow models are often dedicated to ideal, infinite, homogenous systems. However, real canopy systems have physical boundaries, where the flow enters and leaves patches of vegetation, generating a complex pressure field and velocity variations. Here we focus our study on the canopy entry region by examining the terms involved in the double (space and time) averaged momentum equations and their relative contribution to the total momentum balance. The estimation of each term is made possible by particle image velocimetry (PIV) measurements in a model canopy constructed of randomly distributed thin glass plates. The instantaneous velocity fields were used to calculate the mean velocities, pressure, drag, Reynolds stresses, and dispersive stresses. It was found that within the entry region, the pressure gradient, the drag forces, and dispersive stresses are the three most significant terms that affect the balance in the streamwise momentum equation. In the vertical direction, the dispersive stresses are also significant and their contribution to the total momentum cannot be ignored. The study shows that dispersive stresses are initially formed around canopy edges; at both the entry region and the canopy top boundary. They start as a sink term, extracting momentum from the flow, and then become a source term that contributes momentum to the flow until they eventually decay at some short penetration distance into the canopy. These results reveal a new understanding on the evolution of momentum within the entry region, necessary in any closure modeling of flow in real canopies.
[1] The spatial averaging of the momentum equation in obstructed environments generates dispersive stress terms that represent momentum flux induced by the spatial heterogeneity of the time-averaged flow. While previous studies ignored the dispersive stresses, recent evidences indicate that they may be significant, in particular within entry flow regions such as the leading edge of submerged vegetation in rivers and streams. The lack of available closure models makes it almost impossible to include the dispersive stresses in canopy flow models. Based on observations and theoretical considerations, we propose to model the normal component of the dispersive stress as a function of square of the double-averaged velocity. The model was tested by detailed particle image velocimetry measurements that were obtained inside and around a modeled vegetation patch, made of randomly distributed vertical thin glass plates. It was found that the normal dispersive stresses are scaled with two parameters: the relative area covered by wakes and the relative magnitude of the recirculation (negative) velocity inside the wake zones. The results indicate that prediction of the dispersive stresses is more sensitive to the wake area parameterization than that of the recirculation zone velocities. It is therefore concluded that when parameterization of the relative wake area is available, the normal dispersive stresses can be modeled and included in future flow simulations.Citation: Moltchanov, S., and U. Shavit (2013), A phenomenological closure model of the normal dispersive stresses, Water Resour.
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