3D NUMERICAL EXPERIMENTS ON DRAG RESISTANCE IN VEGETATED FLOWS

Transcription

1 3D NUMERICAL EXPERIMENTS ON DRAG RESISTANCE IN VEGETATED FLOWS Dimitris Souliotis (), Panagiotis Prinos () () Hydraulics Laboratory, Department of Civil Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece Phone: ; fax: ; () Hydraulics Laboratory, Department of Civil Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece Phone: ; fax: ; ABSTRACT In this work three dimensional computations, based on the RANS equations and a turbulence model of the k-ε type, have been performed for calculating drag resistance in a vegetated channel for various submergence ratios and vegetation densities. The submergence ratio, H/h (H=flow depth, h=vegetation height) varies from.7 to 5. and the vegetation density α (α= A/V, A= frontal area of the cylinder and V= volume influenced by a single cylinder ) from.48 to 4.3. The vegetation is considered three-dimensional, rigid, simulated as cylindrical roughness and arranged in a staggered or a non-staggered pattern according to experimental data (Garcia and Lopez,, Poggi et al., 4 and Huthoff et al., 6). Four methods have been used for the calculation of the drag coefficient, : (a) direct calculation from the computed distribution of the respective forces and (b) three indirect methods based on the simplified momentum equation (by including and excluding the gradients of the significant shear stress) and on the force balance between gravity force and drag force. The effect of submergence ratio (H/h), Reynolds number (Re) and array density (αd r ) on is investigated. Keywords: drag resistance, submerged vegetation, drag coefficient, RANS. INTRODUCTION Discharge capacity of channels with vegetation is dependent on the drag resistance exerted by the vegetation. The drag resistance is the combination of the form drag resulting from the shape and structure of the vegetation elements and the skin friction because of the flow around and through the stems. The accurate calculation of drag resistance is based on the value of the drag coefficient. The latter, and in general the drag resistance depends on the vegetation and flow characteristics (vegetation density, submergence ratio H/h, H=flow depth, h=vegetation height and Reynolds number). Numerical, macroscopic models, for calculating the flow characteristics in flows with vegetation, use the Volume-Averaged Reynolds Navier Stokes equations (VARANS) which require the a priori knowledge of (Finnigan, ). Direct estimation of, based on its definition, is difficult in laboratory studies and indirect methods, based on the simplified momentum equation, are used for estimating the local drag coefficient within the vegetation region as well as the bulk. In the most recent study on the effects of H/h on for flow over the coral species Porites compressa (McDonald et al, 6) the was calculated based on the total flow depth, H, the channel slope, S o,and the bulk velocity of the channel U b. was found to depend on both Reynolds number and H/h for low Re numbers, indicating laminar or transitional flow within the coral canopy. For higher Re numbers showed signs of becoming Re-independent, was inversely proportional to H/h and their relationship was described by a power law.

2 In other experimental studies (Poggi et al, 4, Stone and Shen, Nepf and Vinoni,, Nepf, 999, Wu et al 999, Dunn et al. 996) the drag coefficient has been evaluated implicitly through a simplified momentum equation or a force balance for uniform flow in the flow direction. Poggi et al. (4), in their study on the effect of vegetation density on canopy sub-layer turbulence, computed indirectly through measured profiles of shear stress and velocity and they found that, generally, local decreases with increasing distance from the wall and increases with increasing α (vegetation density). However, the increase with α is inconsistent with the wind-tunnel experiments reported by Novak et al () who found the opposite trend. Stone and Shen () conducted a laboratory study with emergent and submerged cylindrical stems of various sizes and concentrations. They found that the stem drag resistance experienced by the flow through the stem layer is best expressed in terms of the maximum depth-averaged velocity between the stems. The use of this velocity allows the use of an average of.5 regardless of the size and density of the stems. Nepf and Vinoni () and Nepf (999), among others, estimated indirectly the local for both emergent and submerged conditions for various H/h and αdr (dr=diameter of vegetation stems) through measurements of shear stress and velocity. For emergent conditions she found that the bulk is constant for αd r up to approximately. and a steady decline is observed beyond αd r =.. The distribution of the local within vegetation, for H/h varying from. to.75, and the values of (.-3.) were found similar to values observed for terrestrial canopies and for other models of submerged vegetation (.5-.) presented by Dunn et al (996). Wu et al. (999) conducted an experimental study using artificial roughness to investigate the variation of vegetation resistance for both non-submerged and submerged conditions. Their results showed a consistent variation for the drag coefficient versus the Reynolds number. They also represented this trend using a vegetative characteristic number. Dunn et al. (996) used also an experimental procedure and they computed local drag coefficients using measurements of bed slope, vegetation density and horizontally averaged velocities. The analysis of those horizontally averaged profiles showed that the was not constant throughout the canopy. They also defined various bulk drag coefficients and they examined the influence of the flow and channel parameters on the values of those bulk drag coefficients. In this study drag coefficient for various vegetation configurations (Dunn et al 996, Poggi et al, 4, Huthoff and Augustijn, 6) is calculated directly based on three dimensional numerical experiments. The microscopic approach is followed (Souliotis and Prinos, 6) by which the 3D-RANS equations, in conjunction with a turbulence model of the k-ε type, are solved numerically in a computational domain around a single vegetal element with periodic conditions at the inlet and outlet of the domain. Based on the 3-D flow characteristics (velocities, pressure etc.) around the vegetation Fpr elements the local drag coefficient can be calculated directly ( C = d,pr.5ρa < U > ) at various locations within the vegetation height and subsequently the bulk (F pr =pressure force, ρ=fluid density, A=projected area in flow direction, <U>=spatial-averaged velocity). In a similar way the drag due to viscous forces is calculated. In the following sections (a) the methods for estimating the drag coefficient for submerged vegetation are described briefly, (b) the microscopic approach for the computation of the 3D flow characteristics around submerged, rigid, vegetation are presented

3 briefly together with the cases studied and (c) results are analyzed and the effect of flow and vegetation characteristics on is investigated.. METHODS FOR ESTIMATION DRAG RESISTANCE In most of the experimental studies (Dunn et al 996, Poggi et al, 4, Huthoff and Augustijn, 6) the estimation of (local and bulk) is based on a simplified form of the volume-averaged momentum equation. The latter is as follows (Finnigan, ): < Ui > < P > < Ui > < Uj > = + v + < uiu j > xj ρ xj x j x j xj < UU i j > Cdα < Uj >< Uj > < Ui > () x j where the symbol < > means volume (spatial) average over a fluid (volume) area, U i =velocity in the i direction, P=effective pressure, u u i j = Reynolds stresses, U ~ U ~ =dispersive stresses (U i j i =< U i >+ U ), = drag coefficient and α=vegetation density. i Cd In general, volume- averaged (macroscopic) quantities are calculated using the relationships: ψ = V f V f ψdv where ψ=fluid parameter and V f =volume occupied by fluid. In the case of spatial averaged quantities the volume is basically a plane, parallel to the channel bed, extensive enough to eliminate plant-to-plant variations in vegetation structure but thin enough the preserve the characteristic variation of properties in the vertical. If the convection terms are assumed negligible as well as the dispersive stresses the x momentum equation is simplified as < P> = + < uv> Cdα< U > () ρ dx y by which the can be calculated through the calculation of the pressure gradient < P > / x and the shear stress gradient < uv > / y. Although the first term can be easily estimated in open channel flows ( < P > / x = ρgso for uniform flows or g H / x for non-uniform flows) the second term is difficult to be estimated accurately, especially close to the top of vegetation where high gradients of < uv > occur. The variable < uv > has to be assessed from point measurements of shear stress which are rather limited. Poggi et al (4) have used horizontal positions for measuring the variation of mean and turbulence characteristics within the flow depth while Dunn et al (996) have only used 4 profiles for assessing the spatialaveraged quantities at various planes parallel to the channel bed. For the estimation of the bulk the momentum equation is simplified into a balance between the forces acting as a control volume with vegetation and the is estimated through a characteristic velocity (usually the depth averaged within the vegetation) and the slope of the channel. (Stone and Shen, ). McDonald et al (6) have determined experimentally the bulk as a function of the overall flow depth H and the bulk flow velocity <U tot > as follows: ghso Cd = (3) < Utot > while the Re number was defined as Re =< Utot > R / ν, where R is the hydraulic radius, arguing that <U tot > is the most appropriate velocity scale for Re since it accounts for the shear

4 layer effects occurring near the top of vegetation. However, values of based on equation (3) are much lower than those determined by the previous methods (equation ()) mainly due to the bulk velocity. Stone and Shen () have calculated values of between.95 and. using as a characteristic velocity the depth averaged velocity at the constricted section in the stem layer while Macdonald et al (6) have found values between.68 and.3 using the overall bulk velocity. In this study the local drag coefficient,,pr is calculated directly through the computed variation of pressure forces around the vegetation elements C d,pr = Fpr.5ρA < U > (4) where F pr = pressure force, A=frontal area <U>= spatial-averaged velocity Also the local drag coefficient C, v due to viscous forces, is calculated as C = F vis d,v.5ρ Aw < U > d where F vis = viscous drag force and A w = wetted area of the vegetation element. The contribution of viscous drag for the Re numbers examined is found to be insignificant in relation with the pressure drag and hence the due to pressure drag was used subsequently in analyzing the results. 3. MICROSCOPIC APPROACH-CASES STUDIED The 3D RANS equations for steady, incompressible flow in conjunction with the standard k-ε model (Rodi,, 98), for calculating the turbulent stresses, are solved by the FLUENT 6.. CFD code (FLUENT Inc., ). The microscopic RANS equations and the transport equations for k and ε are not presented here for the sake of brevity. Details can be found in Souliotis and Prinos (6). The segregated solver approach is used by which the governing equations are solved sequentially. Because the equations are non-linear (and coupled), several iterations of the solution loop must be performed before a converged solution is obtained. Each iteration consists of five steps. The procedure continues until the convergence criteria are satisfied. In the segregated solution method the discrete, non-linear governing equations are linearized to produce a system of equations for the dependent variables in every computational cell. The resultant linear system is then solved to yield an updated flow-field solution. The cases studied use the implicit form of linearization. For a given variable, the unknown value in each cell is computed using a relation that includes both existing and unknown values from neighbouring cells. Therefore each unknown will appear in more than one equation in the system, and these equations must be solved simultaneously to give the unknown quantities. The CFD code uses a control-volume technique to convert the governing equations to algebraic equations that can be solved numerically. This technique consists of integrating the governing equations about each control volume, yielding discrete equations that conserve each quantity on a control volume basis. All equations were discretized using the second order upwind scheme. Pressure coupling was carried out using the SIMPLE algorithm. Under-relaxation factors are used for controlling the change of φ. Typical values of underrelaxation factors, used in the study, are.3 for pressure,.7 for velocities and for turbulence quantities. (5)

5 The cases studied are based on experimental arrangements of Dunn et al (996), Poggi et al. (4) and Huthoff and Augustijn (6). In Figure, the plan view, the boundary conditions and the computational domain for the arrangement of Poggi et al (4) are shown. For each one of the above arrangements, flows with different submergence depth ratio were simulated while the plant density and bed slope were constant. The vegetation and flow characteristics are shown in Table. The value of porosity φ (=V f /V tot ) is very high for all cases (Table ). For the construction of the grids the GAMBIT program was used. The grids were threedimensional, structured and the shape of the cells was hexahedral. Also, the density of the grids was higher near the solid surfaces for estimating accurately the near-wall flow and turbulence characteristics. Finally the density of the grids inside the vegetative region was exactly the same and the computational cells varied between 33. and 86. depending on the value of H/h. FLOW symmetry periodic Figure : Plan view, boundary conditions and computational domain (Poggi et al., 4) Dunn et al. (996) Huthoff and Augustijn Case H (m) h (m) H/h α (m - ) So φ Ε Ε Ε Ε Ε (6) Ε Ε Poggi et al. (4) Ε Ε Ε Ε Ε Table : Vegetation and flow characteristics 4. ANALYSIS OF RESULTS Figure shows typical distributions of <U> <k> and <-uv > within the vegetation region and above it for three different H/h ranging from. to 4.4. These spatial averaged quantities have been calculated from the 3-D computed results of the respective quantities. The computed 3-D flow field is not shown in this work for the sake of brevity. Detailed 3-D results are presented elsewhere (Souliotis and Prinos, 6).

6 The velocity has been made dimensionless with the shear velocity at the top of vegetation ( U * = g( H h) ) and hence the velocity distribution above the vegetation can be S o compared with that over an impermeable (solid) wall. For all H/h the velocity distribution is similar qualitatively. For the greater part of the vegetation layer, away from the channel bed and the top of vegetation, the distribution is uniform with a value equal to that derived from the balance between gravity forces and drag forces. Close to the top of vegetation, the velocity distribution is exponential with high velocity gradients while in the region above the vegetation the velocity seems to follow a logarithmic profile. The velocities are lower than those over a solid wall indicating the reduced discharge capacity of channels with vegetation. In the same figure, the distribution of <k> is shown for the aforementioned cases together with experimental results and the semi empirical distribution of Nezu and Nakagawa (993) for the distribution of k over a solid wall. The <k> levels at the top of vegetation (made dimensionless with U * ) are lower than the asymptotic value, D=4.7, of the semi-empirical relationship while above the vegetation the <k> values are similar to those of the semiempirical relationship. Below the top of vegetation the <k> values remain high close to the top and fall suddenly to low values in the central part of the layer where turbulence is not considerable. Also, the distribution of < uv > within the flow depth is shown in figure. The shear U * stress has been made dimensionless with the and hence a comparison of the distribution of < uv > above the vegetation with that over a solid wall is possible. It is shown that, with increasing H/h from. to 4., the maximum dimensionless shear stress at the top of vegetation decreases from. to.7 indicating the significant effect of the shear layer, developed near the top of vegetation, on the distribution of the shear stress. Within the vegetation layer the shear stress takes values close to zero indicating that in many cases, especially in highly dense vegetation the flow may be laminar in the central part of the layer. Figure 3 shows the variation of the ratio F pr /F vis within the vegetation region (y/h) for the configurations of Huthoff and Augustijn (6) and Poggi et al (4). The forces F pr and F vis have been calculated directly from computed pressure and boundary shear stress on the vegetal element respectively. For all cases considered the ratio takes high values indicating that the drag due to viscous forces can be assumed negligible in comparison with the drag due to pressure. The ratio keeps a constant value up to y/h=.6 and then increases as the top of vegetation is approached. The effect of H/h on the magnitude of the forces and in particular on the F pr is not significant for the conditions examined (H/h between.6 and 5.). The effect of the channel slope and in consequence the Reynolds number on the magnitude of the forces is significant since the gravity force increases and therefore its reaction, due to vegetation, increases as well. In figure 4 the variation of the local within the vegetation layer is shown for two characteristic H/h, equal to. and 4.. The values have been calculated by three different methods as described earlier, namely (a) directly through the computed forces (b) from equation () and (c) from equation () ignoring the contribution of the shear stress gradients. It is shown that, up to y/h=.65, all three methods give similar results. In this region the has a constant value equal to approximately. Above this region the developed shear layer has as an effect the significant increase of up to.8 (depending on the ration H/h) at y/h equal to.9 and close to the top of vegetation, is decreasing. The distribution agrees qualitatively with that calculated by Nepf and Vinoni () from experimental measurements and disagrees with that of Poggi et al (4). The latter found a continuous decrease of from the bed channel to the top of vegetation which can not be explained from a physical point of view. It is observed that the method based on equation () gives similar

7 results with the direct method. However, some differences do exist due to the inaccurate calculation of the shear stress gradients (d<-uv /dy) in a region with high variation of <-uv > as shown in figure 3. The (c) method (ignoring these gradients) gives much lower values of in this region. This is expected since in this region the contribution of the shear stress gradients is significant (y-h)/(h-h) vegetation Huthoff and Augustijn (6) case 34- H/h=., S o =.6E-3 case 36- H/h=3.33,S o =.4E-3 case 38- H/h=4.4, S o =.4E-3 <U>/U * =(/κ)*ln[(y-h)u * /v]+5.5 (y-h)/(h-h) vegetation Huthoff and Augustijn (6) case 34- H/h=., S o =.6E-3 case 36- H/h=3.33, S o =.4E-3 case 38- H/h=4.4, S o =.4E-3 <k>/u * =D*e [-(y-h)/(h-h)] (y-h)/(h-h) vegetation Huthoff and Augustijn (6) case 34- H/h=., S o =.6E-3 case 36- H/h=3.33, S o =.4E-3 case 38- H/h=4.4, S o =.4E-3 <-uv>/u * =[-(y-h)/(h-h)] <U>/U * <k>/u *..4.6 <-uv>/u * Figure : Distribution of <U>, <k> and <- uv > within and above vegetation. Non-Staggered (Huthoff and Augustijn (6)) Non-Staggered (Poggi et al. (4)).6.6 y/h y/h.4. α=.48(m - ) case 34 -H/h=., S o =.6E-3 case 36-H/h=3.33, S o =.4E-3 case 38-H/h=4.4, S o =.4E-3.4. α=4.3(m - ) H/h=.66- S o=.6e-4 H/h=.66- S o =E-3 H/h=.66- S o=8e-3 H/h=3.33- S o =.6E-4 H/h=5- S o=.6e-4 H/h=5- S o=e F pr /F visc 3 F pr /F visc Figure 3: Variation of F pr /F vis with y/h Non-Staggered (Huthoff and Augustijn (6)) Non-Staggered (Huthoff and Augustijn (6)).6.6 y/h.4 H/h=., α=.48(m - ), S o=.6e-3 =(F D)/(ρA<U veg> ) =(/α<u veg > )(gs o ) =(/α<u veg > (gs o +d<-uv>/dy) y/h.4 H/h=4.4, α=.48(m - ), S o =.4E-3 =(F D )/(ρa<u veg > ) =(/α<u veg > )(gs o ) =(/α<u veg > (gs o +d<-uv>/dy) Figure 4: Variation of local with y/h In the following paragraphs the variation of the bulk (estimated as the average value of the previously described distributions of the local ) with parameters affecting the flow

8 hydrodynamics within and above vegetation is presented. These parameters are: (α) The Reynolds number Re, defined as Re =< Uveg > d r / ν where < U veg >=volume-averaged velocity within vegetation, dr=diameter of the cylinders (vegetation elements), (b) the ratio H/h, indicative of the degree of vegetation submergence (i.e. H/h=., emergent vegetation) and (c) the parameter αd r, representing the plant density. In addition, the bulk has been calculated from equation (3), as proposed by MacDonald et al. (6), and its variation with the above parameters is also presented. In this case the Re number has been calculated in a different way ( Re =< Utot > H / ν ). Figure 5 shows the variation of the bulk (calculated from equation (3)) with H/h, Re and αd r, for both computed results and experimental measurements of Dunn et al. (996), Poggi et al. (4) and Huthoff and Augustijn (6). Also, the empirical relationship proposed by Macdonald et al. (6) for Porites compress, is shown for comparison purposes. Most values follow the empirical trend but fall below the proposed relationship since the vegetation (cylindrical elements) used in these studies has smaller values of αd r. The values of the experimental for the cases of Huthoff and Augustijn (6) are higher than the proposed relationship but this may be due to the non-full development of flow over vegetation in these large scale experiments (H= m and h=.45 m). The variation of with the Re number shows a large scatter indicating that depends not only on the Re number but on other flow parameters. Similar conclusions can be derived from the variation of with αd r. For approximately the same αd r the varies from.5 to.5. In figure 6 the variation of the bulk, computed directly through the computed forces on the vegetation elements, with the same parameters is shown. The variation of with H/h does not show any clear relationship as previously, however most of the computed, take values between.9 and., indicating that an average value of. is a good approximation for to be used when needed. Similarly, the relationship of with Re number (Re varies between and ) is unclear, however again the values of are between.9 and. for Re between and. The variation of with αd r is similar as before, however the range of values is smaller than that of figure 5. Finally, figure 7 shows the variation of, calculated from equation (), with the three parameters H/h, Re and αd r, for both computed results of this study and experimental data of other studies. With this method is found to be increased with regard to that of figure 6 (values of are between. and.3). The low values of for the experiment of Huthoff and Augustijn (6) are due to no calculation the gradient of shear stress..6.4 =(ghs o )/<U tot > Poggi et al. (4)-simulation Huthoff and Augustijn (6)-simulation Dunn et al. (996)-simulation =.(H/h) Poggi et al. (4)-exp. Huthoff and Augustijn (6)-exp. Dunn et al. (996)-exp =(ghs o )/<U tot > Poggi et al. (4)-simulation Huthoff and Augustijn (6)-simulation Dunn et al. (996)-simulation Poggi et al. (4)-exp Huthoff and Augustijn (6)- exp Dunn et al. (996)- exp.5..5 =(ghs o )/<U tot > Poggi et al. (4)-simulation Huthoff and Augustijn (6)-simulation Dunn et al. (996)-simulation Poggi et al. (4)-exp. Huthoff and Augustijn (6)-exp. Dunn et al. (996)-exp H/h Re=(<U tot >H)/v.. αd r Figure 5: Variation of bulk (equation (3)) with H/h, Re and αd r

9 .. =(F D )/ρa<u veg > Poggi et al. (4)-simulation Huthoff and Augustijn (6)-simulation Dunn et al. (996)-simulation. =(F D )/ρa<u veg > Poggi et al. (4)-simulation Huthoff and Augustijn (6)-simulation Dunn et al. (996)-simulation....9 =(F D )/ρa<u veg > Poggi et al. (4)-simulation Huthoff and Augustijn (6)-simulation Dunn et al. (996)-simulation H/h Re=(<U veg >d r )/v.. αd r Figure 6: Variation of bulk (direct calculation) with H/h, Re and αd r.5.4 =(/α<u veg > )(gs o +d<-uv>/dy) Poggi et al. (4)-simulation Huthoff and Augustijn (6)-simulation Dunn et al. (996)-simulation Huthoff and Augustijn (6)-exp. Dunn et al. (996)-exp..4.5 =(/α<u veg > )(gs o +d<-uv>/dy) Poggi et al. (4)-simulation Huthoff and Augustijn (6)-simulation Dunn et al. (996)-simulation Huthoff and Augustijn (6)-exp =(/α<u veg > )(gs o +d<-uv>/dy) Poggi et al. (4)-simulation Huthoff and Augustijn (6)-simulation Dunn et al. (996)-simulation Huthoff and Augustijn (6)-exp. Dunn et al. (996)-exp H/h Re=<U veg >d r /v.. αd r Figure 7: Variation of bulk (equation ()) with H/h, Re and αd r 5. CONCLUSIONS Three dimensional computations, based on the RANS equations and a turbulence model of the k-ε type, have been performed for investigating flow characteristics within and over submerged, rigid vegetation. Emphasis is given on the calculation of drag resistance with direct and indirect methods. Four methods have been used for the calculation of : (a) direct calculation from the computed distribution of the respective forces and (b) three indirect methods based on the simplified momentum equation (by including and excluding the gradients of the significant shear stress) and on the force balance between gravity force and drag force. The following conclusions can be derived: (a) Distributions of <U>, <k> and <-uv > within and above submerged vegetation show the significant effects of vegetation on the flow characteristics and the discharge capacity of channels with submerged vegetation. (b) The distribution of local, calculated directly from the computed forces, indicated three layers in which (i) up to y/h=.7 the is constant, (ii) for y/h between.7 to.9 the is increasing and (iii) for y/h between.9 to. the is decreasing. (c) The indirect method of calculating, through the simplified momentum equation, slightly underestimates the values of when compared with the direct method. (d) For calculated based on the bulk velocity and the total flow depth there is a clear relationship between and H/h, however there is a significant scatter when is related to Re and αd r. (e) An average value of (calculated directly) equal to. was found for the condition examined (.<H/h<5. and Re numbers between and ).

10 REFERENCES Dunn, C., López, F. and García, M.H., (996), Mean flow and turbulence in a laboratory channel with simulated vegetation, Civil Engineering Studies, Hydraulic Engineering Series No. 5. Finnigan, J., (), Turbulence in Plant Canopies, Annual Review of Fluid Mechanics, 3, pp Fluent Inc. (), Fluent 6. Documentation, Lebanon, USA. Huthoff, F. and Augustijn, D. (6), Hydraulic resistance of vegetation: Predictions of average flow velocities based on a rigid-cylinders analogy, Final Project Report, Planungsmanagement fur Auen ( Project no.u/43.9/468). McDonald, C.B., Koseff, J.R., Monismith, S.G. (6), Effects of the depth to coral height ratio on drag coefficients for unidirectional flow over coral, Limnological Oceanography, 5(3), pp Nepf, H.M. (999), Drag, turbulence, and diffusion in flow through emergent vegetation, Water Resources Research, 35(), pp Nepf, H.M. and Vivoni, E.R., (), Flow structure in depth-limited, vegetated flow, Journal of Geophysical Research, 5(C), pp Nezu, I. and Nakagawa, H., (993), Turbulence in open-channel flows, IAHR Monograph, Balkema, Rotterdam, The Netherlands Novak, M. D., Warland, J. S., Orchansky, A.L., Ketler, R., and Green, S. (), Comparison between wind tunnel and field measurements of turbulent flow. Part : Uniformly Thinned Forests, Boundary-Layer Meteorology, 95, pp Poggi, D., Porporato, A., Ridolfi, L., Albertson, D.-J. and Katul G.-G. (4), The effect of vegetation density on canopy sub-layer turbulence, Boundary-Layer Meteorology,, pp Souliotis, D. and Prinos, P., (6), Vegetation Turbulence: From RANS Micro-computations to Macro-analysis, 7 th International Conference of Hydroscience and Engineering (ICHE 6), (Proceedings to be published) Stone, B.M. and Shen, H.T. (), Hydraulic resistance of flow in channels with cylindrical roughness, Journal of Hydraulic Engineering, ASCE, 8 (5), pp Wu, F.C, Shen, H.W and Chou, Y.J. (999), Variation of roughness coefficients for unsubmerged and submerged vegetation, Journal of Hydraulic Engineering, ASCE, 5(9), pp