A Simple Model for Spatially-averaged Wind Profiles Within and Above an Urban Canopy

This paper deals with the modelling of the flow in the urban canopy layer. It critically reviews a well-known formula for the spatially-averaged wind profile, originally proposed by Cionco in 1965, and provides a new interpretation for it. This opens up a number of new applications for modelling mean wind flow over the neighbourhood scale. The model is based on a balance equation between the obstacle drag force and the local shear stress as proposed by Cionco for a vegetative canopy. The buildings within the canopy are represented as a canopy element drag formulated in terms of morphological parameters such as λ _f and λ _p (the ratios of plan area and frontal area of buildings to the lot area). These parameters can be obtained from the analysis of urban digital elevation models. The shear stress is parameterised using a mixing length approach. Spatially-averaged velocity profiles for different values of building packing density corresponding to different flow regimes are obtained and analysed. The computed solutions are compared with published data from wind-tunnel and water-tunnel experiments over arrays of cubes. The model is used to estimate the spatially-averaged velocity profile within and above neighbourhood areas of real cities by using vertical profiles of λ _f .     

Estimating Aerodynamic Parameters of Urban-Like Surfaces with Heterogeneous Building Heights

There are many geometrical factors than can influence the aerodynamic parameters of urban surfaces and hence the vertical wind profiles found above. The knowledge of these parameters has applications in numerous fields, such as dispersion modelling, wind loading calculations, and estimating the wind energy resource at urban locations. Using quasi-empirical modelling, we estimate the dependence of the aerodynamic roughness length and zero-plane displacement for idealized urban surfaces, on the two most significant geometrical characteristics; surface area density and building height variability. A validation of the spatially-averaged, logarithmic wind profiles predicted by the model is carried out, via comparisons with available wind-tunnel and numerical data for arrays of square based blocks of uniform and heterogeneous heights. The model predicts two important properties of the aerodynamic parameters of surfaces of heterogeneous heights that have been suggested by experiments. Firstly, the zero-plane displacement of a heterogeneous array can exceed the surface mean building height significantly. Secondly, the characteristic peak in roughness length with respect to surface area density becomes much softer for heterogeneous arrays compared to uniform arrays, since a variation in building height can prevent a skimming flow regime from occurring. Overall the simple model performs well against available experimental data and may offer more accurate estimates of surface aerodynamic parameters for complex urban surfaces compared to models that do not include height variability.     

Large-Eddy Simulation of Flows over Random Urban-like Obstacles

Further to our previous large-eddy simulation (LES) of flow over a staggered array of uniform cubes, a simulation of flow over random urban-like obstacles is presented. To gain a deeper insight into the effects of randomness in the obstacle topology, the current results, e.g. spatially-averaged mean velocity, Reynolds stresses, turbulence kinetic energy and dispersive stresses, are compared with our previous LES data and direct numerical simulation data of flow over uniform cubes. Significantly different features in the turbulence statistics are observed within and immediately above the canopy, although there are some similarities in the spatially-averaged statistics. It is also found that the relatively high pressures on the tallest buildings generate contributions to the total surface drag that are far in excess of their proportionate frontal area within the array. Details of the turbulence characteristics (like the stress anisotropy) are compared with those in regular roughness arrays and attempts to find some generality in the turbulence statistics within the canopy region are discussed.     

Numerical Simulation of Drag Partition over Rough Surfaces

We present a numerical simulation of drag partition over rough surfaces. A computational fluid dynamics model is applied with high resolution to simulatingturbulent flows over arrays of roughness elements positioned on asmooth surface. The skin drag on the surface and the pressure drag on the roughnesselements are computed. The simulated drag partition compares well with wind-tunnelmeasurements and theoretical estimates for similar rough surfaces. This confirms that the computational approach offers an alternative to wind-tunnel and field experiments in studying drag and drag partition. The model is then applied to studying drag partition over rough surfaces with various roughness configurations. It is shown that drag partition depends not only on the magnitude of the roughness frontal area but also on the sizes and arrangement of roughness elements, because (1) the pressure drag coefficient is sensitive to roughness-element dimensions and (2) the arragement of roughness elements lead to different interferences of turbulent wakes. The impact ofthe latter factor is not insignificant.     

Modelling The Mean Velocity Profile In The Urban Canopy Layer

A simple model originally derived for meanwind speed profiles in vegetative canopy flows ismodified for application to arrays ofthree-dimensional surface obstacles (cubes), whichcould be representative of a simple urban-typesurface. It is shown that for cube arrays that arenot too densely packed, the predicted exponentialvelocity profile provides an adequate fit to thespatially averaged velocity profile (u(z))within the obstacle canopy. Application of the model to a set of wind-tunnel dataallows for the evaluation of an empirical fittingparameter called the attenuation coefficient. This isrelated to the turbulence length scale, which can befound by manipulating the results of thegradient-diffusion model used to derive the velocityprofile. The results show a reduction of theturbulence length scale with increasing obstaclepacking density. By assuming a linear transition fromthis length scale at the top of the canopy to theclassical Prandtl length scale in the overlyinginertial sublayer, an acceptable model is obtained forthe full velocity profile within simple obstaclearrays, from the ground up to the overlyingsemi-logarithmic region.     

A New Aerodynamic Parametrization for Real Urban Surfaces

This study conducted large-eddy simulations (LES) of fully developed turbulent flow within and above explicitly resolved buildings in Tokyo and Nagoya, Japan. The more than 100 LES results, each covering a 1,000 $\times $ 1,000 m $^{2}$ area with 2-m resolution, provide a database of the horizontally-averaged turbulent statistics and surface drag corresponding to various urban morphologies. The vertical profiles of horizontally-averaged wind velocity mostly follow a logarithmic law even for districts with high-rise buildings, allowing estimates of aerodynamic parameters such as displacement height and roughness length using the von Karman constant $=$ 0.4. As an alternative derivation of the aerodynamic parameters, a regression of roughness length and variable Karman constant was also attempted, using a displacement height physically determined as the central height of drag action. Although both the regression methods worked, the former gives larger (smaller) values of displacement height (roughness length) by 20–25 % than the latter. The LES database clearly illustrates the essential difference in bulk flow properties between real urban surfaces and simplified arrays. The vertical profiles of horizontally-averaged momentum flux were influenced by the maximum building height and the standard deviation of building height, as well as conventional geometric parameters such as the average building height, frontal area index, and plane area index. On the basis of these investigations, a new aerodynamic parametrization of roughness length and displacement height in terms of the five geometric parameters described above was empirically proposed. The new parametrizations work well for both real urban morphologies and simplified model geometries.     

A comparative study of some mathematical models of the mean wind structure and aerodynamic drag of plant canopies

A semi-analytical method for describing the mean wind profile and shear stress within plant canopies and for estimating the roughness length and the displacement height is presented. This method incorporates density and vertical structure of the canopy and includes simple parameterizations of the roughness sublayer and shelter factor. Some of the wind profiles examined are consistent with first-order closure techniques while others are consistent with second-order closure techniques. Some profiles show a shearless region near the base of the canopy; however, none displays a secondary maximum there. Comparing several different analytical expressions for the canopy wind profile against observations suggests that one particular type of profile (an Airy function which is associated with the triangular foliage surface area density distribution) is superior to the others. Because of the numerical simplicity of the methods outlined, it is suggested that they may be profitably used in large-scale models of plant-atmosphere exchanges.     

Aerodynamic Parameters of Regular Arrays of Rectangular Blocks with Various Geometries

The aerodynamic effects of various configurations of an urban array were investigated in a wind-tunnel experiment. Three aerodynamic parameters characterising arrays—the drag coefficient (C _d ), roughness length (z _o) and displacement height (d)—are used for analysis. C _d is based on the direct measurement of the total surface shear using a floating element, and the other two parameters are estimated by logarithmic fitting of the measured wind profile and predetermined total drag force. The configurations of 63 arrays used for measurement were designed to estimate the effects of layout, wind direction and the height variability of the blocks on these parameters for various roughness packing densities. The results are summarised as follows: (1) The estimated C _d and z _o of the staggered arrays peak against the plan area index (λ _p ) and frontal area index (λ _f ), in contrast with values for the square arrays, which are less sensitive to λ _p and λ _f . In addition, the square arrays with a wind direction of 45° have a considerably larger C _d , and the wind direction increases z _o/H by up to a factor of 2. (2) The effect of the non-uniformity of roughness height on z _o is more remarkable when λ _f exceeds 20%, and the discrepancy in z _o is particularly remarkable and exceeds 200%. (3) The effect of the layout of tall blocks on C _d is stronger than that of short blocks. These results indicate that the effects of both wind direction and the non-uniformity of the heights of buildings on urban aerodynamic parameters vary greatly with λ _p and λ _f ; hence, these effects should be taken into account by considering the roughness packing density.     

Determination of Roughness Lengths for Heat and Momentum Over Boreal Forests

The roughness length for momentum (z_0m), zero-plane displacementheight (d), and roughness length for heat (z_0h) are importantparameters used to estimate land-atmosphere energy exchange. Although many different approaches have been developed to parameterizemomentum and heat transfer, existing parameterizations generally utilizehighly simplified representations of vegetation structure. Further, a mismatch exists between the treatments used for momentum and heat exchange and those used for radiative energy exchanges. In this paper, parameterizations are developed to estimate z_0m, d, and z_0h for forested regimes using information related to tree crown density and structure. The parameterizations provide realistic representationfor the vertical distribution of foliage within canopies, and include explicit treatment for the effects of the canopy roughness sublayer and leaf drag on momentum exchange. The proposed parameterizationsare able to realistically account for site-to-site differences in roughness lengths that arise from canopy structural properties.Comparisons between model predictions and field measurements show good agreement, suggesting that the proposed parameterizations capture the most important factors influencing turbulent exchange of momentumand heat over forests.     

Using Temperature Fluctuation Measurements to Estimate Meteorological Inputs for Modelling Dispersion During Convective Conditions in Urban Areas

We examine the performance of several methods to estimate meteorological inputs for modelling dispersion in urban areas during convective conditions. Sensible heat flux, surface friction velocity and turbulent velocities are estimated from measurements of mean wind speed and the standard deviation of temperature fluctuations at a single level on a tower at two suburban sites and at one urban site in Riverside, California. These estimates are compared with observations made at these sites during a field study conducted in 2007. The sensible heat flux is overestimated in the urban area, while it is underestimated at a suburban site when temperature fluctuations are used in the free convection formulation to estimate heat flux. The bias in heat flux estimates can be reduced through a correction that depends on stability. It turns out that the bias in heat flux estimates has a minor effect on the prediction of surface friction velocity and turbulent velocities. Estimates of sensible heat flux, surface friction velocity and turbulent velocities are sensitive to estimates of aerodynamic roughness length, and we suggest estimating the aerodynamic roughness length through detailed micrometeorological measurements made during a limited field study. An examination of the impact of the uncertainty in estimating surface micrometeorology on concentrations indicates that, at small distances from a surface release, ground-level concentrations computed using estimates of heat flux and surface friction compare well with the those based on observed values: the bias is small and the 95% confidence interval of the ratio of the two concentrations is 1.7. However, at distances much larger than the Obukhov length, this confidence interval is close to 2.3 because errors in both friction velocity and heat flux affect plume spread. Finally, we show that using measurements of temperature fluctuations in estimating heat flux is an improvement on that based on the surface energy balance, even when net radiation measurements are available.     

Roughness-Length Model for Organized Rough Walls

A model for the roughness length and its correlation with the roughness shear stress on organized rough walls of varying geometry are presented and verified. The roughness length is nondimensionalized by the characteristic roughness length and is expressed as a function of roughness density with a wake-interference parameter. The dimensionless roughness length is independent of Reynolds number. When the model is applied to the whole range of roughness densities, the rough walls can be smooth, transitionally rough, and fully rough. A large number of data from classical experiments and recent simulations are analyzed to evaluate the proposed correlations, which are found to be consistent with the analyzed datasets. The proposed expression for the dimensionless roughness length and the expression for the dimensionless roughness shear stress, proposed previously by the author (Boundary-Layer Meteorology, 2020, Vol. 174, 393–410), are found to be identical in form. Numerous extant measurements of the two roughness parameters can be reproduced when the wake-interference parameters in the two models are treated as identical. The parameters of the roughness-length model are closely related to the geometry of the roughness elements. Different types of roughness elements can be distinguished by the values of the parameters. These results provide the foundation for constructing the unified roughness model for organized rough walls of varying geometry.

     

Wind-Direction Effects on Urban-Type Flows

Practically all extant work on flows over obstacle arrays, whether laboratory experiments or numerical modelling, is for cases where the oncoming wind is normal to salient faces of the obstacles. In the field, however, this is rarely the case. Here, simulations of flows at various directions over arrays of cubes representing typical urban canopy regions are presented and discussed. The computations are of both direct numerical simulation and large-eddy simulation type. Attention is concentrated on the differences in the mean flow within the canopy region arising from the different wind directions and the consequent effects on global properties such as the total surface drag, which can change very significantly—by up to a factor of three in some circumstances. It is shown that for a given Reynolds number the typical viscous forces are generally a rather larger fraction of the pressure forces (principally the drag) for non-normal than for normal wind directions and that, dependent on the surface morphology, the average flow direction deep within the canopy can be largely independent of the oncoming wind direction. Even for regular arrays of regular obstacles, a wind direction not normal to the obstacle faces can in general generate a lateral lift force (in the direction normal to the oncoming flow). The results demonstrate this and it is shown how computations in a finite domain with the oncoming flow generated by an appropriate forcing term (e.g. a pressure gradient) then lead inevitably to an oncoming wind direction aloft that is not aligned with the forcing term vector.     

Large-Eddy Simulations on the Effects of Surface Geometry of Building Arrays on Turbulent Organized Structures

Turbulent organized structures (TOS) above building arrays were investigated using a large-eddy simulation (LES) model for a city (LES-CITY). Square and staggered building arrays produced contrasting behaviour in terms of turbulence that roughly corresponded to the conventional classification of ‘D-type’ and ‘K-type’ roughness, respectively: (1) The drag coefficients (referred to the building height) for staggered arrays were sensitive to building area density, but those for square arrays were not. (2) The relative contributions of ejections to sweeps (S₂/S₄) at the building height for square arrays were sensitive to building area density and nearly equalled or exceeded 1.0 (ejection dominant), but those for staggered arrays were insensitive to building area density and were mostly below 1.0 (sweep dominant). (3) Streaky patterns of longitudinal low speed regions (i.e., low speed streaks) existed in all flows regardless of array type. Height variations of the buildings in the square array drastically increased the drag coefficient and modified the turbulent flow structures. The mechanism of D-type and K-type urban-like roughness flows and the difference from vegetation flows are discussed. Although urban-like roughness flows exhibited mixed properties of mixing layers and flat-wall boundary layers as far as S₂/S₄ was concerned, the turbulent organized structures of urban-like roughness flows resembled those of flat-wall boundary layers.     

Estimation of Roughness Parameters Within Sparse Urban-Like Obstacle Arrays

We conduct wind-tunnel experiments on three different uniform roughness arrays composed of sparsely distributed rectangular cylinders for the estimation of surface parameters. Roughness parameters such as the roughness length z ₀ and zero-plane displacement d are extracted using a best-fit approximation of the measured wind velocity. We also perform a large-eddy simulation (LES) to confirm that four sampling points are sufficient to surrogate a space average above the canopy layer of the sparse roughness arrays. We propose a new morphological model from a systematic analysis of experimental data on the arrays. The friction velocity predicted by the proposed model agrees well with the peak value of the measured Reynolds shear stress ${(-\left<\overline{u'w'}\right>)^{0.5}}${(-\left<\overline{u'w'}\right>)^{0.5}}. The proposed model is further validated in an additional wind-tunnel experiment conducted on a scaled configuration of a real urban area exposed to four wind directions. The proposed model is found to perform very well particularly in the estimation of the friction velocity, readily leading to a better estimation of turbulence, which is essential for an accurate prediction of pollutant dispersion.     

Computational parameter estimation for a maize crop

During a whole growing season, the evolution of the displacement height, d, and roughness length, z ₀, of a maize crop has been estimated by a measurement programme. The results have been used to check different types of existing models to calculate these parameters from canopy characteristics only; a simple geometric model and two matching models have been investigated. A geometric model is based on geometric features of the surface only. After a simple modification, the geometric model gives good results for the displacement height as well as for the roughness length.A matching model, based on gradient-diffusion theory, yields good results for the displacement height. The roughness parameter, however, is overestimated by 17%. By a simple modification, the model results could be improved considerably.A matching model, based on a second-order closure procedure, yields excellent results for the displacement height and good results for the roughness length. But it appears that, when applying this model, the plant density index and plant area density distribution as a function of height must be well known.     

Upscaling of Land-Surface Parameters Through Inverse Stochastic SVAT-Modelling

A novel approach for upscaling land-surface parameters based on inverse stochastic surface-vegetation-atmosphere transfer (SVAT) modelling is presented. It allows estimation of effective parameters that yield scale invariant outputs e.g. for sensible and latent heat fluxes and evaporative fraction. The general methodology is used to estimate effective parameters for the Oregon State University Land-Surface Model, including surface albedo, surface emissivity, roughness length, minimum stomatal resistance, leaf area index, vapour pressure deficit factor, solar insolation factor and the Clapp–Hornberger soil parameter. Upscaling laws were developed that map the mean and standard deviation of the distributed land-surface parameters at the subgrid scale to their corresponding effective parameter at the grid scale. Both linear and bi-parabolic upscaling laws were obtained for the roughness length. The bi-parabolic upscaling law fitted best for the remaining land-surface parameters, except surface albedo and emissivity, which were best fitted with linear upscaling laws.     

Aerodynamic Parameters of Urban Building Arrays with Random Geometries

It is difficult to describe the flow characteristics within and above urban canopies using only geometrical parameters such as plan area index (λ _p ) and frontal area index (λ _f ) because urban surfaces comprise buildings with random layouts, shapes, and heights. Furthermore, two types of ‘randomness’ are associated with the geometry of building arrays: the randomness of element heights (vertical) and that of the rotation angles of each block (horizontal). In this study, wind-tunnel experiments were conducted on seven types of urban building arrays with various roughness packing densities to measure the bulk drag coefficient (C _d ) and mean wind profile; aerodynamic parameters such as roughness length (z _o ) and displacement height (d) were also estimated. The results are compared with previous results from regular arrays having neither ‘vertical’ nor ‘horizontal’ randomness. In vertical random arrays, the plot of C _d and z _o versus λ _f exhibited a monotonic increase, and z _o increased by a factor of almost two for λ _f = 48–70%. C _d was strongly influenced by the standard deviation of the height of blocks (σ) when λ _p ≥ 17%, whereas C _d was independent of σ when λ _p = 7%. In the case of horizontal random arrays, the plot of the estimated C _d against λ _f showed a peak. The effect of both vertical and horizontal randomness of the layout on aerodynamic parameters can be explained by the structure of the vortices around the blocks; the aspect ratio of the block is an appropriate index for the estimation of such features.     

Landscape Roughness Parameters for Sherwood Forest – Experimental Results

The aim of this work is to present experimentally evaluated effective roughnesses (zoe) of a partly forested landscape. Although the ratio of boundary-layer height to obstacle size was only of the order of 50, there still seemed to exist a height range of 75–200 m where surface-layer similarity was approximately valid. Attempts were made to use conventional wind profile analysis to evaluate zoe, but the small height range and the large number of variables initially led to unacceptable uncertainties. Fixing the displacement height zd, rather than fitting it, reduced the data scatter to an acceptable level. The profile-derived roughness lengths zop obtained in this way were in good agreement with previous work, and with an alternative roughness length estimate zof for which flux-derived profile parameters u* and * were used. This implies that the profile-derived roughnesses were consistent with the measured surface-layer momentum flux. Comparison of both roughness estimates also yielded an improved estimate of the displacement height. Besides this, the authors tested a landscape roughness evaluation method which makes use of the gustiness parameter Tu = u/U in the surface layer. The results obtained by this method were in fair agreement with the profile-derived data. In previous work, the gustiness method was advocated because it could be used at relatively low levels, perhaps even within the roughness sub-layer. At the present measuring site, this was not the case as the gustiness method was only valid in an approximate way, and for a limited height range.     

Shear stress partitioning in large patches of roughness in the atmospheric inertial sublayer

Drag partition measurements were made in the atmospheric inertial sublayer for six roughness configurations made up of solid elements in staggered arrays of different roughness densities. The roughness was in the form of a patch within a large open area and in the shape of an equilateral triangle with 60 m long sides. Measurements were obtained of the total shear stress (τ) acting on the surfaces, the surface shear stress on the ground between the elements (τ_S) and the drag force on the elements for each roughness array. The measurements indicated that τ_S quickly reduced near the leading edge of the roughness compared with τ, and a τ_S minimum occurs at a normalized distance (x/h, where h is element height) of (downwind of the roughness leading edge is negative), then recovers to a relatively stable value. The location of the minimum appears to scale with element height and not roughness density. The force on the elements decreases exponentially with normalized downwind distance and this rate of change scales with the roughness density, with the rate of change increasing as roughness density increases. Average τ_S : τ values for the six roughness surfaces scale predictably as a function of roughness density and in accordance with a shear stress partitioning model. The shear stress partitioning model performed very well in predicting the amount of surface shear stress, given knowledge of the stated input parameters for these patches of roughness. As the shear stress partitioning relationship within the roughness appears to come into equilibrium faster for smaller roughness element sizes it would also appear the shear stress partitioning model can be applied with confidence for smaller patches of smaller roughness elements than those used in this experiment.     

Wind profiles,momentum fluxes and roughness lengths at Cabauw revisited

We describe the results of an experiment focusing on wind speed and momentum fluxes in the atmospheric boundary layer up to 200 m. The measurements were conducted in 1996 at the Cabauw site in the Netherlands. Momentum fluxes are measured using the K-Gill Propeller Vane. Estimates of the roughness length are derived using various techniques from the wind speed and flux measurements, and the observed differences are explained by considering the source area of the meteorological parameters. A clear rough-to-smooth transition is found in the wind speed profiles at Cabauw. The internal boundary layer reaches the lowest k-vane (20 m) only in the south-west direction where the obstacle-free fetch is about 2 km. The internal boundary layer is also reflected in the roughness lengths derived from the wind speed profiles. The lower part of the profile (< 40 m) is not in equilibrium and no reliable roughness analysis can be given. The upper part of the profile can be linked to a large-scale roughness length. Roughness lengths derived from the horizontal wind speed variance and gustiness have large footprints and therefore represent a large-scale average roughness. The drag coefficient is more locally determined but still represents a large-scale roughness length when it is measured above the local internal boundary layer. The roughness length at inhomogeneous sites can therefore be determined best from drag coefficient measurements just above the local internal boundary layers directly, or indirectly from horizontal wind speed variance or gustiness. In addition, the momentum and heat fluxes along the tower are analysed and these show significant variation with height related to stability and possibly surface heterogeneity. It appears that the dimensionless wind speed gradients scale well with local fluxes for the variety of conditions considered, including the unstable cases.