Turbulence modelling over two-dimensional hills using an Algebraic Reynolds Stress Expression

We carry out model studies of turbulence quantities for flow over two-dimensional hills using a non-hydrostatic version of the Regional Atmospheric Modeling System (RAMS). We test two turbulence closure models: the first one is an explicit Algebraic Reynolds Stress Model (ARSM) and the second one is a combination of the ARSM and a transport equation for the shear stress {ovuw}. Model predictions for the turbulent stresses are compared with data from a wind-tunnel experiment containing isolated two-dimensional hills of varying slope. From the comparison, it is concluded that the first model can only predict the normal stresses adequately while the second model provides satisfactory predictions for the normal stresses as well as giving an improved result for the shear stress {ovuw}.     

A comparison of wind-tunnel experiments and numerical simulations of neutral and stratified flow over a hill

A comparison is made of numerical and experimental results for flow over two-dimensional hills in both neutral and stably stratified flow. The numerical simulations are carried out using a range of one-and-a-half order and second-order closure schemes. The performance of the various turbulence schemes in predicting both the mean and turbulent quantities over the hill is assessed by comparing the results with new wind-tunnel measurements. The wind-tunnel experiments include both neutral and stably stratified flow over two different hills with different slopes, one of which is steep enough to induce flow separation. The dataset includes measurements of the mean and turbulent parts of the flow using laser Doppler anemometry. Pressure measurements are also made across the surface of the hill. These features make the dataset an excellent test of the model performance. In general second-order turbulence schemes provide the best agreement with the experimental data, however, they can be numerically unstable for steep hills. Some modifications can be made to the standard one-and-a-half order closure scheme, which results in improved performance at a fraction of the computation cost of the second-order schemes.     

Numerical simulation of flow data over two-dimensional hills

We have studied a neutrally-stratified flow over two-dimensional hills using a two-dimensional, non-hydrostatic version of the Regional Atmospheric Modeling System (RAMS). We have implemented three different turbulence closure models: the standardE- model, an Algebraic Reynolds Stress Model (ARSM) and a new model. Model predictions for the mean and turbulence flows using different closure schemes are compared with the data of a wind tunnel experiment containing isolated two-dimensional hills of varying slope. From the comparison, it is concluded that all three models predict the mean flow velocities equally well while only the new closure model accurately predicts the turbulence data statistics.The research reported in this paper was conducted while the first author held a National Research Council (NRC) Associateship.     

An Investigation of Higher-Order Closure Models for a Forested Canopy

Simultaneous triaxial sonic anemometer velocity measurements vertically arrayed at six levels within and above a uniform pine forest were used to examine two parameterization schemes for the triple-velocity correlation tensor employed in higher-order closure models. These parameterizations are the gradient-diffusion approximation typically used in second-order closure models, and the full budget for the triple-velocity correlation tensor typically employed in third-order closure models. Both second- and third-order closure models failed to reproduce the measured profiles of the triple-velocity correlation within and above the canopy. However, the Reynolds stress tensor profiles (including velocity variances) deviated greatly from the measurements only within the lower levels of the canopy. It is shown that the Reynolds stresses are most sensitive to the parameterization of the triple-velocity correlation in these lower canopy regions where local turbulent production is negligible and turbulence is mainly sustained by the flux transport term. The failure of the third-order closure model to reproduce the measured third moments in the upper layers of the canopy-top contradicts conclusions from a previous study over shorter vegetation but agrees with another study for a deciduous forest. Whether the third-order closure model failure is due to the zero-fourth-cumulant closure approximation is therefore considered. Comparisons between measured and predicted quadruple velocity correlations suggest that the zero-fourth-cumulant approximation is valid close to the canopy-atmosphere in agreement with recent experiments.     

Numerical Modelling of the Turbulent Flow Developing Within and Over a 3-D Building Array, Part I: A High-Resolution Reynolds-Averaged Navier—Stokes Approach

A study of the neutrally-stratified flow within and over an array of three-dimensional buildings (cubes) was undertaken using simple Reynolds-averaged Navier—Stokes (RANS) flow models. These models consist of a general solution of the ensemble-averaged, steady-state, three-dimensional Navier—Stokes equations, where the k-ε turbulence model (k is turbulence kinetic energy and ε is viscous dissipation rate) has been used to close the system of equations. Two turbulence closure models were tested, namely, the standard and Kato—Launder k-ε models. The latter model is a modified k-ε model designed specifically to overcome the stagnation point anomaly in flows past a bluff body where the standard k-ε model overpredicts the production of turbulence kinetic energy near the stagnation point. Results of a detailed comparison between a wind-tunnel experiment and the RANS flow model predictions are presented. More specifically, vertical profiles of the predicted mean streamwise velocity, mean vertical velocity, and turbulence kinetic energy at a number of streamwise locations that extend from the impingement zone upstream of the array, through the array interior, to the exit region downstream of the array are presented and compared to those measured in the wind-tunnel experiment. Generally, the numerical predictions show good agreement for the mean flow velocities. The turbulence kinetic energy was underestimated by the two different closure models. After validation, the results of the high-resolution RANS flow model predictions were used to diagnose the dispersive stress, within and above the building array. The importance of dispersive stresses, which arise from point-to-point variations in the mean flow field, relative to the spatially-averaged Reynolds stresses are assessed for the building array.     

Large-eddy Simulations of Flow Over Forested Ridges

Large-eddy simulations (LES) of flow over a series of small forested ridges are performed, and compared with numerical simulations using a one-and-a-half order mixing length closure scheme. The qualitative and quantitative similarity between these results provides some confidence in the results of recent analytical and numerical studies of flow over forested hills using first-order mixing length schemes. Time series of model velocities at various locations within the canopy allow the application of various experimental techniques to study the turbulence in the LES. The application of conditional analysis shows that the structure of the turbulence over a forested hill is broadly similar to that over flat ground, with sweeps and ejections dominating. Differences are seen across the hill, particularly associated with regions of mean flow separation and recirculation near the summit and in the lee of the hill. Detailed comparison of derived mixing lengths from the LES with the assumed values used in mixing-length closure schemes show that the mixing length varies with location across the hill and with height in the canopy. This is consistent with previous wind-tunnel measurements, and demonstrates that a constant mixing-length assumption is not strictly valid within the canopy. Despite this, the first-order mixing-length schemes do give similar results both for the mean flow and the turbulence in such situations.     

Scalar Transport over Forested Hills

Numerical simulations of scalar transport in neutral flow over forested ridges are performed using both a 1.5-order mixing-length closure scheme and a large-eddy simulation. Such scalar transport (particularly of CO₂) has been a significant motivation for dynamical studies of forest canopy–atmosphere interactions. Results from the 1.5-order mixing-length simulations show that hills for which there is significant mean flow into and out of the canopy are more efficient at transporting scalars from the canopy to the boundary layer above. For the case with a source in the canopy this leads to lower mean concentrations of tracer within the canopy, although they can be very large horizontal variations over the hill. These variations are closed linked to flow separation and recirculation in the canopy and can lead to maximum concentrations near the separation point that exceed those over flat ground. Simple scaling arguments building on the analytical model of Finnigan and Belcher (Q J Roy Meteorol Soc 130:1–29, 2004) successfully predict the variations in scalar concentration near the canopy top over a range of hills. Interestingly this analysis suggests that variations in the components of the turbulent transport term, rather than advection, give rise to the leading order variations in scalar concentration. The scaling arguments provide a quantitative measure of the role of advection, and suggest that for smaller/steeper hills and deeper/sparser canopies advection will be more important. This agrees well with results from the numerical simulations. A large-eddy simulation is used to support the results from the mixing-length closure model and to allow more detailed investigation of the turbulent transport of scalars within and above the canopy. Scalar concentration profiles are very similar in both models, despite the fact that there are significant differences in the turbulent transport, highlighted by the strong variations in the turbulent Schmidt number both in the vertical and across the hill in the large-eddy simulation that are not represented in the mixing-length model.     

On the Parameterisation of the Effective Roughness Length for Momentum Transfer over Heterogeneous Terrain

We develop a parameterisation for the effective roughness length of terrain that consists of a repeating sequence of patches, in which each patch is composed of strips of two roughness types. A numerical model with second-order closure in the turbulent stress is developed and used to show that: (i) the normalised Reynolds stress develops as a self-similar profile; (ii) the mixing-length parameterisation is a good first-order approximation to the Reynolds stress. These findings are used to characterise the blending layer, where the stress adjusts smoothly from its local surface value to its effective value aloft. Previous studies have assumed that this adjustment occurs abruptly at a single level, often called the blending height. The blending layer is shown to be characterised by height scales that arise naturally in linear models of surface layer flow over roughness changes, and calculations with the numerical model show that these height scales remain appropriate in the nonlinear regime. This concept of the blending layer allows the development of a new parameterisation of the effective roughness length, which gives values for the effective roughness length that are shown to compare well with both atmospheric measurements and values determined from the second-order model.     

Note on the Growth Rate of Water Waves Propagating at an Arbitrary Angle to the Wind

The dependence of the turbulent airflow over water waves on the angle,, between mean wind and wavedirections is investigated. To this end,an existing semi-analytical model is extended. In this model, the main simplification of the problem is obtained by using the well-established divisionof the wave boundary layer into inner and outer regions for modelling turbulence. The effect of waves on turbulence is restricted to the thin inner region. Simulations show that the influence of the wind speed component transverse to the wave direction on the air flow, and hence on the growth rate of the waves, is small. This is confirmed by calculations with a numerical model that solves the full Reynolds equations using a second-order turbulence closure scheme. The growth rate of slowly moving waves (as compared to the wind speed) is then proportional to cos², whereas, for faster waves, it has a narrower angular distribution.     

Eddy exchange coefficients in numerical models of the planetary boundary layer

Specification of the eddy exchange coefficients is perhaps one of the most difficult problems in the numerical modeling of the planetary boundary layer. These coefficients have been computed from finite-difference analogs to analytical expressions associated with surface boundary-layer similarity theory, which is based on observations in an equilibrium surface layer. This procedure leads to erroneous results in the region above the surface layer and in a non-equilibrium surface layer. In addition, differencing problems arise in regions of small vertical wind shear. A new turbulence transport model has been obtained through the closure procedures for the transport equations of the Reynolds stress and the turbulent length scale. The new approach could be used to calculate Reynolds stresses and eddy exchange coefficients throughout a non-neutral planetary boundary layer under non-equilibrium conditions.     

Large-Eddy Simulation Of Neutral Turbulent Flow Over Rough Sinusoidal Ridges

The requirements for a credible large-eddy simulation of neutral, turbulent flow over hills with an aerodynamically rough surface are discussed, in order to select a suitable case for simulation. As well as providing adequate resolution within the dynamically important inner region, obtaining a realistic upstream or undisturbed mean velocity profile is also of critical importance. A distributed drag canopy formulation has been introduced to the model to allow it to obtain such a profile while resolving very close to the rough surface. Simulations have then been performed of flow over ridges of varying heights. The results from the steepest case, which is just on the verge of separation, are compared with wind-tunnel observations. It is shown that the large-eddy simulation results are in much better agreement with the experimental data than are the results from a simple first-order mixing-length closure model. An encouraging lack of sensitivity of the simulation results to numerical resolution is also demonstrated.     

An assessment of mixing-length closure schemes for models of turbulent boundary layers over complex terrain

The effectiveness of closure assumptions implemented in turbulent boundary-layer models is rather uncertain over complex terrain. Different closure schemes for Reynolds shear stress based on the mixing-length concept are compared with data from wind tunnel experiments over complex terrain and the results are analysed on the basis of second-order moment equations. A good estimation of the vertical momentum flux velocity scale turns out to be given by the standard deviation of the vertical velocity while the turbulent kinetic energy scaling gives less satisfactory results in regions where turbulence anisotropy is large. Fairly good results are given by closure models implementing a shear-limited mixing-length already proposed for non-logarithmic wind profiles, while large errors characterize traditional mixing-length formulations.     

Neutral Flow Over A Series Of Rough Hills: A Laboratory Experiment

This paper presents laboratory experiments of aerodynamically fully rough, neutral flow over a series of sinusoidal hills. Two sets of hills, with maximum gradients (slopes) of 0.2 (10°) and 0.4 (20°), were considered.The flow remained attached in the former case while separation occurredin the latter. Characteristics of the mean flow and turbulence statistics are discussed and compared with profiles over a flat surface covered with the same roughness as the hills. Comparisons are made with linear theory predictions for the flow in the inner region and aloft. Accurate measurements of the surface pressure were also made, enabling the comparison between the measured pressure drag and predictions from theoretical and computational work with different turbulent closure schemes. Organised secondary flow in the spanwise direction, observed previously in both experimental and computational studies, was also observed here over the small hills.     

Comparison of Speed-Up Over Hills Derived from Wind-Tunnel Experiments,Wind-Loading Standards,and Numerical Modelling

We evaluate the accuracy of the speed-up provided in several wind-loading standards by comparison with wind-tunnel measurements and numerical predictions, which are carried out at a nominal scale of 1:500 and full-scale, respectively. Airflow over two- and three-dimensional bell-shaped hills is numerically modelled using the Reynolds-averaged Navier–Stokes method with a pressure-driven atmospheric boundary layer and three different turbulence models. Investigated in detail are the effects of grid size on the speed-up and flow separation, as well as the resulting uncertainties in the numerical simulations. Good agreement is obtained between the numerical prediction of speed-up, as well as the wake region size and location, with that according to large-eddy simulations and the wind-tunnel results. The numerical results demonstrate the ability to predict the airflow over a hill with good accuracy with considerably less computational time than for large-eddy simulation. Numerical simulations for a three-dimensional hill show that the speed-up and the wake region decrease significantly when compared with the flow over two-dimensional hills due to the secondary flow around three-dimensional hills. Different hill slopes and shapes are simulated numerically to investigate the effect of hill profile on the speed-up. In comparison with more peaked hill crests, flat-topped hills have a lower speed-up at the crest up to heights of about half the hill height, for which none of the standards gives entirely satisfactory values of speed-up. Overall, the latest versions of the National Building Code of Canada and the Australian and New Zealand Standard give the best predictions of wind speed over isolated hills.     

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 (C_D) 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.     

Measurements and predictions of turbulent recirculating flow over a rectangular depression

Measurements of mean velocity and turbulence intensity components are reported for flow over a two-dimensional rectangular depression; these include measurements in the highly turbulent regions of recirculating flow. Predictions of the mean-flow variables were obtained from three finite-difference models: (1) a vorticity stream-function model using constant effective viscosity, (2) a primitive variable model using constant effective viscosity, and (3) a primitive variable model in which effective viscosity is computed from a turbulence model. The turbulent kinetic energy was also predicted by the last of these models. These predictions were compared with the measurements in order to evaluate what accuracy can be expected when state-of-the-art finite-difference models are applied to complex flow situations in the atmospheric environment. Some areas are noted where improvement of modeling capabilities for complex flows is still needed.     

Boundary-layer flow over analytical two-dimensional hills: A systematic comparison of different models with wind tunnel data

Two mass consistent models (MATHEW and MINERVE) and two dynamic linearized models (MS3DJH/3R and FLOWSTAR) are used to simulate the mean flow over two-dimensional hills of analytical shape and of varying slope. The results are compared with detailed wind tunnel data (RUSHIL experiment at US EPA). Different numerical experiments have been performed, varying input data and control parameters, to test the data-processing methodology and to evaluate the minimum input data (for mass consistent models only) necessary to obtain a reliable flow field. The models behave differently according to the physical assumptions made and numerical procedure used: an assessment is then made in order to identify the proper solution for the different conditions of topography and wind data.     

Boundary-layer flow over topography: Impacts of the Askervein study

One of the objectives of the Askervein Hill Project was to obtain a comprehensive and accurate dataset for verification of models of flow and turbulence over low hills. In the present paper, a retrospective of the 1982 and 1983 Askervein experiments is presented. The field study is described in brief and is related to similar studies conducted in the early 1980s. Data limitations are discussed and applications of numerical and wind-tunnel models to Askervein are outlined. Problems associated with model simulations are noted and model results are compared with the field measurements.     

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.     

A Method of Analysis for Turbulent Flows Using the Streamline Coordinate System

A procedure is described for analysing the transport equations for Reynolds stresses written in a streamline coordinate system, starting from the fields of first- and second-order moments of wind velocity measured in a terrain-following system over topography. In the analysis, the equations are split into two parts: the first contains the terms that can be calculated directly from measurements; the second contains third-order moments that are parameterized using suitable models. To evaluate the error associated with both parts, a Monte-Carlo technique that takes into account the experimental errors is proposed. An example of the application of this method for the Reynolds shear stress equation, using wind-tunnel data for non-separating flow over a two-dimensional valley, is reported. The comparison between the measured and modelled parts is fair near the surface, while at higher levels, the modelled part can be shown to miss a correct treatment of the third-order moments. In the frame of this analysis, the use of the correct derivative transformation has been found to be significant even for moderately sloping topography.