The flow in a turbulent boundary layer upstream of a change in surface roughness

Modification of a turbulent flow upstream of a change in surface roughness has been studied by means of a stream function-vorticity model.A flow reduction is found upstream of a step change in surface roughness when a fluid flows from a smooth onto a rough surface. Above that layer and above the region of flow reduction downstream of a smooth-rough transition, a flow acceleration is observed. Similar flow modification can be seen at a rough-smooth transition with the exception that flow reduction and flow acceleration are reversed. Within a fetch of –500 < x/z ₀< + 500 (z ₀ is the maximum roughness length, the roughness transition is located at x/z ₀ = 0), flow reduction (flow acceleration) upstream of a roughness transition is one order of magnitude smaller than the flow reduction (flow acceleration) downstream of a smooth-rough (rough-smooth) transition. The flow acceleration (flow reduction) above that layer is two orders of magnitude.The internal boundary layer (IBL) for horizontal mean velocity extends to roughly 300z ₀ upstream of a roughness transition, whereas the IBL for turbulent shear stress as well as the distortion of flow equilibrium extend almost twice as far. For the friction velocity, an undershooting (overshooting) with respect to upstream equilibrium is predicted which precedes overshooting (undershooting) over new equilibrium just behind a roughness transition.The flow modification over a finite fetch of modified roughness is weaker than over a corresponding fetch downstream of a single step change in roughness and the flow stays closer to upstream equilibrium. Even in front of the first roughness change of a finite fetch of modified roughness, a distortion of flow equilibrium due to the second, downwind roughness change can be observed.     

Direct Numerical Simulation of Stable Channel Flow at Large Stability

We consider a model for the stable atmospheric boundary at large stability, i.e. near the limit where turbulence is no longer able to survive. The model is a plane horizontally homogeneous channel flow, which is driven by a constant pressure gradient and which has a no-slip wall at the bottom and a free-slip wall at the top. At the lower wall a constant negative temperature flux is imposed. First, we consider a direct numerical simulation of the same channel flow. The simulation is computed with the neutral channel flow as initial condition and computed as a function of time for various values of the stability parameter h/L, where h is the channel height and L is related to the Obukhov length. We find that a turbulent solution is only possible for h/L < 1.25 and for larger values turbulence decays. Next, we consider a theoretical model for this channel flow based on a simple gradient transfer closure. The resulting equations allow an exact solution for the case of a stationary flow. The velocity profile for this solution is almost linear as a function of height in most of the channel. In the limit of infinite Reynolds number, the temperature profile has a logarithmic singularity at the upper wall of the channel. For the cases where a turbulent flow is maintained in the numerical simulation, we find that the velocity and temperature profiles are in good agreement with the results of the theoretical model when the effects of the surface layer on the exchange coefficients are taken into account. Frans Nieuwstadt, a recently retired member of the BLM Editorial Board and a well-known member of the boundary-layer/turbulence community, died unexpectedly on 18 May 2005. An obituary will appear in a later issue of BLM.     

Similarity of structure-function parameters in the stably stratified boundary layer

The structure-function parametersC _T ² andC _v ² of temperature and velocity, respectively, from the 1973 Minnesota experiments and from large-eddy and direct numerical simulations show a smooth transition from M–O similarity to the local scaling hypothesized by Nieuwstadt for the outer regions of the stable boundary layer. Under that hypothesis, turbulence statistics aloft depend on the local vertical fluxes of momentum and temperature, so these results suggest that remote-sensing measurements ofC _T ² andC _v ² could be used to infer vertical profiles of those fluxes. We argue that the sensitivity of the fluxes to unsteadiness, baroclinity, terrain slope, and breaking gravity waves precludes the universality of the vertical profiles of structure-function parameters in the stable PBL. We find that theC _T ² profile is particularly sensitive to these effects, which is consistent with observations that it varies considerably from case to case.     

Neutrally Stratified Turbulent Ekman Boundary Layer: Universal Similarity for a Transitional Rough Surface

The geostrophic Ekman boundary layer for large Rossby number (Ro) has been investigated by exploring the role played by the mesolayer (intermediate layer) lying between the traditional inner and outer layers. It is shown that the velocity and Reynolds shear stress components in the inner layer (including the overlap region) are universal relations, explicitly independent of surface roughness. This universality of predictions has been supported by observations from experiment, field and direct numerical simulation (DNS) data for fully smooth, transitionally rough and fully rough surfaces. The maxima of Reynolds shear stresses have been shown to be located in the mesolayer of the Ekman boundary layer, whose scale corresponds to the inverse square root of the friction Rossby number. The composite wall-wake universal relations for geostrophic velocity profiles have been proposed, and the two wake functions of the outer layer have been estimated by an eddy viscosity closure model. The geostrophic drag and cross-isobaric angle predictions yield universal relations, which are also supported by extensive field, laboratory and DNS data. The proposed predictions for the geostrophic drag and the cross-isobaric angle compare well with data for Rossby number Ro ≥ 10⁵. The data show low Rossby number effects for Ro < 10⁵ and higher-order effects due to the mesolayer compare well with the data for Ro ≥ 10³.     

光滑海面上风、温、湿廓线的层结订正模式及海气整体交换系数

详细介绍了光滑面标量粗糙度ＺＴ,Ｚｑ与风速粗糙度Ｚ０的相似表达式,论述了把Ｍｏｎｉｎ－Ｏｂｕｋｈｏｖ相似理论推广到光滑面上湍流气层的合理性,从而得到光滑面风、温、湿层结订正廓线与粗糙面廓线相统一的形式。总结了用该模式处理的三个海上梯度观测资料的计算结果,揭示出了微风时通过光滑海面的海气通量及整体交换系数受层结影响远大于风速影响的特征。并给出光滑界面上不同层结的整体交换系数随风速变化的拟合公式。     

A theory for the scalar roughness and the scalar transfer coefficients over snow and sea ice

Although the bulk aerodynamic transfer coefficients for sensible (C _H ) and latent (C _E ) heat over snow and sea ice surfaces are necessary for accurately modeling the surface energy budget, they have been measured rarely. This paper, therefore, presents a theoretical model that predicts neutral-stability values of C _H and C _E as functions of the wind speed and a surface roughness parameter. The crux of the model is establishing the interfacial sublayer profiles of the scalars, temperature and water vapor, over aerodynamically smooth and rough surfaces on the basis of a surface-renewal model in which turbulent eddies continually scour the surface, transferring scalar contaminants across the interface by molecular diffusion. Matching these interfacial sublayer profiles with the semi-logarithmic inertial sublayer profiles yields the roughness lengths for temperature and water vapor. When coupled with a model for the drag coefficient over snow and sea ice based on actual measurements, these roughness lengths lead to the transfer coefficients. C _E is always a few percent larger than CH. Both decrease monotonically with increasing wind speed for speeds above 1 m s^–1, and both increase at all wind speeds as the surface gets rougher. Both, nevertheless, are almost always between 1.0 × 10^–3 and 1.5 × 10^–3.     

A parameterization of the boundary layer of a tropical cyclone

In an earlier paper by one of the authors (Smith, 1968), a momentum integral method was developed to parameterize the gross constraint imposed by the surface boundary layer of a steady, axisymmetric, tropical cyclone on the meridional circulation within the vortex itself. Specifically, the method provides an effective means of estimating the radial variation of mean upflow/downflow induced by the boundary layer, compatible with a prescribed radial variation of azimuthal velocity just above the boundary layer,V _gr. However, it relies on a judicious choice of vertical profiles of radial and azimuthal velocity components within the boundary layer. An especially suitable set of profiles is discussed herein; these are Ekman-like profiles in which turbulent mixing is characterized by a vertically constant eddy diffusivityK _M , matched to a constant stress sublayer just above the sea surface. An attractive feature of the formulation is that a suitable value forK _M as a function of radius, which is extremely difficult to extract from observational data, can be calculated when the state of the sea surface, described by a roughness lengthZ ₀, is prescribed. Although observations ofZ ₀ at high wind speeds are not yet available, the effect of radial variations in sea surface roughness can be assessed and it is shown that these affect the upflow to a significant degree.     

Aerodynamic Roughness Length of Fresh Snow

This study presents the results from a series of wind-tunnel experiments designed to investigate the aerodynamic roughness length z ₀ of fresh snow under no-drift conditions. A two-component hot-film anemometer was employed to obtain vertical profiles of velocity statistics in a zero pressure gradient turbulent boundary layer for flow over naturally deposited snow surfaces. The roughness of these snow surfaces was measured by means of digital photography to capture characteristic length scales that can be related to z ₀. Our results show that, under aerodynamically rough conditions, the mean value of the roughness length for fresh snow is \({\langle{z}_{0}\rangle= 0.24}\) mm with a standard deviation σ(z ₀) = 0.05 mm. In this study, we show that variations in z ₀ are associated with variations in the roughness geometry. The roughness measurements suggest that the estimated values of z ₀ are consistent with the presence of irregular roughness structures that develop during snowfalls that mimic ballistic deposition processes.     

Flux–Profile Relationships Over a Fetch Limited Beech Forest

The influence of an internal boundary layer and a roughness sublayer on flux–profile relationships for momentum and sensible heat have been investigated for a closed beech forest canopy with limited fetch conditions. The influence was quantified by derivation of local scaling functions for sensible heat flux and momentum (_h and _m) and analysed as a function of atmospheric stability and fetch. For heat, the influences of the roughness sublayer and the internal boundary layer were in agreement with previous studies. For momentum, the strong vertical gradient of the flow just above the canopy top for some wind sectors led to an increase in _m, a feature that has not previously been observed. For a fetch of 500 m over the beech forest during neutral atmospheric conditions, there is no height range at the site where profiles can be expected to be logarithmic with respect to the local surface. The different influence of the roughness sublayer on _h and _m is reflected in the aerodynamic resistance for the site. The aerodynamic resistance for sensible heat is considerably smaller than the corresponding value for momentum.     

A new nonlinear analytical model for canopy flow over a forested hill

A new nonlinear analytical model for canopy flow over gentle hills is presented. This model is established based on the assumption that three major forces (pressure gradient, Reynolds stress gradient, and nonlinear canopy drag) within canopy are in balance for gentle hills under neutral conditions. The momentum governing equation is closed by the velocity-squared law. This new model has many advantages over the model developed by Finnigan and Belcher (Quart J Roy Meteorol Soc 130: 1–29 2004, hereafter referred to as FB04) in predicting canopy wind velocity profiles in forested hills in that: (1) predictions from the new model are more realistic because surface drag effects can be taken into account by boundary conditions, while surface drag effects cannot be accounted for in the algebraic equation used in the lower canopy layer in the FB04 model; (2) the mixing length theory is not necessarily used because it leads to a theoretical inconsistency that a constant mixing length assumption leads to a nonconstant mixing length prediction as in the FB04 model; and (3) the effects of height-dependent leaf area density (a(z)) and drag coefficient (C _d ) on wind velocity can be predicted, while both a(z) and C _d must be treated as constants in FB04 model. The nonlinear algebraic equation for momentum transfer in the lower part of canopy used in FB04 model is height independent, actually serving as a bottom boundary condition for the linear differential momentum equation in the upper canopy layer. The predicting ability of the FB04 model is largely restricted by using the height-independent algebraic equation in the bottom canopy layer. This study has demonstrated the success of using the velocity-squared law as a closure scheme for momentum transfer in forested hills in comparison with the mixing length theory used in FB04 model thus enhancing the predicting ability of canopy flows, keeping the theory consistent and simple, and shining a new light into land-surface parameterization schemes in numerical weather and climate models.     

Effects of Wall Heating on Flow Characteristics in a Street Canyon

We develop a large-eddy simulation (LES) model based on a meteorological numerical model for a real scale street-canyon flow with rough building facets heated by a given temperature. The model is applied to a canyon with the aspect ratio of unity for two idealized heating scenarios: (1) the roof and the entire upstream wall are heated, named as ‘assisting cases’, and (2) the roof and the entire downstream wall are heated, named as ‘opposing cases’. These facets were heated up to 15 K above the air temperature. A wall function for temperature is proposed for a rough facet with an assumption that the thermal roughness length, z _0T, is much smaller than the aerodynamic roughness length, z ₀. It is demonstrated that the sensible heat flux and canyon-air temperature are significantly influenced by the near-facet process that is parametrized by z _0T as the primary factor; other processes such as in-canyon mixing and roof-level exchange are secondary. This new finding strongly suggests that it is vital to choose an appropriate value of z _0T in a numerical simulation of street-canyon flows with the facet-air exchange processes of heat or any scalar. The finding also raises an awareness of the demand for carefully designed laboratory or field experiments of quantifying z _0T values for various urban surfaces. For the opposing cases, an unsteady penetrating narrow updraft zone appears occasionally along the heated wall and this feature is consistent field observations. The unique result indicates the superior capability of LES. The results of this study can be used to guide the parametrization of turbulent processes inside the urban canopy layer.     

Subfilter-scale Fluxes over a Surface Roughness Transition. Part I: Measured Fluxes and Energy Transfer Rates

A wind-tunnel experiment was designed and carried out to study the effect of a surface roughness transition on subfilter-scale (SFS) physics in a turbulent boundary layer. Specifically, subfilter-scale stresses are evaluated that require parameterizations and are key to improving the accuracy of large-eddy simulations of the atmospheric boundary layer. The surface transition considered in this study consists of a sharp change from a rough, wire-mesh covered surface to a smooth surface. The resulting magnitude jump in aerodynamic roughnesses, M = ln(z ₀₁/z ₀₂), where z ₀₁ and z ₀₂ are the upwind and downwind aerodynamic surface roughnesses respectively, is similar to that of past experimental studies in the atmospheric boundary layer. The two-dimensional velocity fields used in this study are measured using particle image velocimetry and are acquired at several positions downwind of the roughness transition as well as over a homogeneous smooth surface. Results show that the SFS stress, resolved strain rate and SFS transfer rate of resolved kinetic energy are dependent on the position within the boundary layer relative to the surface roughness transition. A mismatch is found in the downwind trend of the SFS stress and resolved strain rate with distance from the transition. This difference of behaviour may not be captured by some eddy-viscosity type models that parameterize the SFS stress tensor as proportional to the resolved strain rate tensor. These results can be used as a benchmark to test the ability of existing and new SFS models to capture the spatial variability SFS physics associated with surface roughness heterogeneities.     

Wind structure near a city centre

Wind speed and temperature profiles to a height of 8 m were recorded for 30-, 60-, and 90-min averaging times over a striated snow surface at the geographic South Pole during the austral winter of 1975. A gradient Richardson number was calculated for each averaging time to determine conditions of neutral stability under which the logarithmic wind law would hold. A log-linear regression technique was used to determine values of aerodynamic roughness height (Z ₀) for those profile averages recorded in conditions of neutral stability. A plot of Z ₀ as a function of average wind direction revealed a variation in Z ₀ of almost three orders of magnitude, from 0.01 to 7 cm, over 120 deg of wind direction. A simple model is presented to justify the fact that aerodynamic roughness is a function of wind direction and erosion history.     

A simple unified theory for flow in the canopy and roughness sublayer

The mean flow profile within and above a tall canopy is well known to violate the standard boundary-layer flux–gradient relationships. Here we present a theory for the flow profile that is comprised of a canopy model coupled to a modified surface-layer model. The coupling between the two components and the modifications to the surface-layer profiles are formulated through the mixing layer analogy for the flow at a canopy top. This analogy provides an additional length scale—the vorticity thickness—upon which the flow just above the canopy, within the so-called roughness sublayer, depends. A natural form for the vertical profiles within the roughness sublayer follows that overcomes problems with many earlier forms in the literature. Predictions of the mean flow profiles are shown to match observations over a range of canopy types and stabilities. The unified theory predicts that key parameters, such as the displacement height and roughness length, have a significant dependence on the boundary-layer stability. Assuming one of these parameters a priori leads to the incorrect variation with stability of the others and incorrect predictions of the mean wind speed profile. The roughness sublayer has a greater impact on the mean wind speed in stable than unstable conditions. The presence of a roughness sublayer also allows the surface to exert a greater drag on the boundary layer for an equivalent value of the near-surface wind speed than would otherwise occur. This characteristic would alter predictions of the evolution of the boundary layer and surface states if included within numerical weather prediction models.     

Momentum Absorption in Rough-Wall Boundary Layers with Sparse Roughness Elements in Random and Clustered Distributions

A wind-tunnel experiment has been used to investigate momentum absorption by rough surfaces with sparse random and clustered distributions of roughness elements. An unusual (though longstanding) method was used to measure the boundary-layer depth δ and friction velocity u _* and thence to infer the functional relationship z ₀/h = f(λ) between the normalised roughness length z ₀/ h and the roughness density λ (where z ₀ is the roughness length and h the mean height of the roughness elements). The method for finding u _* is based on fitting the velocity defect in the outer layer to a functional form for the dimensionless velocity-defect profile in a canonical zero-pressure-gradient boundary layer. For the conditions investigated here, involving boundary layers over sparse roughness with strong local heterogeneity, this velocity-defect-law method is found to be more robust than several alternative methods for finding u _* (uw covariance, momentum integral and slope of the logarithmic velocity profile).The experimental results show that, (1) there is general agreement in the relationship z ₀/h = f(λ) between the present experiment with random arrays and other wind-tunnel experiments with regular arrays; (2) the main effect of clustering is to increase the scatter in the z ₀/h = f(λ) relationship, through increased local horizontal heterogeneity; (3) this scatter obscures any trend in the z ₀/h = f(λ) relationship in response to clustering; and (4) the agreement between the body of wind-tunnel data (taken as a whole) and field data is good, though with scatter for which it is likely that a major contribution stems from local horizontal heterogeneity in the field.     

Interaction between the atmospheric and oceanic boundary layers

The two-layer system of an atmosphere over water bodies is reduced to a single-layer problem. Values of the interfacial quantities, such as the friction velocity, the surface velocity, the angles, and , between the surface shear stress and the geostrophic wind velocity and the surface wind velocity, respectively, and the surface roughness, all of which depend upon external parameters, such as the geostrophic wind and stratifications, are obtained. The geostrophic drag coefficient C _d, the geostrophic wind coefficient C _f, and the angles , and , of the turbulent flow at the sea-air interface are functions of a dimensionless number, mfG/kg, with S ₁ and S ₂ as two free stratification parameters. The surface roughness is uniquely determined from the geostrophic wind rather than from the wind profile in the boundary layer.Formerly Visiting Research Associate, Applied Physics Branch, Earth Observations Division, NASA-Manned Spacecraft Center, Houston, Texas.     

Incorporation of a high-roughness lower boundary into a mesoscale model for studies of dry deposition over complex terrain

For flow over natural surfaces, there exists a roughness sublayer within the atmospheric surface layer near the boundary. In this sublayer (typically 50z ₀ deep in unstable conditions), the Monin-Obukhov (M-O) flux profile relations for homogeneous surfaces cannot be applied. We have incorporated a modified form of the M-O stability functions (Garratt, 1978, 1980, 1983) in a mesoscale model to take account of this roughness sublayer and examined the diurnal variation of the boundary-layer wind and temperature profiles with and without these modifications. We have also investigated the effect of the modified M-O functions on the aerodynamic and laminar-sublayer resistances associated with the transfer of trace gases to vegetation. Our results show that when an observation height or the lowest level in a model is within the roughness sublayer, neglect of the flux-profile modifications leads to an underestimate of resistances by 7% at the most.     

Velocity and Surface Shear Stress Distributions Behind a Rough-to-Smooth Surface Transition: A Simple New Model

A simple new model is proposed to predict the distribution of wind velocity and surface shear stress downwind of a rough-to-smooth surface transition. The wind velocity is estimated as a weighted average between two limiting logarithmic profiles: the first log law, which is recovered above the internal boundary-layer height, corresponds to the upwind velocity profile; the second log law is adjusted to the downwind aerodynamic roughness and local surface shear stress, and it is recovered near the surface, in the equilibrium sublayer. The proposed non-linear form of the weighting factor is equal to ln(z/z ₀₁)/ln(δ _i /z ₀₁), where z, δ _i and z ₀₁ are the elevation of the prediction location, the internal boundary-layer height at that downwind distance, and the upwind surface roughness, respectively. Unlike other simple analytical models, the new model does not rely on the assumption of a constant or linear distribution for the turbulent shear stress within the internal boundary layer. The performance of the new model is tested with wind-tunnel measurements and also with the field data of Bradley. Compared with other existing analytical models, the proposed model shows improved predictions of both surface shear stress and velocity distributions at different positions downwind of the transition.     

Influence of a Two-scale Surface Roughness on a Neutral Turbulent Boundary Layer

Flow in the urban boundary layer is strongly influenced by the surface roughness, which is composed principally of isolated buildings or groups of buildings. Previous research has shown that the flow regime depends on the characteristic height of these obstacles (H), and the spacing between them (W). In reality, the urban boundary layer contains roughness elements with a wide range of length scales; in many practical situations these can be classified into large-scale roughness—buildings, or groups of buildings—and small-scale roughness, such as street furniture and elements on the façades and roofs. It is important to understand how the small-scale roughness might modify mass and momentum transfer in the urban boundary layer, but relatively little information is available concerning the potential interaction between large- and small-scale roughness elements in the different flow regimes. This problem has been studied using wind-tunnel experiments, by measuring vertical velocity profiles over a two-dimensional obstacle array, adding small-scale roughness elements to the top of larger parallel square bars. The experiments were performed for different cavity aspect ratios: the results show that the small-scale roughness increases the turbulence intensities and the momentum transfer when the large-scale obstacles are closely packed (H/W > 1) but it has very little effect for more widely-spaced obstacles (H/W < 1).     

Estimation of the Lagrangian Velocity Structure Function Constant C0 by Large-Eddy Simulation

The inertial subrange Kolmogorov constant C ₀, which determines the effective turbulent diffusion in velocity space, plays an important role in the Lagrangian modelling of pollutants. A wide range of values of the constant are found in the literature, most of them determined at low Reynolds number and/or under different assumptions. Here we estimate the constant C ₀ by tracking an ensemble of Lagrangian particles in a planetary boundary layer simulated with a large-eddy simulation model and analysing the Lagrangian velocity structure function in the inertial subrange. The advantage of this technique is that it easily allows Reynolds numbers to be achieved typical of convective turbulent flows. Our estimates of C ₀ is C ₀=4.3±0.3 consistent with values found in the literature