Reynolds-Number Dependence of Gas Dispersion Over a Wavy Wall

Direct numerical simulations of an Ekman layer are performed to study flow evolution during the response of an initially neutral boundary layer to stable stratification. The Obukhov length, L, is varied among cases by imposing a range of stable buoyancy fluxes at the surface to mimic ground cooling. The imposition of constant surface buoyancy flux , i.e. constant-flux stability, leads to a buoyancy difference between the ground and background that tends to increase with time, unlike the constant-temperature stability case where a constant surface temperature is imposed. The initial collapse of turbulence in the surface layer owing to surface cooling that occurs over a time scale proportional to \(L/u_*\), where \(u_*\) is the friction velocity, is followed by turbulence recovery. The flow accelerates, and a “low-level jet” (LLJ) with inertial oscillations forms during the turbulence collapse. Turbulence statistics and budgets are examined to understand the recovery of turbulence. Vertical turbulence exchange, primarily by pressure transport, is found to initiate fluctuations in the surface layer and there is rebirth of turbulence through enhanced turbulence production as the LLJ shear increases. The turbulence recovery is not monotonic and exhibits temporal intermittency with several collapse/rebirth episodes. The boundary layer adjusts to an increase in the surface buoyancy flux by increased super-geostrophic velocity and surface stress such that the Obukhov length becomes similar among the cases and sufficiently large to allow fluctuations with sustained momentum and heat fluxes. The eventual state of fluctuations, achieved after about two inertial periods (\(ft \approx 4\pi \)), corresponds to global intermittency with turbulent patches in an otherwise quiescent background. Our simplified configuration is sufficient to identify turbulence collapse and rebirth, global and temporal intermittency, as well as formation of low-level jets, as in observations of the stratified atmospheric boundary layer.     

Stability Dependence and Diurnal Change of Large-Scale Turbulence Structures in the Near-Neutral Atmospheric Boundary Layer Observed from a Meteorological Tower

Based on the analysis of observations from a 213-m tall meteorological tower at Tsukuba, Japan, we have investigated the favourable conditions for the predominant existence of large-scale turbulence structures in the near-neutral atmospheric boundary layer (ABL). From the wavelet variance spectrum for the streamwise velocity component ( $u$ ) measured by a sonic anemometer-thermometer at the highest level (200 m), large-scale structures (time-scale range of 100–300 s) predominantly exist under slightly unstable and close to neutral conditions. The emergence of large-scale structures also can be related to the diurnal evolution of the ABL. The large-scale structures play an important role in the overall flow structure of the lower boundary layer. For example, $u$ velocity components at the 200-m and 50-m levels show relatively high correlation with the existence of large-scale structures. Under slightly unstable (near-neutral) conditions, a low-speed region in advance of the high-speed structure shows a positive deviation of temperature and appears as the plume structure that is forced by buoyancy in the heated lower layer. In spite of the difference in buoyancy effects between the near-neutral and unstable cases, large-scale structures are frequently observed in both cases and the same vertical correlation of $u$ components is indicated. However, the vertical wind shear is smaller in the unstable cases. On the other hand, in near-neutral cases, the transport efficiency of momentum at the higher level and the flux contribution of sweep motions are larger than those in the unstable cases.     

The Effects of Thermal Stratification on Clustering Properties of Canopy Turbulence

The intermittent structure of turbulence within the canopy sublayer (CSL) is sensitive to the presence of foliage and to the atmospheric stability regime. How much of this intermittency originates from amplitude variability or clustering properties remains a vexing research problem for CSL flows. Using a five-level set of measurements collected within a dense hardwood canopy, the clustering properties of CSL turbulence and their dependence on atmospheric stability are explored using the telegraphic approximation (TA). The binary structure of the TA removes any amplitude variability from turbulent excursions but retains their zero-crossing behaviour, and thereby isolating the role of clustering in intermittency. A relationship between the spectral exponents of the actual and the TA series is derived across a wide range of atmospheric stability regimes and for several flow variables. This relationship is shown to be consistent with a relationship derived for long-memory and monofractal processes such as fractional Brownian motion (fBm). Moreover, it is demonstrated that for the longitudinal and vertical velocity components, the vegetation does not appreciably alter fine-scale clustering but atmospheric stability does. Stable atmospheric stability conditions is characterized by more fine scale clustering when compared to other atmospheric stability regimes. For scalars, fine-scale clustering above the canopy is similar to its velocity counterpart but is significantly increased inside the canopy, especially under stable stratification. Using simplified scaling analysis, it is demonstrated that clustering is much more connected to space than to time within the CSL. When comparing intermittency for flow variables and their TA series, it is shown that for velocity, amplitude variations modulate intermittency for all stability regimes. However, amplitude variations play only a minor role in scalar intermittency. Within the crown region of the canopy, a ‘double regime’ emerges in the inter-pulse duration probability distributions not observed in classical turbulence studies away from boundaries. The double regime is characterized by a power-law distribution for shorter inter-pulse periods and a log-normal distribution for large inter-pulse periods. The co-existence of these two regimes is shown to be consistent with near-field/far-field scaling arguments. In the near-field, short inter-pulse periods are controlled by the source strength, while in the far-field long inter-pulse periods are less affected by the precise source strength details and more affected by the transport properties of the background turbulence.     

Measurements of internal waves and turbulence in two-dimensional stratified shear flows

A turbulent stratified shear flow is generated in a towing tank by towing a grid or a circular cylinder through a tank of stratified salt water. The internal waves and turbulence generated in these flows are visualized with shadowgraphs and measured with quartz-coated hot-film probes (up to four probes for velocity fluctuations) and single-electrode conductivity probes (up to four probes for salinity fluctuations) which are towed at the same speed as the obstacle. The velocity and salinity signals are recorded on magnetic tapes. A portion of these signals is processed directly-on-line with a digital computer. From these shadowgraphs and probe measurements, we observe that

1. Far downstream of the obstacle where the turbulence has already subsided, the stratified fluid always has a layered structure. This layered structure persists for a long time, and is a result of the convection of turbulently mixed layers by the mean flow. These results indicate that in the regions of a stably stratified atmosphere and ocean where the turbulence has subsided, one could often find layered structure.
2. There are spectral peaks and valleys in the measured velocity and salinity autospectra when the stratifications are sufficiently strong. Under certain conditions, these spectral peaks tend to lift up the spectral curves to show substantialf ^?5/3 subranges, although the turbulence Reynolds numbers are too low for the flows to have recognizable inertial subranges. This anomalousf ^?5/3 subrange demonstrates the pitfalls of using spectral measurements in thef ^?5/3 subrange to predict the turbulent energy dissipation rate through the Kolmogorov hypothesis.
3. A diagnostic method is developed for distinguishing internal waves from turbulence, utilizing their phase characteristics. The phase characteristics can be conveniently examined from the cospectra and quadrature spectra measurements of: (a), two vertically separated velocity probes; (b), two vertically separated density probes; and (c), a velocity probe and a density probe. This method is demonstrated to be useful in the laboratory and can be applied directly to atmospheric and oceanic measurements to distinguish internal waves from turbulence.
4. From the coherency measurements, it is found that the entire turbulent stratified wake is actually whipping up and down at a frequency corresponding to the Brunt-Väisälä frequency. This indicates that similar stratified shear flows in the atmosphere and in the ocean, such as the jet streams in the atmosphere and the Cromwell current in the ocean, may oscillate vertically, which in turn can induce horizontal oscillation and meandering.

     

Energy- and Flux-Budget Turbulence Closure Model for Stably Stratified Flows. Part II: The Role of Internal Gravity Waves

We advance our prior energy- and flux-budget (EFB) turbulence closure model for stably stratified atmospheric flow and extend it to account for an additional vertical flux of momentum and additional productions of turbulent kinetic energy (TKE), turbulent potential energy (TPE) and turbulent flux of potential temperature due to large-scale internal gravity waves (IGW). For the stationary, homogeneous regime, the first version of the EFB model disregarding large-scale IGW yielded universal dependencies of the flux Richardson number, turbulent Prandtl number, energy ratios, and normalised vertical fluxes of momentum and heat on the gradient Richardson number, Ri. Due to the large-scale IGW, these dependencies lose their universality. The maximal value of the flux Richardson number (universal constant ≈0.2–0.25 in the no-IGW regime) becomes strongly variable. In the vertically homogeneous stratification, it increases with increasing wave energy and can even exceed 1. For heterogeneous stratification, when internal gravity waves propagate towards stronger stratification, the maximal flux Richardson number decreases with increasing wave energy, reaches zero and then becomes negative. In other words, the vertical flux of potential temperature becomes counter-gradient. Internal gravity waves also reduce the anisotropy of turbulence: in contrast to the mean wind shear, which generates only horizontal TKE, internal gravity waves generate both horizontal and vertical TKE. Internal gravity waves also increase the share of TPE in the turbulent total energy (TTE = TKE + TPE). A well-known effect of internal gravity waves is their direct contribution to the vertical transport of momentum. Depending on the direction (downward or upward), internal gravity waves either strengthen or weaken the total vertical flux of momentum. Predictions from the proposed model are consistent with available data from atmospheric and laboratory experiments, direct numerical simulations and large-eddy simulations.     

The Analysis of Turbulence Intensity and Reynolds Shear Stress in Wall-Bounded Turbulent Flows at High Reynolds Numbers

We investigated the turbulent intensities and Reynolds shear stress at high Reynolds number $({Re_\tau = 5 \times 10^{6}})$ in the atmosphere surface layer (ASL) through analyzing observations in near-neutral stratified conditions. The results show that with increasing Reynolds number the streamwise turbulent intensity increases linearly, and the peak of the Reynolds shear stress extends to a higher non-dimensional height, which means that the thickness of the logarithmic region increases. Furthermore, our results provide evidence for treating the ASL as a canonical turbulent boundary layer, the results of which can be extended and applied to higher Reynolds number wall turbulence in the ASL.     

Towards Reconciling the Large-Scale Structure of Turbulent Boundary Layers in the Atmosphere and Laboratory

A collaborative experimental effort employing the minimally perturbed atmospheric surface-layer flow over the salt playa of western Utah has enabled us to map coherence in turbulent boundary layers at very high Reynolds numbers, \({Re_{\tau}\sim\mathcal{O}(10^6)}\) . It is found that the large-scale coherence noted in the logarithmic region of laboratory-scale boundary layers are also present in the very high Reynolds number atmospheric surface layer (ASL). In the ASL these features tend to scale on outer variables (approaching the kilometre scale in the streamwise direction for the present study). The mean statistics and two-point correlation map show that the surface layer under neutrally buoyant conditions behaves similarly to the canonical boundary layer. Linear stochastic estimation of the three-dimensional correlation map indicates that the low momentum fluid in the streamwise direction is accompanied by counter-rotating roll modes across the span of the flow. Instantaneous flow fields confirm the inferences made from the linear stochastic estimations. It is further shown that vortical structures aligned in the streamwise direction are present in the surface layer, and bear attributes that resemble the hairpin vortex features found in laboratory flows. Ramp-like high shear zones that contribute significantly to the Reynolds shear-stress are also present in the ASL in a form nearly identical to that found in laboratory flows. Overall, the present findings serve to draw useful connections between the vast number of observations made in the laboratory and in the atmosphere.     

On atmospheric vortex streets in the wake of large islands

Summary Results from laboratory experiments on stably stratified flows over three-dimensional obstacles are related to atmospheric vortex streets formed in the lee of large islands. A quasi horizontal flow around the island can be expected if stable stratification favours the formation of a so-called dividing streamline below the islands top. The subsequent shedding of vortices with vertical axis from islands may then be due to viscous boundary layer separation, but also other possible mechanisms of vortex shedding in stably stratified flows are discussed.With 6 Figures     

E − ε − 〈θ 2〉 turbulence closure model for an atmospheric boundary layer including the urban canopy

A modified three-parameter model of turbulence for a thermally stratified atmospheric boundary layer (ABL) is presented. The model is based on tensor-invariant parameterizations for the pressure–strain and pressure–temperature correlations that are more complete than the parameterizations used in the Mellor–Yamada model of level 3.0. The turbulent momentum and heat fluxes are calculated with explicit algebraic models obtained with the aid of symbol algebra from the transport equations for momentum and heat fluxes in the approximation of weakly equilibrium turbulence. The turbulent transport of heat and momentum fluxes is assumed to be negligibly small in this approximation. The three-parameter $E - \varepsilon - {\left\langle {\theta ^{2} } \right\rangle }$ model of thermally stratified turbulence is employed to obtain closed-form algebraic expressions for the fluxes. A computational test of a 24-h ABL evolution is implemented for an idealized two-dimensional region. Comparison of the computed results with the available observational data and other numerical models shows that the proposed model is able to reproduce both the most important structural features of the turbulence in an urban canopy layer near the urbanized ABL surface and the effect of urban roughness on a global structure of the fields of wind and temperature over a city. The results of the computational test for the new model indicate that the motion of air in the urban canopy layer is strongly influenced by mechanical factors (buildings) and thermal stratification.     

A numerical study of the convective boundary layer

Computations of the buoyantly unstable Ekman layer are performed at low Reynolds number. The turbulent fields are obtained directly by solving the three-dimensional time-dependent Navier-Stokes equations (using the Boussinesq approximation to account for buoyancy effects), and no turbulence model is needed. Two levels of heating are considered, one quite vigorous, the other more moderate. Statistics for the vigorously heated case are found to agree reasonably well with laboratory, field, and large-eddy simulation results, when Deardorff's mixed-layer scaling is used. No indication of large-scale longitudinal roll cells is found in this convection-dominated flow, for which the inversion height to Obukhov length scale ratio –z _i /L _*=26. However, when heating is more moderate (so that –z _i /L _*=2), evidence of coherent rolls is present. About 10% of the total turbulent kinetic energy and turbulent heat flux, and 20% of the Reynolds shear stress, are estimated to be a direct consequence of the observed cells.     

A Boundary-Layer Scaling for Turbulent Katabatic Flow

Scaling relationships are proposed for the turbulent katabatic flow of a stably stratified fluid down a homogeneously cooled planar slope—the turbulent analogue of a Prandtl-type slope flow. The \(\Pi \) Theorem predicts that such flows are controlled by three non-dimensional parameters: the slope angle, the Prandtl number, and a Reynolds number defined in terms of the surface thermal forcing (surface buoyancy or surface buoyancy flux), Brunt-Väisälä frequency, slope angle, and molecular viscosity and diffusivity coefficients. However, by exploiting the structure of the governing differential equations in a boundary-layer form, scaled equations are deduced that involve only two non-dimensional parameters: the Prandtl number and a modified Reynolds number. In the proposed scaling framework, the slope angle does not appear as an independent governing parameter, but merely acts as a stretching factor in the scales for the dependent and independent variables, and appears in the Reynolds number. Based on the boundary-layer analysis, we hypothesize that the full katabatic-flow problem is largely controlled by two rather than three parameters. Preliminary tests of the scaling hypothesis using data from direct numerical simulations provide encouraging results.     

Analysis of Turbulence Collapse in the Stably Stratified Surface Layer Using Direct Numerical Simulation

The nocturnal atmospheric boundary layer (ABL) poses several challenges to standard turbulence and dispersion models, since the stable stratification imposed by the radiative cooling of the ground modifies the flow turbulence in ways that are not yet completely understood. In the present work we perform direct numerical simulation of a turbulent open channel flow with a constant (cooling) heat flux imposed at the ground. This configuration provides a very simplified model for the surface layer at night. As a result of the ground cooling, the Reynolds stresses and the turbulent fluctuations near the ground re-adjust on times of the order of L/u _τ , where L is the Obukhov length scale and u _τ is the friction velocity. For relatively weak cooling turbulence survives, but when Re_L=Lu_t/n <~100{Re_L=Lu_\tau/\nu \lesssim 100} turbulence collapses, a situation that is also observed in the ABL. This criterion, which can be locally measured in the field, is justified in terms of the scale separation between the largest and smallest structures of the dynamic sublayer.     

Mixing efficiency of a stably stratified fluid

Universally valid expressions for the momentum and mass diffusion coefficients in stratified shear flows do not exist. Similarly, the effect of molecular diffusion in the mixing effciency of a stably stratified has not been investigated previously. In this paper the mixing efficiency in stably stratified flow is considered. Numerical investigations of unsheared stratified flows provide qualitative insight into the turbulence characteristics of stably stratified turbulent flows. Different types of stratified shear flows give qualitatively similar results for the mixing efficiency in the gradient Richardson number, Ri, range of 0.01–1.0. Mixing efficciency is proportional to Ri and its magnitude appears to depend on stratification in the fluid column. Suspended solids appear to effect the turbulence exchange processes in a different manner to those of salt and heat. The ratio of the diffusivity of solute (or mass) to the diffusivity of momentum is less than 1 and can be taken as 0.25 for sediment-laden turbulent shear flows. This ratio is about a factor of 2–3 larger when heat- and salt-induced stratification exist, the difference possibly due to the settling nature of suspended solids.     

Turbulence Anisotropy in the Near-Surface Atmosphere and the Evaluation of Multiple Outer Length Scales

Length scales determined by maximum turbulent kinetic energy (TKE), the integral scale, and two length scales based on Reynolds stress-tensor anisotropy are compared to the often stated outer length scales of boundary-layer depth and distance from the earth’s surface, $z$ . The scales are calculated using sonic anemometer data from two elevations, 5 and 50 m above the ground at the main tower site of the CASES-99 field campaign. In general, none of these scales agrees with the other, although the scale of maximum TKE is often similar to the boundary-layer depth during daytime hours, and the length scales derived from anisotropy characteristics are sometimes similar to $\kappa \!z, z$ , and $2z$ depending on scale definition and thermal stability. Except for the scale with the strictest isotropy threshold, the turbulence is anisotropic for each of the various candidates for the outer scale. Length scales for maximum buoyancy flux and temperature variance are evaluated and the turbulence characteristics at these scales are almost always found to be anisotropic.     

On the Prediction of the Kurtosis of Velocity Derivatives in a Turbulent Field

The prediction of the values of non-dimensional fourth-order moment (kurtosis) of the velocity derivative in a turbulent field is made under the assumption that the values of kurtosis depend on both the turbulence Reynolds number and the intermittency factor. The method consists of modeling a suitable probability density of the variable in a given turbulence Reynolds number and the intermittency factor. A crude model of the probability density function is derived, and the numerical calculations based on the model show excellent agreement with many of the experimental data. The analysis shows that the values of kurtosis depend strongly on the intermittency factor, and that depending on the value of the intermittency factor, it is entirely possible to have values of kurtosis as low as five in a flow with a turbulence Reynolds number of 5000.     

Particle Image Velocimetry Measurements of Turbulent Flow Within Outdoor and Indoor Urban Scale Models and Flushing Motions in Urban Canopy Layers

We investigate the spatial characteristics of urban-like canopy flow by applying particle image velocimetry (PIV) to atmospheric turbulence. The study site was a Comprehensive Outdoor Scale MOdel (COSMO) experiment for urban climate in Japan. The PIV system captured the two-dimensional flow field within the canopy layer continuously for an hour with a sampling frequency of 30 Hz, thereby providing reliable outdoor turbulence statistics. PIV measurements in a wind-tunnel facility using similar roughness geometry, but with a lower sampling frequency of 4 Hz, were also done for comparison. The turbulent momentum flux from COSMO, and the wind tunnel showed similar values and distributions when scaled using friction velocity. Some different characteristics between outdoor and indoor flow fields were mainly caused by the larger fluctuations in wind direction for the atmospheric turbulence. The focus of the analysis is on a variety of instantaneous turbulent flow structures. One remarkable flow structure is termed ‘flushing’, that is, a large-scale upward motion prevailing across the whole vertical cross-section of a building gap. This is observed intermittently, whereby tracer particles are flushed vertically out from the canopy layer. Flushing phenomena are also observed in the wind tunnel where there is neither thermal stratification nor outer-layer turbulence. It is suggested that flushing phenomena are correlated with the passing of large-scale low-momentum regions above the canopy.     

Helicopter-Borne Flux Measurements in the Nocturnal Boundary Layer Over Land – a Case Study

This case study introduces measurements of turbulent fluxes in a nocturnal boundary layer in North Germany with the new helicopter-borne turbulence measurement system HELIPOD, a detailed data analysis and examination in regard of systematic errors of the instrument, and some comparison with local similarity theory and experiments of the past, in order to confirm the occurrence of small vertical turbulent fluxes. The examined nocturnal boundary layer offered excellent conditions to analyse the quality of the measurement system. In this connection, a detailed look at a strong ground-based inversion disclosed small turbulent fluxes with a spectral maximum at ten metres wavelength or less, embedded in intermittent turbulence. For verification of these fluxes, the measurements were compared with well established results from past experiments. Local similarity theory was applied to calculate dimensionless variances of the turbulent quantities, which were found in good agreement with other observations. Since shear and stratification varied significantly on the horizontal flight legs due to global intermittency, a method was developed to determine vertical gradients on a horizontal flight pattern, by use of small fluctuations of the measurement height. With these locally determined gradients, gradient transport theory became applicable and the turbulent diffusivities for heat and momentum, the Richardson number, and the flux Richardson number were estimated within isolated strong turbulent outbursts. Within these outbursts the flux Richardson number was found between 0.1 and 0.2. The functional relationship between the gradient Richardson number and the turbulent Prandtl number agreed well with observations in past experiments and large eddy simulation. The impact of the stratification on the vertical turbulent exchange, as already described for the surface layer using Monin–Obukhov similarity, was analogously observed in the very stably stratified bulk flow when local scaling was applied.     

Turbulence and diffusion over inhomogeneous terrain

This paper provides an overview of some aspects of atmospheric boundary-layer dispersion processes over homogeneous and complex terrain. Special emphasis is placed on a discussion of the boundarylayer scaling regimes over homogeneous terrain and the characteristics of the dispersion processes associated with each of these regimes. The paper points out that vertical concentration profiles usually deviate substantially from a Gaussian distribution. The mean flow and turbulence over a low hill is dealt with, and in the inner layer the turbulence levels are increased due to the mean flow speed-up. In the outer layer the turbulence is modified by the rapid distortion effect. In a middle layer the turbulence is reduced due to the effect of a hill-induced streamline curvature. The paper concludes that the flow perturbations introduced by large-scale hills and valleys invalidate the use of simple approximations for describing atmospheric dispersion processes, and that it is necessary to utilize the full set of equations of motion.     

A New Sensitivity Analysis and Solution Method for Scintillometer Measurements of Area-Averaged Turbulent Fluxes

Scintillometer measurements of the turbulence inner-scale length $l_\mathrm{o }$ l o and refractive index structure function $C_n^2$ C n 2 allow for the retrieval of large-scale area-averaged turbulent fluxes in the atmospheric surface layer. This retrieval involves the solution of the non-linear set of equations defined by the Monin–Obukhov similarity hypothesis. A new method that uses an analytic solution to the set of equations is presented, which leads to a stable and efficient numerical method of computation that has the potential of eliminating computational error. Mathematical expressions are derived that map out the sensitivity of the turbulent flux measurements to uncertainties in source measurements such as $l_\mathrm{o }$ l o . These sensitivity functions differ from results in the previous literature; the reasons for the differences are explored.     

On extension of Malkus' theory of turbulence to stably stratified shear flow

Malkus' theory of turbulent shear flow and turbulent convection is extended to stably stratified shear flow. The local Richardson number is found to be close to its critical value throughout the turbulent flow. Assuming a strong interaction between velocity and density fields requires that their profiles depend linearly on height. From this, for stratified shear flow at sufficiently high Reynolds numbers, a combination of logarithmic and linear profiles is expected. The purely linear and the log-linear profile are both observed in laboratory channel flow and in atmospheric boundary layer flow.Work undertaken while at the Department of Meteorology and Physical Oceanography, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.