Outer Length Scale Research Articles

On a Correlation Model for Laser Scanners: A Large Eddy Simulation Experiment

Large Eddy Simulations (LES) allow the generation of spatio-temporal fields of the refractivity index for various meteorological conditions and provide a unique way to simulate turbulence-distorted phase measurements as those from geodetic sensors. This approach enables a statistical quantification of the von Kármán model’s adequacy in describing the phase spectrum and the assessment of the validity of common assumptions such as isotropy or the Taylor frozen hypothesis. This contribution shows that the outer scale length, defined using the Taylor frozen hypothesis as the saturation frequency of the phase spectrum, can be statistically estimated, along with an error fit factor between the model and its estimation. It is found that this parameter strongly varies with height and meteorological conditions (convective or wind-driven boundary layer). The simulations further highlight the linear dependency with the variance of the turbulent phase fluctuations but no dependency on the local outer scale length as defined by Tatarskii. An application of these results within a geodetic context is proposed, where an understanding and solid estimation of the outer scale length is mandatory in avoiding biased decisions during statistical deformation analysis. The LES presented in this contribution support derivations for an improved stochastic model of terrestrial laser scanners.

Evidence of quasiequilibrium in pressure-gradient turbulent boundary layers

Two sets of measurements utilizing hot-wire anemometry and oil-film interferometry for flat-plate turbulent boundary layers, exposed to various controlled adverse and favourable pressure gradients, are used to evaluate history effects of the imposed and varying free-stream gradients. The results are from the NDF wind tunnel at Illinois Tech (IIT) and the MTL wind tunnel at KTH, over the range $800 < Re_\tau < 22\,000$ (where $Re_{\tau }$ is the friction Reynolds number). The streamwise pressure-gradient parameter $\beta \equiv (-\ell /\tau _{w}) \cdot (\partial P_{e}/\partial x)$ varied between $-2 < \beta < 7$ , where $\ell$ is an outer length scale for boundary layers equivalent to the half-height of channel flow and the radius of pipe flow, and is estimated for each boundary-layer profile; note that $\tau_w$ is the wall-shear stress and $P_e$ is the free-stream static pressure. Extracting from each profile the three parameters of the overlap region, following the recent work of Monkewitz & Nagib (J. Fluid Mech., vol. 967, 2023, p. A15) that led to an overlap region of combined logarithmic and linear parts, we find minimum history effects in the overlap region. Thus, the overlap region in this range of pressure-gradient boundary layers appears to be in ‘quasiequilibrium’.

Outer scaling of the mean momentum equation for turbulent boundary layers under adverse pressure gradient

A new scaling of the mean momentum equation is developed for the outer region of turbulent boundary layers (TBLs) under adverse pressure gradient (APG). The maximum Reynolds shear stress location, denoted as $y_{m}$ , is employed to determine the proper scales for the outer region of an APG TBL. An outer length scale is proposed as $\delta _e - y_{m}$ , where $\delta _e$ is the boundary layer thickness. An outer velocity scale for the mean streamwise velocity deficit is proposed as $U_e - U_{m}$ , where $U_e$ and $U_m$ are the mean streamwise velocities at the boundary layer edge and $y_{m}$ , respectively. An outer velocity scale for the mean wall-normal velocity deficit is proposed as $V_e - V_{m}$ , where $V_e$ and $V_{m}$ are the wall-normal velocities at $\delta _e$ and $y_{m}$ , respectively. The maximum Reynolds shear stress is found to scale as $(\delta _e - y_{m}) U_e \,{\rm d}U_e/{{\rm d}x}$ . The new outer scaling collapses well the experimental and numerical data on APG TBLs over a wide range of Reynolds numbers and strengths of pressure gradient. Approximations of the new scaling are developed for TBLs under strong APG and at high Reynolds numbers. The relationships between the new scales and previously proposed scales are discussed.

Impact of spanwise effective slope upon rough-wall turbulent channel flow

Whereas streamwise effective slope ( $ES_{x}$ ) is accepted as a key topographical parameter in the context of rough-wall turbulent flows, the significance of its spanwise counterpart ( $ES_{y}$ ) remains largely unexplored. Here, the response of turbulent channel flow over irregular, three-dimensional rough walls with systematically varied values of $ES_{y}$ is studied using direct numerical simulation. All simulations were performed at a fixed friction Reynolds number 395, corresponding to a viscous-scaled roughness height $k^{+}\approx 65.8$ (where $k$ is the mean peak-to-valley height). A surface generation algorithm is used to synthesise a set of ten irregular surfaces with specified $ES_{y}$ for three different values of $ES_{x}$ . All surfaces share a common mean peak-to-valley height and are near-Gaussian, which allows this study to focus on the impact of varying $ES_{y}$ , since roughness amplitude, skewness and $ES_{x}$ can be eliminated simultaneously as parameters. Based on an analysis of first- and second-order velocity statistics, as well as turbulence co-spectra and the fractional contribution of pressure and viscous drag, the study shows that $ES_{y}$ can strongly affect the roughness drag penalty – particularly for low- $ES_{x}$ surfaces. A secondary observation is that particular low- $ES_{y}$ surfaces in this study can lead to diminished levels of outer-layer similarity in both mean flow and turbulence statistics, which is attributed to insufficient scale separation between the outer length scale and the in-plane spanwise roughness wavelength.

Revisiting and revising Tatarskii’s formulation for the temperature structure parameter (C_T^2) in atmospheric flows

In this paper, we revisit a well-known formulation of temperature structure parameter (C_T^2), originally proposed by V. I. Tatarskii. We point out its limitations and propose a revised formulation based on turbulence variance and flux budget equations. Our formulation includes a novel physically-based outer length scale which can be estimated from routine meteorological data.

Exploring the outskirts of the EAGLE disc galaxies

ABSTRACT Observations show that the surface brightness of disc galaxies can be well-described by a single exponential (TI), up-bending (TIII), or down-bending (TII) profiles in the outskirts. Here we characterize the mass surface densities of simulated late-type galaxies from the eagle project according to their distribution of mono-age stellar populations, the star formation activity, and angular momentum content. We find a clear correlation between the inner scale lengths and the stellar spin parameter, λ, for all three disc types with λ > 0.35. The outer scale lengths of TII and TIII discs show a positive trend with λ, albeit weaker for the latter. TII discs prefer fast rotating galaxies. With regards to the stellar age distribution, negative and U-shape age profiles are the most common for all disc types. Positive age profiles are determined by a more significant contribution of young stars in the central regions, which decrease rapidly in the outer parts. TII discs prefer relative higher contributions of old stars compared to other mono-age populations across the discs whereas TIII discs become progressively more dominated by intermediate age (2–6 Gyr) stars for increasing radius. The change in slope of the age profiles is located after the break of the mass surface density. We find evidence of larger flaring for the old stellar populations in TIII systems compared to TI and TII, which could indicate the action of other processes. Overall, the relative distributions of mono-age stellar populations and the dependence of the star formation activity on radius are found to shape the different disc types and age profiles.

Elucidating the atmospheric boundary layer turbulence by combining UHF radar wind profiler and radiosonde measurements over urban area of Beijing

The knowledge of atmospheric turbulence over urban environment remains limited albeit its great importance for air pollution and convection initiation. This study investigates turbulence characteristics in boundary layer based on concurrent radar wind profiler (RWP) and radiosonde observations for the year 2019 at an urban site in Beijing, China. The eddy dissipation rate (ϵ) is retrieved based on Doppler spectral width measurements from RWP. The radiosonde measurements are utilized for estimating the Brunt–Väisälä frequency which is further collocated with RWP based ϵ retrievals for the estimation of vertical eddy diffusivity, outer scale length and inner scale length profiles. Contrasting seasonality is revealed in diurnal cycle of height-resolved turbulence parameters over Beijing. Within the lowermost 1 km altitude, ϵ is estimated to be more than 10−2 m2s−3 during all seasons (except for winter), which possibly represents the highly turbulent state of the atmosphere near ground surface due to the local mountain-plain flow variation. However, above 1 km altitude, the ϵ magnitude varies between 10−4 and 10−2 m2s−3. The vertical eddy diffusivity is found to be higher in the spring and summer seasons while being approximately an order of magnitude lower in the autumn and winter seasons, which agrees with few existing studies.

Stable channel flow with spanwise heterogeneous surface temperature

Recent studies have demonstrated that large secondary motions are excited by surface roughness with dominant spanwise length scales of the order of the flow's outer length scale. Inspired by this, we explore the effect of spanwise heterogeneous surface temperature in weakly to strongly stratified closed channel flow (at$Ri_\tau =120$, 960;$Re_\tau = 180$, 550) with direct numerical simulations. The configuration consists of equally sized strips of high and low temperature at the lower and upper boundaries, while an overall stable stratification is induced by imposing an average temperature difference between the top and bottom. We consider the influence of the width of the strips (${\rm \pi} /8 \leq \lambda /h \leq 4{\rm \pi} $), Reynolds number, stability and upper boundary condition on the mean flow structure, skin friction and heat transfer. Results indicate that secondary flows are excited, with alternating high- and low-momentum pathways and vortices, similar to the patterns induced by spanwise heterogeneous surface roughness. We find that the impact of the surface heterogeneity on the outer layer depends strongly on the spanwise heterogeneity length scale of the surface temperature. Comparison to stable channel flow with uniform temperature reveals that the heterogeneous surface temperature increases the global friction coefficient and reduces the global Nusselt number in most cases. However, for the high-Reynolds cases with$\lambda /h \geq {\rm \pi} /2$, we find a reduction of the friction coefficient. At stronger stability, the vertical extent of the vortices is reduced and the impact of the heterogeneous temperature on momentum and heat transfer is smaller.

The logarithmic variance of streamwise velocity and conundrum in wall turbulence

The logarithmic dependence of streamwise turbulence intensity has been observed repeatedly in recent experimental and direct numerical simulation data. However, its spectral counterpart, a well-developed $k^{-1}$ spectrum ( $k$ is the spatial wavenumber in a wall-parallel direction), has not been convincingly observed from the same data. In the present study, we revisit the spectrum-based attached eddy model of Perry and co-workers, who proposed the emergence of a $k^{-1}$ spectrum in the inviscid limit, for small but finite $z/\delta$ and for finite Reynolds numbers ( $z$ is the wall-normal coordinate, and $\delta$ is the outer length scale). In the upper logarithmic layer (or inertial sublayer), a reexamination reveals that the intensity of the spectrum must vary with the wall-normal location at order of $z/\delta$ , consistent with the early observation argued with ‘incomplete similarity’. The streamwise turbulence intensity is subsequently calculated, demonstrating that the existence of a well-developed $k^{-1}$ spectrum is not a necessary condition for the approximate logarithmic wall-normal dependence of turbulence intensity – a more general condition is the existence of a premultiplied power-spectral intensity of $O(1)$ for $O(1/\delta ) < k < O(1/z)$ . Furthermore, it is shown that the Townsend–Perry constant must be weakly dependent on the Reynolds number. Finally, the analysis is semi-empirically extended to the lower logarithmic layer (or mesolayer), and a near-wall correction for the turbulence intensity is subsequently proposed. All the predictions of the proposed model and the related analyses/assumptions are validated with high-fidelity experimental data (Samie et al., J. Fluid Mech., vol. 851, 2018, pp. 391–415).

Mean thermal energy balance analysis in differentially heated vertical channel flows

Direct numerical simulations of differentially heated vertical channel (DHVC) flows were performed for Ra=105–109 to investigate the characteristics of the streamwise mean momentum and mean thermal energy equations. The log law for mean temperature was observed for Ra≥108 at y+>50, where y+ is the wall-normal distance normalized by the viscous wall unit. From the mean momentum equation, negligible viscous force and logarithmically increasing Reynolds shear stress were observed in the region where the log law for mean temperature occurred. The streamwise mean velocity did not exhibit a linear relationship with y+ close to the wall and did not show logarithmic development far from the wall due to the buoyancy force. In the mean thermal energy equation, a constant heat flux layer was observed, and the turbulent heat flux contribution was scaled by the inverse of wall-normal distance to satisfy the log law of mean temperature. For a high Rayleigh number (Ra=109), the turbulent heat flux spectra contained scale-separated inner and outer sites with linearly growing energetic structures along the wall-normal distance, which was not observed for a low Rayleigh number (Ra=106). The flow structures of turbulent heat flux originated from the upward wall-normal velocity fluctuations that triggered the non-directional structures of the temperature. These results suggest that the scale separation between the viscous and outer length scales with the wall-attached energetic structures resulted in the log law for mean temperature. These findings could serve as the basis of scaling formulations for the mean velocity and temperature in DHVC flows.

Predictive accuracy of wall-modelled large-eddy simulation on unstructured grids

The predictive accuracy of wall-modelled LES is influenced by a combination of the subgrid model, the wall model, the numerical dissipation induced primarily by the convective numerical scheme, and also by the density and topology of the computational grid. The latter factor is of particular importance for industrial flow problems, where unstructured grids are typically employed due to the necessity to handle complex geometries. Here, a systematic simulation-based study is presented, investigating the effect of grid-cell type on the predictive accuracy of wall-modelled LES in the framework of a general-purpose finite-volume solver. Following standard practice for meshing near-wall regions, it is proposed to use prismatic cells. Three candidate shapes for the base of the prisms are considered: a triangle, a quadrilateral, and an arbitrary polygon. The cell-centre distance is proposed as a metric to determine the spatial resolution of grids with different cell types. The simulation campaign covers two test cases with attached boundary layers: fully-developed turbulent channel flow, and a zero-pressure-gradient flat-plate turbulent boundary layer. A grid construction strategy is employed, which adapts the grid metric to the outer length scale of the boundary layer. The results are compared with DNS data concerning mean wall shear stress and profiles of flow statistics. The principle outcome is that unstructured simulations may provide the same accuracy as simulations on structured orthogonal hexahedral grids. The choice of base shape of the near-wall cells has a significant impact on the computational cost, but in terms of accuracy appears to be a factor of secondary importance.

Accurate Computation of Scintillation in Tropospheric Turbulence With Parabolic Wave Equation

Parabolic wave equation (PWE) propagation models using multiple phase screens (MPS) are widely used to predict the log-amplitude variance of the signal propagating in tropospheric turbulence. Previous literature demonstrates the limitations of the MPS technique, usable only when the signal frequency is low (typically below 5 GHz for radio-wave propagation in the troposphere). In this article, we demonstrate that the accuracy of the MPS technique depends not only on the signal frequency but also on the outer scale length ( L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">os</sub> ) of the turbulent eddy. For example, when L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">os</sub> is on the order of tens of meters (e.g., in most terrestrial communication links), the MPS method is indeed unusable at high frequencies, but, when L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">os</sub> is on the order of hundreds of meters (e.g., in air-ground and satellite links), the method can be used even at millimeter-wave frequencies. Next, a correlated phase screen (CPS) implementation within the PWE is introduces, which avoids the condition of statistical noncorrelation of the phase screens in the MPS technique, and as such, it can be used for any signal frequency. Finally, an ad hoc error metric is provided, allowing for the selection of the appropriate phase screen model based on the signal frequency, turbulence outer scale length, and the electromagnetic (EM) link distance. The simulated signal log-amplitude variances as a function of range at microwave and millimeter-wave frequencies using the CPS method are reported here for the first time and are shown to be in excellent agreement with the analytically obtained values.

Multifractal Model of Atmospheric Turbulence Applied to Elastic Lidar Data

This paper shall present a multifractal interpretation of turbulent atmospheric entities, considering them a complex system whose dynamics are manifested on continuous yet non-differentiable multifractal curves. By bringing forth theoretical considerations regarding multifractal structures through non-differentiable functions in the form of an adaptation of scale relativity theory, the minimal vortex of an instance of turbulent flow is considered. In this manner, the spontaneous breaking of scale invariance becomes a mechanism for atmospheric turbulence generation. This then leads to a general equation for the non-differentiable vortex itself, with its component velocity fields, and to a vortex turbulent energy dissipation—all of which are plotted and studied. Once the structure of the non-differentiable multifractal structure is mathematically described, an improved phenomenological turbulence model and relations between turbulent energy dissipation and the minimal vortex are employed together, exemplifying the codependency of such models. Using turbulent medium wave propagation theory, certain relations are then extrapolated which allow the obtaining of the inner and outer length scales of the turbulent flow using lidar data. Finally, these altitude profiles are compiled and assembled into timeseries to exemplify the theory and to compare the results with known literature. This model is a generalization of our recent results published under the title “On a Multifractal Approach of Turbulent Atmosphere Dynamics”.

On the determination of the atmospheric outer scale length of turbulence using GPS phase difference observations: the Seewinkel network

Microwave electromagnetic signals from the Global Navigation Satellite System (GNSS) are affected by their travel through the atmosphere: the troposphere, a non-dispersive medium, has an especial impact on the measurements. The long-term variations of the tropospheric refractive index delay the signals, whereas its random variations correlate with the phase measurements. The correlation structure of residuals from GNSS relative position estimation provides a unique opportunity to study specific properties of the turbulent atmosphere. Prior to such a study, the residuals have to be filtered from unwanted additional effects, such as multipath. In this contribution, we propose to investigate the property of the atmospheric noise by using a new methodology combining the empirical mode decomposition with the Hilbert–Huang transform. The chirurgical “designalling of the noise” aims to filter both the white noise and low-frequency noise to extract only the noise coming from tropospheric turbulence. Further analysis of the power spectrum of phase difference can be performed, including the study of the cut-off frequencies and the two slopes of the power spectrum of phase differences. The obtained values can be compared with theoretical expectations. In this contribution, we use Global Positioning System (GPS) phase observations from the Seewinkel network, specially designed to study the impact of atmospheric turbulence on GPS phase observations. We show that (i) a two-slope power spectrum can be found in the residuals and (ii) that the outer scale length can be taken to a constant value, close to the physically expected one and in relation with the size of the eddies at tropospheric height.

Roughness-induced secondary flows in stably stratified turbulent boundary layers

Spanwise heterogeneous roughness, more specifically, spanwise-adjacent strips of relatively high and low roughness, is known to cause large-scale secondary flows in the vertical domain in neutral turbulent boundary layers. In this work, we study the response of secondary vortices to thermal stratification with focus on the stably stratified case. We systematically vary the Monin–Obukhov (MO) length scale from L/h = 0.3, which corresponds to relatively strong stratification, to ∞, which corresponds to the neutral condition. Here, L is the MO length and h is an outer length scale. The results show that stable stratification suppresses the vertical motions and reduces the vertical size and strength of the first off-wall secondary vortex. Moreover, the reduced vertical extent of the first off-wall vortex and the shear near its top together give rise to a second vortex, and the second to a third vortex. This leads to a stack of secondary vortices in a stably stratified boundary layer flow with decreasing strength when moving away from the wall.

Heat Transport through Diurnal Warm Layers

AbstractPenetration of solar radiation in the upper few meters of the ocean creates a near-surface, stratified diurnal warm layer. Wind stress accelerates a diurnal jet in this layer. Turbulence generated at the diurnal thermocline, where the shear of the diurnal jet is concentrated, redistributes heat downward via mixing. New measurements of temperature and turbulence from fast thermistors on a surface-following platform depict the details of this sequence in both time and depth. Temporally, the sequence at a fixed depth follows a counterclockwise path in logϵ–logN parameter space. This path also captures the evolution of buoyancy Reynolds number (a proxy for the anisotropy of the turbulence) and Ozmidov scale (a proxy for the outer vertical length scale of turbulence in the absence of the free surface). Vertically, the solar heat flux always leads to heating of fluid parcels in the upper few meters, whereas the turbulent heat flux divergence changes sign across the depth of maximum vertical temperature gradient, cooling fluid parcels above and heating fluid parcels below. In general, our measurements of fluid parcel heating or cooling rates of order 0.1°C h−1 are consistent with our estimates of heat flux divergence. In weak winds (&lt;2 m s−1), sea surface temperature (SST) is controlled by the depth-dependent absorption of solar radiation. In stronger winds, turbulent mixing controls SST.

On the dissipation rate of temperature fluctuations in stably stratified flows

In this study, we explore several integral and outer length scales of turbulence which can be formulated by using the dissipation of temperature fluctuations (chi) and other relevant variables. Our analyses directly lead to simple yet non-trivial parameterizations for both spatially-averaged overline{chi } and the structure parameter of temperature (C_T^2). For our purposes, we make use of high-fidelity data from direct numerical simulations of stratified channel flows.

On a Multifractal Approach of Turbulent Atmosphere Dynamics

This paper aims to present a multifractal approach of the turbulent atmosphere, by proposing that it can be considered a complex system whose structural units support dynamics on continuous but non-differentiable multifractal curves. Implementing the theoretical framework of multifractality through non-differentiable functions in the form of scale relativity theory with arbitrary and constant fractal dimension, the minimal vortex of an instance of turbulent flow is considered. The results of this assumption lead to an equation that describes the minimal vortex itself, and the velocity fields that compose it, the vortex and turbulent energy dissipation derived from the vortex being plotted and studied. With its structure mathematically described, while employing a classical dynamical turbulence model and relations between turbulent energy dissipation and the minimal vortex, relations are then extrapolated to allow for the solving of multiple turbulent parameters using the inner and outer length scales of the turbulent flow. These equations are then solved as altitude profiles with the necessary length scales obtained from processing lidar data. Finally, profiles are taken periodically and assembled into timeseries, in order to exemplify the method and to compare the results with known literature.

Inner, meso, and outer scales in a differentially heated vertical channel

Reynolds shear stress and Reynolds normal stress in a differentially heated vertical channel were shown in a previous paper to scale by a mixed scale of the friction velocity uτ and the maximum mean vertical velocity Umax. This scale was empirically determined; this work presents a physical understanding of the new scaling. A new dimensional analysis is developed that involves a simple modification of the traditional inner–outer scaling. The functional dependencies in the new dimensional analysis are determined using direct numerical simulation (DNS) data. The DNS data confirm the new physical reasoning and reveal, as follows: (i) The velocity fluctuation variance at the centerline scales with an outer scale, which is the mixed scale uτUmax. (ii) The location of the maximum mean vertical velocity scales as an Obukhov-style length scale, which, in turn, scales as the geometric mean of the inner length scale and the outer length scale. (iii) The temperature fluctuation variance at the centerline scales with an outer temperature scale. (iv) The temperature fluctuation variance peaks in the inner layer, and its peak value scales with an inner temperature scale. The new outer scales obtained from the dimensional analysis are shown to be consistent with the properties of the mean momentum balance equation.

The role of the critical layer in the channel flow transition revisited

The focus of this paper is on the development of a novel and simple linear modeling to interpret the (asymptotic) instability conditions for laminar channel (Poiseuille) flow, thereby giving insight on the major length and time scales of the subsequent mean turbulent regime. Although this is an old problem, recently there has been a renewed effort to understand how information on the wall turbulence scalings can be obtained by studying the flow linear dynamics, among other approaches by using the resolvent analysis. Here the classic asymptotic stability analysis is reformulated by means of a pseudo two-fluid model, a central inviscid flow motion, and a wall viscous one, forcing each other on their common interface streamline. The best agreement with the results of the celebrated Orr–Sommerfeld equation is obtained for the interface located around the elevation of the base flow average value, equal to 2/3 times the maximum velocity (frequency scale). It is argued that, in the early stage of perturbations growth, while the mechanism leading to instability (and that is related to the extraction of energy from the base flow via the Reynolds stress), is located near the wall in the critical layer region (already known result for inner length scale), the central region around the flow average velocity (i.e., the outer length scale is of order $${\mathcal {O}}(h)$$, with h being the channel half-height) is also deeply involved in the onset of the instability. The initial linear amplification of the wall viscous modes of the present analysis agrees with the major features of the wall functions driving the full turbulent profile, found by means of the resolvent analysis, in the subsequent nonlinear stage.