Effect of finite sampling time on estimation of Brownian fluctuation

2015 ◽  
Vol 767 ◽  
pp. 65-84 ◽  
Author(s):  
Shahram Pouya ◽  
Di Liu ◽  
Manoochehr M. Koochesfahani
Keyword(s):  
Diffusion Coefficient ◽  
Scaling Laws ◽  
Solid Wall ◽  
Exact Analytical Solution ◽  
Scaling Law ◽  
Time Interval ◽  
Free Diffusion ◽  
Mean Velocity ◽  
Hindered Diffusion ◽  
Near Wall

AbstractWe present a study of the effect of finite detector integration/exposure time $E$, in relation to interrogation time interval ${\rm\Delta}t$, on analysis of Brownian motion of small particles using numerical simulation of the Langevin equation for both free diffusion and hindered diffusion near a solid wall. The simulation result for free diffusion recovers the known scaling law for the dependence of estimated diffusion coefficient on $E/{\rm\Delta}t$, i.e. for $0\leqslant E/{\rm\Delta}t\leqslant 1$ the estimated diffusion coefficient scales linearly as $1-(E/{\rm\Delta}t)/3$. Extending the analysis to the parameter range $E/{\rm\Delta}t\geqslant 1$, we find a new nonlinear scaling behaviour given by $(E/{\rm\Delta}t)^{-1}[1-((E/{\rm\Delta}t)^{-1})/3]$, for which we also provide an exact analytical solution. The simulation of near-wall diffusion shows that hindered diffusion of particles parallel to a solid wall, when normalized appropriately, follows with a high degree of accuracy the same form of scaling laws given above for free diffusion. Specifically, the scaling laws in this case are well represented by $1-((1+{\it\epsilon})(E/{\rm\Delta}t))/3$, for $E/{\rm\Delta}t\leqslant 1$, and $(E/{\rm\Delta}t)^{-1}[1-((1+{\it\epsilon})(E/{\rm\Delta}t)^{-1})/3]$, for $E/{\rm\Delta}t\geqslant 1$, where the small parameter ${\it\epsilon}$ depends on the size of the near-wall domain used in the estimation of the diffusion coefficient and value of $E$. For the range of parameters reported in the literature, we estimate ${\it\epsilon}<0.03$. The near-wall simulations also show a bias in the estimated diffusion coefficient parallel to the wall even in the limit $E=0$, indicating an overestimation which increases with increasing time delay ${\rm\Delta}t$. This diffusion-induced overestimation is caused by the same underlying mechanism responsible for the previously reported overestimation of mean velocity in near-wall velocimetry.

Similarity in non-rotating and rotating turbulent pipe flows

1999 ◽  
Vol 379 ◽  
pp. 1-22 ◽  
Author(s):  
MARTIN OBERLACK
Keyword(s):  
Reynolds Number ◽  
Scaling Laws ◽  
Stokes Equations ◽  
Scaling Law ◽  
Navier Stokes ◽  
Invariant Solutions ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
Pipe Flows ◽  
The Mean

The Lie group approach developed by Oberlack (1997) is used to derive new scaling laws for high-Reynolds-number turbulent pipe flows. The scaling laws, or, in the methodology of Lie groups, the invariant solutions, are based on the mean and fluctuation momentum equations. For their derivation no assumptions other than similarity of the Navier–Stokes equations have been introduced where the Reynolds decomposition into the mean and fluctuation quantities has been implemented. The set of solutions for the axial mean velocity includes a logarithmic scaling law, which is distinct from the usual law of the wall, and an algebraic scaling law. Furthermore, an algebraic scaling law for the azimuthal mean velocity is obtained. In all scaling laws the origin of the independent coordinate is located on the pipe axis, which is in contrast to the usual wall-based scaling laws. The present scaling laws show good agreement with both experimental and DNS data. As observed in experiments, it is shown that the axial mean velocity normalized with the mean bulk velocity um has a fixed point where the mean velocity equals the bulk velocity independent of the Reynolds number. An approximate location for the fixed point on the pipe radius is also given. All invariant solutions are consistent with all higher-order correlation equations. A large-Reynolds-number asymptotic expansion of the Navier–Stokes equations on the curved wall has been utilized to show that the near-wall scaling laws for at surfaces also apply to the near-wall regions of the turbulent pipe flow.

Scaling laws for fully developed turbulent shear flows. Part 1. Basic hypotheses and analysis

1993 ◽  
Vol 248 ◽  
pp. 513-520 ◽  
Author(s):  
G. I. Barenblatt
Keyword(s):  
Reynolds Number ◽  
Velocity Distribution ◽  
Scaling Laws ◽  
Scaling Law ◽  
Turbulent Shear ◽  
Power Exponent ◽  
Turbulent Shear Flow ◽  
Mean Velocity ◽  
Friction Law ◽  
The Mean

The present work consists of two parts. Here in Part 1, a scaling law (incomplete similarity with respect to local Reynolds number based on distance from the wall) is proposed for the mean velocity distribution in developed turbulent shear flow. The proposed scaling law involves a special dependence of the power exponent and multiplicative factor on the flow Reynolds number. It emerges that the universal logarithmic law is closely related to the envelope of a family of power-type curves, each corresponding to a fixed Reynolds number. A skin-friction law, corresponding to the proposed scaling law for the mean velocity distribution, is derived.In Part 2 (Barenblatt & Prostokishin 1993), both the scaling law for the velocity distribution and the corresponding friction law are compared with experimental data.

scholarly journals Asymmetric Unimodal Maps with Non-universal Period-Doubling Scaling Laws

2020 ◽  
Vol 379 (1) ◽  
pp. 103-143
Author(s):  
Oleg Kozlovski ◽  
Sebastian van Strien
Keyword(s):  
Scaling Laws ◽  
Scaling Law ◽  
Period Doubling ◽  
Cantor Sets ◽  
Unimodal Maps ◽  
Renormalization Theory

Abstract We consider a family of strongly-asymmetric unimodal maps $$\{f_t\}_{t\in [0,1]}$$ { f t } t ∈ [ 0 , 1 ] of the form $$f_t=t\cdot f$$ f t = t · f where $$f:[0,1]\rightarrow [0,1]$$ f : [ 0 , 1 ] → [ 0 , 1 ] is unimodal, $$f(0)=f(1)=0$$ f ( 0 ) = f ( 1 ) = 0 , $$f(c)=1$$ f ( c ) = 1 is of the form and $$\begin{aligned} f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)&{} \text{ for } x<c, \\ 1-K_+|x-c|^\beta + o(|x-c|^\beta ) &{} \text{ for } x>c, \end{array}\right. \end{aligned}$$ f ( x ) = 1 - K - | x - c | + o ( | x - c | ) for x < c , 1 - K + | x - c | β + o ( | x - c | β ) for x > c , where we assume that $$\beta >1$$ β > 1 . We show that such a family contains a Feigenbaum–Coullet–Tresser $$2^\infty $$ 2 ∞ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $$2^\infty $$ 2 ∞ map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum–Coullet–Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results.

Law of bounded dissipation and its consequences in turbulent wall flows

2021 ◽  
Vol 933 ◽  
Author(s):  
Xi Chen ◽  
Katepalli R. Sreenivasan
Keyword(s):  
Dissipation Rate ◽  
Scaling Law ◽  
Random Variable ◽  
Energy Dissipation Rate ◽  
Shear Stresses ◽  
Mean Velocity ◽  
Wall Flows ◽  
Maximum Value ◽  
Turbulent Wall ◽  
Promising Perspective

The dominant paradigm in turbulent wall flows is that the mean velocity near the wall, when scaled on wall variables, is independent of the friction Reynolds number $Re_\tau$ . This paradigm faces challenges when applied to fluctuations but has received serious attention only recently. Here, by extending our earlier work (Chen & Sreenivasan, J. Fluid Mech., vol. 908, 2021, p. R3) we present a promising perspective, and support it with data, that fluctuations displaying non-zero wall values, or near-wall peaks, are bounded for large values of $Re_\tau$ , owing to the natural constraint that the dissipation rate is bounded. Specifically, $\varPhi _\infty - \varPhi = C_\varPhi \,Re_\tau ^{-1/4},$ where $\varPhi$ represents the maximum value of any of the following quantities: energy dissipation rate, turbulent diffusion, fluctuations of pressure, streamwise and spanwise velocities, squares of vorticity components, and the wall values of pressure and shear stresses; the subscript $\infty$ denotes the bounded asymptotic value of $\varPhi$ , and the coefficient $C_\varPhi$ depends on $\varPhi$ but not on $Re_\tau$ . Moreover, there exists a scaling law for the maximum value in the wall-normal direction of high-order moments, of the form $\langle \varphi ^{2q}\rangle ^{{1}/{q}}_{max}= \alpha _q-\beta _q\,Re^{-1/4}_\tau$ , where $\varphi$ represents the streamwise or spanwise velocity fluctuation, and $\alpha _q$ and $\beta _q$ are independent of $Re_\tau$ . Excellent agreement with available data is observed. A stochastic process for which the random variable has the form just mentioned, referred to here as the ‘linear $q$ -norm Gaussian’, is proposed to explain the observed linear dependence of $\alpha _q$ on $q$ .

scholarly journals Scaling and Similarity of the Anisotropic Coherent Eddies in Near-Surface Atmospheric Turbulence

2018 ◽  
Vol 75 (3) ◽  
pp. 943-964 ◽  
Author(s):  
Khaled Ghannam ◽  
Gabriel G. Katul ◽  
Elie Bou-Zeid ◽  
Tobias Gerken ◽  
Marcelo Chamecki
Keyword(s):  
Scaling Laws ◽  
Velocity Structure ◽  
Scaling Law ◽  
Longitudinal Velocity ◽  
Structure Functions ◽  
Length Scale ◽  
Near Surface ◽  
Vegetation Canopies ◽  
Velocity Spectra ◽  
Attached Eddies

Abstract The low-wavenumber regime of the spectrum of turbulence commensurate with Townsend’s “attached” eddies is investigated here for the near-neutral atmospheric surface layer (ASL) and the roughness sublayer (RSL) above vegetation canopies. The central thesis corroborates the significance of the imbalance between local production and dissipation of turbulence kinetic energy (TKE) and canopy shear in challenging the classical distance-from-the-wall scaling of canonical turbulent boundary layers. Using five experimental datasets (two vegetation canopy RSL flows, two ASL flows, and one open-channel experiment), this paper explores (i) the existence of a low-wavenumber k−1 scaling law in the (wind) velocity spectra or, equivalently, a logarithmic scaling ln(r) in the velocity structure functions; (ii) phenomenological aspects of these anisotropic scales as a departure from homogeneous and isotropic scales; and (iii) the collapse of experimental data when plotted with different similarity coordinates. The results show that the extent of the k−1 and/or ln(r) scaling for the longitudinal velocity is shorter in the RSL above canopies than in the ASL because of smaller scale separation in the former. Conversely, these scaling laws are absent in the vertical velocity spectra except at large distances from the wall. The analysis reveals that the statistics of the velocity differences Δu and Δw approach a Gaussian-like behavior at large scales and that these eddies are responsible for momentum/energy production corroborated by large positive (negative) excursions in Δu accompanied by negative (positive) ones in Δw. A length scale based on TKE dissipation collapses the velocity structure functions at different heights better than the inertial length scale.

scholarly journals Tectonic Geomorphology and Paleoseismology of the Sharkhai fault: a new source of seismic hazard for Ulaanbaatar (Mongolia)

2021 ◽  
Author(s):  
Abeer Al-Ashkar ◽  
Antoine Schlupp ◽  
Matthieu Ferry ◽  
Ulziibat Munkhuu
Keyword(s):  
Seismic Hazard ◽  
Scaling Laws ◽  
Active Faults ◽  
Seismic Hazard Assessment ◽  
Capital City ◽  
Tectonic Geomorphology ◽  
Time Interval ◽  
Coseismic Slip ◽  
Slip Rates ◽  
Large Earthquakes

Abstract. We present new constraints from tectonic geomorphology and paleoseismology along the newly discovered Sharkhai fault near the capital city of Mongolia. Detailed observations from high resolution Pleiades satellite images and field investigations allowed us to map the fault in detail, describe its geometry and segmentation, characterize its kinematics, and document its recent activity and seismic behavior (cumulative displacements and paleoseismicity). The Sharkhai fault displays a surface length of ~40 km with a slightly arcuate geometry, and a strike ranging from N42° E to N72° E. It affects numerous drainages that show left-lateral cumulative displacements reaching 57 m. Paleoseismic investigations document the faulting and deposition record for the last ~3000 yr and reveal that the penultimate earthquake (PE) occurred between 1515 ± 90 BC and 945 ± 110 BC and the most recent event (MRE) occurred after 860 ± 85 AD. The resulting time interval of 2080 ± 470 years is the first constraint on the Sharkhai fault for large earthquakes. On the basis of our mapping of the surface rupture and the resulting segmentation analysis, we propose two possible scenarios for large earthquakes with likely magnitudes between 6.4 ± 0.2 and 7.1 ± 0.2. Furthermore, we apply scaling laws to infer coseismic slip values and derive preliminary estimates of long-term slip rates between 0.2 ± 0.2 and 1.0 ± 0.5 mm/y. Finally, we propose that these original observations and results from a newly discovered fault should be taken into account for the seismic hazard assessment for the city of Ulaanbaatar and help build a comprehensive model of active faults in that region.

scholarly journals Active control of multiscale features in wall-bounded turbulence

2019 ◽  
Vol 36 (1) ◽  
pp. 12-21 ◽  
Author(s):  
Xiaotong Cui ◽  
Nan Jiang ◽  
Xiaobo Zheng ◽  
Zhanqi Tang
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Turbulent Energy ◽  
Streamwise Velocity ◽  
Discrete Wavelet ◽  
Wall Region ◽  
Mean Velocity ◽  
Pzt Actuator ◽  
The Impact ◽  
Near Wall

Abstract This study experimentally investigates the impact of a single piezoelectric (PZT) actuator on a turbulent boundary layer from a statistical viewpoint. The working conditions of the actuator include a range of frequencies and amplitudes. The streamwise velocity signals in the turbulent boundary layer flow are measured downstream of the actuator using a hot-wire anemometer. The mean velocity profiles and other basic parameters are reported. Spectra results obtained by discrete wavelet decomposition indicate that the PZT vibration primarily influences the near-wall region. The turbulent intensities at different scales suggest that the actuator redistributes the near-wall turbulent energy. The skewness and flatness distributions show that the actuator effectively alters the sweep events and reduces intermittency at smaller scales. Moreover, under the impact of the PZT actuator, the symmetry of vibration scales’ velocity signals is promoted and the structural composition appears in an orderly manner. Probability distribution function results indicate that perturbation causes the fluctuations in vibration scales and smaller scales with high intensity and low intermittency. Based on the flatness factor, the bursting process is also detected. The vibrations reduce the relative intensities of the burst events, indicating that the streamwise vortices in the buffer layer experience direct interference due to the PZT control.

New perspectives on the impulsive roughness-perturbation of a turbulent boundary layer

2011 ◽  
Vol 677 ◽  
pp. 179-203 ◽  
Author(s):  
I. JACOBI ◽  
B. J. McKEON
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Flow Field ◽  
Spectral Energy ◽  
Internal Layers ◽  
Hot Wire ◽  
Mean Velocity ◽  
Composite Maps ◽  
Downstream Flow ◽  
Near Wall

The zero-pressure-gradient turbulent boundary layer over a flat plate was perturbed by a short strip of two-dimensional roughness elements, and the downstream response of the flow field was interrogated by hot-wire anemometry and particle image velocimetry. Two internal layers, marking the two transitions between rough and smooth boundary conditions, are shown to represent the edges of a ‘stress bore’ in the flow field. New scalings, based on the mean velocity gradient and the third moment of the streamwise fluctuating velocity component, are used to identify this ‘stress bore’ as the region of influence of the roughness impulse. Spectral composite maps reveal the redistribution of spectral energy by the impulsive perturbation – in particular, the region of the near-wall peak was reached by use of a single hot wire in order to identify the significant changes to the near-wall cycle. In addition, analysis of the distribution of vortex cores shows a distinct structural change in the flow associated with the perturbation. A short spatially impulsive patch of roughness is shown to provide a vehicle for modifying a large portion of the downstream flow field in a controlled and persistent way.

scholarly journals On the streamwise evolution of turbulent boundary layers in arbitrary pressure gradients

2002 ◽  
Vol 461 ◽  
pp. 61-91 ◽  
Author(s):  
A. E. PERRY ◽  
IVAN MARUSIC ◽  
M. B. JONES
Keyword(s):  
Boundary Layers ◽  
Scaling Laws ◽  
Power Spectra ◽  
Adverse Pressure Gradient ◽  
Turbulent Boundary Layers ◽  
Stream Velocity ◽  
Mean Flow ◽  
Mean Velocity ◽  
Law Of The Wall ◽  
The Mean

A new approach to the classic closure problem for turbulent boundary layers is presented. This involves, first, using the well-known mean-flow scaling laws such as the log law of the wall and the law of the wake of Coles (1956) together with the mean continuity and the mean momentum differential and integral equations. The important parameters governing the flow in the general non-equilibrium case are identified and are used for establishing a framework for closure. Initially closure is achieved here empirically and the potential for achieving closure in the future using the wall-wake attached eddy model of Perry & Marusic (1995) is outlined. Comparisons are made with experiments covering adverse-pressure-gradient flows in relaxing and developing states and flows approaching equilibrium sink flow. Mean velocity profiles, total shear stress and Reynolds stress profiles can be computed for different streamwise stations, given an initial upstream mean velocity profile and the streamwise variation of free-stream velocity. The attached eddy model of Perry & Marusic (1995) can then be utilized, with some refinement, to compute the remaining unknown quantities such as Reynolds normal stresses and associated spectra and cross-power spectra in the fully turbulent part of the flow.

Sign in / Sign up

Export Citation Format

Share Document