Загрязнение ландшафтов и водных объектов при авариях на трубопроводах (на примере месторождений Западной Сибири)

Based on direct numerical simulations of forced turbulence, shear turbulence, decaying turbulence, a turbulent channel flow as well as a Kolmogorov flow with Taylor based Reynolds numbers Reλ between 69 and 295, the normalized probability density function of the length distribution ˜ P( ˜ l ) of dissipation elements, the conditional mean scalar difference at the extreme points as well as the scaling of the two-point velocity difference along gradient trajectories are studied. Using the field of the instantanous turbulent kinetic energy k as a scalar, we find a good agreement between the model equation for ˜ P ( ˜ l ) as proposed by Wang and Peters (2008) and the results obtained in the different DNS cases. This confirms the independance of the model solution from both, the Reynolds number and the type of turbulent flow, so that it can be considered universally valid. In addition, we show a 2/3 scaling for the mean conditional scalar difference. In the second part of the paper, we examine the scaling of the conditional two-point velocity difference along gradient trajectories. In particular, we compare the linear s/τ scaling, where τ denotes an integral time scale and s the separation arclength along a gradient trajectory in the inertial range as derived by Wang (2009) with the s·a_∞ scaling, where a_∞ denotes the asymtotic value of the conditional mean strain rate of large dissipation elements

Geometrical Structure of Small Scales and Wall-bounded Turbulence

Turbulence is observed in most technical and natural environments involving fluid motion. However, the theory behind is still not fully understood. Due to the irregular, complex character of turbulence, it is treated statistically since a deterministic approach is usually not possible. Spatial structures in turbulence, known as eddies, are essential to describe the turbulent flow. In this thesis, a new method proposed by Wang & Peters (2006) is employed to decompose turbulent scalar fields completely and uniquely into small spatial sub-units. The approach is called Dissipation Element method. Gradient trajectories in the scalar field are traced in ascending and descending directions where they inevitably reach a minimum and a maximum point, respectively. All trajectories leading to the same pair of extremal points define a dissipation element (DE). In the present work, DE analysis is extended to the canonical wall-bounded turbulent channel flow. Special focus will be given to the effect of the wall boundaries with respect to the size of the DEs and their distribution along the wall-normal direction of the channel. To obtain data for analysis, Direct Numerical Simulations (DNS) have been conducted at different Reynolds numbers as presented in chapter 2 following a brief introduction in §1. Turbulent channel flow statistics are discussed in §3 which are later addressed to interpret results from DE analysis. In chapter 4, three-dimensional turbulent structures, called vortices, are presented which are obtained with classical methods. Classical turbulent length scales in Poiseuille flow are analyzed in §5 before the DE method is applied in chapter 6. Mean length of DEs and its variation with the distance from the wall will be addressed extensively. The influence of the Reynolds number and the choice of the scalar variable is discussed. Marginal, joint and conditional probability densities (pdf) of the Euclidean distance and scalar difference between extremal points are investigated. Employing Lie symmetry analysis, invariant solutions of the pdf are obtained. Further, a log-normal model for the pdf is derived. In addition to the classical Poiseuille flow, three different channel flows are investigated by means of the DE method, namely channel flows with wall-normal and streamwise rotations and wall transpiration. Finally, streamline segments are examined in chapter 7 with respect to the length and the velocity difference between their ending points. As in the case of DEs, marginal and conditional pdfs, as well as the influence of the wall distance and Reynolds number are discussed

Length scale analysis in wall-bounded turbulent flow by means of Dissipation Elements

The Dissipation Element ( DE ) method is used to analyse the geometric structure of turbulent pattern for several scalar fields in a turbulent plane channel flow obtained by Direct Numerical Simulations ( DNS ) of the Navier-Stokes equations. We show that both the probability density function ( pdf ) and the number of DE exhibit a clear scaling behavior as a function of the wall. Further, a remarkable insensitivity of the pdf is observed with respect to the Reynolds number and the choice of the scalar

Length scale analysis in wall-bounded turbulent flow by means of Dissipation Elements

The Dissipation Element (DE) method is used to analyse the geometric structure of turbulent pattern for several scalar fields in a turbulent plane channel flow obtained by Direct Numerical Simulations (DNS) of the Navier-Stokes equations. We show that both the probability density function (pdf) and the number of DE exhibit a clear scaling behavior as a function of the wall. Further, a remarkable insensitivity of the pdf is observed with respect to the Reynolds number and the choice of the scalar

Dissipation element analysis of scalar fields in wall-bounded turbulent flow

In order to analyze the geometry of turbulent structures in turbulent channel flow, the scalar field obtained by Direct Numerical Simulations (DNS) is subdivided into numerous finite size regions. In each of these regions local extremal points of the fluctuating scalar are determined via gradient trajectory method. Gradient trajectories starting from every material point in the scalar field Φ(x,y, z, t) in the directions of ascending and descending scalar gradients will always reach a minimum and a maximum point where \nabla Φ = 0. The ensemble of all material points belonging to the same pair of extremal points defines a dissipation element 2. They can be characterized statistically by two parameters: namely the linear length connecting the minimum and maximum points and the absolute value of the scalar difference ΔΦ at these points, respectively. Because material points are space-filling, dissipation elements are also space-filling and unique, which means that the turbulent scalar field can be decomposed into such elements. This allows the reconstruction of certain statistical quantities of small scale turbulence. Here special focus will be given to examine if and how critical points and accordingly dissipation elements are in relationship with the characteristic layers of a turbulent channel flow

Extensive strain along gradient trajectories in the turbulent kinetic energy field

Based on direct numerical simulations of forced turbulence, shear turbulence, decaying turbulence, a turbulent channel flow as well as a Kolmogorov flow with Taylor-based Reynolds numbers Reλ between 69 and 295, the normalized probability density function of the length distribution of dissipation elements, the conditional mean scalar difference ⟨Δk∣l⟩ at the extreme points as well as the scaling of the two-point velocity difference along gradient trajectories ⟨Δun⟩ are studied. Using the field of the instantaneous turbulent kinetic energy k as a scalar, we find good agreement between the model equation for as proposed by Wang and Peters (2008 J. Fluid Mech. 608 113–38) and the results obtained in the different direct numerical simulation cases. This confirms the independence of the model solution from both the Reynolds number and the type of turbulent flow, so that it can be considered universally valid. In addition, we show a 2/3 scaling for the mean conditional scalar difference. In the second part of the paper, we examine the scaling of the conditional two-point velocity difference along gradient trajectories. In particular, we compare the linear s/τ scaling, where τ denotes an integral time scale and s the separation arclength along a gradient trajectory in the inertial range as derived by Wang (2009 Phys. Rev. E 79 046325) with the s·a∞ scaling, where a∞ denotes the asymptotic value of the conditional mean strain rate of large dissipation elements

Extensive strain along gradient trajectories in the turbulent kinetic energy field

Based on direct numerical simulations of forced turbulence, shear turbulence, decaying turbulence, a turbulent channel flow as well as a Kolmogorov flow with Taylor-based Reynolds numbers Reλ between 69 and 295, the normalized probability density function of the length distribution P(l) of dissipation elements, the conditional mean scalar difference Δkl at the extreme points as well as the scaling of the two-point velocity difference along gradient trajectories Δun are studied. Using the field of the instantaneous turbulent kinetic energy k as a scalar, we find good agreement between the model equation for P(l) as proposed by Wang and Peters (2008 J. Fluid Mech. 608 113–38) and the results obtained in the different direct numerical simulation cases. This confirms the independence of the model solution from both the Reynolds number and the type of turbulent flow, so that it can be considered universally valid. In addition, we show a 2/3 scaling for the mean conditional scalar difference. In the second part of the paper, we examine the scaling of the conditional two-point velocity difference along gradient trajectories. In particular, we compare the linear s/τ scaling, where τ denotes an integral time scale and s the separation arclength along a gradient trajectory in the inertial range as derived by Wang (2009 Phys. Rev. E 79 046325) with the sa∞ scaling, where a∞ denotes the asymptotic value of the conditional mean strain rate of large dissipation elements

Compressive and extensive strain along gradient trajectories

Based on direct numerical simulations of forced turbulence, shear turbulence, decaying turbulence, a turbulent channel flow as well as a Kolmogorov flow with Taylor based Reynolds numbers Reλ between 69 and 295, the normalized probability density function of the length distribution P̃(l̃) of dissipation elements, the conditional mean scalar difference at the extreme points as well as the scaling of the two-point velocity difference along gradient trajectories are studied. Using the field of the instantanous turbulent kinetic energy k as a scalar, we find a good agreement between the model equation for P̃(l̃) as proposed by Wang and Peters (2008) and the results obtained in the different DNS cases. This confirms the independance of the model solution from both, the Reynolds number and the type of turbulent flow, so that it can be considered universally valid. In addition, we show a 2/3 scaling for the mean conditional scalar difference. In the second part of the paper, we examine the scaling of the conditional two-point velocity difference along gradient trajectories. In particular, we compare the linear s/τ scaling, where τ denotes an integral time scale and s the separation arclength along a gradient trajectory in the inertial range as derived by Wang (2009) with the s · a∞ scaling, where a∞ denotes the asymtotic value of the conditional mean strain rate of large dissipation elements