Velocity field fluctuating

The third application is to velocity field fluctuations. For an equilibrium fluid the velocity field is, on average, zero everywhere but it does fluctuate. The correlations trim out to be... 

Turbulence is generally understood to refer to a state of spatiotemporal chaos that is to say, a state in which chaos exists on all spatial and temporal scales. If the reader is unsatisfied with this description, it is perhaps because one of the many important open questions is how to rigorously define such a state. Much of our current understanding actually comes from hints obtained through the study of simpler dynamical systems, such as ordinary differential equations and discrete mappings (see chapter 4), which exhibit only temporal chaosJ The assumption has been that, at least for scenarios in which the velocity field fluctuates chaotically in time but remains relatively smooth in space, the underlying mechanisms for the onset of chaos in the simpler systems and the onset of the temporal turbulence in fluids are fundamentally the same. 

Velocity Field Fluctuations for a Soft Glassy Material... 

The actual velocity field fluctuates wildly. Reynolds modeled it by a superposition of a Eulerian time mean value v defined by... 

Let Ap, Au and AT denote the deviations of the mass density, p, the velocity field, u, and the temperature, T, fiom their fiill equilibrium values. The fluctuating, linearized Navier-Stokes equations are... 

Although direct numerical simulations under limited circumstances have been carried out to determine (unaveraged) fluctuating velocity fields, in general the solution of the equations of motion for turbulent flow is based on the time-averaged equations. This requires semi-... 

The Kolmogorov velocity field mixes packets of air with different passive scalars a passive scalar being one which does not exchange energy with the turbulent velocity flow. (Potential) temperature is such a passive scalar and the temperature fluctuations also follow the Kolmogorov law with a different proportionality constant... 

In the velocity field of the determining eddies, which is characterized by the turbulent fluctuation velocity the particles experience a dynamic stress according to the Reynolds stress Eq. (2) ... 

The velocity field in turbulent flow can be described by a local mean (or time-average) velocity, upon which is superimposed a time-dependent fluctuating component or eddy. Even in one-dimensional flow, in which the overall average velocity has only one directional component (as illustrated in Fig. 6-3), the turbulent eddies have a three-dimensional structure. Thus, for the flow illustrated in Fig. 6-3, the local velocity components are... 

It follows that the concentration C will also fluctuate as a result of the velocity field so... 

Thus, the mean velocity field - as well as the fluctuation field u - are solenoidal. 

Turbulent mixing (i.e., the scalar flux) transports fluid elements in real space, but leaves the scalars unchanged in composition space. This implies that in the absence of molecular diffusion and chemistry the one-point composition PDF in homogeneous turbulence will remain unchanged for all time. Contrast this to the velocity field which quickly approaches a multi-variate Gaussian PDF due, mainly, to the fluctuating pressure term in (6.47). 

However, the mean velocity field will depend on conditional moments of the fluctuating velocity through the mean velocity transport equation,... 

In the presence of a mean scalar gradient V and a fluctuating (zero-mean) velocity field... 

Earlier it was stated that the structure of a turbulent velocity field may be presented in terms of two parameters—the scale and the intensity of turbulence. The intensity was defined as the square root of the turbulent kinetic energy, which essentially gives a root-mean-square velocity fluctuation U. Three length scales were defined the integral scale l0, which characterizes... 

Let us examine the critical dynamics near the bulk spinodal point in isotropic gels, where K + in = A(T — Ts) is very small, Ts being the so-called spinodal temperature [4,51,83-85]. Here, the linear theory indicates that the conventional diffusion constant D = (K + / )/ is proportional to T — Ts. Tanaka proposed that the density fluctuations should be collectively convected by the fluid velocity field as in near-critical binary mixtures and are governed by the renormalized diffusion constant (Kawasaki s formula) [84],... 

The decomposition of turbulent motion into mean and random fluctuations resulting in the separation of the flux, Eq. 22-27, leaves us a serious problem of ambiguity. It concerns the question of how to choose the averaging interval s introduced in Eq. 22-24. In a schematic manner we can visualize turbulence to consist of eddies of different sizes. Their velocities overlap to yield the turbulent velocity field. When these eddies are passing a fixed point, they cause fluctuation in the local velocity. We expect that some relationship should exist between the spatial dimension of those eddies and the typical frequencies of velocity fluctuations produced by them. Small eddies would be connected to high frequencies and large eddies to low frequencies. 

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