Turbulent Velocity Field

For Ti (z), the principle of the parameterization is that ii,TJLg is constant with height within the canopy (except very close to the ground) and in the roughness sublayer. This is a consequence of the mixing-layer analogy. Constancy of u.TJL leads to two alternative parameterizations 

The height of the roughness sublayer, Zr i), can be estimated by noting that the fiir-field eddy diffusivity Kf is given everywhere by ct 7l (Eq. (19)). In particular, within the roughness sublayer (z Zrun), Eq. (29) implies that Tfp = Cnhcr flu.. Also, Kf in the neutral inertial sublayer (z z, ip) is given by 76p = ku- z — d), where k is the von Karman constant (assumed to be 0.4). Equating these two estimates for Kf at it is seen that z ,j, = d + (a if/K)c-fJt, where Rvii = iT ln. in the inertial sublayer. Typically, Zmj, is about 2h. 

Recent works by Leuning et al. (2000) and Leuning (2000) have led to improvements in these parameterizations in two respects. First, smoothing of the slope discontinuities in (t ,(z) and Tpfz) (implied by Eqs. (28) and (29)) produces more stable behavior in source distributions inferred from concentration profiles by 

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Rielly and Marquis (2001) present a review of crystallizer fluid mechanics and draw attention to the inconsistency between the dependence of crystallization kinetic rates on local mean and turbulent velocity fields and the averaging assumptions of conventional well-mixed crystallizer models. 

Warholic MD, Schmidt GM, Hanratty TJ (1999) The influence of a drag-reducing surfactant on a turbulent velocity field. J Fluid Mech 388 1-20... 

In this study, the flame can be classified as a wrinkled flame throughout most of the flow field. The main findings of [25] are related to both (1) the question of how the turbulent velocity field is affected by the chemical reaction and induced expansion phenomena and (2) the measurements of mean flame surface density and the... 

The transported PDF models discussed so far in this chapter involve the velocity and/or compositions as random variables. In order to include additional physics, other random variables such as acceleration, turbulence dissipation, scalar dissipation, etc., can be added. Examples of higher-order models developed to describe the turbulent velocity field can be found in Pope (2000), Pope (2002a), and Pope (2003). Here, we will limit our discussion to higher-order models that affect the scalar fields. 

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... 

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. 

Figure 11 Schematic diagram of the experimental facility for simultaneous measurement of turbulent velocity field and free-surface wave amplitude in an open channel flow using PIV (Li et al., 2005c).

Figure 13 plots an example of the processed PIV frame. The turbulent velocity field and its boundaries, solid wall, and liquid-free surface are simultaneously shown in Figure 13. The turbulence structures such as the coherent vortical structure near the bottom wall and its modification after release from the no-slip boundary condition near the free surface of the open-channel flow, and the evolvement of the free-surface wave can be seen in Figure 13. This simultaneous measurement technique for free-surface level and velocity field of the liquid phase using PIV has been successfully applied to the investigation of wave-turbulence interaction of a low-speed plane liquid wall-jet flow (Li et al., 2005d), and the characteristics of a swirling flow of viscoelastic fluid with deformed free surface in a cylindrical container driven by the constantly rotating bottom wall (Li et al., 2006c).

Figure 16 An example of PIV/LIF/SIT taken turbulent velocity fields in the wake region of two bubbles together with the bubble shadows at four instants (Tokuhiro...

Chemistry-Hydrodynamics Coupling and Feedback. Explicit energy feedback mechanisms from mixing and reactions to the turbulent velocity field and the macroscopic flow must be formulated... 

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 /q, which characterizes the large eddies the Taylor microscale X, which is obtained from the rate of strain and the Kolmogorov microscale 1, which typifies the smallest dissipative eddies. These length scales and the intensity can be combined to form not one, but three turbulent Reynolds numbers Ri = U lo/v, Rx. = U X/v, and / k = U ly/v. From the relationship between Iq, X, and /k previously derived it is found that / ... 

Prof. Roger Shaw (University of California, Davis) describes new developments in the analysis of the turbulent velocity fields in vegetative canopies, using proper orthogonal decomposition (a technique for eddy identification first introduced by J. Lumley in 1967). He argues that this technique complements the previous studies based on statistical models and large eddy simulation numerical methods. 

In the following we outline a self-consistent Flory-type argument to enable the computation of S. This type of argument has also been used in contexts relating to the wandering exponent of polymers [31] and diffusion in turbulent velocity fields [32] and has provided results reasonably in accord with more rigorous calculations. 

Reynolds [127] provided the fundamental ideas about averaging and was the first to accomplish the formulation of the governing equations for turbulent flows in terms of mean and fluctuating flow quantities rather than instantaneous quantities. Reynolds stated the mathematical rules for forming mean values. That is, he suggested splitting a turbulent velocity field into its mean and fluctuating components, and wrote down the equations of motion for these two velocity quantities. 

The phrases similarity hypothesis and universal form refer to a mathematical consequence of the Kolmogorov h3q)othesis denoting that on the small scales all high-Reynolds-number turbulent velocity fields are statistically similar. That is, they are statistically identical when they are scaled by the Kolmogorov velocity scale ([121], p. 186). 

In parallel to the research on the turbulent velocity field, investigations were made concerning the local structure of the fields of passive scalars like chemical species concentrations and temperature. A thorough review of this theory is given by Baldyga and Bourne [5], so this vast theory will not be repeated in this book. 

However, to solve the heat and mass transfer equations an additional modeling problem has to be overcome. While there are sufficient measurements of the turbulent velocity field available to validate the different i>t modeling concepts proposed in the literature, experimental difficulties have prevented the development of any direct modeling concepts for determining the turbulent conductivity at, and the turbulent diffusivity Dt parameters. Nevertheless, alternative semi-empirical modeling approaches emerged based on the hypothesis that it might be possible to calculate the turbulent conductivity and diffusivity coefficients from the turbulent viscosity provided that sufficient parameterizations were derived for Prj and Scj. 

The subject of hydrodynamic stability theory is concerned with the response of a fluid system to random disturbances. The word hydrodynamic is used in two ways here. First, we may be concerned with a stationary system in which flow is the result of an instability. An example is a stationary layer of fluid that is heated from below. When the rate of heating reaches a critical point, there is a spontaneous transition in which the layer begins to undergo a steady convection motion. The role of hydrodynamic stability theory for this type of problem is to predict the conditions when this transition occurs. The second class of problems is concerned with the possible transition of one flow to a second, more complicated flow, caused by perturbations to the initial flow field. In the case of pressure-driven flow between two plane boundaries (Chap. 3), experimental observation shows that there is a critical flow rate beyond which the steady laminar flow that we studied in Chap. 3 undergoes a transition that ultimately leads to a turbulent velocity field. Hydrodynamic stability theory is then concerned with determining the critical conditions for this transition. 

An equation for the mean flow, defined as an average over many realizations of the turbulent velocity field, can be obtained (Reynolds, 1895) by decomposing the velocity and pressure fields into the sum of an ensemble-averaged and a fluctuating component ... 

Thus, in the Re —> oo limit the turbulent velocity field is a rough non-differentiable function Sv(l) l1/3 for l -h. 0. This expression identifies the so-called Holder exponent as a = 1/3. It is easy to see that the energy transfer rate 8v(1)2/t(1) is independent of the length scale l and applying the first of the above formulas to the integral scale (L) one can relate the energy dissipation rate to the large scale properties of the flow as u3/l. 

This is the Kolmogorov scale that represents the size of the smallest flow structures in the turbulent velocity field. Thus, the range of validity of the scalings in (1.21) and (1.25) is given by the so called inertial range where L l r] (see Fig. 1.3). Note that the... 

As in the case of the turbulent velocity field, the Obukhov scaling (2.119) can be generalized to higher order structure functions. Assuming self-similarity of the scalar field would imply that higher... 

C. Torney and Z. Neufeld. Phototactic clustering of swimming microorganisms in a turbulent velocity field. Phys. Rev. Lett., 101 078105, 2008. 

This Appendix supplements Section 2.4. The problem is to find the mean concentration field a(x, t) for an arbitrary scalar entity, given a turbulent velocity field u(x, f), a specified source density homogeneous initial and boundary conditions a = 0 on the outer boundary Sq of a region V. Hence, all scalar is introduced into the flow by the sources (p within V. We consider an ensemble of realizations of the turbulent flow, denoted by a superscript co, so that a" , u" and (p " =

Peclet number limit, the relationship between a ", u" and (p is given by Eq. (14), here rewritten as... 

Lodes, A., and Macho, V., The influence of poly(vinyl acetate) additive in water on turbulent velocity field and drag reduction, Exp. Fluids, 7, 383-387 (1989). 

Krajewski, B., "Determination of Turbulent Velocity Field in a Recti-linear Duct with Non-Circular Cross Section", Int. J. Heat Mass Transfer, 13, 1819 (1970)... 

The turbulent velocity field can be represented by eddies of different sizes. An eddy can be described as a turbulent motion located within a region of size I, which shows a moderately coherent structure in this region [7], Coherent structure is understood as a region of space that, at a given time, has some sort of organization in relation to any variable related to the flow (velocity, vorticity, pressure, density, temperature, etc.). 

Where x is some location in the flowfield, and Ax is a displacement vector from location x. If the turbulent velocity field was tmly random, then Ru u, ( 1 would be equal to zero for all Ax 7 0. However, in a turbulent flow, Ru iu j x) is not equal to zero, signifying the existence of underlying structure in the turbulent flow. The properties of Ru iu j (Ax x) can be used to discern some characteristics about this underlying flow structure. 

Anderson, M.B., Garard, A.D. and Hassan, U., "Teeter Excursions of a Two-Blade Horizontal-Axis Wind-Turbine Rotor in a Turbulent Velocity Field," Journal of Wind Engineering and Industiral Aerodynamics, Vol. 17, 1984, pp. 71-88. 

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