Turbulence length scale

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

The turbulent kinetic energy is calculated from equation 41. Equation 43 defines the rate of energy dissipation, S, which is related to the length scale via... 

Prandtl mixing length, length scale of turbulence... 

The same results are obtained from Equations (3.38) and (3.39), which apply to the turbulent flow of ideal gases. Thus, tube radius and length scale in the same way for turbulent liquids and gases when the pressure drop is constant. For the gas case, it is further supposed that the large and small reactors have the same discharge pressure. 

In any circumstances, it can be expected that and (5x are algebraic functions of turbulence length scale and kinetic energy, as well as chemical and molecular quantities of the mixture. Of course, it is expedient to determine these in terms of relevant dimensionless quantities. The simplest possible formula, in the case of very fast chemistry, i.e., large Damkohler number Da = (Sl li)/ SiU ) and large Reynolds Re = ( Ij)/ (<5l Sl) and Peclet numbers, i.e., small Karlovitz number Ka = sjRej/Da will be Sj/Sl =f(u / Sl), but other ratios are also quite likely to play a role in the general case. 

Stud5ting the influence of increased operating pressure on Bimsen turbulent flames, Kobayashi and coworkers [38,39] have recently put into evidence possible effects of flamelets instability, including modification of length scales, in particular. Figure 7.1.12 shows this remarkable... 

The ability to resolve the dissipation structures allows a more detailed understanding of the interactions between turbulent flows and flame chemistry. This information on spectra, length scales, and the structure of small-scale turbulence in flames is also relevant to computational combustion models. For example, information on the locally measured values of the Batchelor scale and the dissipation-layer thickness can be used to design grids for large-eddy simulation (LES) or evaluate the relative resolution of LES resulfs. There is also the potential to use high-resolution dissipation measurements to evaluate subgrid-scale models for LES. 

Rate of turbulence production (m s ) Velocity of a turbulent eddy of size X Rate of turbulence dissipation (m s ) Kolmogorov length scale (m)... 

Since in the case of turbulent stress the ratio of particle diameter dp to length scale of turbulence qp is decisive for the stress regime (see Fig. 1) the model particle systems must have properties which guarantee dp/qp values which are in the same range as for the biological particle systems. 

The ratio Zt/Zp is in technical reactors much higher than 1. It becomes, e.g. also for a small scale reactor of V-IOOL (H/D = 2 D = 0.4 m) equipped with three turbines (d/D = 0.3) and working at a average impeller power per mass of only = lmVs in media with water like viscosity to Zt/Zp>36...72. The maximal energy dissipation in the impeller zones, required for the calculation of length scale of turbulence here taken from Eq. (20). 

Cherry and Papoutsakis [33] refer to the exposure to the collision between microcarriers and influence of turbulent eddies. Three different flow regions were defined bulk turbulent flow, bulk laminar flow and boundary-layer flow. They postulate the primary mechanism coming from direct interactions between microcarriers and turbulent eddies. Microcarriers are small beads of several hundred micrometers diameter. Eddies of the size of the microcarrier or smaller may cause high shear stresses on the cells. The size of the smallest eddies can be estimated by the Kolmogorov length scale L, as given by... 

For axial capillary flow in the z direction the Reynolds number, Re = vzmaxI/v = inertial force/viscous force , characterizes the flow in terms of the kinematic viscosity v the average axial velocity, vzmax, and capillary cross sectional length scale l by indicating the magnitude of the inertial terms on the left-hand side of Eq. (5.1.5). In capillary systems for Re < 2000, flow is laminar, only the axial component of the velocity vector is present and the velocity is rectilinear, i.e., depends only on the cross sectional coordinates not the axial position, v= [0,0, vz(x,y). In turbulent flow with Re > 2000 or flows which exhibit hydrodynamic instabilities, the non-linear inertial term generates complexity in the flow such that in a steady state v= [vx(x,y,z), vy(x,y,z), vz(x,y,z). ... 

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