Integral length scale

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

As a consequence of the aforementioned discretization, the number of test points Np, and the linear cutoff Nx in the mode decomposition turn out to be closely related. Denoted by Ax, the typical linear distance between two adjacent test points, Ax must be small enough to ensure that the integral function E is well approximated by the sum E, i. e., Ax must fulfil the condition c(x + Ax) c(x). This is to say that the length scale Ax directly determines the minimal length scale contribution of the Fourier modes, which is Lr/Nx. Consequently, after having fixed N, the spatial variation of c(x) as a function of the number of test points Np has to be carefully monitored to determine the maximum distance Ax ensuring that c(x + Ax) c(x) in the entire box. Typically, for N = 8, we have found that a minimum of 233 test points in a box of unit length is necessary. 

Path integrals can be expressed directly in Cartesian coordinates [1, 2] or can be transformed to Fourier variables [1,2,20,45]. A Fourier path integral method will be used here [20]. The major reason for doing this is that length scales are directly built... 

In the normalized energy spectrum, k— 1 corresponds to the inverse of the local integral length scale and k — Re 4 to the inverse of the local Kolmogorov length scale. The range of wavenumbers in Fig. 1 over which the slope is —5/3 is... 

In the discussion above, we have considered only the velocity field in a turbulent flow. What about the length and time scales for turbulent mixing of a scalar field The general answer to this question is discussed in detail in Fox (2003). Here, we will only consider the simplest case where the scalar field 4> is inert and initially nonpremixed with a scalar integral length scale that is approximately equal to Lu. If we denote the molecular diffusivity of the scalar by T, we can use the kinematic viscosity to define a dimensionless number in the following way ... 

The universal function x(x) obtained by numerical integration and valid for all neutral atoms decreases monotonically. The electron density is similar for all atoms, except for a different length scale, which is determined by the quantity b and proportional to Z. The density is poorly determined at both small and large values of r. However, since most electrons in complex atoms are at intermediate distances from the nucleus the Thomas-Fermi model is useful for calculating quantities that depend on the average electron density, such as the total energy. The Thomas-Fermi model therefore cannot account for the periodic properties of atoms, but provides a good estimate of initial fields used in more elaborate calculations like those to be discussed in the next section. 

The integral-scale turbulence frequency is the inverse of the turbulence integral time scale. The turbulence time and length scales are defined in Chapter 2. 

The auto-correlation functions can be used to define two characteristic length scales of an isotropic turbulent flow. The longitudinal integral scale is defined by... 

These length scales characterize the larger eddies in the flow (hence the name integral ). Other widely used length scales are the Taylor microscales, which are determined by the... 

For isotropic turbulence, the longitudinal integral length scale Ln is related to the turbulent energy spectrum by... 

The longitudinal integral length scale can also be used to define a characteristic integral time scale re ... 

Two important length scales for describing turbulent mixing of an inert scalar are the scalar integral scale L, and the Batchelor scale A.B. The latter is defined in terms of the Kolmogorov scale r] and the Schmidt number by... 

The scalar integral scale characterizes the largest structures in the scalar field, and is primarily determined by two processes (1) initial conditions - the scalar field can be initialized with a characteristic that is completely independent of the turbulence field, and (2) turbulent mixing - the energy-containing range of a turbulent flow will create scalar eddies with a characteristic length scale I.,p that is approximately equal to Lu. 

Like the velocity spatial correlation function discussed in Section 2.1, the scalar spatial correlation function provides length-scale information about the underlying scalar field. For a homogeneous, isotropic scalar field, the spatial correlation function will depend only on r = r, i.e., R,p(r, t). The scalar integral scale L and the scalar Taylor microscale >-,p can then be computed based on the normalized scalar spatial correlation function fp, defined by... 

Thus, E k, t) Ak represents the amount of scalar variance located at wavenumber k. For isotropic turbulence, the scalar integral length scale is related to the scalar energy spectrum by... 

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