Turbulence correlation coefficient

Turbulent Flow in Stirred Vessels Turbulence parameters such as intensity and scale of turbulence, correlation coefficients, and... 

T irbulent Flow in Stirred Vessels Turbulence parameters such as intensity and scale of turbulence, correlation coefficients, and energy spectra nave been measured in stirred vessels. However, these characteristics are not used directly in the design of stirred vessels. 

An idea of the scale of turbulence can be obtained by measuring instantaneous values of velocities at two different points within the fluid and examining how the correlation coefficient for the two sets of values changes as the distance between the points is increased. 

When these are close together, most of the simultaneously measured velocities will relate to fluid in the same eddy and the correlation coefficient will be high. When the points are further apart the correlation coefficient will fall because in an appreciable number of the pairs of measurements the two velocities will relate to different eddies. Thus, the distance apart of the measuring stations at which the correlation coefficient becomes very poor is a measure of scale of turbulence. Frequently, different scales of turbulence can be present simultaneously. Thus, when a fluid in a tube flows past an obstacle or suspended particle, eddies may form in the wake of the particles and their size will be of the same order as the size of the particle in addition, there will be larger eddies limited in size only by the diameter of the pipe. 

The first expression here is very similar to the Damkohler result for A and B equal to 1. Since the turbulent exchange coefficient (eddy diffusivity) correlates well with IqU for tube flow and, indeed, /0 is essentially constant for the tube flow characteristically used for turbulent premixed flame studies, it follows that... 

In the consideration of the statistical aspects of turbulence it was found to be of utility (B6, K4, Rl) to establish the correlation in time (B6) of the velocity vectors as a function of the distance between two points in the turbulent stream. The correlation coefficient is defined by... 

An estimate of the effect of separation of the points upon the correlation coefficient is given in Fig. 3 (C7). Batchelor (B6) has been able to predict many of the basic characteristics of the correlation coefficient shown in Fig. 3 for both the transverse, and longitudinal fluctuating velocities. Much has been written about the characteristics of double, triple, and in a few cases higher correlations (K4, L5). It is beyond the scope of this discussion to consider these more refined measures of the statistical characteristics of turbulence. It suffices to indicate that at present a reasonable beginning has been made in the evaluation of the microscopic characteristics of turbulence but that much more experimental work must be carried out in order to supply the quantitative information required to make the extensive theoretical effort capable of quantitative application. 

Significance of Correlation Coefficient—Let denote the correlation coefficient between velocities at points in an element of turbulent fluid at times differing by /, and let 0C denote the correlation coefficient between deviations of the concentration C of a suspended material from the average of the surrounding concentrations at different points in the element in the same time difference. 

Plant canopies exhibit remarkable turbulence statistics, which makes canopy aerodynamics a topic of substantial scientific interest. Of particular note are the degree to which vertical and streamwise velocities are correlated, and the high degree of skewness in these two velocity components. The correlation coefficient that relates stream-wise and vertical velocities, defined as... 

In this regime the typical distance from the origin of motion increases as the square root of time. Thus, the dispersion in turbulent flows at long times is analogous to molecular diffusion or random walks with independent increments and comparison of Eq. (2.24) with (2.16) relates the turbulent diffusion coefficient, Dt, to the integral of the Lagrangian correlation function, Tl, as... 

The scale of turbulence is based on correlation coefficients such as jR ., measured as a function of the distance between stations. By determining the values of as a function of y, the scale Ly of the eddy in the y direction is calculated by the integral... 

ISOTROPIC TURBULENCE. Although correlation coefficients generally depend upon the choice of component, in some situations this is not true, and the root-mean-square components are equal for all directions at a given point. In this situation the turbulence is said to be isotropic, and... 

REYNOLDS STRESSES. It has long been known that shear forces much larger than those occurring in laminar flow exist in turbulent flow wherever there is a velocity gradient across a shear plane. The mechanism of turbulent shear depends upon the deviating velocities in anisotropic turbulence. Turbulent shear stresses are called Reynolds stresses. They are measured by the correlation coefficients of the type defined in Eq. (3.15). 

Unfortunately, there is no very simple measurerrient we can make of the size, of an individual eddy, nor is there even a very diiject measurement of the average size of the eddies passing a given point. Therefore, two terms (the correlation coefficient and the scale of turbulence) have been defined as expressions of the average size of eddies. The correlation coefficient (borrowed from statistics, where it is widely used) is a measure o f how much of the time two variables coincide with each other. The correlation coefficient of two arbitrary functions of time and < 2(0 is ... 

The correlation coefficient can take values from -l-l to j-1. Here it is illustrated for simple analytic functions, where its value is otivious. In the study of turbulence it is generally applied to randomly fluctuating variables. It can be... 

Correlation coefficient in the direction normal to the airflow in a pipe of 4-in inside diameter at various velocities. [From T. K. Sherwood, Heat transfer, mass transfer and fluid friction relationships in turbulent flow, Ind. Eng. Chem 42 2077 (1950). Reproduced by piermission of the publisher.]... 

Figure 16.4 is typical of correlation coefficient measurements in a pipe. In free turbulence one often measures a curve as sketched in Fig. 16.5. The negative correlation coefficient shown in this figure is not an experimental error. Rather it reflects the fact that in free turbulence generally two adjacent eddies will, on average, be moving in opposite directions. Thus, when the probes are placed far enough apart to be normally present in two adjacent eddies, the correlation coefficient will be negative. ...

Although the correlation coefficient shown here is the most commonly used in the turbulence literature, Eq. 16.7, which coefficient, can be applied to various combinations sometimes see references to the correlation coefficient between v,... 

Typical correlation coefficient measurement for free turbulence. 

It can also be shown [8] that there exists a relatively simple mathematical relationship between the spectrum of turbulence and the correlation coefficient, so that a detailed, accurate measurement of one enables the calculation... 

At zero time delay, the two parts of the signal correlate exactly, but after some long delay there is no coherence at all, and the correlation coefficient reaches zero. Some recent results obtained by Roberts and Williams which illustrate this for an electrostatic probe following ionization in a turbulent diffusion flame of hydrogen containing 1 % of acetylene. The time delay required for the correlation coefficient to reach a value of is known as the e-folding distance and, transformed into... 

Reaction zone of cylindrical design is characterized by the fact that values of pressure at the beginning and at the end are equal (Fig. 4.12). In this case low pressure difference at apparatus ends is observed and it does not exceed 0,03 atm at experimental conditions. Numerical dependences of pressure in tubular turbulent apparatus on volume flow of reaction mixture (R - correlation coefficient) were obtained ... 

A cylindrical device is characterised by similar input and output pressures (Figure 2.36) the input-output pressure drop is low in this case, not exceeding 0.03 atmospheres under experimental conditions. Quantitative correlations between pressure in a tubular turbulent device and the reaction mixture flow rate R is the correlation coefficient) [45, 97] ... 

The semi-empirical analysis of many experimental data collected in two different pilot plants led to satisfactory correlations for the turbulent friction coefficient and to the prediction of laminar-turbulent transition velocity for the flow of viscoplastic suspensions in annuli. 

Bai et al (2012) have carried out DBM simulations to computationally measure the key quantities of the bubbly flow in a square bubble column in terms of the turbulent liquid dispersion coefficient. This was done by releasing two sets of neutrally buoyant tracer particles in the fiquid phase and recording their dispersion with time one at the top of the column (tracer 0) and one set at the bottom. It was found that with increasing superficial gas velocity, the dynamics of the liquid in the column is enhanced, and hence the turbulent dispersion coefficient increases. The obtained dispersion coefficients are shown in Fig. 8 and are well within the spread of literature correlations. Note that this spread can be attributed to differences in geometry of the studied bubble columns. The power of the DBM is that it accounts for the details of the geometry and thereby provides a predictive capability that is hard to match when using empirical correlations. 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

The minimum velocity requited to maintain fully developed turbulent flow, assumed to occur at Reynolds number (R ) of 8000, is inside a 16-mm inner diameter tube. The physical property contribution to the heat-transfer coefficient inside and outside the tubes are based on the following correlations (39) ... 

Mass-Transfer Coefficient Denoted by /c, K, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theoiy, the surface renewal theoiy, and the penetration the-oiy, all of which pertain to ideahzed cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flowthrough banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. 

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