TURBULENCE AND DIFFUSION HUGH L. DRYDEN National Bureau of Standards, Washington, D. C.
science of fluid mechanics is in a state of rapid development partly because of the introduction of new experimental techniques for studying the details of turbulent flow and partly because of new theoretical concepts introduced recently by G. I. Taylor, von Kármán, and others. Advances in fluid mechanics will ultimately be reflected in engineering applications in many fields, but the use of the new knowledge may perhaps be accelerated by the early dissemination of information about the new developments to those who may make use of them. Chemical engineers are interested in the motion of fluids, not only as a means of transporting materials in gaseous or liquid form or in suspension or solution, but also as a factor in many other physical processes used in the chemical industry, such as heat transfer and diffusion. In such processes the state of motion of the fluid—i. e., whether laminar or turbulent—is of primary importance. The process of diffusion has been found very useful in the study of turbulence, and a discussion of turbulence and diffusion forms a suitable framework for setting forth the present state of knowledge of turbulence.
Downloaded via UNIV OF EDINBURGH on January 23, 2019 at 16:32:58 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
THE
Turbulence According to the dictionary, turbulence is a state of violent agitation or commotion, and the individuals in that state may be human beings, molecules, small masses of fluid, or any similar collection. In the field under consideration we may begin with the definition given by G. I. Taylor and subscribed to by von Kármán (10): “Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid, when they flow past solid surfaces or even when neighboring streams of the same fluid flow past or over one another.” Everyone is familiar with the manifestation of turbulence in the irregular rise and fall of the speed of the wind and in the tumultuous course of a rapidly flowing mountain brook. This behavior is characteristic of the flow of fluids in most cases of technical interest. The technician speaks of such a flow as a turbulent flow, considering the actual motion as made up of a steady motion on which a turbulence is superposed. The presence of turbulence profoundly affects the magnitude of the friction between the fluid and objects immersed in it, heat transfer, diffusion, mixing, dissolving, evaporation, etc. Turbulence, then, is a departure from the condition of uniform and steady flow. However, variations of mean speed or mean direction of flow from place to place are not regarded It is fluctuation of the speed at any point as turbulence. from instant to instant about a mean value which is regarded It is also esas one of the essential features of turbulence. sential that the fluctuations be irregular. When a cylinder is immersed in a fluid stream, under certain circumstances a regular vortex system appears, and the vortexes break away from the cylinder with a regular periodicity. In this Kármán “vortex trail” the fluctuations of speed at any point with time are periodic (Figure 1). This vortex motion does not produce turbulent mixing and should not be designated as turbulent motion. With increasing distance behind the cylinder the regular pattern gradually disappears, and the vortex mo416
tion gradually changes into a turbulent motion in which the fluctuations are no longer periodic. One further aspect must be included in this general description. The separation of the actual flow into a steady flow and a superposed turbulence cannot be made in any unique manner. Thus even the so-called laminar flow appears nonturbulent merely because the instruments used do not respond to the motions of individual molecules. The actual molecular motion is very irregular, as can be shown by the proper choice of instruments. The concept of turbulence depends upon the purpose of the study as can best be illustrated by atmospheric turbulence. According to Wehrlé (8), there are at least three different possibilities. The general circulation of the atmosphere may be regarded as the steady flow, in which case the wind movements near the "highs” and “lows” of the weather map constitute the superposed turbulence. The second possibility is to regard the motion on the synoptic scale—i. e., the motion described on the ordinary weather map—as the steady flow. In this case the gusts studied by aerologists constitute the superposed turbulence. The final possibility is to consider the gust studied by the aerologist as the steady flow, in which case the superposed turbulence is on a small scale similar to that studied in wind tunnels. Although the choice has been stated in terms of the scale of the turbulence, it could equally well have been stated in terms of the frequency of the fluctuations of velocity at a point. The flow which is regarded as the basic steady flow is obtained by smoothing over a certain space and time interval. Thus it is necessary to adopt some lower limit to the frequencies of the fluctuations which constitute turbulence in a particular case. In technical applications in chemical engineering the lower limit might be arbitrarily taken as of the order of magnitude of the mean speed of flow divided by the
Modern developments in the experimental and theoretical aspects of turbulence in relation to the The concept of process of diffusion are reviewed. isotropic turbulence, its description in terms of the statistical properties designated “intensity and scale,” and its experimental realization are discussed at some length. Taylor’s theory of diffusion by continuous movements is described and illustrated by his application of the theory to the
diffusion of heat from a line source in an air stream with isotropic turbulence. At distances downstream which are small in comparison with the scale of the turbulence, the diffusion depends only on the intensity of the turbulence; at large distances, the scale of the turbulence is also an important factor. Reynolds’ description of nonisotropic turbulence of mean is given in terms values of products of fluctuations of the velocity components and of the methods now available for measuring these mean
APRIL,
INDUSTRIAL AND ENGINEERING
1939
diameter or equivalent linear dimension of the pipe or channel through which the fluid is flowing. Or we might say that the fluctuations are to be regarded as turbulent only if they are too fast to be indicated by the instruments commonly used for measuring the mean values which are of interest in the problem under study. The difficulty of giving a precise definition of turbulence applicable to a wide variety of problems can now be appreciated. For present purposes we may define turbulence as the irregular random motion, fluctuating with time at rates not detected by the ordinary measuring instruments, which may be regarded as superposed on the steady flow whose average properties are under study. The method of describing this complex phenomenon by quantitative statistical properties is considered in the following sections.
CHEMISTRY
Turbulence ofWlndTunnel,
417
*.0065. Tlme-OAsec.approximately. Relative Amplification-64
nl^iUji N.'i
| nr
ii.jY!,,,
¿iii.M.v.
Center of Wake. 2g Behind cylinder. Relative Amplification Tlme-Appinoximafely 0.3sec.
.
-
1
laterally from Center o-f Wake. 2¡f Behind ri’ Cylinder Relative Amplification-1. Time-Approximately 0.3sec.
Isotropic Turbulence The velocity of a fluid at any point is a vector quantity; it may fluctuate both in magnitude and direction. The simplest kind of irregular fluctuation is one in which the changes in magnitude and in direction are wholly random and in which the fluctuations are statistically the same at every point of the field. Such a uniformly distributed eddying motion in which the components in different directions have the same average magnitude and in which there is no correlation between the components in different directions is called “isotropic turbulence.” It can be experimentally produced by the use of honeycombs or wire screens placed in a fluid stream at a sufficient distance upstream from the place of observation. The turbulence found at the center of a pipe in which the flow is eddying or in the natural wind at a sufficient height above the ground is approximately isotropic. The experimental study of turbulence quantitatively has been greatly facilitated by the development of the hot-wire anemometer (12) in a form suitable for recording accurately the fluctuation of speed in air streams: The sensitive element is a heated wire (Figure 2) of small diameter (0.002 to 0.017 mm.). Speed fluctuations cause changes in the rate of cooling of the wire and hence in its temperature, and for a wire of suitable material such as platinum, in its resistance. The resulting variations of the potential drop across the wire are amplified, passed through a suitable electrical net-
values
in
two-dimensional
flow.
Von Karman’s
description of the scale characteristics of nonisotropic turbulence in terms of correlation tensors is briefly stated. Recent results on the mixing of jets with the surrounding air are briefly reviewed, and the essential difference between diffusion of momentum and diffusion of heat or matter (smoke, dye) is pointed out—namely, that the diffusion of is affected momentum by pressure gradients in the whereas no corresponding factor occurs diffusion of heat or matter. Hence the parallel between skin friction and heat transfer is invalid except in special cases. The practical means of securing large turbulence to promote rapid mixing is discussed as applied to continuous processing-—namely, by turbulent flow in a pipe, by jets emerging in quiescent air or in a and by turbulence in the wake of countercurrent, obstacles.
)i*laterally from Center of Wake. 2¿e Behind Cylinder. Time- Approximately O.Ssec. Relative Amplification 6 -
Air Speed Approx. GOfl/$ec. *
Figure 1. Oscillograms of Speed Fluctuations Wind-Tunnel Turbulence and in the Vortex Trail behind a Cylinder (1)
in
The second and third records are considerably distorted because the electric current in the amplifier output is not proportional to the speed fluctuation for the extremely large fluctuations near the center of the wake. Note the frequency doubling at the center of the wake and the regular character of the records of the vortex trail as compared with the record of wind-tunnel turbulence.
work which compensates for the lag of the wire, and applied either to a cathode-ray oscillograph for recording the actual fluctuation with time or to a thermal-type milliammeter which indicates the intensity of the alternating output current (Figure 3). The wire is affected mainly by the component of the fluctuation parallel to the direction of mean flow, the other components producing only second order effects. For small fluctuations, the output current is proportional to the speed fluctuation, and the oscillograph trace gives directly a record proportional to the variation of the speed with time. The meter gives the rootmean-square values of the fluctuation in current, and from its indication the root-mean-square fluctuation of speed can be computed, a quantity designated as the intensity of the turbulence.
Whatever may be the method of measurement, one statistical property descriptive of isotropic turbulence is the intensity, defined as the root-mean-square fluctuation of the component of the velocity in any direction. Within the last few years it has become evident that the effects of isotropic turbulence cannot be satisfactorily correlated with the single property, intensity. It is necessary to introduce some measure of the statistical scale of the turbulence, involving the relations between fluctuations at neighboring points at the same instant or between fluctuations at the same point at different times. A method of defining the scale in terms of the correlation1 between the fluctuations (Figure 4) at neighboring points was suggested by Taylor (21): “It is clear that whatever we may mean by the diameter of an eddy (scale of the turbulence), a high degree of correlation must exist between the velocities at two points which are close together when compared with this diameter. On the other hand, the correlation is likely to be small between the velocity at two points situated many eddy diameters apart. If, therefore, we imagine that the correlation Ry, between the values of the speed u at two points distant y apart in the direcThe correlation Between two velocity fluctuations, ui and «2, is the value of the product of the instantaneous velocities divided by the product of the root-mean-square values of the fluctuations—i. e., mu%/ 1
mean
INDUSTRIAL AND ENGINEERING CHEMISTRY
418
VOL. 31, NO. 4
information is obtained. A discussion of the physical significance of the observed shape of the correlation and spectrum curves is given elsewhere (8). Isotropic turbulence was experimentally studied in windtunnel air streams (5). Beginning with a well-designed wind tunnel in which the intensity of the turbulence is less than 0,5 per cent of the mean speed and the scale is of the order of 6 mm., the intensity may be increased to about 5 per cent of the mean speed and the scale independently to at least 25 mm. by the aid of wire screens without destroying the isotropy of the turbulence. Measurements of intensity and scale of turbulence at varying distances behind a series of nearly similar wire screens of varying mesh and their effect on the critical Reynolds number of spheres have been described ( ). A general review of the concept of turbulence was published recently (1). new
Figure 2. A.
B. C.
Hot-Wire Anemometer, Compared A Millimeter Scale (I)
with
Complete exploring head, Sewing needles which form the tips of the supporting prongs. A loop of Wollaston wire is soldered to the needles. Loop of Wollaston wire. The coating is etched away for a length of about 1 rmm. and leaves the Toare platinum wire about 0.008 mm. (0.0003 inch) in diameter.
tion of the y coordinate has been determined for various values of y, we may plot a curve of Rv against y, and this curve will represent, from the statistical point of view, the distribution of « along the y axis. If Ra falls to zero at, say, y Y, then a length Z5 can be defined such that —
h
**
/
»
Jo
R^dy
~
IfY Rtdy
....
Power supply at right. Amplifier, compensation circuit, low-frequency oscillator, switches, and meters at left
Jo
may be taken as a possible definition of the " The correlation Rv can readily ‘average size of the eddies.’ be measured by the use of two hot wires (5). A second statistical property of isotropic turbulence is, then, the scale, defined as the area under the curve of correlation between fluctuations at neighboring points plotted against distance between the points, the fluctuations being taken in a direction at right angles to the line joining the points (Figure 5). The description in terms of intensity and scale resembles the description of the molecular motion of a gas by temperature and mean free path. It is possible to study other statistical properties. For example, electrical filters may be introduced in the circuits of the hot wire equipment previously described to determine the “spectrum” of the turbulence (IT, 84). .However, Taylor showed (84) that the spectrum curve may be computed from the correlation curve, so that no
This length
Figure 3. Electrical Equipment Used with HotWire Anemometer for Turbulence Measurements (1)
In the approximately isotropic turbulence at the center of a pipe in which the flow is eddying, Simmons’ measurements (88) show an intensity of about 3 per cent and a scale of about 0.17 times the radius of the pipe. In the atmosphere at some distance from the ground the scanty information available indicates a scale of the order of 30 meters or more.
Molecular Diffusion in Isotropic Turbulence The molecular motion of fluids is an irregular motion whose statistical properties are also often described in terms of the root-mean-square velocity of the molecules and the mean free path. It is therefore useful to begin the study of diffusion in isotropic turbulence by a review of the process of ordinary diffusion in the light of the elementary aspects of the kinetic theory of gases.
APRIL,
INDUSTRIAL AND ENGINEERING CHEMISTRY
1939
419
From experiments on the diffusion of gases it has been assumed that in a steady state of diffusion in a single direction x, the number of molecules n diffusing through a given area A in time t is given by the relation n
=
-DAt
dN
S
w
D depends on the gas or gases under study, on the temperature, and to a slight extent on the concentration. When A is a function of the time and the diffusion takes place in a fluid stream, the equation governing the phenomenon is identical in form with the well-known equation for heat transfer in a moving fluid—namely, ™ + + + bz bt bx by MV b2N\ , Z—
„
V
when D is not
a
\bx2
+,
by2
+
bz2
)
(2)
function of the coordinates and
(» )+ )+ (» )
(3)
when D varies from place to place.
The simplest case of diffusion is that of a gas with molea certain mass diffusing into a gas with molecules of the same mass but distinguishable in some other way—for exThe diffusion of carbon dioxide ample, by chemical means. into nitrogen monoxide approximates this case. Following the treatment given by Loeb (11), consider two large vessels of these gases connected by a long tube, in which diffusion has continued for a sufficiently long time to reach a quasi-steady state such that the change of concentration with time at any given point is very slow. Let the number of molecules per unit volume at some fixed section of the tube be denoted by N. At a distance L on either side the number will be IV + L(dN/dx) and N L (dN/dx), respectively. If Lis the mean free path, one sixth of the molecules moving with velocity In c will pass through the section on their next free path. time dt the total number within a cylinder of length c dt will cross the section. Denoting the area by A, the number of molecules passing in one direction will be 1/
