TURBULENCE AND DIFFUSION HUGH L. DRYDEN National Bureau of Standards, Washington, D. C.
T
HE science of fluid mechanics is in a stateof 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 KfirmBn, 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. 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. I n the field under consideration we may begin with the definition given by G. I. Taylor and subscribed to by von KBrmAn (IO): “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 as turbulence. It is fluctuation of the speed a t any point from instant to instant about a mean value which is regarded as one of the essential features of turbulence. It is also essential 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 KBrmitn “vortex trail” the fluctuations of speed a t 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 WehrlC! (8), there are a t 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 a t 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 process of diffusion are reviewed. The concept of 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. A t distances downstream which are small in comparison w i t h 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 is given in terms of mean values of products of fluctuations of the velocity components and of the methods now available for measuring these mean
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INDUSTRIAL AND ENGINEERING CHEMISTRY
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.
417
ji’Latemlly from Center of WakR. 2:‘Behind ;f’Cyllndcr RuIatIVe Amplificat ion-’i . Time -Approximately 0.3sec.
Isotropic Turbulence The velocity of a fluid a t any point is a vector quantity; it may fluctuate both ih 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 a t 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 a i a sufficient distance upstream from the place of observation. The turbulence found a t the center of a pipe in which the flow is eddying or in the natural wind a t 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 t o 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 potefltial drop across the wire are amplified, passed through a suitable electrical net-
values in two-dimensional flow. Von K&rm&n’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 momentum is affected by pressure gradients whereas no corresponding factor occurs in the 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 countercurrent, and by turbulence in the wake of obstacles.
1~’Latetullyfrom Center of Wake. 24‘ &hind 3’ Cylinder. Relative Amplification 0 limu- Approximately 0.3Sec..
-
-
A i r Speed Approx. 6Ofy‘sec.
FIGURE1. OSCILLOGRAMS OF SPEEDFLUCTUATIONS WIND-TUNNEL TURBULENCE AND IN THE VORTEX (1) TRAILBEHIND A CYLINDER
IN
The second and third records are considerablv considerably distorted because the electric current in the amplifier output is i s hnot o t proportional t o the speed fluctuation for the extremely large fluctuations near the center of t h e wake. Note the frequency doubling a t t h e center of the wake and the regular character of the records of the vortex trail as oompared with the record of wind-tunnel turbulence.
work which compensates for the lag of the wire, and applied either t o 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 t o 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 a t neighboring points a t the same instant or between fluctuations a t the same point a t different times. A method of defining the scale in terms of the correlation1 between the fluctuations (Figure 4) a t 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 a t 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 a t two points situated many eddy diameters apart. If, therefore, we imagine that the correlation R,, between the values of the speed u a t two points distant y apart in the direc1 The correlation between two velocity fluctuations, UI and u1, is t h e mean value of the product of the instantaneous velocities divided by t h e product of the root-mean-square values of the fluotuations-i. e., uluz/
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new information is obtained. A discussion of the physical significance of the observed shape of the correlation and spectrum curves is given elsewhere (3). Isotropic turbulence was experimentally studied in windtunnel air streams (5). Beginning with a welldesigned 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. hfeasurements 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 (6). A general revim of the concept of turbulence was published recently ( 1 ) .
I.’IGURE A.
R. C.
2. HOT-WIHEANEMOMETER, COMPARE31 WIT11 A MILLIMETER SCALE ( 1 )
Complete eaplorins head.
which form the tips of the supporting prongs. A loop of Wollsston wire is soldered t o the needler. LOOP of Wollmton wire. The oostinp is etohed sway for, B length of sbout 1 mm. snd I + Y c ~ the bsre platinum wire about 0.008 aim. (0.000sinoh) an diameter.
Sewing iieedles
tion of the y coordinate has been determined for various values of y, we may plot a curve of R, asainst y, and this curve will represent, from the statistical point of view, the distribution of u along the y axis. If R, falls to zero at, say, y = Y , then a length 2, can be defined such that
FIGURE3. ELECTnICAL EQUIPMENT USED WIT= XOTWIRE ANEMOMETER FOR ‘hT.BULENcE MEASUREMENTS ( 1 ) Power upp ply at right. Amplifier. oornpensation ciionit. low-Irew e n w mciilStOr. switches, and inetem a t left
This length . . . . may be taken as a possible definition of the ‘average size of the eddies.’ ” The correlation R, can readily be measured by the use of two hot wires (6). 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 (17, 344). .However, Taylor showed (3.4)that the spectrum curve may he computed from the correlation curve, so that no
In the approximately isotropic turbulence at the center of B pipe in which the Sow is eddying, Simmons’ measurements ($3)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.
Molccular 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 isotmpie turbulence by a review of the process of ordinary diffusion in the light of the elementary aspects of the kinetic theory of gases.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
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 tis given by the relation
d"
-tlX
n = -DAt
D depends on the gas or gases un'der study, on the temperature, and to a slight extent on the (concentration. When N is a function of the t:ime 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,
when D is not a function of the coordinates and IO
8
6
when D varies from place to place. The simplest case of diffusion is that of a gas with molecules of a certain mass diffusing into a gas with molecules of the same mass but distinguishable in some other way-for example, by chemical means. The diffusion of carbon dioxide into nitrogen monoxide approximates this case. Following the treatment given by Loeb ( I I ) , 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 a t any given point is very slow. Let the number of molecules per unit volume a t some fixed section of the tube be denoted by N . At a distance L on either side the number will be N L(dN/ds) and N - L (dN/dx),respectively. If L i s the mean free path, one sixth of the molecules moving with velocity C will pass through the section on their next free path. In time dt the total number within a cylinder of length c dt will cross, the section. Denoting the area by A , the number of L molecules passing in one direction will be '/E A 2 dt ( N d N / d x ) and in the other 1/6 A dt ( N - L dN/dx). The total net transfer is therefore '/s A c d t L ( d N / d x ) . By comparison with Equation 1defining the coefficient of diffusion D,
+
+
c
D
=
1/3CL
(4)
4
0
FIGURE5. VARIATIONOF CORRELATION COEFFICIENT WITH THE CROSS-STREAM SEPARATION OF THE HOTWIRE8 (5)
-
R' = observed curves; R curves resulting when wire-length correction is applied. Observations taken 40 mesh lengths from 8oreene; wind speed, 40 feet per second
The same expression is obtained when the distribution of free paths and velocities is taken into account. Actually other effects are neglected in this simple analysis which must be considered in any accurate study of molecular diffusion. Most of the refinements of the theory simply modify the numerical factor 1/3, Chapman's value for this factor being 0.57 if L is the mean free path as computed by Clausius. The coefficient of diffusion can also be interpreted in another way in terms of the mean square distance reached by the molecules in a time interval which is long compared with the time required to traverse the mean free path. For such a time interval t, the displacements in successive time intervals t are independent and randomly distributed. The component of the total displacement x in a given direction in a time nt = t' is x1 x2 . . . x,, and the mean square displacement
+ +
cg; m
= PERFECT COR RELATION
MODERATE DEGREE OF
the product terms vanish because of the inde-
1
pendence of successive displacements. For the same reason the quantity 2 will be a constant independent of the particular interval considered, so that
2
2
CORRELATION
-
= nx12
1
: hence
LOW DEGREE OF CORRELATION
MEANINGOF CORRELATION (1) FIQTIRE 4. PHYSICAL
Suppose that the number of molecules &tany place and time is N ( x , t). Of those found between z and (z dx) a t time t,
+
INDUSTRIAL AND ENGINEERING CHEMISTRY
420
+
let the fraction f (2’ - z)dz’ be found between z’ and (2’ dz’) a t a time (t t ’ ) . The number of molecules at x’ a t time ( t t ’ ) will be given by
+
+
N(z’, t
+ 1’)
=L
N ( x ,t )f ( x ’ m
m
- x)dz’
=
N(x’
- X , t)f(X)dX
r m r n
where X is written for (2’ - z). Suppose t’ is small and expand the function N , retaining only the first order term in t’:
turbulent motion of a fluid there are no well-defined identifiable masses of fluid, and no exact meaning can be given to a collision between two fluid masses. The movement of the fluid is continuous. The theory of diffusion by continuous movements has been developed by Taylor (19) in terms of the correIation between the value of the speed of any particle a t any instant and the value of the speed of the same particle after a time interval T . Consider a condition in which the turbulence is uniformly distributed, and fix attention on velocities and displacements of a given particle parallel to a fixed direction, which will be selected as the x axis. Suppose that 2 is constant and that the correlation RT is known as a function of T . Let u,be the value of u a t time t, and uT be the value of u a t time T . Consider the definite integral
If the higher order terms in the expansion a t the right may be neglected, and if account is taken of the fact that the odd powers of x may be assumed to vanish in the integration because of symmetry and that
VOL. 31, NO. 4
l G dT.
RT this integral is equal to
By the definition of
6”
U T ~- TRdT. ~
Hence, since 2
does not vary with 1 and RT is an even function of T ,
f ( X ) d X = l, we find (5)
where by
rzis the weighted mean-square displacement defined
where X
=
displacement of particle in time t
hence
(7)
J--m This is Einstein’s e q u a t i o n for the diffusion of particles in t h e Brownian movement (6), and by comparison with the general diffusion Equation 2, the coefficient of diffusion D is equal to ‘/2 ( F / t ’ ) . KBrmAn ( 1 0 ) gave a more extended and less mathematical discussion of this interpretation and gave Langevin’s t r e a t m e n t of Brownian moveFIGURE 6. APPARATUSFOR MEASURment in which the ING THERMAL DIFFUSIONFROM HOT assumption of a WIRE (16) . , A . Heated wire of platinum-iridium, 7 om. time interval long long, 0.005 cm. in diameter compared with the E , C . Junctions of copper-constantan thermocouple of No. 36 wire time required t o D . Bracket rotating about axis through A traverse the mean free path is not made. Since, however, a frictional drag is introduced, the treatment is not applicable to molecular diffusion, although pertinent to the discussion of turbulent diffusion.
Theory of Diffusion by Continuous Movements In the molecular motion the molecules move with constant speed until a collision occurs, a t which time there is a sudden change of velocity both in magnitude and direction. I n the
The problem of diffusion in a uniform turbulent field is thus reduced to the consideration of a single function, the correlation function RT. When the time is so small that RT does not differ appreciably from unity, Equation 7 becomes
If
R T
is equal to zero for all times greater than some time T,
A length ll may then be defined by the relation
whence
5 = 211
and
v5 t
(9)
In the molecular motion the coefficient of diffusion D was found equal to (rz/t) and to 0.57 L so that except for a numerical constant Zl plays the same part in turbulent diffusion as the mean free path plays in molecular diffusion. Because of the nature of the correlation curves experimentally obtained, there is some interest in the particular case in which R, = e - t / t Q , for which Equation 7 yields
c
F
=2
2 to
[t
- to (1 -
( 10)
When t is small compared to to, Equation 10 reduces to Equation 8. When t i s large compared to to, Equation 10 reduces t o
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INDUSTRIAL AND ENGINEERING CHEMISTRY
From the original defiEquation 9 if lI is set equal to fl&. nition of lI we find to = L r n R T d T . (Equation 10 is identical with Langevin’s equation for Brownian movement previously mentioned.) The turbulence can be characterized by length 11 only if the correlation RT is such that length
12,
Lrn
RTdT is finite. Similarly,
previously defined as the scale, can be used only
whenJrnR,dy
is finite. Even when ll and
12
are finite, no
general. statement can be made as to the relation between them; they will differ by a numerical factor whose value may depend on the circumstances of the motion. For the turbulence behind a grid or honeycomb, Taylor (21) found that 22 was approximately twice lI.
42 1
at = 134.7 6 / U
(12)
Thus the diffusion near the source depends only on the intensity of the turbulence, not on its scale, and diffusionmeasurements may be used to determine the intensity. Schubauer showed that the angle atwas independent of speed over the range 2.5 to 17 meters per second, and also independent of x over the range 12 to 150 mm. Values of the intensity of the u component of the fluctuations were measured independently; it was found that a,was proportional to the u fluctuation, for values of intensity and scale ranging from 0.0085 U and 6.6 mm., respectively, to 0.023 U and 15.7 mm. I n both cases flx/&U was about 0.2 a t the most so that the time of diffusion was small compared to the time required for a particle to traverse the distance 12.
Diffusion of Heat from a Line Source in Isotropic Turbulence As an application of the foregoing theory, we may consider a problem which has been experimentally investigated by Schubauer (16) a t the National Bureau of Standardsnamely, the diffusion of heat from a heated wire (Figure 6) in a field of isotropic turbulence in a wind-tunnel air stream of mean speed U. Observations of the lateral spread of the heated wake a t a small distance x downstream from the wire give data from which the root-mean-square lateral displacement @of the heated particles a t a time t = x / U may be computed. If the stream were completely free from turbulence, there would be a lateral spread of heat produced b y the ordinary molecular conduction. The temperature distribution may be shown (16)to be given by an “error” curve,
-
A0 = Aemax.e
where A0 p , c,
=
y=pc U l y k x
temperature rise in the wake above the free stream
k = density, specific heat at constant pressure, and
thermal conductivity of air, respectively
The turbulence causes an increased spread of heat (Figure 7). Experimentally the temperature distribution in the turbulent stream was found to follow the “error” curve, but the wake was much wider than that produced by molecular diffusion. The observed shape of the curve is in accord with numerous experiments which show that the turbulent velocity fluctuations are distributed according to the error law. Assuming that all particles coming from the source have the same initial high temperature, the temperature distribution in the wake must be the same as the frequency distribution of Y . It is found convenient to characterize the spread by the angle subtended a t the source by the two positions where the temperature rise is half that in the center of the wake. Calling this angle a,it may be shown (21)that ff2
=
cy12
+
(Yo2
The value of a0 in degrees is 190.8 d k / p c U x . Near the source, Equation 8 becomes in the present notation:
For an error curve the value of y a t which the temperature rise is half the maximum is found to ‘be 1.77 whence a, = 134.7 From Equation 11, accordingly,
dF, 0,’~.
DISTRIBUTION CURVESOBTAINED TRAVERSING THE HEATED WAKEWITH THE APPARATUSor FIGURE6 AT A DISTANCE OF 5.08 CM.FROM THE HEATED WIRE (16) FIGURE 7.
BY
Intensity of turbulence: a = 1.1 per cent, b = 3.5 per cent
When allowance is made for the finite length of the wire (6) used in measuring the value of 4 2 by the hot-wire anemometer agreed well with the value of obtained by the thermal diffusion method, showing that the turbulence is in fact isotropic under the conditions of the experiments. Experimental study of the diffusion of heat in a field of isotropic turbulence for times long in comparison with the time required for a particle to traverse the distance l2 offers some difficulty because of the small temperature difference between the wake and the stream outside a t large distances from the source. Taylor (21) discussed some observations of Simmons which show definite deviations from the linear law of Equation 8 toward the parabolic law of Equation 9. Experiments are known to be in progress in various laboratories on the diffusion of smoke or dye in air or water in turbulent motion and on the diffusion of gases introduced into a turbulent air stream. Unfortunately in many of these experiments, provision has not been made for direct measurements of the intensity and scale of the turbulence or for assuring that the turbulence is approximately isotropic. The essential features of diffusion in isotropic turbulence may be summarized as follows:
4,
dz
1. For times small in comparison with the ratio of the scale to the intensity the diffusing quantity spreads at auniform speed proportional t o the intensity of the turbulence, and the diffusion is not dependent on the scale of the turbulence.
’dz
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VOL. 31, NO. 4
gradient of the mean velocity-i. e., an air stream of uniform speed across its section. Reynolds' theory furnishes an explanation of the large shearing stresses found in turbulent flow but does not yield a method of solving the equations of motion in the absence of information about the fluctuations. Until very recently, theories of turbulent flow were based on assumptions as to the relations between the fluctuations and the mean motion. A brief review of these developments was given elsewhere ( 2 ) . Within the last few years techniques have been perfected for direct measurement of five of the six values required to describe completely the intensity tensor. Within the next few years sufficient experimental data should be accumulated on the actual relationships to enable the recasting of the theory of turbulent motion Courtesy, E. L. Luaces Associates on a more sound experimental basis. ADSORPTIONSYSTEM FOR EXTRACTING GASOLINE FRO31 NATURAL GAS In studies of air flow the auantitv ;II" can be measured by the h i t wire ;nemometer in the manner previously de2. For times large in comparison with the ratio of the scale scribed. Both ?I and 3 can be computed from measurements of the temperature distribution in the wake of a small heated 12 to the intensity the diffusing quantity spreads in accordance with the usual diffusionequation with a coefficient of difwire or from some other diffusion measurement; the direction in which the diffusion is measured is parallel to the v and w OD Rtdt. fusion equal to Z I where~ 11 is a length defined by directions, respectively. An alternate technique is to use a special pyramid of threchot wires as a direction sensitive eleFor intermediate times the diffusion depends : 0 the func3. ment. The quantities uv and & can be simply measured by tion Rt-i. e., the average correlation between the speed of any particle a t any instant and the speed of the same particle after a method developed within the last few months by H. K. a time interval t. Skramstad a t the National Bureau of Standards. For this measurement the wire of the hot-wire anemometer, which is placed normal to the direction of mean flow for the measureNonisotropic Turbulence ment of $, is inclined a t 45" so that it is. about equally sensiIsotropic turbulence is a special type of turbulence for tive to u and v when the wire is in the uv plane or to u and w which the relations are simple compared with the more genwhen in the uw plane. By rotating the wire 180" about the eral case of nonisotropic turbulence. In the general case the direction of mean flow, the wire is again in the uv (or uw) single measure of intensity of isotropic turbulence is replaced plane, but the direction of v (or w ) with respect to the wire is by six quantities and the single measure of scale by six; thus effectively reversed. The usual amplifier and associated aptwelve quantities are required to specify the state of turparatus give a reading proportional to (au b ~ for ) ~one bulence in the neighborhood of a single point. for the other. The difference beposition and tween the readings is proportional to 4 ab%. (A descripConsider first the description of the intensity. Designate tion of this apparatus and some of the results obtained with the components of the fluctuations in the directions of three it will be published soon.) Cartesian axes of reference by u,v, and w. The effect of the It is not the intention here to review critically or to defluctuations on the m-eaE motion is completely determined by scribe extensively the experimental methods for measuring the six mean values u2,us, w2,uv, uw,and vw as first shown by the Reynolds stresses but merely to emphasize that it is posReynolds (16). Reynolds observed experimentally the large sible to make such measurements, that such measurements fluctuations and irregularity of the flow of water in a pipe a t are being made, and that the results already obtained are large Reynolds numbers and developed a theory in which the sufficient to show that the current theories can be regarded flow was regarded as consisting of a mean steady motion on only as empirical and approximate. Such a criticism should which fluctuations of velocity were superposed. He showed not lead the engineer to underestimate the great utility that the equations of motion, were the same as for laminar of the Prandtl-K6rmBn developments in practical calculaflow except that additional stress components were introtions. duced. These eddy stresses consisted of three normal stresses The situation with regard to the measurement of scale is pi2, p$,-and f l ,and three shearing stresses -puv, -puw, not so bright. In the general case there are nine correlation I n isotropic turbulence the shearing stresses are and -pvw. factors, each of which is a function of the three coordinates zero, and hence there is no correlation between the several x, y, and z. The equation of continuity was shown by Khrcomponents of the velocity fluctuation a t a point, or between m6n (9) to yield three relations between these nine factors, the variations in magnitude and in direction of the velocity so that only six are independent. No theoretical or experifluctuations. I n addition, the three normal stresses are mental investigations have been carried out utilizing six equal. Isotropic turbulence can exist only in a flow without scale factors in addition to the six intensity factors. shearing stress and hence only in a flow in which there is no
fi
'n
+
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INDUSTRIAL AND ENGINEERING CHEMISTRY
I n the well-known Prandtl mixing length theory, it has been assumed that a single scale factor is sufficient t o characterize the scale of the turbulence-namely, the mixing length E. This length is defined for a two-dimensional parallel flow by the relation:
where d U / d y = gradient of mean velocity
A definition consistent with this relation can be given for the general case. Since the shearing stress is equal to -p%, the method of Skramstad can be used to give direct measurements of Prandtl’s mixing length.
Diffusion in Nonisotropic Turbulence Equation 13 can be interpreted as an equation governing the diffusion of momentum with a coefficient of diffusion equal to P l d U / d y l . It was, in fact, derived by Prandtl by setting the v fluctuation proportional to 1 ( d v l d y ) , and hence the coefficient of diffusion is of the form of Equations 4 and &namely, E f l . The factor p ( d U / d y ) is to be interpreted as the concentration gradient of momentum, and the shearing stress is equal to the rate of transfer of momentum in the v direction. It follows, therefore, that Equation 13 implies diffusion for a time interval long in comparison with 1/43 which is proportional to l / ( d U / d y ) . Thus the equation holds best where d U / d y i s l a r g e , i f ~ i s i n f a c equal t to E(dU/dy). Taylor (20) pointed out that the fluctuating pressure gradients parallel to the main flow, which are associated with the turbulent velocity fluctuations, would influence the transfer of momentum in turbul&nt mixing. He suggested a theory in which the vorticity is taken as the property undergoing diffusion. In a parallel two-dimensional flow the important component of vorticity is equal to l / z ( d U / d y ) , and the concentration gradient of vorticity, l/,(d2U/dy*). From the equations of motion the vorticity transfer per unit area per unit time transverse to the flow can be shown (14) to be equal to - ( I / * p ) ( d p / d z ); this in turn is equal to - ( l / * p ) (&/ &). Using the same coefficient of diffusion as before-namely, P ( d U / d y ) , the fundamental diffusion Equation 1 becomes:
Thus the vortex transport theory also implies diffusion for a time interval long in comparison with the mean free time of the eddies and should be in best agreement where d U / d y is large. The equation for heat transfer in a two-dimensional parallel ffow is
where
e
U
:= ;=
temperature
mean speed at any point
The diffusion constant is taken to be 1 fl.Where a velocity gradient exists, flmay be set equal to Z(dU/dy), according to the Prandtl assumption, and the diffusion constant becomes lzd U/dg. Equation 15 is analogous to Equation 3. We may also state the fundamentals of the heat transfer problem by expressing the heat convection across a unit area per unit time by cpZ21du/dyldO/by, a statement analogous to Equation 1. If Equation 14 is applied to the spread of turbulence in a plane jet in which 1 may be regarded as constant, the equation may be integrated to give the following:
423
Thus the mixing length in the vortex transport theory is l/z times as great as that in the momentum transport theory. Hence for equal rates of spread of momentum, the spread of heat is twice as great in the vortex transport theory as in the momentum transport theory. From the point of view of diffusion, both theories agree in using the same coefficient of diffusion but differ in what quantity is considered as subjected to the simple diffusion process. From the fundamental equations for various processes, it seems clear that the equations relating to momentum have pressure terms which have no counterpart in the equations , for transport of heat or matter. Hence there is no reason to expect an analogy between heat transfer and momentum transfer (or skin friction) except in the single case where the pressure gradient vanishes. This fact was discussed in more detail elsewhere .(4. There have been many recent papers on the application of the momentum transport and vorticity transport theories in conjunction with various assumptions as to the value of the mixing length and its relation to properties of the mean motion and to the geometry of the particular problem. Tollmien’s computation of the mixing of a parallel stream and of a jet with the surrounding air and Prandtl’s computation of the spreading of a wake, all by the momentum transport theory, have been summarized (IS). Some idea of the trend of the present discussion may be obtained from a recent paper by Howarth (7). The general opinion of Prandtl (13) and Taylor (22) appears to be that the momentum transport theory is applicable to flow near a wall and the vortex transport theory to flow not near a wall-i. e., to the spreading of a jet or wake. Let us now return to the type of diffusion problem considered in the case of isotropic turbulence in which there is introduced into a parallel two-dimensional flow a linear source of heat a t right angles to the direction of flow and to the direction of the velocity gradient. If the diffusion is studied over a region in which the diffusion coefficient is sensibly constant and over a region in which U may be regarded as constant, the solution of Equation 15 is the same as that obtained for the case of molecular difFusion of h e a t n a m e l y ,
Hence the same temperature rise would be found along the parabolas y2/x = constant, and the diffusion would be as represented by Equation 9. Skramstad (18) performed this experiment in the two-dimensional parallel flow in the boundary layer of a large plate, the boundary layer being about 8 cm. thick. The temperature distribution was measured behind a heated wire a t distances from 0.5 to 2 cm. the spread of the thermal wake to either side not exceeding 3.5 mm. The variation of mean speed within the wake from an average value did not exceed 1.5per cent, but the variation of f i a m o u n t e d to as much as 10 per cent. Under these conditions the same temperature rise was found not along the parabolas y z / x = constant, but along straight lines y / x = constant (Figure 8). This indicates that, under these conditions as in the case of isotropic turbulence studied by Schubauer (16)’ the mean free time is long compared to the time required to traverse moderate distances. Hence the assumption of a diffusion coefficient equal to 243 in both momentum transport and vortex transport theories can apply only after the diffusion has proceeded for a considerable time. The assumption cannot be true in the region near the opening for the spread of a jet emerging from a small opening into an existing turbulent stream.
424
INDUSTRIAL AN'D ENGINEERING CHEMISTRY
The experiments of Skramstad showed that the spread of heat outward was greater than that inward toward the plate, the temperature distribution being unsymmetrical. The dissymmetry cannot be accounted for by the slight variations of mean speed or by the local values of flwhich vary across the wake. It is probable that a theory with a single scale diffusion coefficient cannot account for such an effect.
v
.08 .04 0 -04 .w .fZ FIGURE8. D I S T R I B U T I O N OF TEMPERATURE AT VARIOUS DISTANCES B E H I N D A HEATED W I R E IN A BOUNDARY LAYER
Spread is nearly linear with z (independent of y/z) and is unsymmetrical.
The situation with regard to diffusion in nonisotropic turbulence may be summarized by the statement that the only theoretical approach is to consider the process as approximately equivalent to diffusion in isotropic turbulence of intensity equal to the single intensity factor d?; v here is the component in the direction in which the diffusion is studied, and of scale defined by mixing length 1.
Practical Applications This paper has dealt mainly with the fundamental concepts and theory of turbulence in relation to diffusion; the theory, however, has remained in very close touch with the experiments. The experiments described are of a simplified type intended to serve as a guide and test for the fundamental concepts. They do not answer directly the questions of the engineer who is concerned with the problem of designing equipment for the mixing of fluids in continuous processing. Although the author has had no practical experience in that field, it may not be out of place to close this review with some suggestions as to the possible application of the principles stated. The diffusion near the point of introduction of the diffusing fluid is determined solely by the intensity of the turbulence, but at a considerable distance the process may be viewed as analogous to molecular diffusion with a coefficient of diffusion equal to the product of intensity and scale. Hence the most rapid diffusion will occur with turbulence of the highest practicable intensity and largest practicable scale. However, it is obvious that if the scale is large in comparison with the dimensions of the apparatus in which the mixing occurs and small scale turbulence is not also present, the initial mixing will not be very intimate. The problem of securing most rapid diffusion thus becomes one of providing means of securing turbulence of high intensity. The designer will also be interested in seeing that the devices used do not give unduly high energy losses in the piping. For a given coefficient
VOL. 31, NO. 4
of diffusion, the energy of the turbulence itself will be least for low-intensity large-scale turbulence. The simplest arrangement utilizes the turbulence existing in a pipe when the speed of flow is such that the flow is turbulent. This occurs when the product of mean speed by diameter of pipe divided by the kinematic viscosity of the fluid is well above 2000. The intensity of the turbulence a t the center of the pipe is about 3 per cent of the mean speed and the scale ( 1 2 ) is about 0.17 times the radius of the pipe. At a distance from the pipe wall of about one sixth the radius, the intensity is about twice that a t the center and the scale about one third that a t the center. The information available suggests that the maximum diffusion coefficient would be obtained by introducing the diffusing fluid in a zone about halfway between the center and the wall, the coefficient being about 20 per cent greater here than a t the center. The least losses of energy would occur if the diffusing fluid were introduced a t a speed equal to the speed of the fluid into which the diffusion occurs a t the location of the entrance nozzle. The external form of the nozzle should be carefully streamlined. A second arrangement for promoting rapid mixing is the introduction of the diffusing fluid through high-speed jets into slow-moving fluid or even into a countercurrent. The total angular width of the mixing zone a t the edge of such a jet is of the order of 15". No data are available on the intensity or scale of turbulence produced under such conditions. In some cases, as in an ordinary gas burner, a part of the energy of the jet may be utilized to produce a flow of the surrounding gas. It is probable that the over-all energy losses are greater with this arrangement than with the first one described. A third method of securing large turbulence is to introduce obstacles in the flow. The diffusing fluid is then introduced in the wake of the obstacle. There is an energy loss associated with the drag of the obstacle. Streamlining reduces the drag but also reduces the turbulence set up by the obstacle. The use of this principle may often permit the accomplishment of the mixing in a smaller space but is likely to give a low efficiency. If the obstacle is in the form of a grid of cylindrical rods, the intensity close behind the grid is distributed in a nonuniform manner; it is very small immediately behind a rod and approaches 100 per cent of the mean speed near the tangent planes of a rod. The scale is relatively small. At a distance of seventy-five times the diameter of a rod, the turbulence is nearly isotropic with intensity of about 5 per cent of the mean speed ands cale (12) of the order of the diameter of the rod. In any actual device all three of these methods of securing a large turbulent diffusion may be operative. The choice of method depends on many factors such as the relative volumes of the fluids to be mixed, the available space, sources of power available, importance of high efficiency, etc. In any practical design problem, it would be necessary to do considerable experimental work to determine the best arrangement. It is hoped that the principles outlined in this paper may be of some assistance in guiding the designer. Nomenclature a = calibration factor for u fluctuation
A = area b = calibration factor for t, fluctuation e - = specific heat of fluid at constant pressure c = mean molecular velocity of gas D = coefficient of diffusion k = thermal conductivity of fluid 1 = mixing length 11 = a length :zing the scale of turbulence
INDUSTRIAL AND ENGINEERING CHEMISTRY
APRIL, 1939
h =
-
a length fixing the scale of turbulence
A 6 = temperature rise p = density of fluid T = shearing stress
RvdY
Jm
L = mean free path n = number of molecules diffusing through area A in time 1; also a numerical factor N - number of molecules per unit volume P = static pressure of the fluid RT = correlation between values of u for the same fluid particle a t times differing by the interval T UtUt+ T
E
R,
dZ d G = correlation between values of u a t points in a line parallel t o the y axis separated by the distance y-i. e., denoting the values of u a t the two points byLand UlUZ 6
1, t ’ ,
2
dZ
T
= time or time interval as indicated by context
to
= time defined by t o =
J-wRrdT
u, v , w = instantaneous velocity fluctuations in the directions of the 4, y, and z axes (the velocity a t any instant has components U u, V 21, W w) Ut, UT= values of u a t times t and T u2 = time average of uz (bars denote time average) u, = components of mean velocity of fluid in the directions of the x,y, and z axes 2, Y, = Cartesian coordinates of any point (if the flow is
+
,
+
425
+
v,w
uniform, the x axis is in the direction of mean flow) Y = displacements of a fluid particle in time t (or t’) in direction of x and y axes c y = total angle of spread of thermal wake (Yo = angle of spread of thermal wake due t o molecular diffusion at = angle of spread of thermal wake due t o turbulent diffusion e = temperature
x,
Literature Cited Dryden, H. L., J . Aeronaut. Sci., 6 , No. 3, 85 (1939). Dryden, H. L., J. Applied Mech., 4, No. 3, A105 (1937). Dryden, H. L., Proc. 6th Intern. Congr. slpplied Mech., t o be published. Dryden, H. L., in W. F. Durand’s “Aerodynamic Theory,” Vol. VI, p. 258, Berlin, Julius Springer, 1935. Dryden, H. L., Schubauer, G. B., Mock, W. C., Jr., and Skramstad, H. K., Natl. Advisory Comm. Aeronaut., Tech. Rept. 581 (1937). Einstein, A., Ann. P h y s i k , 17, 549 (1905); 19, 371 (1906). Howarth. L.. Proc. Cambridae Phil. SOC..34. Pt. 11. 185 (1938). Kamp6 de FBriet, J., J o u k e s Tech. Intern. 1’ Abronautigue, NOV.23-27, 1936. K&rm&n,Th. von, J . Aeronaut. Sci., 4, No. 4, 131 (1937). KLrmh, Th. von, J . Roy. Aeronaut. SOC..41, 1109 (1937). Loeb, L. B., “Kinetic Theory of Gases,” 2nd ed., p. 257, New York, McGraw-Hill Book Co., 1934. Mock, W. C., Jr., Natl. Advisory Comm. Aeronaut., Tech. Rept. 598 (1937); Dryden, H. L., and Kuethe, A. M., Ibid., 320 (1929); Mock, W. C., Jr., and Dryden, H. L., Ibid., 448 (1932). Prandtl. L.. in W. F. Durand’s “Aerodvnamic Theorv.” _ . Vol. 111, p. 162 & ff., Berlin, Julius Springer, 1935. Ibid., pp. 176 and 177. Reynolds, Osborne, Trans. Roy. SOC.(London), 186, Pt. I(1895). Schubauer, G. B., Natl. Advisory Comm. Aeronaut., Tech. Rept. 524 (1935). Simmons, L. F. G., and Salter, C., Proc. Roy. SOC.(London), A165 (920), 73 (1938). Skramstad, H. K., and Schubauer, G. B., Phys. Rev., 53, No. 11, 927 (1938). Taylor, G. I., Proc. London Math. SOC.,20, 196 (Aug., 1921). Taylor, G. I.,Proc. Roy. SOC.(London), A135, 685 (1932). Ibid., A151 (873), 421 11935); ‘A156 (888), 307 (1936). Ibid., A151 (874), 494 (1935). Ibid., A157 (892), 537 (1936). Ibid., A164 (919), 476 (1938). ,
I
~I
Courtesy, Grinnell Company, Ino.
PREFABRICATED WATERPIPING USEDIN
THE
COOLING CHAMBER OF
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