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Lagrangian Correlation and Some Difficulties in Turbulent Diffusion Experiments
LAGRANGIAN CORRELATION AND SOME DIFFICULTIES IN TURBULENT DIFFUSION EXPERIMENTS S. Corrsin Mechanical Engineering Department, Johns Hopkins University, Baltimore, Maryland, U.S.A.
1. THEPROBLEM OF RELATING LAGRANGUN AND EULERU
CORRELATION For turbulent diffusion we are interested in the statistics of at least single “fluid particle’’ displacement; sometimes we need the joint statistics of two or more. Since particle displacement is an integral function of its (Lagrangian) turbulent velocity vc(a, t ) , i.e.
1 t
Xi(a, t ) = a, + vc(a, tl)dt,, 0
it follows that even the simple probability density function of displacement depends upon the full functional probability of vi. a is the “initial” position of the fluid particle, e.g. a, = X,(a,O). The problem is made even more difficult by the fact that the Eulerian velocity field ui(x, t ) is more accessible (both experimentally and theoretically), so we are actually interested in predicting the statistics of X as a function of ui(x,t).x is the space coordinate vector, t is time. Evidently (2)
%[a,tl = u,CX(a,t ) , $1
and an integral equation for displacement results from substitution of (2) into (1) : (3)
6
We are, of course, concerned with only the statistical properties of these random variabIes.
t Vector and Cartesian tensor notations are both used here. Hence, ai is any component of a. 441
442
S. CORRSIN
Since Taylor’s demonstration of the connection between mean square particle displacement [e.g. Xf(a,t ) ] and time auto-correlation of Lagrangian velocity [vl( a,t )vl(a, t + T ) ] , the problem of expressing this correlation in terms of the Eulerian properties has become a very important practical one.? The Lagrangian two-point correlation tensor is where the average is over a suitable ensemble of realizations. For stationary, homogeneous fields this depends on (a, 7 ) only. The correlation introduced by Taylor is Ll1(0,7). The Eulerian two-point correlation (in space-time) for a homogeneous, stationary field is (5)
Ej,(5, T )= q x , t ) U I ( X +5, t + T )
and, in general, there is no reason to expect that Lik and Ej, will be uniquely related. For simplicity, all of the foregoing expressions are set down for a flow with zero mean velocity. I n a turbulence convected with uniform mean speed 0,they correspond to an Eulerian coordinate system moving with the mean flow. The Eulerian space correlation B&, 0) has been measured fairly extensively in the laboratory. A. Favre, J. Gaviglio and R. Dumas (1953) have carried out an appreciable number of measurements which give information on the Eulerian time correlation Ejl(0,T)and on the full Eulerian two-point correlation E&, T).They used time delay with probes separated in the mean-flow direction to achieve the effect of an instrument travelling with the mean flow. In many experiments with non-zero mean velocity (in the direction), the anemometer sits at a fixed space point x, (i.e. zL,yL,zL)in “laboratory” or “ground ” coordinates. Such an instrument gives an Eulerian time signal which is related to the turbulence field ui(x,t) by
m
- Y,2, t ) . x,, YLY ZL) The time auto-correlation function of 81 is therefore a space-time correlation of the turbulence viewed in the coordinates x moving with the mean flow : (6) ai(t; X , J S l ( t + T ;
xL)
N
N
u , [ x - U t , y , ~ , t ] ~ i [Us (- t + T ) , y , z , t + T I Bll(BT,0, 0, T).
t For a homogeneous field,
is a function o f t only (“I‘ayIor,1921).
LAGRANQIAN CORRELATION AND DIFFUSION EXPERIMENTS
443
If the turbulence level is sufficiently low,? the turbulence pattern is virtually “frozen ” during the time interval required for this correlation to drop to zero. Then the pertinent “Taylor hypothesis ”, on the equivalence of space and time correlations, is valid. Formally, this happens if the whole flow is such that
1
(7) at the same time as
Of course E,,(o,0,O , O ) = $. Equation (4) shows explicitly that Likis the average over an ensemble of random trajectories X(a,t ) in the ensemble of random Eulerian fields u(x,t). Little mathematical work appears to have been done on the properties of such statistical functions, even in the degenerate case in which the trajectories are statistically independent of the fields.$ The immense complexity of our problem is finally brought out by the realization that each trajectory in the ensemble is related to the field it traverses [Equation (3)]. 2. DIFFICULTIES IN DETERMINING LAGRANGIAN CORRELATION
FUNCTION BY DIFFUSION MEASUREMENTS For a homogeneous, stationary turbulence Taylor showed that the mean square particle displacement in the continuum is, for one component,
where a has been chosen ati zero for simplicity. a is zero because we follow a single particle.
t The explicit dynamic conditions are more complex.See, for exampIe. C. C. Lin (1963) or M. S. Uberoi and S. Corrsin (1952). $ If they were independent, equations like (6) of ‘‘ProgressReport on Some Turbulent Diffusion Research’’(p. 182 of this volume)would apply. 0 This particular form is due to Kamp6 de FBriet (1939).
444
9. CORRSIN
Measurements of x f ( t ) should permit calculation of the Lagrangian auto-correlation by double differentiation. Such differentiation of anecessarily uncertain empirical function yields a still more uncertain result, although independent information on asymptotic and integral values of Ll1(O,7)can improve the result (Uberoi and Corrsin, 1952). The common procedure for such experiments is to introduce either heat or a chemical contaminant from a “point” or “line” source into a flowing turbulent fluid. Heat is ordinarily cheapest, most easily supplied and controlled, often the most easily detected. If molecular diffusion is negligible, the principal requirements for “tagging” are that (a)the source be steady in time (and uniform along its length if it is a “line ”), (b) that its pertinent dimension be much smaller than the smallest significant length scale of the turbulence, presumably the “Kolmogoroff microscale”, (.9/~)“4. Y is kinematic viscosity, E is turbulent energy dissipation rate. The only approximately homogeneous turbulence set up to date appears to be that generated by a grid spanning a uniform duct flow. Since this turbulence has no way of continuously extracting energy from the mean stream, it decays with time, i.e. with downatream distance. Townsend (1954) has shown that the resulting inhomogeneity can be approximately allowed for by the assumption (known to be good over limited distances) of simple similarity of all pertinent statistical functions. At least in gases, it is found that molecular diffusion is not negligible. For small times it may be dominant because (from the continuum viewpoint) molecular diffusion causes a parabolic initial spread of a hot sheet ( 4%) whereas turbulent convection causes a linear initial spread ( - 5 ) . This is evidently the case in the photographs in N.A.C.A. Rep. 1142 ;the grid-generated turbulence (v’/U NN 0 * 04) simply causes a slight waving in the relatively thick sheet of warm air.? Farther downstream, in the interesting region where the Lagrangian auto-correlation is no longer nearly unity, the molecular diffusion effect is increasingly amplified by the turbulent strain field, and we cannot correct for its effect by simply subtracting the spread of a steady molecular wake. A first estimate of this phenomenon has been made by A. A. Townsend (1954). It is still possible, however, to determine experimentally the correction for turbulence-accelerated molecular spreading. Downstream of a heated “line ”, for example, the temperature fluctuation correlation function
-
t Fortunately, as pointed out by Sir Geoffrey Taylor during the Discussion of Session C, this part of the hot sheet has not yet been affected by turbulent strain, SO the mean square turbulent displacements and mean square non-turbulent molecular diffusion are superposable.
LAGRANGIAN CORRELATION AND DIFFUSION EXPERIMENTS
445
measured by two fast-response thermometers will give the average cross-section shape of the warm sheet (Townsend, 1954). It is the random convection of this statistically thickened sheet which chiefly indicates the Lagrangian displacements. I n liquid streams the molecular diffusion effect can be considerably smaller. I n these diffusion experiments with local sources, it is especially important that the detecting devices have a linear response to concentration level. It is well known that non-linear devices in general give erroneous average output in the presence of fluctuations. For a given instrument the error tends to increase with ratio of input fluctuation to mean input above the ambient level; the random waving of such a thin sheet corresponds to very high ratio indeed (Uberoi and Corrsin, 1952). The complicating effects of molecular diffusion can be avoided by using small solid particles to “tag” the fluid. These, however, bring with them a new set of difficulties. It is possible to obtain particles smaller than the “Kolmogoroff microscale ”. It is also possible to select particles whose size, density, etc., will permit them to follow the local fluid motion to any desired accuracy asymptotically (Lumley, 1967). The principal trouble lies in proper launching. Unless the “ideal” solid particle is launched into the flow with exactly the local instantaneous fluid velocity, it will not follow a true Lagrangian path until after some finite response time. A launching device which sensed the local velocity vector and performed accordingly would evidently be very complex. Two simpler approaches may be practicable : (a)Introduce the ideal solid particles far enough upstream to permit them to reach their asymptotic (Lagrangian) behaviour before they reach the observation region. Then follow the meandering of only those few that chance to pass a fixed small “initial” observation “point” or “line”. ( b ) Use particles whose “relaxation time” after launching is much smaller than the smallest significant characteristic time in the turbulence, the “Kolmogoroff time ” (u/e)lI2.The relaxation time
3
of a sphere with Reynolds number low enoughis - 2
P
+ 1 ,where
)
d is diameter, p is fluid density, pp is particle density (Lumley, 1957; Tchen, 1947). For a sphere with density equal to fluid density this turns out to be equivalent to the requirement that its diameter be much less than the Kolmogoroff microscale.
446
9. CORRSIN
REFERENCES Favre, A., Gaviglio, J., and Dumas, R. (1953). Rech. uCO.,MarsfAvriI. Paper to 8th Int. Cong. Theor. and App. Mech., Istanbul, 1952 (translated &B NACA, T N 1370, 1955). Kamp6 de FBriet, J. (1939). Am. SOC. Sc. BruxeZZes 50. Lin, C. C. (1953). Quart. app. Math. 10, 4. Lumley, J. L. (1957). Ph.D. Thesis, Johns Hopkins University. Taylor, G. I. (1921). Proc. London math. Soc. 20, 196. Tchen, C. M. (1947). Pub. 51 of Lab. for Aero- and Hydro-dynamics,Delft. Tomsend, A. A. (1954). Proc. roy. SOC.A 224, 487. Uberoi, M. S., and Cornin, S. (1952). NACA Rep., 1142, 1953 (originally NACA, TN 2710, 1962).
DISCUSSION REPORTER: R. A. PANOFSKY R.
w. DAVIES. The g r o u p at Cambridge and Johns Hopkins seem t o have the
small-scale experiments well under control; what large-scale experiments should be performed? F. N. FRENKIEL. We should endeavour to study diffusion over urban areas. P. A. SHEPPARD. A very powerful approach might be to observe and analyse the relative motion of two particles with effectively zero terminal velocity. This might be easier and more rewarding than concentration measurements. Could Corrsin or Batchelor tell us what could be deduced from such measurements on all scales from metres to thousands of kilometres? a. K. BATCHELOR. I shall discuss this in my later paper. 0. a. SUTTON. I n the early days, much attention was paid to the diurnal temperature wave. It is easy to measure the change of phase with height. This change is slow and is proportional t o zlla. On the other hand, the eddy coefficient of heat conduction varies as z p where pis about one or less. With this condition, the heat conduction equation yields a solution for the phase lag varying as zlla or more rapidly. Nobody has yet explained this discrepancy. It should be considered by the theoreticians. H. LETTAU. For a linear variation of the exchange coefficient with height, the phase varies as the logarithm of height. I n the atmosphere we need t o allow for radiation effects in the theory. Is this necessary in the wind tunnel? s. CORRSIN. No; the temperature difference is small, of order 2°C a short distance downstream of the source. a. D. ROBINSON. One can easily measure time correlations. With much labour, we could measure space correlations. But Lagrangian correlations can only be measured with great difficulty. Can we do anything with the measurable correlations? s. CORRSIN. You underestimate the possibilities of tracing the motions of neutral balloons or other tracers. You can also measure the “true” Eulerian correlations (following the mean wind). This is the minimum one needs for a theory of Lagrangian correlation.
LAORANQIAN CORRELATION A N D DIFFUSION EXPERIMENTS
447
o. D. ROBINSON. The mean wind does not really exist. 5. CORRSIN. Perhaps arrays of instruments in several directions would be required. J. 2. HOLLAND. One difficulty in the definition of the mean wind is that the wind changes with height aa well as time. There is no point of vanishing gradient. How do you define the true Eulerian frame of reference? S . CORRSIN. We may have to be content with less than optimum conditions. a. I. TAYLOR. The turbulence-induced molecular diffusion varies as t3. It also varies with molecular diffusivity. Why not use different gases with different difisivities t o study this effect? s. CORRSM. I agree that this could be a useful set of experiments. 0. 0. SUTTON. Some difficulties are very apparent to meteorologists. If a thunderstorm occurs, the observation is rejected in most experiments. But in large-scale diffusion, thunderstorms, or sudden wind changes are part of the system. When is this interruption irrelevant? Only ideal situations have been studied so far. The Windscale accident occurred with a oold-front passage, and we must treat situations of this type. Turbulence may contain “eddies” of any size. E. M. FOURNIER D’ALBE. The failure to define turbulence is disturbing. If Maxwell’s demon were to witness molecular motion, would he not describe it as we here descyibe turbulence? F. IF FORD. If one regards the atmosphere as a large machine that is generating random numbers, then the experience of the people working on the problem of making random-number tables is instructive. If a random number machine starts to generate, say, recurrent sequences of sixes, the operator would, no doubt, call in the repair man and have it fixed ; the particular mechanism causing the trouble would be changed, and no more runs of sixes would occur. I n the air this might correspond t o a thunderstorm. It was pointed out by Spencer Brown (I believe of Oxford) that such operations signiiicantly bias L‘random” series ;runs of sixes of arbitrary length are bound t o occur. The moral for us seems to be that we have t o make a distinction between what we call turbulence, and ra.ndom motion. Turbulence seems to be what is left over when all the determined motions (thunderstorms, gravity, waves, etc.) are removed from atmospheric flow. Consequently, we can expect any definition of turbulence we might agree on temporarily to change as more is learned about air motions. H. E. CRAMER. If I remember correctly, Sheppard and Priestley8 some years ago emphasized that most meteorological phenomena may be considered within the general framework of turbulence ; and that turbulence techniques restricted in the past to micrometeorologicalscales could be extended to the larger-scale atmospheric processes. We now have considerable empirical evidence that the meteorological spectrum is reasonably continuous over a broad range of scales (centimetres t o thousands of kilometres). On this basis, there is a simple operational definition of turbulence; turbuIence is the ensemble of all Auctuations included within the time or length scale of any particular meteorological phenomenon
448
9. CORRSIN
and is necessarily a function of the scale. Spectral analysis of time or space sequences of the va,rious parameters wilI tall us the reIative contributions of various frequenciesto the t o d ,variances or mean square departures from the ensemble averages. P.A. SIIEPPARD. We are here manufacturing a problem. Turbulence is largely what you choose to make it and arises in fluid flow by formulating problems in a particular way. If a problem is posed in statistical terms so that certain details are excluded a p*m.,i.e. if the problem is not completely determined and has random components, we deal with a problem of turbulence. The difficulty is that the laws governing turbulent motions are not the same for the different scales of more or less arbitrarily defbed “mean ” motions. 0 . 0 . BUTTON. I disagree. The Reynolds phenomenon of change from laminar to turbulent flow is a real phenomenon, changing the deviations from mean 00w from U t e s i m a l to finite quantities. We should be able to distinguish mathematically between turbulent and molecular states. Brownian motions are independent of the boundary conditions ; likewise, the turbulent motions are controlled by the intrinsic nature of the fluid. F. I. BADQLEY. Is turbulence-accelerated molecular diffusion important for meteorological problems? a. CORRSIN. It may be negligible in large-scale motion.
* Priestby, C. H. B.,and Sheppard, P. A. (1952). Quart. J . R. met. Boo. 78,488.
1. THEPROBLEM OF RELATING LAGRANGUN AND EULERU
CORRELATION For turbulent diffusion we are interested in the statistics of at least single “fluid particle’’ displacement; sometimes we need the joint statistics of two or more. Since particle displacement is an integral function of its (Lagrangian) turbulent velocity vc(a, t ) , i.e.
1 t
Xi(a, t ) = a, + vc(a, tl)dt,, 0
it follows that even the simple probability density function of displacement depends upon the full functional probability of vi. a is the “initial” position of the fluid particle, e.g. a, = X,(a,O). The problem is made even more difficult by the fact that the Eulerian velocity field ui(x, t ) is more accessible (both experimentally and theoretically), so we are actually interested in predicting the statistics of X as a function of ui(x,t).x is the space coordinate vector, t is time. Evidently (2)
%[a,tl = u,CX(a,t ) , $1
and an integral equation for displacement results from substitution of (2) into (1) : (3)
6
We are, of course, concerned with only the statistical properties of these random variabIes.
t Vector and Cartesian tensor notations are both used here. Hence, ai is any component of a. 441
442
S. CORRSIN
Since Taylor’s demonstration of the connection between mean square particle displacement [e.g. Xf(a,t ) ] and time auto-correlation of Lagrangian velocity [vl( a,t )vl(a, t + T ) ] , the problem of expressing this correlation in terms of the Eulerian properties has become a very important practical one.? The Lagrangian two-point correlation tensor is where the average is over a suitable ensemble of realizations. For stationary, homogeneous fields this depends on (a, 7 ) only. The correlation introduced by Taylor is Ll1(0,7). The Eulerian two-point correlation (in space-time) for a homogeneous, stationary field is (5)
Ej,(5, T )= q x , t ) U I ( X +5, t + T )
and, in general, there is no reason to expect that Lik and Ej, will be uniquely related. For simplicity, all of the foregoing expressions are set down for a flow with zero mean velocity. I n a turbulence convected with uniform mean speed 0,they correspond to an Eulerian coordinate system moving with the mean flow. The Eulerian space correlation B&, 0) has been measured fairly extensively in the laboratory. A. Favre, J. Gaviglio and R. Dumas (1953) have carried out an appreciable number of measurements which give information on the Eulerian time correlation Ejl(0,T)and on the full Eulerian two-point correlation E&, T).They used time delay with probes separated in the mean-flow direction to achieve the effect of an instrument travelling with the mean flow. In many experiments with non-zero mean velocity (in the direction), the anemometer sits at a fixed space point x, (i.e. zL,yL,zL)in “laboratory” or “ground ” coordinates. Such an instrument gives an Eulerian time signal which is related to the turbulence field ui(x,t) by
m
- Y,2, t ) . x,, YLY ZL) The time auto-correlation function of 81 is therefore a space-time correlation of the turbulence viewed in the coordinates x moving with the mean flow : (6) ai(t; X , J S l ( t + T ;
xL)
N
N
u , [ x - U t , y , ~ , t ] ~ i [Us (- t + T ) , y , z , t + T I Bll(BT,0, 0, T).
t For a homogeneous field,
is a function o f t only (“I‘ayIor,1921).
LAGRANQIAN CORRELATION AND DIFFUSION EXPERIMENTS
443
If the turbulence level is sufficiently low,? the turbulence pattern is virtually “frozen ” during the time interval required for this correlation to drop to zero. Then the pertinent “Taylor hypothesis ”, on the equivalence of space and time correlations, is valid. Formally, this happens if the whole flow is such that
1
(7) at the same time as
Of course E,,(o,0,O , O ) = $. Equation (4) shows explicitly that Likis the average over an ensemble of random trajectories X(a,t ) in the ensemble of random Eulerian fields u(x,t). Little mathematical work appears to have been done on the properties of such statistical functions, even in the degenerate case in which the trajectories are statistically independent of the fields.$ The immense complexity of our problem is finally brought out by the realization that each trajectory in the ensemble is related to the field it traverses [Equation (3)]. 2. DIFFICULTIES IN DETERMINING LAGRANGIAN CORRELATION
FUNCTION BY DIFFUSION MEASUREMENTS For a homogeneous, stationary turbulence Taylor showed that the mean square particle displacement in the continuum is, for one component,
where a has been chosen ati zero for simplicity. a is zero because we follow a single particle.
t The explicit dynamic conditions are more complex.See, for exampIe. C. C. Lin (1963) or M. S. Uberoi and S. Corrsin (1952). $ If they were independent, equations like (6) of ‘‘ProgressReport on Some Turbulent Diffusion Research’’(p. 182 of this volume)would apply. 0 This particular form is due to Kamp6 de FBriet (1939).
444
9. CORRSIN
Measurements of x f ( t ) should permit calculation of the Lagrangian auto-correlation by double differentiation. Such differentiation of anecessarily uncertain empirical function yields a still more uncertain result, although independent information on asymptotic and integral values of Ll1(O,7)can improve the result (Uberoi and Corrsin, 1952). The common procedure for such experiments is to introduce either heat or a chemical contaminant from a “point” or “line” source into a flowing turbulent fluid. Heat is ordinarily cheapest, most easily supplied and controlled, often the most easily detected. If molecular diffusion is negligible, the principal requirements for “tagging” are that (a)the source be steady in time (and uniform along its length if it is a “line ”), (b) that its pertinent dimension be much smaller than the smallest significant length scale of the turbulence, presumably the “Kolmogoroff microscale”, (.9/~)“4. Y is kinematic viscosity, E is turbulent energy dissipation rate. The only approximately homogeneous turbulence set up to date appears to be that generated by a grid spanning a uniform duct flow. Since this turbulence has no way of continuously extracting energy from the mean stream, it decays with time, i.e. with downatream distance. Townsend (1954) has shown that the resulting inhomogeneity can be approximately allowed for by the assumption (known to be good over limited distances) of simple similarity of all pertinent statistical functions. At least in gases, it is found that molecular diffusion is not negligible. For small times it may be dominant because (from the continuum viewpoint) molecular diffusion causes a parabolic initial spread of a hot sheet ( 4%) whereas turbulent convection causes a linear initial spread ( - 5 ) . This is evidently the case in the photographs in N.A.C.A. Rep. 1142 ;the grid-generated turbulence (v’/U NN 0 * 04) simply causes a slight waving in the relatively thick sheet of warm air.? Farther downstream, in the interesting region where the Lagrangian auto-correlation is no longer nearly unity, the molecular diffusion effect is increasingly amplified by the turbulent strain field, and we cannot correct for its effect by simply subtracting the spread of a steady molecular wake. A first estimate of this phenomenon has been made by A. A. Townsend (1954). It is still possible, however, to determine experimentally the correction for turbulence-accelerated molecular spreading. Downstream of a heated “line ”, for example, the temperature fluctuation correlation function
-
t Fortunately, as pointed out by Sir Geoffrey Taylor during the Discussion of Session C, this part of the hot sheet has not yet been affected by turbulent strain, SO the mean square turbulent displacements and mean square non-turbulent molecular diffusion are superposable.
LAGRANGIAN CORRELATION AND DIFFUSION EXPERIMENTS
445
measured by two fast-response thermometers will give the average cross-section shape of the warm sheet (Townsend, 1954). It is the random convection of this statistically thickened sheet which chiefly indicates the Lagrangian displacements. I n liquid streams the molecular diffusion effect can be considerably smaller. I n these diffusion experiments with local sources, it is especially important that the detecting devices have a linear response to concentration level. It is well known that non-linear devices in general give erroneous average output in the presence of fluctuations. For a given instrument the error tends to increase with ratio of input fluctuation to mean input above the ambient level; the random waving of such a thin sheet corresponds to very high ratio indeed (Uberoi and Corrsin, 1952). The complicating effects of molecular diffusion can be avoided by using small solid particles to “tag” the fluid. These, however, bring with them a new set of difficulties. It is possible to obtain particles smaller than the “Kolmogoroff microscale ”. It is also possible to select particles whose size, density, etc., will permit them to follow the local fluid motion to any desired accuracy asymptotically (Lumley, 1967). The principal trouble lies in proper launching. Unless the “ideal” solid particle is launched into the flow with exactly the local instantaneous fluid velocity, it will not follow a true Lagrangian path until after some finite response time. A launching device which sensed the local velocity vector and performed accordingly would evidently be very complex. Two simpler approaches may be practicable : (a)Introduce the ideal solid particles far enough upstream to permit them to reach their asymptotic (Lagrangian) behaviour before they reach the observation region. Then follow the meandering of only those few that chance to pass a fixed small “initial” observation “point” or “line”. ( b ) Use particles whose “relaxation time” after launching is much smaller than the smallest significant characteristic time in the turbulence, the “Kolmogoroff time ” (u/e)lI2.The relaxation time
3
of a sphere with Reynolds number low enoughis - 2
P
+ 1 ,where
)
d is diameter, p is fluid density, pp is particle density (Lumley, 1957; Tchen, 1947). For a sphere with density equal to fluid density this turns out to be equivalent to the requirement that its diameter be much less than the Kolmogoroff microscale.
446
9. CORRSIN
REFERENCES Favre, A., Gaviglio, J., and Dumas, R. (1953). Rech. uCO.,MarsfAvriI. Paper to 8th Int. Cong. Theor. and App. Mech., Istanbul, 1952 (translated &B NACA, T N 1370, 1955). Kamp6 de FBriet, J. (1939). Am. SOC. Sc. BruxeZZes 50. Lin, C. C. (1953). Quart. app. Math. 10, 4. Lumley, J. L. (1957). Ph.D. Thesis, Johns Hopkins University. Taylor, G. I. (1921). Proc. London math. Soc. 20, 196. Tchen, C. M. (1947). Pub. 51 of Lab. for Aero- and Hydro-dynamics,Delft. Tomsend, A. A. (1954). Proc. roy. SOC.A 224, 487. Uberoi, M. S., and Cornin, S. (1952). NACA Rep., 1142, 1953 (originally NACA, TN 2710, 1962).
DISCUSSION REPORTER: R. A. PANOFSKY R.
w. DAVIES. The g r o u p at Cambridge and Johns Hopkins seem t o have the
small-scale experiments well under control; what large-scale experiments should be performed? F. N. FRENKIEL. We should endeavour to study diffusion over urban areas. P. A. SHEPPARD. A very powerful approach might be to observe and analyse the relative motion of two particles with effectively zero terminal velocity. This might be easier and more rewarding than concentration measurements. Could Corrsin or Batchelor tell us what could be deduced from such measurements on all scales from metres to thousands of kilometres? a. K. BATCHELOR. I shall discuss this in my later paper. 0. a. SUTTON. I n the early days, much attention was paid to the diurnal temperature wave. It is easy to measure the change of phase with height. This change is slow and is proportional t o zlla. On the other hand, the eddy coefficient of heat conduction varies as z p where pis about one or less. With this condition, the heat conduction equation yields a solution for the phase lag varying as zlla or more rapidly. Nobody has yet explained this discrepancy. It should be considered by the theoreticians. H. LETTAU. For a linear variation of the exchange coefficient with height, the phase varies as the logarithm of height. I n the atmosphere we need t o allow for radiation effects in the theory. Is this necessary in the wind tunnel? s. CORRSIN. No; the temperature difference is small, of order 2°C a short distance downstream of the source. a. D. ROBINSON. One can easily measure time correlations. With much labour, we could measure space correlations. But Lagrangian correlations can only be measured with great difficulty. Can we do anything with the measurable correlations? s. CORRSIN. You underestimate the possibilities of tracing the motions of neutral balloons or other tracers. You can also measure the “true” Eulerian correlations (following the mean wind). This is the minimum one needs for a theory of Lagrangian correlation.
LAORANQIAN CORRELATION A N D DIFFUSION EXPERIMENTS
447
o. D. ROBINSON. The mean wind does not really exist. 5. CORRSIN. Perhaps arrays of instruments in several directions would be required. J. 2. HOLLAND. One difficulty in the definition of the mean wind is that the wind changes with height aa well as time. There is no point of vanishing gradient. How do you define the true Eulerian frame of reference? S . CORRSIN. We may have to be content with less than optimum conditions. a. I. TAYLOR. The turbulence-induced molecular diffusion varies as t3. It also varies with molecular diffusivity. Why not use different gases with different difisivities t o study this effect? s. CORRSM. I agree that this could be a useful set of experiments. 0. 0. SUTTON. Some difficulties are very apparent to meteorologists. If a thunderstorm occurs, the observation is rejected in most experiments. But in large-scale diffusion, thunderstorms, or sudden wind changes are part of the system. When is this interruption irrelevant? Only ideal situations have been studied so far. The Windscale accident occurred with a oold-front passage, and we must treat situations of this type. Turbulence may contain “eddies” of any size. E. M. FOURNIER D’ALBE. The failure to define turbulence is disturbing. If Maxwell’s demon were to witness molecular motion, would he not describe it as we here descyibe turbulence? F. IF FORD. If one regards the atmosphere as a large machine that is generating random numbers, then the experience of the people working on the problem of making random-number tables is instructive. If a random number machine starts to generate, say, recurrent sequences of sixes, the operator would, no doubt, call in the repair man and have it fixed ; the particular mechanism causing the trouble would be changed, and no more runs of sixes would occur. I n the air this might correspond t o a thunderstorm. It was pointed out by Spencer Brown (I believe of Oxford) that such operations signiiicantly bias L‘random” series ;runs of sixes of arbitrary length are bound t o occur. The moral for us seems to be that we have t o make a distinction between what we call turbulence, and ra.ndom motion. Turbulence seems to be what is left over when all the determined motions (thunderstorms, gravity, waves, etc.) are removed from atmospheric flow. Consequently, we can expect any definition of turbulence we might agree on temporarily to change as more is learned about air motions. H. E. CRAMER. If I remember correctly, Sheppard and Priestley8 some years ago emphasized that most meteorological phenomena may be considered within the general framework of turbulence ; and that turbulence techniques restricted in the past to micrometeorologicalscales could be extended to the larger-scale atmospheric processes. We now have considerable empirical evidence that the meteorological spectrum is reasonably continuous over a broad range of scales (centimetres t o thousands of kilometres). On this basis, there is a simple operational definition of turbulence; turbuIence is the ensemble of all Auctuations included within the time or length scale of any particular meteorological phenomenon
448
9. CORRSIN
and is necessarily a function of the scale. Spectral analysis of time or space sequences of the va,rious parameters wilI tall us the reIative contributions of various frequenciesto the t o d ,variances or mean square departures from the ensemble averages. P.A. SIIEPPARD. We are here manufacturing a problem. Turbulence is largely what you choose to make it and arises in fluid flow by formulating problems in a particular way. If a problem is posed in statistical terms so that certain details are excluded a p*m.,i.e. if the problem is not completely determined and has random components, we deal with a problem of turbulence. The difficulty is that the laws governing turbulent motions are not the same for the different scales of more or less arbitrarily defbed “mean ” motions. 0 . 0 . BUTTON. I disagree. The Reynolds phenomenon of change from laminar to turbulent flow is a real phenomenon, changing the deviations from mean 00w from U t e s i m a l to finite quantities. We should be able to distinguish mathematically between turbulent and molecular states. Brownian motions are independent of the boundary conditions ; likewise, the turbulent motions are controlled by the intrinsic nature of the fluid. F. I. BADQLEY. Is turbulence-accelerated molecular diffusion important for meteorological problems? a. CORRSIN. It may be negligible in large-scale motion.
* Priestby, C. H. B.,and Sheppard, P. A. (1952). Quart. J . R. met. Boo. 78,488.