Turbulence remains an important unsolved problem of classical physics. It is important in pipe flow, ship design, aeronautics and, of course, in meteorology. Many things around us show the results of turbulence, and we rely on it to perform many useful duties, such as dissipating pollution and evaporating water. Turbulence is invisible, so it is often overlooked, even when most active.

We can describe the state of a fluid flowing slowly and placidly in a capillary by its flow velocity, which is a function of the radial distance from the center of a circular capillary. This is *laminar flow*, which is easily handled by the equations of hydrodynamics. Flows like this can be described by a small number of *degrees of freedom*. The property of a fluid, a liquid or a gas, that supports laminar flow is *viscosity*, the ability to support a shear stress by a rate of change of velocity. The velocity is zero at the surface of the capillary, and increases inward, supporting a shear stress on infinitesimal cylinders that retards the flow, and makes the rate of discharge of fluid proportional to the pressure difference.

If the diameter of the capillary is increased, or the flow velocity, a change takes place more or less suddenly to a different, chaotic state of motion called *turbulent flow*. Now a large number of degrees of freedom is necessary to describe the flow, which is dominated by inertial forces. Viscosity still rules near to the walls, however, and a *laminar boundary layer* is formed that makes the transition between v = 0 and the average flow velocity V. In 1883 Osborne Reynolds showed that flow became turbulent when the dimensionless parameter R = ρDV/μ surpassed a certain number, which for circular tubes is about 2000. This parameter is now called the Reynolds number. Flows can be scaled to different sizes if the Reynolds number is the same.

Reynolds studied flow in transparent pipes where a thin thread of potassium permanganate solution was introduced. While the flow was laminar, the thread persisted. When the flow changed to turbulent, the purple thread twisted and dissolved to fill the pipe. On a calm autumn evening, a small fire may make a thin sheet of smoke or a thin column that can be followed for some distance, while by day the smoke dissipates this way and that in gusty breezes. The smoke from a cigarette may rise in a thin laminar plume for a while, then twist this way and that as the flow becomes turbulent.

We may assume that in turbulent flow the actual flow velocity is equal to the average velocity V plus the fluctuating turbulent velocities u,v,w in the x,y,z directions, respectively. The x-axis is taken in the direction of V, so the instantaneous flow velocities are V + u, v, w. The y-direction is across the wind, and the z-direction is upwards. It is not possible to specify the values of u,v,w as functions of time, but from their randomness a statistical description is possible. By definition, the average velocities are zero, but their variances, the average values of their squares, are not. It is usually assumed that the variances of each of the three components are equal, or that the turbulence is *isotropic*. Even when this is not exactly true, it may still be a useful approximation, and is probably true above 25 m. The distribution of the velocities does not appear to be gaussian on small scales, but large-scale turbulence is approximately gaussian.

The average value of the product ρuw is called the *Reynolds stress* on a horizontal plane. It is a shear stress like that caused by viscosity, but is due to turbulence. It depends on the *correlation* of velocity fluctuations in the x and z directions.

The correlations, average values of the products of two velocities, are not in general zero for turbulence. This implies the existence of *eddies* of rotating and swirling fluid, which have very important consequences. When the gustiness of the wind is measured, the frequency power spectrum of the variances of the velocities can be determined. A frequency component of frequency f corresponds to an eddy size L = V/f, which we can call the *scale* of the turbulence. At the other end of the spectrum of size is an eddy of size d, where viscosity acts to suppress the random motion rapidly. The number of degrees of freedom in a given volume varies as (L/d)^{3}, which is normally very large. Over the range of frequencies from 4 to 400 per hour, the variance of a turbulent velocity is proportional to V^{2}. The ratio of the rms turbulent velocity to V, called the *gustiness*, is about 0.2 by day, and 0.1 by night, as a rough estimate for air close to the surface. Gustiness is not isotropic near the surface, but is at 25 m and above. *Fully developed turbulence* is turbulence in which the statistical properties have ceased to vary with time or position.

The most important example of turbulence in meteorology is the surface turbulence created by thermal convection and by the wind. A layer of strong turbulence extends from the surface to perhaps 1 km when it is well-developed, and the surface layer up to 100 m has special properties, mostly due to turbulence. The troposphere as a whole, 11 km thick, is subject to turbulence up to the tropopause, which is primarily responsible for the roughly linear lapse rate in the troposphere. Strong wind shears in the jet stream region create turbulence mechanically. Cumulus clouds show thermal turbulence graphically, making it quite evident to the eye, and one of the rare phenomena when turbulence becomes visible. We can also see turbulence in the effect of the wind on a field of grain, or in the path of a rising column of smoke.

It will be useful to review the concept of fluid viscosity. If there is a velocity gradient in a fluid, then there is a shear stress in a plane perpendicular to the direction of the gradient that is proportional to the velocity gradient. The proportionality constant is called the *dynamic viscosity*, or just the viscosity. We write S = η(du/dz). If u increases upwards, then S acting on the fluid below an imaginary plane boundary is in the direction of the velocity. This is illustrated in the diagram at the right. This flow pattern is called Couette flow, and laminar flow in pipes is very similar.

If S acts for a time t on unit area of the fluid below the boundary, then the momentum of the fluid increases by the amount dP = St. Therefore, S = P/t can also be interpreted as a *downward flux of momentum*. If the velocity gradient is constant, then this flux is constant, and no momentum ends up in the intermediate fluid. If we consider a small thickness dz of fluid, then the momentum transferred to it will be ηd^{2}v/dt^{2}dz, so we find ∂v/∂t = (η/ρ)d^{2}v/dt^{2}. This can be recognized as a diffusion equation, with the η/ρ as the diffusion constant for momentum. This ratio is called the kinematic viscosity ν The kinematic viscosity of air under standard conditions is about 0.15 cm^{2}/s.

Viscosity controls the creation of the *boundary layer* when fluid flows past a surface. The relative velocity between the fluid and the surface increases from zero at the surface to the free stream velocity, and a shearing force is exerted on the surface in the direction of the fluid velocity. The thickness of a laminar boundary layer is usually denoted by δ. All surfaces are rough to some degree, but if the roughness is of a height ε < δ, it has no effect on the flow, and the surface is said to be *aerodynamically smooth*. In this case, the velocity profile has the 1/7th power dependence shown in the figure at the left. If the irregularities project through the laminar boundary layer, then a thicker turbulent boundary layer is formed, and the surface is *aerodynamically rough*. In this case, the velocity profile becomes logarithmic, as described below. The earth's surface is almost always to be considered aerodynamically rough, but a laminar boundary layer may be found over water. In some cases, the 1/7-power law can be used anyway, and it gives a reasonable description of wind velocity at small heights.

An example of random motions that can be described statistically that is much easier to work with than turbulence is the chaotic motion of molecules in a gas. Let us take V = 0 here. Then the average values of u,v,w are zero, while their variances, the mean square speeds, are equal to 3RT/M, so the chaos is isotropic. In air at STP, we can take 485 m/s as an estimate of the rms molecular velocity V (not the average flow velocity here). The mean free path λ in the standard atmosphere at the surface is 6.63 x 10^{-6} cm.

In their random motion, a gradient in molecular concentration dn/dz causes a molecular *flux* of F = -(Vλ/3)dn/dz, as given by an approximate gas-kinetic argument. This is usually written F = -Ddn/dz, where D is the *diffusion coefficient* with dimensions cm^{2}/s. This can be expressed more exactly in three dimensions by **F** = -D grad n, but we can be happy here with one dimension. If we consider a volume V, then the rate of change of the number of molecules in V, nV is equal to the influx of molecules across the surface S of V. This, by the usual analysis, gives the differential equation ∂n/∂t = D(∂^{2}n/∂z^{2}), which is the very familiar diffusion equation.

This random movement of molecules can carry, or *transport* molecular quantities along with it. We have just describe the case where the transported quantity can be considered to be 1 per molecule. A molecule can also transport momentum, of mu g-cm/s per molecule, or an amount of heat mc_{v} per molecule. The mass of one molecule is m. The momentum flux is P = ηdu/dz, and the heat flux is q = -kdT/dz, in terms of the dynamic viscosity η and the heat conductivity k. For molecules, η = ρD, and k = ρc_{v}D. All the transport fluxes are proportional to the same diffusion coefficient D = Vλ/3. With the above values for V and λ, we find D = 0.107 cm^{2}/s, not far from actual value. Then, we have ∂u/∂t = D(∂^{2}u/∂z^{2}) and ∂T/∂t = D(∂^{2}nT/∂z^{2}) for the diffusion of velocity (momentum) and temperature (kinetic energy) in addition to the diffusion of the density, since ρ = mn.

If the values for the rms velocity and the mean free path are substituted in the expression for the diffusion coefficient, a value of 0.107 cm^{2}/s is found. This shows that molecular diffusion is a very slow process on a macroscopic scale. When there is an imbalance in concentration, or momentum, or temperature, nature seeks for a quicker way to even things out, and turbulence provides the mechanism. If we suppose that turbulence acts as a diffusion mechanism, then we can postulate an eddy diffusion coefficient K that will take the place of D. When this suggestion is pursued, measurements show that for the meteorological turbulence near the surface, K is about 10^{5} cm^{2}/s over land, and 10^{3} cm^{2}/s over the sea. Of course, it will also vary with the time of day, becoming larger in the morning and smaller in the night. This figure is much greater than the diffusion coefficient for molecular diffusion, which is quite negligible.

Another illustration from daily life of the difference between molecular and turbulent mixing is in putting cream in coffee. If you are very careful and pour the cream in slowly, it will float on the coffee and remain separate, as in an Irish Coffee (invented at Jury's Hotel in Dublin for American tourists). Eventually, of course, the cream will diffuse into the coffee and make a uniform light brown mixture, but you may have to wait a month or two. Or, you can pick up your spoon and give the contents of the cup a stir, when the mixing will take place in seconds.

The same value of K can be used for turbulent momentum transport and heat transport, just as D was used in molecular diffusion. When warm air moves over a cold sea, the cold temperature diffuses upwards, creating a strong inversion, and fog if the temperature falls below the dew point. The height z of the change in temperature gradient as a function of t can be found from the solution of the diffusion equation as z = √(4Kt). If we assume K = 10^{3} cm^{2}/s, then the cooling will just reach a height of 100 m in 7 hours. If K is a hundred times larger, then the time will be only 42 minutes. In this and similar ways the values of K can be estimated.

Turbulence is only one mechanism of heat tranfer. Radiation must also be considered. Air may be cooled by radiation to a cold surface below, increasing the effect of turbulence, or may be heated by radiation received from above, opposing the effect of turbulence. That the average lapse rate is 6 K/km instead of the dry adiabatic rate 9.8 K/km may partly be due to radiation, as well as to condensation of moist air.

The instantaneous momentum flux can be written in terms of the turbulent velocities as S = ∫∫ ρw(V + u) dxdy = ∫∫ ρVw dxdy + ∫∫ ρwu dxdy. When we take the average, the first term vanishes because the average upward velocity is zero. The second integral, containing the product of turbulent velocities, need not vanish. If we write u = (dV/dz)(z - z') to bring in the velocity gradient, then S = (dV/dz)∫∫ Avg[ρw(z - z') dxdy]. Comparing with earlier results, the eddy diffusion coefficient is then K = Avg[∫∫ w(z - z') dxdy]. Theoretical formulas are not very useful in finding K, however.

This theory, which assumes a constant value of K applying to all turbulent transport phenomena, does indeed explain qualitatively a large number of observations. However, under critical examination it begins to break down here and there, and can be saved only by assuming that K is a function of z, usually a power law variation K = K'(z/z')^{p}. Also, K is a function of time of day because of the diurnal variation in the strength of turbulence. Therefore, this K-theory is best used as a general explanation, and empirically adjusted to fit experimental results. A more general theory of turbulence could replace it, and some advances have been made, but the theory is very difficult, and exact answers are still not available.

In very turbulent flows, as with a large lapse rate, the wind velocity is approximately constant with height and the velocity gradient is zero, meaning that there is little momentum flow to the surface, and so little frictional force on the wind. When there is a strong inversion, the flow is more likely to be laminar, and the velocity will increase linearly with height. The constant velocity gradient will transport momentum to the surface, acting as a frictional drag on the wind. Although these pure cases may be rare, actual conditions may approach one or the other closely. With an adiabatic gradient, the wind velocity usually obeys the formula V = (1/k)(τ/ρ)^{1/2} ln (z/z'), where von Kármán's constant k = 0.4, τ is the shear stress at the surface, ρ the density, and z' a characteristic roughness length. Over ordinary grass or downland, z' is about 1-2 cm. Under the same conditions, (τ/ρ)^{1/2} will be about 40 cm/s.

The "speed of smell" depends mainly on turbulence. The molecular mean free path is so small that it would take seemingly forever for a smellicule to diffuse from source to nose. Turbulence, however, does it quickly. Water would evaporate very slowly if it were not for turbulence. Turbulence cools things for us much more rapidly than radiation and conduction would do.

Turbulence near the surface of the earth is very important for such applications as the dissipation of pollution, the use of chemical warfare agents and military smokes, rates of evaporation, and similar things. Nevertheless, it is a very difficult study which can by no means be regarded as solved. The amount of turbulence depends primarily on the temperature lapse rate and the surface velocity gradient, but there are also other factors that may enter.

Conditions at the surface exhibit a regular diurnal change, caused by solar heating, which is modified by clouds and wind. At night, the lapse rate is small, and is often an inversion, with warmer air above a radiationally cooled ground. The winds are low, and the air is stable and laminar. When the sun rises, the heated earth becomes much hotter than the air, so the lapse rate becomes larger, encouraging instability and turbulence, which usually establishes a dry adiabatic lapse rate of close to 9.86°C/km. As the earth becomes even hotter, large bubbles of hot air leave the surface at intervals, and convection cells are established.

Whether turbulence grows or diminishes depends on the dimensionless Richardson number, Ri = g(dT/dz + Γ)/T(du/dz)^{2}. dT/dz, normally negative, is the lapse rate, and Γ is the dry adiabatic lapse rate. The critical value of Ri is not well established, but if Ri is greater than this value, turbulence will decrease, while if it is less, then turbulence will increase. In the original theory, the critical value was Ri = 1. Turbulence is, therefore, encouraged by a superadiabatic lapse rate and rapid increase of wind velocity with height. In the morning, the velocity gradient increases to a critical point, when turbulence grows. The turbulence rapidly establishes at least an adiabatic lapse rate, and so it will continue to grow.

The cooling of the sun-heated surface can occur by conduction, radiation or convection. The first two are negligible compared with the third. Forced convection occurs when there is a wind, which is, indeed, a powerful cooling agent. If there is no wind, natural convection occurs, with the buoyant rising of bubbles of hot air from the surface. This is a very difficult problem which up to now has been treated largely empirically.

E. W. Hewson and R. W. Longley, *Meteorology Theoretical and Applied* (New York: John Wiley & Sons, 1944). Chapter 9, pp. 144-160.

O. G. Sutton, *Atmospheric Turbulence*, 2nd ed. (London: Methuen, 1955).

U.S. Air Force, *Handbook of Geophysics* (New York: Macmillan Co., 1960). pp. 5-9 to 5-12.

S. P. Parker, ed., *McGraw-Hill Encyclopedia of Physics* (New York: McGraw-Hill, 1993). Art. *Turbulent Flow*, pp. 1476-1479.

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Composed by J. B. Calvert

Created 13 July 2003

Last revised 15 July 2003