From PROCEEDINGS, Auckland University Engineering Society, 1970
C. F. L. Morris
With the world population increasing as rapidly as it is, and the consequent exponential demand on existing resources, there is an increasing demand for new resources and for an increase in the efficiency of utilisation of present resources.
Further mineral resources are being searched out on the land masses at a great rate, as witness the phenomenal growth of the Australian mining industry. Food is another resource where the demand will increase with expanding population. Land farming is the main food producer for the human race, and will probably continue to be so for the foreseeable future. Increases in the efficiency of production of the food supply will depend on three broad factors. (1) The soil chemistry; (2) new grain types; and (3) the weather.
For this reason, and because meteorological phenomena will be of interest in pollution and oceanographic studies, the study of the atmosphere is likely to attract increasing interest.
Although the atmosphere is quite thick, the portion of greatest effect and most immediate interest is that layer closest to the ground, up to 10 km. or so. The movement of this part of the atmosphere is to a large degree affected by what happens in the air on its lower surface, that portion in contact with ground where evaporation, heating and cooling occur. The rest of this paper will be concerned with this interface region, the atmospheric boundary layer, which is generally described as that region of the atmosphere where the air velocity is modified by the presence of the ground.
The atmospheric boundary layer, except in the tropics, is a secondary flow phenomenon, that is, it is the boundary layer of a rotating fluid. Its thickness is therefore a function of rotation of the fluid and the free stream velocity. A. S. Monin has derived the relation for the thickness of the atmospheric boundary layer as:
h = Vg/(100*f)
where
f = 2 w sin q. {Here q is latitude and w is the earth�s rotation.
For example, for a value of q of 45o and geostrophic wind velocity ( Vg ) of 10 meters per second, h = 1 kilometre.
The lower region of the atmospheric boundary layer, where the value of friction velocity is fairly constant, is termed the surface layer. Vertical variations of stress, humidity and beat flux in this region are minimal, and the Lettau Length is essentially constant. The characteristic change in flow direction with distance from the ground is also negligible in this region. Typically, in neutral conditions the surface layer has a thickness of about 50 meters.
For this region, and generally for the whole boundary layer, velocity is considered to vary in the neutral boundary layer according to the partial DE;
dU / dZ = A / Z
Integrating & substituting evaluated constants; U = U*/K ln (Z + Zd)/Zo
where
In practice, evaluation of Zo, Zd and U* are calculated from field measurements of wind velocity at different heights.U = velocity at ht z
U* = friction velocity = squareroot (????
Zo = roughness length,
Zd = displacement height ,
Z = height
K = Von Karman's constant ~ 0.42
A = a constant
Of increasing interest is the wind velocity profile for different stabilities. The various stability numbers describe the temperature gradient of the air. Broadly speaking, if the energy level of. the air decreases with height, so that conditions exist where the lower layers of air would like to displace the upper layers (by convection) the air is called unstable. If the reverse condition exists, so that (after allowing for the Lapse Rate) air at any level has colder air below it, warm air above it, conditions are stable. When the energy is constant with height, (i.e., the temperature distribution follows the Lapse Rate) the air is described as neutrally stable.
There are three non-dimensional numbers in common use that are used to describe stability. All adopt positive values for stable air, zero values for neutral air and negative values for unstable air.
The flux Richardson Number is defined as the rate of consumption of energy by buoyancy forces divided by the rate of production of energy by wind shear.
Rf = b H / (Cp * dV / dZ )
where b is a buoyancy parameter g/t. & H is vertical heat flux.
The Richardson Number is
Ri = Rf * Km / Kh
Where Km / Kh is the turbulent Prandtl Number.
The third parameter is the Lettau Length.
L = - ( Cp ro V*3 / (K b H )
this is used in the form x = Z/L = f ( Rf )
In the surface layer, L is essentially constant: this illustrates the fact that stability is practically a linear function of Height.
The velocity profile of air in conditions near neutral has been proposed to be
dV / dZ = ( V* / Kz) f ( G )
Where G is a stability parameter ( e.g. Ri, Z/L or Rf ).
Integration gives
V = V* / K [ ln Z/Zo + f ( G ) ]
At the moment this equation has been shown to determine the wind profile with reasonable accuracy between values of -0.5 < z/L < 0.1. These conditions are those which exist in fairly high winds in the surface layer. Of more interest in pollution studies, for instance, are the conditions that could exist in low winds, where z/L can have considerably more variation.
Recently in England and America attempts have been made to duplicate thermal gradients in a wind tunnel. Difficulties are that the turbulence level and the velocity profile existing in the atmosphere should also be reasonably approximated. The velocity profile is generally generated in shorter wind tunnels by a grid which mechanically forces the flow into the required velocity profile. In England an attempt was made to heat the air at different heights with radiator bars. Unfortunately the correct turbulence level was not obtained, although further work could solve this problem. The Americans attacked the problem by building a very long wind tunnel and growing a natural boundary layer. By heating' and cooling the floor of this tunnel, a thermal stratified boundary layer was produced. Published work indicates that good turbulence levels are obtained by this technique. Unfortunately, such large (100 metre) wind tunnels are expensive, and shorter tunnels cannot easily he adapted.
In the engineering school at Auckland University the largest section wind tunnel is too short to grow a natural boundary layer of any great thickness.
An attempt is currently being made to simulate a thermal boundary layer by introducing air through the wall of the tunnel. This air can be preheated or precooled. It is hoped that the method of introducing the air can be made to produce approximately the required velocity profile without the use of a grid or screen, although results to date have been inconclusive.