Typical examples of turbulent flows:
Above: Chaotic
character of a
turbulent sub-
merged jet.


Left: Turbulent
vortices in
a cross section
of the turbulent
submerged jet visu-
alised by laser
light sheet.
Turbulent flows are characterised by chaotic rotational, vortical motion of fluid particles. Turbulence is generated and usually exists only in shear regions (it may be present for a certain period of time in flow without transverse velocity gradient, but there it is gradually damped and finally disappers ). Mainly because of its chaotic character, turbulence is a very complicated subject - it is, if fact, often described as one of the least understood subjects in science.
Origins of turbulence The tendency towards rotational, vortical motion is present even at lowReynolds numbers in laminar flows - but it is effectively opposed there by stabilising effects of viscosity. nevertheless, manifests its presence by causing the shear region being not stable. Disturbances, to which any real flow is exposed, may be amplified by the action of this instability.
Fig.I-25
This amplification is extremely strong .. even distrubances as weak as thermal motion of molecules may be amplified into large-scale vortical motions. The problem of stability and amplification was studied by Rayleigh in series of papers published between 1878 and 1917, initially in an attempt to explain sensitivity to acoustic disturbance of laminar jets, which were observed to suffice as causes for transition to turbulence as early as 1858 by Leconte ( who observed flicker of gas lamps, synchronised with changes of music intensity during Beethoven's concert for piano, volin and violincello especially during pizzicatos - the combustion process was soon found to be unimportant, just making easier observation and keeping Reynolds number low ). The basic equation that describes this amplification process derived independently Orr in 1907 and Sommerfeld in 1908. The first solution of this equation, for the case of boundary layer on plane wall, obtained Tollmien in 1929 - the result, describing amplified wave motion in the layer, is described as the Tollmien-Schlichting waves. They are subject to a selective resonance effect:
Fig.I-26 Hairpin vortices are formed from the simple roller vortices
by elongation of bends, which appear due to instability. The bend is
lifted away from the wall and is carried away by faster flowing fluid.
there is a particular frequency (- frequency in Fig. I-20) at which the motions are most amplified. The resultant oscillations carried away by flowing fluid appear as waves having a particular wavelength. In another co-ordinate system, however, the same motions may appear as vortical rotation (- roller votices, Fig.26). The Orr-Sommefeld equation, however extremely difficult to solve, is just a linearised approximation, capable to describe only the initial stages of the process. In the subsequent stages, as the wave amplitude reaches a certain magnitude, nonlinearities
Fig.I-27 Formation of turbulent spots
by secondary deformation of hairpin vortices.
cause spectral dispersion and appearance of other spectral components. In the final stages, three-dimensional effects become important: lifting of "hairpin" vortices into faster flowing layers away from the wall. Vortices cannot be torn into parts, but they are elongated. The higher spectral components appear as small secondary motions, which in the end disintegrate into turbulent spots (Fig.I-27).
The final stage of this process, turbulence, is characterised by a continuous spectrum consisting of infinite number of spectral components. The elongation of vortices continues even in there. Because of (approximate) concervation of derformed vortex momentum, its kinetic energy must increase. It is taken from motions of larger scale, which are responsible for
Transition to turbulence in a submerged water jet issuing from
a nozzle (at the right-hand side). Reynolds number Re = 3 600.

deformation of smaller vortices. Energy in turbulence is thus continuously transferred from large votices to smaller ones. This process is described as a cascade, because energy is most efficiently transferred to nearest neighbours in spectrum (very small vortices are carried away as a whole and not much deformed). The process ends at vortices so small their motion is effectively converted into heat by viscous friction - Reynolds number of their motion is near to one.





An example of a numerical solution of transition to turbulence with clearly recognisable hairpin vortex. This still from a movie made at Group for Software Technology, Deutsches Zentrum fur Luft- und Raumfahrt e.V., Germany, may be compared with the hand-drawn idealised picture shown in Fig.I-26.


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This is page Nr. I08 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS"
Any comments and suggestions concerning this text may be mailed to the author to his address tesar@fsid.cvut.cz

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