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Explore the random nature of wind gusts, turbulence levels, and the energy cascade among eddies in viscous turbulent winds. Learn about Kolmogorov's scale and the impact of different eddy sizes on wind behavior.
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Analysis of Real (Viscous Turbulent) Winds P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Description of Truly Available Wind…
A Model for Mean Wind Speed Profile U* = friction velocity (m/s) = von Karman constant, approximately equal to 0.4 z = elevation above ground level (m) z0= empirical surface roughness length (m)
Random Nature of Wind • The wind in the atmospheric boundary layer is known to be distinctively turbulent and unsteady. • As a consequence the wind speed varies rather randomly on many different time scales. • These time scales range from long-term variations (years) to very short ones (minutes down to less than a second). • The latter are commonly considered to correspond to small-scale (microscale) turbulence. • These small-scale fluctuations are superimposed to the mean velocity varying on diurnal or even larger scales.
Large Scale Wind Variations Local Instantaneous Wind
Wind Gusts • A wind gust is a sudden, short burst of high speed wind that's followed by a lull. • A gust of given magnitude typically lasts for a particular time period. • When wind travels through mountain passes, alleys, or tunnels, the same amount of air is forced through a smaller pathway which also causes an increase in speed, or gusts. • Wind shear also lead to gusting.
Characterization of Turbulence in Physical Space Mean velocity component Turbulence Level
Concept of Wind Gust Local Instantaneous Wind V v(t)
Spectral Characterization of Turbulent Flow • Turbulent flows contain a wide range of eddies of different sizes (scales). • An eddy of a given size is the swirling of a fluid with a fixed size and kinetic energy. • A turbulent velocity field contains eddies of many sizes, • from eddies that are essentially large enough to fill the space available, in our case the engine cylinder, • down to eddies often substantially below a millimeter in size. • The size of the largest eddy can be guessed by asking for the diameter of the largest sphere that will fit in the available space since turbulent eddies are approximately the same size in all directions.
Energy Transactions among Eddies • These eddies pass energy sequentially from the larger eddies gradually to the smaller ones. • This process is known as the turbulent energy cascade.
Energy Transactions Vs Size of Eddy • In STFs the rate of energy transfer from one scale to the next must be the same for all scales. • The rate at which energy is supplied at the largest possible scale (dmax) is equal to that dissipated at the shortest scale (dmin). • Let us denote by this rate of energy supply/dissipation, per unit mass of fluid. d
Kolmogrov’s Quantification of Eddy Energy • Kolmogorov, hypothesized that the characteristics of the turbulent eddies of size d depend solely on d itself and on the energy cascade rate . • This is to mean that the eddies know; • how big they are, • at which rate energy is supplied to them and • at which rate they must supply it to the next smaller eddies in the cascade. • Mathematically, uO depends only on d and . • uO = LT−1, [d] = L and [] = L2T−3. • The only dimensionally acceptable possibility is:
Size of the Smallest Eddy • The shortest eddy scale is set by viscosity, because the shorter the eddy scale, the stronger the velocity shear and the more important the effect of viscosity. • Consequently, the shortest eddy scale can be defined as the length scale at which viscosity becomes dominant. • Viscosity, denoted by ν, has for dimensions = L2T-1. • It is reasonable to assume that dmin depends only on , the rate at which energy is supplied to that scale and on . • Then the only dimensionally acceptable relation is: d This is called the Kolmogorov scale , is typically on the order of a few millimeters or shorter.
