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Hydraulics Prof. B.S. Thandaveswara 42.1.1 Self Aerated Flow – Definitions of Terms and Instrumentation Introduction In any open channel studies the necessary basic parameters to describe the flow phenomenon are velocity and depth. In aerated flow, in addition to these two quantities, information regarding air concentration is also essential. The presence of air in aerated flow has necessitated the development of new measuring techniques to be adopted and the formulation of new definitions of aerated flow quantities. Definition of terms Straub and Anderson have defined some of the essential terms like concentration and depth in aerated flows a brief description of which is given. Also terms such as air water velocity, density of air water mixture as defined by Gangadharaiah, Lakshmana Rao et al. for self aerated flows are also presented. Air concentration, C; is defined as the ratio of the volume of air per unit volume of air water mixture. Upper limit of flow can be defined based on (i) air concentration (ii) velocity distribution. They are a) Upper limit of flow, du this is an upper boundary of air entrained flow and may be defined as the value of y where air concentration is 99 %. b) Upper limit of flow, d u v : This is an upper boundary of the velocity distribution and is defined as the value of y where the velocity is zero in the upper region. Mean depth of flow, d : The depth d represents a mean depth of flow that would exist when all the entrained air is removed up to the highest point where water is found. It corresponds to depths of non - aerated flow of a given discharge with velocity equal to that of the aerated flow. It is defined as ∞ d = ∫ ( 1- C )dy 0 Indian Institute of Technology Madras Hydraulics Prof. B.S. Thandaveswara in which y is the normal distance measured from the bed, and C is the local air concentration. Transitional depth, dT; is defined on the basis of air concentration distribution as that depth which represents the value of y where the transition from the distribution in the lower region to that in the upper region occurs. In other words, it is the value of y where the concentration gradient, dc / dy, is maximum. Mean air concentration, C ; the mean air concentration over the whole range of air concentrations measured at a section is defined as du ∑ C dy 1 du 0 C= ∫ C dy ≈ du 0 du Transitional mean air concentration, C T : It is defined as the mean air concentration in the region below the transitional depth which applies to that air which is being transported by the flow i. e., dT d ∑ Cdy 1 T 0 CT = ∫ C dy ≈ dT 0 dT Velocity of air water mixture, vaw: Lakshmana Rao et al. developed a simple mathematical model for the air water velocity ( vaw ) based on the continuity equation. Assuming va and vw to be the velocities of air and water respectively a relation may be written in terms of concentration given by vaw = ( 1 − C ) vw + C va Mean velocity of flow, V : mean velocity of the flow may be defined as duv 1 V= ∫ vdy duv 0 in which v is the measured local velocity at any depth y. Density of air water mixture, ρ aw : Gangadharaiah developed a definition based on the assumption that the resulting mass density of air water mixture depends on the individual masses and he correlated it with the mean air concentration of the flow as Indian Institute of Technology Madras Hydraulics Prof. B.S. Thandaveswara ρaw =1 - θ C ρw in which ρ w is the mass density of water, θ is a constant found to be equal to 1.1 from an empirical fit. This relationship is valid upto 85 % mean air concentration. This relationship may be used wherever correction for density has to be made. Inception number, I; is defined as the ratio of kinetic energy to surface tension energy for inception to occur. The critical inception number at which air entrainment begins may be taken as approximately equal to 56. Entrainment constant, Ec; the velocity of inflow of the ambient fluid (i.e., air) in to the turbulent region must be proportional to the velocity scale of the layer and the constant of proportionality is called the ' Entrainment constant'. This may be written as 1 d Ec = V dx (du V ) V and d are chosen as the velocity and length scales. u Above equation may be rewritten in the form 3 d (du ) d u dF 1 du d ρ E = + + c 2 dx F dx 2 ρ dx 1 du in which ρ is the characteristic nondimensional density and equal to ∫ ρaw vdy d u Vρ w 0 V and Froude number F = 1 ⎡ ρ g d u cos α ⎤ 2 ⎣ ⎦ Indian Institute of Technology Madras