Hydrosystems Hydrulics open channel flow iit nptel _138_ by hemlalb


									Hydraulics                                                                                          Prof. B.S. Thandaveswara

                42.1.1 Self Aerated Flow – Definitions of Terms and

                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

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


                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
                                                                  ∑ 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.,

                                                        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

                                                     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

                                                                       =1 - θ C

                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.
                      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
                      and Froude number F =                        1
                                                 ⎡ ρ g d u cos α ⎤ 2
                                                 ⎣               ⎦

Indian Institute of Technology Madras

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