Momentum and energy transfer in turbulent pipe flow: The penetration model revisited

Abstract

The extreme difficulties involved in trying to describe mathematically heat, mass, and momentum transfer in turbulent pipe flow are well-known, and for practical calculations one is forced to fall back on the empirical correlations. Our qualitative understanding of the turbulent exchange process has increased significantly in recent years, however, due to the careful studies of Corino and Brodkey (1969), Kline et al. (1967), Popovich and Hummel ( 1967), and many others. These researchers have clearly described the unsteady nature of the viscous sublayer and identified the bursting process as a key event in turbulent flow.Corino and Brodkey (1969) offer a lucid picture of the wall region in turbulent pipe flow reminiscent in many ways of the early penetration model of Einstein and Li (1956). In this model, a lump of fluid with axial velocity uo suddenly penetrates into the region immediately adjacent to the pipe wall. The velocity profile then decays according tofor a time period 8, following which the process begins anew. It is important to emphasize that the picture suggested by Equation (1) represents an extreme idealization of the process described by Corino and Brodkey. In spite of this, the model accurately describes a large variety of experimental data. The quantity uo in Equation ( 1 ) was originally chosen by Einstein and Li (1956) as the velocity at a distance equal to three times the displacement thickness from the wall. More recently, Meek and Baer (1970) have evaluated u o by coupling the velocity profile in the sublayer to the velocity profile in the core obtained via a mixing length formulation. Integration over the cross section of the pipe then yields a relation between uo and the average fluid velocity u b . This approach does eliminate the arbitrariness associated with the definition of uo but complicates the mathematical details of the theory and distracts from its conceptual simplicity. The studies of Corino and Brodkey suggest that uo is of order ub, as would be anticipated on purely physical grounds. It is therefore natural, and consistent with the crude nature of the model, to simply equate these two velocities. This simplification greatly reduces the algebraic and numerical difficulties associated with the model and leads to predictions equivalent to and in some instances superior to the Meek and Baer (1970) formulation. Furthermore, the model as thus formulated is useful in interpreting various aspects of turbulent drag reduction. MOMENTUM TRANSFEREquation ( l ) , with the condition uo = ub, yields the following alternative expressions for the friction factor:where is a dimensionless contact time. Equation (2) indicates that at constant bulk velocity, the energy dissipation in turbulent pipe flow is an inverse function of the square root of the contact time, or time between turbulent bursts. Current theories of drag reduction in dilute polymer solutions suggest that the primary function of the macromolecule is to reduce the bursting frequency and hence increase B+. This is a consequence o...

The influence of polymer conformation on turbulent drag reduction

“…This statement must be qualified somewhat due to the pronounced viscosity increase of the polymer in deionized water. This may be seen more clearly by considering the simplified model of turbulent drag reduction recently presented by Katsibas and Gordon (1974). Using a modification of the Einstein and Li penetration model (1956) gave the following expression for the friction factor:…”

“…is a constant, while (BIB,) ' ' 2 decreases continuously with increasing N R~. [For maximum drag reduction, ( e/ep) ''* -N Re -0.30 (Katsibas and Gordon, 1974). ] At low NRe, if V , / V is sufficiently large, it may offset the modest reduction in e/ep, leading to low or insignificant levels of drag reduction.…”