Two-parameter Turbulence Model Research Articles

Flows in the initial sections of channels with permeable walls

Calculations of two types of flows in the initial sections of channels with permeable walls are carried out on the basis of semiempirical turbulence theories during fluid injection only through the walls and during interaction of the external flow with the injected fluid. Experimental studies of the first type [1–3] show that at least within the limits of the lengths L/h 100 (Re0=v0h/Ν or Re0=v0a/Ν, where v0 is the injection velocity), and small fluid compressibility, the axial velocity component is described by the relations for ideal eddying motion: u=(π/2)x× cos (πy/2) in a flat channel and u=πx cos (πy2/2) in atube (the characteristic values for the coordinates are, respectively, h anda). Measurements indicate the existence of a segment of laminar flow; its length depends on the Reynolds number of the injection [3]. In the turbulent regime the maximum generation of turbulent energy occurs significantly farther from the wall than in parallel flow. Flows of the second type in tubes were studied in [5–7]. These studies disclosed that for Reynolds numbers of the flow at the entrance to the porous part of the tube Re=u0a/Ν 0.01 leads to suppression of turbu lence in the initial section of the tube. An analogous phenomenon was observed in the boundary layer with v0/u0>0.023 [8, 9]. Laminar-turbulent transition in flows with injection was explained in [10, 11] on the basis of hydrodynamic instability theory, taking into account the non-parallel character of these flows. The mechanisms for the development of turbulence and reverse transition in channels with permeable walls are not theoretically explained. Simple semiempirical turbulence theories apparently are insufficient for this purpose. In the present work results are given of calculations with two-parameter turbulence models proposed in [12, 13] for describing complex flows. Due to the sharp changes of turbulent energy along the channel length, a numerical solution of the complete system of equations of motion was carried out by the finite-difference method [14].

The Prediction of the Axisymmetric Turbulent Jet Issuing into a Co-Flowing Stream

SummaryThree analyses are presented for predicting the development of an axisymmetric turbulent jet issuing into a co-flowing external air stream. The first analysis is analogous to a method used by Patel to predict the growth of a two-dimensional jet in an external air stream. The method is found to be inadequate when the excess velocity on the axis of the jet becomes small compared with the external stream velocity. The second analysis assumes that the turbulence structure is similar at different streamwise stations but it breaks down when the advection of turbulent energy becomes comparable with the turbulent energy production. In the third approach, a two-parameter model of turbulence developed by Rodi and Spalding, which uses two differential equations for the turbulent energy and the length scale of the turbulence respectively, is found to predict closely the experimental results of Antonia and Bilger for a ratio of jet to external stream velocity of 3.0. The success of this last method emphasises the non-similar character of turbulence.

A two-parameter model of turbulence, and its application to free jets

A model of turbulence is investigated in which the Reynolds stress appearing in the momentum equation is calculated from the expression $$\overline {u'\upsilon '} = - \sqrt k L(\partial U/\partial y)$$ ; the kinetic energy, k, and the length scale,L, of turbulence are determined from differential transport equations for these quantities. These equations are solved for various free-jet situations, and the empirical constants involved are adjusted so as to give best agreement between predictions and experimental results. The plane mixing layer, the plane jet and the radial jet can be predicted with a single set of constants; for the round jet a different set has to be used. This suggests a dependence of the otherwise universal constants on the ratio ofL to the radiusr. Comparisons are presented of predicted and measured rates of spread, profiles forU, k and $$\overline {u'\upsilon '} $$ , and energy balances. For most cases the agreement is within the experimental accuracy.