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This paper continues the description of investigations started in [1], where we substantiate the gradient correlation between the second and third two-point moments of isotropic turbulence and give the results of calculations of its degeneration, as well as the results of calculation of turbulence in a uniform shear flow, with the turbulence assumed to be isotropic. A semiempirical model of turbulence in a uniform shear flow is constructed which is based on the equations for the second two-point moments assuming the axial symmetry of the statistical characteristics of turbulence. Use is made of the theory of axisymmetric turbulence [2]. The derived equations are closed using the gradient formula suggested in [1]. The suggested model enables one to find the kinetic energy of pulsation motion and its components (on the assumption of axial symmetry), two-point correlation moments, and all known scales of turbulence that are determined on the basis of two-point moments. The calculation results are in adequate agreement with the experimental data.

Thek-kl-model of turbulence based on the equation of second two-point moments (pointsA andB) of the fluctuating velocity field is presented. The second and third moments entering into this equation are expressed using polynomials whose terms are products of the tensor components characterizing a given turbulent motion and scalar functions of the distanceAB. ForAB=0 the equation obtained gives thek-turbulence energy balance equation and, on being integrated overAB from 0 to ∞, the transport equation for thekl-quantity (l is the integral turbulence scale). The model is used for calculating mixing layers, plane and circular jets, the wake behind a cylinder, tube and channel flows, and the boundary layer on a plate. The results of all the calculations agree well with the experimental data for a single set of empirical coefficients.

In this study some of the assumptions introduced in [1] in developing a closed system of equations for a turbulent boundary layer will be simplified. With the aid of the system of equations of [1], a theoretical solution is found for the problem of a jet in an accompanying flow, it being assumed that the structure of the jet turbulence depends solely on local conditions. Experiment has shown that the turbulence in such a jet does depend also on the prehistory of the flow. At large distances from the source, the theoretical characteristics of the jet agree well with the experimentally determined characteristics of the wake beyond a body. Also examined is the problem of the boundary layer between two homogeneous flows, flowing with different velocities.

Processes of turbulent mixing of hot nitrogen and carbon dioxide gas (CO2) in a supersonic nozzle are investigated in connection with gasdynamic lasers with selective thermal excitation. The actual flow pattern in the nozzle and cavity is simulated by the system of equations of a turbulent boundary layer. The results of calculations of the gain profile, the CO2 concentration, and the laser power agree satisfactorily with experimental data.

The article makes a comparison between some of the characteristics of a turbulent boundary layer, calculated in [1, 2], with experimental data. The comparison shows, specifically, that the “mixing path” entering into the definition of the coefficient of turbulent viscosity, for different flows is not proportional to the integral scale of the turbulence, defined with respect to the correlation function.

The effect of external turbulent agitation on jet development has been investigated in [1–3]. The difference of the method employed in the present work lies in the assumption that the turbulence scale of the external flow is substantially larger than the turbulence scales in either the jet or the mixing layer. Utilizing this assumption, it becomes possible to solve separately the energy equations for the turbulence of the external flow and of the jet. Solutions obtained on the basis of this assumption are found to be in qualitative agreement with experimental data.

To characterize the turbulence of boundary layers in the energy-bearing interval of wave numbers several turbulence scales are sometimes used (for example, [1, 2]). In particular, the universality of the semiempirical model of turbulence [2] can be extended in this way. A turbulence model with one equation (energy balance of the turbulence) has been constructed and used [3–6] and it has been established that the number of problems that can be solved for a universal choice of the values of the empirical coefficients increases appreciably if not one but two turbulent scales are used. In the present paper, it is shown that the introduction of a second scale makes it possible to take into account the interaction of shear layers in flows with two shear layers (for example, a channel or jet), and also to take into account the influence of turbulence of an external flow on a boundary layer. The interaction of shear layers is taken into account in theories containing a transport equation for the turbulent frictional stress τt (for example, [7]), in which the essence of the interaction reduces to diffusion of τt from layer to layer. In the present paper, a predominant volume interaction effect is assumed. It takes the form of a difference between the interaction of large-scale vortices with a shear deformation motion in flows with one and two shear layers, and also in the presence of turbulence in an external flow.

In [1, 2] turbulence of the external flow was taken into account by specifying the turbulent energy at the external boundary of the boundary layer on integrating the energy-balance equation for the turbulence. In [3] a special correction that allowed the turbulence of the external flow to be taken into account was introduced in determining the mixture path. In [4, 5] the turbulent energy calculated from the energy-balance equation of the turbulence was added to the energy induced by turbulence of the external flow, the energy distribution of the induced turbulence being specified using an empirically selected function. In [6, 7] a method of taking into account the effect of turbulence of the external flow on a layer of mixing and a jet was proposed. In the present work, this method is applied to the boundary layer at a plate.

The turbulent flow of an incompressible fluid is considered in a plane channel, a circular tube, and the boundary layer on a flat plate. The system of equations describing the motion of the fluid consists of the Reynolds equations and the mean kinetic energy balance equation for turbulent fluctuations. On the basis of an analysis of experimental data, hypotheses are formulated with respect to the eddy kinematic viscosity e and lengthl entering into the expression for specific dissipation of turbulent energy into heat. It is assumed that in the central (outer) region of the flow in a channel, e andl are constants, and expressions are taken for them which are used for a free boundary layer; near the walll varies linearly and e almost linearly. Results of calculations of the turbulent energy distribution, the mean velocity, and the drag coefficient are in good agreement with the existing experimental data. The values of two empirical coefficients, which enter into the system of equations as the result of the hypotheses, are close to those obtained for a free boundary layer.

We examine the discharge of a turbulent jet from a round opening into an infinite uniform crossflow and find the form of the jet centerline and the distribution of the maximum velocities in the jet along the axis. It is shown that the calculated jet axes and velocity distributions agree well with the experimental values for different ratios of the velocity at the source exit to the crossflow velocity and for different angles of entry of the jet into the crossflow. The study [1] formed the basis for the proposed semiempirical theory.

Semiempirical expressions are proposed for the coefficient of turbulent viscosity and for the scale of turbulence in the equations for the free turbulent boundary layer in an incompressible fluid, these equations consisting of the equation of continuity, the equations of motion, and the equation for the average energy balance in the turbulent pulsations. The advantage of the expressions over the existing ones is that the two empirical constants in the equations have nearly the same values for circular and plane turbulent streams and also for a turbulent boundary layer at the edge of a semiinfinite homogeneous flow with a stationary fluid. The mean-energy distribution and the mean energy of the turbulent pulsations computed in this paper agree well with the experimental values.

It is shown that the problem of the mixing of a jet flowing out of an infinitely long slit-type source, with a homogeneous stream flowing parallel to the source of a jet, is analogous to the problem of the propagation of heat in a plane jet.