Two-scale semiempirical theory of turbulent boundary layers and jets

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 _t (for example, [7]), in which the essence of the interaction reduces to diffusion of _t 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.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 8, pp. 17–25, November–December, 1982.     

Energy-balance equation for turbulent pulsations in the theory of the free turbulent boundary layer

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.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 75–79, November–December, 1970.     

Boundary layer in the vicinity of the stagnation point of a body of axial symmetry in a turbulent stream

The flow in the boundary layer in the vicinity of the stagnation point of a flat plate is examined. The outer stream consists of turbulent flow of the jet type, directed normally to the plate. Assumptions concerning the connection between the pulsations in velocity and temperature in the boundary layer and the average parameters chosen on the basis of experimental data made it possible to obtain an isomorphic solution of the boundary layer equations. Equations are obtained for the friction and heat transfer at the wall in the region of gradient flow taking into account the effect of the turbulence of the impinging stream. It is shown that the friction at the wall is insensitive to the turbulence of the impinging stream, while the heat transfer is significantly increased with an increase in the pulsations of the outer flow. These properties are confirmed by the results of experimental studies [1–4].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 83–87, September–October, 1973.     

Calculation of the loss of the total pressure in channels with variable cross sectional area in the presence of turbulent mixing

The method frequently used to estimate the loss of the total pressure, i.e., summation of the losses associated with the individual elements (struts, expansion of the channel, etc.) is not sufficient, since it does not take into account the fact that the nonuniformlty of the flow resulting from flow separation at a particular element and the associated losses may be changed by subsequent profiling of the channel. The refinement based on integration of the equations of continuity and motion in conjunction with an equation for the turbulent viscosity, as was done, for example, by Lanyuk [1] for nonuniform flow in nozzles, is impossible in many cases of practical interest because of the complexity of the channel configurations and the boundary layer separations that occur. Sekundov [2] has given a method for calculating the mixing in channels used for definite purposes based on a decisive role in the mixing being played by the initial turbulence characteristics. In many cases, however, the turbulence characteristics and the mixing are determined by the local flow conditions. Under such an assumption, a simple method is derived in the present paper for calculating the loss of the total pressure with allowance for mixing in a channel of variable cross sectional area.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 38–45, September–October, 1982.     

Multicomponent chemically-reacting turbulent viscous shock layer on a catalytic surface

Turbulent flows past blunt bodies at high supersonic speeds are mainly investigated within the framework of the boundary layer model. However, even at large Reynolds numbers owing to the strong entropy gradient on the lateral surface it becomes necessary to take boundary layer corrections into account in the higher approximations [1]. The use of viscous shock layer theory makes it possible to obtain fairly accurate results over a broad interval of variation of the Reynolds numbers without organizing iterations with respect to vorticity and displacement thickness. The nonequilibrium nature of both homogeneous and heterogeneous catalytic reactions is taken into account. The results obtained are compared with the experimental data [2, 3]. Previously, in [4, 5] turbulent flow was investigated within the framework of viscous shock layer theory in the case of equilibrium homogeneous reactions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 144–149, March–April, 1989.     

Effect of external turbulence on the development of a turbulent jet

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.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 24–29, January–February, 1977.     

Method of mean-mass parameters for a three-dimensional boundary layer in a flow with vorticity

When a gas flows with hypersonic velocity over a slender blunt body, the bow shock induces large entropy gradients and vorticity near the wall in the disturbed flow region (in the high-entropy layer) [1]. The boundary layer on the body develops in an essentially inhomogeneous inviscid flow, so that it is necessary to take into account the difference between the values of the gas parameters on the outer edge of the boundary layer and their values on the wall in the inviscid flow. This vortex interaction is usually accompanied by a growth in the frictional stress and heat flux at the wall [2, 3]. In three-dimensional flows in which the spreading of the gas on the windward sections of the body causes the high-entropy layer to become narrower, the vortex interaction can be expected to be particularly important. The first investigations in this direction [4–6] studied the attachment lines of a three-dimensional boundary layer. The method proposed in the present paper for calculating the heat transfer generalizes the approach realized in [5] for the attachment lines and makes it possible to take into account this effect on the complete surface of a blunt body for three-dimensional laminar, transition, or turbulent flow regime in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 80–87, January–February, 1981.     

Distribution of radiant thermal flux over the surface of a sphere for hypersonic flow of a nonviscous radiating gas past the sphere

In the present study using the Newtonian approximation [1] we obtain an analytical solution to the problem of flow of a steady, uniform, hypersonic, nonviscous, radiating gas past a sphere. The three-dimensional radiative-loss approximation is used. A distribution is found for the gasdynamic parameters in the shock layer, the withdrawal of the shock wave and the radiant thermal flux to the surface of the sphere. The Newtonian approximation was used earlier in [2, 3] to analyze a gas flow with radiation near the critical line. In [2] the radiation field was considered in the differential approximation, with the optical absorption coefficient being assumed constant. In [3] the integrodifferential energy equation with account of radiation was solved numerically for a gray gas. In [4–7] the problem of the flow of a nonviscous, nonheat-conducting gas behind a shock wave with account of radiation was solved numerically. To calculate the radiation field in [4, 7] the three-dimensional radiative-loss approximation was used; in [5, 6] the self-absorption of the gas was taken into account. A comparison of the equations obtained in the present study for radiant flow from radiating air to a sphere with the numerical calculations [4–7] shows them to have satisfactory accuracy.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 44–49, November–December, 1972.In conclusion the author thanks G. A. Tirskii and É. A. Gershbein for discussion and valuable remarks.     

Use of the turbulence energy equation in the theory of jet flows

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.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 75–81, March–April, 1973.     

Generation and direct viscous dissipation of turbulent energy

Existing information about the generation and viscous dissipation of turbulent energy is based, as a rule, on the Laufer test data obtained for fluid flow in circular tubes at two Reynolds numbers (5 · 10⁵ and 5 · 10⁴). Computational dependences are presented herein for the generation and viscous dissipation of turbulent energy, common over the whole stream section and for the whole range of variation of the Reynolds number. The equation of the average energy balance during fluid flow in a circular tube and a flat channel is solved taking account of the equation of motion and the turbulent friction profile obtained by the author [1]. The computational dependences satisfy all the evident boundary conditions, agree with the Laufer test results [2] and yield a well-founded passage to the limit modes of average turbulent motion.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 30–36, November–December, 1973.     

Occurrence of Tolman-Schlichting waves in the boundary layer under the effect of external perturbations

Under small external perturbations, the initial stage of the laminar into turbulent flow transition process in boundary layers is the development of natural oscillations, Tolman-Schlichting waves, which are described by the linear theory of hydrodynamic stability. Subsequent nonlinear processes start to appear in a sufficiently narrow band of relative values of the perturbation amplitudes (1–2% of the external flow velocity) and progress quite stormily. Hence, the initial linear stage of relatively slow development of perturbations is governing, in a known sense, in the complete transition process. In particular, the location of the transition point depends, to a large extent, on the spectrum composition and intensity of the perturbations in the boundary layer, which start to develop according to linear theory laws, resulting in the long run in destruction of the laminar flow mode. In its turn, the initial intensity and spectrum composition of the Tolman-Schlichting waves evidently depend on the corresponding characteristics of the different external perturbations generating these waves. The significant discrepancy in the data of different authors on the transition Reynolds number in the boundary layer on a flat plate [1–4] is probably explained by the difference in the composition of the small perturbing factors (which have not, unfortunately, been fully checked out by far). Moreover, it is impossible to expect that all kinds of external perturbations will be transformed identically into the natural boundary-layer oscillations. The relative role of external perturbations of different nature is apparently not identical in the Tolman-Schlichting wave generation process. However, how the boundary layer reacts to small external perturbations, under what conditions and in what way do external perturbations excite Tolman-Schlichting waves in the boundary layer have practically not been investigated. The importance of these questions in the solution of the problem of the passage to turbulence and in practical applications has been emphasized repeatedly recently [5, 6], Only the first steps towards their solution have been taken at this time [4, 7–10], Out of all the small perturbing factors under the real conditions of the majority of experiments to investigate the flow stability and transition in the case of smooth polished walls, three are apparently most essential, viz.: the turbulence of the external flow, acoustic perturbations, and model vibrations. In principle, all possible mechanisms for converting the energy of these perturbations into Tolman-Schlichting waves can be subdivided into two classes (excluding the nonlinear interactions which are not examined here): 1) distributed wave generation in the boundary layer; and 2) localized wave generation at the leading edge of the streamlined model. Among the first class is both the possibility of the direct transformation of the external flow perturbations into Tolman-Schlichting waves through the boundary-layer boundary, and wave excitation because of the active vibrations of the model wall. Among the second class are all possible mechanisms for the conversion of acoustic or vortical perturbations, as well as the vibrations of the streamlined surface, into Tolman-Schlichting waves, which occurs in the area of the model leading edge.Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 5, pp. 85–94, September–October, 1978.     

Calculation of principal characteristics of turbulent streams in a state of structural equilibrium

A review of articles on the study of turbulent streams having transverse displacement, in which a turbulent energy balance equation is used, is contained in [1]. Levin [2] proposed a certain development of Rotta's method [3] making it possible to determine the characteristics of the average flow and the radial distribution of pulsation magnitudes. However, in this article the scale of the turbulence (the quantityl) was given as an empirical function of the coordinates. At the same time it is clear that the distribution of the turbulence scale depends on the conditions of the problem. A special differential equation proposed in [4,5] describing the variation in time and space of the quantityl has the drawback that in deriving this equation it is necessary to invoke additional hypotheses which are difficult to test experimentally. In the present article, along with the velocity of the average flow, the pressure, and the pulsation magnitudes, the scale of the turbulence is considered as an important characteristic of the stream, determined by the reference system which consists of the Reynolds equations, continuity equations, and equations for the component of the Reynolds stress tensor. Rotta's approximate semiempirical relations and an experimental relation for the single-point correlation coefficient between the turbulent pulsations in velocity are used for closure of the system obtained. An approximate calculation is given for the principal average and pulsation characteristics of the flow for the region of the stream where the turbulence is in a state of structural equilibrium [6]. A comparison of the calculated and experimental data is presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 95–99, January–February, 1973.     

Influence of turbulent Prandtl number on heat transfer of a flat plate

In computations involving heat transfer in turbulent flow past bodies it is necessary to assume turbulent Prandtl number distribution across the boundary layer. A review and comparison of results obtained by different authors are given, e.g., in [1–5]. Unfortunately, the existing data are so contradictory that, at present, it does not appear to be possible to establish reliably a function that determines turbulent Prandtl number distribution across the boundary layer. The absence of sufficiently reliable and general results on the distribution of turbulent Prandtl number led to the result that in the majority of studies conducted in earlier years its value was assumed a constant and either close to or equal to one. The effect of turbulent Prandtl number on the intensity of heat transfer from a flat plate is numerically investigated in the present paper. The thermal, turbulent boundary layer equation is integrated for this purpose at different values of turbulent Prandtl number and results are compared with experimental data. Results from [6], where the thermal boundary layer was numerically integrated with Pr_t=1 and compared with experimental data, were used for comparison in the present paper. The same numerical integration procedure as in [6] was used here.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 81–85, July–August, 1984.     

Theoretical investigation of coherent structures in stratified turbulent channel flow

A theoretical model is constructed of turbulent stratified flow in a flat horizontal channel with allowance for coherent structures that arise in it. The ordered part is separated from the turbulence of the flow and to describe the Reynolds-type equations are derived. The remaining part of the turbulence is taken into account parametrically in the form of an effective exchange coefficient. The flow is divided into a core, in which the ordered structures are manifested quite clearly, and wall regions, in which ordered large-scale structures are weakly manifested. To study the coherent structures in the core of the flow, an approach analogous to one already used to model ordered structures in open flows [4] is used. Monin-Obukhov scaling theory is used to describe the turbulence in the wall region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 32–38, May–June, 1991.     

Model of two-dimensional vortex motion with an entrainment mechanism

A model for describing the vertically averaged vortex motions of an incompressible viscous fluid with an arbitrary vertical structure of the bottom Ekman boundary layer is proposed. An approach similar to that adopted in [1] is used: the second moments of the deviations from the average velocities required in order to close the vorticity equation are calculated by means of the Ekman solution for gradient flows, which makes it possible to take the integral bottom boundary layer effect into account. As a result, these terms lead to a specific form of nonlinear friction with a coefficient that depends on the vorticity of the average flow. In the particular case of a constant vertical turbulent transfer coefficient the inaccuracies of the model described in [1] can be eliminated. The generalized vorticity equation obtained has solutions of the vorticity spot type with a uniform internal vorticity distribution, which can be effectively investigated by means of appropriate algorithms [2]. The mechanism of entrainment at the vorticity front is illustrated with reference to the example of the evolution of vorticity spots. An exact solution of the problem of the evolution of an elliptic vortex (generalized Kirchhoff vortex), which in the case of fairly strong anticyclonic vorticity degenerates first into a line segment (vortex sheet) and then into a point vortex, is constructed. Equations describing the dynamics of an elliptic vorticity spot in an external field with a linear dependence of the velocity on the horizontal coordinates and generalizing the classical Chaplygin-Kida model [3, 4] are constructed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 49–56, November–December, 1992.     

Excitation of Tollmien-Schlichting waves py a vibrating section of a surface with flow round it

In the context of the problem of describing the transition of a laminar boundary layer to a turbulent, great interest attaches to the study of susceptibility, i.e., of the reaction of the flow to various external influences, such as acoustic perturbations, surface roughness, vibration of the wall, turbulence of the unperturbed flow, etc. A general property of the effect of the factors mentioned above on the flow in a laminar boundary layer was discovered in experimental and numerical studies and is noted in [1]: in all cases an external forcing perturbation leads to the excitation of normal modes of oscillation in the boundary layer which propagate downstream, namely, Tollmien-Schlichting waves. There is an analytical calculation in [2, 3] of the amplitude of a wave excited by harmonic oscillations of a narrow band on the surface of a plane plate, the Reynolds number having been assumed to be infinitely large, and the frequency of the vibrator corresponding to the neighborhood of the lower branch of the neutral cuirve [4], In [5] the amplitude of the wave of instability generated is calculated by the method of expansion of the solution in a biorthogonal system of eigenfunctions. The amplitudes of the Tollmien-Schlichting waves are calculated below by means of a generalization of the method of [2] for the whole range of Reynolds numbers and frequencies of the vibrator corresponding to the region of instability: for moderate Reynolds numbers the problem is solved numerically, while for large Reynolds numbers an asymptotic solution is constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 46–51, July–August, 1987.The author is grateful to M. N. Kogan and V. V. Mikhailov for useful discussions of the results of the study.     

Interaction of an elastic boundary with a viscous sublayer of a turbulent boundary layer

The boundary region of a turbulent boundary layer contributes greatly to the drag. Intense turbulence is generated in this region. Below we investigate the interaction of an elastic boundary with a viscous sublayer for a decrease in the Reynolds stresses, and for a corresponding decrease in the drag. It does not seem possible to investigate the general case. Therefore, the problem is solved within the framework of the limitations made by Sternberg [1] for the theory of a viscous sublayer in a turbulent flow near a solid smooth wall.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 58–62, May–June, 1971.The authors thanks G. S. Migirenko for advice and remarks given during a discussion of the work.     

Calculation of a turbulent two-phase isobaric jet

A numerical calculation is carried out by the finite-difference method based on proposed equations for a turbulent submerged jet containing an admixture of solid particles. The relative longitudinal particle velocity and the influence of particles on the turbulence intensity are taken into account. The calculated results adequately agree with available experimental data. A turbulent two-phase jet is examined in [1] on the basis of the theory for a variable density jet, assuming equal mean velocities for the gas and particles and not considering the influence of particles on the turbulence intensity. Particles are analogously taken into account by a noninertial gas mixture in [2, 3], and a particle Schmidt number of 1.1 is assumed in [4]. A model is proposed in [5] which takes into account the influence of particles on the turbulence intensity of the gas phase. Problems concerning the initial and main sections of a submerged jet were solved in [6] by the integral method on the basis of this model and the assumed equality of the mean velocities of the gas and particles. Turbulent mixing of homogeneous two-phase flows with allowance made for dynamic nonequilibrium of the phases is considered in [7]. However, the neglect of turbulent transfer of particle mass and momentum led to a physically unrealistic solution for the particle concentration in the far field of the mixture. A two-phase jet is considered in the present work on the basis of the theory of a two-velocity continuous medium [8, 9] with allowance made for turbulent transfer of particle mass and momentum. The influence of particles on the turbulence intensity of the gas phase is taken into account with the model of [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 57–63, September–October, 1976.The author acknowledges useful comments and discussion.of the work by G. N. Abramovich and participants of his seminar. The author sincerely thanks I. N. Murzinov for scientific supervision of the work.     

Characteristics of an unstably localized disturbance in a compressed boundary layer

Experiments have demonstrated [1] that the transition of streamline-type flow into turbulent flow in a boundary layer occurs as a result of the formation and development of turbulent spots apparently arising from small natural disturbances. A study of the nonlinear evolution and interaction of localized disturbances requires knowledge of their characteristics to a linear approximation [2]. In the current work, results are presented of calculations of such characteristics for the first two unstable modes in a supersonic boundary layer on a two-dimensional plate (M = 4.5, T_w = 4.44).Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 50–53, January–February, 1976.     

一类平衡大气边界层边界条件研究

基于标准k-ε湍流模型,首先利用湍流粘度方程和剪切应力在整个边界层内恒定的假设,推导出一类耗散率表达式,并根据常用的湍动能入口剖面方程以及平均风速剖面方程,计算获得相应的耗散率方程;然后在输运方程中添加自定义源项,通过已经确定的平均速度方程、湍动能方程、耗散率方程计算得到相应输运方程的自定义源项表达式,并进行空风洞数值模拟,从而得到了一类满足平衡大气边界层的来流边界条件．通过将这种边界条件与由湍流平衡条件得到的边界条件进行比较,表明本方法获得的边界条件更适用．并且,本方法无需考虑修正壁面函数和修正湍流模型常数,因而计算更为简单,可为平衡大气边界层的研究提供一种新的思路．