Turbulent friction in the flow of liquid in tubes and channels

A calculating relationship is presented for turbulent flow; it takes a unique form over the whole cross section of the flow. A relationship is also derived between turbulent friction and the mean velocity profile on the basis of the equation for the maximum turbulent friction, which follows directly from the equation of motion. The proportionality factor in this relationship is obtained with due allowance for twelve boundary conditions relating to the turbulent flow, the mean velocity, and their derivatives. The resultant turbulent-friction profiles agree with the experimental data of Laufer. The profile parameters may be related to the Reynolds number.Leningrad. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 140–145, March–April, 1972.     

Experimental investigation of the probability density of temperature fluctuations in nonisothermal turbulent flows

The results of measuring the probability density of the temperature fluctuations in a weakly heated submerged turbulent jet are given for the range of Reynolds numbers extending from 7.7·10³ to 5·10⁵. It is shown that for large positive fluctuations the probability density is independent of the Reynolds number and is formed by bursts with a characteristic length of the order of the integral scale.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 59–68, September–October, 1996.     

Use of turbulent energy balance equation in the theory of turbulent flows along walls

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 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, andl are constants, and expressions are taken for them which are used for a free boundary layer; near the walll varies linearly and 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.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–33, May–June, 1973.     

Numerical investigation of the flow past axisymmetric bodies with surface injection

Incompressible viscous gas flow past axisymmetric bodies with surface injection is numerically investigated on the range of Reynolds numbers from 10 to 2·10⁵.Izhevsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 162–166, July–August, 1996.     

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.     

An application of the k-ɛ model with variable Prandtl number to heat transfer computations in air flows

The two-equation `low Reynolds number' k-? model of turbulence with a set of universal constants suggested by Launder and Sharma is modified in the present paper. The variability of the turbulent Prandtl number Pr_t in the energy equation is assumed along with a change of a constant in the dissipation term of the turbulent kinetic energy equation. The turbulent heat transfer is computed for an air flow in a circular pipe for the Reynolds number within the range of 10⁴?4. The modification considerably improves the agreement between the numerical results and the experiment data published in the available literature.     

基于涡粘性模型的翼型湍流流场熵产率计算

利用有限体积法实现了基于非正交同位网格的SIMPLE算法。基于熵分析方法,采用涡粘性模型求解湍流熵产方程,系统研究了湍流模型对二维翼型绕流流场熵产率的影响。通过计算NACA0012翼型在来流雷诺数为2.88×10⁶时,0°攻角~16.5°攻角范围内的翼型表面压力系数分布和升阻力特性,验证了算法及程序的正确性。结果表明,选择不同湍流模型时,翼型流场熵产的计算结果存在差异,湍流耗散是引起流场熵产的主要原因;翼型流场的熵产主要发生在翼型前缘区、壁面边界层和翼型尾流区域,流场熵产率与翼型阻力系数线性相关;当产生分离涡时,粘性耗散引起的熵产下降。     

Angular vibrations of an ellipsoid of revolution in a confined viscous fluid

Vibrations of bodies in confined viscous fluids have been studied on a number of occasions, transverse vibrations of rods being the main subject of investigation [1–3]. The present authors [4] have considered the general problem of translational vibrations of an axisymmetric body in an axisymmetric region containing a low-viscosity fluid. The present paper follows the same approach and considers the problem of small angular vibrations of an ellipsoid of revolution in a circular cylinder with flat ends. In the general case, the hydrodynamic coefficients in the equation of motion of the ellipsoid are determined numerically for different values of the dimensionless geometrical parameters using Ritz's method. In the case of an unconfined fluid, analytic dependences in terms of elementary functions are obtained for the hydrodynamic coefficients. The theoretical results agree well with experimental investigations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 34–39, March–April, 1984.     

Simulation of two-dimensional turbulence in a planar channel

A study is made of the two-dimensional flow of an incompressible viscous fluid in a planar channel at supercritical Reynolds number Re = 10⁴. Calculation of the flow over an appreciable time interval leads to the establishment of a statistically steady flow regime and stabilization of its averaged characteristics (the profile of the mean velocity, the mean pressure gradient, the pulsation energy, etc.). The calculations show that numerical simulation of turbulent flow of an incompressible fluid on the basis of the Navier-Stokes equations leads to qualitatively correct characteristics of this flow regime.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 26–35, November–December, 1982.We are very grateful to Yu. L. Levitan, B. D. Moiseenko, V. K. Sidorova, and I. N. Simakin, collaboration with whom helped us to do this work.     

Numerical investigation of the hydrodynamic drag of a circular cylinder coated with a thin layer of magnetic fluid

In order to reduce the drag of bodies in a viscous flow it has been proposed to apply to the surface exposed to the flow a layer of magnetic fluid, which can be retained by means of a magnetic field and thus act as a lubricant between the external flow and the body [1, 2]. In [1] the hydrodynamic drag of a current-carrying cylindrical conductor coated with a uniform layer of magnetic fluid was theoretically investigated at small Reynolds numbers. In order to simplify the equations of motion, the Oseen approximation was introduced for the free stream and the Stokes approximation for the magnetic fluid [3]. This approach has led to the finding of an exact analytic solution from which it follows that at Reynolds numbers Re 1 the drag of the cylinder can be considerably reduced if the viscosity of its magnetic-fluid coating is much less than the viscosity of the flow. The main purpose of the present study is to investigate, with reference to the same problem, how the magnetic-fluid coating affects the hydrodynamic drag at Reynolds numbers 1 Re 10²–10³, i.e., under separated flow conditions. In this case the simplifications associated with neglecting the nonlinear inertial terms in the Navier—Stokes equation are inadmissible, so that a solution can be obtained only by numerical methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 11–16, May–June, 1986.     

Anisotropic Organised Eddy Simulation for the prediction of non-equilibrium turbulent flows around bodies

The unsteady turbulent flow around bodies at high Reynolds number is predicted by an anisotropic eddy-viscosity model in the context of the Organised Eddy Simulation (OES). A tensorial eddy-viscosity concept is developed to reinforce turbulent stress anisotropy, that is a crucial characteristic of non-equilibrium turbulence in the near-region. The theoretical aspects of the modelling are investigated by means of a phase-averaged PIV in the flow around a circular cylinder at Reynolds number 1.4×10⁵. A pronounced stress–strain misalignment is quantified in the near-wake region of the detached flow, that is well captured by a tensorial eddy-viscosity concept. This is achieved by modelling the turbulence stress anisotropy tensor by its projection onto the principal matrices of the strain-rate tensor. Additional transport equations for the projection coefficients are derived from a second-order moment closure scheme. The modification of the turbulence length scale yielded by OES is used in the Detached Eddy Simulation hybrid approach. The detached turbulent flows around a NACA0012 airfoil (2-D) and a circular cylinder (3-D) are studied at Reynolds numbers 10⁵ and 1.4×10⁵, respectively. The results compared to experimental ones emphasise the predictive capabilities of the OES approach concerning the flow physics capture for turbulent unsteady flows around bodies at high Reynolds numbers.     

Optimization of body shape at small reynolds numbers

The problem of the optimization of the shape of a body in a stream of viscous liquid or gas was treated in [1–5]. The necessary conditions for a body to offer minimum resistance to the flow of a viscous gas past it were derived in [1], The necessary optimality conditions when the motion of the fluid is described by the approximate Stokes equations were derived in [2], The shape of a body of minimum resistance was found numerically in [3] in the Stokes approximation. The optimality conditions when the motion of the fluid is described by the Navier—Stokes equations were derived in [4, 5], and in [4] these conditions were extended to the case of a fluid whose motion is described in the boundary-layer approximation. The necessary optimality conditions when the motion of the fluid is described by the approximate Oseen equations were derived in [5] and an asymptotic analysis of the behavior of the optimum shape near the critical points was performed for arbitrary Reynolds numbers.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp, 87–93, January–February, 1978.     

Spectrum of developed turbulence of incompressible viscoelastic fluids

It is shown that fine-scale turbulent motions of a viscoelastic fluid damp out as in a viscous fluid with some effective viscosity dependent on the scale of the motion. The elasticity of deformation results in a diminution in the dissipativity of the turbulence, and hence, to an elongation of the high-frequency tail of the spectrum for a given energy influx.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 23–33, January–February, 1972.     

Experimental investigation of the heat exchange in separation zones ahead of cylindrical obstacles

Results of an experimental investigation of the heat exchange in turbulent boundary layer separation zones ahead of cylindrical obstacles at a subsonic air stream velocity are elucidated. The investigation was conducted for changes in the ratio between the obstacle diameter and altitude between 0.25 and 4, between the boundary layer thickness at the separation point and the obstacle altitude between 0.09 and 0.7, between the Reynolds number computed by means of the free stream parameters and the obstacle height between 10⁴ and 4·10⁵. The Mach number reached 0.85. The temperature factor was 0.7. It is shown that the distribution of the heat transfer coefficients in the separation zone depends on the Reynolds and Euler numbers, the ratio between the boundary layer displacement thickness and the diameter (or altitude) of the obstacle, and the ratio between the diameter and the altitude. Criterial dependences are obtained which extend the heat-exchange results at characteristic points of the separation zones, as are also dimensionless distributions of the heat transfer coefficients to determine the heat fluxes on a plate in the plane of symmetry of the separation zone ahead of obstacles.Translated from Zhurnal Prikladnoi Mekhanikii Tekhnicheskoi Fiziki, No. 6, pp. 83–89, November–December, 1972.The authors are grateful to V. S. Avduevskii for discussing the research results.     

Turbulent boundary layer at the initial portion of a pipe with rough walls

The development of a turbulent boundary layer at the initial portion of a pipe with rough walls is considered in the framework of the boundary-layer theory. It is shown that the consideration of roughness can be carried out by introducing into the standard law of friction a function which takes into account this factor. An experimental investigation is carried out on a test portion of a pipe with natural roughness whose relative value equals 10^–3. The range of Reynolds numbers is 5.1 · 10⁴-3.4 · 10⁵. The method of calculation proposed here leads to results which satisfactorily agree with the data of the experimental investigation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 109–116, September–October, 1971.     

A new method for probing the turbulence scalar spectrum by ultrasonic sonar scanning

 A new method is proposed to obtain a turbulent scalar spectrum and the energy dissipation rate in turbulent flow from ultrasonic frequency scanning data. A scanning sonar with frequency varying from 0.5 MHz to 5 MHz has been used to directly probe the energy dissipation rate ɛ and the three-dimensional scalar spectrum E _θ(k). Experiments were conducted in a laboratory open-channel flow in clear water with Reynolds numbers varying from 1.2×10⁵ to 6.9×10⁵. Good agreement is found between measured spectra and those predicted by the Batchelor theory. The energy dissipation rates compare favourably with those obtained from acoustic Doppler velocimeter measurements. Received: 20 March 1997/Accepted: 27 September 1997     

Higher approximations in the problem of axisymmetric jet flow on a circular cylinder

A study is made of a laminar jet of incompressible fluid moving along the generator of an infinite circular cylinder at moderate Reynolds numbers. An asymptotic solution is constructed that takes into account the influence of the curvature of the surface and the interaction of the boundary layer with the outer flow. Comparison with experimental data indicates that the obtained solution is applicable for 0.8·10³ Re_x* 10⁴.Translated from Izvestiya Akademli Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 21–27, September–October, 1979.     

Direct numerical simulation of the turbulent energy balance and the shear stresses in power-law fluid flows in pipes

The results of direct numerical simulation of turbulent flows of non-Newtonian pseudoplastic fluids in a straight pipe are presented. The data on the distributions of the turbulent stress tensor components and the shear stress and turbulent kinetic energy balances are obtained for steady turbulent flows at the Reynolds numbers of 10⁴ and 2×10⁴. As distinct from Newtonian fluid flows, the viscous shear stresses turn out to be significant even far from the wall. In power-law fluid flows the mechanism of the energy transport from axial to transverse component fluctuations is suppressed. It is shown that with decrease in the fluid index the turbulent transfer of the momentum and the velocity fluctuations between the wall layer and the flow core reduces, while the turbulent energy flux toward the wall increases. The earlier-proposed models for the average viscosity and the non-Newtonian one-point correlations are in good agreement with the data of direct numerical simulation.     

Subsonic Radiation Gas Dynamics in a Laser Plasma Generator Channel

The radiation gasdynamic processes in the channel of an air laser plasma generator operating at atmospheric pressure are analyzed. In the multigroup approximation a numerical radiation gasdynamic model is formulated on the basis of the equations of motion of a viscous heat-conducting gas and the selective thermal radiation transport equation. Laminar and turbulent subsonic generator operation regimes are considered.For the purpose of approximately describing the turbulent gas and plasma mixing the Navier-Stokes equations averaged after Reynolds and the k-ε turbulence model are used. The problem is solved in the time-dependent two-dimensional axisymmetric formulation.Strong radiation-gasdynamic interaction regimes are investigated. In these regimes the energy losses due to radiation from the high-temperature region of the laser plasma and the absorption of its thermal self-radiation by the surrounding plasma and gas layers (radiation reabsorption) appreciably affect the gasdynamic flow structure. Two methods of integrating the selective thermal radiation transport equation in the generator channel are discussed. In one of these the thermal radiation transport is calculated inside the heated volume and in the other the radiation heat fluxes are calculated on the surfaces bounding the volume. The results of calculating the spectral and integral radiation heat fluxes on the inner surface of the generator are given.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 126–143.Original Russian Text Copyright © 2005 by Surzhikov.     

Effect of viscous dissipation on mixed convection heat transfer in a vertical tube with uniform wall heat flux

 The laminar and parallel flow of a Newtonian fluid in a vertical cylindrical duct with circular cross section has been analysed. Both the viscous dissipation effect and the buoyancy effect have been taken into account. The momentum balance equation and the energy balance equation have been solved by means of a perturbation method, in the case of a uniform heat flux prescribed at the wall of the duct. The velocity distribution, the temperature distribution, the Nusselt number and the Fanning friction factor have been evaluated analytically. Moreover, the velocity and temperature of the fluid have been compared with those obtained in two special cases: forced convection with viscous dissipation (i.e. negligible buoyancy effect); mixed convection with negligible effects of viscous dissipation. Received on 26 June 2000