Some issues of turbulence statistics

The issue of dropping the random forcef _i and the arbitrariness of choosing the basic variable in the variational approach to turbulence closure problem, pointed out recently by the Russian scientists Bazdenkov and Kukharkin, are discussed. According to the mean-square estimation method, the random forcef _i should be dropped in the error expression of the LFP (Langevin-Fokker-Planck) model. However,f _i is not neglected, its effect has been taken into account by the variational approach. In order to optimize the perturbation solution of the Liouville equation, the LFP model requires that the basic variable is as near to Gaussian as possible. Hence, the velocity, instead of the vorticity, should be chosen as the basic variable in the three-dimensional turbulence. Although the LFP model and the zero-order Gaussian term of PDF (probability density function) imply whiteness assumption (zero correlation time off _i ), the higher-order non-Gaussian terms of PDF correspond to the nonwhiteness of turbulence dynamics, the variational approach does calculate the nonwhiteness effect properly. The work is supported by the National Basic Research Program “Non-linear Sciences” and the National Natural Science Foundation of China     

Some nonlinear problems in servomechanisms

Summary A mathematical model for a hydraulic servomechanism is constructed. It is shown that the model, in general, reduces to a nonlinear third-order equation of the formxx+(₁+xx+–₂ x=p(t). Under certain conditions imposed on the constants involved, it is proved that above equation possesses a periodic solution.     

Some problems in the Z-C-X space

The new concepts of the Z-C-X space and excellent cone are introduced. Some problems of random semiclosed 1-set-contractive operator are investigated in the Z-C-X space. At first, an important inequality is proved. Secondly, several new conclusions are proved by means of random fixed point index in the theory of random topological degree. A random solution of a class of random operator equations under conditions of imitating the parallelogram law is obtained, famous Altman’s theorem is generalized in partially ordered Z-C-X space. Therefore, some new results are obtained.     

Some problems in metallurgical fluid mechanics

Application of computational fluid mechanics in two areas of metallurgy is considered: solidification of liquid alloys and MHD turbulence. In the first class of problems, where the technical issue is to obtain a reasonably homogeneous composition of the cast, one has to consider not only the completely molten and solidified regions but also the “mushy zone” that is made up of small-scale dendrites, which appear between the first two regions. In the completely molten region, the composition is practically constant and the fluid is set into motion due to the inhomogeneous temperature field. In the mushy zone, on the other hand, solutal convection often dominates strongly over thermal convection. It is shown that laminar convection is of prime importance for the composition of the solidified alloy. In the second class of problems, two cases of turbulent MHD flows in cylindrical containers are considered: an electromagnetic furnace and an electromagnetic stirrer. In the electromagnetic furnace, the mean flow consists of two toroidal vortices. The mean motion in the electromagnetic stirrer is a swirling motion that is accompanied by a weak meridional circulation, which is reminiscent of that occuring in spin down phenomena. The MHD flows are computed by using large eddy simulation methodology with a new subgrid model of the Smagorinsky type that accounts for a variable mesh. Predictions from the all model computations are compared with experimental observations. In general, the agreement between theory and experiments is satisfactory.     

Some problems in plane elastic dielectrics

Representations of Galerkin type are obtained for the displacement vector, polarization vector and the potential fields in the static plane theory of elastic dielectrics using the method of associated matrices. Fundamental matrix solutions of an infinite elastic dielectric plane subjected to a concentrated body force, electric force and charge density are derived from the singular solutions of harmonic, biharmonic and Helmholtz equations. Using boundary operatorsY, Z, M, the fundamental matrix solutions, and Betti's formulae, a matrix (x, y) is constructed and an integral representation for (u ₁,u ₂,P ₁,P ₂, ) is obtained. Discontinuity theorems are stated for the double layer potential andQ operator of the single layer potential. By means of these theorems, the solutions of interior and exterior boundary value problems are reduced to the solution of a system of five singular integral equations. The index of one of the systems is shown to be zero and it is concluded that Fredholm theorems and its alternatives hold.

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Zusammenfassung Durch Anwendung der Assoziativ-Matrizen-Methode werden Galerkinische Darstellungen für den Verschiebungsvektor, den Polarisierungsvektor und die potentiellen Prüffelder einer statischen ebenen Theorie des elastischen Dielektrikums entwickelt. Von den singulären Lösungen der harmonischen, biharmonischen und Helmholtz-Gleichungen werden grundlegende Matrizenlösungen für das unendliche, ebene und elastische Dielektrikum, das durch konzentrierte Raumkräfte, elektrische Kräfte und Ladungsdichte beansprucht wird, abgeleitet. Unter Verwendung der Randwert-OperatorenY, Z, M, der grundlegenden Matrizen-Lösungen und Betti's Formel, wird eine Spezial-Matrize (x, y) konstruiert und eine Integraldarstellung für (u ₁,u ₂,P ₁,P ₂, ) erhalten. Unstetigkeitssätze werden für das Doppelschicht-Potential und denQ-Operator eines einschichtigen Potentials angeführt. Durch Anwendung dieser Sätze werden die Lösungen der inneren und äusseren Grenzwertprobleme zur Lösung eines Systems von fünf singulären Integralgleichungen reduziert. Der Index eines der Systeme wird als Null bewiesen und es wird die Schlussfolgerung gezogen, dass der Fredholmsche Satz und seine Alternativen fur diese Theorie anwendbar sind.

     

Application of the Reichardt model in problems of turbulence near the wall

A method is proposed for calculating turbulent boundary layers near the wall, based on the Reichardt semiempirical model of turbulent mixing. The article considers the problem of the turbulent boundary layer of a plate, including the case of supersonic flow around a plate, as well as the problem of the turbulent boundary layer with the nonisothermal flow of turbulent jets around a surface. Here there are introduced several almost self-similar solutions of the differential transfer equations, based on the assumption of the conservative nature of the profiles of the parameters with respect to a change in the sublayer. The results of the calculation are compared with experimental data.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 15–24, March–April, 1972.     

Some expériences and considerations on measuring turbulence in water with hot films

Results are presented of a four year experience with conical hot-film sensors for measurements of the longitudinal component of turbulence in water. Data are presented on the reduced sensitivity by fouling. Special consideration has been given to the amplitude response of these sensors. For scientific purposes careful calibration in the lower frequency range is required; RMS values may be obtained that are accurate to within 5%. With “technical clean” water, as has been used in rigs for industrial research, this accuracy may be within 10%. If dynamic calibration is not possible, existing theories may be used as a first good approximation if the geometry of the sensor is specified in detail.     

Some physical problems of cumulation

A review and analysis of some results related to the phenomenon of cumulation are given. The various models used to describe this phenomenon are considered. Lavrent’ev Institute of Hydrodynamics, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 10–26, September–October, 2000.     

Some problems for viscoelastic rods

Using an approach proposed in [3],we consider some linear and nonlinear problems forviscoelastic rods.Results in [1]for linear case are affirmed more elamentarily and simplyhere.We also treat linear problem with damping term and nonlinear problem with powernonlinearity Pu~3. Some new results are established.     

Some results of variational inclusion problems and fixed point problems with applications

This paper introduces a general iterative algorithm to approximate a common element in the solution set of quasi-variational inclusion problems and the common fixed point set of an infinite family of nonexpansive mappings. It is proven that the iterative sequences generated in the proposed iterative algorithm converge strongly to some common element in the framework of the real Hilbert spaces.     

Some issues associated with the intermediate space in single-crystal plasticity

In this study we show that some discussions of finite-deformation single-crystal plasticity are conceptually flawed in their focus on a set referred to as the intermediate configuration. Specifically, we prove that what is usually referred to as the intermediate configuration is not a configuration but instead a vector space that we term the intermediate space. We argue that when applied to single crystals this intermediate space represents the lattice.     

Some recent results on a shell model for 3D turbulence

In this note we report a series of recently published results (by R. Benzi, M.H. Jensen, G. Paladin, G. Parisi, A. Vulpiani and myself), on a Shell Model for the three dimensional cascade of energy in Fully Developed Turbulence. We present numerical and analytical computations of the intermittent corrections to the K41 scaling of the velocity increments in the inertial range.     

Some basic issues in traditional Eulerian formulations of finite elastoplasticity

In traditional Eulerian formulations of finite elastoplasticity, there are some basic issues that still need clarification and further investigation. Among them are the characterization of initial anisotropy for scalar- and tensor-valued constitutive functions, plastic consistency conditions involving time differentiation of anisotropic yield functions, a unified loading criterion for hardening, softening, and perfectly plastic behaviour, etc. Sometimes, it is thought that a satisfactory and complete treatment for these issues could not be achieved within a framework of traditional Eulerian formulation. In this work, efforts are made towards explaining and clarifying these basic issues. Introducing the notion of an Eulerian type rotation-conjugate group of the initial material symmetry group, we show that consistent Eulerian formulations of scalar- and tensorvalued constitutive functions may be achieved for any given type of initial anisotropy. With such consistent Eulerian formulations, we derive plastic consistency conditions in a corotating frame associated with the foregoing rotation-conjugate group. As to the loading conditions for hardening, softening, and perfectly plastic behaviour, we recall and study the unified criterion proposed by Hill [J Mech Phys Solids 6 (1958) 236]. It is pointed out that the tangential elastic stiffness tensor in this unified criterion has to fulfill rather complicated integrability conditions for the elastic rate equation with an objective stress rate. It appears to be far from being a simple or even trivial matter to give an explicit form of the tangential elastic stiffness tensor meeting these conditions. It is indicated that this issue may be resolved by using the logarithmic stress rate. With this rate, an explicit form of Hill's unified loading criterion, together with an explicit form of tangential elastic stiffness tensor, is presented in terms of a general complementary hyperelastic potential.