Some problems in boundary element programming of axisymmetric elasticity

Axisymmetric problems in elasticity can be reduced to two dimensional ones, but they are a little more complicated than plane problems. Therefore, some special problems will be encountered in the boundary element programming of axisymmetric elasticity. In this paper, the methods to treat these problems and some remarks are given according to our experience in programming. Numerical examples are presented for the checking of these treatments.     

Some axisymmetric problems for an elastic inhomogeneous thick-walled tube

An equilibrium differential equation for an axisymmetric problem is reduced to an integrable form under the assumption that the shear modulus is continuously differentiable and Poisson’s ratio is constant. A procedure of successive approximations is proposed for the case of a compressible material, and the Lamé problem is solved exactly for the case of an incompressible material. A piecewise continuous variation of the Lamé parameter as a function of radius is considered. Several examples of determining the stress-tensor components are given for various cases of inhomogeneity.     

On the obtaining of axisymmetric flows from plane-parallel flows

The obtaining of axisymmetric flows of incompressible and compressible fluids from plane-parallel flows by means of integral transformations relating harmonic and p-harmonic functions [1] is considered. A transformation is found that carries plane-parallel flows from elementary singularities into axisymmetric flows. It is shown that this transformation makes it possible to obtain the general form of the solution of axisymmetric problems of flow past bodies from the solution of plane-parallel problems.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 113–121, September–October, 1982.     

Computing axisymmetric,laminar internal flows

A simple computational scheme is developed to compute laminar flows inside axisymmetric ducts. It is based on the Keller box method where the equations are approximated at the centre of the downstream face of each computational box. The coupling between the pressure gradient and the velocities for internal flow has been observed to introduce stability problems for the Keller box method that are not present for external, boundary layer flow problems. The difference scheme for the velocities is coupled to an iterative scheme to solve for the pressure gradient at each axial step. Example results for developing flow in a pipe and in a 2° conical diffuser are presented.     

A class of steady axisymmetric incompressible flows

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 28–34, November–December, 1990.     

Wake development in turbulent subsonic axisymmetric flows

The development of the wake velocity and turbulence profiles behind a cylindrical blunt based body aligned with a subsonic uniform stream was experimentally investigated as a function of the momentum thickness of the approaching boundary layer and the transfer of mass into the recirculating region. Measurements were made just outside of the recirculating region at distances of 1.5, 2 and 3 diameters downstream of the cylinder. Results indicate that, even at these short distances from the cylinder base, the velocity profiles are similar. They also show that the width of the wake increases with the thickness of the boundary layer while the velocity at the centerline decreases. Near wake mass transfer was found to alter centerline velocities while the width of the wake was not significantly altered. Wake centerline velocity development as a function of boundary layer thickness is presented for distances up to three diameters from the base. This work was supported in part by the ‘Xunta de Galicia’ under Project No. XUGA20611B93.     

Coherent structures in countercurrent axisymmetric shear flows

The dynamical behaviors of coherent structures in countercurrent axisymmetric shear flows are experimentally studied.The forward velocity U1 and the velocity ratio R=(U1-U2)/(U1+U2),where U2 denotes the suction velocity,are consldered as the control parameters.Two kinds of vortex structures,i.e.,axisymmetric and helical structures,were discovered with respect to different reginmes in the R versus U1 diagram .In the case of U1 rangjing from 3 to 20m/s and R from 1 to 3,the axisymmetric structures plan an important role.Based on the dynamical behaviors of axisymmetric structures,a critical forward velocity U1cr=6.8m/s was defined,subsequently,the subcritical velocity regime:U1>U1cr and the supercritical velocity regime:U1<U1er,In the subcritical velocity regine,the flow system contains shear layer self-excited oscillations in a certain range of the velocity ratio with respect to any forward velocity.In the supercritical velocity regime,the effect of the velocity ratio could be explained by the relative movement and the spatial evolution of the axisymmetric structure undergoes the following stages:(1) Kelvin-Helmholtz instability leading to vortex rolling up,(2) first time vortex agglomeration.(3) jet colunn self-excited oscillation,(4) shear layer self-excited oscillation,(5)“ordered tearing“,(6) turbulence in the case of U1<4m/s (the “ordered tearing“ does not exist when U1>4m/s),correspondingly,the spatial evolution of the temporal asymptotic behavior of a dynamical system can be described as follows:(1) Hopt bifurcation,(5) chaos(“weak turbulence“)in the case of U1<4m/s(superharmonic bifurcation does not exist when U1>4m/s).The proposed new terms,superharmonic and reversed superbarmonic bifurcations,are characterized of the frequency doubling rather than the period doubling.A kind of unfamiliar vortices referred to as the helical structure was discovered experimentally when the forward velocity around 2m/s and the velocity range from 1.1 to 2.3,There are two base frequencies contained in the flow system and they could coexist as indicated by the Wigner-Ville-Distribution and the temporal asymptotic behavior of the dynamical system corresponding to the helical vortex could be described as 2-torus as indicted by the 3D reconstructed phase trajectory and correlation dimension.The scenario of the spatial evolution of helical structures could be described as follows:the jet column is separated into two parts at a certain spatial location and they entangle each other to form the helical vortex until the occurrence of those separated jet columns to reconnect further downstream with the result that the flow system evolves into turbulence in a catastrophic form.Correspondingly,the dynamical system evolves directly into 2-tiorus through the supercritical Hopf bifurcation followed by a transition from a quasi-periodic attractor to a strange attractor.In the case of U1=2m/s,the parametric evolution of the temporal asymptotic behavior of the dynamical system as the velocity ratio increases from 1 to 3 could be described as follows:(1)2-torus(Hopf bifurcation),(2) limit cycle(reversed Hopf bifurcation),(3) strange attractor (subbarmonic bifurcation).     

Some geometric acoustic problems of one-dimensional unsteady and two-dimensional steady flows

The behavior of discontinuities (weak shocks) of the parameters of a disturbed flow and their interaction with the discontinuities of the basic flow in the geometric acoustics approximation, when the variation of the intensity of such shocks along the characteristics or the bicharacteristics is described by ordinary differential equations, has been investigated by many authors. Thus, Keller [1] considered the case when the undisturbed flow is three-dimensional and steady, and the external inputs do not depend on the flow parameters. An analogous study was made by Bazer and Fleischman for the MGD isentropic flow of an ideal conducting medium [2], while Lugovtsov [3] studied the three-dimensional steady flow of a gas of finite conductivity for small magnetic Reynolds numbers and no electric field. Several studies (for example, [4]) have considered the behavior of discontinuities of the solutions from the general positions of the theory of hyperbolic systems of quasilinear equations. Finally, the interaction of weak shocks (or the equivalent continuous disturbances) with shock waves was studied in [5–11].In what follows we consider one-dimensional (with plane, cylindrical, and spherical waves) and quasi-one-dimensional unsteady flows, and also plane and axisymmetric steady flows. Two problems are investigated: the variation of the intensity of weak shocks in the presence of inputs which depend on the stream parameters, and the interaction of weak shocks with strong discontinuities which differ from contact (tangential) discontinuities.The thermodynamic properties of the gas are considered arbitrary. We note that the resulting formulas for the interaction coefficients of the weak and strong discontinuities are also valid for nonequilibrium flow.     

Calculation of cavity flows in an axisymmetric channel

The results are presented of a calculation of cavity flow in an axisymmetric channel with an annular obstacle. The problem was suggested to the author by G. B. Tsvetnov.The problem is solved by the method published in [1, 2].     

Two-layer flows of gas in supersonic axisymmetric nozzles

This article describes two methods for calculating two-layer flows. The first is a generalization of a numerical method for solving the inverse problem [1] for the case of two-layer flows, without taking mixing into account. The second is a method of characteristics, for calculating a two-layer flow in a supersonic nozzle. In this case, the usual method of characteristics is changed in such a way that it is possible to calculate a point on the interface between two layers having different adiabatic indices, and different total pressures and temperatures. This article also gives the results of calculation of two-layer flows in nozzles with different adiabatic indices and different ratios of the mass flow rates of the gas in the layers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 76–81, July–August, 1970.The calculations were programmed and carried out by G. D. Vladimirova and M. F. Tamarovskii, to whom the author expresses his thanks.     

Near wake properties of axisymmetric bluff body flows

Experiments have been made to measure some of the near wake properties of axisymmetric bluff body flows with fixed points of separation, including the detention or residence time of fluid borne scalar entities, base pressure coefficient, wake bubble length parameter and shape parameter. Measurements were made in smooth and turbulent air flow for Reynolds number in the range 2×10³<Re<4×10⁴. The results for a given bluff body were found to be uniquely controlled by a free-stream turbulence parameter. The data for all the shapes of bluff body in the class under consideration were found to collapse into unique inter-relationships by the introduction of the face pressure coefficient as a quantitative measure of “bluffness”. This paper was originally presented at the 14th International Congress of Theoretical and Applied Mechanics, Delft (August–September, 1976).     

An improved axisymmetric lattice Boltzmann flux solver for axisymmetric isothermal/thermal flows

In this work, an improved axisymmetric lattice Boltzmann flux solver (LBFS) is proposed for simulation of axisymmetric isothermal and thermal flows. This solver globally resolves the axisymmetric Navier-Stokes (N-S) equations through the finite volume strategy and locally reconstructs numerical fluxes with solutions to the lattice Boltzmann equation. Compared with previous axisymmetric LBFS, some novel strategies are adopted in this work to simplify the formulations and improve the accuracy. First, the macroscopic equations are reformulated to reduce the number of source terms and remove spatial derivatives involved in the source terms. Second, the local reconstruction of numerical fluxes utilizes relationships given by the Chapman-Enskog analysis and combines the radial coordinate (r) with the local solution to the standard LB equation. By adopting these two modifications, the present axisymmetric LBFS avoids the fractional-step formulation and the finite-difference approximation adopted in the previous solver, which reduces the complexity of implementation. Moreover, an alternative way of predicting intermediate pressure is proposed, which could effectively fix the inaccurate resolution of the pressure field in previous axisymmetric LBFS. Further extensions are made to enrich the applicability of the present solver in thermal axisymmetric flows. Validations on various benchmark tests are carried out for comprehensive evaluation of the robustness and flexibility of the proposed solver.     

Integrated photoelasticity for axisymmetric problems

A new nondestructive method for analyzing axisymmetric problems is presented. The method uses the integrated optical effects through the whole transverse section of the body, along with the strain-displacement and equilibrium equations to give the separate internal stresses on that section. The method is reasonably general and may be applied to thermoelastic and residual stress problems. Some experimental results are presented and discussed.     

Modeling of developed cavitation flows in water tunnels

An approximate dependence between cavitation numbers in an unbounded flow and in an experimental section of a water tunnel, at which the equality of the maximum transverse dimensions of the cavities formed behind identical cavitators is ensured, is obtained in the framework of a model of a viscous, weightless, incompressible liquid. On the basis of an analysis of the well-known numerical calculations of developed cavitation flows for cavitators of different shape in the two-dimensional and axisymmetric cases, and those carried out by the authors, an estimate is made showing that when the found relation between these cavitation numbers is satisfied the relative lengths, the relative maximum transverse dimensions, and the elongations of the cavities are also equal in unbounded and bounded flows. These values are equal in the considered cases, correct to 6%, for all the cavitation numbers in the tunnel which differ from the limiting values by not less than 5%. This conclusion is verified by experiments of the authors and other investigators.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 73–80, March–April, 1987.     

Quasihomogeneous model of cavitation flows in diffuser channels

Starting with the experiments carried out by Reynolds in 1894, the flow in Venturi tubes has traditionally been used to study and demonstrate various forms of cavitation. Numerous authors have carried out experimental research on the various flow regimes in diffuser channels [1–7] or have investigated theoretical models of such flows [6, 8]. The occurrence and development of cavitation is closely associated with the phenomenon of turbulent separation complicated by the presence of two-phase flow in the dissipation zone. For a long time these effects were considered separately, until Gogish and Stepanov [9] proposed a single model of cavitation and separation based on the theory of intense interaction of an incompressible potential flow and a turbulent cavitation layer of variable density and embracing the various stages of cavitation. The object of this study is to demonstrate the possibilities of this model with reference to the simple example of flows accompanied by cavitation and separation in plane and axisymmetric diffuser channels of the Venturi tube type with straight and curved walls. The dissipative flow near the walls is described by a quasihomogeneous model of turbulent two-phase flow, in which the presence of two phases is taken into account only by varying the mean density. The potential core of the flow is considered in the one-dimensional formulation. The displacement thickness serves as the flow interaction parameter. The conditions of ocurrence and development of circulatory flows are determined. Examples of symmetrical and nonsymmetrical flows are presented.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 47–54, September–October, 1986.     

Pressure effects on separation in axisymmetric stokes flows

Two Stokes flows which are known to lead to separation are reconsidered from a more dynamic perspective, and it is found that within the region of separated flow there is an extremum for the pressure. A simple argument is presented which indicates that this is true under reasonable conditions.