Crack energy density in arbitrary direction for piezoelectrics and its characteristic features in electromechanical mixed mode crack

The concepts of crack energy density (CED) and its derivatives in arbitrary direction were established for piezoelectric material and, keeping their application to mixed mode fracture in mind, the characteristic features of them as fracture parameters were investigated based on the approximate equations for CED and its derivatives. That is, CED and its derivatives in arbitrary direction are defined first and separation into their each mode contribution is made. Subsequently, path independent integral expressions of them are derived, and then using them, approximate equations of each mode contribution of CED are obtained concretely for the case where linear singular solution is known. The resulting equations are then used to investigate the effects of electric field and electrical boundary condition on CED and its derivatives. An infinite piezoelectric plane with a crack inclined with respect to the poling direction is considered as a numerical example. Mode I contribution of mechanical CED is mainly employed as a possible fracture parameter for the study and it was shown that applied electric field significantly influences on fracture parameters especially for the impermeable crack perpendicular to the poling direction. The effect of electric field has the tendency to decrease as crack inclination angle increases. It was also found that, even for the impermeable crack perpendicular to the poling direction, crack propagation could be deviated from self-similar direction under a strong negative electric field, and this fact is qualitatively consistent with an existing experimental observation. For the ideally sharp crack with no width, impermeable and Hao and Shen type boundary conditions are admissible showing qualitative agreement with experimental results, but exact boundary condition is not suitable and finally consistent with permeable boundary condition.     

On the numerical computation of laminar boundary layers at a phase-changing,gas-liquid interface

The two-dimensional, laminar boundary-layer equations of heat, mass and momentum at a smooth, phase-changing, gas-liquid interface are solved numerically by the Keller Box method. The gas and liquid regimes are embedded in a single marching scheme which computes interfacial parameters implicitly. Results of both self-similar and non-similar boundary-layer computations are presented and effects of mild pressure gradient, a mean current in the liquid, and free-stream vapour concentration on the interfacial parameters are analysed. In order to assess the accuracy of the method, several self-similar problems are solved by Runge-Kutta integration and results are compared to those obtained by the finite-difference scheme. Agreement is excellent in all cases.     

Self-similar regimes of liquid-layer spreading along a superhydrophobic surface

Within the Stokes film approximation, unsteady spreading of a thin layer of a heavy viscous fluid along a horizontal superhydrophobic surface is studied in the presence of a given localized mass supply in the film. The forced (induced by the mass supply) spreading regimes are considered, for which the surface tension effects are insignificant. Plane and axisymmetric flows along the principal direction of the slip tensor of the superhydrophobic surface are studied, when the corresponding slip tensor component is either a constant or a power function of the spatial coordinate, measured in the direction of spreading. An evolution equation for the film thickness is derived. It is shown that this equation has self-similar solutions of a source type. The examples of self-similar solutions are constructed for power and exponential time dependences of mass supply. In the final part of the paper, some of the solutions constructed are generalized to the case of a weak dependence of the flow on the second spatial coordinate, caused by a slight variability of the slip coefficient in the direction normal to that of spreading. The constructed self-similar solutions can be used for experimental determination of the parameters important for hydrodynamics, e.g. the slip tensor components of commercial superhydrophobic surfaces.     

Exact solutions to an evolution equation of plastic layer flow on a plane

The flow of a thin plastic layer between two rigid plates approaching each other in the normal direction is considered. The kinematics of plastic layer flow is studied. An evolution equation describing the free boundary of the flow region is derived. The similarity solutions to this equation are analyzed. It is shown that the evolution equation can be reduced to a particular case of the nonlinear heat conduction equation. New exact particular solutions to the evolution equation are obtained using the variable separation method and the method of self-similar transformations.     

The AGE method for direct numerical simulation of turbulent shear flow

The advected grid explicit (AGE) method for direct numerical simulation of ‘incompressible’ turbulent shear flows is presented. The Navier–Stokes equations are used for momentum in a velocity–pressure formulation. Mass continuity and an equation of state link pressure with density (which is not assumed identically constant). Time advancement is entirely explicit, and spatial representation is localized (e.g. finite difference) and centred. Magnitudes of non-linear terms are reduced on advected grid(s), and numerical instabilities are efficiently reduced by ‘targeted diffusion’. Computation time scales directly on the number of grid points (virtual memory issues aside), and is very short for a DNS method. A spatially developing two-stream mixing layer was simulated as an example, reaching a vorticity thickness Reynolds number >20 000. Comparison with experimental results from self-similar mixing layers is satisfactory in terms of growth rate and Reynolds stress profiles. Turbulent vortical structures are visualized by means of pressure surfaces. © 1998 John Wiley & Sons, Ltd.     

A unified approach to direct and inverse boundary layer solutions

A unified approach is presented for solving the two-dimensional incompressible boundary layer equations. Solutions are obtained for direct and inverse options using the same equation formulation by a simple interchange of boundary conditions. A modified form of the mechul function scheme obtains inverse solutions with specification of transformed wall shear, skin friction coefficient or displacement thickness distributions. Direct solutions may be obtained without altering the block tridiagonal structure of the system by simply requiring no corrections on the streamwise pressure gradient parameter. Fourth-order spline discretization approximates normal derivatives with two- and three-point backward differences approximating streamwise derivatives, yielding a fully implicit solution method. The resulting spline/finite difference equations are solved by Newton-Raphson iteration together with partial pivoting. The results of the study demonstrate the importance of proper linearization of all equations. The successful use of spline discretization is also tied to the use of strong two-point boundary conditions at the wall for cases involving reversed flow. Numerical solutions are presented for several non-similar flows and compared with published results.     

A spectral–finite difference solution of the Navier–Stokes equations in three dimensions

A new computational code for the numerical integration of the three-dimensional Navier–Stokes equations in their non-dimensional velocity–pressure formulation is presented. The system of non-linear partial differential equations governing the time-dependent flow of a viscous incompressible fluid in a channel is managed by means of a mixed spectral–finite difference method, in which different numerical techniques are applied: Fourier decomposition is used along the homogeneous directions, second-order Crank–Nicolson algorithms are employed for the spatial derivatives in the direction orthogonal to the solid walls and a fourth-order Runge–Kutta procedure is implemented for both the calculation of the convective term and the time advancement. The pressure problem, cast in the Helmholtz form, is solved with the use of a cyclic reduction procedure. No-slip boundary conditions are used at the walls of the channel and cyclic conditions are imposed at the other boundaries of the computing domain. Results are provided for different values of the Reynolds number at several time steps of integration and are compared with results obtained by other authors. © 1998 John Wiley & Sons, Ltd.     

Self-similar problem of the motion of a plane layer of heated material with an arbitrary equation of state

The self-similar problem of the nonstationary motion of a plane layer of material in which energy from an external source is released for values of the flux density q₀ on the boundary which are constant in time is considered. The self-similar variable is = m/t, where m is the Lagrangian mass coordinate and t is the time. The characteristic values of the velocity, density, and pressure do not vary with time. For a self-similar problem the energy flux density q must also depend only on the self-similar variable. In this case q() can be an arbitrary function of its argument and can be given by a table. Examples are presented of actual physical processes in which the mass of the energy-release zone increases linearly with time. The equation of state can have an arbitrary form, including specification by a table. The gaseous state of matter for an arbitrary variable adiabatic exponent, the condensed state, and a two-phase state can be described. A solution of the self-similar problem is presented for the heating of a half-space bounded by a vacuum for a certain specific equation of state and various flux densities q₀ and velocities M of the advance of the energy-release zone.Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskoi Fiziki, No. 5, pp. 136–145, September–October, 1975.     

Conical Mach reflection of moving shock waves

Conical Mach reflections differ from those of the equivalent plane, two-dimensional Mach reflection because in axisymmetry, the disturbances generated at the reflecting surface are modified by their more rapidly increasing or decreasing area as they move towards or away from the centerline. Equations for conical Mach reflection cases have now been developed using a simplified ray-shock theory formulation based on the initial assumption that the stem is straight and normal to the wall. These are in a form that applies generally. Their simple structure provides an easy conceptual understanding of self-similarity and non-self-similarity as well as a clear mathematical approach for the development of the curved triple-point locus of the latter by integration. They provide a quick and direct solution in all cases and can easily incorporate the Mach stem curvature by progressively calculating the new ray direction. A range of cases has been considered and results are presented for converging and diverging, self-similar and non-self-similar cases.     

Time dependent shear stress and temperature distribution over an insulated flat plate moving at hypersonic speed

The laminar two-dimensional flow over a stepwise accelerated flat plate moving with hypersonic speed at zero angle of attack is analysed. The governing equations in the self-similar form are linearized and solved numerically for small times. The solutions obtained are the deviations of the velocity and the temperature profiles from those of steady state. The presented results may be used to find the first order boundary layer induced pressure on the plate.     

Transonic expansion of a penny-shaped crack

The problem of a penny-shaped crack propagating in an unbounded isotropic linear elastic body is solved. The crack expands from a zero radius in a self-similar manner and is assumed to have speed larger than the S-wave speed and less than the P-wave speed. A tensile load is directed normal to the crack at infinity. The method of self-similar potentials and rotational superposition are applied. Attention is given to the stress singularity at the crack border.     

BEHAVIOR OF OBSTRUCTED SQUARE BUOYANT VERTICAL JETS IN STATIC AMBIENT (Ⅱ)-ANALYSIS ON BEHAVIOR OF FLOW FIELD

Based on a series of numerical calculations, the behavior of flow field in obstructed square buoyant vertical jet is summarized and analyzed. Based on the axial line velocity distribution, the flow after the disc can be divided into three regions, i.e.,recirculation region, transitional region and self-similar region The characteristic of selfsimilarity of upright velocity was validated. The three regions can also be distinguished based on the axial velocity. The axial velocity in self-similar region was found to obey the same law and the formula presented by introducing the velocity expression used by Chen and Rodi. The isolines of pressure on cross-sections of different heights were displayed and the production, expansion, breaking and disappearing of negative pressure regions were found.     

Pressure distribution in the neighborhood of a growing fracture with a constant wedge force

Exact solutions of the problem of the pressure field in the neighborhood of a hydraulic fracture developing in accordance with a square root law in a permeable porous medium with a constant wedge force acting on the fracture edges are constructed. A particular case admitting a self-similar formulation and an exact solution and, as a result, the fairly complete investigation, is considered. The solution constructed holds for an arbitrary self-similar pressure distribution over the fracture edges. The problem considered reduces to the solution of a mixed boundary-value problem for the Helmholtz equation. The solution found can be useful both in itself and for testing more universal numerical algorithms.     

BEHAVIOR OF OBSTRUCTED SQUARE BUOYANT VERTICAL JETS IN STATIC AMBIENT （Ⅱ）——ANALYSIS ON BEHAVIOR OF FLOW FIELD

Based on a series of numerical calculations, the behavior of flow field in obstructed square buoyant vertical jet is summarized and analyzed. Based on the axial line velocity distribution, the flow after the disc can be divided into three regions, i.e., recirculation region, transitional region and self-similar region The characteristic of self-similarity of upright velocity was validated. The three regions can also be distinguished based on the axial velocity. The axial velocity in self-similar region was found to obey the same law and the formula presented by introducing the velocity expression used by Chen and Rodi. The isolines of pressure on cross-sections of different heights were displayed and the production, expansion, breaking and disappearing of negative pressure regions were found.     

激励小尺度模式在湍流圆管射流中的应用

采用非涡黏性的激励小尺度（Stimulated Small Scale)模式对空间发展的轴对称湍流圆管射流进行了大涡模拟。以雷诺数为10000的流动为例,考证了激励小尺度模式在自由剪切流模拟中的可行性,描述了湍流强度、雷诺应力和湍流耗散量的变化,同时与标准的Smagorinsky涡黏性模式的计算结果进行了比较。数值结果显示,激励小尺度模式能够更为合理地描述湍流的耗散特性和能量传输特性,从而较为准确地展示出空间发展射流中由于流动不稳定而出现的旋涡产生、发展、破碎及合并等过程。     

On the steady motion of two immiscible fluids in an undulating porous reservoir

We describe steady two-dimensional flows of two immiscible fluids through an undulating porous medium of constant thickness, with impermeable or slightly permeable boundaries. Flows in the same or opposite directions are called, respectively, direct or counter flows. Three special classes of flow are determined:

1. The pressure dominated case occurs for high direct flows and has the interface approximately a constant vertical distance from the impermeable boundaries.
2. The gravity dominated case occurs for low direct flows and has the interface very close to the lower (upper) boundary for downward (upward) sloping boundaries except at crossovers.
3. Counter flows require the interface to decrease in the direction of flow of the lower fluid.

Numerical examples illustrate the three classifications above. For incompressible flows the interface and pressure equations uncouple. A stability analysis shows that the direction of integration of the differential equation for the interface must be opposite to the flow direction for direct flows; for counter flows the direction of integration depends on whether the interface is above or below a critical height. Direct flows through cyclic geometries are asymptotically cyclic upstream. If the reservoir is ‘leaky’, asymptotically self-similar flows result when the (small) permeability ratio is scaled to the dynamical flow parameters.     

定态火焰在可燃预混气中产生的压力波

火焰在可燃预混气中传播时，在火焰面前方产生一道压力波。忽略点火及火焰的初期加速，仅考虑火焰达到稳定传播速度的情况。用Ｏｐｅｎｈｅｉｍ自相似解分析流场，得到相应的控制方程及定解条体；用自适应步长的四阶Ｒｕｎｇｅ－Ｋｕｔｔａ法对方程积分，讨论了流场压力波结构及弱激波近似声波解；认为火焰为间断面，能量释放在火焰面后瞬时完成。利用火焰面两侧的能量关系，得到了火焰位置、燃速及对应Ｃ－Ｊ条件的火焰位置、Ｃ－Ｊ燃速。     

激波-涡干扰声场的数值研究

采用高精度差分格式求解非定常可压缩Navier-Stokes方程,对激波－单涡／双涡相互干扰产生的声场进行了直接数值。详细研究了波－涡干扰声场结构的早期发展阶段,将激波－单涡的计算结果和相应实验进行 对比,并给出近场声压的衰减规律。在此基础上模拟较为复杂的激波－双涡干扰,给出不同旋涡旋转方向下的声场结构。     

Self-similar hypervelocity shock interactions with oblique contact discontinuities

The two-dimensional inviscid refraction of a shock wave at an oblique contact discontinuity is self-similar; i.e. it depends only upon the variables , . We transform the compressible Euler equations into the self-similar () coordinates and solve the resulting boundary-value problem by an implicit equilibrium flux method. We present results for strong shock ( where M is the incident shock Mach number) interactions with an oblique contact discontinuity separating an inert gas from a gas which exhibits high-temperature gas chemistry effects. To model high-temperature effects we employ Lighthill's ideal dissociating gas (IDG) model. Comparison between the frozen and equilibrium limits, both of which are self-similar, indicate large changes in peak density and temperatures. Significant differences in the overall flow pattern between the frozen and equilibrium limits are observed for interfaces with low negative Atwood ratio and high positive Atwood ratio. Results from a local analysis are presented when the shock refraction is regular. The critical angle at which the transition from regular to irregular refraction occurs is slightly larger for the equilibrium chemistry case. Received 6 September 1996 / Accepted 20 May 1997