Calculation of aerodynamic characteristics of a wing profile with discharge of a controllable jet into the external flow

A wing profile of infinite span, whose lower surface is replaced by a system of guide vanes, is placed in a flow of an ideal incompressible fluid. Fluid flows out through the system of guide vanes from the internal cavity of the wing into the external stream, forming a jet in the wake (Fig. 1). The total pressure in the wing cavity and in the jet differs from the total pressure in the outer free stream. The jet boundaries are streamlines extending to infinity, along which there is a discontinuity of the velocity value. The flow of fluid in the internal wing cavity is simulated by a flow caused by a system of suitably located sources, and the system of guide vanes is replaced by discrete vortices.The form of the profile arc is selected so that the fluid flow from the sources in the direction which is nearly opposite the direction of the freestream velocity is restrained by the segment of the contour with high curvature in the vicinity of the leading edge. We consider the flow regime about the profile with an exhausting jet for which the two ends of the arc the points of detachment of the stream and the velocity discontinuity line (profile arc, jet boundary) is a smooth curve, which imposes an additional condition on the magnitude of the circulation. As the model for the study of the flow about a profile with jet blowing we take the arc of a logarithmic spiral.Formulas are obtained for determining the over-all characteristics of the stream forces acting on the profile in the presence of the jet and the total pressure discontinuity. On the basis of the calculations made for a thin wing a qualitative analysis is made for the stream force acting on the profile.The authors wish to thank S. A. Khristianovich for formulating the problem and for his advice.     

内部压力作用下矩形板中源于椭圆孔的分支裂纹应力强度因子的一种数值分析

应用一种边界元方法来研究内部压力作用下矩形板中源于椭圆孔的分支裂纹。该边界元方法由Crouch与Starfied建立的常位移不连续单元和笔者最近提出的裂尖位移不连续单元构成。在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界。本数值结果进一步证实这种数值方法对计算有限大板中复杂裂纹的应力强度因子的有效性,同时该数值结果可以揭示裂纹体几何对应力强度因子的影响。     

Thermal stress analysis of a three-dimensional anticrack in a transversely isotropic solid

A three-dimensional analysis is performed for an infinite transversely isotropic elastic body containing an insulated rigid sheet-like inclusion (an anticrack) in the isotropy plane under a remote perpendicularly uniform heat flow. A general solution scheme is presented for the resulting boundary-value problems. Accurate results are obtained by constructing suitable potential solutions and reducing the thermal problem to a mechanical analog for the corresponding isotropic problem. The governing boundary integral equation for a planar anticrack of arbitrary shape is obtained in terms of a normal stress discontinuity. As an illustration, a complete solution for a rigid circular inclusion is obtained in terms of elementary functions and analyzed. This solution is compared with that corresponding to a penny-shaped crack problem.     

Flow of a supersonic electrogasdynamic stream,with formation of an electroconducting gaseous shock layer

Surfaces of strong discontinuity in electrogasdynamics were considered in [1, 2]. An investigation was done for the case when a gas has the properties of a unipolar charged medium on both sides of a surface of discontinuity. However, with sufficiently high supersonic gas flow over bodies the gas becomes electroconducting and acquires the properties of a low-temperature plasma in the compressed layer between the shock wave and the body, because of the temperature increase. Therefore, there is great interest in investigating type S* Shockwaves dividing a unipolar charged medium and a low-temperature plasma. The S* waves separating the uncharged medium and a gas with high electrical activity in the presence of an electrical field were studied in [3]. Below we examine the general properties of S* waves (physicalmodel, relations at the wave, conditions for development, shock adiabats, and polars). We formulate the problem of flow of a supersonic electrogasdynamic stream over bodies, with formation of S* waves. A perturbation method is proposed for solution of the problem, using a small parameter to describe electrogasdynamic interaction. By way of example a complete solution for flow over a wedge is constructed.     

Calculation of stationary inviscid flow over three-dimensional bodies

The results of the calculation of inviscid supersonic flow of a perfect gas over a blunt three-dimensional configuration are considered. An explicit finite-difference scheme of second order of accuracy [1] was used for the numerical integration of the hyperbolic system of equations, which was written in divergence form. The region of integration is situated between the body and the outer shock wave. The internal discontinuity surfaces were not separated out and the calculation was made through them. The points on the surface of the body were calculated using relations on characteristics written in a form that makes it possible to calculate flows with strong entropy layers. The results of calculation of flow over a three-dimensional configuration at an angle of attack are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 4, pp. 172–175, July–August, 1980.     

A Numerical Approach for Arbitrary Cracks in a Fluid-Saturated Medium

A finite element method is proposed that can capture arbitrary discontinuities in a two-phase medium. The discontinuity is described in an exact manner by exploiting the partition-of-unity property of finite element shape functions. The fluid flow away from the discontinuity is modelled in a standard fashion using Darcy’s relation, while at the discontinuity a discrete analogon of Darcy’s relation is proposed. The results of this finite element model are independent of the original discretisation, as is demonstrated by an example of shear banding in a biaxial, plane-strain specimen.     

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e. , a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.     

Thermodynamic restrictions and wave features of a non-linear Maxwell model

A non-linear rate-type constitutive equation, established by Rajagopal, provides a generalization of the Maxwell fluid. This note embodies such a constitutive equation within the scheme of materials with internal variables thus allowing also for solids with both dissipative and thermoelastic mechanisms. The compatibility with the second law of thermodynamics, expressed by the Clausius–Duhem inequality, is examined and the restrictions on the evolution equations are determined. Next the propagation condition of discontinuity waves is derived, for shock waves and acceleration waves, by regarding the body as a definite conductor. Infinitesimal shock waves and acceleration waves show similar effects. The effective acoustic tensor proves to be the sum of a thermoelastic tensor and a tensor arising from the rate-type equation.     

Faults simulations for three-dimensional reservoir-geomechanical models with the extended finite element method

Faults are geological entities with thicknesses several orders of magnitude smaller than the grid blocks typically used to discretize reservoir and/or over-under-burden geological formations. Introducing faults in a complex reservoir and/or geomechanical mesh therefore poses significant meshing difficulties. In this paper, we consider the strong-coupling of solid displacement and fluid pressure in a three-dimensional poro-mechanical (reservoir-geomechanical) model. We introduce faults in the mesh without meshing them explicitly, by using the extended finite element method (X-FEM) in which the nodes whose basis function support intersects the fault are enriched within the framework of partition of unity. For the geomechanics, the fault is treated as an internal displacement discontinuity that allows slipping to occur using a Mohr–Coulomb type criterion. For the reservoir, the fault is either an internal fluid flow conduit that allows fluid flow in the fault as well as to enter/leave the fault or is a barrier to flow (sealing fault). For internal fluid flow conduits, the continuous fluid pressure approximation admits a discontinuity in its normal derivative across the fault, whereas for an impermeable fault, the pressure approximation is discontinuous across the fault. Equal-order displacement and pressure approximations are used. Two- and three-dimensional benchmark computations are presented to verify the accuracy of the approach, and simulations are presented that reveal the influence of the rate of loading on the activation of faults.     

Axisymmetric nonstationary flow around an obstable in a cylindrical tube

Questions associated with the interaction between a gas stream and a body in a launching tube, especially the high-speed propulsion of a body by a gas stream [1], have become of great interest in recent years. Partial destruction of the body and the formation of a gap between the body and the launching tube, through which the working gas will flow, inevitably occurs at high velocities. In this case it is possible to consider the ejection of a free body which does not come into contact with the walls of the launching tube as it is accelerated. An analogous problem occurs in the transportation of containers in a tube under the effect of a compressed gas [2], as well as in a gas-dynamic analysis of piston apparatus with different kinds of gas flow through the orifice inmoving or fixed pistons. The interaction between the gas stream and the body or the obstacle in the launching tube must be known for a theoretical investigation of all these problems. The solution is obtained by numerical integration of the nonstationary gas-dynamic equations by means of a through-computation difference scheme [3]. Values of the blockage factors are found for different freestream Mach numbers, for which the reflected shock stands off at infinity upstream. A comparison is given with the one-dimensional approximation obtained under the assumption that the body being streamlined is replaced by two jumps of a strong discontinuity on which the mass, momentum, and energy conservation conditions are satisfied. The results obtained are used in the problem of ejection of a free body from a launching tube under the effect of an unsteady gas flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 81–86, July–August, 1978.     

Asymptotic model of the development of separations inside a boundary layer under the action of a traveling pressure wave

A model of the nonlinear interaction between a pressure perturbation traveling at a constant velocity and an incompressible boundary layer is constructed when its near-wall part is described by the “inviscid boundary layer” equations. A steady-state solution is managed to obtain in the finite form under the assumption that it exists in a moving coordinate system. It is shown that the boundary layer can easily overcome pressure perturbations whose amplitude is not higher than the dynamic pressure calculated from the velocity of the pressure perturbation. At the higher pressure perturbation amplitudes a vortex sheet sheds from the body surface to the boundary layer. The vortex sheet represents an unstable surface of the tangential discontinuity which separates the regions of the direct and reverse separation flows. In the case of an arbitrary shape of the pressure perturbation the surface of the tangential discontinuity sheds from the body surface at a finite angle with the formation of a stagnation point. An example of the pressure perturbation in which the vortex sheet sheds from the body surface along the tangent is constructed.     

Experimental Study on the Pore Structure Fractals and Seepage Characteristics of a Coal Sample Around a Borehole in Coal Seam Water Infusion

In this study, a two-dimensional fully coupled computational model is developed for simulation of proppant settlement in hydro-fractures with the use of the extended finite element framework. The porous domain is governed by the well-known \((\mathbf{u}-p)\) formulation, which consists of the momentum balance equation of the bulk, in conjunction with the momentum balance and continuity equations of the pore fluid. The hydro-fracture inflow is modeled as a 1D flow on the basis of the Darcy law, in which fracture permeability is incorporated by means of the cubic law. Contact constraints are elaborated to eliminate the overlap of fracture edges and the leak-off flow. Proppant settlement is conducted on the basis of Stokes’ law for particle terminal velocity, in which the effects of fracture walls, concentration, viscosity and bridging are incorporated into the model. A fixed-point algorithm is introduced to achieve the optimum combination for the proppant injection. Using the extended finite element method, the strong discontinuity in the displacement field due to crack body, as well as the weak discontinuity in the pressure field due to leakage, is included in the model with the use of the Heaviside and modified level set enrichment functions, respectively. The robustness and versatility of the proposed numerical algorithm in determining the optimum proppant injection is examined through several numerical simulations.     

Problem of the decay of a small-amplitude discontinuity in two-layer shallow water: First approximation

A method is proposed for the approximate solution of the problem of the decay of a small-amplitude discontinuity in the case of a strictly hyperbolic system of conservation laws which has linearly degenerate characteristic fields. The method is used to analyze the qualitatively different flow regimes occurring in the solution of the problem of the decay of a small-amplitude discontinuity for two-layer shallow water model with a free upper boundary. The most detailed study is made for the special case — the dam-break problem.     

Unsteady one-dimensional water and steam flows through a porous medium with allowance for phase transitions

Fluid flow through a porous medium is considered with allowance for heat conduction and phase transition processes. The one-dimensional problem of the breakdown of an arbitrary discontinuity is solved with reference to the processes of combined nonisothermal water and steam flow through the porous medium. It is assumed that there are two-phase zones of water and steam flow through the porous medium to the left and right of the initial discontinuity. Six qualitatively different discontinuous solutions with internal single-phase water or steam zones are constructed and domains corresponding to each of the solutions are found in the determining parameter space. For the parameters considered a solution of the breakdown problem exists and is unique when the requirements for the existence of a discontinuity structure are satisfied [{xc1}].     

Extension of a V-notched bar and failure of plastic bodies

The effect of plastic strain localization near the domains of sharp variation in shape and transverse cross-section of bodies is well known. But such processes have not yet been studied analytically well enough. On the basis of the model of an ideally rigid-plastic body, we propose an approach for determining the strain fields near the concentrators on the basis the motion of the displacement velocity field (near surfaces or discontinuity lines in the form of rigid-plastic boundaries and centers of the fan of slip lines under plane strain). We consider the problem on plastic flow with failure for a V-notched bar. We show that the plastic flow is not unique (in the framework of the solution completeness).We propose to use the strain criterion for choosing the preferable plastic flow. On the basis of the solutions thus obtained, we state an approach to studying failure processes for more complicated models of bodies.     

A numerical analysis of cracks emanating from a surface elliptical hole in infinite body in tension

This paper deals with such a kind of surface crack problem with an approximately same depth, which is called a liked-plane crack problem. Based on the previous investigations on internal rectangular crack and surface rectangular crack in infinite solid in tension and a hybrid displacement discontinuity method (a boundary element method) proposed recently by Yan, a numerical approach for the liked-plane crack problem in hand is presented. Numerical examples are given to illustrate the numerical approach is simple, yet accurate for calculating the SIFs of a liked-plane crack. Specifically, a pair of cracks emanating from a surface elliptical hole in infinite body in tension are investigated in detail.     

Horizontal Redistribution of Two Fluid Phases in a Porous Medium: Experimental Investigations

Classical models for flow and transport processes in porous media employ the so-called extended Darcy’s Law. Originally, it was proposed empirically for one-dimensional isothermal flow of an incompressible fluid in a rigid, homogeneous, and isotropic porous medium. Nowadays, the extended Darcy’s Law is used for highly complex situations like non-isothermal, multi-phase and multi-component flow and transport, without introducing any additional driving forces. In this work, an alternative approach by Hassanizadeh and Gray identifying additional driving forces were tested in an experimental setup for horizontal redistribution of two fluid phases with an initial saturation discontinuity. Analytical and numerical solutions based on traditional models predict that the saturation discontinuity will persist, but a uniform saturation distribution will be established in each subdomain after an infinite amount of time. The pressure field, however, is predicted to be continuous throughout the domain at all times and is expected to become uniform when there is no flow. In our experiments, we also find that the saturation discontinuity persists. But, gradients in both saturation and pressure remain in both subdomains even when the flow of fluids stops. This indicates that the identified additional driving forces present in the truly extended Darcy’s Law are potentially significant.     

Dynamics of discontinuity formation in a cavitating liquid layer under shock wave loading

The problem of experimental modeling of discontinuity formation in a cavitating liquid layer under shock wave loading is considered. It is shown that the discontinuity takes the shape of a spherical segment and retains it up to the closure instant. The discontinuity surface becomes covered with a dynamically growing thin boundary layer consisting of bubbles, which transforms to a ring-shaped vortex bubble cluster at the instant of the discontinuity closure, generating a secondary shock wave. Specific features of the structure of the cavitating flow discontinuity arising at loading intensities lower than 0.1 and 5 kJ are discussed.     

Numerical solution of the problem of wave diffraction by a wedge

The systematic development of the theory of shock reflection from a solid wall started in [1]. Regular reflection and a three-shock configuration originating in Mach reflection were considered there under the assumption of homogeneity of the domains between the discontinuities and, therefore, of rectilinearity of these latter. The difficulties of the theoretical study include the essential nonlinearity of the process as well as the instability of the tangential discontinuity originating during Mach reflection. Analytic solutions of the problem in a linear formulation are known for a small wedge angle or a weak wave (see [2–4], for example). The solution in a nonlinear formulation has been carried out numerically in [5, 6] for arbitrary wedge angles and wave intensities. Since the wave was nonstationary, the internal flow configuration is difficult to clarify by means of the constant pressure and density curves presented. A formulation of the problem for the complete system of gasdynamics equations in self-similar variables is given in [7] and a method of solution is proposed but no results are presented. Difficulties with the instability of the contact discontinuity are noted. The problem formulation in this paper is analogous to that proposed in [7]. However, a method of straight-through computation without extraction of the compression shocks in the flow field is selected to compute the discontinuous flows. The shocks and contact discontinuities in such a case are domains with abrupt changes in the gasdynamics parameters. The computations were carried out for a broad range of interaction angles and shock intensities. The results obtained are in good agreement with the analytical solutions and experimental results. Information about the additional rise in reflection pressure after the Mach foot has been obtained during the solution.     

Two-dimensional self-similar flows generated by the interaction between a shock and low-density gas regions

A plane time-dependent flow generated by the interaction between a normal shock and a low-density gas region occupying a quarter of the plane is theoretically investigated. Numerical simulation is performed on the basis of the Euler equations. It is established that after the shock has come in contact with the low-density region two-dimensional self-similar flows of different type can develop. On regular interaction the original shock is refracted on the low-density region with the matching of the accelerated and original shock and the refracted contact discontinuity at a common point. On irregular interaction a complicated flow occurs; it includes curved and oblique shocks, a contact discontinuity with points of inflection, multiple matching points, a high-pressure jet, and a layered vortex. The jet and vortex structures are investigated in detail. The tendency of the gasdynamic structure development with variation in the control parameters of the problem is determined. A simplified, near-analytical technique for estimating the slopes of the main shocks and the gas parameters behind them is proposed.