Regular versus Mach reflection for converging polygonal shocks

The onset of Mach reflection or regular reflection at the vertices of a converging polygonal shock wave was investigated experimentally in a horizontal annular shock tube. The converging shock waves were visualized by schlieren optics. Two different types of polygonal shock convergence patterns were observed. We compared the behavior during the focusing process for triangular and square-shaped shocks. It is shown that once a triangular shaped shock is formed, the corners in the converging shock will undergo regular reflection and consequently the shape will remain unaltered during the focusing process. A square-shaped shock suffers Mach reflections at the corners and hence a reconfiguring process takes place; the converging shock wave alternates between a square and an octagon formation during the focusing process.      

The formation of hydrodynamic slugs by the interaction of waves in gas–liquid two-phase pipe flow

This paper examines the effects of wave interaction on the formation of hydrodynamic slugs in two-phase pipe flow at relatively low gas and liquid superficial velocities. The experiments were conducted using a horizontal 31 m long, D = 10 cm internal diameter transparent pipe at atmospheric pressure. High resolution photography allowed the location of the gas–liquid interface to be measured with a high degree of accuracy at 5 Hz. Image analysis allowed individual waves to be tracked over a 14D section of the pipe. Regular waves having similar properties such as speed, amplitude and length were seen far from the region of slug formation. However, near the transition region, where hydrodynamic slugs were formed, significant differences between wave properties were observed which resulted in wave interaction leading to a type of sub-harmonic resonance and slug formation. The formation of hydrodynamic slugs due to wave interaction differs from predictions for slug formation using long wavelength stability theory. The properties of the waves were quantified which gave detailed information on the resonance mechanism found near the transition to slugging.     

Discontinuous wave interaction with interfaces between anisotropic elastic media

The problem about dynamic interaction of discontinuous waves with interfaces between anisotropic elastic media is considered. To investigate this phenomenon accompanied by formation of reflected and refracted quasi-longitudinal and quasi-shear discontinuous waves, a technique based on joint usage of the zero approximation of the ray theory and method of stereomechanical impact is proposed. It is used for the analysis of the wave front transformation, scattering and focusing. The setup problem solutions can be applied to discovering the most seismically hazardous zones in the earth’s crust, interpretation of geophysical data about geological rock structures and the analysis of the causes of dynamic delamination of layered composite and nanomaterials.     

椭球罩作用下的水下爆炸冲击波反射聚焦模型

基于冲击波传播、非线性反射和聚焦理论,建立水下爆炸冲击波在椭球罩作用下的反射聚焦模型。讨论自由传播、壁面反射和定向聚焦阶段的冲击波特性和压力计算方法,利用波前和波法线的近似方程构建压力场的数值计算域,进而模拟聚焦过程,并与现有实验结果进行对比,结果表明:所建模型可为正聚焦压力提供满足工程精度的预测,并能描述水下冲击波及产生的拉伸波聚焦过程中的一些细节;椭球罩能有效地聚焦水下冲击波,在动力学焦点附近获得有效增益区,在近轴方向上明显削弱冲击波压力衰减;理想条件下的动力学焦点一般位于几何焦点之前,但实际的反射罩变形和背向位移将使其发生后迁,甚至能越过几何焦点。     

Nonlinear wave motions on liquid surfaces

A study is made of the formation of a shock wave (bore), produced by the movement of an initially weak discontinuity in the spatial derivatives of velocity and liquid depth in an area of stationary current in a channel of constant inclination. The formation of shock waves from compression waves was first studied by Riman [1]. Frictional resistance was considered in the Chezy form. The equations obtained therein for determination of the moment in time and spatial coordinates of the point at which the shock wave is formed, as well as the laws for propagation of shock waves are applicable to the problem of one-dimensional transient motion in a gas, the pressure of which is dependent on density. Instantaneous collapse of waves, as well as formation and movement of bores in rivers for an idealized flow model in a channel with horizontal bottom, neglecting friction, were described by Khristianovich, Mikhlin, and Devison [2], and Stoker [3]. Recently in the work of Sachdev and Bhatnagar [4], using numerical integration of the equation for bore intensity, the problem of shock wave propagation in a channel of constant inclination with consideration of fluid resistance in the Chezy form was studied. Gradual wave collapse and the bore formation mechanism were studied by Stoker [3] on the basis of the shallow-water theory. Neglecting friction on the horizontal channel bottom, he calculated the moment of time and coordinates of the point at which the shock wave is formed in the case where the initial disturbance is sinusoidal. The dependence of these values on wave amplitude for a channel of constant inclination was obtained by Jeffrey [5], who also neglected friction on the channel bottom and considered the initial disturbance to be sinusoidal. Lighthill and Whitham [6] discovered that for Froude numbers greater than two, the linear theory led to unlimited growth in the intensity of the flood wave. We note that the studies of flood-wave motion in the region of the first characteristic, performed in [3, 6], differ only in the forms of the resistance laws and dependences of the unknown functions on the variables. Physical peculiarities of various liquid wave motions were also examined by Lighthill in [7].Saratov. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 62–66, March–April, 1972.     

Wave front analysis of a plane compressional pulse scattered by a cylindrical elastic inclusion

The plane-strain problem of a stress pulse striking an elastic circular cylindrical inclusion embedded in an infinite elastic medium is treated. The method used determines dominant stress singularities that arise at wave fronts from the focusing of waves refracted into the interior. It is found that a necessary and sufficient condition for the existence of a propagating stress singularity is that the incident pulse has a step discontinuity at its front. The asymptotic wave front behavior of the first few P and SV waves to focus are determined explicitly and it is shown that the contribution from other waves are less important. In the exterior, it is found that in most composite materials the reflected waves have a singularity at their wave front which depends on the angle of reflection. Also the wave front behavior of the first few singular transmitted waves is given explicitly.The analysis is based on the use of a Watson-type lemma, developed here, and Friedlander's method [5]. The lemma relates the asymptotic behavior of the solution at the wave front to the asymptotic behavior of its Fourier transform on time for large values of the transform parameter. Friedlander's method is used to represent the solution in terms of angularly propagating wave forms. This method employs integral transforms on both time and θ, the circumferential coordinate. The θ inversion integral is asymptotically evaluated for large values of the time transform parameter by use of appropriate asymptotics for Bessel and Hankel functions and the method of stationary phase. The Watson-type lemma is then used to determine the behavior of the solution at singular wave fronts.The Watson-type lemma is generally applicable to problems which involve singular loadings or focusing in which wave front behavior is important. It yields the behavior of singular wave fronts whether or not the singular wave is the first to arrive. This application extends Friedlander's method to an interior region and physically interprets the resulting representation in terms of ray theory.     

Experimental and numerical studies of shock wave focusing in water

The focusing of plane shock waves and spherical blast waves in water by parabolic and ellipsoidal reflectors is studied experimentally and numerically. The numerical computations are performed by a second order method based on an extension of Godunov's method, which is called piecewise-linear method. Tests of this method and a comparison with pressure measurements show a strong dependence of the maximum pressure attained in the focal region on the mesh size. Outside this focal region, however, good agreement between the pressure measurements and the result of the numerical computation is found. Furthermore it is shown by numerical computation, that non-linear effects in the focusing process may be partly avoided by an optimized reflector contour which depends on the strength of the incident shock wave. This modification provides that the maximum pressure is attained at the geometric focus and not at a location between reflector surface and geometrical focus.     

A holographic interferometric study of shock wave focusing in a circular reflector

A holographic interferometric study was made of the focusing of reflected shock waves from a circular reflector. A diaphragmless shock tube was used for incident shock Mach numbers ranging from 1.03 to 1.74. Hence, the process of reflected shock wave focusing was quantitatively observed. It is found that a converging shock wave along the curved wall undergoes an unsteady evolution of mach reflection and its focusing is, therefore, subject to the evolution of the process of shock wave reflections. The collision of triple points terminates the focusing process at the geometrical focus. In order to interprete quantitatively these interferograms, a numerical simulation using an Eulerian solver combined with adaptive unstructured grids was carried out. It is found numerically that the highest density appears immediately after the triple point collision. This implies that the final stage of focusing is mainly determined by the interaction between shock waves and vortices. The interaction of finite strength shock waves, hence, prevents a curved shock wave from creating the infinite increase of density or pressure at a focal point which is otherwise predicted by the linear acoustic theory.     

Formation of a spall cavity in a dielectric during electrical explosion

The wave dynamics of the stress-strain state of a solid dielectric during electrical explosion near its surface is analyzed. A quantitative model of an electrical explosion is developed which describes the operation of a high-voltage generator, the expansion of the discharge channel, and the generation and distribution of shock-wave perturbations. Two mechanisms of formation of a spall cavity on the surface of the solid are considered: the less energetic mechanism implemented by means of the waves reflected from the surface, and the more energetic mechanism in which result from the action of a direct wave of compressive stresses. The effects of the reflection surface shape and the mode of energy input into the channel on the possible fracture pattern are estimated.     

Breaking of Faraday waves and jet launch formation

Irregular and breaking Faraday waves are experimentally investigated. Among the irregular waves those with a small depression in the wave crest and periodic triplets are distinguished. In the case of breaking waves the mechanism of jet launch formation on the wave crest is considered. It is experimentally demonstrated that the breaking of standing waves in a rectangular reservoir starts with cavity collapse on the wave crest in process of formation. It is shown that jet launch from the wave crest is preceded by the initiation, development, and collapse of a cavity. A universal power-law dependence governing cavity collapse is obtained. A comparison of the experimental data with an analytical model suggests that cavity initiation is due to the nonlinearity of the waves themselves, namely, the presence of two small disturbances of the free surface traveling counter to one another and forming a cavity. The results obtained underline the importance of the initial stage of wave breaking.     

环形激波绕射, 反射和聚焦的数值模拟研究

应用频散可控耗散格式对环形激波在圆柱形激波管内绕射、反射和聚焦的问题进行了数值模拟研究，研究结果表明环形激波形成强烈聚焦的关键因素是环形激波在圆柱形管道中向对称轴运动时，绕射激波就不断加速而不作通常情况下的衰减；不同马赫数的环形激波绕射也产生不同马赫数及形状的准柱形激波，导致聚焦效果和位置的差异；另外，环形激波聚焦于一个点而圆柱形激波聚焦于一条线，两者有本质不同。     

Numerical simulation of the onset of slug initiation in laminar horizontal channel flow

Results are presented for the initiation of slug-type structures from stratified 2D, two-layer pressure-driven channel flow. Good agreement is obtained with an Orr–Sommerfeld-type stability analysis for the growth rate and wave speed of very small disturbances. The numerical results elucidate the non-linear evolution of the interface shape once small disturbances have grown substantially. It is shown that relatively short waves (which are the most unstable according to linear theory) saturate when the length of the periodic domain is equally short. In longer domains, coalescence of short waves of small-amplitude is shown to lead to large-amplitude long waves, which subsequently exhibit a tendency towards slug formation. The non-uniform distribution of the interfacial shear stress is shown to be a significant mechanism for wave growth in the non-linear regime.     

An experimental investigation of roll waves in high pressure two-phase inclined pipe flows

This paper examines the effects of small upward inclinations on the formation of roll waves and the properties of fully developed roll waves at high pressure conditions. A total of 984 experiments were conducted at six positive pipe inclinations θ = 0.00°, 0.10°, 0.25°, 1.00°, 2.50° and 5.00° using a 25 m long 10 cm i.d. pipe. Sulfur hexafluoride (SF₆) was used at 8 bara giving a gas density of 50 kg/m³. Two independent mechanisms for the formation of roll waves were identified; (1) interaction between 2D shallow water waves and (2) a visible long wavelength instability of the stratified layer. Viscous long wavelength linear stability analysis predicted the critical liquid flow rate and liquid height for the initiation of roll waves when roll waves were formed due to the second mechanism. A simple equation from shallow water wave theory agreed with measurements for critical liquid flow rate when roll waves were formed due to the first mechanism. Shallow water wave speed agreed with critical wave speeds at transition and nonlinear wave speeds for fully developed roll waves in certain cases. The increase in interfacial friction due to the presence of large waves was compared with models from the literature.     

Formation of Regions with High Energy and Pressure Gradients at the Free Surface of Liquid Dielectric in a Tangential Electric Field

The nonlinear dynamics of the free surface of an ideal incompressible non-conducting fluid with a high dielectric constant subjected to a strong horizontal electric field is simulated using the method of conformal transformations. It is shown that in the initial stage of interaction of counter-propagating periodic waves of significant amplitude, there is a direct energy cascade leading to energy transfer to small scales. This results in the formation of regions with a steep wave front at the fluid surface, in which the dynamic pressure and the pressure exerted by the electric field undergo a discontinuity. It has been demonstrated that the formation of regions with high gradients of the electric field and fluid velocity is accompanied by breaking of surface waves; the boundary inclination angle tends to 90?, and the surface curvature increases without bound.     

Fluid–structure interaction in water-filled thin pipes of anisotropic composite materials

The effects of elastic anisotropy in piping materials on fluid–structure interaction are studied for water-filled carbon-fiber reinforced thin plastic pipes. When an impact is introduced to water in a pipe, there are two waves traveling at different speeds. A primary wave corresponding to a breathing mode of pipe travels slowly and a precursor wave corresponding to a longitudinal mode of pipe travels fast. An anisotropic stress–strain relationship of piping materials has been taken into account to describe the propagation of primary and precursor waves in the carbon-fiber reinforced thin plastic pipes. The wave speeds and strains in the axial and hoop directions are calculated as a function of carbon-fiber winding angles and compared with the experimental data. As the winding angle increases, the primary wave speed increases due to the increased stiffness in the hoop direction, while the precursor wave speed decreases. The magnitudes of precursor waves are much smaller than those of primary waves so that the effect of precursor waves on the deformation of pipe is not significant. The primary wave generates the hoop strain accompanying the opposite-signed axial strain through the coupling compliance of pipe. The magnitude of hoop strain induced by the primary waves decreases with increasing the winding angle due to the increased hoop stiffness of pipe. The magnitude of axial strain is small at low and high winding angles where the coupling compliance is small.     

On the approximation of local efflux/influx bed discharge in the shallow water equations based on a wave propagation algorithm

In cities, flood waves may propagate over street surfaces below which lie complicated pipe networks used for storm drainage and sewage. The flood and pipe flows can interact at connections between the underground pipes and the street surface. The present paper examines this interaction, using the shallow water equations to model the flood wave hydrodynamics. Sources and sinks in the mass conservation equation are used to model the pipe inflow and outflow conditions at bed connections. We consider the problem reduced to one dimension. The shallow water equations are solved using a Godunov‐type wave propagation scheme. Wave speeds are modified in the wave propagation algorithm to enable flows to be simulated over nearly dry beds and dry states. First, the model is used to simulate vertical flows through finite gaps in the bed. Next, the interaction of the vertical flows with a dam break flow is considered for both dry and wet beds. An efflux number, En, is defined based on the vertical efflux velocity and the gap length. Comparisons are made with numerical predictions from STAR‐CD, a commercial Navier–Stokes solver that models the free‐surface motions, and a parameter study is undertaken to investigate the effect of the one‐dimensional approximation of the present model, for a range of non‐dimensional efflux numbers. It is found that the shallow flow model gives sensible predictions at all time provided En<0.5, and for long durations for En>0.5. Dam break flow over an underground connecting pipe is also considered. Copyright © 2010 John Wiley & Sons, Ltd.     

Controlling the level of the sonic boom generated by a flying vehicle by means of cryogenic forcing. 1. Cooling of the vehicle surface

The possibility of controlling the sonic boom level by means of cooling the surface of a flying vehicle is discussed. The effect of surface cooling on the formation of the perturbed flow structure at large distances from the vehicle is demonstrated by an example of a modified power-law body of revolution. The intensity of the intermediate shock wave and the perturbed pressure pulse near the body are seen to decrease, which expands the altitude range of the region where the sonic boom is reduced (down to 50%). At larger distances from the body, cryogenic forcing ensures a 12% decrease in the bow shock wave intensity. The possibility of controlling the process of formation of wave structures near the surface, such as barrel shock waves, is demonstrated. An explanation of the cryogenic forcing mechanism is offered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 6, pp. 88–98, November–December, 2008.     

Specific gasdynamical features of spontaneous condensation in an unsteady rarefaction wave

On the basis of numerical modeling, the formation of an unsteady shock wave induced by a condensation shock in a rarefaction wave moving in the high-pressure channel of a shock tube filled with moist air is demonstrated. It is shown that in a fairly long channel a periodic structure consisting of an alternating sequence of condensation shocks and the shock waves they generate may be formed. This structure is a linear unsteady analog of the self-oscillation regime of type IV in the classification [1] for condensing medium flows in the subsonic section of a Laval nozzle. The specific features detected are important for planning and interpreting experiments aimed at investigating spontaneous condensation using a “condensation shock tube”.     

Effect of viscosity on the propagation of nonlinear Alfvén waves along a tangential magnetohydrodynamic discontinuity in an incompressible fluid

It is proposed to consider the propagation of surface waves along a tangential magnetohydrodynamic discontinuity in the particular case where the fluid velocities on both sides of the interface are equal to zero. In [1] it was shown that waves called surface Alfvén waves may be propagated along the surface separating a semi-infinite region without a field from a region with a uniform magnetic field. The linear theory of surface Alfvén waves in a compressible medium was considered in [2]. In [3] the damping of surface Alfvén waves as a result of viscosity and heat conduction was investigated. The propagation of low-amplitude nonlinear surface Alfvén waves in an incompressible fluid in the absence of dissipative processes is described by the integrodifferential equation obtained in [4]. By means of a numerical solution of this equation it was shown that a perturbation initially in the form of a sinusoidal wave will break. The breaking time was determined. In this paper the equation derived in [4] is extended to the case of a viscous fluid. It is shown that the equation obtained does not have steady-state solutions. The propagation of periodic disturbances is investigated numerically. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 94–104, November–December, 1986. The author wishes to thank L. S. Fedorov for assisting with the calculations.