Critical Wedge Angle Research Articles

On supersonic combustion and hypersonic propulsion

The advanced engine has been the core technology of the aviation industry for several decades. The air-breathing hypersonic propulsion is the top problem for future aerospace flight. The engine's performance depends on its energy conversion method and combustion mode, and its relevant theory is of fundamental and revealing significance. In this paper, the supersonic combustion is discussed first since it is the theoretical basis for the research and development of scramjet engines. Then, by reviewing related research progresses, three criteria of the air-breathing hypersonic ramjet propulsion are established. The first one can be used to determine the local subsonic or supersonic flow states of combustion products in supersonic reacting gas flows, revealing the mechanism of the upstream-traveling shock wave. The second one defines the critical Mach number for hypersonic ramjet operation for all the combustion modes, and is a necessary condition that needs to be considered in the engine design under the equivalent ratio combustion. The last one gives a critical wedge angle corresponding to the CJ oblique detonation, and its physical basis is the critical initiation state of detonation. Finally, the experimental research progress on the stationary oblique detonation ramjet (Sodramjet) engine is summarized, and its feasibility as a hypersonic engine for future aerospace flight is demonstrated.

Mach reflection of a H2-O2-Ar detonation wave on the rough wedge based on soot track measurements

Gaseous detonation wave reflection on the rough wedge was experimentally investigated by soot track measurements. Various wedge angles, step numbers and initial pressures were taken into account. The results show that the Mach reflection can form on the rough surface due to the three-dimensional structure of the cellular detonation wave or the pseudo triple-point configuration. The wedge angle is still a significant parameter to affect the Mach stem height and the triple-point trajectory. The Mach stem on the rough surface is weaker and the triple-point is easily deflected due to the triple-point disappearance from the Mach stem. The wedge with smaller step number leads to the triple-point trajectory starting in the downstream region, instead of its starting from the wedge apex. As the step number increases, the Mach stem height on the rough wedge gradually approaches the one on the smooth wedge. The initial pressure has a more significant effect on the Mach reflection on the rough surface compared to that on the smooth surface. Moreover, the Mach reflection on the rough wedge keeps localized self-similarity, and satisfies the frozen limit in the near field and the equilibrium limit in the far field. The critical wedge angle of the transition from regular reflection to Mach reflection is dependent on the step number.

Experimental and numerical studies on detonation reflections over cylindrical convex surfaces

The detonation reflection over a cylindrical convex surface was investigated experimentally and numerically by focusing on the length-scale effect on the reflection process, such as the triple-point trajectory and the critical wedge angle at which a transition occurs from regular reflection to Mach reflection. The results show that the critical wall angle plots exhibit significant scatter because of the cellular properties of the detonation front. If the transverse spacing is large as compared to the radius of curvature, the scatter range extends. If the transverse spacing is small as compared to the radius of curvature, the scattering is dramatically reduced. The critical wall angle is found to mainly depend on the scaled length i.e., the radius of curvature (R) over the cell size λ (or the reaction zone thickness Δ). Moreover, the critical wall angle increases with the decrease in the detonation thickness or with the increase in the radius. As R/λ increases to approximately ten, the critical wall angle approaches a value calculated using the non-reactive two-shock theory for pseudo-steady flows. The numerical results reveal that the transition to Mach reflection occurs earlier in the case of a ZND detonation than in the case of an inert shock wave because of the higher sound speed due to the release of chemical heat.

The Peseudo-Static Displacement Calculation of Reinforced and Unreinforced Soil Walls in Frictional Cohesive Soils Using Horizontal Slices Method

Horizontal slices method is a simple and efficient tool for analysis of retaining walls. In this paper, using some of the principles of equilibrium and horizontal slices method, an analytical approach is presented to determine the displacement of concrete walls in the reinforced and unreinforced soils. The formulation of the so-called method is able to calculate wall displacement and critical wedge angle displacement for frictional cohesive soils and draw the distribution of displacement in the wall height. Based on the proposed method, a simple equation is presented to calculate the wedge angle of rupture in the frictional cohesive soils in seismic condition. Comparison of the results with previous methods and numerical methods shows that horizontal slices are able to analyze the concrete wall displacement, while the proposed method has the advantage of analyzing walls via considering soil parameters such as adhesion, seismic coefficients, and the number of reinforcements. In contrast with previous methods that ignored the cohesion concrete wall characteristics, the offered method considers all wall and soil parameters. In addition, design charts are presented for different soils and walls. Considering wall displacement, maximum displacement of wall, displacement of wall apex, and maximum place of wall displacement are obtained, revealing that critical wedge angle displacement is different from rupture wedge angle of pressure imposed on wall.

Length scale effect on Mach reflection of cellular detonations

An experimental study of the Mach reflection of cellular detonations over the wedge is reported in this paper. Throughout the experiments, high-quality smoked foil is obtained to record cell pattern variation. The initial pressures are varied to yield cellular detonations with varied transverse wave spacing (or cell size). This paper focuses on length scales controlling the deviation and recovery of self-similarity in an unsteady Mach reflection process. The results show that the frozen limit and equilibrium limit both exist for all the mixture compositions. The Mach reflection undergoes a frozen condition in the near field, and then goes through a transition process before asymptotically approaching an equilibrium state in the far field. The cell size variation in the Mach stem region also confirms the transient process. Although the final triple-point trajectory angle in the equilibrium state disagrees with classical three-shock theories, it is in agreement with the reactive three-shock theory when the wedge angle is less than 30°. The triple-point trajectory angle asymptotically approaches zero as the wedge angle increases to approximately 50°, indicating the critical wedge angle from Mach to regular reflection. The transition length associated with the equilibrium limit is found to be dependent on the mixture composition and has the same order of a hydrodynamic thickness, which is approximately a few cell lengths. This means that the hydrodynamic thickness is the characteristic length scale that most significantly dominates the Mach reflection process of cellular detonations.

Insights into the velocity-dependent geometry and internal strain in accretionary wedges from analogue models

AbstractAlthough the brittle material in analogue models is characterized by a linear Navier-Coulomb behaviour and rate-independent deformation, the geometry and style of deformation in accretionary wedges is sensitive to shortening velocity. In this study we have constructed a series of analogue models with various shortening velocities in order to study the influence of shortening velocity on the geometry and kinematics of accretionary wedges. Model results illustrate how shortening velocity has an important influence on the geometry and kinematics of the resulting wedge. In general, for models having similar bulk shortening, the accretionary wedges with higher velocities of shortening are roughly steeper, higher and longer, as well as having larger critical wedge angles and height. It accommodates a number of foreland-vergent thrusts, larger fault spacing and displacement rates than those of low- to medium-velocity shortening, which indicates a weak velocity-dependence in geometry of the wedge. Moreover, models with a high velocity of shortening undergo larger amounts of volumetric strain and total layer-parallel shortening than models with low- to medium-velocity shortening. The former accommodate a greater development of back thrusts and asymmetric structures; a backwards-to-forwards style of wedge growth therefore occurs in the frontal zone under high-velocity shortening.

Mach reflection of a ZND detonation wave

The Mach reflection of a ZND detonation wave on a wedge is investigated numerically. A two-step chain-branching reaction model is used giving a thermally neutral induction zone followed by a chemical reaction zone for the detonation wave. The presence of a finite reaction zone thickness renders the Mach reflection process non-self-similar. The variation of the height of the Mach stem with distance of propagation does not correspond to a straight curve from the wedge apex as governed by self-similar three-shock theory. However, the present results indicate that in the near field around the wedge apex, and in the far field where the reaction zone thickness is small compared to the distance of travel of the Mach stem, the behavior appears to be self-similar. This corresponds to the so-called frozen and equilibrium limit pointed out by Hornung and Sanderman for strong discontinuity shock waves and by Shepherd et al. for cellular detonations. The critical wedge angle for the transition from regular to Mach reflection is found to correspond to the value determined by self-similar three-shock theory, but not by reactive three-shock theory for a discontinuous detonation front.

Pseudo-dynamic evaluation of passive response on the back of a retaining wall supporting c-Φ backfill

The study presents a rational analytical approach to obtain the seismic passive response of an inclined retaining wall backfilled with horizontal c-Φ soil. Pseudo-dynamic analysis is carried out to obtain the seismic passive response. Here in this analysis, the critical wedge angle is a single one irrespective of weight, surcharge and cohesion and this fact satisfies the field situation in a more realistic manner. A planer failure surface is considered in the analysis. The effect of soil and wall friction angle, wall inclination, horizontal and vertical earthquake acceleration on the passive resistance and the variation of passive earth pressure along the height of the wall have been explored. A comparison to pseudo-static and other available methods have been made to highlight the non-linearity of seismic passive earth pressure distribution.

Numerical Study on Critical Wedge Angle of Cellular Detonation Reflections

The critical wedge angle (CWA) for the transition from regular reflection (RR) to Mach reflection (MR) of a cellular detonation wave is studied numerically by an improved space-time conservation element and solution element method together with a two-step chemical reaction model. The accuracy of that numerical way is verified by simulating cellular detonation reflections at a 19.3° wedge. The planar and cellular detonation reflections over 45°–55° wedges are also simulated. When the cellular detonation wave is over a 50° wedge, numerical results show a new phenomenon that RR and MR occur alternately. The transition process between RR and MR is investigated with the local pressure contours. Numerical analysis shows that the cellular structure is the essential reason for the new phenomenon and the CWA of detonation reflection is not a certain angle but an angle range.

Numerical study of the oscillations induced by shock/shock interaction in hypersonic double-wedge flows

In this paper, the shock pattern oscillations induced by shock/shock interactions over double-wedge geometries in hypersonic flows were studied numerically by solving 2D inviscid Euler equations for a multi-species system. Laminar viscous effects were considered in some cases. Temperature-dependent thermodynamic properties were employed in the state and energy equations for consideration of the distinct change of the thermodynamic state. It was shown that the oscillation results in high-frequency fluctuations of heating and pressure loads over wedge surfaces. In a case with a relatively lower free-stream Mach number, the shock/shock interaction structure maintains a seven-shock configuration during the entire oscillation process. On the other hand, the oscillation is accompanied by a transition between a six-shock configuration (regular interaction) and a seven-shock configuration (Mach interaction) in a case with a higher free-stream Mach number. Numerical results also indicate that the critical wedge angle for the transition from a steady to an oscillation solution is higher compared to the corresponding value in earlier numerical research in which the perfect diatomic gas model was used.

Numerical analysis of coupled wedge plasmons in a structure of two metal wedges separated by a gap

This paper presents the results of the numerical finite-difference time-domain analysis of a strongly localized antisymmetric plasmon, coupled across a nanogap between two identical metal wedges. Dispersion, dissipation, field structure, and existence conditions of such coupled wedge plasmons are determined and investigated on an example of the fundamental coupled mode. It is shown that in the general case there exist three critical wedge angles and a critical gap width (separation between the wedge tips). If the gap width is larger than the critical separation, then the antisymmetric wedge plasmons can exist only in the ranges between the first and the second critical angles, and between the third critical angle and 180°. If the gap width is smaller or equal to the critical separation, then the third and the second critical angles merge, leaving only one interval of wedge angles within which the antisymmetric coupled wedge plasmons can exist. The effect of rounded wedge tips is also investigated and is shown to be similar to that of different wedge angles. Feasibility of using these plasmons for the design of efficient subwavelength waveguides is discussed.

Specimen Design for IFSS Measurement in Fiber Pullout Test

Since stress singularity was found at the interface end in current specimen of pullout test, interface shear strength (IFSS) obtained from the tests loses its rationality [2]. But a useful conclusion [2] is that when the wedge angle of the matrix is less than a critical angle, the singularity of stress field at the interface end of the specimen in micro-debond test nearly disappears. Following this conclusion, a conic specimen shown in Fig. 1 is presented, in which the wedge angle of the specimen is designed to be less than a critical angle in order to prevent the singular stress field occurred at the interface end. The conic specimen is designed for pullout test to avoid disadvantages inherent in the micro-debond test [3]. An axisymmetric model of fiber/matrix system with arbitrary wedge angles at the interface end is used for the determination of critical wedge angle. With the aid of asymptotic analysis and variable separation, eigenvalue, λ, could be determined by a characteristic determinant. For a given fiber-matrix system, a curve representing the relationship between the stress singularity index and wedge angle could be obtained by solving the characteristic determinant. We define the critical wedge angle, θcr, as the corresponding singularity index of – 0.005. The design of a conic pullout specimen is also discussed. FEM analysis is adopted to calculate the distribution of interfacial stresses near the interface end with different wedge angle. The results verify the rationality of the principle of the design of conic pullout specimen for IFSS measurement.

Computation of capillary surfaces for the Laplace–Young equation

Summary A novel hybrid flnite-element/flnite-volume numerical method is developed to determine the capillary rise of a liquid with a free surface (under surface tension and gravitational forces). The few known exact analytical solutions are used to verify the numerical computations and establish their accuracy for a range of liquid contact angles. The numerical method is then used to ascertain the limitations of a number of theoretical approximations to solutions for the capillary rise in the linearized limit, for special geometries such as plane walls, concentric cylinders and in a wedge of arbitrary included angle. The existence of a critical wedge angle for a given contact angle is verifled. However, the efiect of slight practical rounding of wedge corners dramatically reduces the theoretical corner height.

Design free of stress singularities for bi-material components

The disappearance of a stress singularity near the apex in a bi-material interface is investigated based on an eigenvalue investigation in conjunction with finite element analysis. Contours of the combination of critical wedge angles, from which the disappearance conditions of the stress singularity can be found, are obtained. Emphasis is also placed on how to apply these conditions to design components or specimens. Finite element analysis of various joint geometries with both normal and shear loadings is used to demonstrate the possible effects of geometry factors on the stress distributions along the interface. The focus of this paper is the problems associated with elimination of the stress singularities, with the goal of providing an approach for designing singularity-free bi-material joints.

Coulomb wedge analysis of cuts in steep slopes

Coulomb's trial wedge method for estimating the active force on retaining walls is applied to situations where there are steep slopes at limiting equilibrium behind the walls. The analysis is applied initially to slopes of dry cohesionless material; in this case the slope angle is equal to the friction angle of the material. The analysis produces two interesting, and perhaps surprising, results. The first is that the critical wedge angle equals the slope angle, and the critical wedge becomes a "slab" extending an unlimited distance up the slope above the wall. The second is that as the inclination of the slope and friction angle (which are equal) increase the active force on the wall decreases. The method is then applied to more general slopes involving cohesion, friction, and seepage. Back-analysis is first used to obtain combinations of cohesion and friction corresponding to limiting equilibrium; these are then used in the wedge analysis. The results obtained are similar to those for the dry cohesionless slope. The method provides a means of estimating required retaining forces for cuts made in steep slopes of residual soil or partly weathered rock where estimation of strength parameters is often very difficult.Key words: slopes, steep, back-analysis, Coulomb, wedge, analysis.

On the absolute instability of the triple–deck flow over humps and near wedged trailing edges

The triple-deck equations for the flow over a hump, a corner and a wedged trailing edge are solved numerically using a novel method based on spectral collocation. It is found that for the flow over a corner, separation begins at a scaled angle β of 2.09, and for the wedged trailing edge for a wedge angle of 2.56. Here β is defined in terms of the small physical angle π by β= Re 1/4λ-1/2π, λ= 0.3320, and Re is the Reynolds number. The absolute instability of the nonlinear mean flows computed is investigated. It is found that the flow over a hump is inviscidly absolutely unstable with the maximum absolute unstable growth rate occurring near the maximum height of the hump, and increasing with hump size. The wake region behind the wedged trailing edge is also found to be absolutely unstable beyond a critical wedge angle, and the extent of the region of absolute instability increases with increasing wedge angle and separation.

Flow near a sharp corner in a nematic liquid crystal

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Díez M. and Atkinson C. 2000Flow near a sharp corner in a nematic liquid crystalProc. R. Soc. Lond. A.45663–95http://doi.org/10.1098/rspa.2000.0509SectionRestricted accessFlow near a sharp corner in a nematic liquid crystal M. Díez M. Díez Escuela Superior de Ingenieros Industriales, Universidad de Navarra, P. Manuel Lardizábal 13, 20018 San Sebastián, Spain Google Scholar Find this author on PubMed Search for more papers by this author and C. Atkinson C. Atkinson Department of Mathematics, Imperial College of Science, Technology and Medicine, 180Que en's Gate, London SW7 2BZ, UK Google Scholar Find this author on PubMed Search for more papers by this author M. Díez M. Díez Escuela Superior de Ingenieros Industriales, Universidad de Navarra, P. Manuel Lardizábal 13, 20018 San Sebastián, Spain Google Scholar Find this author on PubMed Search for more papers by this author and C. Atkinson C. Atkinson Department of Mathematics, Imperial College of Science, Technology and Medicine, 180Que en's Gate, London SW7 2BZ, UK Google Scholar Find this author on PubMed Search for more papers by this author Published:08 January 2000https://doi.org/10.1098/rspa.2000.0509AbstractSolutions are presented for the orientation and flow of a positive nematic liquid crystal near a corner between two planes on which a variety of boundary conditions may be imposed.The flow near the corner is induced by a general motion at a large distance from the corner, which can make the flow symmetric or antisymmetric with respect to the bisector of the angle. The real situation may be a combination of both.The viscosity, usually a damping mechanism, may be responsible for the generation of a geometrical progression of eddies. The smaller the angle between plates, the more probable the existence of eddies. The critical wedge angle for the appearance of eddies is also investigated.The conditions considered are rigid walls and no–slip conditions at the boundaries, which have been treated in order to get a perfect alignment of the directors. For the two specific examples analysed, PAA near 125°C and MBBA near 25°C, eddies only appear in MBBA for an angle between plates smaller than 60°, when the flow induced near the corner is antisymmetric with respect to the bisector of the plates. In this case, the flow near the corner consists of a sequence of eddies whose size and intensity fall off in geometric progression with a ratio depending only on the angle between plates.PAA and MBBA present this qualitative change in behaviour because the viscosity coefficients are much smaller in magnitude in PAA than in MBBA. We construct a nematic liquid crystal with Leslie coefficients, which vary monotonically between the two sets of coefficients for PAA and MBBA. Writing the six viscosity coefficients as μ = μiP + (1 − x)μiM, with i = 1, 6, where the superscript ‘M’ is for MBBA and ‘P’ is for PAA, we find that the critical wedge angle at which eddies would appear (in antisymmetric flow) decreases as x increases from zero to one. The critical wedge angle is zero for x > 0.974 approximately, so no eddies exist in these cases. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Walker A and Stewart I (2011) Wave-induced perturbations in cylindrically layered smectic A liquid crystals, Zeitschrift für angewandte Mathematik und Physik, 10.1007/s00033-011-0155-y, 63:2, (357-371), Online publication date: 1-Apr-2012. Walker A and Stewart I (2007) Periodic Disturbances in Cylindrically Layered Smectic A, Molecular Crystals and Liquid Crystals, 10.1080/15421400701731995, 478:1, (33/[789]-43/[799]), Online publication date: 14-Dec-2007. This Issue08 January 2000Volume 456Issue 1993 Article InformationDOI:https://doi.org/10.1098/rspa.2000.0509Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/01/2000Published in print08/01/2000 License: Citations and impact KeywordsPAAcornerviscosity coefficientsMBBAnematicliquid crystal

The remarkable nature of cylindrically orthotropic elastic materials under plane strain deformations

In a recent paper by the author, a wedge of cylindrically orthotropic elastic material under anti-plane deformations was studied. The solution depends on one non-dimensional material parameter γ, which is the square root of the ratio of the two shear moduli. For any given wedge angle 2α (no matter how small), one can choose a γ so that the stress at the wedge apex is infinite. In the special case of a crack (2α = 2π) there may be more than one stress singularity at the wedge apex. Some of them can be larger than the square-root singularity. On the other hand, one can also choose a γ so that there is no stress singularity at the wedge apex for any wedge angle, including the special case of a crack. In this paper we show that the same remarkable nature for anti-plane deformations prevails for the more complicated plane strain deformations. (The phrase 'remarkable nature' was used in the title of a paper by Antman and Negron-Marrero who considered the pressuring of radially symmetric nonlinear elastic bodies.) The solution now depends on two non-dimensional material parameters η and γ. The γ here is the square root of the ratio of the two principal elastic stiffnesses. The existence or non-existence of a singularity at the wedge apex depends on both η and γ. The classical paradox of Levy also appears here. The existence of a critical wedge angle depends entirely on η, not on γ. A critical wedge angle can be any angle, and there may be more than one critical wedge angle.

Non-linear solutions for smectic C liquid crystals in wedge and cylinder geometries

This paper discusses some non-linear problems for smectic C liquid crystals based on the continuum theory proposed by Leslie et al. New restrictions on the nine elastic constants are also derived. Attention is restricted to samples involving concentric cylindrical layers in which both the layer thickness and the tilt angle are assumed to be constant. Non-linear solutions are presented for a sample contained in a wedge with an electric field applied across the bounding plates, extending earlier work by Carlsson et al., and for a sample between two coaxial concentric circular cylinders to which an azimuthal magnetic field is applied. Freedericksz thresholds, which may lead to the experimental determination of some of the elastic constants, are deduced. In the absence of an applied field it is found that, under suitable restrictions on the elastic constants, there is a critical wedge angle (or critical radius ratio in the concentric cylinder case) above which a variable non-linear symmetric solution satisfying the zero boundary conditions is energetically more favourable than the zero solution.

The Remarkable Nature of Cylindrically Anisotropic Elastic Materials Exemplified by an Anti-Plane Deformation

A material is cylindrically anisotropic when its elastic moduli referred to a cylindrical coordinate system are constants. Examples of cylindrically anisotropic materials are tree trunks, carbon fibers [1], certain steel bars, and manufactured composites [2]. Lekhnitskii [3] was the first one to observe that the stress at the axis of a circular rod of cylindrically monoclinic material can be infinite when the rod is subject to a uniform radial pressure (see also [4]). Ting [5] has shown that the stress at the axis of the circular rod can also be infinite under a torsion or a uniform extension. In this paper we first modify the Lekhnitskii formalism for a cylindrical coordinate system. We then consider a wedge of cylindrically monoclinic elastic material under anti-plane deformations. The stress singularity at the wedge apex depends on one material parameter γ. For a given wedge angle α, one can choose a γ so that the stress at the wedge apex is infinite. The wedge angle 2α can be any angle. It need not be larger than π, as is the case when the material is homogeneously isotropic or anisotropic. In the special case of a crack (2α=2π) there can be more than one stress singularity, some of them are stronger than the square root singularity. On the other hand, if γ < \( - \frac{1}{2}\) there is no stress singularity at the wedge apex for any wedge angle, including the special case of a crack. The classical paradox of Levy [6] and Carothers [7] for an isotropic elastic wedge also appears for a cylindrically anisotropic elastic wedge. There can be more than one critical wedge angle and, again, the critical wedge angle can be any angle.