Numerical investigation of the onset of instability of triple shock configurations in steady supersonic gas flows

Triple configurations of shock waves that arise in a steady supersonic flow of a real gas have been numerically simulated. Previously, we have theoretically predicted that a new triple shock configuration with a negative reflection angle can appear in a gas flow at large Mach numbers and small adiabatic indices. This configuration is now obtained for the first time in numerical experiments. It is shown that the formation of this triple shock configuration leads to instability of the entire flow pattern.     

On reflection of shock waves and shock-wave configurations

A new approach to the problem of unsteady reflection of shock waves is proposed. It is shown that the type and parameters of the shock wave reflection configuration depend on the type of an object (single shock versus shock-wave configuration) incident on the reflecting surface. The concept of the reflection of a shockwave configuration is introduced, and data are presented to illustrate the fact that theoretically established boundaries of the existence of various types of configurations do not represent the transitions lines if shockwave configurations rather than single shocks interact with the surface.     

Finite element computation of unsteady viscous transonic flows past stationary airfoils

Results are presented for computations of unsteady viscous transonic flows past a stationary NACA0012 airfoil at various angles of attack. The Reynolds number, based on the chord-length of the airfoil, is 10,000 and the Mach number is 0.85. Stabilized finite-element formulations are employed to solve the compressible Navier-Stokes equations. The equation systems, resulting from the discretization, are solved iteratively by using the preconditioned GMRES technique. Time integration of the governing equations is carried out for large values of the non-dimensional time to understand the unsteady dynamics and long-term behavior of the flows. The results show interesting flow patterns and a complex interaction between the boundary/shear layers, shock/expansion waves and the lateral boundaries of the computational domain. For transonic flow past an airfoil at various angles of attack in a narrow channel/wind-tunnel one can observe solutions that are qualitatively different from each other. At low angles of attack an unsteady wake is observed. At moderate angles of attack the interaction between the shock system and the lateral walls becomes significant and the temporal activity in the wake decreases and eventually disappears. At high angles of attack a reflection shock is formed. Hysteresis is observed at an angle of attack 8^∘. For the flow in a domain with the lateral boundaries located far away, the unsteadiness in the flow increases with an increase in the angle of attack. Computations for a Mach 2, Re 106 flow past an airfoil at 10^∘ angle of attack compare well with numerical and experimental results from other researchers     

On the computation of steady‐state compressible flows using a discontinuous Galerkin method

Computation of compressible steady‐state flows using a high‐order discontinuous Galerkin finite element method is presented in this paper. An accurate representation of the boundary normals based on the definition of the geometries is used for imposing solid wall boundary conditions for curved geometries. Particular attention is given to the impact and importance of slope limiters on the solution accuracy for flows with strong discontinuities. A physics‐based shock detector is introduced to effectively make a distinction between a smooth extremum and a shock wave. A recently developed, fast, low‐storage p‐multigrid method is used for solving the governing compressible Euler equations to obtain steady‐state solutions. The method is applied to compute a variety of compressible flow problems on unstructured grids. Numerical experiments for a wide range of flow conditions in both 2D and 3D configurations are presented to demonstrate the accuracy of the developed discontinuous Galerkin method for computing compressible steady‐state flows. Copyright © 2007 John Wiley & Sons, Ltd.     

Grazing reflection of Gaussian beams

The reflectivities of most surfaces are higher for grazing or near-90-deg angles of incidence than for more perpendicular or near-zero-deg angles. Grazing-incidence configurations are especially important in the development of lasers and optical systems that operate in the far-ultraviolet and soft-x-ray regions of the spectrum, where transparent or highly reflecting media are almost unknown. Analytical solutions of the paraxial wave equation are obtained for the grazing reflection and complex interference effects that take place when a Gaussian beam interacts at shallow angles with a reflecting surface.     

Film cooling over a concave surface through two Staggered rows of compound angle holes

Abstract

Experimental results of film cooling effectiveness over a concave surface with two staggered rows are presented by employing transient liquid crystal thermography. Four different discrete hole configurations are used for the injection of jet flow, including a straight circular hole configuration with a spanwise injection angle (β) of 0° and three forward‐expanded hole configurations with β = 0°, 45° and 90° respectively. In all test models there are two staggered rows of discrete holes with streamwise injection angle (γ ) of 35°. Blowing ratios (M) are 0.5, 1.0, and 2.0. The effects of blowing ratio, hole expanded angle, and injection angle orientation on film cooling performance are investigated. The jet flow with M = 0.5 is fairly uniform along the wall surface. The lift‐off phenomenon can be found in the jet flow with β = 0° for both cases of simple and compound angles at M = 1.0 and also exists among all test cases at M = 2.0 except for β = 90°. The lift‐off effect results in a decrease in both η and h /h0. At β = 0°, the jet flow with a forward‐expanded hole gives higher η and lower h /h0 than a simple angle hole. At a fixed blowing ratio, the jet flow with compound angle holes has lower q /q0 and thus provides better wall protection than that with simple angle holes. In the present study, the compound angle with β = 0° at M = 2.0 provides the best film‐cooling protection over the concave surface among all the test configurations.     

Angle of Total Refraction in the Interaction of a Plane Skew Shock Wave with the Interface of Two Media

The solution of practical problems associated with determination of the shockwave configuration that results from the interaction of a stationary plane skew shock wave (SSW) with the interface of two media, is invariably related to determination of the critical angles of the interaction. The author suggests a method of analytical calculation of one of these angles within the framework of the applicability of the hydrodynamic theory of shock waves, namely, the angle of total refraction, denoted by _t in some of the indicated works.     

On the form of the elastic potential relative to different reference states

Summary Using the method of wavefront analysis the paper presents an analysis of shock wave formation in a two-dimensional steady supersonic flow of a radiating gas past plane and axisymmetric bodies such as a beak and sharp edged ring. Transport equations are derived which lead to the determination of the shock formation distance and also to conditions which insure that no shock will ever evolve on the wavefront. It is assessed as to how the shock formation distance is influenced by the presence of thermal radiation, the initial body curvature and upstream flow Mach numberM ₀>1.With 1 Figure     

Boundaries of the ambiguity area upon reflection of compression shock waves

Oblique shock waves can be reflected from hard walls, the axis, or the plane of symmetry, as well as from other counterpropagating shock waves with the formation of regular and Mach shock wave configurations. The specific form of shock wave structures is determined by the parameters of the problem: Mach number and intensity of incident shock waves. On the plane of parameters, there exists an ambiguity area in which laws of conservation admit both the regular and Mach reflection of shock waves. The boundaries of this region have been determined.     

The structure of gasdynamic flow in a supersonic separator of natural gas

Results are given of calculations of flow within the two-dimensional Euler model of supersonic swirling flow of gas in a supersonic separator of natural gas. The formulation of the problem is given, numerical experiment is performed, and the basic parameters of gas flow (velocity components, pressure, and so on) are obtained as functions of radius. The process of relaxation of flow to steady state with the formation of shock wave is considered, and the shock wave structure is determined. The behavior of gasdynamic parameters is analyzed under conditions of separation in the region of shock wave and behind it.     

Flutter of a viscoelastic strip

An approximate solution of the problem of flutter of a cantilever clamped viscoelastic strip is obtained under conditions where the flow velocity vector is parallel to the plane of the strip and forms an angle with its edges, which can take arbitrary values. The approximate solution in all the cases is based on the linear combinations of polynomials, which satisfy the boundary conditions. Approximate estimates of the values of the critical flutter speed are obtained by using the Laplace transform in time and a Galerkin expansion for the spatial structures. The nature of the changes in the critical speed with respect to the value of the angle of flow is examined. It is shown that for any non-negative angle of flow with increasing velocity of flow there is an instability in the form of flutter. For negative angles of the flow, either flutter or a cylindrical bending of the strip is observed. Principally, a new mechanical effect is found: there is a whole sector of directions for which increase in flow rate leads to a cylindrical bending.     

Shock wave reflection and diffraction on a convex double wedge

We have numerically simulated the interaction of a shock wave with a convex double angle within the framework of a model of inviscid non-heat-conducting gas. The main attention is paid to the stage of a two-shock diffraction configuration on the second face of the wedge. Special features of flow under various condi-tions of diffraction are revealed. We also propose an explanation of the appearance and behavior of a purely gasdynamic layer formally resembling the viscous boundary layer.     

Interference of nonstationary oblique shock waves

Special features of calculation of the flow parameters behind a nonstationary oblique shock wave moving in a stream of absolutely nonviscous gas are considered. The wave intensity at which the stream behind the shock wave may exhibit singularities is determined. The problem of calculating a nonstationary shock wave configuration formed during the interaction of a supersonic jet with an obstacle is solved.     

Rotating Inclined Cylinder and the Effect of the Tilt Angle on Vortices

We study numerically some possible vortex configurations in a rotating cylinder that is tilted with respect to the rotation axis and where different numbers of vortices can be present at given rotation velocity. In a long cylinder at small tilt angles the vortices tend to align along the cylinder axis and not along the rotation axis. We also show that the axial flow along the cylinder axis, caused by the tilt, will result in the Ostermeier-Glaberson instability above some critical tilt angle. When the vortices become unstable the final state often appears to be a dynamical steady state, which may contain turbulent regions where new vortices are constantly created. These new vortices push other vortices in regions with laminar flow towards the top and bottom ends of the cylinder where they finally annihilate. Experimentally the inclined cylinder could be a convenient environment to create long lasting turbulence with a polarization which can be adjusted with the tilt angle.     

Coefficients of transformation of long-wavelength perturbations of a supersonic incident flow around a wedge into pressure fluctuations on its surface

The interaction of different types of long-wavelength perturbations of a supersonic incident flow around a wedge with an arising shock wave is simulated numerically. The results of parametric calculations are compared with the data of the linear theory for the interaction of perturbations with a shock wave in the absence of a body. Data on the structure of pressure fluctuations behind the shock wave on the wedge and data on the coefficients of transformation of external perturbations represented in the form of analytical dependences on the angle of inclination of the shock wave, the angle of propagation of external perturbations, and the incident Mach number are obtained.     

某跨音速高压涡轮非定常流动数值模拟研究

通过求解三维非定常雷诺平均N-S方程模拟某跨音速高压涡轮非定常流场,研究涡轮内非定常流动特征。通过对静子尾迹及静子尾缘激波和转子叶排之间的相互干涉过程进行详细分析,发现定常/非定常模拟方法获得的涡轮总体性能参数基本一致但流场存在较大差异。静子尾迹是导致涡轮流场非定常性的重要因素之一:在转子叶栅通道中部和下部,静子尾迹和转子叶片附面层及下通道涡发生明显干涉,并导致通道中下部损失周期性波动幅度较大,此外尾迹和下通道涡间的干涉作用在转子尾缘处诱导出高频脱落涡。静子尾缘激波也是导致涡轮流场非定常性的原因之一,激波和转子叶片作用形成复杂的波系结构,对涡轮流场影响显著:一方面激波/附面层干涉导致转子和静子的吸力面产生周期性变化的高温区域;另一方面激波撞击叶片导致叶片表面的气流在激波后出现分离,对转子静压分布产生影响,使得转子叶片表面载荷出现明显的非定常性,进而导致涡轮输出功的周期性波动十分剧烈。     

Water wave diffraction by a small deformation of the ocean bottom for oblique incidence

Summary In this paper, the problem of oblique water wave diffraction by a small deformation of the bottom of a laterally unbounded ocean is considered using linear water wave theory. It is assumed that the fluid is incompressible and inviscid, and the flow irrotational. A perturbation analysis is employed to obtain the velocity potential, reflection and transmission coefficients up to the first order in terms of integrals involving the shape functions c(x) representing the bottom deformation by using Green's integral theorem. Two particular forms of the shape function are considered, and the integrals for the reflection and transmission coefficients are evaluated for these two different functions. Among those cases, for the particular case of a patch of sinusoidal ripples at the bottom, the reflection coefficient up to the first order is found to be an oscillatory function in the quotient of twice the wave number along the x-axis and the ripple wave number. When this quotient becomes one, the theory predicts a resonant interaction between the bed and free surface, and the reflection coefficient becomes a multiple of the number of ripples, and high reflection of the incident wave energy occurs if this number is large. Known results for the normal incidence are recovered as special cases. The numerical solutions for the reflection and transmission coefficients are also evaluated against wave numbers and angles of incidence.     

Self-collimated waveguide bends and partial bandgap reflection of photonic crystals with parallelogram lattice

The photonic crystal structure with parallelogram lattice, capable of bending a self-collimated wave with free angles and partial bandgap reflection, is presented. The equifrequency contours show that the direction of the collimation wave can be turned by tuning the angle between the two basic vectors of the lattice. Acute, right, and obtuse angles of collimating waveguide bends have been realized by arc lattices of parallelogram photonic crystals. Moreover, partial bandgap reflection of the parallelogram lattice photonic crystals is validated from the equifrequency contours and the projected band structures. A waveguide taper based on this partial bandgap reflection is also designed and proved to have above 85% transmittance over a very wide operating bandwidth of 180 nm.     

Reflection and transmission of strongly focused light beams at a dielectric interface

The reflection and transmission of tightly focused azimuthally, radially and linearly polarized electromagnetic wave beams with subwavelength spot size and wavefront curvature at a dielectric interface are investigated. For a given wavefront radius, the existence of the optimal radius of beam spot corresponding to a minimal reflectance and maximal transmittance is shown. Significant lateral shift in the transmitted intensity peak is revealed for strongly focused azimuthally and radially polarized beams that are normally incident to an interface. The reflection and transmission of transverse-electric- and transverse-magnetic- polarized extremely narrow wave beams which are obliquely incident on the dielectric interface is analysed. Disappearance of the Brewster angle and total internal reflection effects for the strongly focused beams are predicted. The change in beam profile after reflection and transmission for different polarizations, incident beam spots and incidence angles are analysed.     

Contours of the constant third Brewster angle in the complex plane of a dielectric constant

Three special angles associated with the external reflection of light at the surface of an absorbing medium are well known: the principal angle, the pseudo-Brewster angle, and the second Brewster angle. Another significant angle has been identified recently by Azzam and El-Saba [Appl. Opt. 28, 1365 (1989)]. At this angle, the slope of the differential reflection phase shift with respect to the angle of incidence is maximum negative. For convenience it is called the third Brewster angle. The nature of the contours of the constant third Brewster angle in a complex dielectric constant plane is considered both analytically and graphically in comparison with the other constant special-angle contours.