A soap film shock tube to study two-dimensional compressible flows

 A new experimental approach to the study of the two-dimensional compressible flow phenomena is presented. In this technique, a variety of compressible flows were generated by bursting plane vertical soap films. An aureole and a “shock wave” preceding the rim of the expanding hole were clearly observed using traditional high-speed flash photography and a fast line-scan charge coupled device (CCD) camera. The moving shock wave images obtained from the line-scan CCD camera were similar to the x–t diagrams in gas dynamics. The moving shock waves cause thickness jumps and induce supersonic flows. Photographs of the supersonic flows over a cylinder and a wedge are presented. The results suggest clearly the feasibility of the “soap film shock tube”. Received: 11 May 2000/Accepted: 2 November 2000     

Nonlinear Strong Shock Interactions: A Shock-Fitted Approach

This paper addresses nonlinear effects which result from the interaction of shock waves with vortices. A series of experiments are carried out, which involve the interaction of a strong shock wave with a single plane vorticity wave and a randomly distributed wave system. These experiments are first conducted in the linear regime to obtain a mutual verification of theory and computation. They are subsequently extended into the nonlinear regime. A systematic study of the interaction of a plane shock wave and a single vortex is then conducted. Specifically, we investigate the conditions under which nonlinear effects become important, both as a function of shock Mach number, M ₁, and incident vortex strength (characterized by its circulation Γ). The shock Mach number is varied from 2 to 8, while the circulation of the vortex is varied from infinitesimally small values (linear theory) to unity. Budgets of vorticity, dilatation, and pressure are obtained. They indicate that nonlinear effects become more significant as both the shock Mach number and the circulation increase. For Mach numbers equal to 5 and above, the dilatation in the vortex core grows quadratically with circulation. An acoustic wave propagates radially outward from the vortex center. As circulation increases, its upstream-facing front steepens at low Mach numbers, and its downstream-facing front steepens at high Mach numbers. A high Mach number asymptotic expansion of the Rankine--Hugoniot conditions reveals that nonlinear effects dominate both the shock motion and the downstream flow for ΓM ₁ > 1. Received 28 June 1997 and accepted 25 November 1997     

Shock wave triple-point morphology

The existence and characteristics of shock wave triple points are examined. The analysis utilizes a single flow plane for the three shocks and is local to the triple point. It applies when the flow is unsteady, three-dimensional, and the upstream flow is nonuniform. Under more restrictive conditions, a relation is also derived for the ratio of the curvature of the Mach stem to that of the reflected shock. For given values of the ratio of specific heats, γ, and the upstream Mach number, M ₁, a solution window is established. A parametric set of solutions is generated within the window for γ = 1, 1.4, and 5/3 and for 16 values of M ₁ ranging from solution onset to M ₁ = 6.A solution can be one of three types, these stem from the velocity tangency condition along the slip stream. Topics are addressed such as solution multiplicity, shock wave and slip stream orientation, the nature of the reflected wave (weak, strong, inverted, normal), the nature of the Mach stem (weak, strong, normal), and differences due to changes in γ and M ₁.     

Vorticity and its rate of change just downstream of a curved shock

A theory is developed for the vorticity and its substantial derivative just downstream of a curved shock wave, the resulting formulas are exact, algebraic, and explicit. Analysis is for a cylinder-wedge or sphere-cone body, at zero incidence, whose downstream half-angle is θ_b. Derived formulas directly depend only on the ratio of specific heats, γ, the freestream Mach number, M ₁, the local slope and curvature of the shock, and the dimensionality parameter, σ, which is zero for a two-dimensional shock and unity for an axisymmetric shock. In turn, the slope and curvature depend on γ, M ₁, and θ_b. Numerical results are provided for a bow shock in which θ_b is 5°, 10°, or 15°, M ₁ is 2, 4, or 6, and γ = 1.4. There is little dependence on the half angle but a strong dependence on the freestream Mach number and on dimensionality. For vorticity and its substantial derivative, the dimensionality dependence gradually decreases with increasing Mach number. In comparison to the two-dimensional case, an axisymmetric shock generates considerable vorticity in a region relatively close to the symmetry axis. Moreover, the magnitude of the vorticity, in this region, is further enhanced in the flow downstream of the shock. This dimensionality difference in vorticity and its substantial derivative is attributed to the three-dimensional relief effect in an axisymmetric flow.

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Type of reflection of a near-wall shock wave in the process of diffraction from a square channel

The results of an experimental and numerical investigation of the process of diffraction of shock waves from a square channel at a ninety-degree convex corner are presented for various incident shock wave Mach numbers M₀ (1.4<M₀<7). The type of reflection of the near-wall fragment of the diffracting shock wave from the wall and the wave velocity are determined as functions of M₀, direction, and time. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 145–151, January–February, 2000. The work was carried out with partial support from the Russian Foundation for Basic Research (project No. 96-02-16170a).     

The criterion of the existence or inexistence of transverse shock wave at wedge supported oblique detonation wave

A simplified theoretic method and numerical simulations were carried out to investigate the characterization of propagation of transverse shock wave at wedge supported oblique detonation wave.After solution validation,a criterion which is associated with the ratio Φ (u 2 /u CJ) of existence or inexistence of the transverse shock wave at the region of the primary triple was deduced systematically by 38 cases.It is observed that for abrupt oblique shock wave (OSW)/oblique detonation wave (ODW) transition,a transverse shock wave is generated at the region of the primary triple when Φ < 1,however,such a transverse shock wave does not take place for the smooth OSW/ODW transition when Φ > 1.The parameter Φ can be expressed as the Mach number behind the ODW front for stable CJ detonation.When 0.9 < Φ < 1.0,the reflected shock wave can pass across the contact discontinuity and interact with transverse waves which are originating from the ODW front.When 0.8 < Φ < 0.9,the reflected shock wave can not pass across the contact discontinuity and only reflects at the contact discontinuity.The condition (0.8 < Φ < 0.9) agrees well with the ratio (D ave /D CJ) in the critical detonation.     

Numerical simulation of receptivity of a hypersonic boundary layer to acoustic disturbances

Direct numerical simulations of the evolution of disturbances in a viscous shock layer on a flat plate are performed for a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵. Unsteady Navier-Stokes equations are solved by a high-order shock-capturing scheme. Processes of receptivity and instability development in a shock layer excited by external acoustic waves are considered. Direct numerical simulations are demonstrated to agree well with results obtained by the locally parallel linear stability theory (with allowance for the shock-wave effect) and with experimental measurements in a hypersonic wind tunnel. Mechanisms of conversion of external disturbances to instability waves in a hypersonic shock layer are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 84–91, May–June, 2007.     

Design of a tunnel-entrance hood with multiple windows and variable cross-section

A theory is proposed for the design of a uniform tunnel-entrance hood whose cross-sectional area exceeds the tunnel area . A train entering the tunnel produces a low-frequency compression wave that can be subject to nonlinear steepening in a long tunnel. An optimized hood of length ℓ_h increases the initial thickness of the compression wave front from R/M to ℓ_h/M, where Rℓ_h is the nominal radius of the tunnel and M is the train Mach number. In addition, the pressure rise should be linear across the wave front to obtain an overall minimum value of the subjectively important pressure gradient. This is achieved in a uniform hood by distributing windows along the hood wall to vent away high-pressure air displaced by the train. We consider the problem of determining the distribution and sizes of these windows and the magnitude of the area ratio to ensure that the hood behaves optimally at low-train Mach numbers (M<0.2), when the hood can be regarded as being acoustically compact. At the projected higher Mach numbers of advanced high-speed trains (0.4, say) recent analysis for hoods of uniform cross-section by Howe in 2003 indicates that a hood optimized for low Mach number operations continues to produce an essentially linear pressure rise across a compression wave of thickness ℓ_h/M except for a low-amplitude oscillation at the very front of the wave.     

Experimental study and direct numerical simulation of the evolution of disturbances in a viscous shock layer on a flat plate

The evolution of disturbances in a hypersonic viscous shock layer on a flat plate excited by slow-mode acoustic waves is considered numerically and experimentally. The parameters measured in the experiments performed with a free-stream Mach number M _∞ = 21 and Reynolds number Re _L = 1.44 · 10⁵ are the transverse profiles of the mean density and Mach number, the spectra of density fluctuations, and growth rates of natural disturbances. Direct numerical simulation of propagation of disturbances is performed by solving the Navier-Stokes equations with a high-order shock-capturing scheme. The numerical and experimental data characterizing the mean flow field, intensity of density fluctuations, and their growth rates are found to be in good agreement. Possible mechanisms of disturbance generation and evolution in the shock layer at hypersonic velocities are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 3–15, September–October, 2006.     

Optimization of Conical Wings in Hypersonic Flow

A method of calculation is presented to determine conical wing shapes that minimize the coefficient of (wave) drag, C _D, for a fixed coefficient of lift, C _L, in steady, hypersonic flow. An optimization problem is considered for the compressive flow underneath wings at a small angle of attack δ and at a high free-stream Mach number M _∞ so that hypersonic small-disturbance (HSD) theory applies. A figure of merit, F=C _D/C _L ^3/2, is computed for each wing using a finite volume discretization of the HSD equations. A set of design variables that determine the shape of the wing is defined and adjusted iteratively to find a shape that minimizes F for a given value of the hypersonic similarity parameter, H= (M _∞δ)⁻², and planform area. Wings with both attached and detached bow shocks are considered. Optimal wings are found for flat delta wings and for a family of caret wings. In the flat-wing case, the optima have detached bow shocks while in the caret-wing case, the optimum has an attached bow shock. An improved drag-to-lift performance is found using the optimization procedure for curved wing shapes. Several optimal designs are found, all with attached bow shocks. Numerical experiments are performed and suggest that these optima are unique. Received 1 May 1998 and accepted 14 October 1998     

Effects of Mach number on turbulent separation behaviours induced by blunt fin

 An experimental study of the interaction between shock wave and turbulent boundary layer induced by blunt fin has been carried out at M _∞=7.8 using oil flow visualization and simultaneous measurements of fluctuating wall pressure and heat transfer. This paper presents the effects of Mach number on turbulent separation behaviours induced by blunt fin. Received: 21 July 1996/Accepted: 4 February 1998     

Numerical investigation of inviscid shock wave dynamics in an expansion tube

Highly complicated shock wave dynamics has been numerically calculated by solving the Euler equations for a circular shock tube suddenly expanded three times of the original tube diameter atx=0. Shock waves of different shock Mach number,M _s =1.5 and 2.0, have produced remarkably distinct blast jet structures. A planar shock wave took its final form after the blast by repeated Mach reflections of the blast wave: the first one at the wall and the second one at the central axis. The central Mach disc overtook and merged with the annular Mach stem before the planar shock wave was formed. In contrast to the blast wave which would propagate spherically in an open space, the present blast wave undergoes complex morphological transformation in the restricted flow passage, resulting in an unstable and oscillatory blast jet structure of highly rotational nature. The slipstream tube emanating from the shock tube exit corner decomposed into a chain of small vortex rings that interacted with the barrel shock of the jet, which caused periodic collapse of the jet structure. The finite volume-FCT formulation equipped with the time-dependenth-refinement adaptive unstructured triangular mesh technique in the present paper has contributed to resolution of the intricate physical discontinuities developing in the blast flow fields.     

Shock–vortex interactions in a soap film

This work experimentally visualizes the interaction of a quasi-one-dimensional moving shock wave with a two-dimensional vortex in a soap film for the first time. A vertical soap film shock tube was used to generate a quasi-one-dimensional moving shock wave and a NACA-0012 airfoil intruded into the soap film was towed to shed the starting vortex. The interesting interaction phenomena were then visualized using a traditional high-speed flash photography. The concentration of sodium dodecyl sulfate (SDS) used was 0.5 CMC (critical micelle concentration) to keep the surfactant molecules behave as two-dimensional gases. A sequence of pictures shows that the shock is distorted non-symmetrically as it passes through the spiral vortex flow field and the vortex structure is compressed in the direction normal to the shock. These flow features observed in soap films are qualitatively similar to their counterparts in gases. In addition, the visualization of the interactions of a quasi-one-dimensional moving shock wave with a Kármán vortex street are presented.      

Normal shock interaction with a yawed wedge moving at supersonic speed

In this article, the interaction of a normal shock with a yawed wedge moving at supersonic speed has been considered. The vorticity distribution of a particle over the diffracted shock wave for various combinations of yawed angles, Mach number of the shock wave and Mach number of the moving wedge have been obtained. Further triple point angle χ in Mach reflection has been calculated for the various parameters.      

Mach reflection of a plane shock wave passing a mountain of 45° inclination

The transition from regular reflection (RR) to Mach reflection (MR) as a plane shock wave diffracts around a triangular mountain of 45° inclination is analysed in this paper, both by optical measurement in a shock tube and by numerical simulation the numerical method developed by Li Yingfan^[1] is of the FLIC type with triangular mesh. The dependence of the critical transition point Lk ofRR→MR on shock Mach numberM _i is analyzed and the variations of the incidence angle ω _i of the impinging shock and the reflection angle ω _r with the distanceL ^* are investigated. Our experimental and numerical results agree well with the theoretical results of Iton and Italya.     

Experimental study of the shock–bubble interaction with reshock

The interaction of a planar shock wave with a spherical density inhomogeneity is studied experimentally under reshock conditions. Reshock occurs when the incident shock wave, which has already accelerated the spherical bubble, reflects off the tube end wall and reaccelerates the inhomogeneity for a second time. These experiments are performed at the Wisconsin Shock Tube Laboratory, in a 9m-long vertical shock tube with a large square cross section (25.4×25.4 cm²). The bubble is prepared on a pneumatically retracted injector and released into a state of free fall. Planar diagnostic methods are used to study the bubble morphology after reshock. Data are presented for experiments involving two Atwood numbers (A = 0.17 and 0.68) and three Mach numbers (1.35 < M < 2.33). For the low Atwood number case, a secondary vortex ring appears immediately after reshock which is not observed for the larger Atwood number. The post-reshock vortex velocity is shown to be proportional to the incident Mach number, M, the initial Atwood number, A, and the incident shock wave speed, W _i.     

Experimental investigation of a twice-shocked spherical gas inhomogeneity with particle image velocimetry

Results are presented from an experimental investigation into the interaction of a planar shock wave with a vortex ring. A free-falling spherical soap bubble is traversed by the incident shock wave and develops into a vortex ring as a result of baroclinically deposited vorticity (?r×?p 1 0{\nabla\rho\times\nabla p \neq 0}). The vortex ring translates with a velocity relative to the particle velocity behind the shock wave due to circulation. After the shock wave reflects from the tube end wall, it traverses the vortex ring (this process is called “reshock”) and deposits additional vorticity. Planar Mie scattering is used to visualize the atomized soap film at high frame rates (up to 10,000 fps). Particle image velocimetry (PIV) was performed for an argon bubble in nitrogen accelerated by a M = 1.35 shock wave. Circulation was determined from the PIV velocity field and found to agree well with Kelvin’s vortex ring model.     

Correlation of the separation region length in shock wave/channel boundary layer interaction

 Results of an experimental investigation of the characteristics of a separation region induced by the interaction of an externally generated oblique shock with the turbulent boundary layer formed in a rectangular half channel are discussed. The experiments were carried out in the supersonic wind tunnel of the Institute of Theoretical and Applied Mechanics SB RAS at a free-stream Mach number M _∞=3.01 over a range of Reynolds numbers Re ₁=(9.7–47.5)×10⁶ m^-1 and at zero incidence and zero yaw of the model. Particular attention is paid to the size of the zone of the upstream propagation of disturbances (upstream influence region) under different experimental conditions: with varied values of the shock wave strength, half channel width, and Reynolds number. It is shown, in particular, that the normalized upstream influence region length as a function of inclination angle of the shock generator in a rectangular half channel is readily approximated by a simple exponential function. In support of the known reference data obtained for supersonic numbers M _∞ and moderate Re in other configurations, it is also shown that the upstream influence region length decreases with increasing Reynolds number. Generalization of experimental data on the length of the upstream influence region formed in similar geometric configurations is possible using an additional reference linear scale which is the distance from the leading edge of the shock generator to the exposed surface. A substantial dependence of the reference dimensions of separation region on the half channel width is also established. Received: 20 January 1997/Accepted: 30 June 1997     

Motion of tracer particles in supersonic flows

 Factors that may act on particle motion in high-speed flow are investigated. The classical expressions of drag coefficient C _D for a sphere are reviewed. Then, a drag expression is proposed, extending Cunningham’s method to higher velocities and Knudsen numbers. This law, valid from continuum to free molecule conditions, for Re≲200 and M≲1 (where Re and M are, respectively, the Reynolds and Mach numbers based on relative velocity), is used to compare calculated and experimental values of the drag coefficient, as well as the particle velocities across an oblique shock wave. Calculated results are found to be in agreement with experiments. Received: 3 June 1997/Accepted: 16 August 1998     

Characteristics of counter-rotating vortex rings formed ahead of a compressible vortex ring

Characteristics of high Mach number compressible vortex ring generated at the open end of a short driver section shock tube is studied experimentally using high-speed laser sheet-based flow visualization. The formation mechanism and the evolution of counter rotating vortex ring (CRVR) formed ahead of the primary vortex ring are studied in details for shock Mach number (M) 1.7, with different driver section lengths. It has been observed that the strength of the embedded shock, which appears at high M, increases with time due to the flow expansion in the generating jet. Strength of the embedded shock also varies with radius; it is strong at smaller radii and weak at larger radii; hence, it creates a velocity gradient ahead of the embedded shock. At critical Mach number (M _c ≥ 1.6), this shear layer rolls up and forms a counter rotating vortex ring due to Biot-Savart induction of the vortex sheet. For larger driver section lengths, the embedded shock and the resultant shear layer persists for a longer time, resulting in the formation of multiple CRVRs due to Kelvin–Helmholtz type instability of the vortex sheet. CRVRs roll over the periphery of the primary vortex ring; they move upstream due to their self-induced velocity and induced velocity imparted by primary ring, and interact with the trailing jet. Formation of these vortices depends strongly upon the embedded shock strength and the length of the generating jet. Primary ring diameter increases rapidly during the formation and the evolution of CRVR due to induced velocity imparted on the primary ring by CRVR. Induced velocity of CRVR also affects the translational velocity of the primary ring considerably.