Search

Ballistic range experiments were performed for a hemisphere-flare-cylinder model at supersonic Mach numbers in the transitional Reynolds number range at Nagoya University. The free-flight portion was modelled as axisymmetric in ANSYS Fluent® V.19.0. Projectile deceleration was included in the simulation as a function of the drag force over an approximate flight Mach number range 2.0–1.90. The projectile deceleration magnitude averaged approximately 700g and 1550g (g = 9.81 m s $$^{-2}$$ ) for two experiment cases. The Reynolds number (Re $$_\mathrm{d}$$ ) for each case based on the initial flight Mach number was Re $$_\mathrm{d}$$ = 90,000 and 177,000 respectively. The flow field, separation shock angle, averaged deceleration magnitude and averaged drag coefficient were compared between experiment and simulation. Agreement of these parameters was consistent for the Re $$_\mathrm{d}$$ = 177,000 case. This result contributed towards validation of the numerical acceleration technique. Differences for the Re $$_\mathrm{d}$$ = 90,000 case are explained with reference to experiment and simulation data.

Increasingly agile manoeuvre is an advantage in the flight of aircraft, missiles and aerial vehicles, but the principles of accelerating aerodynamics in the transonic regime are only now being fully investigated. This study contributes to the understanding of shock and separation effects on drag during axial acceleration, using a simple geometric configuration. Unsteady shock wave behavior was numerically investigated for an axisymmetric cone-cylinder using a commercial solver and the Moving Reference Frame acceleration technique. This acceleration technique was validated using unsteady numerical and experimental methods. The cone-cylinder was accelerated from Mach number 0.6 to Mach number 1.2 at 100g constant and deceleration was from Mach number 1.2 until Mach number 0.6 at –100g constant. Three cone angles were tested for the cone-cylinder with uniform cylinder diameter. Acceleration through the transonic Mach regime was characterised by a delayed and gradual shock wave development when compared to steady state, demonstrating a clear flow history effect. Deceleration through the transonic Mach regime was characterised by shock wave propagation from the base to the nose. New flow structures appeared during deceleration that do not have counterparts in the steady state, including shock interactions and propagating expansion-compression features. Gross changes in the unsteady drag coefficient curves for each cone-angle are explained with reference to unsteady shock wave behaviour for accelerating and decelerating motion.

The flow field characteristics that form for a shock wave propagating through a membrane-like termination at the exit of a shock tube are studied. The strength of the shock wave reflected back into the tube, as well as the strength of the shock wave transmitted, is examined. Six different materials are used, ranging from copper and aluminium foils to a variety of elastic and plastic sheets, in a few cases with different initial pressure differentials. High-speed shearing interferometry imaging is done of the external flow. Three principal characteristics are present in the transmitted flow: a diffracted shock wave, an expansion wave and a re-compression shock wave. It is found that the prominence of these features varies depending on the material type. For the later flow development and material rupture, there are a number of principal characteristics: small vortices, secondary shock waves, a vortex ring, oblique waves and a Mach disc.

Projectile accelerations above $$500~\hbox {ms}^{-2}$$ are commonly encountered in aerodynamic applications, but suitable validation data are rare in this regime. Experimental transonic velocity range data for a sphere decelerating under its own drag have been used to validate a numerical model for decelerating objects. The validated model is then used to explore the flow field ahead of objects decelerating from supersonic to subsonic Mach numbers. To model the non-inertial frame of the projectile, source terms were included in the momentum and energy equations in a computational fluid dynamics model. In decelerating cases, the bow shock formed in supersonic flight persists in the subsonic regime. The differences in the flow field between the steady and unsteady cases are explained using the concept of flow history. In the experiment, a tubular insert was present near the observation window in the ballistic range. The insert was numerically modelled, and it is shown that the resulting bow shock behaviour can be explained in terms of the Kantrowitz criterion, in conjunction with flow history. The RAE2822 aerofoil was used to explore the effects of shock overtaking and propagation during deceleration from supersonic to low subsonic Mach numbers. In this case, the bow shock wave persists from the initial supersonic speed to projectile Mach numbers lower than 0.4. The expansion wave and tail shock are shown to overtake the decelerating projectile and propagate forward, behind the bow shock.

A discussion on the diffraction of convex cylindrical shock wave segments is presented. Cylindrical shock waves are considered diffracting around several convex sharp corners with different wall angles ( $$27.5^{\circ }, 45^{\circ }$$ ). It is shown that the behaviour of cylindrical shocks is qualitatively similar to that of plane shocks but quantitatively different. These differences can be attributed to the cylindrical shock’s non-constant Mach number and its varying orientation along its profile. The analysis follows the concept of a first disturbance on the shock front and its locus in the laboratory’s frame of reference. Using the concept of disturbances propagating on the shock front introduced by Whitham, a method for determining the locus of the first disturbance is presented. A comparison is made between the loci calculated using this method and those obtained from CFD simulations. Finally, an analytical method for determining the locus of the inflection point is also introduced. The results from the methods presented show good correspondence with calculations from CFD.

There is a dearth of information on the reflection of very weak shock waves on curved surfaces; and particularly a distinct lack of experimental data at Mach numbers below 1.03. A previous study at Mach numbers between 1.03 and 1.05 showed that there are variations in the accepted reflection evolution on curved surfaces. That work indicated the existence of an apparent regular refection pattern in the curved shock wave reflection evolution which is also predicted for very much weaker waves in acoustic studies. To improve spatial resolution, the current study is performed in a large shock tube with a test section height of 450 mm using a 520 mm radius curved test piece with zero initial ramp angle. Imaging is primarily using shadowgraphy and high-speed photography. Experiments concentrated on tests in a range of $$1.007 \le M \le 1.09$$ with some tests at higher values. No shear layer is evident at the lower Mach numbers as for von Neumann and weak Mach reflection patterns. Transitioned regular reflection appears to cease at incident Mach numbers below 1.07. A reflection pattern resembling a regular reflection exists in the transition between the three-shock patterns and transitioned regular reflection.

Three-dimensional shock wave reflection comprises flow physics that is significantly different from the well-documented two-dimensional cases in a number of aspects. The most important differentiating factor is the sweep of the shock system. In particular, this work examines the nature of flow fields in which there is a transition of shock reflection configuration in three-dimensional space. The flow fields investigated have been made to exist in the absence of edge effects influencing the shock interaction and transition, as found previously to exist in conventional double-wedge studies. In general, the shock configurations are those with central regular and peripheral Mach reflection portions. It is shown that the sweep angle of the portions on either side of the transition point is subject to a cusp, as per an analytical model that is developed. This is confirmed with the use of numerical models with additional evidence provided by experimental results using oblique shadow photography. Further application of the principles of three-dimensional shock analysis and those pertaining to the sweep cusp model yield important insights regarding the overall shock geometry and that at transition.

The use of a Sanderson prism for the visualization of density gradients in a compressible flow is shown to be an inexpensive and versatile substitute for Wollaston prisms in the application of shearing interferometry. Experimentation using the Sanderson prism in a schlieren optical setup was performed to examine the effectiveness of the prism’s use for flow visualization. A range of polycarbonate prisms were tested for a range of prism heights and different physical deflections to produce a range of divergence angles. For each prism height and deflection, tests were done using a hair dryer, helium jet, a soldering iron and a moving shock wave interaction to generate flow density gradients. The light which passes through the prism was also focused at different distances to determine the effects that different fringe arrangements have on the visibility of the flow. Tests were also done comparing horizontal and vertical fringe arrangements. It was found that the colour of the infinite fringe selected had an influence on the results because the larger the difference in colour between that of the adjacent fringes, the better the contrast in the final result. It was also found that infinite fringe use showed better results due to flow features such as shocks and vortices being more easily identified against a uniform background. It is shown that the technique produces good visualization of the density gradients that form in the flow.

The flow field which results from an expansion wave entering a cavity from an upstream tube, and the focusing effect which occurs, is investigated. Different cavity geometries, different expansion wave pressure ratios and different expansion wave widths are explored. As the expansion wave propagates into the cavity it induces flow in the opposite direction and back down the walls. The flow experiences compression as it flows out back into the tube because of the concave surface of the cavity it encounters. This can result in the formation of shock waves which can propagate back up into the cavity. Very low pressure and temperature regions can develop because of the focusing action of the expansion. A convenient way of generating an expansion wave numerically and/or experimentally is in a shock tube. This consists of a tube divided into two compartments, one at high pressure and one at low pressure separated by a frangible diaphragm. On bursting the diaphragm, a shock wave travels in one direction and an expansion in the other towards the cavity. Whilst ideal boundary conditions can be imposed in numerical simulation laboratory experiments are complicated by the diaphragm being curved and having a finite opening time. The effect of an initially curved diaphragm is briefly considered. The expansion wave pressure ratio was altered by changing the initial pressure ratio across the diaphragm. For an initially high pressure ratio, supersonic flow can occur behind the trailing edge of the expansion wave which has a marked influence on the flow. The width of the wave is dependent on the distance of the diaphragm from the cavity and also has a significant influence on the flow. As the width of the wave increases and the density gradient decreases, focusing effects becomes significantly weaker. Correspondence between experiment and simulation is examined.

Streamlines behind axisymmetric curved shock waves were used to predict the internal surfaces that produced them. Axisymmetric ring wedge models with varying internal radii of curvature and leading-edge angles were used to produce numerical results. Said numerical simulations were validated using experimental shadowgraph results for a series of ring wedge test pieces. The streamlines behind curved shock waves for lower leading-edge angles are examined at Mach 3.4, whereas the highest leading-edge angle cases are explored at Mach 2.8 and 3.4. Numerical and theoretical streamlines are compared for the highest leading-edge angle cases at Mach 3.6. It was found that wall-bounding theoretical streamlines did not match the internal curved surface. This was due to extreme streamline curvature curving the streamlines when the shock angle approached the Mach angle at lower leading-edge angles. Increased Mach number and internal radius of curvature produced more reasonable results. Very good agreement was found between the theoretical and numerical streamlines at lower curvatures before the influence of the trailing edge expansion fan.

An experimental and numerical study was conducted to examine the effects of internal surface curvature and leading-edge angle on the shock waves and steady flow fields produced by axisymmetric ring wedges. Test models with leading-edge-radius-normalised internal radii of curvature of $R_{c}=\{1,1.5,2\}$ and leading-edge angles of $\unicode[STIX]{x1D6FC}=\{0^{\circ },4^{\circ },8^{\circ }\}$ were manufactured and tested. Experimental shadowgraph and schlieren results were obtained for Mach numbers ranging from 2.8 to 3.6 using a blowdown supersonic wind tunnel with accompanying numerical results for additional insight. The higher the internal surface curvature and leading-edge angle, the greater the flow fields were impacted. As a result, steeper compression waves were formed, thus curving the shock wave more noticeably. The internal surface curvature and leading-edge angle were both found to have an effect on the trailing-edge expansion fans. This altered the shape of downstream shock wave structures. The highest curvature models produced steady double reflection patterns due to the imposed internal surface curvature. The effects of conical and curved internal surfaces were explored for the presence of flow-normal curvature and the curving of the attached shock waves.

Previous work has shown that heat transfer into a plane conducting surface affects shock wave reflection properties. In the more complex reflection case of a curved surface the situation would be different because of the increased number of reflection patterns, with a rapid change in reflection geometry as the wave propagates up the surface. Test pieces of different thermal conductivities (0.19 W/mK and 401 W/mK) and hydraulically smooth surfaces are used on both convex and concave cylindrical surfaces. The test pieces are placed in identical positions on either side of a plane of symmetry and at the same incident angles in a shock tube. Tests are performed at incident shock Mach numbers of 1 . 22 M 1 . 5 and imaged using high-speed shadowgraph photography. The images are analyzed both quantitatively through reflection angle measurements and qualitatively by searching for asymmetry. Both the qualitative and quantitative measurements clearly indicate that the thermal conductivity of the surface affects the reflection patterns due to the complex transient thermal environment. Asymmetry in the reflection patterns is found at all Mach numbers. In contrast to the plane wall case the lack of theory limits comparison, where in that case it was shown that the lower the thermal conductivity the closer the experimental results approached the ideal adiabatic case. Similar comparisons in the relationship of reflection types between different materials are found in the current case. The findings could be refined by using higher imaging frame rates and spatial resolution, and sophisticated numerical modeling.

A complex interference flowfield consisting of multiple shocks and expansion waves is produced when high-speed slender bodies are placed in close proximity. The disturbances originating from a generator body impinge onto the adjacent receiver body, modifying the local flow conditions over the receiver. This paper aims to uncover the basic gas dynamics produced by two closely spaced slender bodies in a supersonic freestream. Experiments and numerical simulations were used to interpret the flowfield, where good agreement between the predictions and measurements was observed. The numerical data were then used to characterise the attenuation associated with shock wave diffraction, which was found to be interdependent with the bow shock contact perimeter over the receiver bodies. Shock-induced boundary layer separation was observed over the conical and hemispherical receiver bodies. These strong viscous-shock interactions result in double-reflected, as well as double-diffracted shock wave geometries in the interference region, and the diffracting waves progress over the conical and hemispherical receivers’ surfaces in “lambda” type configurations. This gives evidence that viscous effects can have a substantial influence on the local bow shock structure surrounding high-speed slender bodies in close proximity.

Most computational fluid dynamics simulations are, at present, performed in a body-fixed frame, for aeronautical purposes. With the advent of sharp manoeuvre, which may lead to transient effects originating in the acceleration of the centre of mass, there is a need to have a consistent formulation of the Navier–Stokes equations in an arbitrarily moving frame. These expressions should be in a form that allows terms to be transformed between non-inertial and inertial frames and includes gravity, viscous terms, and linear and angular acceleration. Since no effects of body acceleration appear in the inertial frame Navier–Stokes equations themselves, but only in their boundary conditions, it is useful to investigate acceleration source terms in the non-inertial frame. In this paper, a derivation of the energy equation is provided in addition to the continuity and momentum equations previously published. Relevant dimensionless constants are derived which can be used to obtain an indication of the relative significance of acceleration effects. The necessity for using computational fluid dynamics to capture nonlinear effects remains, and various implementation schemes for accelerating bodies are discussed. This theoretical treatment is intended to provide a foundation for interpretation of aerodynamic effects observed in manoeuvre, particularly for accelerating missiles. Funding agencies: DRDB [KT466921, KT528944, KT470887]

Understanding the shape of curved axisymmetric shock waves in supersonic intake-type conditions allows one to better design for downstream compression and heating requirements. Numerical results were obtained for ring wedge models with varying internal surface curvatures and wedge entry angles at different flow Mach numbers. Experimental results were obtained to validate the numerical method. A general power law fit which describes the shape of continuously curved axisymmetric shock wave segments was determined via curve fitting numerical results. The relationships between the curve fit constants in the general curved shock wave shape equation were shown for the models with a wedge entry angle of α = 4° between Mach 2.9 and 3.6. The curve fit constants were found to be self-similar in nature which promoted the process of characterising them with respect to normalised internal radius of curvature and flow Mach number. This resulted in a proposed empirical model which could predict the shape of the continuously curved axisymmetric shock wave segments within particular justified parameter limitations.

The bow shock stand-off distances of a sphere decelerating in air under its own drag were determined through numerical simulations using Fluent. Three cases were investigated with three different initial velocities. These results were then compared to steady-state numerical results obtained from the same CFD code in order to identify differences between the steady and unsteady cases at given Mach numbers. The initial Mach numbers used were 1.13, 1.19 and 1.25. A two-dimensional axisymmetric model was used in conjunction with a viscous turbulence model suitable for analysing transonic external aerodynamic problems. Numerically determined shock stand-off distances were compared to previous experimental results from literature for the steady and unsteady cases. There is a very good agreement between the steady-state numerical and experimental results, which confirms that a suitable numerical model was used. In the unsteady scenario, the numerical results follow the same trend as the experimental results, but the agreement is not as good. Some explanation for this is given. It was found that the shock stand-off distance for the unsteady cases was generally smaller than for the steady-state cases. Also, for the unsteady cases, a bow shock that was formed in supersonic flight transforms into a wave and persists well into the subsonic regime. In the steady-state cases, it is of course well known that no such phenomenon exists in flow at sonic and subsonic Mach numbers. The differences in the flow field and consequently in the drag in the steady and unsteady cases are explained using the concepts of flow history and fluid inertia.

A computational analysis of expansion wave diffraction was conducted using a large eddy simulation solver. The investigation was aimed at resolving particular flow features following a diffracting expansion wave which were previously identified experimentally through shadowgraph imaging but unresolved in the accompanying RANS computational analysis. The features included large-scale turbulent structures within the separation bubble at the apex, a large wake region downstream of the bubble, shear layer instability and vortex shedding. The analysis was made feasible by combining the embedded LES and wall-modelled LES hybrid RANS-LES techniques in ANSYS Fluent. The larger-scale turbulence within the separation bubble was resolved, and vortex shedding was identified in the LES solution. Evidence of the wake region was noted but remained largely unresolved due to the relatively small-scale turbulent motion. An oblique shock, found towards the rear end of the separation bubble, resolved at higher initial diaphragm pressure ratios (PR > 9) in the RANS analysis, was observed from PR = 7.7 in the LES analysis. A strong indication of this shock was seen in the shadowgraph images for PR = 7.7.

This study numerically investigates the compressible flow field near an axisymmetric body that is influenced by constant acceleration at 100 g where the motion is straight and level flight. Selected, instantaneous acceleration and deceleration results are compared to steady-state flow at corresponding projectile Mach numbers. The geometries tested were sharp 10° and 30° half-angle cone-cylinders with constant cylindrical aft section. Significant differences in the behavior of the flow field were found for the transonic Mach number range when compared with steady-state results. This was largely due to relative movement of the shock systems and their influence on the near flow field around the projectile. In particular, acceleration was found to delay the complete development of the bow shock and alter the expansion region at the cone-cylinder interface. Deceleration caused all shock structures to propagate upstream relative to the body and influence the nature of the flow field in a different manner compared to acceleration. The flow history concept was evident in both geometries and is demonstrated using simulation results from acceleration cases.

Surface effects resulting from the impact of an underwater shock wave have previously mainly been studied as a consequence of underwater chemical or nuclear explosions. Of the many complex features that occur, one is the development of surface jets or plumes. This particular aspect is examined by applying a shock wave impulse to the bottom of a circular disk submerged in a water bath. Different strength shocks are used with different depths of water. It is demonstrated that a wide variety of jets and plumes may be generated.

A preliminary experimental and numerical investigation into the interaction between two independent shock waves emerging perpendicular to each other into a common space is presented. It is arranged that two shock tubes have a common diffracting edge, so that the two waves arrive at the edge simultaneously. The shock Mach number was 1.31. The merging three-dimensional diffracting shocks reflect regularly off each other, but as they become more curved due to diffraction the angle between them changes and Mach reflection develops. L-shaped vortices are shed at the two free edges of each tube exit. As they meet, they merge and interact in a complex manner with each other.

In the conventional von Neumann theoretical treatment of two-dimensional shock wave reflection off a surface, it is assumed that the flow is inviscid and that the reflecting surface is perfectly smooth, rigid, non-porous, and adiabatic. These theoretical predictions have been found to be good predictions of reflection over a significant range where regular reflection exists and for a limited range around Mach 2 for strong shocks in the case of Mach reflection. However, experiments on regular reflection have shown that this pattern persists to a small extent beyond what the theory predicts. This effect has been ascribed to the development of a viscous boundary layer behind the point of reflection, and some studies have been done on the effect of surface roughness on reflection topology. The possibility of thermal effects and heat transfer from the shock-heated gas to the wall and on the boundary layer has, on the other hand, been almost totally neglected. To study this, two surfaces of different conductivities have been placed at the same angle, symmetrically in a shock tube, and impacted by a single plane shock wave and the reflection patterns examined. Tests were conducted over a range of Mach numbers between 1.28 and 1.4, and incident shock wave angles between \(36^\circ \) and \(70^\circ \) covering both regular and Mach reflection. Both quantitative and qualitative tests show that there is a difference in the angles between the reflected waves and the reflecting surfaces based on the material thermal conductivity. In the quantitative tests the value of this angle was larger for materials with a lower thermal conductivity, and vice versa. A material, such as aluminium, with mid-range thermal conductivity had angles that lay within the limits of the two extreme values for glass and copper. The qualitative images supported these findings, showing asymmetry in reflection topography, with the intersection of the two reflected shock waves lying closer to the material with a higher thermal conductivity.

This study numerically and experimentally examines the resulting flow field of a shock wave passing through a pipe gap. The effects of gap geometry and shock Mach number variation are investigated. Incident shock Mach numbers of 1.3, 1.4, and 1.5 and gap widths of 25 and 50 mm were used, which correspond to 0.5 and 1.0 pipe inner diameters, respectively. For both cases, the incident shock wave propagated into the downstream pipe at much reduced strength. A strong expansion propagated into the upstream pipe causing a significant pressure drop from the initial post-shock pressure. Expansion waves at the outflow resulted in supersonic speeds as the flow entered the gap for Mach 1.4 and 1.5. A notable feature was the formation of a standing shock at the inlet to the downstream pipe for the higher two Mach numbers in both cases. Decreasing the gap width moved the standing shock closer to the downstream pipe. For the lowest Mach number of 1.3, no standing shock system was set up. The propagation conditions in the downstream pipe showed that the pressure is initially unsteady, but becomes more uniform, controlled by the developed wave system in the gap. For the flanged gap case, the flow within the gap is confined for much longer and hence produces more intense and complex flow feature interactions and an earlier transition to turbulence. The induced shock strength in the downstream pipe is independent of gap geometry and separation distance examined in this paper as verified by experimental pressure traces.

The diffraction of an initially one-dimensional plane expansion wave over a 90° corner was explored using experiment and numerical simulation. Unlike studies of shock diffraction, expansion wave diffraction has hardly been documented previously. The planar expansion wave was produced in a shock tube by bursting a diaphragm. Two independent parameters were identified for study: (i) the initial diaphragm shock tube pressure ratio, which determines the strength (pressure ratio) of the expansion, and (ii) the position of the diaphragm from the apex of the 90° corner, which determines the width of the wave. The experimentation only considered variation in the shock tube pressure ratio whereas the simulation varied both parameters. A Navier–Stokes solver with Menter’s shear stress transport $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}k\mbox{--}\omega $ turbulence model was found to adequately model the overall flow field. A number of major flow features were identified occurring in the vicinity of the corner. These were: a shear layer that originated by flow separation at the apex of the corner; a vortex within a separation bubble that remained attached to the wall, in sharp contrast to what happens in the shock wave diffraction case, where the vortex convects downstream; and a reflected compression wave arising from perturbation signals generated by the diffraction. For a narrow-width expansion wave existing prior to diffraction, the reflected compression wave steepens into an outwardly propagating, weak cylindrical shock wave. Regions of supersonic flow are identified surrounding the bubble and can extend downstream depending on the pressure ratio. Another major flow feature identified in some cases was an oblique shock located near the separation bubble. A large wake region is evident immediately downstream of the bubble and appears to consist of two distinct layers. The experimental results showed large-scale turbulent structures within the separation bubble, and shear layer instability and vortex shedding from the separation bubble were also evident.

The Kelvin–Helmholtz instability (KHI) is an instability that takes the form of repeating wave-like structures which forms on a shear layer where two adjacent fluids are moving at a relative velocity to one another. Such a shear layer forms in the Mach reflection of shock waves. This work focuses on experimentally visualising the presence of the KHI in Mach reflection as well as its evolution. Experimentation was performed at shock Mach numbers of 1.34, 1.46 and 1.61. Plane test pieces and parabolic profiled pieces followed by a plane section having wedge angles of 30 $$^\circ $$ and 38 $$^\circ $$ were tested. Flow field visualisation was performed with a schlieren optical system. The KHI was best visualised with the camera-side knife edge perpendicular to the shear layer (i.e. the axis of sensitivity along the length of the shear layer). The structure and growth of the instability were readily identified. The KHI forms more readily with increasing Mach number and wedge angle. Second-order Euler, and Navier–Stokes numerical simulations of the flow field were also conducted. It was found that the Euler and laminar Navier–Stokes solvers achieved very similar results, both producing the KHI, but at a much less developed state than the experimental cases. The k $$-\epsilon $$ solver, however, did not produce the instability.

The steady, two-dimensional transition criteria between regular and Mach reflection are well established. Little has been done on the dynamic effect on transition due to a rapidly rotating wedge. Results from experiments and computations done on steady and unsteady shock wave transition from Mach reflection to regular reflection, MR $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\rightarrow $ RR, are described. The measured motion and the initial shock incidence was used to mimic the experiment with a two-dimensional numerical code. The maximum rotation speed achieved at transition was approximately $2500^\circ \ {\mathrm{s}}^{-1}$. Rapid wedge rotation was shown to have a significant measurable effect on transition. The code was also applied to the dependence of dynamic MR $\rightarrow $ RR transition on other variables in the parameter space. These include rotation about the leading or trailing edge, initial incidence and rotation speed at two free-stream conditions. Impulsively started rotation in these cases was used with the rotation specified by $M_E = \omega c/a_{\infty }$ where $\omega $ is constant angular velocity (negative anticlockwise), $c$ the distance from the edge considered to the pivot point and $a_{\infty }$ the free-stream sound speed. For the Mach numbers and range of rotation speeds tested, both the wedge and shock angle at transition decreased with increased rotation speed. The sensitivity of the transition angle to changing the rotation point from the trailing edge to the experimental model pivot point was investigated briefly at a free-stream Mach number of $M = 2.98$ with $M_E = -0.1$. The wedge angle at transition increased by 1.5° and the shock angle at transition decreased by 1.5°, a significant variation. The effect of the initial incidence was also investigated. By reducing the initial wedge angle from 24.5 to 23.5° for these initial conditions the shock angle at transition decreased by approximately 1.8°, also a marked sensitivity. The flow field development for impulsive rotation about the wedge trailing and leading edges at $M = 1.93$ for $M_E = -0.075$ was analysed in some detail. The flow field development is very sensitive to the rotation centre, more especially at large rotation rates. Four phases of the Mach stem development were identified in both cases. For rotation about the wedge leading edge the Mach stem height remains constant until the expansion waves arrive at the triple point. This is followed by an increase in Mach stem height, which then remains constant for a short period after which it decreases until transition to RR. For rotation about the wedge trailing edge the impulsive start generates a disturbance on the incident wave which propagates down the wave, through the triple point and down the Mach stem. The stem height is constant until the arrival of the incident wave disturbance. This causes a sudden decrease in Mach stem height. Subsequently, the Mach stem height remains constant for a short time, before it decreases until transition to RR. Similar effects in the variation of stem height with wedge angle occur at the higher Mach number of 2.98. It was demonstrated that MR can be maintained for a while at zero wedge incidence with a sufficiently large rotation rate of $M_E = -0.1$, with $M=1.93$, for both leading and trailing edge pivot points.

When a planar shock wave propagating down a channel encounters a decrease in cross-sectional area, not only is the shock strengthened but the shock shape and post-shock flow are disturbed. The current research investigates how various area reduction profiles affect the shock strength and shape, as well as the uniformity of the post-shock flow for planar shocks. Bird [1] and Russell [2] investigated experimentally the effects of wall shaping and strengthening by convergence respectively using strong incident shock waves; however the current study looks at relatively low Mach numbers of 1 < M < 2. By optimising the wall shape an area reduction can be used to increase the strength of a shock significantly without compromising on the quality of the post-shock flow, which is of particular importance in the design of experimental shock tube testing. More importantly, by analysing different profiles using numerical studies, the technique could then potentially be generalized to examine to what extent a shock shape may be purposefully manipulated to a required profile by suitable wall shaping. In order to provide a comprehensive study of the topic, numerical, analytical and experimental analyses are conducted. A comparison of computational fluid dynamic (CFD) simulations and Milton’s corrected ray-shock theory [3] is examined in detail.

The unsteady separation of the compressible flow field behind a diffracting shock wave was investigated along convex curved walls, using shock tube experimentation at large length and time scales, complemented by numerical computation. Tests were conducted at incident shock Mach numbers of \(M_{\hbox {s}} =\) 1.5 and 1.6 over a 100 mm radius wall over a dimensionless time range up to \(\tau \le \) 6.45. The development of the near wall flow at \(M_{\hbox {s}} =\) 1.5 has been described in detail and is very similar to that observed for slightly lower \(\tau \)’s at \(M_{\hbox {s}} =\) 1.6. Computations were performed at wall radii of 100 and 200 mm and for incident shock Mach numbers from 1.5 up to and including Mach 2.0. Comparing dimensionless times for different size walls shows that for a given value of \(\tau \) the flow field is very similar for the various wall radii published to date and tested in this study. Previously published results that were examined alongside the results from this study had typical values of \(1.6 < \tau < 3.2\). At the later times presented here, flow features were observed that previously had only been observed at higher Mach numbers. The larger length scales allowed for a degree of Reynolds number independence in the results published here. The effect of turbulence on the numerical and experimental results could not be adequately examined due to limitations of the flow imaging system used and a number of questions remain unanswered.

In order to examine the transition between regular and Mach reflection in a three-dimensional flow, a range of special geometry test pieces, and inlets, were designed. The concept is to have a geometry consisting of two plane wedges which results in regular reflection between the incident waves off the top and bottom of the inlet capped by two curved end sections causing Mach reflection. The merging of these two reflection patterns and the resulting downstream flow are studied using laser vapor screen and shadowgraph imaging supported by numerical simulation. An angled Mach disc is formed which merges with the line of regular reflection. A complex wave pattern results with the generation of a bridging shock connecting the reflected wave from the Mach reflection with the reflected waves from the regular reflection. In order to experimentally access the flow within the duct, a number of tests were conducted with one end cap removed. This resulted in a modified flow due to the expansive flow at the open end the influence of which was also studied in more detail.

When a planar shock wave moving into stationary fluid encounters a straight wedge inclined at some angle, the resulting reflection configuration will be either a regular reflection (RR) or a Mach reflection (MR)(Ben-dor, Shock Wave Reflection Phenomena. Springer, Berlin, 2007). Hornung et al. (J. Fluid Mech. 90:541–560, 1979) note that in order for an MR to form, information regarding the length scale of the system needs to be transmitted to the reflection point. For pseudosteady reflections, this can only happen if the flow behind the reflected shock is subsonic relative to the reflection point. The reflection configuration resulting from a planar shock encountering an inclined wedge can be therefore be predicted from the Mach number and angle between the incident shock and the reflecting surface.

The study of interfering bodies in supersonic airflow is important to the development of a number of flight vehicles, since the aerodynamic loads experienced by the body of interest (the receiver) can be influenced significantly by the interference created when in close proximity to a disturbance generator. Of particular significance is the three-dimensional bow shock emanating from the disturbance generator, striking the cylindrical surface of the slender, disturbance receiver. The impinging forebody bow shock not only causes local elevations in the receiver’s surface pressure but modifies the flow angularity over the body as well. In addition, altering the attitude of the receiver to the free stream flow introduces additional complexity to the interaction, since the impinging disturbance shock wave undergoes a mutual interaction with the separated flow on the leeside of the inclined receiver. Many researchers have reported integrated force and moment characteristics of bodies in interference flow fields, but in most cases, limited explanation of the underlying flow physics and wave dynamics of the interaction is given.

In the past, the Paul Vieille lecture has taken one of two formats: either the exposition on a single topic, such as by Ben-Dor in 2013, or an overview of a range of topics covering work from a single institution or group, such as that by Takayama in 2011. This paper is similar to the latter and will cover a range of experimental studies, some successful, some not, and some started and never completed, done over a number of years in what is now the Flow Research Unit at the University of Witwatersrand. The work covers a number of different aspects covering studies in liquids and deformation of solids using liquid shock waves, but the primary focus has been investigation of the behaviour of gas dynamic shock waves. In a number of cases, it has led to the development of unusual and unique experimental facilities and diagnostic techniques.

Following the original numerical work by Hornung, [1], an experimental study of the behaviour of imploding conical shock waves was undertaken by Skews et al [2] in which good correlation was exhibited with the patterns of reflection for waves of various strengths. This study extends that study to various incident wave strengths.

An experimental study shows that the Guderley reflection (GR) of shock waves can be produced in a standard shock tube. A new technique was utilised which comprises triple point of a developed weak Mach reflection undergoing a number of reflections off the ceiling and floor of the shock tube before arriving at the test section. Both simple perturbation sources and diverging ramps were used to generate a transverse wave in the tube which then becomes the weak reflected wave of the reflection pattern. Tests were conducted for three ramp angles (10°, 15°, and 20°) and two perturbation sources for a range of Mach numbers (1.10–1.40) and two shock tube expansion chamber lengths (2.0 and 4.0 m). It was found that the length of the Mach stem of the reflection pattern is the overall vertical distance traveled by the triple point. Images with equivalent Mach stem lengths in the order of 2.0 m were produced. All tests showed evidence of the fourth wave of the GR, namely the expansion wave behind the reflected shock wave. A shocklet terminating the expansion wave was also identified in a few cases mainly for incident wave Mach numbers of approximately 1.20.

Results of numerical and experimental investigations of steady state supersonic viscous corner flows in simplified geometries that resemble typical wing body–fuselage intersections are presented. These flows are numerically calculated using the computational fluid dynamics code of Fluent and include the use of standard turbulence models. Experimental work includes the use of the oil film flow visualization technique to visualize the surface flow patterns. The flows considered are symmetric about the corner bisector. This article focuses on the development of the three shock structure along the various geometries, specifically focusing on the formation and growth of the shear layers that result from the Mach reflection. The resultant effects of this shear layer on pressure distributions, shear stress, and changes in Mach numbers are discussed. The shock wave boundary layer interaction is also presented.

There have been numerous studies on the steady-state transition criteria between regular and Mach reflection of shock waves generated by a stationary, two-dimensional wedge in a steady supersonic flow, since the original shock-wave reflection research by Ernst Mach in 1878. The steady, two-dimensional transition criteria between regular and Mach reflection are well established. There has been little done to consider the dynamic effect of a rapidly rotating wedge on the transition between regular and Mach reflection. This paper presents the results of an investigation on the effect of rapid wedge rotation on regular to Mach reflection transition in the weak- and strong-reflection ranges with the aid of experiment and computational fluid dynamics. The experimental set-up includes a novel facility to rotate a pair of large aspect ratio wedges in a 450 mm × 450 mm supersonic wind tunnel at wedge rotation speeds up to 11000 deg s−1. High-speed images and measurements are presented. A numerical solution of the inviscid governing flow equations was used to mimic the experimental motion and to extend the investigation beyond the limits of the current facility to explore the influence of variables in the parameter space. There is good agreement between experimental measurements and numerical simulation. This paper includes the first experimental evidence of the regular to Mach reflection transition beyond the steady-state detachment condition in the weak- and strong-reflection ranges. It also presents results of simulations for the dynamic regular to the Mach reflection transition which show a difference between the sonic, length-scale and detachment conditions. This paper includes experimental evidence of the Mach to regular reflection transition below the steady-state von Neumann condition.

The aim of this work was to investigate the flow evolution with time of fluid between two parallel disks and the corresponding pressure variations at the centre of the lower disk that occur subsequent to an impact-loading situation arising from dropping a mass onto the upper disk from a chosen height. During the event a fixed amount of energy is dissipated in the fluid between the disks through the action of friction. Therefore, this fundamental system may be regarded as a constant energy one, as distinct from one in which the upper disk is moving at a constant velocity, or is acted upon by a constant force. A test cell was set up to conduct the investigation, for which the separation between the disks was monitored, together with the pressure at the centre of the lower disk, over the duration of the experiment (about 8–10 ms). Glycerine was used as the test fluid. The equation of motion, based on a self-similarity approach, was reduced to a simpler (quasi-steady linear or QSL) form. Measured values of disk separation, velocity and acceleration were substituted as inputs into the full QSL model and two limiting cases, namely an inviscid and a viscous model. The QSL model provided excellent comparisons between the pressure measurements and data generated by a commercial computational fluid dynamics software package, throughout the duration of a typical experiment. The inviscid and viscous models achieved good correlations with measurements for the initial impact (during which disk accelerations exceeding 2 km s−2 occurred) and towards the end of the event, that were characterized by a small and much larger pressure rise, respectively. The former feature appears not to have been previously reported and is likely to typify that which would be observed in impact systems involving squeeze films.

The two-dimensional diffraction of a shock wave over a wall made up of a series of plane and/or curved sections is considered. The analysis is based on the theory presented by, for the interaction of an originally plane shock wave with a corner. A method is presented by which the shock profile may be determined for a wall of any shape and for any incident Mach number, in regions where the characteristics form a simple wave. Comparisons are made between experimental measurements and theoretical predictions for convex walls consisting of a number of facets, and for circular arcs, for a range of incident shock wave Mach numbers. It is shown that the theory gives a satisfactory prediction of the wave shape, which improves as the Mach number increases. Modifications in the flow field behind the shock, compared to that for a simple corner made up of two plane walls is discussed, particularly relating to flow separation. For circular arc concave walls a inverse Mach reflection results experimentally, leading to regular reflection, for which the theory is of no use.

Previous detailed studies of the interaction of a shock wave with a perforated sheet considered the impact of a shock wave on a plate with regularly spaced slits giving area blockages of 60 and 67%, at various angles of incidence, and resulting in both regular and Mach reflection. The current work extends this study to a much wider variety of plate geometries. Blockage ratios of 20, 25, 33, 50, and 67 and inclinations of 45, 60, 75, and 90° to the shock wave were tested. Four different thicknesses of plate were tested at the same frontal blockage in order to assess the effects of gap guidance. Tests were conducted at two shock Mach numbers of 1.36 and 1.51 (inverse pressure ratios of 0.4 and 0.5). It is found that secondary reflected and transmitted waves appear due to the complex interactions within the grid gaps, and that the vortex pattern which is generated under the plate is also complex due to these interactions. The angle of the reflected shock, measured relative to the plate, decreases with plate blockage and the angle of inflow to the plate reduces with increasing blockage. By analysing the flow on the underside of the plate the pseudo-steady flow assumption is found to be a reasonable approximation. Both the pressure difference and the stagnation pressure loss across the plate are evaluated. It is found that over the range tested the plate thickness has a minimal effect.

The deformation of metal plates and tubes achievable through the use of liquid shock waves generated in a shock tube is studied, with reference to both free-forming and forming the metal into dies, as well as to imprinting detailed features. The process is highly controllable, in terms of the magnitude and duration of the applied pressure pulse. A projectile is fired into a liquid column producing a high-pressure liquid shock wave which impinges on the testpiece. Different projectile materials, driving pressures and impact velocities are used to alter the energy and impulse transmitted. A particular attraction of its use in a laboratory is the application of high-speed photography to the deformation process. Illustration of the application of the facility to slamming studies and to fracture of brittle materials is included. It is concluded that the techniques employed offer a useful and versatile tool for many studies of material deformation.

Transition criteria for the steady case are derived by consideration of local flow conditions at the reflection point and were published by Ben-Dor [1]

In the past, much attention was given to the diffraction of a shock wave over various geometries; extensively explored through experimentation and simulation [1, 2], and by many others

Vortices resulting from shock wave diffraction have been studied in detail over the last several decades. These studies have concentrated on two dimensional flows. The study of three dimensional vortices as a result of shock wave diffraction is limited

Consider the pressure wave interaction that occurs between two slender bodies in close proximity, in a steady supersonic freestream. The pressure wave systems produced by the bodies creates an interference flowfield that is dominated by shock and expansion waves

The behaviour of a planar shock wave that encounters an inclined wedge is well known, and Ben-Dor [1] gives a detailed description of the various reflection patterns that form under these circumstances. If the angle of incidence between the shock and the wedge is low enough, then the two shock pattern known as regular reflection (RR) will occur