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We develop a theoretical framework that predicts and fully characterizes the diverse experimental observations of the nonlinear, combustion wave propagation in a rotating detonation engine (RDE), including the nucleation and formation of combustion pulses, the soliton-like interactions between these combustion fronts, and the fundamental, underlying Hopf bifurcation to time-periodic modulation of the waves. In this framework, the mode-locked structures are classified as autosolitons, or stably-propagating nonlinear waves where the local physics of nonlinearity, dispersion, gain, and dissipation exactly balance. We find that the global dominant balance physics in the RDE combustion chamber are dissipative and multi-scale in nature, with local fast scale (nano- to microseconds) combustion balances generating the fundamental mode-locked autosoliton state, while slow scale (milliseconds) gain-loss balances determine the instabilities and structure of the total number of autosolitons. In this manner, the global multi-scale balance physics give rise to the stable structures - not exclusively the frontal dynamics prescribed by classical detonation theory. Experimental observations and numerical models of the RDE combustion chamber are in strong qualitative agreement with no parameter tuning. Moreover, numerical continuation (computational bifurcation tracking) of the RDE analog system establishes that a Hopf bifurcation of the steadily propagating pulse train leads to the fundamental instability of the RDE, or time-periodic modulation of the waves. Along branches of Hopf orbits in parameter space exist a continuum of wave-pair interactions that exhibit solitonic interactions of varying strength., Comment: 18 pages, 13 figures

A rotating detonation engine with two unique features, a wave generator and radial injection of fuel and oxidizer, has been designed and tested. The wave generator, consisting of a number of equidi...

Direct observation of a Rotating Detonation Engine combustion chamber has enabled the extraction of the kinematics of its detonation waves. These records exhibit a rich set of instabilities and bifurcations arising from the interaction of coherent wave fronts and global gain dynamics. We develop a model of the observed dynamics by recasting the Majda detonation analog as an autowave. The solution fronts become attractors of the engine; i.e., mode-locked rotating detonation waves. We find that detonative energy release competes with dissipation and gain recovery to produce the observed dynamics and a bifurcation structure common to driven-dissipative systems, such as mode-locked lasers., 10 pages, 11 figures

Interestingly, the Chapman–Jouguet detonation velocity ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}D_{CJ}$) based on a one-dimensional and steady model compares well with the measured data. For the spinning detonation, in particular, this agreement is particularly notable, since the flow is highly three-dimensional and unsteady; perpendicular to the leading shock front, a transverse detonation wave (TDW) spins periodically. In the wake of this TDW, a secondary flow, called here the cross-current, appears which is orthogonal to the leading shock. Despite the presence of these cross-currents, the $D_{CJ}$ agreement remains remarkably satisfactory, and we investigate the reason for this, for spinning detonation in a tube. First, we focus on the origin of the cross-current. The cross-current, driven by the shock pressure, arises initially across a warped shock frontal surface, and both its sign and magnitude depend on the local slopes of the shock surface. The cross-current undergoes further pressure-driven transitions, with its magnitude eventually diminishing downstream and reducing the flow to a quasi-one-dimensional one. Second, regarding the unsteadiness, under the assumptions that the TDW spins at constant angular wave speed and the flow is steady in the frame rotating with it, the unsteady energy equation becomes integrable, resulting in the invariance of the so-called rothalpy. Also, in the integral forms of the mass and momentum balance, the unsteady terms drop out. Taken together, in the far field the governing equations are reduced to being one-dimensional and steady. From these the $D_{CJ}$ follows immediately, which appears to be the reason for the enduring usefulness of the $D_{CJ}$. The results of the analysis are confirmed with computational fluid dynamics (CFD). Additionally, the area-averaged flow profiles are found to display more than a passing resemblance to the Zeldovitch–Von Neumann–Doering (ZND) model.

In order to investigate the connection between the bubble and the spiral form of vortex breakdown, experiments were conducted: an external disturbance in the form of an azimuthally spinning waveform was imposed in a pipe. The azimuthal wave number was varied by adjusting the phase difference among four oscillating pistons mounted circumferentially on the pipe. By imposing a disturbance of zero azimuthal wave number, a spiral was transformed into a bubble, and this occurred only for selective piston frequencies; the vortex breakdown which altered from the spiral to the bubble moved upstream, where it remained as a bubble as long as the external disturbance remained. Once the disturbance was removed, the bubble returned to a spiral. By imposing a disturbance of azimuthal wave number +1 (the first circumferential mode rotating in the same direction as the mean swirl), a bubble was transformed into a spiral for selective piston frequencies, and the spiral moved downstream. These preferred frequencies were found to be the same as the unexcited frequencies observed in the spiral in its natural state. As long as the external disturbance was imposed, the breakdown altered from the bubble to the spiral remained as a spiral; once the disturbance was removed, the spiral reverted to a bubble. By imposing a disturbance with azimuthal wave number –1 (the first circumferential mode rotating in the opposite direction to the mean swirl), no change was detected in either a bubble or a spiral. By imposing a disturbance with azimuthal wave number 2 (the second circumferential mode), for selective piston frequencies a bubble was transformed into what appears to be the so-called two-tailed type. Thus, it appears that hydrodynamic instability plays a role in interchanging vortex breakdown types, and a comparison with available stability theories is discussed.

Water tunnel experiments were conducted to examine the effect of hole exit geometry on the near-field characteristics of crossflow jets. Hole shapes investigated were round, elliptical, square, and rectangular, all having the same cross-sectional area. Laser-induced fluorescence (LIF) and particle image velocimetry (PIV) were used.The vorticity around the circumference of the jet was tracked to identify its relative contributions to the nascent streamwise vortices, which evolve eventually into kidney vortices downstream. The distinction between sidewall vorticity and that from the leading and trailing edges, though blurred for a round hole, became clear for a square or a rectangular hole. The choice of non-circular holes also made it possible to reveal the unexpected double-decked structures of streamwise vortices and link them to the vorticity generated along the wall of the hole.The lowermost vortex pair of the double-decked structures, located beneath the jet, is what we call a ‘steady’ vortex pair. This pair is always present and has the same sense of rotation as the kidney vortices. The origin of these lower-deck vortices is the hole sidewall boundary layer: as the jet emanates from the hole, the crossflow forces the sidewall boundary layer to roll up into nascent kidney vortices. Here, hole width sets the lateral separation of these steady sidewall vortices.The vortices comprising the upper deck ride intermittently over the top of the ‘steady’ lower pair. The sense of rotation of these upper-deck vortices depends on hole geometry and can be the same as, or opposite to, the lower pair. The origin of the upper deck is the hole leading-edge boundary layer. This vorticity, initially aligned transverse to the crossflow direction, is realigned by the entrainment of crossflow momentum and thus induces a streamwise component of vorticity. Depending on hole geometry, this induced streamwise vorticity can be opposite to the lower-deck vortex pair. The opposing pair, called the ‘anti-kidney’ pair, competes with the nascent kidney-vortices and affects the jet lift-off. The hole trailing-edge boundary layer can likewise be turned toward the streamwise direction. In this case, the turning is caused by the strong reverse flow just downstream of the jet.In the present range of parameters, all hole boundary layer vorticity, regardless of its origin along the hole circumference, is found to influence the kidney vortices downstream.

The subject of total temperature separation in jets was treated in Fox et al. (1993) for subsonic jets. When we extended this study to the case of supersonic jets, we found the presence of a different mechanism of cooling, an effect which does not appear to have been known in the past. Named the ‘shock-induced total temperature separation’, this cooling can be of much greater magnitude than the subsonic cooling treated previously; it is caused by the interaction of convected vortical structures near the jet exhaust with the shock structure of the supersonic jet.In studying this phenomenon, we focus our attention on overexpanded jets exiting a convergent-divergent nozzle. The theoretical results for the shock-induced cooling which are based on a linearized, unsteady supersonic analysis are shown to agree favourably with experiments.When an impingement plate is inserted, the shock-induced cooling would manifest itself as wall cooling, whose magnitude is significantly larger than the subsonic counterpart. This has implications for heat transfer not only in jets, but wherever vortical structures may interact with shock waves.

In this paper, we propose a technique of suppressing aerofoil flutter using sound waves. Acoustic equipment is used to carry out the technique of active sound control that is known as anti-sound. Firstly, Theodorsen's method was applied to analyze the aerodynamic forces when an aerofoil is oscillating in an airstream with sound-wave disturbances. It is shown that sound waves do affect the maximum energy input in heaving and pitching oscillations. Secondly, we fabricated flutter equipment which has two degrees of freedom of aerofoil oscillation : heaving and pitching oscillations. The acoustic equipment consists of an amplifier, delay equipment and a loudspeaker. The loudspeaker is driven by the amplified and phase shifted-signals from the strain gauge used to measure the heaving or pitching oscillation of the aerofoil. We measured the energy input from the airstream and the relation of amplitude ratio and phase lag for heaving and pitching oscillations with increasing airstream velocity up to flutter speed. It is shown that the effective sound wave for suppressing the aerofoil flutter depends on the amplitude ratio between heaving and pitching oscillations.

An incompressible inviscid unsteady vortex dynamics analysis is presented to investigate the flow details for a class of free and bounded shear flows. Using the unsteady vortex models developed for the starting vortex behind sharp edges and the vortex street wakes, particle dynamics computations have been carried out to determine the pathline and streakline patterns in each of these model flows. Based on these results, the energy equations has been integrated along particle paths. The computations reconfirm that an instantaneous separation of energy occurs in the wake. Another interesting phenomenon, termed the Eckert-Weise effect, where the near-wake region contains essentially a cold flow, has also been computationally re-obtained using the present vortex model. Using a similar model, the wall-bounded vortex street wake problem has also been studied. The computed results highlight the important dynamical effects of convecting unsteady vortex motion close to a wall, which indicates that the inviscid entrainment effects play a vital role on the wall layer fluid eruption.

This paper investigates the characteristics of a particular fluid-structure interaction phenomenon, i.e. the continuous oscillation in rotational mode of an equilateral triangular wedge in the uniform incoming flow in the water tunnel. We propose the explanation of this self-excited oscillation. It is the unbalance force acting on the wedge’s side faces that causes such movement. If the wedge is positioned initially asymmetrically against the freestream, on one side, the flow will be flow-past-flat-plate like, whereas the other side will be flow-past-sharp-edge like. Due to the unbalanced pressure exerting on the two sides, the wedge will rotate. When the wedge moves, these mechanisms switch side interchangeably, and bring the wedge to continuous oscillation. To understand more thoroughly, several experiments were conducted to investigate such behavior by means of Food coloring dye and Laser Induced Fluorescence flow visualization. The oscillating frequency is governed by Strouhal numbers, which appears within limited range of 0.12 < Str < 0.18. Beyond this range, the wedge is either stationary or rotates only in one direction. Accompanied with, the simple wedge’s dynamic behavior is examined, by considering incoming freestream flow’s hydrodynamic force acting on the wedge.

The difference in film cooling effectiveness of three shaped holes is hereby explained by their vortical structures. Highlighted, in particular, are the competing effects of two pairs of vortices present in jets: the so-called kidney-shaped pair of vortices and a newly found anti-kidney pair. Three shaped holes investigated were referred to as TP2, TP3, and TP4. Except for the difference of how the hole area is increased, they are essentially the same. At the higher blowing ratio, the film cooling effectiveness of TP4 was found to be far superior to those of either TP2 and TP3. By flow visualization in water, the cause of these differences was investigated and found to be explainable on the basis of different vortical structures. In particular, for TP4 with a better film cooling effectiveness, the kidney- and anti-kidney pairs were found to cancel each other, leading to the improved jet attachment.

Water tunnel experiments have been conducted to understand the effect that hole exit geometry has on the lift-off characteristics of a jet in a crossflow stream. The following basic hole shapes were investigated, all having approximately the same cross-sectional area: round, elliptical, square, and rectangular. Laser Induced Fluorescence (LIF) was used to visualize the jet structure in the region near the hole exit. Both jet trajectory and jet cross-section views provide insight into the basic characteristics of the jet, which are strongly dependent on the formation of kidney-shaped vortices. Particle Image Velocimetry (PIV) was used to obtain the velocity field in the region immediately downstream of the jet. Based on the LIF and PIV results, hole geometry alone was found to influence the basic character of the jet, therefore, effecting the lift-off behavior.

In film cooling used in gas turbines, coolant from compressors is introduced to the hot gas stream of the turbine as crossflow jets. The interaction of the coolant jet and crossflow results in the formation of a pair of counter-rotating vortices, or kidney vortices. The sense of rotation of the kidney vortices is such that they exert two indesirable effects : 1) hot air is forced down beneath the jet to the turbine wall, and 2) the vortices tend to lift the jet off the surface by the mutual induction between the vortex pair. We report the potential of promoting jet attachment by weakening the kidney vortices through cancellation.

In this investigation we explore the effect of unsteady vortical structures on the adiabatic wall temperature distribution in an impinging jet. Treating first the simpler case of a free jet, we introduce a conceptual model for the separation of the total temperature, appealing to the dynamics of particle pathlines and vortex rings in the jet. The presence of a region of higher total temperature on the inside of the jet and a region of lower total temperature toward the jet periphery, predicted by the model, exhibits good agreement with the experimental data taken at high subsonic Mach number. The results from a numerical simulation further confirm the theoretical expectations.Through a similar argument, we show that when a thermally insulated flat plate is inserted into the jet, the wall temperature distribution is modified by the presence of secondary vortical structures, which are induced near, and swept over, the plate surface. When the plate is near the jet nozzle, a region of lower wall temperature, attributable to these additional vortices, is observed in the experimental data. When the plate is further from the nozzle, no secondary vortices are formed and no region of lowered wall temperature is measured. Self-sustaining acoustic resonance, when it occurs, is found to alter significantly this picture of the wall temperature distribution.Although the scope of this work is limited to free and impinging jets, this present topic, along with the previously reported mechanism of the Eckert–Weise effect, exemplifies the wider family of problems in which unsteady vortical structure strongly affects the wall temperature and heat transfer.

When a bluff body is placed in a crossflow, the total temperature in its wake can become substantially less than the incoming one, as manifested by the fact that the recovery factorRon its rearmost surface takes negative values at high subsonic flow: this is the phenomenon referred to here as the Eckert-Weise effect. Although a vortex street has been a suspected cause, the issue of whether this is so, and what the mechanism is, has remained unsettled. In this experimental and theoretical investigation, we first examine the cause of the Eckert-Weise effect by enhancing the vortex shedding through acoustic synchronization: resonance between the vortex shedding and transversely standing acoustic waves in a wind tunnel. At the lowest synchronization, where a ringing sound emanates from the wind tunnel,Rat the rearmost section of the cylinder is found to become negative even at a Mach number of 0.2; the base pressure (Cpb) takes dips correspondingly, indicative of the intensification of the vortex street. At this lowest acoustic resonance, the decrease ofRandCpb, uniform along the span, agrees with the expectation based on the spanwise uniformity of the lowest standing wave. At the next acoustic resonance where the standing wave now varies along the span, the corresponding dips inRandCpb, non-uniform along the span, reveals an interesting ‘strip-theory’-like behaviour of the vortex intensities in the vortex street. These results correlating the change inRwithCpbconfirm that the Eckert-Weise effect is indeed caused by the vortex shedding, the mechanism of which is examined theoretically in the latter half of the paper.A simple theoretical argument, bolstered by a full numerical simulation, shows that the time-varying static pressure field due to the vortex movement separates the instantaneous total temperature into hot and cold spots located around vortices; once time-averaged, however, the total temperature distribution conceals the presence of hot spots and takes the guise of a colder wake, the Eckert-Weise effect. Therefore the correct explanation of the Eckert-Weise effect, a time-averaged phenomenon, emerges only out of, and only as a superposition of, instantaneous total temperature separation around vortices. Such a separation is not confined to the outside of vortex cores; every vortex in its entirety becomes thermally separated. Nor is it limited to the far downstream equilibrium configuration of the Kármán vortex street but applies to the important near-wake vortices, and to any three-dimensional vortical structure as well. For low subsonic flows in particular, this dynamical explanation also leads to a similar separation of total pressure; these features may thus be potentially exploited as a general marker to identify and quantify vortices.

This paper examines the features of the flow field off the surface of an oscillating flat-plate airfoil immersed in a two-dimensional supersonic flow Although the exact linearized solution for a supersonic unsteady airfoil has been known for a long time, its expression in the form of an integral is not convenient for a physical interpretation. In the present paper, the quintessential features of the flow field are extracted from the exact solution by obtaining an asymptotic expansion in descending powers of a frequency parameter through the repeated use of the stationary-phase and steepest descent methods. It is found that the flow field consists of two dominant and competing signals: one is the acoustic ray or that component arising from Lighthill's ‘convecting slab’ and the other is the leading-edge disturbance propagating as a convecting wavelet. The flow field is found to be divided into several identifiable regions defined by the relative magnitude of the signals, and the asymptotic expansions appropriate for each flow region are derived along with their parametric restrictions. Such intimate knowledge of the flow field in unsteady, supersonic flow is of interest for interference aerodynamics and related acoustic problems.

An experimental study investigated the interactive effect between the heat transfer and the steady transonic flow in a simple convergent-divergent duct. The results show that the position of the shock is considerably affected by the effect of wall cooling. The physical explanation for the shift is also offered. The results may have implications in the other internal flow problems.

Pursuant to Part 1, the analysis of the aerodynamic forces acting on slowly oscillating airfoils in a supersonic cascade with a subsonic leading edge is presented. First the flow field between adjacent airfoils is determined. In the limit of sonic leading edge, the present results for the velocity potential agree with the sonic limit of Lane’s supersonic leading edge analysis. The requirement of the continuity of pressure in the “train” leads to functional equations for the train velocity; their solutions, obtained in closed form, are found to involve arbitrary constants which are related to the back pressure. The effect of the back pressure on the “train” is discussed in detail. For a cascade with zero pressure rise across it, the train velocity is determined completely and the formulas for lift and moment, accurate to the first order of a frequency parameter, are obtained in closed form. Stability criteria for a single-degree-of-freedom motion are examined. A pure bending motion is found to be stable, but a pure torsional motion becomes unstable under certain circumstances. These results are consistent with analogous oscillations of an isolated airfoil. However, the stability boundary for a typical cascade differs significantly from the case of the isolated airfoil, being strongly influenced by such cascade parameters as solidity, blade-to-blade phase difference, and stagger angle.

In this paper we examine how the unsteady flow field radiated from an oscillating body is altered from the result of acoustic theory as the direct consequence of disturbances propagating through the non-uniform flow produced by the presence of the body. Taking the specific example of an oscillating airfoil placed in supersonic flow and having the contour of a parabolic arc, we derive a closed-form representation for the unsteady flow field in terms of the confluent hypergeometric function. The analytical expression reveals explicitly that, though the body shape has a negligible effect in the near field, it inextricably affects the unsteady flow at a large distance, both in its amplitude and phase, and substantially modifies the results of acoustic theory. In addition, we display the relation of this solution to the ‘fundamental solution’ and the other salient physical features connected with disturbances propagating through non-uniform flow. The present results recover Whitham's rule in the limit of zero frequency of oscillation and also include, as another special case, the unsteady solution for a wedge obtained by Carrier and Van Dyke.

An investigation on multiple pure tone (MPT) noise is described. The model fan is operated, in Freon 12, in a closed loop acoustical facility. It has been found experimentally that the rotor relative Mach number and incidence angle are parameters important to the evolution of the MPT sound and the inlet duct length has a significant influence on the MPT sound emission. The experimental results are compared with a previous analysis. From the known blade nonuniformities of the model fan, the MPT distributions are computed. The analysis correctly predicts the frequency of the dominant multiple pure tone and sound pressures of the blade passing frequency noise and the dominant MPT that are in the same range as the experimental values.

A theoretical investigation of multiple pure tone noise is presented. An analysis based on a two-dimensional inviscid flow model is developed to predict the generation and subsequent evolution of multiple pure tone noise from prescribed blade-to-blade nonuniformities in the rotor geometry. The results show that even small nonuniformities within manufacturing tolerances can be a significant source of multiple pure tone noise. Among the nonuniformities investigated, errors in blade spacing are less significant multiple pure tone noise sources than errors in blade stagger or blade contours.

M a recent paper, Verdon and McCune1 presented a inearized analysis of an unsteady supersonic cascade with subsonic axial velocity. It is an interesting advancement on Ver don's previous paper.2 To achieve its objective of computing the pressure distribution, the analysis of Ref. 1 starts out to add the contributions from unsteady disturbances generated at all the oscillating airfoils below the reference airfoil in the cascade. The partial sums of the series were found to oscillate about apparent limiting values in general but, unfortunately, convergence was not proved. Aside from the problem of convergence, Ref. 1 also reports that under certain circumstances the numerical scheme broke down rather inexplicably. Thus it states in the concluding remarks that further work is needed to resolve these questions. Several years ago, the author of the present Note encountered essentially the same series and found that it diverges at a certain number of discrete points, although these results were never published. The appearance of Ref. 1, therefore, seems a fitting opportunity to point out the divergence of the series, to offer it as a possible explanation for the aforementioned breakdown of the numerical scheme and to discuss its physical implications in regard to resonance and other salient points. Consider, for example, the following kernel function K(x) , which appears in the integral equation (23) of Ref. 1 : where S (x) is given by -7

An investigation of the unsteady disturbances of a fixed frequency within a radial duct rotating at a set speed is presented. The flow is assumed to be compressible, inviscid, and of a fluid which is a perfect gas. Equations are developed for the steady and the unsteady parts of the flow in cylindrical coordinates. The unsteady disturbances are expressed by Fourier decomposition in angular position, distance into the duct, and in time. It is found that a resonance is possible when the frequency of flow disturbances is twice the shaft-rotation frequency, considering only the radial and tangential disturbances and not the radial and circumferential disturbances. The particular point at which the resonance occurs indicates the occurrence is due to the Coriolis force, which is only present in the radial and tangential directions. It is noted that the Coriolis force can only be present in open-ended ducts, such as those found in centrifugal compressors.

The objective of the investigation is to acquire fundamental understanding of a phenomenon characterized by violent fluctuations occurring in swirling air flows in aircraft engines; this flow instability, dubbed here as 'vortex whistle', is known to be capable of causing severe fatigue failure in gas turbine components. in the present report period, the analysis started from the compressible unsteady Navier-Stokes equations; this was necessary in order to assess the importance of the so-called higher order effects in the boundary layer upon the streaming. Resorting to the apparatus of a matched asymptotic expansion, the analytical representation of the acoustic streaming was derived anew. The results are in essential agreement with our previous conclusions and we were able to confirm, on firmer ground, the existence of the threshold swirl beyond which the free vortex distribution changes into a forced vortex type; these are written up in a paper form appended to this report. Second, based upon the analytical results, a test rig with two different tangential injection manifolds was designed, constructed and installed; the acquisition of the data from them will form the central effort of the next, Phase II activity.

The Ranque–Hilsch effect, observed in swirling flow within a single tube, is a spontaneous separation of total temperature, with the colder stream near the tube centreline and the hotter air near its periphery. Despite its simplicity, the mechanism of the Ranque–Hilsch effect has been a matter of long-standing dispute. Here we demonstrate, through analysis and experiment, that the acoustic streaming, induced by orderly disturbances within the swirling flow is, to a substantial degree, a cause of the Ranque–Hilsch effect. The analysis predicts that the streaming induced by the pure tone, a spinning wave corresponding to the first tangential mode, deforms the base Rankine vortex into a forced vortex, resulting in total temperature separation in the radial direction. This is confirmed by experiments, where, in the Ranque–Hilsch tube of uniflow arrangement, we install acoustic suppressors of organ-pipe type, tuned to the discrete frequency of the first tangential mode, attenuate its amplitude, and show that this does indeed reduce the total temperature separation.