Detailed information on the flow field structure is often important in numerous industrial applications. Although commercial computational fluid dynamics packages are often capable of providing the required data, they are costly and not universally available. This study was motivated by the operation of an open-pool nuclear research reactor where low radiation levels can be maintained by the installation of a stable purified hot water layer in the upper part of the pool. Maintaining a stable stratification requires a detailed description of the structure of the velocity field. Due to the inherent complications and restrictions of performing accurate measurements in a pool of a real-size operating reactor, either smaller-scale models or oversimplified fluid dynamics computational schemes are routinely used. These methods cannot be validated, and therefore do not necessarily capture the large-scale behavior correctly.We present an alternative approach to evaluate the velocity components in the pool that is based on the potential flow theory. The model results are validated by measurements using particle image velocimetry. The presented potential theory allows for the quick and easy assessment of the global properties of the fluid velocity distribution within the pool, and in particular, close to its surface. The suggested computational models are flexible and allow for easily varying the spatial dimensions of the flow field. The technique thus can be upscaled, and enables the validation of numerical computations in various fluid mechanical installations where the flow field cannot be resolved.

We experimentally and theoretically study the dynamics of a one-dimensional array of pendula with a mild spatial gradient in their self-frequency and where neighboring pendula are connected with weak and alternating coupling. We map their dynamics to the topological Su-Schrieffer-Heeger model of charged quantum particles on a lattice with alternating hopping rates in an external electric field. By directly tracking the dynamics of a wave-packet in the bulk of the lattice, we observe Bloch oscillations, Landau-Zener transitions, and coupling between the isospin (i.e., the inner wave function distribution within the unit cell) and the spatial degrees of freedom (the distribution between unit cells). We then use Bloch oscillations in the bulk to directly measure the nontrivial global topological phase winding and local geometric phase of the band. We measure an overall evolution of 3.1 [Formula: see text] 0.2 radians for the geometrical phase during the Bloch period, consistent with the expected Zak phase of [Formula: see text]. Our results demonstrate the power of classical analogs of quantum models to directly observe the topological properties of the band structure and shed light on the similarities and the differences between quantum and classical topological effects.

Understanding the air velocity profile over random and three-dimensional wind-waves is crucial for evaluating momentum and energy transfer between air and water. Unfortunately, determining accurate air velocity profiles in field conditions is nearly impossible. It is well accepted, though, that the airflow above water waves has a logarithmic profile usually expressed in terms of the effective roughness parameter. However, this parameter cannot be evaluated directly from the wave measurements at the water surface. For young wind waves, it was recently shown that the airflow over the waves maintains wall similarity. In this case, the airflow vertical velocity distribution can be described by the Nikuradse fully rough logarithmic profile, using the root mean square value of the water surface elevation as the equivalent sand grain roughness height. The existence of mean current in water in either co- or counter-wind direction may significantly modify the wave field, the vertical wind-velocity profile and thus the momentum and energy transfer from air to water. The effect of the water current on the spatially developing boundary layer over young wind waves and the wall similarity is examined. Combined laboratory measurements at several fetches of finely resolved mean air velocity profile above the water surface and of the characteristics of the wind-wave field are performed at multiple wind-forcing and mean water current conditions. The shear stress at the air–water interface estimated using two independent approaches is weakly dependent on current, while the resultant wave field differs significantly.

We present a theory of evolution of wind waves in time and space under abruptly applied wind forcing that is experimentally validated in a laboratory wind-wave tank. The model describes qualitatively and quantitatively the complex wave field development from the initial smooth surface to the finite state. The stochastic nature of wind waves is treated by considering an ensemble of coexisting unstable harmonics that grow due to shear flow instability. Breaking limits the wave growth initially; the process is then controlled by fetch-limited growth duration.

Recently, we demonstrated that the temporal fetch-dependent wind-wave growth under abruptly applied wind forcing can be accurately described by considering a stochastic ensemble of multiple unstable harmonics (submitted to PRL). In that study, the two-phase viscous shear flow instability at the air-water interface was examined using the energy growth rates β and the group velocities cg of the unstable harmonics obtained by solving the coupled Orr-Sommerfeld (OS) equations in air and water with appropriate boundary and initial conditions. The predictions of this unidirectional model compare well with measurements of random time-and space-dependent wave field performed in our laboratory (JFM 828, 459, 2017). The eigenvalues of the model equations determine β and cg of each harmonic defined by its wavenumber; the suggested model then allows computation of variation with time and with fetch of the statistical wave field parameters such as the characteristic wave amplitude and the instantaneous dominant frequency. The eigenvalues of the OS system however depend strongly on the adopted mean vertical velocity profile in air and in water. The water velocity is assumed to decay exponentially with depth from the maximum value corresponding to the drift velocity at the interface. In air, we assumed the lin-log suggested by Miles that consists of a linear segment in the viscous sublayer connected smoothly to a logarithmic turbulent velocity profile over smooth water surface. The assumption of smooth water surface is reasonable at the onset of wind. However, emerging wind-waves render the surface rough; the surface roughness becomes more pronounced at higher wind forcing and larger fetches. In the present study, we extend our previous study and apply the developed OS solver to investigate the dependence of the viscous shear-flow stability on the shape of air velocity profile. We take advantage of the detailed wind-velocity profiles measured in our facility at various wind velocities and a number of fetches (JGR 117, C00J19, 2012)that demonstrated the significant deviations of the actual air velocity profiles over waves from the shape corresponding to smooth-surface. The surface drift velocities under different operational conditions were also measured. The effect of the evolving wind-wave field on eigenvalues of the OS system of equation and thus on the domains of instability, the energy growth rates β and the group velocities cg is studied. These results extend our understanding of the interrelation between the varying in time and space wind-wave field and the turbulent airflow above the water surface and shed light on momentum and energy exchange between air and water.

The structure of the steady boundary layer in the airflow over young wind waves is studied in detail using extensive data accumulated in our laboratory as well as in additional laboratory facilities and in field measurements. Following the approach adopted in studies of turbulent flow over solid rough surfaces, the coupling between the spatial evolution of wind waves and wind velocity profiles over the water is analysed. The roughness of the moving water surface under wind is not constant; wind waves that constitute the roughness elements are unsteady, random and three-dimensional. The effective water surface roughness increases with airflow velocity as well as with downstream distance. Nevertheless, the existence of wall similarity as observed in a flow over solid surfaces is demonstrated; a fully rough boundary layer is obtained for young wind waves at diverse operational conditions. This approach enables quantitative study of the coupling between the airflow and local wind wave characteristics.

Variation of temperature in time and space was recorded at multiple vertical locations in the course of initiation of a heated water layer in an open-pool research reactor of the Soreq Nuclear Research Center. The pool was initially filled with warm water, and heavier cooler water was then injected at the bottom of the facility. Different modes of coolant injection were employed in two different experiments. In both cases, a finite width thermocline that separated cool water at the lower part of the pool from the warm water in its upper part was observed. The thermocline gradually moved up eventually attaining a constant raise velocity. In both experiments, the thermocline characteristics were different, but wave-trains with notable temperature fluctuations were observed within the thermocline. The characteristic frequencies of oscillations were below the Brunt–Väisälä frequencies that characterize the density gradient within the thermoclines. The finite dimensions of the tank impose conditions in which standing internal waves with the length commensurate with tank size can be expected. The oscillations were thus associated with resonant internal waves excited by disturbances introduced by the coolant flow at the lower part of the pool. In both experiments, the measured wave spectra agree with the results of linear analysis of two-layer and three-layer stratification models.

Shock-wave reflection over concave surfaces poses a difficulty in its analysis due to the unsteady nature of the reflection process and the occurrence of various types of Mach reflections caused by it. In a pseudo-steady flow, the reflection's configuration is self-similar since the shock wave reflects over a surface with constant inclination. The unsteady Mach reflection introduces an additional complexity as it is affected by the changing inclination of the surface, forcing the reflection to continuously adjust itself to the varying boundary condition. In this study, validated simulations of Mach reflection (MR) over cylindrical concave surfaces with different radii were performed for three inviscid perfect gases with moderate incident shock Mach numbers (M-s) ranging from 1.3 to 1.5. The reflection was investigated up to the point of transition from MR to transitioned regular reflection. A similar behaviour of the configuration and evolution of the Mach stem was observed, one that is independent of the surface radius and type of gas. With regards to different gases, the speed of sound a(0) is a dominant factor since it dictates the propagation of wall disturbances. A universal condition of the rate of surface change was found, accounting for different radii, different gases and M-s variation. Analysis based on shock dynamics is employed to explain how disturbances caused by surface variations play a significant role in the behaviour of the reflection. This method successfully supports the similarity that was demonstrated and facilitates a more informed perception of the MR process.

Experiments and inviscid numerical computations were performed in air at an incident shock wave Mach number of 1.3. The incident shock waves were reflected over cylindrical convex surfaces. The models differed in the radii and initial angles. Great agreement was obtained between the high-resolution computations and the high-resolution experiments. Examination of the flow Mach number distributions revealed a fundamental difference between pseudo-steady and unsteady reflections. As the radius of the surface increases, the orientation of the reflected shock wave, with respect to the incident shock wave, approaches its orientation in a pseudo-steady reflection. This means that the radius of curvature does play a rule in the RR reflection. Therefore, it is reasonable to assume that the RR→MR transition is also affected by the radius of curvature. This observation is in contrast with the claim that variation between the RR→MR unsteady and pseudo-steady transitions is a result of optical limitations as was previously suggested by several studies. The problematic estimation of the RR→MR transition is also discussed. This study is a section of an extensive ongoing research dealing with the unsteady mechanism that leads to transition.

The non-stationary transition from regular reflection (RR) to Mach reflection (MR) over convex segments has been the focus of many recent studies. Until recently, the problem was thought to be very complicated because it was believed that many parameters such as the radius of curvature, initial angle and geometrical shape of the reflecting surface influenced this process. In this study, experiments and inviscid numerical computations were performed in air ($\unicode[STIX]{x1D6FE}=1.4$) at an incident shock-wave Mach number of 1.3. The incident shock waves were reflected over cylindrical and elliptical convex surfaces. The computations were validated by high-resolution experiments, which enabled the detection of features in the flow having characteristic lengths as small as 0.06 mm. Therefore, the RR →MR transition and Mach stem growth were successfully validated in the early stages of the Mach stem formation and closer to the surface than ever before. The evolution of the RR, the transition to MR and the Mach stem growth were found to depend only on the radius of the reflecting surface. The reflected shock wave adjusts itself to the changing angles of the reflecting surface. This feature, which was demonstrated at Mach numbers 1.3 and 1.5, distinguishes the unsteady case from the self-similar pseudo-steady case and requires the formulation of the conservation equations. A modification of the standard two-shock theory (2ST) is presented to predict the flow properties behind a shock wave that propagates over convex surfaces. Until recently, the determination of the time-dependent flow properties was possible solely by numerical computations. Moreover, this derivation explains the controversial issue on the delay in the transition from the RR to the MR that was observed by many researchers. It turns out that the entire RR evolution and the particular moment of transition to MR, are based on the essential ‘no-penetration’ condition of the flow. Therefore, we proposed a simple geometrical criterion for the RR →MR transition.

The reflection pattern over a convex cylindrical surface followed by a concave cylindrical surface is studied using a high spatial resolution and high temporal resolution experimental setup. This fully automated setup enabled the repetition of experiments many times while retaining extremely high repeatability. For the investigated moderate shock strengths (1.2–1.4), the repeatability was less than 0.01 in the incident shock wave Mach number. Each experiment produced a single schlieren image with a pixel size of 0.03 mm. Thus, it was possible to distinguish minuscule flow features of size 0.06 mm. All the images were later combined into a detailed description of a single reflection process. The process was analyzed using an automatic image processing procedure that located the triple point in each image. The tested model enabled studying both the evolution of the RR → MR transition over the convex segment and the evolution of a complicated shock pattern reflection over the concave part. It was found that the RR → MR nonstationary transition is closer to the pseudo-steady criterion in comparison to previous experimental studies. Yet, based on error estimation analysis and in contrast to Kleine et al. (J Fluid Mech 740:47–60, 2014) the pseudo-steady criterion will remain 3o–8o higher than the dynamic transitions investigated. Over the concave segment, we observed the evolution of a newly three-shock configuration established on the Mach stem of the original reflection (MRMR). This double Mach configuration explains the enhancement in the pressure when using blunt entry reflectors. The transition of the newly formed MR to regular reflection (MRRR) was also investigated. The reflection process over the concave segment depends on the first reflection process. Since the later dictates a specific relative angle between the incident shock wave and the original Mach stem. This angle can be later used to adjust the MRMR → MRRR transition angles. Even though this three-shock configuration originated from the Mach stem, it was found that the adjusted transition angles were close to the MR → RR transition angles reported in the literature.

Numerical simulations were conducted to understand the different wave configurations associated with the shock-wave reflections over double-concave cylindrical surfaces. The reflectors were generated computationally by changing different geometrical parameters, such as the radii of curvature and the initial wedge angles. The incident-shock-wave Mach number was varied such as to cover subsonic, transonic and supersonic regimes of the flows induced by the incident shock. The study revealed a number of interesting wave features starting from the early stage of the shock interaction and transition to transitioned regular reflection (TRR) over the first concave surface, followed by complex shock reflections over the second one. Two new shock bifurcations have been found over the second wedge reflector, depending on the velocity of the additional wave that appears during the TRR over the first wedge reflector. Unlike the first reflector, the transition from a single-triple-point wave configuration (STP) to a double-triple-point wave configuration (DTP) and back occurred several times on the second reflector, indicating that the flow was capable of retaining the memory of the past events over the entire process.

Studying the nature of transient reflections of shock waves from surfaces is important in many engineering fields, e.g. blast protection, supersonic flights, shock focusing, medical and industrial applications and more. The recent advancements in this field reveal that the major obstacle in better understanding this phenomenon by means of experimental investigations is the limited temporal and spatial resolution. An alternative approach to commonly used high-speed photography is based on the use of a single-lens reflex (SLR) camera that captures only one image per experiment. Using this method to study a transient reflection process necessitates repeating each experiment many times while retaining extremely high repeatability. In the present study, we present a solution to this obstacle by means of a fully automated shock tube facility, which has been developed in the course of this study. A typical experiment can be executed a few hundred times with a repeatability of less than 0.01 in the incident-shock-wave Mach number at moderate shock strengths ($M=1.2{-}1.4$). The system offers a very high spatial and temporal resolution description of the transient reflection process of a shock wave over a coupled convex–concave surface. The study of this complex configuration using a fully automated shock tube enables one to observe, in greater detail than ever before, both the transient transition from regular reflection, RR, to Mach reflection, MR, and the reverse transient transition from MR to RR. The geometry studied can also be found in blunt leading-edge reflectors in which higher pressures were recorded, and the results presented also describe in detail the shock reflection process inside such a reflector. The results highlight and strengthen the recent understanding of the importance of high spatial and temporal resolution in determining the transition process from RR to MR over a coupled concave–convex surface. However, despite achieving very high statistical certainty in the experimental measurements, the question of the difference between the pseudo-steady transition criterion and the experimental results remains unresolved.

Abstract The non-stationary transition from Mach to regular reflection followed by a reverse transition from regular to Mach reflection is investigated experimentally. A new experimental setup in which an incident shock wave reflects from a cylindrical concave surface followed by a cylindrical convex surface of the same radius is introduced. Unlike other studies that indicate problems in identifying the triple point, an in-house image processing program, which enables automatic detection of the triple point, is developed and presented. The experiments are performed in air having a specific heats ratio 1.4 at three different incident-shock-wave Mach numbers: 1.2, 1.3 and 1.4. The data are extracted from high-resolution schlieren images obtained by means of a fully automatically operated shock-tube system. Each experiment produces a single image. However, the high accuracy and repeatability of the control system together with the fast opening valve enables us to monitor the dynamic evolution of the shock reflections. Consequently, high-resolution results both in space and time are obtained. The credibility of the present analysis is demonstrated by comparing the first transition from Mach to regular reflection ( $\mathrm{MR} \rightarrow \mathrm{RR} $ ) with previous single cylindrical concave surface experiments. It is found that the second transition, back to Mach reflection ( $\mathrm{RR} \rightarrow \mathrm{MR} $ ), occurs earlier than one would expect when the shock reflects from a single cylindrical convex surface. Furthermore, the hysteresis is observed at incident-shock-wave Mach numbers smaller than those at which the dual-solution domain starts, which is the minimal value for obtaining hysteresis in steady and pseudo-steady flows. The existence of a non-stationary hysteresis phenomenon, which is different from the steady-state hysteresis phenomenon, is discovered.