Shock Angle Research Articles

Global instability of the interaction between an oblique shock and a laminar boundary layer

This study investigates the oblique shock-wave/boundary-layer interaction with a Mach number of 2.15 and a Reynolds number of 1×105. Both global stability analysis and direct numerical simulation are used to reveal the global instability characteristics and three-dimensional details of the incident shock flow. The results of global stability analysis indicate that stationary global instability occurs when the shock angle exceeds the critical angle 31.8°. At a shock angle equal to 33°, an additional unstable mode appears, which is oscillatory at large wavelength and gradually dominant when the wavelength decreases. As the wavelength is further reduced, the mode and its conjugation evolve into two stationary modes with different growth rates. A global instability criterion for incident shock flow is established based on the triple deck theory, which determines the instability only through free-stream conditions and shock angles. A direct numerical simulation is performed for the 32° shock angle case. It is found that secondary separation occurs during the nonlinear growth, which is absent in the two-dimensional base flow. Moreover, the separated flow undergoes a secondary perturbation growth, during which the dominant spanwise wavelength is doubled and the flow structures change significantly. The flow oscillates around a quasi-steady state in the end, indicating that a stationary unstable mode can develop unsteadiness without external disturbances.

Non-locality of ion reflection at the shock front: dependence on the shock angle

In typical heliospheric collisionless shocks most of the mass, momentum and energy are carried by ions. Therefore, the shock structure should be most affected by ions. With the increase of the Mach number, ion reflection becomes more and more important, and reflected ions participate in shaping the shock profile. Ion reflection at the collisionless shock is a non-local process: the reflected–transmitted ions re-enter the shock front far from the reflection point. The direction and the magnitude of this shift depend on the shock angle. The distance between the reflection point and the re-entry point is of the order of the upstream ion convective gyroradius and exceeds the shock width. The non-locality of ion reflection may have implications for shock rippling since reflected ions may carry perturbations along the shock front.

Decomposition of mean skin friction in incident shock wave/turbulent boundary layer interaction flows at Mach 2.25

The evolution characteristics of the mean skin friction beneath the supersonic turbulent boundary layer that interacts with incident shock waves at Mach 2.25 are analyzed using Direct Numerical Simulation (DNS). The separated and attached boundary layers in the interaction region that respectively correspond to 33.2° and 28° incident shock angles are considered. The mean skin friction recovery rate for the separated boundary layer is much gentler and distinctly less than that for the attached case where the skin friction completes its recovery within one boundary layer thickness. The novel mean skin friction decomposition method for compressible flows proposed by the recent research is applied in the interaction region to investigate the internal evolution characteristics quantitatively. The results reveal that the three decomposition components are distinctly unequal between the two cases. The contributions of the turbulent motions at different scales to the associated term are focused on using empirical mode decomposition technology. It indicates that the outer large-scale structures dominate separation and reattachment regions, while contributions from inner small-scale structures are limited. In contrast, contributions from the outer large-scale structures are dramatically reduced in the attached case, which results in the outer large-scale and inner small-scale motions being of equal importance.

Flow structure and parameter evaluation of conical convergent–divergent nozzle supersonic jet flows

Supersonic gas jets in conical convergent–divergent nozzles are studied numerically using the OpenFOAM rhoCentralFoam solver. The spatiotemporal evolution of the jet flow field is analyzed. The influence of the operating conditions on the flow field is studied parametrically, including the nozzle pressure ratio (NPR), area ratio, and throat position. The behaviors and mechanisms of the double-diamond structure, throat wave, and exit wave are interpreted in detail. The results show that a conical convergent–divergent nozzle always generates shock waves. The throat shock reaches its maximum length during the initial stage, then becomes slightly shorter before becoming stationary, dominated by the exit velocity. Furthermore, it is shown that the jet flow changes from overexpansion to underexpansion with increasing NPR. With an increasing area ratio, the trend is the opposite. The throat position affects the jet divergence angle at the nozzle exit, consequently causing a variation in the core radius of the jet. It is further shown that the double-diamond structure does not always appear. The throat shock angle, exit wave angle, and shear layer width directly affect the shape of the double-diamond structure. The favorable pressure gradient of the nozzle ultimately dominates the changes in the length of the throat wave and exit wave.

Numerical Study of Shock Wave Boundary Layer Interaction Based on an Anisotropic Turbulence Model

This paper presents numerical simulations of incident oblique shock wave impinging on a flat plate with a turbulent boundary layer using a nonlinear anisotropic turbulence model. Three cases with different incident shock angles under the same free-stream Mach number are studied, including attachment flow, incipient separation flow and outstanding separation flows. Convection terms and diffusion terms are calculated using the third-order Roe’s flux difference splitting scheme and the second-order central difference scheme in the code, respectively. The Runge-Kutta time marching method is employed to solve the governing equations for steady state solutions. Comparisons between the simulated results and experimental data are carried out, which include density contours, pressure distribution, skin friction coefficients distribution and streamline vectors. To capture the essential behavior of this complex anisotropic turbulence separation flow caused by the shock-wave-boundary-layer-interaction (SWBLI), an anisotropic eddy viscosity tensor and a nonlinear Reynolds stress-strain constitutive relationship are used in this model. The anisotropic eddy viscosity tensor is composed of turbulent fluctuation velocity vector and characteristic length vector which are obtained by statistical partial average scheme. With proper modeling of anisotropic eddy viscosity tensor, the numerical results are in good agreement with the experimental data. It is worthy to point it out that the computation accuracy decreases with increasing interaction intensity.

Analysis of shock train leading shock structure under oscillatory backpressure

The leading shock in a steady shock train takes one of two forms: oblique or normal. However, the phenomenon and mechanism of the normal-to-oblique transition of the leading shock in a forced shock train have not been widely reported. In this study, leading normal and oblique shocks are experimentally observed at the same position and similar velocities in a forced oscillation shock train with an incoming Mach number of 1.83. The normal-to-oblique transition is found to be closely related to the self-excited oscillation of the shock train. In addition, we find that the direct use of free-interaction theory cannot accurately predict the leading shock angle in a moving shock train. Thus, free-interaction theory requires appropriate correction for this scenario.

Multi-fidelity parametric sensitivity estimation for large eddy simulation with the Spalart–Allmaras model

ABSTRACT A computationally affordable approach to estimate parametric sensitivities of engineering relevant quantities of interest for a large eddy simulation (LES) is explored. The method is based on defining a Reynolds-averaged Navier–Stokes (RANS) problem that is constrained to reproduce the LES mean flow field. The proposed method is described and assessed for a shock/boundary layer interaction problem, where the shock angle and wall temperature are considered variable or uncertain. In the current work, we show that the proposed method offers improved sensitivity predictions for certain flow features as compared to standalone RANS simulations, while using a fraction of the LES cost.

Flow visualisation of water channel setup for supersonic flow analogy

In the present study, flow over aerodynamics models is analysed in water channel flow to understand flow physics at affordable cost. The water channel flow visualization setup is constructed to analyse the analogy between Froude number and Mach number and flow visualization over various models is observed. In order to conduct the study, the shallow water speed is varied from 0.4 m/s to 0.6 m/s in the test section of water channel which will corresponds to the Froude number, Fr= 2. The water channel results will be finally validated to wind tunnel at Mach number, M=2. The velocity of water channel can be found by tracing the motion of suspended particles over shallow water at defined length and also by the volumetric flow rate. Several tests were conducted over a spike model to understand the flow visualization and videos were captured. The hydraulic jump and divergent angle in the water surface will be observed and validated with the corresponding shock angle in air medium. Failure analysis throughout the constructs are discussed to understand and resolve different problems faced with the specimen, test section along with camera setup. The present setup can be used multiple times for long duration to observe the flow visualization for undergraduate students.

Experimental investigation of mini supersonic wind tunnel for flow visualization

The present study proposes to design, fabricate and investigate a miniature supersonic wind tunnel at an affordable cost, safe to operate, and reliable for education purposes. The configuration consists of a high pressure reservoir, and mini tunnel which will be used for providing the supersonic flow over aerodynamic models. To design the mini supersonic wind tunnel, the method of characteristics is used to ensure that any shocks that arise in contact with the surface do not impact the uniformity and velocity of the flow through the duct. To achieve the supersonic flow, the nozzle contour of the supersonic portion of the wind tunnel is built in such a way that the shocks nullify each other and have zero effect on the velocity of the flow. For compressible flow visualization, an affordable Schlieren setup consisting of a torch, mirrors, and a mobile camera is used for flow visualization. Finally, the visualization results of the present study is in good relation with the theoretical calculations. The present facility is also calibrated by measuring the oblique shock angle in the test section, in the range of Mach number 1.6 to 1.8.

Post-shock turbulence recovery in oblique-shock/turbulent boundary layer interaction flows

The oblique shock impinging on the supersonic turbulent boundary layer leads to a mixing layer and the emergence of large-scale coherent structures within the interaction zone which leave significant velocity defect and turbulence amplification downstream. In the present study, we investigate the turbulence recovery in the post-shock region by exploiting direct numerical simulation data of the oblique-shock/turbulent boundary layer interaction flow at the incoming Mach number of $2.28$ and the shock angle of $33.2^\circ$ , with special attention paid to the contribution of the mixing layer and large-scale structures to flow dynamics. For that purpose, we propose to split the mean velocity, Reynolds stresses and spanwise spectra into a canonical portion that is constructed according to the statistics of canonical turbulent boundary layers, and a mixing-layer-induced portion. We found that the hidden mixing layer grows with the boundary layer thickness and that the induced mean shear and Reynolds stresses decay at different rates. The mean velocity recovers to the canonical profiles at a distance of 13 boundary layer thicknesses downstream where the mixing-layer-induced mean shear ceases to have strong impacts. The recovery of Reynolds stresses requires 10 boundary layer thicknesses in the near-wall region but a much longer streamwise extent in the outer region due to the slow decay of large-scale motions. These large-scale motions superpose on the near-wall turbulence, intensifying the turbulent fluctuations, yet having a trivial impact on the skin friction, for the contribution of the mixing-layer-induced mean shear and Reynolds shear stress are balanced by the advection term. We further establish a simple physical model capable of approximately predicting the streamwise evolution of mixing-layer-induced mean shear and turbulent kinetic energy. This model suggests that the complete recovery of turbulence in the outer region requires a streamwise extent of approximately 50 boundary layer thicknesses.

Reflection and transition of planar curved shock waves

In this paper, the reflection of curved shock waves over a symmetry plane in planar supersonic flow is studied. This includes stable Mach reflection (MR) and the regular reflection (RR) to MR transition process. Curved shock theory (CST) is applied to derive the high-order parameters in front of and behind the shock wave. The method of curved shock characteristics is used to establish an analytical model to predict the wave configurations. The shock structures provided by the proposed model agree well with the numerical results. Flow structures, such as the height of the Mach stem and the shape of the shock wave and slip line, are studied by applying the analytical model. Isentropic waves generated from a curved wall are found to significantly influence the flow patterns. It appears that the compression waves obstruct the formation of the sonic throat and increase the Mach-stem height. The expansion waves have the opposite effect. The evolution mechanism of the Mach stem is found in conjunction with the RR-to-MR transition process. The CST is extended to a moving frame and used to model the transition. The time history of the moving triple point illustrates the effects of the incident shock angle and isentropic waves on the transition process.

Study on shock vibration analysis and foundation reinforcement of large ball mill

The vibration of the ball mill foundation will seriously affect the operation stability of the ball mill. According to the connection relationship between the ball mill and the foundation, the mill-foundation transient shock model is established. The effects of shock force, concrete grade and shock angle on the transient response of the foundation were studied. And based on the problem of the mill foundation, the reinforcement scheme of the foundation was put forward. The following conclusions are drawn: with the increase of shock force and shock angle, the displacement of foundation joints increases, and the maximum stress also increases; concrete grade has little effect on foundation stability; after the foundation reinforcement scheme is used, the vibration caused by the shock of the ball mill is effectively reduced, and the stability of the foundation is increased. Finally, the correctness of the model and the scheme is verified through the comparison with the test data. The calculation results provide a reference for the improvement of the ball mill foundation.

The Efficiency of Electron Acceleration by ICME-driven Shocks

We present a study of the efficiency of the acceleration of suprathermal electrons at collisionless shock waves driven by interplanetary coronal mass ejections (ICMEs), with the data analysis from both the spacecraft observations and test-particle simulations. The observations are from the 3DP/EESA instrument on board Wind during the 74 shock events listed in Yang et al., and the test-particle simulations are carried out through 315 cases with different shock parameters. A total of seven energy channels ranging from 0.428–4.161 keV are selected. In the simulations, using a backward-in-time method, we calculate the average downstream flux in the 90° pitch angle. On the other hand, the average downstream and upstream fluxes in the 90° pitch angle can also be directly obtained from the 74 observational shock events. In addition, the variation in the event number ratio with the downstream to upstream flux ratio above a threshold value in terms of the shock angle (the angle between the shock normal and upstream magnetic field), upstream Alfvén Mach number, and shock compression ratio is statistically obtained. It is shown from both the observations and simulations that a large shock angle, upstream Alfvén Mach number, and shock compression ratio can enhance the efficiency of the shock acceleration. Our results suggest that shock drift acceleration is more efficient in the electron acceleration by ICME-driven shocks, which confirms the findings of Yang et al.

Computing pressure effect on projectile at an altitude 4Km above the sea level

In this paper the projectile was on altitude 4 Km above sea level ,Mach number change from 2-6 ,shock angle changing from 30 to 60 degree this angle depend on deflection angle which uses it .
 Shock angle studying as oblique ,a smallest mach ,shock angle was 2 Mach and 30 degree because this value it is true for supersonic (normal component ) so that this value is true for supersonic .by mathlab program increasing pressure with increasing Mach number ,shock angle ,by this know pressure values in figures on homing head can know suitable material type for it and altitude ,Mach number deflection angle .

Effect of interaction strength on recovery downstream of incident shock interactions

Direct numerical simulations of a supersonic turbulent boundary layer on a flat plate interacting with an impinging shock wave are carried out with two different incident shock angles at Mach 2.25. The effect of the interaction strength on the recovery process in the downstream region is systematically studied, including the turbulence evolution, the statistical and structural properties of wall pressure fluctuations, and the generation of mean skin friction and wall heat flux. The variations of the Reynolds stress components, the anisotropy tensor, and the turbulent kinetic energy budget in the two flow cases highlight a slow reversal tendency and an increasingly pronounced importance of the outer-layer large-scale structures in the relaxation region of the strong interaction. We find that the effect of increasing the interaction strength on the fluctuating wall pressure is reflected by a decrease in the characteristic frequencies, an increase in the spatial extent, and a decrease in the convection velocity. We decompose the mean skin friction and wall heat flux into different physically informed contributions and reveal that the mean wall heat flux generation is the same regardless of the interaction strength; in contrast, the generation mechanism of mean skin friction is found to be fundamentally changed. A novel scale-decomposition method is used to quantify the effect of the increased interaction strength on the leading components, and it is demonstrated that the energetic outer-layer large-scale structures are the dominant contributor in the recovery process as the interaction strength is increased.

Oblique shock train motion based on schlieren image processing

In this paper, shock train motion in a Mach number 2.7 duct is studied experimentally, and large numbers of schlieren images are obtained by a high-speed camera. An image processing method based on Maximum Correlation Detection (MCD) is proposed to detect shock train motion from the schlieren images, based on which the key structures, e.g., separation positions and separation shock angles on the top and bottom walls, can be analysed in detail. The oscillations of the shock train are generated by rhombus and ellipse shafts at various excitation frequencies. According to the analysis of MCD results, the distributions of the frequency components of shock train oscillation generated by the two shafts are distinctly different, in which the motion generated by the ellipse shaft is much smoother; shock train motion is mainly characterized by the oscillation of separation position while the separation shock strength is not so sensitive to downstream disturbance; there is a hysteresis loop relation between the downstream pressure and separation position.

Combining Rankine–Hugoniot relations, ion dynamics in the shock front, and the cross-shock potential

Rankine–Hugoniot relations (RH) connect the upstream and downstream plasma states. They allow us to determine the magnetic compression, the density compression, and the plasma heating as functions of the Mach number, shock angle, and upstream temperature. RH are based on the conservation laws in the hydrodynamical form. In collisionless shocks, the ion distributions behind the shock transition are determined by ion dynamics in the macroscopic fields of the shock front. The ion parameters upon crossing the shock are directly related to the magnetic compression and the cross-shock potential. For given upstream parameters, RH provide the magnetic compression. If there is no substantial overshoot, an analytical estimate provides the cross-shock potential as a function of the magnetic compression and the Mach number. Numerical tracing of ions across a shock profile with the derived parameters provides the ion pressure, which is in good agreement with the combination of the two theoretical approaches.

Experimental study of wall temperature effect on shock wave/turbulent boundary layer interaction in hypersonic aircraft

The wall temperature effects on turbulent boundary layer and shock wave/turbulent boundary layer interaction are studied experimentally at Ma 2.7 with wall-to-recovery temperature ratio Tw/Tr from 0.67 to 1.2. The high-resolution turbulent flow structures are captured and analyzed by applying the nano-tracer planar laser scattering and particle image velocimetry techniques. The influence mechanisms of wall temperature on the vortices and interaction flow field are explained. The results indicate that under the heating wall condition, the increase of gas viscosity and volume causes the turbulent boundary layer to develop thicker and faster. The thickness increases 0.79 mm as Tw/Tr increases 0.2. The boundary layer shape presents intermittence with large-scale vortices increasing in size and number. The shock wave interaction point moves 1.43 mm upstream as Tw/Tr increases 0.2 due to the increasing boundary layer thickness. More intensive separation of the boundary layer leads to the height and length extension of the separation region and the increase of the separation shock angle. Under the wall cooling effect, the boundary layer thickness decreases, and the global vorticity reduces. The velocity field becomes more uniform. The present study can provide an important experimental reference for the design of hypersonic aircraft.

Thermal emission from bow shocks

The nearby, massive, runaway star ζ Ophiuchi has a large bow shock detected in optical and infrared light and, uniquely among runaway O stars, diffuse X-ray emission detected from the shocked stellar wind. Here we make the first detailed computational investigation of the bow shock of ζ Ophiuchi, to test whether a simple model of the bow shock can explain the observed nebula, and to compare the detected X-ray emission with simulated emission maps. We reanalysed archival Chandra observations of the thermal diffuse X-ray emission from the shocked wind region of the bow shock, finding total unabsorbed X-ray flux in the 0.3–2keV band corresponding to a diffuse X-ray luminosity of LX = 2.33−1.54+1.12 × 1029 erg s−1, consistent with previous work. The diffuse X-ray emission arises from the region between the star and the bow shock. Three-dimensional magnetohydrodyanmic simulations were used to model the interaction of the star’s wind with a uniform interstellar medium (ISM) using a range of stellar and ISM parameters motivated by observational constraints. Synthetic infrared, Hα, soft X-ray, emission measure, and radio 6 GHz emission maps were generated from three simulations, for comparison with the relevant observations. Simulations where the space velocity of ζ Ophiuchi has a significant radial velocity produce infrared emission maps with the opening angle of the bow shock in better agreement with observations than for the case where motion is fully in the plane of the sky. All three simulations presented here have X-ray emission fainter than observed, in contrast to results for NGC 7635. The simulation with the highest pressure has the closest match to X-ray observations, with a flux level within a factor of two of the observational lower limit, and emission weighted temperature of log10(TA/K) = 6.4, although the morphology of the diffuse emission appears somewhat different. The observed X-ray emission is from a filled bubble that is brightest near the star, whereas simulations predict brightening towards the contact discontinuity as density increases. This first numerical study of the bow shock and wind bubble around ζ Ophiuchi uses a relatively simple model of a uniform ISM and ideal-magnetohydrodynamics, and can be used as a basis for comparing results from models incorporating more physical processes, or higher resolution simulations that may show more turbulent mixing.

Statistical Study of Favorable Foreshock Ion Properties for the Formation of Hot Flow Anomalies and Foreshock Bubbles

AbstractHot flow anomalies (HFAs) and foreshock bubbles (FBs) are frequently observed in Earth's foreshock, which can significantly disturb the bow shock and therefore the magnetosphere‐ionosphere system and can accelerate particles. Previous statistical studies have identified the solar wind conditions (high solar wind speed and high Mach number, etc.) that favor their generation. However, backstreaming foreshock ions are expected to most directly control how HFAs and FBs form, whereas the solar wind may partake in the formation process indirectly by determining foreshock ion properties. Using Magnetospheric Multiscale mission and Time History of Events and Macroscale Interactions during Substorms mission, we perform a statistical study of foreshock ion properties around 275 HFAs and FBs. We show that foreshock ions with a high foreshock‐to‐solar wind density ratio (>∼3%), high kinetic energy (>∼600 eV), large ratio of kinetic energy to thermal energy (>∼0.1), and large ratio of perpendicular temperature to parallel temperature (>∼1.4) favor HFA and FB formation. We also examine how these properties are related to solar wind conditions: high solar wind speed and oblique bow shock (angle between the interplanetary magnetic field and the bow shock normal ) favor high kinetic energy of foreshock ions; foreshock ions have large ratio of kinetic energy to thermal energy at large (>30°); small (<30°), high Mach number, and closeness to the bow shock favor a high foreshock‐to‐solar wind density ratio. Our results provide further understanding of HFA and FB formation.