Curved Shock Wave Research Articles

A study of conical and curved shock waves via the streamline transformation method

The axisymmetric conical and curved shock waves are studied theoretically via the streamline transformation method (STM), which integrates the differential geometries of streamlines into the conservation laws and solves the flow field geometrically. Under the condition of a straight shock wave, the governing equation is significantly simplified, from which the assumptions of the conical flow and geometric self-similarity are mathematically proved. The simplified equation is also demonstrated as equivalent to the classical Taylor–MacColl equation. The flow mechanism after a conical shock wave is discussed from a geometric perspective, where the key factor is the dimensionless lateral distance between symmetric planes. The corresponding geometric and flow parameters, e.g., the distance between streamlines, the streamline curvature, and the Mach number, are computed for the demonstration of this mechanism. In the flow field over a circular cone, the isentropic compression is modeled geometrically. Due to the isentropic compression, the conical shock wave remains approximately a Mach wave at small semi-angles and the detachment is suspended at large semi-angles. For the axisymmetric curved shock wave, the flow field is computed directly by the streamline transformation method, where both the concave and convex shock waves are considered. Along the wall streamline, an explicit proportional relation between the shock curvature and the orthogonal derivatives of flow parameters is also provided and verified numerically. These results are applicable in the theoretical analysis and inverse design of axisymmetric shock waves.

Analytical reconstruction of axisymmetric curved shock wave/boundary layer interactions

In practical aerodynamic scenarios, interactions between curved shock waves and boundary layers with composite shock curvatures in both streamwise and spanwise directions are encountered more frequently than those induced by planar oblique shocks. To gain a more systematic understanding of this phenomenon in physics, internal and external axisymmetric curved shock wave/boundary layer interactions (AC-SBLI) are being studied analytically by establishing the underlying relations within the flow structures. Specifically, based on reconstructing the dominant structures of AC-SBLI as a semi-analytical model, this study focuses on the impacts of shock spanwise curvatures on separation and the mechanism of their action. It reveals that spanwise curvatures affect the separation in both internal and external AC-SBLI, but each exhibits a different influence pattern, emphasizing apparent axisymmetric characteristic. The spanwise curvatures exert influence indirectly by altering the compression ratio and directly by affecting the distribution of streamwise shock intensities. These findings propose a new approach to estimating the flow structures in AC-SBLI, where analyzing the spanwise curvatures will assist with comprehending other more intricate three-dimensional separation phenomena.

Mach reflection of three-dimensional curved shock waves on V-shaped blunt leading edges

Theoretical investigation of the primary Mach reflection (MR) configuration on V-shaped blunt leading edges (VBLEs) forms the focus of this study. By ignoring the secondary interactions, a theoretical method based on a simplified form of the continuity relation is developed to predict the shock configurations, including the detached shock, the Mach stem, the transmitted shock and the triple point. The comparison of the theoretical results with both numerical and previous experimental results shows the reliability of the theoretical approach in predicting shock structures across a wide range of free stream and geometric parameters. The theoretical model provides a detailed comprehension of the occurrence mechanism of inverse MRs on VBLEs and the influence of the free stream and geometric parameters on primary MR configurations. Along with the primary MR configuration, the curved shock or compression waves generated by the crotch are solved and offer insight into the transition from the MR to the regular reflection from the same family (sRR). The increase of the ratio $R/r$ and the free stream Mach number $M_0$ appears to facilitate the transition, while the effect of the half-span angle $\beta$ is non-monotonic. The predicted shock positions allow for the identification of the transition boundary between the primary MR and sRR. It is found that $R/r$ below a threshold (for a set $M_0$ value) produces MR, irrespective of $\beta$ . If this threshold is exceeded, the configuration can transition from the primary MR to sRR and then back to the primary MR as $\beta$ increases.

Investigating the flow characteristics and thermodynamic performance of curved detonation waves

Oblique detonation waves (ODWs) are induced by oblique shock waves (OSWs) and hold significant research value for air-breathing ramjet engines due to their high thermodynamic efficiency and specific impulse. In this research, detonation is induced using a curved shock wave (CSW) that provides greater flexibility compared to an OSW. Curved detonation waves (CDWs) are classified into two categories based on the curvature of the CSW: positive-curvature curved detonation waves (PC-CDWs) and negative-curvature curved detonation waves (NC-CDWs). We conduct a numerical investigation of the flow characteristics and potential thermodynamic performance in curved detonation waves (CDWs) by solving the two-dimensional multi-species Euler equations with a detailed hydrogen combustion mechanism. In contrast to ODWs, we observe continuous compression and expansion effects in the flow fields of PC-CDWs and NC-CDWs, respectively. Continuous compression shortens the initiation length of the detonation wave and intensifies the compression wave preceding the combustion front, whereas continuous expansion produces the opposite outcome. Changes in the curvature of the CSW result in alterations to the flow characteristics of the CDW. These flow characteristic changes are manifested in their thermodynamic and propulsion-related performance. More specifically, a lower overdrive of the detonation wave corresponds to a reduced loss in total pressure.

Analysis of wave converging phenomena inside the shocked two-dimensional cylindrical water column

Due to the curvature of the droplet surface, the propagation of transmitted waves is complex inside a droplet impacted by an incident shock wave. The wave converging phenomena inside a two-dimensional water column impacted by different curved shock waves are explored in this paper by means of theoretical ray analysis and high-resolution numerical simulations. An analytical method describing the wave structure evolution characteristics inside the shocked water column is established. Hence, the morphological pattern and focus locations of these waves are theoretically obtained. The analysis shows that both the first and the second reflected waves focus inside the water column regardless of the convergent, planar or divergent nature of the incident shock wave shape. The dimensionless distances from focusing points to the column centre are derived as ${\kappa }/{( 3\kappa -{{M}_{0}}{{f}_{s}} )}$ for the former and ${\kappa }/{( 5\kappa -{{M}_{0}}{{f}_{s}})}$ for the latter, respectively. Here, $\kappa$ , $M_0$ and $f_s$ represent the sound-speed ratio of the two phases, the incident shock wave strength and a function characterising the shock wave shape effect, respectively. Moreover, highly negative pressures due to the first reflected wave focusing and significant pressure oscillations due to the second reflected wave focusing are numerically tracked for three shapes of the incident shock. The effects of the incident shock wave intensity on the pressure variations at focus points are further studied. As the incident shock wave intensity increases, stronger negative pressure and higher pressure oscillation are induced. The converged incident shock wave can enhance the above phenomena, but the diverged one can weaken them.

Analysis on the radial structure of rotating detonation wave in a hollow combustor

This experimental study discusses the radial structure of the rotating detonation wave (RDW) in a hollow chamber. The head chamber wall could be fabricated with quartz glass for high-speed visualizations or metal cover for high-frequency pressure measurement. The entire radial-stratified reaction zone was directly and continuously captured. The wall-attached bright zone indicated that the main reaction zone propagated along the chamber wall, but some deflagration was still observed in the inner zone. Based on the phase-averaged CH* chemiluminescence image, the radial thickness of the reaction zone was determined to be about 15–20 mm. The diminishing pressure peaks and deteriorating waveforms of the high-frequency pressure signals proved that the intensity of shock waves decreased along the chamber radius. The structure of RDW was reconstructed using radially distributed pressure sensors. Due to the compression effect of the concave chamber wall, a Mach reflection of detonation wave might occur. The detonation wave was connected to a diffracted shock, which stretched in the radial direction with decreasing intensity. The central angles calculated with the time-difference analysis demonstrated that the entire RDW bent forward in the inner zone of the chamber. Moreover, the radial thickness of the reaction zone and the curvature of shock waves increased with decreasing nozzle contraction ratio, which mainly attributed to the increasing expansion effect. This study demonstrated the distribution of reaction zone and the radial structures of shock waves for the RDW in a hollow combustor, contributing to the exploration of the flow and combustion process in a rotating detonation engine.

Baroclinic interaction of forced shock waves with random thermal gradients

Density gradients aligned at an angle to pressure gradients result in baroclinic torque in fluid flows, generating vorticity. In this work, we study the vorticity generated by the baroclinic torque exerted by the interaction of pressure jumps across random two-dimensional shock waves with density gradients. A field of random two-dimensional shock waves has acoustic spectral energy scaling as Êk∼ε2/3ℓ−1/3k−2, where k is the wavenumber, ε is the energy dissipation, and ℓ is the integral length scale of the field. Since the acoustic energy is broadband, pressure and velocity gradients exist in a wide range of length scales. We study the interaction of these broadband gradients with isobaric thermal gradients localized at a length scale in the spectral space. We show that the method of generating shock waves or injection of wave energy in the system governs the baroclinic interactions. For stochastically forced shock waves, baroclinic terms are negligible. Broadband vorticity with energy at least two orders of magnitude smaller is generated due to continuous variation in curvature of shock waves caused by stochastic forcing. On the other hand, shock waves maintained by energy rescaling result in the generation of coherent vorticity. We also discuss the relative magnitude of the baroclinic torque generated due to total density gradients compared to the one generated due to non-isentropic density gradients within the shock waves interacting with the pressure gradients.

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.

On weak stability of shock waves in 2D compressible elastodynamics

By using an equivalent form of the uniform Lopatinski condition for 1-shocks, we prove that the stability condition found by the energy method in [A. Morando, Y. Trakhinin and P. Trebeschi, Structural stability of shock waves in 2D compressible elastodynamics, Math. Ann. 378 (2020) 1471–1504] for the rectilinear shock waves in two-dimensional flows of compressible isentropic inviscid elastic materials is not only sufficient but also necessary for uniform stability (implying structural nonlinear stability of corresponding curved shock waves). The key point of our spectral analysis is a delicate study of the transition between uniform and weak stability. Moreover, we prove that the rectilinear shock waves are never violently unstable, i.e. they are always either uniformly or weakly stable.

Nonequilibrium Factor in the Structure of a Curved Shock Wave Refracted Into an Intensively Heated Medium

The role of thermal nonequilibrium state in the structure of a curved shock wave during its refraction into an intensively heated medium has been studied analytically and numerically. At elevated gas temperatures common for optical discharges and for moderate-to-strong shock intensities, the width of the zone of variable specific heat is on sub- to mm scale, and thus is comparable to the geometrical scale of the problem. The asymptotic temperature gap, the specific temperature distribution in the zone, and the relaxation zone width are discussed in connection with the changes in the contrast level of the shock images obtained with shadow techniques. Numerical examples are presented in spherical and elliptical geometries for the conditions of a laser-induced breakdown in pure nondissociating nitrogen gas.

Experiments in Shock-Vortex Interactions

Studies of shock-vortex interactions in the past have predominantly been numerical, with a number of idealizations such as assuming an isolated vortex and a plane shock wave. In the present case the vortex is generated from flow separation at a corner. A shear layer results which wraps up into a spiral vortex. The flow is impulsively initiated by the diffraction of a shock wave over the edge. The strength of the shock determines the nature of the flow at the corner and that induced behind the diffracted wave. A wide variety of cases are considered using different experimental arrangements such as having two independent shock waves arriving at the corner at different times, to reflecting the diffracting wave off different surfaces back into the vortex, and to examining the flow around bends where the reflection off the far wall reflects back onto the vortex. The majority of studies have shown that the vortex normally retains its integrity after shock transit. Some studies with curved shock waves and numerous traverses have shown evidence of vortex breakup and the development of turbulent patches in the flow, as well as significant vortex stretching. Depending on the direction of approach of the shock wave it refracts through the shear layer thereby changing the strength and direction of both. Of particular note is that the two diffracted waves which emerge from the vortex as the incident wave passes through interact with each other resulting in a pressure spike of considerable magnitude. An additional spike is also identified.

Classification of Filament Formation Mechanisms in Magnetized Molecular Clouds

Abstract Recent observations of molecular clouds show that dense filaments are the sites of present-day star formation. Thus, it is necessary to understand the filament formation process because these filaments provide the initial condition for star formation. Theoretical research suggests that shock waves in molecular clouds trigger filament formation. Since several different mechanisms have been proposed for filament formation, the formation mechanism of the observed star-forming filaments requires clarification. In the present study, we perform a series of isothermal magnetohydrodynamics simulations of filament formation. We focus on the influences of shock velocity and turbulence on the formation mechanism and identified three different mechanisms for the filament formation. The results indicate that when the shock is fast, at shock velocity v sh ≃ 7 km s−1, the gas flows driven by the curved shock wave create filaments irrespective of the presence of turbulence and self-gravity. However, at a slow shock velocity v sh ≃ 2.5 km s−1, the compressive flow component involved in the initial turbulence induces filament formation. When both the shock velocities and turbulence are low, the self-gravity in the shock-compressed sheet becomes important for filament formation. Moreover, we analyzed the line-mass distribution of the filaments and showed that strong shock waves can naturally create high-line-mass filaments such as those observed in the massive star-forming regions in a short time. We conclude that the dominant filament formation mode changes with the velocity of the shock wave triggering the filament formation.

RBF-POD reduced-order modeling of flow field in the curved shock compression inlet

The ramjet/scramjet engines require the control-oriented model to predict the inlet flow field in less than a few seconds. However, it is challenging for these kinds of inlets which utilize curved shock waves to compress the air flow. In this paper, a reduced-order model based on the computational fluid dynamics, the proper orthogonal decomposition theory, and the radial basis function interpolation method is developed. After that, the curved shock waves dominated flow fields of a ramjet inlet under different angles of attack and free stream Mach numbers are predicted with this reduced-order model and compared to the full order computational fluid dynamics solutions. The results show that this reduced-order model can successfully predict the curved shock waves, the curved shock wave/boundary layer interactions, and the shock trains caused by a back pressure with high accuracies. The consumed time is only 0.11 s. The performance parameters are also predicted with the relative errors no more than 2%.

An efficient deep learning framework to reconstruct the flow field sequences of the supersonic cascade channel

Accurate and comprehensive flow field reconstruction is essential for promptly monitoring the flow state of the supersonic cascade. This paper proposes a novel data-driven method for reconstructing the slices of the two-dimensional (2D) pressure field in three-dimensional (3D) flow of the supersonic cascade by using deep neural networks. Considering the complicated spatial effects of 2D pressure field slices, the architecture embeds the convolution into the long short-term memory (LSTM) network to realize the purpose of using the upstream pressure to reconstruct downstream pressure. Numerical simulations of the supersonic cascade under different back pressures are performed to establish the database capturing the complex relationship between the upstream and downstream flow. The pressure of different upstream slices can be used as a spatial-dependent sequence as the input of the model to reconstruct the pressure of different downstream slices. A deep neural network including special convolutional LSTM layers and convolutional layers is designed. The trained model is then tested under different back pressures. The reconstruction results are in good agreement with the computational fluid dynamics, especially for the identification of shock wave position changes and the recognition of complex curved shock waves in 3D flow with high accuracy. Moreover, analyzing the frequency distribution of reconstructed pressure at different positions can clearly distinguish the flow separated zone, which will further improve the accuracy of the state monitoring. Specifically, it is of great significance for identifying the stall of the flow field promptly.

Evolution characteristics of subsonic-supersonic mixing layer impinged by shock waves

The study of interaction between shock wave and mixing layer commonly existed in rocket based combined cycle engine is of profound significance for mixing enhancement in combustor. In this paper, the three-dimensional structures of subsonic-supersonic mixing layer impinged by shock waves induced by wedge were investigated experimentally employing the ice-cluster based planar laser scattering technique. The results showed that the mixing layer would partially deflect to the subsonic stream side and inhibit the development of K-H instability vortices under the interaction of induced shock wave. With the increasing of wedge angle (β), the interaction of shock waves could enlarge the global deflection angle of mixing layer and reduce the scale of large-scale vortices. Moreover, the T-shaped structures and horseshoe vortices were existed as usual topological structures in small-scale turbulent transport. Besides, the curved shock waves were generated under the influence of large-scale vortices. In addition, the comparison of mixing layer thickness indicated that the evolution of mixing layer could be promoted for β=5°, while it would be suppressed for β=15° in general.

Very weak shock wave reflection off curved surfaces

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.

Design method for non-axisymmetric generalized internal conical flowfield based on double 3D curved shock waves

To provide ease of integration design of forebody/hypersonic inward turning inlet as well as to extend the domain of option for basic flowfield of the inlet, a design method for axisymmetric basic flowfield with controllable incident/reflected shock wave and certain post reflected shock parameter distribution is developed by using the method of characteristics and optimization algorithm. On this basis, a design method for non-axisymmetric generalized internal conical (NAGIC) basic flowfield is proposed by using the theory of osculating axisymmetric flows to incident and reflected shock wave in parallel. Numerical simulations are performed to verify the design methods. Detailed flow characteristics of NAGIC flowfield with an elliptical entrance that is assigned somewhat arbitrarily are presented. The incident and reflected shock wave geometries are controllable as given. In cross sections, the shock wave geometries are similar to the entrance shape of the inner cone. In osculating planes, the incident and reflected shock waves attach accurately to the lip and the shoulder, respectively. In addition, the crossflow in the 3D flowfield is negligible, and the flow parameters in the exit plane are almost uniform. Thus, the NAGIC flowfield is suitable for inward turning inlet design with complex configurations.

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ОБТЕКАНИЯ СПУСКАЕМОГО АППАРАТА ПРИ СВЕРХЗВУКОВЫХ И ГИПЕРЗВУКОВЫХ СКОРОСТЯХ. СРАВНЕНИЕ ДВУХ СПУСКАЕМЫХ АППАРАТОВ. ЧАСТЬ 1

This study presents flow simulation over the reentry capsule at supersonic and hypersonic speeds. Numerical algorithms solve for the CFD method, which is produced using help ANSYS Fluent 19.2. The using GPU core to get a solution faster. The main purpose – flow simulation and numerical analysis reentry capsule; understand the behavior of supersonic and hypersonic flow and its effect on the reentry capsule; compare temperature results for the range Mach numbers equals 2–6. This study showed results on velocity counters, on temperature counters and vector of velocity for range Mach numbers equals 2–6. This study demonstrates the importance of understanding the effects of shock waves and illustrates how the shock wave changes as the Mach number increases. For every solves, the mesh had adapted for pressure gradient and velocity gradient to get the exact solution. As a result of the obtained solution, it is found that a curved shock wave appears in front of the reentry capsule. The central part of which is a forward shock. An angular expansion process is observed, which is a modified picture of the Prandtl- Mayer flow that occurs in a supersonic flow near the sharp edge of the expanding region. It is revealed that with an increase in the Mach number, the shock wave approaches the bottom of the reentry capsule, and there is also a slope of the shock to the flow direction, with an increase in the Mach number. The relevance and significance of this problem for the design of new and modernization of old reentry capsules.

The Relationship between Discharge Power and Shock Wave Curvature Phenomena

The Relationship between Discharge Power and Shock Wave Curvature Phenomena

CFD Analysis of Dynamics of Interaction of ShockWave -Vortex Core over Flapped Wing of Supersonic Aircraft at different angle of attack & Mach number

The objective of this paper was to analysis the condition for the appearance of the many types of interaction of a vortex core with shock wave over a flapped wing of a supersonic aircraft. A five digit NACA 23012 aerofoil was selected for this work. Structured Mesh was generated by Quadrilaterals Method. Steady-state density based implicit solver and K-ω SST turbulent model was selected. Q criterion method with vorticity magnitude was used to calculate the vortex core. NACA aerofoil Scaled model was manufactured by using NACA profile for experimental work and CFD results were validated by pressure coefficient calculated by wind tunnel setup. Finally, concluded that weak interaction with no vortex breakdown was observed at M= 1.4 and a strong interaction with a bubble-like vortex breakdown formed at M= 1.8 and It found that when a shock wave interact with vortex core, disturbance is generated, which expands along the shock wave and deformed into many small vortices. The flow field is compressed behind the curved shock wave which is reason of acoustic waves. This principle are related to the shock–turbulence interaction which is one of major source of noise. Also concluded that initially at low angle of attack,it observed a strong organized flow field in the downstream region which is due to less strength of the shock. The development of a transmitted shock wave across the vortex core was observed because of shock scattering phenomenon. The moderate breakdown of the vorticity field that occurs after a very strong shock at M =1.4 also observed and the breakdown was more intense when increased Mach No. up to 1.8. Weak and strong interaction region were observed and three stages of interaction found by the flow field over aerofoil at high Mach No. =1.8.