Conical 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.

Numerical simulation of gas dynamics of the inflow into a channel located behind a conical or plane shock wave

Numerical simulation of gas dynamics of the inflow into a channel located behind a conical or plane shock wave

Comparative Analysis of Supersonic Flow in Atmospheric and Low Pressure in the Region of Shock Waves Creation for Electron Microscopy.

This paper presents mathematical-physics analyses in the field of the influence of inserted sensors on the supersonic flow behind the nozzle. It evaluates differences in the flow in the area of atmospheric pressure and low pressure on the boundary of continuum mechanics. To analyze the formation of detached and conical shock waves and their distinct characteristics in atmospheric pressure and low pressure on the boundary of continuum mechanics, we conduct comparative analyses using two types of inserted sensors: flat end and tip. These analyses were performed in two variants, considering pressure ratios of 10:1 both in front of and behind the nozzle. The first variant involved using atmospheric pressure in the chamber in front of the nozzle. The second type of analysis was conducted with a pressure of 10,000 Pa in front of the nozzle. While this represents a low pressure at the boundary of continuum mechanics, it remains above the critical limit of 113 Pa. This deliberate choice was made as it falls within the team's research focus on low-pressure regions. Although it is situated at the boundary of continuum mechanics, it is intentionally within a pressure range where the viscosity values are not yet dependent on pressure. In these variants, the nature of the flow was investigated concerning the ratio of inertial and viscous flow forces under atmospheric pressure conditions, and it was compared with flow conditions at low pressure. In the low-pressure scenario, the ratio of inertial and viscous flow forces led to a significant reduction in the value of inertial forces. The results showed an altered flow character, characterized by a reduced tendency for the formation of cross-oblique shockwaves within the nozzle itself and the emergence of shockwaves with increased thickness. This increased thickness is attributed to viscous forces inhibiting the thickening of the shockwave itself. This altered flow character may have implications, such as influencing temperature sensing with a tipped sensor. The shockwave area may form in a very confined space in front of the tip, potentially impacting the results. Additionally, due to reduced inertial forces, the cone shock wave's angle is a few degrees larger than theoretical predictions, and there is no tilting due to lower inertial forces. These analyses serve as the basis for upcoming experiments in the experimental chamber designed specifically for investigations in the given region of low pressures at the boundary of continuum mechanics. The objective, in combination with mathematical-physics analyses, is to determine changes within this region of the continuum mechanics boundary where inertial forces are markedly lower than in the atmosphere but remain under the influence of unreduced viscosity.

Numerical investigations on interactions between 2D/3D conical shock wave and axisymmetric boundary layer at Ma=2.2

Numerical simulation and analysis are carried out on interactions between a 2D/3D conical shock wave and an axisymmetric boundary layer with reference to the experiment by Kussoy et al. [AIAA J, 1980, 18(12): 1477–1487], in which the shock was generated by a 15° half-angle cone in a tube at 15° angle of attack (AOA). Based on the Reynolds-averaged Navier–Stokes equations and Menter's SST turbulence model, the present study uses the newly developed WENO3−PRM1,12 scheme and the PHengLEI CFD platform for the computations. First, computations are performed for the 3D interaction corresponding to the conditions of the experiment by Kussoy et al., and these are then extended to cases with AOA = 10° and 5° For comparison, 2D axisymmetric counterparts of the 3D interactions are investigated for cones coaxial with the tube and having half-cone angles of 27.35°, 24.81°, and 20.96° The shock wave structure, vortex structure, variable distributions, and wall separation topology of the interaction are computed. The results show that in 2D/3D interactions, a new Mach reflection-like event occurs and a Mach stem-like structure is generated above the front of the separation bubble, which differs from the model of Delery for 2D planar shock wave/boundary layer interaction. A new interaction model is established to describe this behavior. The relationship between the length of the circumferentially unseparated region in the tube and the AOA of the cone indicates the existence of a critical AOA at which the length is zero, and a prediction of this angle is obtained using an empirical fit, which is verified by computation. The occurrence of side overflow in the windward meridional plane is analyzed, and a quantitative knowledge is obtained. To further elucidate the characteristics of the 3D interaction, the scale and structure of the vortex and the pressure and friction force distributions are presented and compared with those of the 2D interaction.

Numerical Simulation of Flow Field Over a Sharp-Tipped Double Cone at High Speed

The flowfield over a sharp-tipped double cone axisymmetric configuration is carried out by solving time dependent axisymmetric laminar compressible Navier-Stokes equations for freestream Mach number range of 2.0 - 6.0. The fluid dynamics equations are discretized in spatial coordinates using integral formulation in conjunction with finite volume method which reduces the governing equations to semi-discretized ordinary differential equations. Temporal integration is performed employing multistage Runge-Kutta time-stepping scheme. A local time step is used to achieve steady-state solution. The numerical computation is carried out on a mono-block with structured grids. The flowfield features over the sharp-tipped double cone configuration such as conical shock wave, separation region, separation and reattachment shock wave and bow shock wave in the cone region, expansion fan, and recirculation flow in the base region are well captured at Mach number 3 and 6 which are identical to the Edney VI type shock interaction. The velocity vector, Mach contours plots and variations of surface pressure coefficient along the sharp-tipped double cone configuration are analyzed at various Mach numbers. The fore-body aerodynamic drag is calculated employing computed pressure distribution. The paper presents the influence of the freestream Mach number on the flows with shock interactions over the sharp-tipped double cone geometry.

On wall pressure fluctuations in conical shock wave/turbulent boundary layer interaction

The structure and the frequency spectra of wall pressure fluctuations beneath a planar turbulent boundary layer interacting with a conical shock wave at Mach number $M_\infty =2.05$ and Reynolds number $\textit {Re}_\theta \approx 630$ (based on the upstream boundary layer momentum thickness) are examined to elucidate the effects of pressure gradient and flow separation on the characteristics of the wall pressure fluctuations, by exploiting a direct numerical simulation database. Upstream of the interaction, in the zero pressure gradient region, wall pressure statistics compare well with canonical compressible boundary layers in terms of fluctuation intensities and frequency spectra. Across the main interaction zone (APG1), the root-mean-square of wall pressure fluctuations becomes very large (corresponding to approximately 173.3 dB), with maximum increase approximately 12.7 dB from the incoming level. In the second adverse pressure gradient zone (APG2), the root-mean-square of wall pressure fluctuations attains a second peak (corresponding to $164.7$ dB), with an increase of 8.4 dB from the upstream level. Both the APG1 and APG2 regions feature a substantial fraction of flow reversal events, which are, however, scattered and interspersed with regions of attached flow. The wall pressure power spectral density exhibits a broadband and energetic low-frequency component associated with the global unsteadiness of the separation bubble/conical shock system. Analysis of the two-point correlations and wavenumber/frequency spectra of wall pressure fluctuations further suggests that the typical eddies become more elongated along the spanwise direction, as the flow in the separated region tends to escape the centreline, and the convection velocity is significantly reduced.

Three‐dimensional stationary supersonic flows with axial symmetry in relativistic hydrodynamics

AbstractThe study of supersonic flows in relativistic hydrodynamics (RHDs) is of some interest for cosmology. This paper studies two types of three‐dimensional (3D) stationary supersonic flows with axial symmetry in RHDs: supersonic flows expanding in vacuum and supersonic flows against an infinite curved cone. To study supersonic flows expanding in vacuum, we consider a Cauchy problem for the 3D steady RHD equations with initial data that are a combination of an axisymmetric supersonic flow in a circular domain and vacuum in the remaining domain. Global existence of a piecewise smooth solution expanding in vacuum to the Cauchy problem is obtained, provided that the initial mass‐energy density is sufficiently small. When a supersonic flow arrives and hits a curved cone, a shock wave will arise around the cone. Moreover, if the vertex angle of the cone is less than a critical angle, then the shock will be attached to the vertex of the cone. We obtain the global existence of attached conical shock waves, provided that the curved cone satisfies some suitable conditions and the speed of the incoming flow is close to a limit speed.

Numerical Investigation of Asymmetric Mach 2.5 Turbulent Shock Wave Boundary Layer Interaction

Supersonic shock wave boundary layer interactions are common to inlet flows of supersonic and hypersonic vehicles. This paper reports on wall-resolved implicit large-eddy simulations of a canonical Mach 2.5 turbulent shock wave boundary layer interaction experiment at the NASA Glenn Research Center. The boundary layer upstream of the interaction was nominally axisymmetric and two-dimensional. A conical centerbody with a 16 deg half-angle and a maximum radius of 0.147D of the test section diameter was employed to generate a conical shock wave, where D is the test section diameter. Asymmetric (swept) interactions were obtained by displacing the shock generator away from the test section centerline. The present simulation is for a shock generator displacement of D/6. Results from the asymmetric simulation are compared with results from an earlier simulation of a corresponding axisymmetric interaction. The experimental Reynolds number based on test section diameter was ReD=4×106. For the simulations, the Reynolds number was lowered to ReD=4×105 to keep the computational expense of the simulations within limits. Compared to the axisymmetric interaction, the streamwise extent of the separation varies considerably in the azimuthal direction for the asymmetric interaction. The separation is strongest at the azimuthal location that is closest to the shock generator. The streamwise extent of the separated flow regions is noticeably reduced and substantial crossflow is observed between the locations that are closest and farthest from the shock generator. A Fourier analysis of the unsteady flow data indicates low-frequency content for the separated region that is closest to the shock generator. Away from this region, with increasing sweep angle and cross-flow, the low-frequency content is diminished. A proper orthogonal decomposition captures spanwise coherent structures for the more two-dimensional parts of the interaction.

Drop Interactions with the Conical Shock Structure Generated by a Mach 4.5 Projectile

This work presents measurements of liquid drop deformation and breakup time behind approximately conical shock waves and evaluates the predictive capabilities of low-order models and correlations developed using planar shock experiments. A conical shock was approximated by firing a bullet at Mach 4.5 past a vertical column of water drops with a mean initial diameter of . The time-resolved drop position and maximum transverse dimension were characterized using backlit stereo images taken at 500 kHz. The gas density and velocity fields experienced by the drops were estimated using a Reynolds-averaged Navier–Stokes simulation of the bullet. Classical correlations predict drop breakup times and deformation in error by a factor of 3 or more. The Taylor analogy breakup (TAB) model predicts deformed drop diameters that agree within the confidence bounds of the ensemble-averaged experimental values using a dimensionless constant compared to the accepted value . Results demonstrate existing correlations are inadequate for predicting the drop response to the three-dimensional relaxation of the flowfield downstream of a conical-like shock and suggest the TAB model results represent a path toward improved predictions.

Numerical Investigation of Mach 2.5 Axisymmetric Turbulent Shock Wave Boundary Layer Interactions

Shock wave boundary layer interactions are common to both supersonic and hypersonic inlet flows. Wall-resolved implicit large-eddy simulations of a canonical Mach 2.5 axisymmetric shock wave boundary layer interaction experiment at Glenn Research Center were carried out. A conical shock wave was generated with axisymmetric centerbodies with 16 deg half-angle cone. The centerbody radii were 9.2% and 14.7% of the test section diameter. The conical shock wave interacted with the turbulent boundary layer on the inside of the cylindrical test section. The experimental Reynolds number based on diameter was six million. For the simulations, the Reynolds number was reduced by a factor of 10 to lower the computational expense. The turbulent boundary layer separates for both centerbody radii and the separation is stronger for the larger centerbody radius. Frequency spectra of the spanwise-averaged wall-pressure coefficient reveal low-frequency content at Strouhal numbers based on separation length between 0.02 and 0.05 in the vicinity of the separation shock and mid-frequency content between 0.1 and 0.2 downstream of separation. A proper orthogonal decomposition captures spanwise coherent structures with a Strouhal number of 0.03–0.04 over the interaction region and streamwise coherent structures inside and downstream of the interaction with a Strouhal number of 0.1–0.4.

Smooth Transonic Flows Around Cones

<p style='text-indent:20px;'>We consider a conical body facing a supersonic stream of air at a uniform velocity. When the opening angle of the obstacle cone is small, the conical shock wave is attached to the vertex. Under the assumption of self-similarity for irrotational motions, the Euler system is transformed into the nonlinear ODE system. We reformulate the problem in a non-dimensional form and analyze the corresponding ODE system. The initial data is given on the obstacle cone and the solution is integrated until the Rankine-Hugoniot condition is satisfied on the shock cone. By applying the fundamental theory of ODE systems and technical estimates, we construct supersonic solutions and also show that no matter how small the opening angle is, a smooth transonic solution always exists as long as the speed of the incoming flow is suitably chosen for this given angle.</p>

Mathematical and Physical Analysis of the Effect of Conical and Detached Shock Waves at the Tip of a Static Probe in an Experimental Chamber

As part of the research in the field of pumping vacuum chambers in the Environmental Electron Microscope, research on supersonic flow through apertures is being carried out at the Department of Electrical and Electronic Technology of the Brno University of Technology in cooperation with the Institute of Scientific Instruments of the CAS. This paper deals with the influence of the shape of the static probe cone design for static pressure measurements in the supersonic flow regime in the Experimental Chamber. The cone of the probe has an effect on the shape of the shock wave, which significantly influences the detected static pressure value.

Intensification of non-uniformity in convergent near-conical hypersonic flow

The inevitable formation of a Mach disk at the central axis of a convergent conical shock wave may suffer from fundamental changes when the flow deviates from the axisymmetric condition. In this paper, the behaviours of near-conical shocks, which are generated by a circular ring wedge of $10^{\circ }$ at typical angles of attack (AoAs), are investigated at a free stream Mach number of 6 in a shock tunnel. To reveal the characteristics and mechanism of the flow, numerical analyses are carried out under the same conditions. The results indicate that when the flow deviates from axial symmetry, the circumferential non-uniformity is remarkably intensified as the shock converges downstream. The converging centre shifts against the inclination of the incoming flow and moves to the leeward side. For a sufficiently small AoA, the formation of a Mach disk remains similar to that in the axisymmetric case, although the Mach disk shrinks in size and is slightly flattened. As the circumferential non-uniformity of the shock increases at an AoA of approximately $3^{\circ }$ , a pair of kinks separate the shock surface into two discontinuous segments with the stronger shock segment on the windward side and the weaker shock segment on the leeward side. When the AoA increases further, the shrinkage of the Mach disk continuously occurs, and the Mach disk is eventually replaced by a regular reflection. The discontinuity of a convergent shock with flattening on the separated shock segments and the insufficient strength increase during the subsequent convergence are responsible for the appearance of regular reflection.

Design and analysis of the air-breathing aircraft with the full-body wave-ride performance

Waverider is expected to break the lift-to-drag ratio constraint in the hypersonic flight condition and it is commonly been studied as the forebody of the cruise vehicle. In this paper, we developed an approach to make waverider as the airframe for the air-breathing cruise vehicle. The fuselage that designed based on the cone-derived waverider theory with a unique upper boundary curve generates the same conical shock wave as that generated by the conical nose of the vehicle. Therefore, the vehicle is expected to have a full-body wave-ride performance and suitable for cruising in the hypersonic condition. Also, a gap between the conical nose and the fuselage is designed to be the inlet of the air-breathing engine. In comparison with the traditional cone-derived waverider, this kind of configuration has additional volume so that it can afford enough space for installing the engine system in the airframe. The performance of this kind of configuration is validated by the numerical method in the design Mach number. With the engineering estimation approach used to calculate the aerodynamic performance of the vehicle in hypersonic conditions, we studied the influence of design parameters on the aerodynamic performance. Results show that the increase of the design Mach number does negative impact on the lift-to-drag ratio but positive impact on the volumetric efficiency while the increase of the design direction angle has an opposite influence. However, in comparison with the response of the lift-to-drag ratio, the volumetric efficiency is less affected by the dihedral angle, which means that the dihedral angle can control the aerodynamic performance effectively within a limited volumetric efficiency range.

Energy Harvesting Using Thermocouple and Compressed Air.

In this paper, we describe the possibility of using the energy of a compressed air flow, where cryogenic temperatures are achieved within the flow behind the nozzle, when reaching a critical flow in order to maximize the energy gained. Compared to the energy of compressed air, the energy obtained thermoelectrically is negligible, but not zero. We are therefore primarily aiming to maximize the use of available energy sources. Behind the aperture separating regions with a pressure difference of several atmospheres, a supersonic flow with a large temperature drop develops. Based on the Seebeck effect, a thermocouple is placed in these low temperatures to create a thermoelectric voltage. This paper contains a mathematical-physical analysis for proper nozzle design, controlled gas expansion and ideal placement of a thermocouple within the flow for best utilization of the low temperature before a shockwave formation. If the gas flow passes through a perpendicular shockwave, the velocity drops sharply and the gas pressure rises, thereby increasing the temperature. In contrast, with a conical shockwave, such dramatic changes do not occur and the cooling effect is not impaired. This article also contains analyses for proper forming of the head shape of the thermocouple to avoid the formation of a detached shockwave, which causes temperature stagnation resulting in lower thermocouple cooling efficiency.

The scaling of separation bubble in the conical shock wave/turbulent boundary layer interaction

A direct numerical simulation database (Zuo et al., J. Fluid Mech. vol. 877, 2019, 167–195) is analyzed to investigate the size of the separation bubble and the relation with the pressure rise in the conical shock wave/turbulent boundary layer interaction. The characteristics of the separation bubbles in the supersonic turbulent boundary layer induced by conical shock waves are discussed. The inviscid reflected conical shock theory is analyzed, to conclude that regular reflection transition to Mach reflection will occur somewhere along the spanwise direction in the interaction zone. Then, the typical shape of a turbulent separation bubble in the conical shock wave/turbulent boundary layer interaction is described. The scaling between the size of the turbulent separation bubble and moderate conical shock intensities is established, giving a potential alternative method to rapidly predict the size of turbulent separation bubbles, especially in the supersonic viscous corrections under condition of boundary layer separation.

Investigation of conical shock wave/boundary layer interaction in axisymmetric internal flow

A RANS (Reynolds-averaged Navier-Stokes simulation) study is performed on an internal duct at a free-stream Mach number 4.0, to investigate the internal conical shock wave/boundary layer interaction (CSBLI) developing from an axisymmetric wall surface. An effort is devoted on the analysis of the flow features of the internal CSBLI phenomena through a simple physical model. The work aims to extend the investigation of Zuo et al. [1] on external CSBLI. The numerical strategy is first described and the capability of turbulence models to capture SBLI physics is then assessed by comparison with experiments. In the interacted region a small recirculation bubble is identified, and the foot of the conical shock forms a λ system. Unlike the external CSBLI case, a horseshoe pattern of separation whose strength and size are reduced away from the symmetry plane is observed. The topology of flow separation is similar to the one of two-dimensional closed separation. The pressure in the pipe increases rapidly by the effect of multi-reflected conical shock waves. The wall-surface pressure in the separation bubble continues to increase due to three-dimensional area compression effect.

A novel approach for inverse design of three-dimensional shock waves under non-uniform flows

Traditional osculating axisymmetric flow (OA) theory is widely used in the inverse design of generalized shock waves. However, this theory is only applicable to uniform incoming flows. In this paper, a novel method called micro osculating axisymmetric flow (MOA) method is proposed to solve the inverse problem of the 3D generalized shock wave design under non-uniform incoming flows. This method constructs a series of micro osculating planes (MOPs) along the shock wave surface in spanwise and streamwise directions. Actual 3D flows are then approximated by 2D axisymmetric flows in each MOP. Thus, the new method breaks the restriction of no lateral velocities or pressure gradients in the traditional method. An internal conical shock wave at a 4° angle of attack and the other one in an external conical flow of a 10° cone half-angle are obtained by the novel method to validate its feasibility and applicability. Numerical simulation results of the two cases indicate that the 3D shock wave geometries completely match the target. A streamwise integration of the inward turning wavecatcher inlet and outward turning waverider forebody is also presented. This integration shows the remarkable application prospects of the MOA method in the field of air-breathing hypersonic propulsion.

Self-focusing of UV radiation in 1 mm scale plasma in a deep ablative crater produced by 100 ns, 1 GW KrF laser pulse in the context of ICF

Experiments at the GARPUN KrF laser facility and 2D simulations using the NUTCY code were performed to study the irradiation of metal and polymethyl methacrylate (PMMA) targets by 100 ns UV pulses at intensities up to 5 × 1012 W cm−2. In both targets, a deep crater of length 1 mm was produced owing to the 2D geometry of the supersonic propagation of the ablation front in condensed matter that was pushed sideways by a conical shock wave. Small-scale filamentation of the laser beam caused by thermal self-focusing of radiation in the crater-confined plasma was evidenced by the presence of a microcrater relief on the bottom of the main crater. In translucent PMMA, with a penetration depth for UV light of several hundred micrometers, a long narrow channel of length 1 mm and diameter 30 μm was observed emerging from the crater vertex. Similar channels with a length-to-diameter aspect ratio of ∼1000 were produced by a repeated-pulse KrF laser in PMMA and fused silica glass at an intensity of ∼109 W cm−2. This channel formation is attributed to the effects of radiation self-focusing in the plasma and Kerr self-focusing in a partially transparent target material after shallow-angle reflection by the crater wall. Experimental modeling of the initial stage of inertial confinement fusion-scale direct-drive KrF laser interaction with subcritical coronal plasmas from spherical and cone-type targets using crater-confined plasmas seems to be feasible with increased laser intensity above 1014 W cm−2.

Study of MHD dynamo effect at the outlet from plasma accelerator in the presence of longitudinal magnetic field

AbstractProperties of compressible flows in the quasi‐stationary plasma accelerator have been studied in the presence of an additional longitudinal magnetic field and the arising rotation of plasma flow. Numerical study was carried out within the framework of two‐dimensional magnetic hydrodynamics (MHD) model of the axisymmetric plasma flows taking into account the finite conductivity of medium and radiation transport. Dynamics of compressible plasma flows is accompanied by the MHD dynamo effect or generation of magnetic field on a conical shock wave forming at the outlet from the accelerator.