Steady Mach reflection with two incident shock waves

2018 ◽  
Vol 855 ◽  
pp. 882-909 ◽  
Author(s):  
Xiao-Ke Guan ◽  
Chen-Yuan Bai ◽  
Zi-Niu Wu
Keyword(s):  
Shock Wave ◽  
Shock Waves ◽  
Mach Reflection ◽  
Lower Surface ◽  
Incident Shock ◽  
Incident Shock Wave ◽  
Neumann Condition ◽  
Mach Stem ◽  
Stem Height ◽  
Reflecting Surface

Mach reflection in steady supersonic flow with two incident shock waves is studied. The second incident shock wave is produced by an additional deflection of the wedge lower surface, at some point ensuring that the two incident shock waves would intersect at the reflecting surface in case of normal reflection. Both theory and computational fluid dynamics (CFD) are used to study the flow structure and the influence of the second incident shock wave. The overall flow configuration, in case of Mach reflection, is shown to be composed of a triple shock structure, a shock/shock interaction structure and a shock/slipline reflection structure. Similar phenomenon, triggered by a high downstream pressure, has been observed before numerically, but not studied theoretically. The second incident shock wave reflects over the slipline to deflect the slipline more towards the reflecting surface, increasing thus the Mach stem height, advancing the transition of regular reflection to Mach reflection of the first incident shock wave, and causing an inverted Mach reflection below the usual von Neumann condition. A Mach stem height model built for a weak second incident shock wave is used to study the influence of the second incident shock wave on the Mach stem height. Both theory and CFD predict a maximum of the Mach stem height at some additional wedge deflection angle.

A parametric study of Mach reflection in steady flows

1997 ◽  
Vol 341 ◽  
pp. 101-125 ◽  
Author(s):  
H. LI ◽  
G. BEN-DOR
Keyword(s):  
Present Model ◽  
Mach Reflection ◽  
Incident Shock ◽  
Incident Shock Wave ◽  
Incoming Flow ◽  
Steady Flows ◽  
Mach Stem ◽  
Stem Height ◽  
Von Neumann ◽  
Set Up

The flow fields associated with Mach reflection wave configurations in steady flows are analysed, and an analytical model for predicting the wave configurations is proposed. It is found that provided the flow field is free of far-field downstream influences, the Mach stem heights are solely determined by the set-up geometry for given incoming-flow Mach numbers. It is shown that the point at which the Mach stem height equals zero is exactly at the von Neumann transition. For some parameters, the flow becomes choked before the Mach stem height approaches zero. It is suggested that the existence of a Mach reflection not only depends on the strength and the orientation of the incident shock wave, as prevails in von Neumann's three-shock theory, but also on the set-up geometry to which the Mach reflection wave configuration is attached. The parameter domain, beyond which the flow gets choked and hence a Mach reflection cannot be established, is calculated. Predictions based on the present model are found to agree well both with experimental and numerical results.

Evaluation of assumptions and criteria in pseudostationary oblique shock-wave reflections

1986 ◽  
Vol 406 (1830) ◽  
pp. 75-92 ◽  
Keyword(s):  
Shock Wave ◽  
Shock Waves ◽  
Mach Reflection ◽  
Wedge Angle ◽  
Incident Shock ◽  
Incident Shock Wave ◽  
Oblique Shock ◽  
Oblique Shock Wave ◽  
Reflected Shock ◽  
Wedge Surface

Many experiments in various gases have now been performed on regular and Mach reflection of oblique shock waves in pseudostationary flow. Experimental agreement with the analytical boundaries for such reflec­tions with two- and three-shock theories is reasonable but not precise enough over the entire range of incident shock-wave Mach numbers ( M s ) and compression wedge angle ( θ W ) used in the experiments. In order to improve the agreement, the assumptions and criteria employed in the analysis were critically examined by the use of the experimental data for nitrogen (N 2 ), argon (Ar), carbon-dioxide (CO 2 ), air and sulphurhexa-fluoride (SF 6 ). The assumptions regarding the excitation of the internal degrees of freedom were evaluated based on a relation between the relaxation lengths and a characteristic length of the flow. The ranges in which the frozen-gas and vibrational-equilibrium-gas assumptions can be applied were verified by comparing the experimental and numerical values of δ, the angle between the incident and the reflected shock waves. The deviations of the experimental orientation of the Mach stem at the triple point from a line perpendicular to the wedge surface were considered. A new criterion for the transition from single-Mach to complex-Mach reflection improved the agreement with experiments in the ( M S , θ W )-transition-boundary map. The effects of the shock-induced boundary layer on the wedge surface on the reflected-wave angle and the persistence of regular reflection into the Mach reflection region (‘von Neumann paradox’) were evaluated.

The regular reflection→Mach reflection transition in unsteady flow over convex surfaces

2017 ◽  
Vol 837 ◽  
pp. 48-79 ◽  
Author(s):  
M. Geva ◽  
O. Ram ◽  
O. Sadot
Keyword(s):  
Shock Wave ◽  
Mach Reflection ◽  
Incident Shock ◽  
Stem Growth ◽  
Mach Stem ◽  
Regular Reflection ◽  
Flow Properties ◽  
Numerical Computations ◽  
Convex Surfaces ◽  
Reflecting Surface

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.

Paper 14: On the Reflection of Shock Waves from Deflector Plates

1965 ◽  
Vol 180 (10) ◽  
pp. 245-259
Author(s):  
W. A. Woods
Keyword(s):  
Shock Wave ◽  
Shock Waves ◽  
Incident Shock ◽  
Incident Shock Wave ◽  
Reflected Shock Wave ◽  
Reflected Wave ◽  
Reflected Shock ◽  
Pressure Time ◽  
Wave Travel ◽  
Wave Region

The paper first explains the importance of the reflection of shock waves in the design of certain chemical plant. The theory of the reflection of shock waves is also discussed in the first part of the paper. It is shown that when a shock wave travelling along a pipe containing stationary gas reaches the outlet end of the pipe there may be ( a) a reflected expansion wave, ( b) a reflected shock wave, ( c) a reflected sound wave, ( d) no reflected wave at all, ( e) a standing shock wave situated at the end of the pipe, depending upon the strength of the incident shock wave and the amount of blockage present at the outlet end of the pipe. The conditions for each kind of reflection are determined, and in the case of the reflected shock wave region the strengths and speeds of the reflected shock waves are established throughout the region and the results are presented graphically. In the second part of the paper the results are given of experiments carried out on a shock tube fitted with various kinds of deflector plates. The experiments were performed to study the reflection of shock waves from the deflector plates by measuring pressure/time indicator diagrams near the outlet end of the pipe. The indicator diagrams revealed the approximate pressure amplitudes of the incident and reflected shock waves and also the wave travel times for the shock waves. This information was used in conjunction with the charts given in the first part of the paper to establish the deflector geometry and spacing needed in order to avoid the occurrence of a reflected shock wave.

Transition study for asymmetric reflection between moving incident shock waves

2021 ◽  
Vol 929 ◽  
Author(s):  
Miao-Miao Wang ◽  
Zi-Niu Wu
Keyword(s):  
Shock Waves ◽  
Incident Shock ◽  
Incident Shock Wave ◽  
Neumann Condition ◽  
Plane Shock ◽  
Von Neumann ◽  
Transition Criteria ◽  
Shock Motion ◽  
Reduced Problem

The transition criteria seen from the ground frame are studied in this paper for asymmetrical reflection between shock waves moving at constant linear speed. To limit the size of the parameter space, these criteria are considered in detail for the reduced problem where the upper incident shock wave is moving and the lower one is steady, and a method is provided for extension to the general problem where both the upper and lower ones are unsteady. For the reduced problem, we observe that, in the shock angle plane, shock motion lowers or elevates the von Neumann condition in a global way depending on the direction of shock motion, and this change becomes less important for large shock angle. The effect of shock motion on the detachment condition, though small, displays non-monotonicity. The shock motion changes the transition criteria through altering the effective Mach number and shock angle, and these effects add for small shock angle and mutually cancel for large shock angle, so that shock motion has a less important effect for large shock angle. The role of the effective shock angle is not monotonic on the detachment condition, explaining the observed non-monotonicity for the role of shock motion on the detachment condition. Furthermore, it is found that the detachment condition has a wavefunction form that can be approximated as a hybrid of a sinusoidal function and a linear function of the shock angle.

The influence of the downstream pressure on the shock wave reflection phenomenon in steady flows

1999 ◽  
Vol 386 ◽  
pp. 213-232 ◽  
Author(s):  
G. BEN-DOR ◽  
T. ELPERIN ◽  
H. LI ◽  
E. VASILIEV
Keyword(s):  
Shock Wave ◽  
Wave Reflection ◽  
Numerical Study ◽  
Mach Reflection ◽  
Incident Shock ◽  
Incident Shock Wave ◽  
Steady Flows ◽  
Shock Wave Reflection ◽  
Wave Angle ◽  
Good Agreement

The effect of the downstream pressure (defined here as the wake pressure behind the tail of the reflecting wedge) on shock wave reflection in steady flows is investigated both numerically and analytically. The dependence of the shock wave configurations on the downstream pressure is studied. In addition to the incident-shock-wave-angle-induced hysteresis, which was discovered a few years ago, a new downstream- pressure-induced hysteresis has been found to exist. The numerical study reveals that when the downstream pressure is sufficiently high, an inverse-Mach reflection wave configuration, which has so far been observed only in unsteady flows, can be also established in steady flows. Very good agreement between the analytical predictions and the numerical results is found.

Transmission of elastic disturbances caused by air shock waves in a living body

1961 ◽  
Vol 16 (3) ◽  
pp. 426-430 ◽  
Author(s):  
Carl-Johan Clemedson ◽  
Arne Jönsson
Keyword(s):  
Shock Wave ◽  
Shock Waves ◽  
Lead Zirconate Titanate ◽  
Technical Assistance ◽  
Great Part ◽  
Pressure Rise ◽  
Incident Shock ◽  
Incident Shock Wave ◽  
The Body ◽  
Pressure Curve

Anesthetized rabbits were exposed to air shock waves in a detonation chamber. The pressure wave patterns were recorded by means of a small lead zirconate titanate pressure transducer in the following parts of the body: at and under the skin of the side facing the charge, in the pleural sac and in the lung of that side, in the right and left ventricle of the heart, in the lung and in the pleural sac on the side opposite the charge, under the skin of that side, in the stomach, and in the skull between the bone and the brain. When the incident shock wave is propagated through the body the very steep shock front is converted so that the ascending limb of the pressure peak is much less steep, with a duration up to several hundred microseconds. The longest periods of pressure rise were found in the heart ventricles and stomach. The amplitude of the pressure curve generally diminishes as the wave passes through the body. The changes of the original shock wave are due probably in great part to the inhomogeneous structure of the animal body. Note: (With the Technical Assistance of A.-B. Sundqvist) Submitted on October 24, 1960

Study of asymmetrical shock wave reflection in steady supersonic flow

2019 ◽  
Vol 864 ◽  
pp. 848-875 ◽  
Author(s):  
Jing Lin ◽  
Chen-Yuan Bai ◽  
Zi-Niu Wu
Keyword(s):  
Cfd Simulation ◽  
Mach Reflection ◽  
Vertical Axis ◽  
Lower Surface ◽  
Mach Stem ◽  
Stem Height ◽  
Horizontal Shift ◽  
Reflection Configuration ◽  
The Asymmetry ◽  
Asymmetrical Model

The asymmetrical Mach reflection configuration is studied analytically in this paper, using an asymmetrical model extended from a recent symmetrical model and accounting for the new features related to asymmetry of the two wedges. It is found that the two sliplines do not turn parallel to the incoming flow at the same horizontal location and the sonic throat locates at the position where the difference of slopes of the two sliplines vanishes. This allows us to define a new sonic throat compatibility condition essential to determine the size of the Mach stem. The present model gives the height of the Mach stem, declined angle of the Mach stem from vertical axis, sonic throat location and shape of all shock waves and sliplines. The accuracy of the model is checked by computational fluid dynamics (CFD) simulation. It is found that the Mach stem height is strongly dependent on asymmetry of the wedge angles and almost linearly dependent on the asymmetry of the wedge lower surface lengths. The Mach stem height is shown to be insensitive to the asymmetry of the horizontal positions of the two wedges. The mechanisms for these observations are explained. For instance, it is demonstrated that the Mach reflection configuration remains closely similar when there is horizontal shift of either wedge.

Shock-wave/expansion-wave interactions and the transition between regular and Mach reflection

2007 ◽  
Vol 575 ◽  
pp. 399-424 ◽  
Author(s):  
R. HILLIER
Keyword(s):  
Shock Wave ◽  
Numerical Simulations ◽  
Solid Surface ◽  
Narrow Range ◽  
Mach Reflection ◽  
Incident Shock ◽  
Incident Shock Wave ◽  
Expansion Wave ◽  
Incident Flow ◽  
Wave Interactions

This paper presents numerical simulations for the interaction of an expansion wave with an incident shock wave of the opposite family, the specific aim being to study the resultant reflection of the now-perturbed shock wave from a solid surface. This problem is considered in the context of an incident flow entering a parallel duct, a situation that commonly arises in a range of flow-turning problems including supersonic intake flows. Once the incident shock conditions are such that Mach reflection must occur, it is shown that stabilization of a simple Mach reflection is only possible for a narrow range of Mach numbers and that this depends sensitively on the relative streamwise positioning of the origins of the shock wave and the expansion wave.

scholarly journals The shape of incident shock wave in steady axisymmetric conical Mach reflection

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Yu-xin Ren ◽  
Lianhua Tan ◽  
Zi-niu Wu
Keyword(s):  
Differential Equation ◽  
Shock Wave ◽  
Ordinary Differential Equation ◽  
Mach Reflection ◽  
Internal Flow ◽  
Present Theory ◽  
Incident Shock ◽  
Incident Shock Wave ◽  
Shock Reflection ◽  
Inlet Conditions

Abstract For internal flow with supersonic inflow boundary conditions, a complicated oblique shock reflection may occur. Different from the planar shock reflection problem, where the shape of the incident shock can be a straight line, the shape of the incident shock wave in the inward-facing axisymmetric shock reflection in steady flow is an unknown curve. In this paper, a simple theoretical approach is proposed to determine the shape of this incident shock wave. The present theory is based on the steady Euler equations. When the assumption that the streamlines are straight lines at locations just behind the incident shock is adopted, an ordinary differential equation can be derived, and the shape of the incident shock wave is given by the solution of this ordinary differential equation. The predicted curves of the incident shock wave at several inlet conditions agree very well with the results of the numerical simulations.

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