No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
The effects of thermal conductivity and viscosity of argon on shock waves diffracting over rigid ramps
Abstract
Experiments were done with strong shocks diffracting over
steel ramps immersed in argon. Numerical simulations of the
experiments were done by integrating the Navier–Stokes
equations with a higher-order Godunov finite difference numerical
scheme using isothermal non-slip boundary conditions. Adiabatic, slip
boundary conditions were also studied to simulate cavity-type
diffractions. Some results from an Euler numerical scheme for an
ideal gas are presented for comparison. When the ramp angle θ
is small enough to cause Mach reflection MR, it is found that real
gas effects delay its appearance and that the trajectory of its shock
triple point is initially curved; it eventually becomes straight as
the MR evolves into a self-similar system. The diffraction is a
regular reflection RR in the delayed state, and this is subsequently
swept away by a corner signal overtaking the RR and forcing the
eruption of the Mach shock. The dynamic transition occurs at, or
close to, the ideal gas detachment criterion
θe. The passage of the corner signal is
marked by large oscillations in the thickness of the viscous boundary
layer. With increasing θ, the delay in the onset of MR is
increased as the dynamic process slows. Once self-similarity is
established the von Neumann criterion is supported. While the
evidence for the von Neumann criterion is strong, it is not
conclusive because of the numerical expense. The delayed transition
causes some experimental data for the trajectory to be subject to a
simple parallax error. The adiabatic, slip boundary condition for
self-similar flow also supports the von Neumann criterion while
θ < θe, but the trajectory angle
discontinuously changes to zero at θe, so
that θe is supported by the numerics,
contrary to experiments.
... This discovery of the existence of a Mach reflection occurring in the dual region above the sonic boundary was noted and discussed by Alzamora Previtali et al. [23], for the case of an inviscid flow of a diatomic gas. Furthermore, they also mentioned that a similar result for the case of argon is obtained in the paper by Henderson et al. [24]. ...
... The second objective of the present study is to illustrate that the method used in some previous investigations (e.g., [6,11,13,24]) of extrapolating a set of triple-point angles (χ ) versus the wedge angle (θ w ) for MR patterns, for a given shock Mach number, to the location at which χ → 0 • can yield an incorrect RR-to-MR transition-boundary point, especially for incident shock Mach numbers corresponding to the dual RR and MR region. For the dual region, inappropriate extrapolations can lie near the mechanical-equilibrium boundary of von Neumann, and this sometimes results in the conjecture that the RR-to-MR transition boundary is the mechanical-equilibrium boundary. ...
... Support for the new numerical transition boundary for argon without a boundary layer on the wedge surface by means of CFD studies from other researchers is also scarce. The work of Henderson et al. [24] contains one result of a CFD flow-field simulation that supports the authors' new numerical transition boundary for argon. This result is shown graphically in their Figs. ...
The transition boundary separating the regions of regular and Mach reflections for a planar shock moving in argon and interacting with an inclined wedge in a shock tube is investigated using flow-field simulations produced by high-resolution computational fluid dynamics (CFD). The transition boundary is determined numerically using a modern and reliable CFD algorithm to solve Euler’s inviscid equations of unsteady motion in two spatial dimensions with argon treated as a polytropic gas. This numerically computed transition boundary for inviscid flow, without a combined thermal and viscous boundary layer on the wedge surface, is determined by post-processing many closely stationed flow-field simulations to accurately determine the transition-boundary point when the Mach stem of the Mach-reflection pattern just disappears, and this pattern then transcends into that of regular reflection. The new numerical transition boundary for argon is shown to agree well with von Neumann’s closely spaced sonic and extreme-angle boundaries for weak incident shock Mach numbers from 1.0 to 1.55, but it deviates upward and above the closely spaced sonic and extreme-angle boundaries by almost 2∘ at larger shock Mach numbers from 1.55 to 4.0. This upward trend of the numerical transition boundary for this sequel case with monatomic gases like argon (γ=5/3) and no boundary layer on the wedge surface (inviscid flow) is similar to the previous finding for the case of diatomic gases and air (γ=7/5). An alternative method used to determine one point on the transition boundary between regular and Mach reflections, from a collection of Mach-reflection patterns with a constant-strength shock and different far-field wedge angles, by linear and higher-order polynomial extrapolations to zero for triple-point trajectories versus wedge angle, is compared to the present method of using near-field data that are close to and surround the new transition boundary. Such extrapolation methods are shown to yield a different transition-boundary estimate that corresponds to the mechanical-equilibrium boundary of von Neumann. Finally, the significance of the computed inviscid transition boundary between regular and Mach reflections for monatomic and diatomic gases is explained relative to the case of viscous flow with a combined thermal and viscous boundary layer on the wedge surface.
... In 1994, when the MRS was held for second time in Victoria, Prof. Dewey presented his unreported experimental results. The reflection pattern from a wedge of slightly smaller wedge angle from θ crit is RR near the leading edge and later MR appeared [6]. Audience argued but no one could explain such a strange transition. ...
... Later in 1997, Henderson [6], for the first time, reported numerically the delayed transition in argon and proven that it was created by the displacement effect of the boundary layer developing along the wedge surface. To experimentally confirm the delayed transition, we performed a series of experiments in a 100 mm × 180 mm diaphragm-less shock tube and confirmed that the delayed transition occurred by the viscous effect [7]. ...
... The scheme was VAS2D, two-dimensional vectorizable adoptive solver by Sun (1998) [15] using the Sutherlands formula for viscosity and thermal mvcher@mail.ru 6 K. Takayama et al. ...
This is a summary of shock tube experiments performed at the Tohoku University from 1975 to 2005. Critical transition angles of reflected shock waves over concave and convex walls of radii from 20 to 300 mm were experimentally obtained and compared with the numerical simulation based on the Navier–Stokes equations. The experimental critical transition angles varied with the radii of curved walls and strongly affected by the presence of the boundary layer developing along curved walls. The effects of critical transition on shock wave dynamic, shock wave reflection from roughened wedges, double wedges, and shock focusing from curved walls were presented.
... In the parameter regime (shock strength and incident shock angle) in which either regular or Mach re ection can occur based on shock polars (local conservation about the intersection point of the shock waves), the hysteresis behavior implies that both types of re ection are locally stable. A related phenomena is the delayed transition due to boundary layer e ects in pseudo-steady (shock tube) experiments Henderson et al. (1997). When the observation time is limited, a delayed transition can not be distinguished from a hysteresis e ect. ...
... Thus, for time dependent problems the nonuniqueness of shock interactions is a consequence of the ambiguity of when a wave pattern bifurcates and the nonuniqueness of the possible wave patterns into which another can bifurcate. Both shock tube experiments and numerical experiments have shown that when a shock impacts a wedge leading to a Mach re ection, the path of the triple point can be greatly a ected by a boundary layer due to either viscosity or heat conduction Henderson et al. (1997). Thus, dissipative mechanisms at small scales can lead to local downstream boundary conditions which a ects the bifurcation process. ...
... In the region in which the solution to the shock polars is not unique, hysteresis has been observed both experimentally and numerically for the regular to Mach transition Chpoun et al. (1995), Vuillon et al. (1995), Ivanov et al. (1995a), Ivanov et al. (1995b), Henderson et al. (1997. To set the stage for our discussion of the transition we rst describe the possible wave patterns derived from shock polar analysis. ...
For a convex equation of state, a general theorem on shock waves
is proved: a
sequence of two shocks has a lower entropy than a single shock to the same
final
pressure. We call this the triple-shock
entropy
theorem. This
theorem has important
consequences for shock interactions. In one dimension the interaction of
two shock
waves of the opposite family always results in two outgoing shock waves.
In two
dimensions the intersection of three shocks, such as a Mach configuration,
must have
a contact. Moreover, the state behind the Mach stem has a higher entropy
than
the state behind the reflected shock. For the transition between a regular
and Mach
reflection, this suggests that the von Neumann (mechanical equilibrium)
criterion
would be preferred based on thermodynamic stability, i.e. maximum entropy
subject
to the system constraint that the total specific enthalpy is fixed. However,
to explain
the observed hysteresis of the transition we propose an analogy with phase
transitions
in which locally stable wave patterns (regular or Mach reflection) play
the role of
meta-stable thermodynamic states. The hysteresis effect would occur only
when the
transition threshold exceeds the background fluctuations. The transition
threshold is
affected by flow gradients in the neighbourhood of the shock intersection
point and
the background fluctuations are due to acoustic noise. Consequently, the
occurrence of
hysteresis is sensitive to the experimental design, and only under special
circumstances
is hysteresis observed.
... It was concluded that the surface temperature rise caused by the passage of a shock would be small. On the basis of these studies this was accepted in an important and comprehensive numerical study by Henderson et al. (1997) which included the effects of both thermal conductivity and viscosity in the argon gas tested. This work was primarily concerned with the transition between regular and Mach reflection but made a significant finding with respect to heat flow into the wall. ...
... The situation is complicated in that the velocity and temperature of the gas along the length of the surface are changing as the wave moves along it. High-resolution simulations such as that done in Henderson et al. (1997) would be useful in quantifying the detailed effects on the thermal and viscous boundary layers. ...
In the conventional von Neumann theoretical treatment of two-dimensional shock wave reflection off a surface, it is assumed that the flow is inviscid and that the reflecting surface is perfectly smooth, rigid, non-porous, and adiabatic. These theoretical predictions have been found to be good predictions of reflection over a significant range where regular reflection exists and for a limited range around Mach 2 for strong shocks in the case of Mach reflection. However, experiments on regular reflection have shown that this pattern persists to a small extent beyond what the theory predicts. This effect has been ascribed to the development of a viscous boundary layer behind the point of reflection, and some studies have been done on the effect of surface roughness on reflection topology. The possibility of thermal effects and heat transfer from the shock-heated gas to the wall and on the boundary layer has, on the other hand, been almost totally neglected. To study this, two surfaces of different conductivities have been placed at the same angle, symmetrically in a shock tube, and impacted by a single plane shock wave and the reflection patterns examined. Tests were conducted over a range of Mach numbers between 1.28 and 1.4, and incident shock wave angles between 36 ∘ and 70 ∘ covering both regular and Mach reflection. Both quantitative and qualitative tests show that there is a difference in the angles between the reflected waves and the reflecting surfaces based on the material thermal conductivity. In the quantitative tests the value of this angle was larger for materials with a lower thermal conductivity, and vice versa. A material, such as aluminium, with mid-range thermal conductivity had angles that lay within the limits of the two extreme values for glass and copper. The qualitative images supported these findings, showing asymmetry in reflection topography, with the intersection of the two reflected shock waves lying closer to the material with a higher thermal conductivity.
... A boundary layer which is always present in ramp experiments is not accounted for in the present Euler calculations. Calculations using the Navier-Stokes equations, which simulated a laminar heat-conducting boundary layer (Henderson et al. 1997(Henderson et al. , 2001 showed that both regular and irregular reflections can occur in the dual-solution domain. The simulations also showed that the boundary layer made it possible for the regular reflection to persist into the IR-domain. ...
... The flow passes through a sonic surface behind the refracted wave, and behind the final reflection from the ramp (figures 15 and 16). The sonic surfaces are like corner signals, and are similar to the corner signals found in the Navier-Stokes calculations of Henderson et al. (1997). ...
The jetting effect often appears in the Mach reflection of a shock and in more complicated irregular shock reflections. It is studied numerically using a W-modification of the second order Godunov scheme, to integrate the system of Euler equations. It is shown that there is no correspondence between the shock reflection patterns and the occurrence of jetting. Furthermore there are two kinds of jetting: strong which occurs when there is a stagnation point on the ramp surface where the streamlines divide into an upstream moving jet and a downstream moving slug; and weak which has no stagnation point and may occur at small and large values of the ramp angle. The width of the jet for Mach reflections is determined by the angle of the Mach stem at the shock triple point (the Mach node).
... A boundary layer which is always present in ramp experiments is not accounted for in the present Euler calculations. Calculations using the Navier–Stokes equations, which simulated a laminar heat-conducting boundary layer (Henderson et al. 1997Henderson et al. , 2001) showed that both regular and irregular reflections can occur in the dual-solution domain. The simulations also showed that the boundary layer made it possible for the regular reflection to persist into the IR-domain. ...
... The flow passes through a sonic surface behind the refracted wave, and behind the final reflection from the ramp (figures 15 and 16). The sonic surfaces are like corner signals, and are similar to the corner signals found in the Navier–Stokes calculations of Henderson et al. (1997). In summary, for all the two-node DIR systems in figure 15, there is a Mach node at T and an overtaking node at T 1 , but the T 1 node is degenerate in figures 15(a) and 15(b), because the n 2 -shock is absent. ...
The jetting effect often appears in the Mach reflection of a shock and in more complicated irregular shock reflections. It also occurs in some natural phenomena, and industrially important processes. It is studied numerically using a W-modification of the second-order Godunov scheme, to integrate the system of Euler equations. It is shown that there is no correspondence between the shock reflection patterns and the occurrence of jetting. Furthermore, there are two kinds of jetting: strong which occurs when there is a branch point on the ramp surface where the streamlines divide into an upstream moving jet and a downstream moving slug; and weak which has no branch point and may occur at small and large values of the ramp angle . The width of the jet for Mach and other reflections is determined by the angle of the Mach stem at the triple point (also called the Mach node or three-shock node). Strong jetting is unstable and the primary instability is in the jet itself. The contact discontinuity is also unstable, but its instability is secondary with respect to the jet instability. Two types of irregular reflection are identified in the dual-solution-domain. They are a two-node system comprising a Mach node followed by a four-shock (overtake) node; and another which seems to be intermediate between the previous system and a three-node reflection, which was first hypothesized by Ben-Dor & Glass (1979). An approximate criterion for the jetting no-jetting transition is presented. It is derived by an analysis of the system of Euler equations for a self-similar flow, and has a simple geometrical interpretation.
... One can thus expect that a finite heat transfer, from the hot gas behind the shock wave to the cool shock tube wall, still exists as Re approaches infinity. The results of Henderson et al. (1997) are thus understandable, but the relative contribution of the thermal effects as transition is approached, such as in the extrapolation process of Hornung & Taylor (1982), are still to be clarified. ...
... A second-order polynomial curve fit was found to be the most appropriate, producing an extremely accurate fit with a coefficient of determination of r 2 = 99.93% and M s residuals of the order of 10 −3 . Numerous researchers, such as Henderson et al. (1997), have also found that the triple-point trajectory angle curve can be modelled well by a second-order polynomial and that significant curvature does not occur as transition is approached. The detachment (M s = 1.135) and sonic (M s = 1.128) theoretical transition points as well as the 95% statistical confidence limits (which are a function of the standard deviation and the position of the extrapolated point from the mean of the experimental data and from the experimental points themselves) are also indicated on figure 9. ...
For many years there has been debate regarding why shock wave reflection off a
solid surface has allowed regular reflection to persist beyond the incidence angles
where it becomes theoretically impossible. Theory predicts that at some limiting angle
the reflection point will move away from the wall and Mach reflection will occur.
Previous studies have suggested that the paradox could be due to the presence of the
growing viscous boundary layer immediately behind the point of reflection, and some
numerical studies support this view. This paper takes the approach of establishing
an experimental facility in which the theoretical assumptions regarding the surface of
reflection are met, i.e. that the reflecting surface is perfectly smooth, perfectly rigid,
and adiabatic. This is done by constructing a bifurcated shock tube facility in which
a shock wave is split into two plane waves that are then allowed to reflect off each
other at the trailing edge of wedge. The plane of symmetry between the waves then
acts as the perfect reflection surface.
... Ben-Dor [29] investigated the classical three-shock theory of von Neumann with viscous effects in shock tubes and waves. The effects of thermal conductivity and viscosity on shock waves in argon were studied by Henderson et al. [30]. Simeonides [31] studied viscous effect in the hypersonic flow. ...
This work presents the structure of viscous shock front in a non-ideal gas. The analytical expressions for the particle velocity, temperature, pressure and change-in-entropy within the shock transition region are derived taking into consideration the Landau and Lifshitz equation of state for non-ideal gas. The effects on the structure of shock front due to the variations of the coefficient of viscosity, Mach number, adiabatic exponent and parameter of non-ideality of the gas are investigated. The model developed in the paper is valid only for small values of Mach number M i.e., M < 2:5.
... The hysteresis phenomenon has been proved by experiments when quiet supersonic test facilities became available [6][7][8][9][10][11] and by computations when high-performance computers were available. [12][13][14][15][16][17][18] The research on shock wave interactions is still active in recent times. 1,[19][20][21][22][23][24] The shock wave interactions have significant impacts on the performance and reliability of a high-speed aircraft. ...
In a previous research article [Z. M. Hu et al., Phys. Fluids 21, 011701 (2009)], an overall Mach reflection (oMR) configuration with double inverse Mach reflection patterns was computationally confirmed when a double-wedge geometry interacts with a hypersonic flow. Extended computations are conducted in this paper and compared to analytical solutions based on the classical two- and three-shock theories. A geometric criterion is proposed for the transition between regular reflection and Mach reflection occurring inside the parameter space where a type V interaction of shock wave presents in hypersonic double-wedge flows. An oMR solution is allowed by the geometric criterion, while it is theoretically inadmissible. In the vicinity of symmetric condition, regular to Mach reflection transition can also be triggered prior to the theoretical criterion by disturbance generated by a slight increase in the second wedge angle.
... With increasing wedge angle the RR → MR transition occurs near the detachment condition, while with decreasing wedge angle the MR → RR transition occurs at the von Neumann condition. The hysteresis phenomenon has been proved by experiments (see Chpoun, Passerel, Li & Ben-Dor 1995;Skews 1997Skews , 2000Li, Chpoun & Ben-Dor 1999;Ivanov et al. 2001;Sudani et al. 2002) and computations (see Vuillon, Zeitoun & Ben-Dor 1995;Ivanov et al. 1996Ivanov et al. , 2002Henderson, Crutchfield & Virgona 1997;Ben-Dor, Elperin & Vasilev 1999;Kudryavtsev et al. 2002). Ben-Dor et al. (2002) reviewed the hysteresis process. ...
In this paper, the regular reflection (RR) to Mach reflection (MR) transition of asymmetric shock waves is theoretically studied by employing the classical two- and three-shock theories. Computations are conducted to evaluate the effects of expansion fans, which are inherent flow structures in asymmetric reflection of shock waves, on the RR → MR transition. Comparison shows good agreement among the theoretical, numerical and experimental results. Some discrepancies between experiment and theory reported in previous studies are also explained based on the present theoretical analysis. The advanced RR → MR transition triggered by a transverse wave is also discussed for the interaction of a hypersonic flow and a double-wedge-like geometry.
... They named this new type of reflection the von Neumann reflection, which they introduced as a different regime from Mach reflection (where the slope has a discontinuity at the triple point, as described by three-shock theory). Henderson, Crutchfield & Virgona (1997) provided computational and experimental evidence of the importance of viscosity and heat conductivity in the flow in solving the paradox. The boundary condition on the slipstream separating the two constant states behind the triple point has been questioned by Skews (1972) (see also Ben-Dor 1987;Kobayashi, Adachi & Suzuki 1995), and recently Kobayashi, Adachi & Suzuki (2004) gave experimental evidence for the non-self-similar nature of weak Mach reflection. ...
We study the reflection of acoustic shock waves grazing at a small angle over a rigid surface. Depending on the incidence angle and the Mach number, the reflection patterns are mainly categorized into two types, namely regular reflection and irregular reflection. In the present work, using the nonlinear KZ equation, this reflection problem is investigated for extremely weak shocks as encountered in acoustics. A critical parameter, defined as the ratio of the sine of the incidence angle and the square root of the acoustic Mach number, is introduced in a natural way. For step shocks, we recover the self-similar (pseudo-steady) nature of the reflection, which is well known from von Neumann's work. Four types of reflection as a function of the critical parameter can be categorized. Thus, we describe the continuous but nonlinear and non-monotonic transition from linear reflection (according to the Snell–Descartes laws) to the weak von-Neumann-type reflection observed for almost perfectly grazing incidence. This last regime is a new, one-shock regime, in contrast with the other, already known, two-shock (regular reflection) or three-shock (von Neumann-type reflection) regimes. Hence, the transition also resolves another paradox on acoustic shock waves addressed by von Neumann in his classical paper. However, step shocks are quite unrealistic in acoustics. Therefore, we investigate the generalization of this transition for N-waves or periodic sawtooth waves, which are more appropriate for acoustics. Our results show an unsteady reflection effect necessarily associated with the energy decay of the incident wave. This effect is the counterpart of step-shock propagation over a concave surface. For a given value of the critical parameter, all the patterns categorized for the step shock may successively appear when the shock is propagating along the surface, starting from weak von-Neumann-type reflection, then gradually turning to von Neumann reflection and finally evolving into nonlinear regular reflection. This last one will asymptotically result in linear regular reflection (Snell–Descartes). The transition back to regular reflection is one of two types, depending on whether a secondary reflected shock is observed. The latter case, here described for the first time, appears to be related to the non-constant state behind the incident shock, which prevents secondary reflection.
... There are two candidate causes: viscosity and surface roughness. According to Henderson et al.'s numerical experiment [3], the condition on the solid boundary plays a key role in the phenomenon. The authors' experiment in Karlsruhe found different wave angles from those obtained in Saitama [4]. ...
This paper presents several properties associated with the two-variable extension of the Chebyshev matrix polynomials of the second kind. In particular, we establish a three-term recurrence relation for these two-variable matrix polynomials and show that these two-variable matrix polynomials satisfy some second-order matrix differential equations. We derive their hypergeometric matrix representation and an expansion formula which links these generalized Chebyshev matrix polynomials with the Hermite matrix polynomials and the Laguerre matrix polynomials. We also drive their Volterra integral equation.
In this paper, homotopy analysis method (HAM) is used to obtain the analytic solution for fragmentation population balance equation. Different sample problems are solved using HAM and their series solution is obtained. A detailed analysis of the series solution and the region of convergence of the solution is also studied. It is observed that the convergence region of the series solution can be adjusted with the help of certain parameters involved in HAM.KeywordsHomotopy analysis methodPopulation balance equationAnalytic approximationsFragmentationConvergence
This paper investigates the effect of viscosity on the propagation of spherical shock waves in a dusty gas with a radiation heat flux and a density that grows exponentially. It is assumed that the dusty gas is a blend of fine solid particles and ideal gas. In a perfect gas, solid particles are uniformly distributed. To obtain several significant shock propagation properties, the solid particles are treated as a pseudo-fluid, and the mixture’s heat conduction is neglected. The flow’s equilibrium conditions are expected to be maintained in an optically thick gray gas model, and radiation is assumed to be of the diffusion type. The effects of modifying the viscosity parameter and time are explored, and non-similar solutions are found. The formal solution is determined by assuming that the shock wave’s velocity is variable and its total energy is not constant.
When a shock wave is reflected from a steep wedge, the reflected patter of the incident shock wave, or in short IS, forms a V shaped wave pattern. This pattern is similar to a sound wave reflection from a plane wall and hence is named as a “regular reflection”, and in short RR. A head-on-collision of a shock wave with a plane wall is an extreme case of the RR.
Viscous effects on centreline shock reflection in an axisymmetric flow are studied numerically using Navier-Stokes and direct simulation Monte Carlo solvers. Computations at low Reynolds numbers have resulted in a configuration consisting of two shock waves, in contrast to the inviscid theory. On the other hand, computations at high Reynolds numbers have yielded a three-shock configuration in qualitative agreement with the inviscid theory prediction. This behaviour is explained by the presence of the so-called non-Rankine-Hugoniot zone, which accounts for the deviation of the shock structure from the inviscid paradigm. At Reynolds numbers on the verge of the transition from a two-shock to three-shock configuration, extremely high pressure that would be unattainable with the classical Rankine-Hugoniot relation for any shock configuration may occur. An analogy to the Guderley singularity in cylindrical shock implosion has been deduced for the shock behaviour from a mathematical viewpoint.
A series of experiments on the transition of a shock reflection was conducted using a conventional shock tube. Attention has been focused on the transition phenomenon during the propagation of an incident shock wave whose Mach number ranges from 1.10 to 1.40. The present detailed investigation has helped clarify the transition phenomenon from regular to Mach reflection during the shock propagation. To date, this phenomenon has been observed only in low-pressure experiments, but it was observed even under atmospheric pressure in this experiment. The "take-off" distance, defined as the distance from the wedge tip to the location where the transition takes place, increases with the reflecting wedge angle. Such non-self-similarity should be logically ascribed to the transport properties of the medium. It is also possibly the cause of the well-known von Neumann paradox.
From 1961 to 1972, they were the high light of the Apollo project. Researchers, who were the same age as the author and interested in high-speed gas dynamics, were aware of the importance of the shock wave research. Shock tubes reproduced atmospheric reentry conditions of the Apollo commander modules and shock tunnels simulated the reentry flights. In the late 1960s, Japanese researchers became interested in space technology and science, which initiated the establishment of the Japan Society for Shock Tube Research. Japan was a newcomer to the international shock tube research community.
A new parallel, fully implicit, anisotropic block-based adaptive mesh refinement (AMR) finite-volume scheme is proposed, described and demonstrated for the prediction of laminar, compressible, viscous flows associated with unsteady oblique shock reflection processes. The proposed finite-volume method provides numerical solutions to the Navier-Stokes equations governing the flow of polytropic gases in an accurate and efficient manner on two-dimensional, body-fitted, multi-block meshes consisting of quadrilateral computational cells. The combination of the anisotropic AMR and parallel implicit time-marching techniques adopted is shown to readily facilitate the simulation of challenging and complex shock interaction problems, as represented by the time-accurate predictions of unsteady oblique shock reflection configurations with fully resolved internal shock structures.
Up to the present, the von Neumann paradox has been mentioned only for weak shock waves, and the classical theories have been considered to be valid for strong shock waves. In the present paper, we discovered experimentally that the paradox also occurs in strong shock waves for the same reason as in weak shock waves. We also explain why it has not been observed so far and demonstrate that the essence of the problem lies in the transport properties of the fluid.
We present an unsplit method for the time-dependent compressible Navier-Stokes equations in two and three dimensions. We use a conservative, second-order Godunov algorithm. We use a Cartesian grid, embedded boundary method to resolve complex boundaries. We solve for viscous and conductive terms with a second-order semiimplicit algorithm. We demonstrate second-order accuracy in solutions of smooth problems in smooth geometries and demonstrate robust behavior for strongly discontinuous initial conditions in complex geometries.
For many years there has been debate regarding why shock wave reflection off a solid surface has allowed regular reflection to persist beyond the incidence angles where it becomes theoretically impossible. Theory predicts that at some limiting angle the reflection point will move away from the wall and Mach reflection will occur. Previous studies have suggested that the paradox could be due to the presence of the growing viscous boundary layer immediately behind the point of reflection, and some numerical studies support this view. This paper takes the approach of establishing an experimental facility in which the theoretical assumptions regarding the surface of reflection are met, i.e. that the reflecting surface is perfectly smooth, perfectly rigid, and adiabatic. This is done by constructing a bifurcated shock tube facility in which a shock wave is split into two plane waves that are then allowed to reflect off each other at the trailing edge of wedge. The plane of symmetry between the waves then acts as the perfect reflection surface.
Through a careful set of visualization experiments, and the use of multivariate analysis to take account of the uncertainty in shock Mach number, triple-point trajectory angle, and slightly different shock wave arrival times at the trailing edge, the current work shows that the transition from one type of reflection to the other does indeed occur at the theoretical value. Conventional tests of reflection off a solid wall show significantly different transition results. Furthermore, it is also shown that the thermal boundary layer plays an important role in this regard. It is thus confirmed that viscous and thermal effects are the reason for the paradox. Reasons are also suggested for the counter-intuitive behaviour of the reflected shock wave angle.
This paper experimentally investigates the validity of self-similarity
of strong shock reflection phenomena in a shock tube. The models used
for the shock-tube experiment are ordinary wedges with various
reflecting wedge angles. The triple-point trajectory is approximately a
straight line through the wedge apex for each reflecting wedge. However,
a detailed measurement of the angle made by the incident and reflected
shocks shows that the wave angle varies as the incident shock proceeds.
This means that the shock reflection configuration deviates from
self-similarity. The most remarkable phenomenon is the dynamic
transition from regular to Mach reflection during shock propagation,
where the validity of self-similarity breaks down. The flow-field behind
the Mach stem is subsonic with respect to the triple point, so the
condition on the solid boundary can catch up with the triple point and
affect the flow around it. We also explain why the discrepancy between
theory and experiment has gone unnoticed for strong shock waves and
demonstrate that it is due to the transport properties of the fluid,
such as the viscosity.
In this paper, we consider the phenomenon of unsteady Mach reflection generated by a plane shock wave advancing over a straight wedge surface, with particular attention to the deviation of the flow field from the self-similar nature. We examine the observed change in angle between incident and reflected shocks, which is in contrast to the fact that the angle should remain constant with time in a self-similar flow. The effect of the boundary layer behind the advancing shock wave over the surface of the wedge is considered to cause this, and boundary layer theory is utilized to estimate the thickness of the layer. It is found that the thickness increases as
to the time t compared with t by the overall expansion in the self-similar flow. Assuming that the thicker boundary layer is effectively equivalent to a change in wedge angle, the effect of the boundary layer on the flow field should be less in later stages with larger t values in accordance with the observation above.
Forty years continuous efforts for shock wave research and its recent applications to interdisciplinary topics conducted at the Interdisciplinary Shock Wave Research Center, Tohoku University Sendai Japan are summarized. Development of optical flow visualization for shock wave research, use of micro-explosives for shock generation in air and water, and design and operation of reliable shock tubes and related facilities are described. Lastly applications of shock wave research to medicine and geophysics are briefly demonstrated.
Shock wave interaction with a rotating circular cylinder is numerically and experimentally investigated. It is shown that the transition from regular reflection to Mach reflection on the lower surface of a rotating cylinder where cylinder motion is the same as the incident shock motion, is more quickly promoted than that on the upper surface. Numerical calculations solving the Navier-Stokes equations using extremely fine grids further reveal that the reflected shock transition from RR yields MR is either advanced or postponed depending on whether or not a surface motion favors the incident shock wave.
We report on calculations and experiments with strong shocks diffracting over rigid
ramps in argon. The numerical results were obtained by integrating the conservation
equations that included the Navier–Stokes equations. The results predict that if the
ramp angle θ is less than the angle θe that corresponds to the detachment of a
shock, θ < θe, then the onset of Mach reflection (MR) will be delayed by the initial
appearance of a precursor regular reflection (PRR). The PRR is subsequently swept
away by an overtaking corner signal (cs) that forces the eruption of the MR which
then rapidly evolves into a self-similar state. An objective was to make an experimental
test of the predictions. These were confirmed by twice photographing the diffracting
shock as it travelled along the ramp. We could get a PRR with the first exposure
and an MR with the second. According to the von Neumann perfect gas theory,
a PRR does not exist when θ < θe. A viscous length scale
xint is a measure of the
position on the ramp where the dynamic transition PRR [rightward arrow] MR takes place. It is
significantly larger in the experiments than in the calculations. This is attributed to
the fact that fluctuations from turbulence and surface roughness were not modelled
in the calculations. It was found that xint
[rightward arrow] [infty infinity] as θ [rightward arrow] θe. Experiments were done to find out how
xint depended on the initial shock tube pressure p0. The dependence was
strong but could be greatly reduced by forming a Reynolds number based on xint.
Finally by definition, regular reflection (RR) never interacts with a boundary layer,
while PRR always interacts; so they are different phenomena.
Shock wave reflection over a rotating circular cylinder is numerically and experimentally investigated. It is shown that the transition from the regular reflection to the Mach reflection is promoted on the cylinder surface which rotates in the same direction of the incident shock motion, whereas it is retarded on the surface that rotates to the reverse direction. Numerical calculations solving the Navier-Stokes equations using extremely fine grids also reveal that the reflected shock transition from RR
Þ\Rightarrow
MR is either advanced or retarded depending on whether or not the surface motion favors the incident shock wave. The interpretation of viscous effects on the reflected shock transition is given by the dimensional analysis and from the viewpoint of signal propagation.
This paper is the part 2 of our previous thin film heat transfer measurements. In the first report we measured time variations
of heat flux over a cylinder placed in a shock tube flow and compared experimental results with CFD results, Saito etal.
(Shock Waves 14:327–333, 2004). We report a result of heat transfer measurements over an 86° apex angle cone surface impinged
by a Ms = 2.38 shock wave in air with distributed thin film transfer gauges along cone surface and its comparison with results of
numerical simulations. We performed double exposure holographic interferometric observation, and also from the heat transfer
measurement and numerical simulation, confirmed the presence of delayed transition from regular to Mach reflection over the
cone. The numerical estimation of delayed transition distance from the apex agreed very well with experimental one.
According to standard textbooks on compressible fluid dynamics, a shock wave is formed by an accumulation of compression waves. However, the process by which an accumulated compression wave grows into a shock wave has never been visualized. In the present paper, the authors tried to visualize this process using a model wedge with multiple steps. This model is useful for generating a series of compression waves and can simulate a compression process that occurs in a shock tube. By estimating the triple-point trajectory angle, we demonstrated visually that an accumulated compression wave grows into a shock wave. Further reflection experiments over a rough-surface wedge confirmed the tendency for the triple point trajectory angle to reach the asymptotic value
s in the end.
It is well accepted that the persistence of regular reflection of a shock wave off a wedge beyond the ideal theoretical prediction
is due to viscous and thermal boundary layers induced behind the reflection point. Experiments have been done by reflecting
two shock waves of equal strength off each other so that the plane of symmetry between them becomes an ideal inviscid and
adiabatic reflection plane thereby experimentally mimicking the assumptions of the theory. There is one definitive experiment
done at a wall angle of 40° using a bifurcated shock tube that indicates that the actual transition angle is the theoretical
detachment condition. This paper extends these results to two cases near limiting conditions; one at a very low incidence
shock Mach number and one at a wall angle very close to the theoretical transition limit. The first confirms the reasons for
the von Neumann Paradox but cannot discriminate between sonic and detachment conditions, but is within about 0.5% of them,
and the second shows transition much closer to the sonic than the detachment condition but with both within the experimental
error bounds. In both cases, the results are notably different from transition conditions off a wedge and confirm the effects
of transport properties being the cause of persistence of regular reflection.
Keywordsvon Neumann paradox–Reflection transition–RR persistence
This paper deals with the von Neumann paradox, an as yet unsolved major problem in supersonic gas dynamics. We conduct a series of experiments using a conventional shock tube and focus attention on the flow-field around the triple point at various locations of the incident shock wave. By measuring the position of the triple point and the angle made by the incident and reflected shocks for propagating shock waves, we prove that the flow-field around the triple point is not self-similar. The von Neumann paradox is at least partly ascribed to non-self-similarity because classical theory assumes pseudo-steadiness that results from self-similarity for shock reflection over a wedge in a shock tube. The non-self-similarity revealed here clarifies the classical experiments by Smith (Photographic investigation of the reflection of plane Shocks in air. OSRD Report 6271, Washington, USA, 1945) and Bleakney and Taub (Rev. Mod. Phys. 21 (1949) 584). Specifically, since they implicitly assumed self-similarity, they only measured the wave angles at some particular location for each reflecting wedge angle, and their relations between angles of incidence and reflection were recovered by the present experiment.
In the Mach reflection of weak shock waves, there has been a wide difference between theory and experiment for the shock wave angles at the triple point. Using the electric tank analogy in the hodograph to obtain the shock patterns it is found that the shock waves are so strongly curved near the intersection that the triple point angles cannot be observed in the laboratory. However, it is shown that the visible shock patterns are apparently inconsistent with the hodograph boundaries specified by the existing triple point theory. The effect of shock wave thickness on the boundary conditions at the triple point is then examined. It is shown that at the intersection there must be a non‐Rankine‐Hugoniot shock wave zone separating the three R‐H shock waves. It follows that for weak shock waves where the shock curvatures → ∞ at the triple point, the simple boundary conditions of equal pressure and direction are invalid for a real fluid. A semiquantitative description is then given of a real fluid model which is consistent with the experimental data and with the Navier‐Stokes equations. The examination of this model reveals the formidable difficulties in the way of obtaining a detailed mathematical description of a triple shock wave intersection in a real fluid. The question of the solution for the limiting case as μ → 0 is also discussed.
A local adaptive mesh refinement algorithm for solving hyperbolic systems of conservation laws in three space dimensions is described. The method is based on the use of local grid patches superimposed on a coarse grid to achieve sufficient resolution in the solution. A numerical example computing the interaction of a shock with a dense cloud in a supersonic inviscid regime is presented. Detailed timings are given to illustrate the performance of the method in three dimensions.
The laminar wall boundary layer behind a strong shock moving with nonuniform velocity into a stationary fluid has been investigated. In particular, two‐dimensional and axisymmetric boundary layers behind plane, cylindrical, and spherical shocks which move according to the power law xs = Ctm have been considered. The wall boundary layers associated with blast waves are special cases of the class of problems treated herein. It was assumed that the fluid is a perfect gas, that viscosity is proportional to temperature, and that the wall surface temperature is small relative to the temperature in the free stream. The resulting boundary‐layer equations were simplified by expanding the dependent variables in powers of a nondimensional distance measured from the shock. The zero‐order flow corresponds, at each instant, to a two‐dimensional boundary layer behind a shock wave moving with uniform velocity. Numerical solutions of the first‐order equations have been found for several cases of interest, and the results for wall shear and heat transfer have been tabulated and discussed.
Two-dimensional Riemann problems occurring at the intersection points of discontinuous waves in a compressible, inviscid, polytropic gas are studied from both numerical and theoretical points of view using the front tracking method. The results are compared to experimental ones for two specific test problems. An example of how the motion of a two-dimensional coherent wave is determined numerically is given, and it is shown that in two-dimensional compressible gas dynamics there are only a small number of such two-dimensional coherent waves. Outstanding questions related to Riemann problems are also discussed.
It is shown that the equations of an unsteady compressible flow in the (x, y)-plane, which is expressible in terms of the two variables x/t and y/t only, can be reduced to those of a steady compressible flow with a non-conservative field of external forces and a field of sinks. The steady-flow problems of this type, which correspond to the diffraction or reflexion of a plane shock travelling parallel to a rigid wall and reaching a corner, are discussed qualitatively. It is shown that, under certain conditions, there are regions in the corresponding steady flows which are entirely supersonic and for which a simple solution can be given without determining the whole field of flow. No complete solution for the whole field of flow has yet been given. In the diffraction, at a convex corner, of certain strong shocks, it is shown that there can be an area of Prandtl-Meyer flow, uniformly increasing with time, and that the upper limit to which it can extend is calculable as a characteristic curve in the corresponding steady flow. In the case of regular reflexion beyond a concave corner, or reflexion at a concave corner which gives rise to a reflected shock passing through the corner, it is shown that there can be areas of uniform flow, uniformly increasing with time, and that the upper limits to which they can extend are arcs of circles, which appear as sonic curves in the corresponding steady flows.
Results of theoretical and experimental investigations on shock reflection at a fixed wall are reported in this paper. In the region of regular reflection, agreement of theory and experiment is found to be good except for the fact that experimental points extend a litter bit beyond the theoretical limit. A theoretical investigation of the flow characteristics is made on Mach reflection cases with the conclusion that a singularity would appear at the triple point, and hence, the reflected shock angle would not coincide with the theoretical one when the flow is subsonic behind the reflected shock. This theoretical prediction is fully confirmed by the results of shock angle measurement in schlieren pictures obtained by the experiment. In case of a strong incident shock, angle of the reflected shock decreases discontinuously at the transition from regular to Mach reflection as expected from the theory. In case of a weak incident shock, on the other hand, its change is continuous against the ordinary three-shock theory. It is found in the experiment that the boundary between these two cases takes place approximately at xi{=}0.42, where xi is the pressure ratio across the incident shock. This is in good agreement with the theoretically predicted value, xi{=}0.433, above which the Mach reflection with a singularity at the triple point would appear. It corresponds to the condition of the stationary Mach reflection. Comparisons of theory and experiment are also made in this paper on the condition at which the flow behind the incident shock is just sonic relative to the triple point and on the angle of the Mach stem. Throughout the present study experiment was made by use of a shock tube.
Our 1975 paper reported the results of experiments on shock reflexion in a wind tunnel and a shock tube; further results are presented here. For strong shocks it is shown that transition to Mach reflexion takes place continuously at the shock wave incidence angle ω0 corresponding to the normal shock point ω0 = ωN, unless the downstream boundaries form a throat. In this event transition can be promoted anywhere within the range ω0 [less-than-or-eq, slant] ωN, and it is even possible to suppress regular reflexion altogether! However when ω0 < ωN the transition is discontinuous and accompanied by hysteresis. Again for strong shocks evidence is presented which suggests that the famous persistence of regular reflexion beyond the ωN point ω0 > ωN is spurious. For weak shocks the transition condition is not known but it is found that even for regular reflexion a marked discrepancy between theory and experiment develops as the shocks become progressively weaker. Also when weak shocks diffract over single concave corners there is a somewhat surprising discontinuity in the regular reflexion range. It seems that none of these phenomena can be adequately explained by real gas effects such as viscosity and variation of specific heats.
Detailed experimental data are presented on the transition between regular and Mach reflexion. Data have been obtained for steady, pseudo-steady and un-steady flows, and include a study of the continuous and discontinuous transitions predicted by previous researchers. It is found that the criterion often used to calculate the transition condition is wrong in every case that we have investigated. In its place we propose an alternative criterion which has the property that the system remains always in mechanical equilibrium during transition.
It is shown experimentally that, in steady flow, transition to Mach reflection occurs at the von Neumann condition in the strong shock range (Mach numbers from 2.8 to 5). This criterion applies with both increasing and decreasing shock angle, so that the hysteresis effect predicted by Hornung, Oertel & Sandeman (1979) could not be observed. However, evidence of the effect is shown to be displayed in an unsteady experiment of Henderson & Lozzi (1979).
We present results from our experiments with the irregular reflection of shock waves in argon. We compare the data with the results we obtained numerically; the assumptions for the computational code were that we had unsteady, two-dimensional, compressible, inviscid, flow of a perfect gas. When precautions were taken to reduce the effects of the gas viscosity on the experimental data, we obtained very good agreement between the numerical and the experimental results for the ramp Mach number and the trajectory path triple-point angle, but there were discrepancies with the wave-angle data. The discrepancies were ascribed to the sensitivity of the data to both viscosity and to a singularity. We show that there are actually two weak irregular wave reflections, namely a classic Mach reflection (MR) and a new type, that we call a von Neumann reflection (NR). The structure of the NR is discussed in some detail, and so are the transition criteria for the various wave systems.
This paper examines, as completely as possible, the wave systems that may occur when a plane shock wave i diffracts over a concave corner. For a perfect gas, a particular wave system is specified by three parameters ( gamma , xi //i, omega //0), which are respectively the ratio of the specific heats gamma , the inverse strength xi //i of i, and its angle of incidence omega //0. The topology of the ( xi //i omega //0) plane is studied for a given gamma (gas), and regions or domains corresponding to each wave system are delineated. Exact expressions are given for the boundaries of many of these regions. Although it is plausible that the boundaries represent the transition conditions between the wave systems, a detailed analysis shows that the situation is sometimes more complicated than might be expected. A new transition criterion, which may be valid for diffraction over flexible surfaces, is presented. For very weak shocks ( xi //i greater than 0. 8328 in air), it is found that Mach reflexion cannot exist, and in its place a continuous wave system is indicated.
The signal speed, namely the local sound speed plus the flow velocity, behind the reflected shocks produced by the interaction of weak shock waves (M
i
< 1.4)="" with="" rigid="" inclined="" surfaces="" has="" been="" measured="" for="" several="" shock="" strengths="" close="" to="" the="" point="" of="" transition="" from="" regular="" to="" mach="" reflection.="" the="" signal="" speed="" was="" measured="" using="" piezo-electric="" transducers,="" and="" with="" a="" multiple="" schlieren="" system="" to="" photograph="" acoustic="" signals="" created="" by="" a="" spark="" discharge="" behind="" a="" small="" aperture="" in="" the="" reflecting="" surfaces.="" both="" methods="" yielded="" results="" with="" equal="" values="" within="" experimental="" error.="" the="" theoretical="" signal="" speeds="" behind="" regularly="" reflected="" shocks="" were="" calculated="" using="" a="" non-stationary="" model,="" and="" these="" agreed="" with="" the="" measured="" results="" at="" large="" angles="" of="" incidence.="" as="" the="" angle="" of="" incidence="" was="" reduced,="" for="" the="" same="" incident="" shock="" mach="" number,="" so="" as="" to="" approach="" the="" point="" of="" transition="" from="" regular="" to="" mach="" reflection,="" the="" measured="" values="" of="" the="" signal="" speed="" deviated="" significantly="" from="" the="" theoretical="" predictions.="" it="" was="" found,="" within="" experimental="" uncertainty,="" that="" transition="" from="" regular="" to="" mach="" reflection="" occurred="" at="" the="" experimentally="" observed="" sonic="" point,="" namely,="" when="" the="" signal="" speed="" was="" equal="" to="" the="" speed="" of="" the="" reflection="" point="" along="" the="" reflecting="" surface.="" this="" sonic="" condition="" did="" not="" coincide="" with="" the="" theoretical="">
An adaptive method based on the idea of multiple component grids for the solution of hyperbolic partial differential equations using finite difference techniques is presented. Based upon Richardson-type estimates of the truncation error, refined grids are created or existing ones removed to attain a given accuracy for a minimum amount of work. The approach is recursive in that fine grids can contain even finer grids. The grids with finer mesh width in space also have a smaller mesh width in time, making this a mesh refinement algorithm in time and space. We present the algorithm, error estimation procedure, and the data structures, and conclude with numerical examples in one and two space dimensions.
We present a class of second-order conservative finite difference algorithms for solving numerically time-dependent problems for hyperbolic conservation laws in several space variables. These methods are upwind and multidimensional, in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge, and that all fluxes are evaluated and differenced at the same time; in particular, operator splitting is not used. Correct behavior at discontinuities is obtained by the use of solutions to the Riemann problem, and by limiting some of the second-order terms. Numerical results are presented, which show that the methods described here yield the same high resolution as the corresponding operator split methods.
The aim of this work is the development of an automatic, adaptive mesh refinement strategy for solving hyperbolic conservation laws in two dimensions. There are two main difficulties in doing this. The first problem is due to the presence of discontinuities in the solution and the effect on them of discontinuities in the mesh. The second problem is how to organize the algorithm to minimize memory and CPU overhead. This is an important consideration and will continue to be important as more sophisticated algorithms that use data structures other than arrays are developed for use on vector and parallel computers.