Non-steady Wave Research Articles

Nonsteady Shock Wave Attenuation Due to Leakage

Leakage flow between the rotor and the stator can cause serious performance degradation of wave rotors which utilize nonsteady shock waves to directly transfer energy from burned gases to precompressed air. To solve this problem, primary flow features relevant to leakage are extracted and it was found that the leakage-attributed performance degradation could be abstracted to a special initial-boundary value problem of one-dimensional Euler equations. Then, a general loss assessment method is proposed to solve the problem of nonsteady flow loss prediction. Using the above method, a reasonable physical hypothesis of the initial-boundary value problem depicting the nonsteady leakage flow process is proposed and further, a closed-form leakage loss analytical model combined with an empirical correction method for the discharge coefficient is established. Finally, with the experimentally verified CFD method, comprehensive numerical verification is conducted for the loss prediction model; it is proved that the physical hypothesis of the proposed model in this paper is reasonable and the model is capable of predicting nonsteady shock wave attenuation due to leakage exactly within the range of parameter variations of wave rotors.

Wave adjustment: general concept and examples

We formulate a general theory of wave adjustment applicable to any physical system (not necessarily a hydrodynamic one), which, being linearized, possesses linear invariants and a complete system of waves harmonically depending on the time$t$. The invariants are determined by the initial conditions and are zero for the waves, which, therefore, do not transport and affect the invariants. The evolution of such a system can be represented naturally as the sum of a stationary component with non-zero invariants and a non-steady wave part with zero invariants. If the linear system is disturbed by a small perturbation (linear or nonlinear), then the state vector of the system is split into slow balanced and fast wave components. Various scenarios of the wave adjustment are demonstrated with fairly simple hydrodynamic models. The simplest scenario, called ‘fast radiation’, takes place when the waves rapidly (their group speed$c_{gr}$greatly exceeds the slow flow velocity$U$) radiate away from the initial perturbation and do not interact effectively with the slow component. As a result, at large times, after the waves propagate away, the residual flow is slow and described by a balanced model. The scenario is exemplified by the three-dimensional non-rotating barotropic flow with a free surface. A more complicated scenario, called ‘nonlinear trapping’, occurs if oscillations with small group speed$c_{gr}\leqslant U$are present in the wave spectrum. In this case, after nonlinear wave adjustment, the state vector is a superposition of the slow balanced component and oscillations with small$c_{gr}$trapped by this component. An example of this situation is the geostrophic adjustment of a three-dimensional rotating barotropic layer with a free surface. In the third scenario, called ‘incomplete splitting’, the wave adjustment is accompanied by non-stationary boundary layers arising near rigid and internal boundaries at large times. The thickness of such a layer tends to zero and cross-gradients of physical parameters in the layer tend to infinity as$t\rightarrow \infty$. The layer is an infinite number of wave modes whose group speed tends to zero as the mode number tends to infinity. In such a system, complete splitting of motion into fast and slow components is impossible even in the linear approximation. The scenario is illustrated by an example of stratified non-rotating flow between two rigid lids. The above scenarios describe, at least, the majority of known cases of wave adjustment.

Hugoniot experiments with unsteady waves

Recent development of transparent shock wave standard materials, such as quartz, enables continuous tracking of shock waves using optical velocimetry, providing information on shock wave steadiness and pressure perturbations in the target. From a first order perturbation analysis, we develop a set of analytical formulas that connect the pressure perturbations at the drive surface to the shock velocity perturbations observed in measurements. With targets that incorporate a calibrated transparent witness material, such as quartz, and with the analytical formulas describing the perturbation response, it is possible to determine the sound speed and Grüneisen coefficient of an unknown sample by using evolution of the non-steady perturbations as a probe. These formulas can also be used to improve the accuracy of traditional shock wave impedance match Hugoniot experiments of opaque samples driven with non-steady waves.

The α→ϵ phase transition in iron at strain rates up to ∼109 s−1

We have used a table-top scale laser to dynamically compress iron at strain rates in excess of 109 s−1. Using an embedded ultrafast interferometer, we have measured corresponding free surface histories with a time resolution of approximately 10 ps. We have analyzed the surface histories using a method that accounts for nonsteady wave propagation and time-dependent material behavior. We show that at these strain rates, the α→ϵ polymorphic transition begins within 100 ps after an initial very large (∼10 GPa) and mostly elastic compression and appears largely complete within a similar time thereafter. The corresponding deviatoric stress before the transition begins can exceed 3 GPa, while the transition stress itself is up to 25 GPa, nearly twice the value measured at low strain rates. We use these results to propose a systematic variation with loading time of the normal-stress/relative-volume curve followed by iron during rapid compression.

Effect of compositional variation on the shock wave response of bulk amorphous alloys

The objective of this work was to determine how variations in composition influence the shock wave response of bulk amorphous alloys (BAAs). Toward this end, (Hf,Zr)-based bulk amorphous alloy specimens (nominal composition of (Hf0.5Zr0.5)56.7Cu15.3Ni12.5Nb5.0Al10.0Y0.5) were subjected to peak stresses of 4-16 GPa in plate impact experiments and their response was compared to a previously studied Zr-based BAA (nominal composition of Zr56.7Cu15.3Ni12.5Nb5.0Al10.0Y0.5). The (Hf,Zr)-based BAA displayed a Hugoniot elastic limit (HEL) of ∼7.4 GPa corresponding to an elastic strain of 4.3%. (Hf,Zr)-based BAAs shock compressed above the HEL exhibited distinct two wave structures, small elastic precursor relaxation, non-steady plastic waves, and strength loss. All of these features are qualitatively similar to previous observations in various Zr-based BAAs. One dimensional wave propagation simulations incorporating a strain-softening strength model, developed previously for a Zr-based BAA, showed excellent agreement between measured and simulated particle velocity histories for the (Hf,Zr)-based BAA. The only significant differences in the shock wave responses of the (Hf,Zr)-based alloy and the Zr-based alloy are elastic shock velocity and plastic shock velocity differences which are due to the different ambient densities and different elastic moduli for the two alloy compositions. These findings demonstrate that, apart from differences related to ambient density and elastic stiffness, the substitution of Hf for 50% of the Zr did not significantly alter the shock compression response of the (Hf,Zr)-based BAA as compared to the previously examined Zr-based BAA. Based on the results of this study and other relevant data in the literature, it is expected that monolithic BAAs displaying brittle quasi-static compression behavior will likely display shock compression response that is comparable to Zr-based and (Hf,Zr)-based BAAs.

Quantifying spatial and temporal discharge dynamics of an event in a first order stream, using distributed temperature sensing

Abstract. Understanding the spatial distribution of discharge can be important for water quality and quantity modeling. Non-steady flood waves can, particularly as a result of short high intensity summer rainstorms, influence small headwater streams significantly. The aim of this paper is to quantify the spatial and temporal dynamics of stream flow in a headwater stream during a summer rainstorm. These dynamics include gains and losses of stream water, the effect of bypasses that become active and hyporheic exchange fluxes that may vary over time as a function of discharge. We use an advection-dispersion model coupled with an energy balance model to simulate in-stream water temperature, which we compare with high resolution temperature observations obtained with Distributed Temperature Sensing. This model was used as a learning tool to stepwise unravel the complex puzzle of in-stream processes subject to varying discharge. Hypotheses were tested and rejected, which led to more insight in the spatial and temporal dynamics in discharge and hyporheic exchange processes. We showed that, for the studied stream infiltration losses increase during a small rain event, while gains of water remained constant over time. We conclude that, eventually, part of the stream water bypassed the main channel during peak discharge. It also seems that hyporheic exchange varies with varying discharge in the first 250 m of the stream; while further downstream it remains constant. Because we relied on solar radiation as the main energy input, we were only able to apply this method during a small summer storm and low flow conditions. However, when additional (artificial) energy is available, the presented method is also applicable in larger streams, during higher flow conditions or longer storms.

Thermal and mechanical analysis of material response to non-steady ramp and steady shock wave loading

Ramp wave experiments on the Sandia Z accelerator provide a new approach to study the rapid compression response of materials at pressures, temperatures and stress or strain rates not attainable in conventional shock experiments. Due to its shockless nature, the ramp wave experiment is often termed as an isentropic (or quasi-isentropic) compression experiment (ICE). However, in reality there is always some entropy produced when materials are subjected to large amplitude compression even under shockless loading. The entropy production mechanisms that cause deformation to deviate from the isentropic process can be attributed to mechanical and thermal dissipations. The former is due to inelasticity associated with various deformation mechanisms and the rate effect that is inherent in all the deformation processes and the latter is due to irreversible heat conduction. The main purpose of the current study is to gain insights into the effects of ramp and shock loading on the entropy production and thermomechanical responses of materials. Another purpose is to investigate the role of heat conduction in the material response to both the non-steady ramp wave and steady shock. Numerical simulations are used to address the aforementioned research objectives. The thermomechanical response associated with a steady shock wave is investigated first by solving a set of nonlinear ordinary differential equations. Using the steady wave solutions as the reference, the material responses under non-steady ramp waves are then studied with numerical wave propagation simulation. It is demonstrated that the material response to ramp and shock loading is essentially a manifestation of the interaction between the time scale associated with the loading and the intrinsic time scales associated with mechanical deformation and heat transfer. At lower loading rates as encountered in ramp loading, the loading path is closer to an isentrope and results in lower entropy production. The reasonable ramp rate to obtain a quasi-isentropic state depends on the intrinsic time scales of the dissipation mechanisms which are strongly material dependent. Thus shockless loading does not necessarily produce an isentropic response. Between two equilibrium states, heat conduction was shown to have significant effect on the temperature history but it contributes little to the overall temperature change if the specific heat remains constant. It also affects the history of entropy, but only the irreversible part of heat conduction contributes to the net entropy change. The various types of thermomechanical responses of materials would manifest themselves more significantly in terms of the thermal history than the mechanical history. Thus temperature measurement appears to be an important experimental tool in distinguishing the various mechanisms for the thermomechancial responses of the materials.

ω-phase and solitary waves induced by shock compression of bcc crystals

Molecular dynamics simulations show that shock waves in single crystals are frequently accompanied by solitary waves. Here we present results for bcc crystals where the atoms interact via the Dzugutov potential [M. Dzugutov, J. Non-Cryst. Solids 131, 62 (1991)]. In general the shock waves cause a phase transformation to a close-packed phase. In addition, special phenomena occur if bcc is shocked along the threefold ⟨111⟩ direction. We observe an intermediate phase, which is identified as the so-called $\ensuremath{\omega}$-phase, and closely related nonsteady solitary waves in the regime of overdriven shock waves. The phenomena are described in detail and compared to solitary waves in other materials. The origin and stability of the $\ensuremath{\omega}$-phase are studied. We find that a single solitary wave mode exists parallel to the ⟨111⟩ direction, and solitary waves along other directions are derived from it. A simple collision model is presented that captures the essential properties of the solitary waves.

A numerical method for the solution of the wave model and convection dominated diffusion type models for catalytic packed bed reactors

A general numerical method for the solution of non-steady state wave and convection dominated dispersion packed bed reactor models is presented. Several numerical problems specific for this type of equations are addressed. Special care is taken to accurately and stably discretize the convection terms. The stiffness of the model equations is tackled by a simple but robust method based on the Newton–Kantorovich procedure and accompanied with an automatic time stepping mechanism. A new technique is proposed to automatically select and solve the minimum possible number of equations. The focus of this article is on the development of simple but robust approximation techniques for dealing with particular numerical problems and their coupling in an overall algorithm. All the methods described in this article have been implemented in a software package for the mathematical modeling of catalytic packed bed reactors.

Shock structure formation in dusty plasmas

The problem of the evolution of a perturbation in a dusty plasma and its transformation into a nonlinear wave structure is considered. A computational method that allows one to solve the set of nonlinear evolutionary equations describing variable-charge dust grains, Boltzmann electrons, and inertial ions is developed. Exact steady-state solutions corresponding to ion-acoustic shock structures associated with anomalous dissipation originating from dust grain charging are found taking into account the effect of electron and ion charge separation. The role of this effect increases with the speed of the shock. The evolutions of an initial soliton (which is a steady-state wave solution in a plasma containing dust grains with a constant charge) and an initially immobile perturbation with a constant increased ion density are investigated. In a charge-varying dusty plasma, the soliton evolves into a nonsteady shock wave structure that propagates at a constant speed and whose amplitude decreases with time. The initially immobile perturbation with a constant increased ion density evolves into a shock structure similar to a steady-state shock wave. In the latter case, the compression shock wave is accompanied by a rarefaction region (dilatation wave), which finally leads to the destruction of the shock structure. The solution of the problem of the evolution of a perturbation and its transformation into a shock wave in a charge-varying dusty plasma opens up the possibility of describing real phenomena (such as supernova explosions) and laboratory and active space experiments.

Evolutionary equation of inertial waves in 3‐D multiply connected domain with Dirichlet boundary condition

We study initial‐boundary value problem for an equation of composite type in 3‐D multiply connected domain. This equation governs nonsteady inertial waves in rotating fluids. The solution of the problem is obtained in the form of dynamic potentials, which density obeys the uniquely solvable integral equation. Thereby the existence theorem is proved. Besides, the uniqueness of the solution is studied. All results hold for interior domains and for exterior domains with appropriate conditions at infinity.

Heat Transfer in Bubbly Liquids: Fundamentals and Waves

The theory of non-steady wave processes in two-phase gas-liquid media with bubble structure, including hierarchy structure of mathematical models of bubbly liquids dynamic behavior, has been developed. The models differ in the degree of accounting the effects of pressure, temperature and phase velocities non-coincidence in the process of dynamic deformation. Estimations of the influence of dynamic and heat interaction between gas and liquid on shock waves evolution have presented. The technique of the worked out models program realization has been proposed. All models have been realized on a computer. A considerable number of computational experiments has been carried out, resulting in defining the principal mechanisms determining shock waves evolution in bubbly liquids in a wide range of parameters. In particular, it has been shown that shock waves evolution in low-viscous liquids with bubble size about 1 mm is determined by heat dissipation resulting from non-equilibrium heat transfer between the gas in the bubbles and surrounding liquid. Thermophysics characteristics of the gas, insignificant in volume and still more insignificant in mass, play the decisive role on evolution and formation of oscillar and monotonous wave configuration. It has been discovered that at waves propagation there may be observed their amplification as a result of the presence of locative deformation inertia characteristics in the bubbly system. The mechanism and peculiarity of shock waves amplification have been studied. The analysis of other experimental data contained in literature has been carried out. The comparison of the calculated and experimental data has proved correctness of the worked-out theory.

Synthesis of Ti5Si3-x Nb composites by the field-activated combustion method

The influence of an electric field on the combustion synthesis of Ti5Si3-x Nb composites (0≤ x ≤0.8) was investigated. Composites where x ≥ 0.8 can only be synthesized in the presence of a field. In the absence of a field, those systems where x=0.8 can result in a non-steady combustion wave causing an incomplete reaction. That is, an unstable wave propagates to the middle of the sample and then becomes extinguished. The wave velocity of the Ti5Si3-x Nb composites slightly increases with the imposition of an external electric field across the sample.

Electric field-activated combustion synthesis of Ti5Si3-Nb and Ti5Si3-ZrO2 composites

The field-activated combustion synthesis of the composites Ti5Si3−x Nb (0≤x≤0.35) and Ti5Si3−y ZrO2 (0≤y≤0.3) has been investigated. Composites with x≥0.35 and y≥0.2 can only be synthesized in the presence of an electric field. Although in the absence of a field the systems with x=0.35, y=0.2 and y=0.3 can sustain a non-steady combustion wave, the reaction is not complete. An unstable wave propagates to the middle of the sample and then becomes extinguished. The wave velocity of the Ti5Si3-Nb and Ti5Si3-ZrO2 composites increased slightly with the application of a field.

Synthesis of MoSi 2 xNb and MoSi 2 yZrO 2 composites by the field-activated combustion method

The effect of an electric of field on the combustion synthesis of MoSi 2 xNb (0 ⩽ x ⩽ 0.45) and MoSi 2 yZrO 2 (0 ⩽ y ⩽ 0.15) was investigated. The velocity of the combustion wave in both composite systems increased nearly linearly with an increase in the applied field. Composites with x ⩾ 0.45 and y ⩾ 0.15 can only be synthesized in the presence of a field. Although, in the absence of a field, the system with x = 0.45 and y = 0.15 can sustain a non-steady combustion wave, the reaction is not complete and only partial conversion to MoSi 2 is evident. For the synthesis of pure MoSi 2, the field was shown to affect the particle size of the product. The average particle size decreased when the synthesis was carried out under the effect of the field.

On the behavior near the caustic of a nonsteady wave field with singularity (A homogeneous generalized function) at the initial front

The Cauchy problem for the wave equation in the case of discontinuity on the initial front is investigated. The discontinuity is described by a homogeneous generalized function of degree λ. The transformation of the initial front while passing the space-time caustic is studied. The structure of the wave front and the space-time rays near the caustic is considered. Bibliography: 4 titles.

Weakly disturbed steady transonic flows of a vibrationally relaxing gas

The formation of shock waves in transonic flows is a problem which is far from being resolved. Conventional gas dynamics contains numerous examples of the construction of transonic flows free of discontinuities and flows in which discontinuities occur in the transition through the speed of sound (see [I], for example). Experiments have confirmed the existence of both continuous [2] transonic flows and transonic flows containing discontinuities [3]. The exceptional nature of nonshock flow in a local supersonic region was shown in [4], which suggested to one author that continuous stagnation of a transonic flow is not possible [5]. In the opinion Kuo and Sirs [6], such a flow is either unstable or contains shock waves or both. One possible approach to solving this problem is the study of transonic flows for their stability in relation to steady and nonsteady disturbances that might develop in the flow due to irregularities on the walls bounding the flow or the arrival of weak nonsteady waves at the sonic line. Such an investigation is complex in the general case, with approximate methods usually being used to obtain a solution. The stability of a transonic flow relative to small nonsteady disturbances in a small neighborhood of the sonic line was examined in [7, 8] on the basis of a quasi-unidimensional approximation. An attempt to perform an analysis in general form was made in [9], where results agreeing with the data in [7] were obtained after certain simplifying assumptions were made. Kuz'min [i0] analyzed the stability of a transonic flow on the basis of linearization of the Linn-Reissner-Tsien equation. It was found that transonic flow is stable in relation to a small change in the shape of the walls of the nozzle and the conditions at the inlet if the acceleration of the specified flow is everywhere positive. The examples of steady continuous transonic flows that have been constructed exist only near certain types of solid boundaries, thus giving rise to the view that such flows are the exception. A study of the behavior of steady perturbations of such flows due to irregularities in the walls bounding the flow will make it possible to study the development of the features of the flow. Aspects of the formation of shock waves and the conditions for nonshock flow in transonic flows were examined in [II]. It was shown that shock waves form if the slope of the walls of the channel near the mouth of the nozzle is too gentle. Discontinuities may form at a point on the sonic line or downstream, in the supersonic region. Numerical calculations of plane transonic flow with a local supersonic zone have shown that a shock wave is formed inside the region of supersonic flow [12]. Below, we examine the behavior of steady and nonsteady perturbations of steady transonic flows of a vibrationally relaxing gas. Use of the methods in [8] makes it possible to reduce the problem of determining the stability of a transonic flow against small nonsteady distortions to the study of a certain nonlinear partial differential equation. Analysis of the solutions of this equation for specific states of the medium shows that flow with a transition from the supersoni c regime to the subsonic regime is stable relative to small distortions if the gas is in the equilibrium state or if it is nonequilibrium with unexcited vibrational degrees of freedom. In this case, transonic flow will also be stable in the course of the transition from the subsonic to the supersonic regime. In our study of the behavior of steady distortions, we examined the supersonic region of transonic flow. We derive the equations of the transonic approximation as in traditional

Nonshock initiation of detonation in vacuum with unitary-fuel particles

The nonshock initiation of detonation in vacuum with unitary-fuel particles is considered. The dynamics of reaction-zone formation of a nonsteady detonation wave with disperse structure is discussed. The approach to steady Chapman—Jouquet detonation is considered. The critical initiation energies of a plane detonation wave with variation in length of the initiation section and the energy density supplied are calculated, and may be used in experimental investigations.

Behavior of singularities of a nonsteady wave field near a caustic in a medium with a variable velocity

The Cauchy problem for the wave equation is considered. A discontinuity in the form of a homogeneous generalized function of degree λ is given at the initial front. An asymptotic solution is found in the vicinity of the space-time caustic in the interaction of the discontinuity surface with the caustic. The solution is represented in the form of a certain combination of the generalized function with associated Legendre polynomials.

Determination of spall strength from surface motion studies

An expression in the literature has been used to determine the spall strength of materials from measurements of the surface velocity on reflection of a nonsteady shock wave. It uses initial density, the sound speed, and the difference between the initial surface jump-off velocity and the minimum in the velocity seen when the spall shock reaches the surface. Workers using the expression have found it necessary to choose the sound speed carefully in order to get reasonable results. It is shown that this is due to the inadequacy of the acoustic approximation used to represent the release isentrope. Approximation of the release isentrope by the Hugoniot makes it unnecessary to use special velocities and gives better agreement with experiment.