Gasdynamic Wave Research Articles

A Comparative Study of the Explicit Finite Difference Method and Physics-Informed Neural Networks for Solving the Burgers’ Equation

The Burgers’ equation is solved using the explicit finite difference method (EFDM) and physics-informed neural networks (PINN). We compare our numerical results, obtained using the EFDM and PINN for three test problems with various initial conditions and Dirichlet boundary conditions, with the analytical solutions, and, while both approaches yield very good agreement, the EFDM results are more closely aligned with the analytical solutions. Since there is good agreement between all of the numerical findings from the EFDM, PINN, and analytical solutions, both approaches are competitive and deserving of recommendation. The conclusions that are provided are significant for simulating a variety of nonlinear physical phenomena, such as those that occur in flood waves in rivers, chromatography, gas dynamics, and traffic flow. Additionally, the concepts of the solution techniques used in this study may be applied to the development of numerical models for this class of nonlinear partial differential equations by present and future model developers of a wide range of diverse nonlinear physical processes.

Shock wave structures in an isentropically unstable heat-releasing gas

In this work, we analytically and numerically investigate the types of stationary gasdynamic waves formed in a heat-releasing medium with isentropic (acoustic) instability. As the mathematical model, the system of one-dimensional gasdynamic equations is used, in which the heating and cooling processes are taken into account using the generalized heat-loss function. Our analysis reveals that the type of stationary structures depends on their velocity W and heating/cooling processes acting in the medium. In an isentropically unstable medium, it is shown that the type of structures depends on whether they propagate faster or slower than the critical velocity Wcr. If W>Wcr, a shock wave is formed, in which, after the shock-wave compression, the gas expands to a stationary value. The characteristic size of the expansion region depends on the characteristic heating time, which is determined by the specific type of the heat-loss function. If W<Wcr, the shock wave turns out to be unstable and decays into a sequence of autowave (self-sustaining) pulses. The amplitude and velocity (W=Wcr) of the autowave pulse, found analytically in the article, are also determined by the type of the heat-loss function. The comparison of analytical predictions of the developed method with the results of nonlinear equation previously obtained using the perturbation theory, as well as with the numerical simulations, confirms the high accuracy of the method.

The effect of initial amplitude and convergence ratio on instability development and deposited fluctuating kinetic energy in the single-mode Richtmyer–Meshkov instability in spherical implosions

This paper investigates the growth of perturbations on the inner surface of a dense imploding spherical shell due to hydrodynamic instabilities. The perturbations change in amplitude due to Richtmyer–Meshkov instability and Rayleigh-Taylor instability, geometric convergence, and compressibility. Two mode numbers (ℓ=5,50) and three different perturbation amplitudes (a0=0.1λ,0.01λ,0.001λ) are applied to the surface of the dense inner shell. Two independent codes were used to perform simulations with these six perturbation profiles at four convergence ratios, ranging from 3.6 to 7.3, for a total of 48 cases. The mixing layer amplitudes show good agreement between the two simulation codes across the range of convergence ratios, mode numbers, and initial amplitudes. The growth of the mixing layer is employed to validate an extension of a recently proposed Bell-Plesset model, showing good agreement across the range of convergence ratios. Persistent and substantial shock-deposited fluctuating kinetic energy is observed within the light core, away from the perturbed interface. Temporal evolution of fluctuating kinetic energy indicates a time delay between the peak radial and theta directions, consistent with a build-up of vortical motion. A “jet” like phenomena is observed at the peaks and troughs of the initial perturbations in both simulation codes across a variety of cases. It is postulated that these occur due to the shape singularity shown to develop in spatially periodic perturbed planar shock waves in ideal gas dynamics. Significant anisotropy of kinetic energy components is present at all times.

Detonation propagation for shock-driven, subsonic and supersonic confiner flow

We study the compressible flow dynamics of two-dimensional, steady detonation wave propagation in a high explosive (HE) confined by aluminium (Al) or stainless steel (SS), outside of which is an air layer. We examine how the thickness of the confinement affects the subsonic detonation driving zone structure (DDZ) and the detonation speed , demonstrating a strong dependence on whether the oblique shock-driven flow in the confiner is supersonic, as for SS, or subsonic, as for Al. A characteristic path analysis is used to examine the information flow from the material boundaries through the supersonic flow regions in both the HE and confiner that can impact the sonic surfaces bounding the subsonic flow regions. It is shown that the nature of gas-dynamic wave reflection off the SS–air or Al–air boundary can significantly influence the DDZ and .

Structural stability of shock waves in 2D compressible elastodynamics

We study the two-dimensional structural stability of shock waves in a compressible isentropic inviscid elastic material in the sense of the local-in-time existence and uniqueness of discontinuous shock front solutions of the equations of compressible elastodynamics in two space dimensions. By the energy method based on a symmetrization of the wave equation and giving an a priori estimate without loss of derivatives for solutions of the constant coefficients linearized problem we find a condition sufficient for the uniform stability of rectilinear shock waves. Comparing this condition with that for the uniform stability of shock waves in isentropic gas dynamics, we make the conclusion that the elastic force plays stabilizing role. In particular, we show that, as in isentropic gas dynamics, all compressive shock waves are uniformly stable for convex equations of state. Moreover, for some particular deformations (and general equations of state), by the direct test of the uniform Kreiss–Lopatinski condition we show that the stability condition found by the energy method is not only sufficient but also necessary for uniform stability. As is known, uniform stability implies structural stability of corresponding curved shock waves.

Performance modeling of pulse detonation engines using the method of characteristics

A quasi-one-dimensional method of characteristics (MOC) model is developed to evaluate the single-cycle gasdynamic flow field and associated propulsive performance of a fully- and partially-filled pulse detonation engine (PDE), and PDEs equipped with diverging nozzles. A detailed description of the PDE thrust chamber flow field is provided to highlight the dominant gasdynamic wave processes encountered in a general single-cycle operation. The MOC model is developed using a simplified unit process approach with an explicit inverse time marching algorithm in order to readily construct the complex thrust chamber flow field along a predefined grid. A grid dependency study is carried out to determine the appropriate grid resolution for the purposes of minimizing computational expense and maximizing numerical accuracy. Lastly, the MOC model is validated with existing numerical and experimental performance data for PDEs operating with oxyhydrogen and hydrocarbon mixtures in both atmospheric and sub-atmospheric environments. It is shown that the current MOC model has good agreement with existing numerical and experimental data for a variety of PDE configurations and operating conditions.

Квазиодномерная модель гиперболического типа гидроразрыва пласта

Представлена квазиодномерная модель гиперболического типа развития трещины гидроразрыва пласта в предположении, что коэффициент интенсивности напряжений намного превосходит коэффициент трещиностойкости породы. Рассматриваемая модель, учитывающая конвективные и нестационарные слагаемые в уравнении движения жидкости, является обобщением локальной модели Перкинса--Керна--Нордгрена. Доказано, что полученная система дифференциальных уравнений является квазилинейной строго гиперболической системой, для которой найдены характеристики и соотношения на характеристиках. В случае пренебрежения поправкой Кориолиса найдены инварианты Римана. При пренебрежении фильтрационной утечкой и вязкостью закачиваемой жидкости определены волны Римана, аналогичные простым плоским волнам в газовой динамике, и изучены их свойства. Исследована эволюционность границ трещины гидроразрыва пласта. Поставлена начально-краевая задача развития трещины гидроразрыва пласта. Показано, что при пренебрежении диссипативными слагаемыми в представленной модели можно построить теорию простых волн, аналогичную теории одномерной газовой динамики с изоэнтропическими плоскими волнами.

A geometric singular perturbation approach for planar stationary shock waves

The non-linear non-equilibrium nature of shock waves in gas dynamics is investigated for the planar case. Along each streamline, the Euler equations with non-equilibrium pressure are reduced to a set of ordinary differential equations defining a slow–fast system, and geometric singular perturbation theory is applied. The proposed theory shows that an orbit on the slow manifold corresponds to the smooth part of the solution to the Euler equation, where non-equilibrium effects are negligible; and a relaxation motion from the unsteady to the steady branch of the slow manifold corresponds to a shock wave, where the flow relaxes from non-equilibrium to equilibrium. Recognizing the shock wave as a fast motion is found to be especially useful for shock wave detection when post-processing experimental measured or numerical calculated flow fields. Various existing shock detection methods can be derived from the proposed theory in a rigorous mathematical manner. The proposed theory provides a new shock detection method based on its non-linear non-equilibrium nature, and may also serve as the theoretical foundation for many popular shock wave detection techniques.

Interaction of Flame Propagation and Pressure Waves During Knocking Combustion in Spark-Ignition Engines

Knocking combustion in spark-ignition (SI) engines is a typical abnormal combustion phenomenon that severely limits engine performance and thermal efficiency. However, its mechanism has not so far been completely revealed, such as the origin of pressure oscillation with tremendous amplitude and broken mechanisms of the engine body when knock occurs. This article systematically reviews the series of physical and chemical phenomena involved in knocking combustion, including auto-ignition, gas-dynamic waves, cavity resonance, and auto-ignited flame propagation. Significant research has shown that the rapid heat release of end-gas auto-ignition and subsequent strong pressure waves play a crucial role during knocking combustion. A methodology of the interaction of flame propagation and pressure waves has been proposed to reveal knock formation in terms of positive feedback between strengthened pressure waves and auto-ignited flame propagation. Also, new suggestions on the weakness in prior research of knocking combustion have also been discussed in the present article.

Disturbances Propagation in Supersonic Boundary Layers

Disturbances propagation processes are investigated in two-dimensional boundary layers for the case of strong viscous-inviscid interaction. The speed of upstream disturbances propagation as a function of specific heat ratio and Prandtl number is determined. Formula for speed propagation is developed on the basis of characteristics and subcharacteristics analysis corresponding to the gasdynamic wave processes and processes of convection and diffusion.

Real Gas Effect on Unsteady Flow of Natural Gases in Shock Tubes

The unsteady flow behaviors in devices like gas wave machines, wave rotor refrigerators and so on are complex due to real gas effect at high operational pressure and low temperature. In this work, a detail computational model for unsteady flow analysis of real natural gases is established. The real effect on unsteady behaviors of natural gases in shock tubes have been studied extensively. Results show that the non-classical flow of the gases will not exist. The discipline of reflection and refraction of various gas waves or discontinuities remain unchanged for natural gases. Attention should be paid only to the deviations between perfect gas model and real gas model for gasdynamic waves.

Unsteady Behavior of Real Gas in Wave Rotor Refrigerators

Real gas effect has great impact on unsteady flow behaviors in wave rotor refrigerator (WRR). In this work, the main thermal separation effect and propagation speed of gasdynamic wave of real gases are briefly studied. Numerical results have shown the large deviations between perfect gas model and real gas model for gasdynamic waves. A corrected wave diagram of WRR has been achieved based on real gas PVT relations, through which the problem of offset design of port timing can be completely solved.

On the modified dispersion-controlled dissipative (DCD) scheme for computation of flow supercavitation

In the present study, a robust scheme initially proposed for capturing shock waves in gasdynamics, the dispersion-controlled dissipative (DCD) scheme is modified to simulate supercavitating flows in the framework of one-fluid model. Due to the stiffness of the equations of state (EOS), the Steger-Warming method becomes inapplicable and the Lax–Friedrichs method should be used for the numerical flux splitting. Following the formulation, the updated scheme is validated by a series of comparisons with theoretical and experimental data. The validation shows that the modified DCD scheme performs very well for supercavitating flows in the presence of steep gradients of both density and EOS. Several one-fluid cavitation models and speed of sound (SoS) models are compared for the numerical investigation of supercavitating flows.

Conditional Stability of Unstable Viscous Shock Waves in Compressible Gas Dynamics and MHD

Extending our previous work in the strictly parabolic case, we show that a linearly unstable Lax-type viscous shock solution of a general quasilinear hyperbolic–parabolic system of conservation laws possesses a translation-invariant center stable manifold within which it is nonlinearly orbitally stable with respect to small L 1 ∩ H 3 perturbations, converging time asymptotically to a translate of the unperturbed wave. That is, for a shock with p unstable eigenvalues, we establish conditional stability on a codimension-p manifold of initial data, with sharp rates of decay in all L p . For p = 0, we recover the result of unconditional stability obtained by Mascia and Zumbrun. The main new difficulty in the hyperbolic–parabolic case is to construct an invariant manifold in the absence of parabolic smoothing.

Analysis of Gas-Dynamic Effects in Explosively Actuated Valves

A quasi-one-dimensional model is formulated to assess unsteady gas dynamics occurring within axisymmetric explosively actuated valves. The model accounts for pressure-dependent explosive burn within an actuator, compressible product-gas flow through a narrow port connecting the actuator to a gas expansion chamber, and piston motion due to the combined effects of gas-dynamic forces within the expansion chamber and structural deformation forces between the piston and valve bore. The initial boundary-value problem is posed in terms of generalized coordinates to facilitate numerical computations on a domain that volumetrically expands due to combustionandpistonmotion.Predictionsforabaselineconfigurationthatisrepresentative ofaconventionalvalve indicate that gas-dynamic waves do not result in irregular operation, implying that spatially homogeneous models may be adequate for describing its performance. However, small changes in valve geometry and explosive mass can produce large variations in gas-dynamic fields that significantly affect both the magnitude and frequency of the pyrotechnic shock transmitted to the valve’s supporting structure. Port diameter is shown to control the rates of explosive energy release andacoustic energy transport, which can significantly affect pistonmotion andstroke time.

Hopf Bifurcation of Viscous Shock Waves in Compressible Gas Dynamics and MHD

Extending our previous results for artificial viscosity systems, we show, under suitable spectral hypotheses, that shock wave solutions of compressible Navier–Stokes and magnetohydrodynamics equations undergo Hopf bifurcation to nearby time-periodic solutions. The main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in H s for data in H s . We accomplish this by a novel energy estimate carried out in Lagrangian coordinates; interestingly, this estimate is false in Eulerian coordinates. At the same time, we greatly sharpen and simplify the analysis of the previous work.

Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics

A Bose-Einstein condensate (BEC) is a quantum fluid that gives rise to interesting shock wave nonlinear dynamics. Experiments depict a BEC that exhibits behavior similar to that of a shock wave in a compressible gas, eg. traveling fronts with steep gradients. However, the governing Gross-Pitaevskii (GP) equation that describes the mean field of a BEC admits no dissipation hence classical dissipative shock solutions do not explain the phenomena. Instead, wave dynamics with small dispersion is considered and it is shown that this provides a mechanism for the generation of a dispersive shock wave (DSW). Computations with the GP equation are compared to experiment with excellent agreement. A comparison between a canonical 1D dissipative and dispersive shock problem shows significant differences in shock structure and shock front speed. Numerical results associated with the three dimensional experiment show that three and two dimensional approximations are in excellent agreement and one dimensional approximations are in good qualitative agreement. Using one dimensional DSW theory it is argued that the experimentally observed blast waves may be viewed as dispersive shock waves.

Modelling of Unsteady Gas-Dynamic Flow in a Pipe and at Its Exit using CFD

Traditionally the simulation of the thermodynamic aspects of the internal combustion engine has been undertaken using one-dimensional gas-dynamic models to represent the intake and exhaust systems. Computational fluid dynamics (CFD) analysis of engines has been restricted to modelling of in-cylinder flow structures. With the increasing accessibility of CFD software it is now worth considering its use for complete gas-dynamic engine simulation. This paper appraises the accuracy of various CFD models in comparison with a one-dimensional gas-dynamic simulation. All of the models are compared to experimental data acquired on an apparatus that generates a single gas-dynamic pressure wave. The progress of the wave along a constant area pipe and its subsequent reflection from the open pipe end are recorded with a number of high-speed pressure transducers. It was found that there was little to choose between the accuracy of the one-dimensional model and the best CFD model. The CFD model did not require experimentally derived loss coefficients to represent accurately the open pipe end; however, it took several hundred times longer to complete its analysis. The best congruency between the CFD models and the experimental data was achieved using the RNG k-∊ turbulence model. The open end of the pipe was most effectively represented by surrounding it with a relatively small volume of cells connected to the rest of the environment using a pressure boundary.

Construction of uniform in time asymptotics for interaction of shock waves in gas dynamics

We construct uniform in time asymptotics which describes the interaction of two isothermal shock waves.

Dissipative Symmetrizers of Hyperbolic Problems and Their Applications to Shock Waves and Characteristic Discontinuities

By introducing the notations of dissipative and strictly dissipative p-symmetrizers of initial-boundary-value problems for linear hyperbolic systems we formalize the dissipative integrals technique [A. Blokhin, Yu. Trakhinin, in Handbook of Mathematical Fluid Dynamics, Vol. 1, North-Holland, Amsterdam, 2002, pp. 545-652] applied earlier to shock waves and characteristic discontinuities for various concrete systems of conservation laws. This enables us to prove the local in time existence of shock-front solutions of an abstract symmetric system of hyperbolic conservation laws, provided that the corresponding constant coefficients linearized problem has a strictly dissipative p-symmetrizer. Our result does not, in particular, require the fulfillment of Majda's block structure condition. A p-symmetrizer is, in some sense, a "secondary" Friedrichs symmetrizer for the symmetric system for the vector of p-derivatives of unknown functions, and the structure of p-symmetrizer takes into account (if applicable) the set of divergent constraints for the original system. After applying a p-symmetrizer, which is in general a set of matrices and vectors, the boundary conditions for a resulting symmetric system are dissipative (or strictly dissipative). We give concrete examples of p-symmetrizers. Our main examples are strictly dissipative 2-symmetrizers for shock waves in gas dynamics and magnetohydrodynamics. A general procedure for constructing a p-symmetrizer does not however exist. But, if it was somehow constructed, then we do not need to test the Lopatinski condition that is often connected with insuperable technical difficulties. As an illustration, we refer to the author's recent result [Yu. Trakhinin, Arch. Ration. Mech. Anal., 177 (2005), pp. 331-366] for compressible current-vortex sheets for which the construction of a dissipative 0-symmetrizer has first enabled the finding of sufficient conditions for their weak linearized stability.