Equations Of One-dimensional Gas Dynamics Research Articles

Boundary proportional control of thermoacoustic instability in a Rijke tube

Thermoacoustic instability has always been a critical concern of low pollution combustion chambers and combustion components of ground gas turbines. It is also a severe challenge for dynamic system design. In order to reveal the thermoacoustic instability, a typical Rijke tube based on partial differential equations (PDEs) cascaded with the Heckl heat release is studied. A boundary proportional feedback control to stabilize the thermoacoustic instability in the Rijke tube is proposed. First of all, one-dimensional gas dynamic equations coupled with the Heckl heat release are presented to describe the thermoacoustic unstable dynamic system in the Rijke tube. Then by using the small deviation linearization and characteristic coordinate transformation, the thermoacoustic coupling inside the domain is transferred into a group of 4 × 4 PDEs with the boundary coupling. Next, a boundary proportional feedback control, characterized by its practical implementation feasibility, is proposed to ensure the exponential stabilization of the thermoacoustic unstable system. A solvability condition is also provided to theoretically design the boundary proportional controller. Furthermore, the optimal control parameters are given by solving the nonlinear programming problem to achieve the fastest convergence rate of the system. Finally, the effectiveness of the boundary proportional feedback control is verified by simulations, and its robustness to the uncertain time delay and the nonlinearity of the heat release is also discussed.

Construction of Exact Solutions of the System of One-Dimensional Gas Dynamics Equations without Gradient Catastrophe

Construction of Exact Solutions of the System of One-Dimensional Gas Dynamics Equations without Gradient Catastrophe

Group analysis tasks. Conservation laws

Differential equations of mechanics are derived from the laws of conservation of mass, momentum and energy. Other conservation laws can be obtained using the symmetries of these equations. Any conservation law can be derived from one using canonical operators, special cases of which are symmetry operators. The calculation of canonical conservation law operators is equivalent to a direct method of finding a divergent form of equations that are valid by virtue of the equations of mechanics. Using the example of one-dimensional equations of gas dynamics, all zero-order conservation laws are obtained by the direct method. For special equations of state, an infinite set of conservation laws have been obtained, many of which are new.

Construction of Exact Solutions of the System of One-Dimensional Gas Dynamics Equations without Gradient Catastrophe

The system of equations that describes one-dimensional polytropic gas flows is considered. The invariants up to the second order of characteristics of the considered system of equations are classified. The method of reducing the Cauchy problems to systems of ordinary differential equations is proposed. Examples of the solutions without gradient catastrophe are constructed using invariants of characteristics supplementary to the Riemann invariants.

Group classification of one-dimensional equations of relativistic gas dynamics

Analysis of the one-dimensional relativistic gas dynamics equations is presented in the paper. The equations contain the internal energy as an arbitrary element. Depending on the form of this function the equations have different symmetries. Complete group classification of the one-dimensional relativistic gas dynamics equations with respect to an admitted Lie group is obtained in the present paper. The classification divides the equations into several classes. Analysis of invariant solutions is also provided.

A Semi-Lagrangian Godunov-Type Method without Numerical Viscosity for Shocks

One of the most important and complex effects in compressible fluid flow simulation is a shock-capturing mechanism. Numerous high-resolution Euler-type methods have been proposed to resolve smooth flow scales accurately and to capture the discontinuities simultaneously. One of the disadvantages of these methods is a numerical viscosity for shocks. In the shock, the flow parameters change abruptly at a distance equal to the mean free path of a gas molecule, which is much smaller than the cell size of the computational grid. Due to the numerical viscosity, the aforementioned Euler-type methods stretch the parameter change in the shock over few grid cells. We introduce a semi-Lagrangian Godunov-type method without numerical viscosity for shocks. Another well-known approach is a method of characteristics that has no numerical viscosity and uses the Riemann invariants or solvers for water hammer phenomenon modeling, but in its formulation the convective terms are typically neglected. We use a similar approach to solve the one-dimensional adiabatic gas dynamics equations, but we split the equations into parts describing convection and acoustic processes separately, with corresponding different time steps. When we are looking for the solution to the one-dimensional problem of the scalar hyperbolic conservation law by the proposed method, we additionally use the iterative Godunov exact solver, because the Riemann invariants are non-conserved for moderate and strong shocks in an ideal gas. The proposed method belongs to a group of particle-in-cell (PIC) methods; to the best of the author’s knowledge, there are no similar PIC numerical schemes using the Riemann invariants or the iterative Godunov exact solver. This article describes the application of the aforementioned method for the inviscid Burgers’ equation, adiabatic gas dynamics equations, and the one-dimensional scalar hyperbolic conservation law. The numerical analysis results for several test cases (e.g., the standard shock-tube problem of Sod, the Riemann problem of Lax, the double expansion wave problem, the Shu–Osher shock-tube problem) are compared with the exact solution and Harten’s data. In the shock for the proposed method, the flow properties change instantaneously (with an accuracy dependent on the grid cell size). The iterative Godunov exact solver determines the accuracy of the proposed method for flow discontinuities. In calculations, we use the iteration termination condition less than 10−5 to find the pressure difference between the current and previous iterations.

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.

Singularities in one-dimensional Euler flows

In this paper, a system of one-dimensional gas dynamics equations is considered. This system is a particular case of Jacobi type systems and has a natural representation in terms of 2-forms on 0-jet space. We use this observation to find a new class of multivalued solutions for an arbitrary thermodynamic state model and discuss singularities of their projections to the space of independent variables for the case of an ideal gas. Caustics and discontinuity lines are found.

НЕЯВНЫЙ МЕТОД ХАРАКТЕРИСТИК НА РЕГУЛЯРНОЙ СЕТКЕ (ГАЗОДИНАМИЧЕСКИЕ ЗАДАЧИ)

In this work, an implicit method is proposed to numerically solve a system of the onedimensional nonstationary equations of gas dynamics transformed by the method of characteristics. Internal points of the channel for a solid-propellant charge are considered at a preignition period of the solid-propellant rocket engine operation. The use of the implicit method makes it possible to calculate the values of gas-dynamic parameters at nodal points of the regular coordinate grid. Calculations of the gas-dynamic parameters both when integrating over time and along the spatial coordinate are performed with the second order of accuracy. Both subsonic and supersonic flows are studied. It is shown that, when predicting the expected pressure value during the transition from one time layer to another with the second order of accuracy, the twenty-fold efficiency of the implicit method is achieved in comparison with the explicit difference method. The trial calculation is performed.

Об инвариантных конечно-разностных схемах для уравнений одномерных течений политропного газа для задач с пространственными симметриями

One-dimensional polytropic gas dynamics equations for plane, radially symmetric, and spherically symmetric flows are considered. Invariant properties of equations are discussed, local conservation laws are derived. Additional conservation laws are written, which take place only in case of special values of adiabatic exponent. Classical difference scheme of Samarsky-Popov for gas dynamics has all difference analogs of conservation laws, except for additional ones. In difference schemes additional conservative laws take place in case of special state equation approximation. Scheme of Samarsky-Popov with special state equation was initially suggested by V.A. Korobitsyn. He described it as ‘thermodynamically consistend’ In current paper group properties, and conservation laws of thermodynamically consistent schemes are discussed, and numerical implementation for plane, cylinder, and spherical flows is perfomed.

The Existence of a Bounded Invariant Region for Compressible Euler Equations in Different Gas States

In this article, by the mean-integral of the conserved quantity, we prove that the one-dimensional non-isentropic gas dynamic equations in an ideal gas state do not possess a bounded invariant region. Moreover, we obtain a necessary condition on the state equations for the existence of an invariant region for a non-isentropic process. Finally, we provide a mathematical example showing that with a special state equation, a bounded invariant region for the non-isentropic process may exist.

Conservation laws of the one-dimensional equations of relativistic gas dynamics in Lagrangian coordinates

The present paper is focused on the analysis of the one-dimensional relativistic gas dynamics equations. The studied equations are considered in Lagrangian description, making it possible to find a Lagrangian such that the relativistic gas dynamics equations can be rewritten in a variational form. Such a Lagrangian is found in the paper. Complete group analysis of the Euler–Lagrange equation is performed. The found Lagrangian and the symmetries are used to derive conservation laws in Lagrangian variables by means of Noether’s theorem. The analogs of the newly found conservation laws in Eulerian coordinates are presented as well.

Study of Entropy Properties of a Linearized Version of Godunov’s Method

The ideas of formulating a weak solution for a hyperbolic system of one-dimensional gas dynamics equations are presented. An important aspect is the examination of the scheme for the fulfillment of the nondecreasing entropy law, which must hold for weak solutions and is obligatory from a physics point of view. The concept of a weak solution is defined in a finite-difference formulation with the help of the simplest linearized version of the classical Godunov scheme. It is experimentally shown that this version guarantees an entropy nondecrease. As a result, the growth of entropy on shock waves can be simulated without using any correction terms or additional conditions.

Group Analysis of the One-Dimensional Gas Dynamics Equations in Lagrangian Coordinates and Conservation Laws

A group analysis of the second-order equation including the one-dimensional gas dynamics equations in Lagrangian coordinates as a particular case is performed. The use of Lagrangian coordinates makes it possible to consider the one-dimensional gas dynamics equations as a variational Euler-Lagrange equation with an appropriate Lagrangian. Conservation laws are derived with the use of the variational presentation and Noether’s theorem. A complete group classification of the Euler-Lagrange equation is obtained; as a result, 18 different classes can be identified.

Group Analysis of One-Dimensional Gas Dynamics Equations in Lagrangian Coordinates and Conservation Laws

Group Analysis of One-Dimensional Gas Dynamics Equations in Lagrangian Coordinates and Conservation Laws

Comparison of the numerical and self-similar solutions of Sedov’s problem on a point explosion in gas

Comparative analysis of solutions of Sedov’s problem of a point explosion in gas for the plane case, obtained by the analytical method and using the open software package of computational fluid dynamics OpenFOAM, is carried out. A brief analysis of methods of dimensionality and similarity theory used for the analytical self-similar solution of point explosion problem in a perfect gas (nitrogen) which determined by the density of uncompressed gas, magnitude of released energy, ratio of specific heat capacities and by the index of geometry of the explosion is given. The system of one-dimensional gas dynamics equations for a perfect gas includes the laws of conservation of mass, momentum, and energy is used. It is assumed that at the initial moment of time there is a point explosion with instantaneous release of energy. Analytical self-similar solutions for the Euler and Lagrangian coordinates, mass velocity, pressure, temperature, and density in the case of plane geometry are given. The numerical simulation of considered process in sonicFoam solver of OpenFOAM package built on the PISO algorithm was performed. For numerical modeling the system of differential equations of gas dynamics is used, including the equations of continuity, Navier-Stokes motion for a compressible medium and conservation of internal energy. Initial and boundary conditions were selected in accordance with the obtained analytical solution using the setFieldsDict, blockMeshDict, and uniformFixedValue utilities. The obtained analytical and numerical solutions have a satisfactory agreement.

L2-Dissipativity Criteria for Linearized Explicit Finite Difference Schemes for Regularization of One-Dimensional Gas Dynamics Equations

We obtain criteria for the L2-dissipativity of finite-difference schemes based on regularizations of 1D barotropic and full gas dynamics systems of equations that are linearized at a constant solution.

Влияние индуцированных волн в камере на динамику процесса газодинамического разгона поршня в трубе

The purpose of the paper was to study the effect of induced gas perturbations in the reservoir on the ballistic characteristics of single-stage gas guns. Two ways of waves generation at the initial moment are considered: by means of a non-uniform initial distribution of gas parameters and by creating a shock wave propagating from the bottom of the reservoir by external forces. The study is based on the numerical solution of one-dimensional gas-dynamic equations on a moving grid. Findings of research show that for relatively large (compared to the accelerating gas mass) mass of the projectiles, the wave processes induced by inhomogeneous filling lead to an increase in the muzzle velocity 1.4 times, and the shock waves generation by an external source — 1.8 times.

One-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates: Symmetry classification, conservation laws, difference schemes

A comprehensive analysis of the one-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates is performed. One of the representations of these equations in Lagrangian coordinates is given by a single second-order partial differential equation. Symmetries of this equation are analyzed using the entropy for the group classification. Noether’s theorem is applied for constructing conservation laws. The obtained conservation laws are represented in the gas dynamics variables in Lagrangian coordinates and in Eulerian coordinates as well. Invariant and conservative difference schemes are discussed for the basic adiabatic case.

Numerical simulation of single vapor bubble dynamics in a liquid in an intense acoustic field

The possibility of increasing the calculation efficiency by the joint use of two models of the dynamics of a single weakly-nonspherical vapor bubble under its strong collapse in liquid is studied. In both models the motion of liquid and vapor is split into a spherical component and its small nonspherical perturbation. The models differ in the description of the spherical component. In the first (simplified) model, it is described by a system of ODE together with partial differential equations in temperature, derived under the assumption of weak compressibility of liquid and bubble homobaricity. In the second model, one-dimensional gas dynamics equations are applied. The advantage of the simplified model consists in determining a numerical solution with much-less computer time costs in comparison with what is required for the numerical integration of gas dynamics equations. The assumptions used in the simplified model in the final stage of collapse become incorrect, and, as a result, the numerical solution errors increase. Therefore, the simplified model is applied at the beginning of bubble collapse, whereas the gas dynamics equations are used at its end. Within this approach, the numerical solution in the final stage of collapse is dependent on the moment of transition to the gas dynamics equations. It is shown that satisfactory description of evolution of bubble sphericity distortion is achieved when the transition is made under the condition that the Mach number M of vapor in the vicinity of the bubble surface is less than 0.4. Satisfactory resolution of the shock wave in the bubble is attained when the transition is performed at M<0.2.