One-dimensional Compressible Flow Research Articles

A high-order physical-constraints-preserving arbitrary Lagrangian–Eulerian discontinuous Galerkin scheme for one-dimensional compressible multi-material flows

In this paper, a high-order physical-constraints-preserving arbitrary Lagrangian–Eulerian (ALE) discontinuous Galerkin scheme is proposed for one-dimensional compressible multi-material flows. Our scheme couples a conservative equation related to the volume-fraction model with the Euler equations for describing the dynamics of fluid mixture. The mesh velocity in the ALE framework is obtained by using an adaptive mesh method that can automatically concentrate the mesh nodes near the regions with large gradient values and greatly reduce the numerical dissipation near material interfaces. Using this adaptive mesh, the resolution of solution near some special regions such as material interfaces can be improved effectively by our scheme. With the appropriate time step condition and using a bound-preserving and positivity-preserving limiter, our scheme can ensure the positivity of density and pressure and the boundness of volume-fraction, which further ensures the computational robustness and degree of confidence of simulations under large density or pressure ratios and so on. In general, our scheme can be applied to the simulations of compressible multi-material flows efficiently with the essentially non-oscillatory property and physical-constraints-preserving (bound-preserving and positivity-preserving) property, and its steps are more concise compared to some other methods such as the indirect ALE methods. Some examples are tested to demonstrate the accuracy, essentially non-oscillatory property and physical-constraints-preserving property of our scheme.

Propagation Characteristics of Initial Compression Wave Induced by 400 km/h High-Speed Trains Passing through Very Long Tunnels

When high-speed trains enter tunnels, an initial compression wave is generated. As the compression wave propagates at the local speed of sound to the tunnel exit, it radiates into the surrounding environment, forming micro-pressure waves (MPWs). MPWs create sonic booms, resulting in significant environmental issues. The magnitude of the micro-pressure waves is directly proportional to the pressure gradient of the compression wave at the tunnel exit. The nonlinear effects of the initial compression wave during propagation lead to a significant increase in pressure gradient. Therefore, the propagation characteristics of the initial compression wave during the tunnel are the crucial factor affecting the amplitude of MPWs. Based on the one-dimensional compressible unsteady non-isentropic flow model and the improved generalized Riemann variable characteristic method, this paper researched the propagation and evolution characteristics of an initial compression wave generated when 400 km/h high-speed trains enter tunnels with three portal shapes: (no tunnel entrance hood (no hood), an oblique, enlarged tunnel entrance hood (type A), an enlarged equal-section non-uniform opening hole tunnel entrance hood (type B)). The results show that when the initial compression wave propagates inside very long tunnels, the pressure gradient of the compression wave exhibits a trend of initially increasing and then decreasing with the increase in propagation distance. When the pressure gradient of the compression wave reaches its maximum value, the corresponding propagation distance is the steepening critical distance. For no tunnel entrance hoods, type A tunnel entrance hoods, and type B tunnel entrance hoods, the steepening critical distances are 5 km, 6 km, and 16 km, respectively. The steepening critical distance shortens with increasing train speed. Steady friction and unsteady friction effects mainly affect the pressure amplitude and pressure gradient during compression wave propagation, respectively. At lower ambient temperatures, the nonlinear effects in compression wave propagation are significantly enhanced. The mitigation effects of type A tunnel entrance hoods and type B tunnel entrance hoods on pressure gradient reduction are mainly concentrated within 4 km and 12 km, respectively. It is necessary to determine the optimal matching relationship between the tunnel entrance hood and tunnel length based on the characteristics of compression wave propagation to ensure their mitigating performance is maximized.

A high-order limiter-free arbitrary Lagrangian–Eulerian discontinuous Galerkin scheme for compressible multi-material flows

In this paper, a high-order limiter-free arbitrary Lagrangian–Eulerian discontinuous Galerkin (ALE-DG) scheme is proposed for simulating one-dimensional compressible multi-material flows. The system is discretized by the DG method, and a kind of Taylor’s expansion basis functions in the general cell is used to construct the interpolation polynomials of the variables. The mesh velocity at the node recognized as the material interface is set to approximate fluid velocity, which makes our scheme can capture the material interface accurately and clearly. For controlling the oscillations and reducing the numerical dissipation, a reconstruction based on the boundary variation diminishing (BVD) algorithm and Tangent of Hyperbola for INterface Capturing (THINC) function is employed to minimize the jump between the values at the left and right sides of cell boundaries. Due to the essentially monotone and bounded properties of THINC function, the difficulties caused by solving discontinuous solutions and complexities of designing limiters can be avoided. Finally, several examples are presented to demonstrate the accuracy and good performance of our scheme such as the essentially non-oscillatory property and so on for simulating the compressible multi-material flows.

Modelling Air Flow through Pneumatic Valves: A Brief Review with an Experimental Case Study

Compressible flow models are commonly used for describing air flow through pneumatic valves. Because of the difficulties in predicting viscous losses, these models ultimately rely on experimental determination of coefficients. Different equations have been proposed for different fluid speeds, having the sonic fluid velocity as a reference mark. However, one might question whether a much simpler approach, where the fluid is considered as incompressible, would still give good results within the typical range of industrial applications. Moreover, practically all models presuppose that the valve output pressure decreases in time, as in a discharge process. This paper reviews some representative one-dimensional compressible flow models and discusses the appropriateness of using equations based solely on discharging flows. Two experimental circuits, where an air reservoir is pressurized and, subsequently, decompressed, are used for comparison between different flow models. It is shown that a simpler set of equations still produces acceptable results for practical pneumatic applications.

Symmetry and scaling in one-dimensional compressible two-phase flow

Investigations of shock compression of heterogeneous materials often focus on the shock front width and overall profile. The number of experiments required to fully characterize the dynamic response of a material often belie the structure–property relationships governing these aspects of a shock wave. Recent observations measured a pronounced shock-front width on the order of 10 s of ns in particulate composites. Here, we focus on particulate composites with disparate densities and investigate whether the mechanical interactions between the phases are adequate to describe this emergent behavior. The analysis proceeds with a general Mie–Grüneisen equation of state for the matrix material, a general drag force law with general power-law scaling for the particle-matrix coupling of the phases, and a volume fraction-dependent viscosity. Lie group analysis is applied to one-dimensional hydrodynamic flow equations for the self-consistent interaction of particles embedded in a matrix material. The particle phase is characterized by a particle size and volume fraction. The Lie group analysis results in self-similar solutions reflecting the symmetries of the flow. The symmetries lead to well-defined scaling laws, which may be used to characterize the propagation of shock waves in particle composites. An example of the derived scaling laws for shock attenuation and rise time is shown for experimental data on shock-driven tungsten-loaded polymers. A key result of the Lie analysis is that there is a relationship between the exponents characterizing the form of the drag force and the exponent characterizing the shock velocity and its attenuation in a particulate composite. Comparison to recent experiments results in a single exponent that corresponds to a conventional drag force.

The Riemann problem for one-dimensional dusty gas dynamics with external forces

This article concerns the study of the Riemann Problem (RP) for a non-homogeneous system governing the one-dimensional compressible flow of dusty gas, where the external forces are assumed to be continuous function of time. The elementary wave solutions such as rarefaction wave, shock wave, and contact discontinuity are rigorously determined. The effect of dust particles on the density and flow velocity and shock speed is examined, and their consequences on the solution of the Riemann Problem are discussed. Moreover, we discussed the condition for the existence of vacuum appears in the solution of the Riemann Problem.

The sharp time-decay rates for one-dimensional compressible isentropic Navier-Stokes and magnetohydrodynamic flows

The sharp time-decay rates for one-dimensional compressible isentropic Navier-Stokes and magnetohydrodynamic flows

Leading-to-trailing edge theoretical design of a generic scramjet

The aerospace vehicle flying at hypersonic speed, using an airbreathing propulsion system based on supersonic combustion (scramjet) technology, requires a highly integrated system without moving parts, making the propulsion system and vehicle shape indistinguishable. A scramjet uses shock waves, generated by its inlet during the hypersonic flight to provide the temperature and velocity of atmospheric air to burn the hydrogen, at the combustor at supersonic speed. The divergent exhaust nozzle accelerates the combustion products providing the thrust. A two-dimensional hydrogen-powered generic scramjet has been designed to demonstrate supersonic combustion in atmospheric flight at a Mach number of 6.8 and an altitude of 30 km. Temperature and velocity at the combustion chamber are the most important key parameters for a preliminary design. The inlet configuration must provide a temperature higher than the ignition temperature of the fuel to guarantee spontaneous combustion and the velocity must remain supersonics. The nozzle exit velocity should be higher than the vehicle flight velocity to obtain sustained flight and produce enough thrust. For this theoretical analysis, the air is considered a calorically perfect gas without viscous effects. The compressible flow is analyzed based on oblique shock waves and one-dimensional compressible flow with friction and heat addition and Prandtl–Meyer coupled to the area ratio to describe each key component of the scramjet, such as compression, combustor, and nozzle sections, respectively. The inlet mass flow and hydrogen mass flow are also critical for the scramjet design to ensure stoichiometrically combustion at the combustion chamber, generating the high flow velocity at the nozzle to produce thrust.

Riemann problem for constant flow with single-point heating source

This work focuses on the Riemann problem of Euler equations with global constant initial conditions and a single-point heating source, which comes from the physical problem of heating one-dimensional inviscid compressible constant flow. In order to deal with the source of Dirac delta-function, we propose an analytical frame of double classical Riemann problems (CRPs) coupling, which treats the fluids on both sides of the heating point as two separate classical Riemann problems and then couples them. Under the double CRPs frame, the Riemann solution is self-similar, and three types of structures are found. The theoretical analysis is also supported by the numerical simulation. Furthermore, the uniqueness of the Riemann solution is established with some restrictions on the Mach number of the initial condition.

Exact Riemann solutions for the drift-flux equations of two-phase flow under gravity

The exact Riemann solutions are constructed in completely explicit forms for the drift-flux equations of the one-dimensional inviscid, compressible and isentropic liquid-gas two-phase flow model in an inclined pipeline under gravity. It is found that the curved delta shock wave and vacuum state are involved in the Riemann solutions for the pressureless situation with the adiabatic exponent being one. It is proved rigorously that this curved delta shock wave solution satisfies the pressureless drift-flux equations in the sense of distributions. Finally, the asymptotic limits of Riemann solutions are investigated in detail when the adiabatic exponent tends to one, in which the concentration phenomenon can be observed when the non-self-similar Riemann solution consisting of curved 1-shock wave, 2-contact discontinuity and 3-shock wave converges to this curved delta shock wave solution. In addition, the cavitation phenomenon can also be inspected that all the states in the two rarefaction wave fans become the vacuum states in the limiting situation.

Existence and uniqueness of solution to one-dimensional compressible biaxial nematic liquid crystal flows

Existence and uniqueness of solution to one-dimensional compressible biaxial nematic liquid crystal flows

Global solution to the compressible non-isothermal nematic liquid crystal equations with constant heat conductivity and vacuum

This paper considers the initial-boundary value problem of the one-dimensional full compressible nematic liquid crystal flow problem. The initial density is allowed to touch vacuum, and the viscous and heat conductivity coefficients are kept to be positive constants. Global existence of strong solutions is established for any H^{2} initial data in the Lagrangian flow map coordinate, which holds for both vacuum and non-vacuum case. The key difficulty is caused by the lack of the positive lower bound of the density. To overcome such difficulty, it is observed that the ratio of frac{rho _{0(y)}}{rho (t,y)} is proportional to the time integral of the upper bound of temperature and vector director field, along the trajectory. Density weighted Sobolev type inequalities are constructed for both temperature and director field in terms of frac{rho _{0(y)}}{rho (t,y)} and small dependence on their dissipation estimates. Besides this, to deal with cross terms arising due to liquid crystal flow, higher order priori estimates are established by using effective viscous flux.

One dimensional steam ejector model based on real fluid property

The one-dimensional ejector model based on the real fluid property is proposed to design the steam ejector geometries and to predict its performance. This model is developed from the one-dimensional compressible flow of steam-water which was firstly proposed by Stoecker (1958). The ejector geometry and performance determined by the proposed model are validated with the experimental results to ensure that the proposed model can be used efficiently for designing the steam ejector. Also, the results obtained from the one-dimensional model based on the ideal gas are compared with those obtained from the proposed model. It is found that both models can efficiently be used to design the ejector geometry and to predict the performance (in terms of entrainment ratio and critical condenser pressure). However, the one-dimensional model based on the ideal gas provides relatively low temperatures at each state compared to the proposed model. The calculated temperature is well below the freezing point while that calculated by the proposed model provides fluid properties at equilibrium condition. In addition, the condensation process of the supersonic flow is involved in many processes inside the ejector. This indicates that the ideal gas assumption for the wet fluid (steam is classified as the wet fluid) might not suitable for analysing the process of the ejector.

Code development and analysis on the operation of liquid metal high temperature heat pipes under full condition

Heat pipe cooled reactor (HPR) is an innovative solid-core reactor which utilizes heat pipes to extract heat from the core to the energy conversion system. In this paper, a mathematical model for high-temperature heat pipes from a frozen state to a steady state is developed based on experimental data. The model is simplified and solved numerically. The two-dimensional transient heat conduction model of the wall and wick and the one-dimensional quasi-steady compressible vapor flow model are coupled to simulate the operation of a heat pipe when vaporization and condensation occur at the liquid–vapor interface. And the equivalent heat capacity model and the rarefied vapor model are introduced to simulate the process of heat pipe startup from a frozen state. Numerical results indicate that the model of heat pipe can predict the operation of high-temperature heat pipe with an error of less than 1.04% at steady state and 23.4% during the startup. Based on the simulation results, the velocity of vapor decreases with the heat transfer power in the steady state and the velocity of potassium vapor is 58.4% of sodium vapor velocity under the same operation condition. During the startup, when the rarefied-continuum interface located in the end of the evaporator section, the maximum vapor velocity occurs which is 1.3 times of steady velocity. This work makes it possible to predict the operation of a high-temperature heat pipe and provides a reference for the design and application of HPR.

Analytical solutions for one-dimensional diabatic flows with wall friction

Abstract

Lie group solutions of advection-diffusion equations

Transport phenomena in homogeneous and inhomogeneous media are commonly encountered in many practical and industrial applications, which are modeled by advection-diffusion equations (ADEs) with constant or variable diffusivities, respectively. This paper provides a new perspective on how to solve advection-diffusion equations that model different transport phenomena in low Reynolds number flows. A mathematical description of the Lie group method is conducted first and then its potential in solving advection-diffusion equations for passive scalars transport with no-slip and no-flux boundary conditions is explored. The key step is to recast advection-diffusion equations as homogeneous diffusion processes on unimodular matrix Lie groups. Consequently, an approximate solution can be obtained from mean and covariance propagation techniques developed for diffusion equations on these Lie groups. The motivation to transform the advection-diffusion equation from Euclidean space to Lie groups is to exploit the available solutions of diffusion equation on these Lie groups so that the original equation can be solved in a simple way. In this paper, methodological details have been illustrated in solving ADEs modeling three kinds of transport phenomena. Two of them govern homogeneous transport and the solutions from mean and covariance propagation on the Lie group agree well with available results in published papers. We also use this method to solve more complicated ADEs governing inhomogeneous transport in one-dimensional compressible flows with spatially varying diffusivity, which is beyond the capabilities of existing approaches. The three real problems solved by the Lie group method illustrate the potential of this method. Instead of numerical calculations, the proposed closed-form method provides a simple alternative to study mass transfer encountered in various complex physical and industrial processes.

Numerical investigation on startup characteristics of high temperature heat pipe for nuclear reactor

Heat pipe cooled reactor (HPR) has a good adaptability to portable power system, which is popular in recent years. Due to the particularity of alkali-metal working fluid, the deep understanding of startup characteristics of high temperature heat pipe from frozen state is essential for the development of HPRs. In this paper, a three-stage frozen startup model is developed to describe the thermal behavior of heat pipe during the startup process, and the continuum flow in the vapor space is modeled as a one-dimensional compressible flow. A numerical code is carried out, in which the governing equations are discretized by Finite Element Method (FEM), and then the code is used to simulate the startup performance of a NaK heat pipe in HPRs. Numerical results indicate that the heat pipe startup behavior can be well described by the three-stage model. The NaK heat pipe is successfully started with a final consistency temperature of 834 K, although the entire second stage is restricted by the sonic limitation. The startup lasts 1550 s in total, and enters the second and third stage at 230 s and 650 s, respectively. After 1500 s, the maximum Mach number of vapor flow is lower than 0.1, which verifies the rationality of the one-dimensional compressible flow model of vapor. This work could provide a reference for the design and application of HPRs.

Exact solutions for quasi-one-dimensional compressible viscous flows in conical nozzles

Abstract

Development of a method of extended cells for the formulation of boundary conditions in numerical integration of gas dynamics equations in the domains of a curvilinear shape

A method of extended cells for the formulation of boundary conditions in the numerical integration of the Euler equations in domains with curvilinear boundaries for the cases of one-dimensional and two-dimensional compressible gas flows has been proposed in this study. The proposed method is based on the use of the explicit Godunov type finite volume method on a regular rectangular Cartesian grid. The essence of the extended cell method is that when integrating the basic equations of gas dynamics in a fractional cell, numerical fluxes are calculated through the sides of the new extended cell. This new cell is constructed tangentially to the curved boundary and has a size no less than the cell size in the regular domain. Parameters inside the new cell are calculated as mean integral values over the area of the neighboring regular cells included in it. In this case, when choosing a time step in accordance with the Courant condition, the stability of the method in the main computational domain is preserved both when integrating fractional cells and when integrating regular cells. Thanks to this approach, the proposed method has low requirements for computing resources, the ability to generalize for three-dimensional space and increase the order of accuracy without major modifications of the algorithm. To test the proposed method, solutions were obtained for the generally accepted test problems of gas dynamics: normal and double Mach reflection of a shock wave from a plane wall. The choice of the time step was made in accordance with the Courant condition in regular finite volumes. The results obtained have made it possible to assess the convergence of the proposed method and their comparison with the results of calculations using other methods have shown a good quantitative and qualitative agreement

Solution of a fractal energy integral operator without body force using measure of noncompactness

In this paper, we study the solution of fractal energy integral equation for one-dimensional compressible flows without body force using measure of noncompactness. We also discuss the solution of the local fractal equation of losing energy system using the notion of local fractal differential idea. For this, a new notion of χ-Δ-set contraction condition under simulation function is defined and two main fixed point and coupled fixed point results are obtained.