Solutions Of Gas Dynamics Research Articles

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

Abstract

The comparison of the Riemann solutions in gas dynamics

In this paper, we compare two Riemann solution constructions of isentropic and non-isentropic gas dynamic equations, with the same initial values. We obtain that when rarefaction waves occur in the non-isentropic solution, the same family rarefaction waves occur in the isentropic solution, however, when advancing or reflected shock waves occur in the isentropic case, the same kind shock waves occur in the non-isentropic case. Moreover, by numerical calculation, it can be found that the wave intensities of the two solutions are quite close. Our conclusion reflects that almost non-isentropic gas dynamic process can be simulated by isentropic gas dynamic process in practice.

Electromagnetic Waves from TNT Explosions

Experimental observations indicate that electromagnetic (EM) radiation is emitted after the detonation of high explosives (HE) charges. The movement of ionized atoms, particles and electrons seems to be the underlying cause. Expansion of the detonation products (DP) drives a strong (~1 kb) shock in surrounding air. This forms an intense thermal wave (T ~11,000 K) with duration of ~20 microseconds. Such temperatures create significant ionization of the air. According to Ohm’s Law, movement of ionized patches generates current; and according to the Biot-Savart Law, such currents induce electric and magnetic fields. We investigate these effects through numerical simulations of TNT explosions. A high-order Godunov scheme is used to integrate the one-dimensional conservation laws of gasdynamics. An extremely fine grid (10 microns) was needed to get converged temperature and conductivity profiles. The gasdynamic solution provided a source current, which was fed into a time-domain Green’s function code to predict three-dimensional electromagnetic waves emanating from the TNT explosion. This analysis clearly demonstrates one mechanism—the Boronin current—as the source of EM emissions from TNT explosions, but other mechanisms are also possible.

Partially invariant solutions in gas dynamics and implicit equations

This paper studies a nonbarochronic, regular, partially invariant solution (submodel) of rank one and defect two to the equations of gas dynamics which describes spatial unsteady gas motion. The equations of gas dynamics are reduced to an implicit ordinary differential equation of the first order for an auxiliary function and to an integrable system. A complete classification of the irregular singular points of the key equation according to a parameter characterizing the gas flow is given, and transformations of the irregular singular points with variation in the parameter are obtained. Qualitative properties of the solution are investigated and physically interpreted in terms of gas motion. It is shown that there are two modes of motion, one of which is supersonic, and in the second modes, a continuous transition through the speed of sound is possible.

Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics

Progress in computer technology now allows the application of numerical methods to some practical multidimensional steady and unsteady gas flows. While selecting a numerical method, preference should be given to higher-order schemes, since they allow the study of the fine features of the solution and the reduction of the computation time. In general, however, solutions of non-stationary nonlinear equations of gas dynamics are not smooth and may include strong discontinuities, such as shock waves or contact discontinuities. This fact is important for choosing an appropriate numerical scheme. If the strong discontinuities are extracted from the simulation domain and traced (shock fitting), while the rest of the flow is smooth, it is possible to use numerical schemes with approximation close to the second order [1-3]. However, if the number of discontinuities increases with time leading to a complicated pattern, the application of the shock-fitting approach becomes difficult. In this case it is preferable to apply shock-capturing schemes [4-6], which have a lower order of approximation but allow the integration of the governing equations through the discontinuities without extracting their surfaces. Shock-capturing schemes of a higher order of approximation are currently under development [7-9]. In this paper, certain aspects of constructing a numerical scheme applicable to discontinuous flows of gas dynamics are discussed using the idea of minimizing the derivatives of the solution. A numerical scheme that is second-order accurate in space and first-order accurate in time is developed for the model equation ut+ux=0. The properties of this scheme are analyzed and the conditions for its stability and monotonicity are formulated. These results are then used in devising a similar scheme for the non-stationary one-dimensional equations of gas dynamics. The performance of the new scheme is compared to that of other schemes, using the break-up of an initial discontinuity (Riemann problem) as an example.

A historical oversight: Vladimir P. Kolgan and his high-resolution scheme

The Russian landmark paper “Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics” by Vladimir P. Kolgan is discussed in a historical–technical perspective. The 1972 paper already featured a Godunov-type scheme for the Euler equations with second-order spatial accuracy, and a limiter to make it monotonicity-preserving. The work remained little known within and completely unknown outside the USSR, largely because of Kolgan’s untimely death in 1978. To honor Kolgan’s contribution to CFD, an English translation of his paper is included in this issue of the Journal of Computational Physics. Facts about Kolgan’s life are presented here, and an appraisal is given of the originality of his ideas.

THE K-VERSION OF FINITE ELEMENT METHOD FOR NONLINEAR OPERATORS IN BVP

In the companion papers [1,2], authors introduced the concepts of k-version of finite element method and k, hk, pk, hkp-processes of the finite element method for boundary value problems described by self-adjoint and non-self adjoint operators using Ĥk,p(Ω) spaces with specific details including numerical studies for weak forms and least square processes. It was demonstrated that a variationally consistent (VC) weak form is possible when the differential operator is self-adjoint, however, in case of non-self-adjoint operators the weak forms are variationally inconsistent (VIC) which lead to degenerate computational processes that can produce spurious oscillations in the computed solutions. In this paper we demonstrate that when the boundary value problems are described by non-linear differential operators, Galerkin processes and weak forms can never be variationally consistent and hence result in degenerate computational processes and suffer from same problems as in the case of non-self-adjoint operators ...

New modelling of a large long-duration X-ray flare on UX Ari

We have carried out a new gas-dynamic simulation of the physical processes occurring in a fixed amount of plasma confined in coronal loops heated in the top. We chose the solution which explains both the time behaviour of the temperature and the energetics of the X-ray emission of a long-duration X-ray flare observed by BeppoSAX on UX Ari. This analysis makes it possible to obtain the length of the loop and the distribution of the temperature and 'density along the loop (with T max ≈ 100 MK and n e = (1 – 3) · 10 11 cm −3), as well as to estimate the X-ray flare area S ≈ 1022 cm 2. Franciosini et al. (2001) have developed a time-dependent model of magnetic reconnection for the decay phase of the same flare. Our gas-dynamic solution corresponds to reconnection of middle-scale magnetic fields which in the model by Franciosini et al. (2001) corresponds to a degree n of the Legendre polynomial close to 5. The magnetic field strength is estimated to be about 100 G near the loop top. We argue that the shrinkage-effect plays a more important role during large long-duration X-ray stellar flares than in solar two-ribbon flares.

A spectral Chebyshev method for linear stability analysis of one-dimensional exact solutions of gas dynamics

We present a spectral numerical method for solving one-dimensional systems of partial differential equations (PDEs) which arise from linearization of the Euler equations about an exact solution depending on space and time. A two-domain Chebyshev collocation method is used. Matching of quantities is performed in the space of characteristic variables as suggested by Kopriva [Appl. Numer. Math. 2 (1986) 221; J. Comput. Phys. 125 (1996) 244]. Time-dependent boundary conditions are handled following an approach proposed by Thompson [J. Comput. Phys. 68 (1987) 1; 89 (1990) 439]. An exact numerical stability analysis valid for any explicit three-step third-order non-degenerate Runge–Kutta scheme is provided. The numerical method is tested against exact solutions for the three fundamental modes of a compressible flow (entropy, vorticity and acoustic modes).

Observations of bow shock and magnetopause crossings from geosynchronous orbit on 31 March 2001

On 31 March 2001, significant enhancements in both the strength of the interplanetary magnetic field and the solar wind dynamic pressure pushed the magnetopause and bow shock inside of geosynchronous orbit. We present here observations of the unshocked solar wind observed with the Los Alamos magnetospheric plasma analyzer on geosynchronous satellite 1994‐084 near local noon. The magnetosheath was observed intermittently at geosynchronous satellites 1991‐080, 1994‐084, and LANL‐01A, across much of the dayside magnetosphere. Comparisons with theory of the observed bow shock and magnetopause locations at geosynchronous orbit near noon are in reasonable agreement. Using nearly simultaneous observations of the magnetopause and bow shock locations at 0451 UT, we have tested models for the thickness of the magnetosheath and flaring of the magnetopause at that time. Using the least flared of the magnetopause models, the estimated thickness of the magnetosheath would be thicker than would be predicted by either the gas dynamic solution of Spreiter et al. [1966] or the MHD simulations of Cairns and Lyon [1996]. Alternatively, placing the location of the magnetopause at the nose of the magnetosphere in conformance with the thickness of the magnetosheath from Cairns and Lyon [1996] requires less flaring on the flanks of the magnetosphere than currently present in the empirical models. The models also have difficulty predicting the other observed magnetopause crossings far from local noon apparently due to excessive flaring of the models for the conditions on this day.

Adaptive multiresolution for finite volume solutions of gas dynamics

Multiresolution analysis is used to improve the CPU and memory performance of a finite volume scheme. Departing from Harten's original scheme we present a fully adaptive scheme in the sense that at a given time, the solution is represented in a compressed form by a set of significant wavelet coefficients. Numerical benchmarks for Euler's system of compressible gas dynamics are performed on triangular meshes.

Non‐weak/strong solutions in gas dynamics: a C11p‐version STLSFEF in Eulerian frame of reference using ρ, u, p primitive variables

AbstractThis paper presents a space–time least squares finite element formulation of one‐dimensional transient Navier–Stokes equations (governing differential equations: GDE) for compressible flow in Eulerian frame of reference using ρ, u, p as primitive variables with C11 type p‐version hierarchical interpolations in space and time. Time marching procedure is utilized to compute time evolutions for all values of time. For high speed gas dynamics the C11 type interpolations in space and time possess the same orders of continuity in space and time as the GDE. It is demonstrated that with this approach accurate numerical solutions of Navier–Stokes equations are possible without any assumptions or approximations. In the approach presented here SUPG, SUPG/DC, SUPG/DC/LS operators are neither used nor needed. Time accurate numerical simulations show resolution of shock structure (i.e. shock speed, shock relations and shock width) to be in excellent agreement with the analytical solutions. The role of diffusion i.e. viscosity (physical or artificial) and thermal conductivity on shock structure is demonstrated. Riemann shock tube is used as a model problem. True time evolutions are reported beginning with the first time step until steady shock conditions are achieved. In this approach, when the computed error functionals become zero (computationally), the computed non‐weak solutions have characteristics as those of the strong solutions of the gas dynamics equations. Copyright © 2001 John Wiley & Sons, Ltd.

Non‐weak/strong solutions in gas dynamics: a C11p‐version STLSFEF in Lagrangian frame of reference using ρ, u, p primitive variables

AbstractIn this paper, we present computations of the non‐weak solutions of class C11 of transient Navier–Stokes equations for compressible flow in Lagrangian frame of reference using space‐time least squares finite element formulation with primitive variables ρ, u, p. For high speed compressible flows the solutions reported here possess the same orders of continuity as the governing differential equations. The role of diffusion i.e. viscosity (physical or artificial) and thermal conductivity on shock structure is demonstrated. Compression of air in a rigid cylinder by a rigid, massless and frictionless piston is used as a model problem. True time evolutions of class C11 are reported beginning with the first time step until steady shock conditions are achieved. Comparisons with analytical solutions are presented when possible. Copyright © 2001 John Wiley & Sons, Ltd.

Numerical consideration of quasilinear electron cloud dynamics in plasma

The dynamics of a hot electron cloud in the solar corona-like plasma based on the numerical solution of kinetic equations of weak turbulence theory is considered. Different finite difference schemes are examined to fit the exact analytical solutions of quasilinear equations in hydrodynamic limit (gas-dynamic solution). It is shown that the scheme suggested demonstrates correct asymptotic behavior and can be employed to solve initial value problems for an arbitrary initial electron distribution function.

Self-similar solutions in gas dynamics with exponential time dependence

Self-similar gas flow with similarity coordinate ξ=re∓t, exponential in time, is investigated for plane, cylindrical and spherical geometry. The underlying Lie group symmetry is pointed out and also how it is obtained from flow with power-law self-similarity (ξ=r/|t|α) in the limit α→∞. Six distinct types of solutions are derived and plotted in r−t coordinates. They are characterized by steep density gradients. Also, solutions with strong shock boundaries are discussed. A completely analytical solution is presented, related to Sedov’s solution of a strong point explosion.

On the proper Mach number and ratio of specific heats for modeling the Venus bow shock

When considering local solutions for the jump in plasma conditions across planetary bow shocks, a magnetohydrodynamic (MHD) formalism of the Rankine‐Hugoniot relations is used. However, when planetary shocks are modeled on a global scale, gas dynamic (GD) treatments are frequently employed which neglect the magnetic field effect on the shock structure and the flowing plasma. These treatments require the specification of a free‐stream Mach number and a ratio of specific heats. Herein observations of the Venus bow shock are used to illustrate how closely one can approximate the observed magnetic field jump with MHD and GD treatments if one chooses a particular Mach number and ratio of specific heats. As expected theoretically, it is found that the observed magnetosonic Mach number (and not the sonic Mach number) when used in the GD calculations gives the best approximate solutions to the MHD problem using reasonable value for the ratio of specific heats. At Venus we find that the best value for the ratio of specific heats (γ) is 1.85 (rather than 5/3 or 2), although this ratio depends on the Alfvenic Mach number MA and the angle between the shock normal and the upstream field, θBN. We cannot determine how well the GD solution reproduces the shock shape at Venus, because of the possible influence of both MHD and mass‐loading effects. In fact, the Venus bow shock is much farther away from the planet at the terminator than the GD model would predict for either γ = 5/3 or 2. Thus we must depend on terrestrial studies to determine how best to approximate shock shape with GD modes.

Planetary Mach cones: Theory and observation

This study uses observations by a number of spacecraft to investigate the asymptotic behavior of planetary bow shocks. Toward this end a single standard method has been used to model distant bow shock position and shape. Mach cone angles of 13.9±2°, 11.4±3°, and 8.1±4° at Venus, Earth, and Mars, respectively, were determined from the observational shock models. These cone angles and their decrease with growing distance from the sun are consistent with downstream bow shock position being limited by the MHD fast mode Mach number. Gas dynamic solutions for solar wind flow about Venus, Earth, and Mars were computed up to 50 ROB (i.e., obstacle radii) behind each planet and compared with observed bow shock location. In each case the position of the shock was well predicted up to a certain distance downstream: −4 ROB at Venus, −6 ROB at Earth, and −10 ROB at Mars. Beyond this point the observed shock position lies farther from the aberrated sun‐planet line than the gas dynamic model with the discrepancy greatest at Venus and least at Mars. The better agreement between gas dynamic theory and observation with growing distance from the sun is attributed to an increase in the accuracy of the gas dynamic approximation with decreasing IMF strength.

Hydraulic Analogy for Isentropic Flow Through a Nozzle

Modelling aspects of isentropic compressible gas flow using hydraulic analogy are discussed. Subsonic and supersonic flows through a typical nozzle are simulated as free surface incompressible water flow in an equivalent 2-D model on a water table. The results are first compared for the well known classical analogy in order to estimate experimental errors. Correction factors for pressure and temperature, to account for non-ideal compressible gas flow are presented and the results obtained on the water table are modified and compared with gas dynamic solution. Within the experimental errors, it is shown that the hydraulic analogy can be used as an effective tool for the study of two dimensional isentropic flows of gases.

Aligned magnetohydrodynamic solution for solar wind flow past the earth's magnetosphere

Exact numerical solutions of the magnetohydrodynamic equations for a perfect dissipation-less gas with aligned magnetic field are given for conditions representative of steady supersonic solar wind flow past an axisymmetric model of the earth's magnetosphere. The solution is based on use of a transformation that relates without approximation the equations of magnetohydrodynamics to those of gasdynamics of a pseudo gas that has an unusual equation of state. The results confirm the applicability of the previously existing gasdynamic solutions for the typically modest intensities of the interplanetary magnetic field that lead to Alfvén Mach numbers of about 10 or greater. For smaller values, however, significant effects are indicated with the flanks of the bow wave moving away from the earth and the nose moving toward the earth. The results are consistent with direct observations in space.

Kinetic Theory of the Piston Problem by the Bhatnagar-Gross-Krook Equation

The formation of shock and expansion waves by a moving piston is studied according to the kinetic theory of gases. A piston is impulsively moved at a constant but arbitrary speed with a diffuse reflection on either isothermal or adiabatic piston wall. The Bhatnagar-Gross-Krook equation has been solved by applying the numerical method proposed by Chu for the shock tube problem. The waves are developed through intermolecular collisions for various piston speeds (Mach number 0.438, 0.876, and 1.314). The properties across the shock wave agree quite well with the Rankine-Hugoniot relations. A kinetic layer of several mean free paths near the isothermal piston wall is clearly observed. A temperature jump at the isothermal wall is also developed. Solutions for the adiabatic wall with zero temperature gradient are obtained and compared very well with the gas dynamic solutions. Temperature overshoots (Tx) inside the shock waves have been observed for higher piston speeds (Mach number 0.876 and 1.314).