Four-Quadrant Riemann Problem for a 2×2 System II

In previous work, we considered a four-quadrant Riemann problem for a 2×2 hyperbolic system in which delta shock appears at the initial discontinuity without assuming that each jump of the initial data projects exactly one plane elementary wave. In this paper, we consider the case that does not involve a delta shock at the initial discontinuity. We classified 18 topologically distinct solutions and constructed analytic and numerical solutions for each case. The constructed analytic solutions show the rich structure of wave interactions in the Riemann problem, which coincide with the computed numerical solutions.

Decomposition of the switchback boundary on MHD wave modes.

<p>Switchback boundaries separate two plasmas moving with different velocities, that may have different temperatures and densities and typically manifest sharp magnetic field deflections through the boundary. They may be analyzed similarly to MHD discontinuities. The first step of their characterization consists of analysis in terms of MHD discontinuities. Such an analysis was performed by Larosa et al., (2021) who has found that 32% of them may be attributed to rotational discontinuities, 17% to tangential, about 42% to the group of discontinuities that are difficult to unambiguously define whether they are tangential or rotational, and 9% that do not belong to any of these two groups. We describe and apply hereafter for two events another approach for the characterization of the boundaries based on classification of the general type discontinuity in MHD approximation. It is based on the problem of the decay of the general type of discontinuity. It is well known [Kulikovsky and Lyubimov, 1962, Gogosov, 1959} that general type MHD discontinuity decays on 7 separate discontinuities belonging to different types of MHD waves, namely, entropic wave, two slow mode waves, two Alfvenic waves, and two fast mode waves. Entropic wave is standing in the reference frame of the discontinuity; other wave modes are supposed to run in the opposite directions from the initial discontinuity with their characteristic velocities. Making use of plasma parameters from two sides of the boundary one can evaluate the fraction of each wave mode present in the discontinuity. We apply this method to two boundary crossings. This repartition of the discontinuity allows characterizing the deviation from Alfvenicity quantitatively.</p><p>References</p><p>Larosa, A., et al., A&A, 2021, (accepted)</p><p>Kulikovsky, Lyubimov, Magnetohydrodynamics, (1962)</p><p>Gogosov, V.V., Decay of the MHD discontinuity, (1959)</p>

Four-Quadrant Riemann Problem for a 2×2 System Involving Delta Shock

In this paper, a four-quadrant Riemann problem for a 2×2 system of hyperbolic conservation laws is considered in the case of delta shock appearing at the initial discontinuity. We also remove the restriction in that there is only one planar wave at each initial discontinuity. We include the existence of two elementary waves at each initial discontinuity and classify 14 topologically distinct solutions. For each case, we construct an analytic solution and compute the numerical solution.

Second-Order Symmetric Smoothed Particle Hydrodynamics Method for Transient Heat Conduction Problems with Initial Discontinuity

To eliminate the numerical oscillations appearing in the first-order symmetric smoothed particle hydrodynamics (FO-SSPH) method for simulating transient heat conduction problems with discontinuous initial distribution, this paper presents a second-order symmetric smoothed particle hydrodynamics (SO-SSPH) method. Numerical properties of both SO-SSPH and FO-SSPH are analyzed, including truncation error, numerical accuracy, convergence rate, and stability. Experimental results show that for transient heat conduction with initial smooth distribution, both FO-SSPH and SO-SSPH can achieve second order convergence rate, which is consistent with the theoretical analysis. However, for one- and two-dimensional conduction with initial discontinuity, the FO-SSPH method suffers from serious unphysical oscillations, which do not disappear over time, and hence it only achieves a first-order convergence rate; while the present SO-SSPH method can avoid unphysical oscillations and has second-order convergence rate. Therefore, the SO-SSPH method is a feasible tool for solving transient heat conduction problems with both smooth and discontinuous distributions, and it is easy to be extended to high dimensional cases.

Buoyancy-Driven Rayleigh–Taylor Instability in a Vertical Channel

Suppose that a vertical tube is composed of two chambers that are separated by a retractable thermally insulated thin membrane. The upper and lower chambers are filled with an incompressible fluid and maintained at temperatures {T_{c}} and {T_{h}}>{T_{c}}, respectively. Upon removal of the membrane, the two fluid masses form an unstably stratified Rayleigh–Taylor-type configuration with cold and heavy fluid overlying a warmer and lighter fluid and separated by an interface across which there is a discontinuity in the density. Due to the presence of an initial discontinuity between two homogeneous states, this problem is mathematically homologous to that of the shock tube problem with the thermal expansion playing the role of pressure. When the two fluid regions are brought directly into contact with each other and the transient interfacial fluctuations have subsided, we show the emergence of a stationary state of convection through a supercritical bifurcation provided a threshold value for the temperature difference is exceeded. We suggest a possible way for the experimental testing of the theoretical results put forth in this paper.

A modification of the CABARET scheme for the computation of multicomponent gaseous flows

Предложен явный численный алгоритм для расчета одномерного движения смеси идеальных газов. Приведены физическая модель и уравнения движения смеси в консервативной и характеристической формах. Дискретизация уравнений движения выполнена по методике Кабаре. Предложенный численный алгоритм испытан на решении задачи о распаде разрыва с различными газами по разные стороны разрыва. Произведено сравнение численных решений с аналитическим, а также с решениями, полученными по другим численным методикам. Показано, что предложенный алгоритм демонстрирует высокую точность решений на рассмотренном классе задач. An explicit numerical algorithm for computing one-dimensional motion of multicomponent gaseous mixtures is proposed. A physical model and the equations of motion are presented in conservative and characteristic forms. The discretization of the governing equations is made in accordance with the CABARET (Compact Accurately Boundary Adjusting-REsolution Technique) approach. The proposed algorithm is tested on the Riemann problem with different gases on the different sides of the initial discontinuity. The resulting numerical solutions are compared with the analytical one and with those obtained by other numerical approaches. It is shown that the proposed algorithm is of high accuracy on the class of problems being considered.

Study on the Initial Discontinuity State Distribution of LY12CZ Aluminum Alloy

The initial discontinuity state (IDS) concept was developed just several years ago in an attempt to describe the as-manufactured or as-produced state of material. As a geometric and material characteristic, IDS is the major parameter of holistic life assessment methodology which establishes the initial analysis condition. In this paper, a number of IDS values were determined for several types of smooth specimens. A crack growth analysis software was used to develop the IDS. The results obtained indicate that the calculated IDS values scatter widely. A statistical analysis shows that the distribution of the calculated IDS values fit well with Weibull distribution.