Self-similar generalized Riemann problems for the 1-D isothermal Euler system
Self-similar generalized Riemann problems for the 1-D isothermal Euler system
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DOI：
10.1007/s00033-021-01505-x
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发表时间：
2021-03
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影响因子：
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通讯作者：
H. Jenssen;Yushuang Luo
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作者：
H. Jenssen;Yushuang Luo
本文研究一维等温可压缩气体动力学欧拉方程组的自相似解。对于每个β ∈ R β∈ R，系统存在形式为ρ（t，x）= t^ β Ω（Ω）\qquad u（t，x）= U（Ω）\qquad\qquad\textstyle Ω = x t，ρ（t，x）= t β Ω（Ω）u（t，x）= U（Ω）Ω = xt的解，其中ρ ρ和u表示密度场和速度场。Ω Ω和U的常微分方程可以隐式求解，并得到初始值为ρ（0，x）=\left {R_l}的广义Riemann问题的解|X| ^ β & x<0\R_0 ^ β & x> 0。\ qquad u（0，x）=| left {U_l & x< 0\U_r & x> 0，.ρ（0，x）= R l| X| β x< 0 R ∈ β x> 0 u（0，x）= Ulx < 0 U rx>0，其中R_1，I，R_r> 0 R_1，R_r> 0，U_1，I U_r U_1，U_r是任意常数.对于β ∈（-1，0）β∈（-1，0），数据在x= 0 x= 0处局部可积但无界，而对于β ∈（0，1）β∈（0，1），数据在x= 0 x= 0处局部有界连续但梯度无界.通过选择足够大的β> 1 β> 1和U_l= U_r U_l = U_r，数据的任何（有限）光滑度都是可能的。(Theβ ≤-1 β≤-1是非物理的，因为初始密度不是局部可积的，在本工作中没有处理。β= 0 β= 0的情况对应于标准黎曼问题，其解是向后和向前激波以及稀疏波的组合。相反，对于β ∈（-1，∞）\smallsetminus {0\} β∈（-1，∞）{0，我们构造了自相似解，并证明了它总是包含两个激波.这些必须在时间0+ 0+生成，并沿沿着直线分开。我们提供了一个物理解释的解决方案的结构和描述的行为的解决方案中出现的楔之间的冲击波。
We consider self-similar solutions to the 1-dimensional isothermal Euler system for compressible gas dynamics. For each β ∈ R β∈ R, the system admits solutions of the form ρ (t, x)= t^ β Ω (ξ)\qquad u (t, x)= U (ξ)\qquad\qquad\textstyle ξ= x t, ρ (t, x)= t β Ω (ξ) u (t, x)= U (ξ) ξ= xt, where ρ ρ and u denote the density and velocity fields. The ODEs for Ω Ω and U can be solved implicitly and yield the solution to generalized Riemann problems with initial data ρ (0, x)=\left {R_l| x|^ β & x< 0\R_rx^ β & x> 0.\qquad u (0, x)=\left {U_l & x< 0\U_r & x> 0,. ρ (0, x)= R l| x| β x< 0 R rx β x> 0 u (0, x)= U lx< 0 U rx> 0, where R_l,\, R_r> 0 R l, R r> 0 and U_l,\U_r U l, U r are arbitrary constants. For β ∈ (-1, 0) β∈(-1, 0), the data are locally integrable but unbounded at x= 0 x= 0, while for β ∈ (0, 1) β∈(0, 1), the data are locally bounded and continuous but with unbounded gradients at x= 0 x= 0. Any (finite) degree of smoothness of the data is possible by choosing β> 1 β> 1 sufficiently large and U_l= U_r U l= U r.(The case β ≤-1 β≤-1 is unphysical as the initial density is not locally integrable and is not treated in this work.) The case β= 0 β= 0 corresponds to standard Riemann problems whose solutions are combinations of backward and forward shocks and rarefaction waves. In contrast, for β ∈ (-1, ∞)\smallsetminus {0\} β∈(-1,∞){0, we construct the self-similar solution and show that it always contains exactly two shock waves. These are necessarily generated at time 0+ 0+ and move apart along straight lines. We provide a physical interpretation of the solution structure and describe the behavior of the solution in the emerging wedge between the shock waves.