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The limits of Riemann solutions to Euler equations of compressible fluid flow with a source term
Journal of Engineering Mathematics ( IF 1.3 ) Pub Date : 2020-10-22 , DOI: 10.1007/s10665-020-10066-3
Shouqiong Sheng , Zhiqiang Shao

In this paper, we investigate the limits of Riemann solutions to the Euler equations of compressible fluid flow with a source term as the adiabatic exponent tends to one. The source term can represent friction or gravity or both in Engineering. For instance, a concrete physical model is a model of gas dynamics in a gravitational field with entropy assumed to be a constant. The body force source term is presented if there is some external force acting on the fluid. The force assumed here is the gravity. Different from the homogeneous equations, the Riemann solutions of the inhomogeneous system are non self-similar. We rigorously proved that, as the adiabatic exponent tends to one, any two-shock Riemann solution tends to a delta shock solution of the pressureless Euler system with a Coulomb-like friction term, and the intermediate density between the two shocks tends to a weighted $$\delta $$ -mesaure which forms the delta shock; while any two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution of the pressureless Euler system with a Coulomb-like friction term, whose intermediate state between the two contact discontinuities is a vacuum state. Moreover, we also give some numerical simulations to confirm the theoretical analysis.

中文翻译:

具有源项的可压缩流体流动欧拉方程的黎曼解的极限

在本文中,我们研究了可压缩流体流动欧拉方程的黎曼解的极限,其中源项为绝热指数趋于 1。源项可以表示工程中的摩擦或重力或两者。例如,具体的物理模型是假设熵为常数的引力场中的气体动力学模型。如果有一些外力作用在流体上,则出现体力源项。这里假定的力是重力。与齐次方程不同的是,非齐次系统的黎曼解是非自相似的。我们严格证明,由于绝热指数趋于 1,任何二次激波黎曼解都趋于具有类库仑摩擦项的无压欧拉系统的 δ 激波解,并且两次冲击之间的中间密度趋向于加权 $$\delta $$ -mesaure,形成了 delta 冲击;而任何二稀疏波黎曼解都趋于具有类库仑摩擦项的无压欧拉系统的二接触不连续解,其两个接触不连续点之间的中间状态是真空状态。此外,我们还给出了一些数值模拟来证实理论分析。
更新日期:2020-10-22
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