Riemann problem for constant flow with single-point heating source

TL;DR

This paper analyzes the Riemann problem for Euler equations with a point heating source, proposing a novel double Riemann problems coupling method to find self-similar solutions and establish uniqueness under certain conditions.

Contribution

It introduces a new analytical framework using double Riemann problems to handle point heating sources in Euler equations, providing a self-similar solution structure and uniqueness results.

Findings

Three types of solutions identified

Solution is self-similar under the proposed framework

Uniqueness established with Mach number restrictions

Abstract

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 classic Riemann problems(CRPs) coupling, which treats the fluids on both sides of the heating point as two separate Riemann problems and then couples them. Under the double CRPs frame, the solution is self-similar, and only three types of solution 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.