Exact solutions for the transcritical Riemann problem of two-parameter fluids

doi:10.1007/s10483-025-3324-7

摘要/Abstract

Abstract:

Transcritical and supercritical fluids widely exist in aerospace propulsion systems, such as the coolant flow in the regenerative cooling channels of scramjet engines. To numerically simulate the coolant flow, we must address the challenges in solving Riemann problems (RPs) for real fluids under complex flow conditions. In this study, an exact numerical solution for the one-dimensional RP of two-parameter fluids is developed. Due to the comprehensive resolution of fluid thermodynamics, the proposed solution framework is suitable for all forms of the two-parameter equation of state (EoS). The pressure splitting method is introduced to enable parallel calculation of RPs across multiple grid points. Theoretical analysis demonstrates the isentropic nature of weak waves in two-parameter fluids, ensuring that the same mathematical properties as ideal gas could be applied in Newton’s iteration. A series of numerical cases validate the effectiveness of the proposed method. A comparative analysis is conducted on the exact Riemann solutions for the real fluid EoS, the ideal gas EoS, and the improved ideal gas EoS under supercritical and transcritical conditions. The results indicate that the improved one produces smaller errors in the calculation of momentum and energy fluxes.

Key words: two-parameter fluid, Riemann problem (RP), exact solution, supercritical fluid, pressure splitting, transcritical fluid

中图分类号:

O351.2

. [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(12): 2385-2406.

Haotong BAI, Yixin YANG, Wenjia XIE, Dejian LI, Mingbo SUN. Exact solutions for the transcritical Riemann problem of two-parameter fluids[J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(12): 2385-2406.

图/表 16

Chemical formula	$R$ / $(J ⋅ kg − 1 ⋅ K − 1)$	$W$ / $(g ⋅ mol − 1)$	$T c$ /K	$p c$ /kPa	$ρ c$ / $(kg ⋅ m − 3)$	$ω$
$CO 2$	188.923	44.02	304.12	7 377.3	467	0.228
$N 2$	296.838	28.01	126.192	3 395.8	313	$− 0.220$
$H 2$	3949.85	2.015	33.145	1 296.4	31.26	0.040

Chemical formula, state	$p L$ /Pa	$p R$ /Pa	$ρ L$ / $(kg ⋅ m − 3)$	$ρ R$ / $(kg ⋅ m − 3)$	$u L$ / $(m ⋅ s − 1)$	$u R$ / $(m ⋅ s − 1)$
$CO 2,$ gas	$5 × 105$	$1 × 105$	5	1	0	0
$N 2,$ transcritical	$1.1 × 107$	$2 × 105$	180	7.4	150	50
$H 2,$ supercritical	$5 × 107$	$1 × 107$	32.26	7.81	0	0

Parameter	$p L$ /Pa	$p R$ /Pa	$ρ L$ / $(kg ⋅ m − 3)$	$ρ R$ / $(kg ⋅ m − 3)$	$T L$ /K	$T R$ /K	$u L$ / $(m ⋅ s − 1)$	$u R$ / $(m ⋅ s − 1)$
Case 1	$7 × 105$	$1 × 105$	7	1	531.5	529.6	0	0
Case 2	$1.5 × 107$	$5 × 106$	200	60	457.9	465.4	0	0
Case 3	$1.5 × 107$	$1.5 × 107$	200	200	457.9	457.9	50	$− 50$
Case 4	$1.5 × 107$	$1.0 × 107$	400	800	389.5	296.3	0	0

Case	EoS	$ρ u$ / $(kg ⋅ m ⋅ s − 1)$	$(ρ u 2 + p)$ /kPa	$(ρ e t + p) u$ / $(kJ ⋅ m − 2 ⋅ s − 1)$	Shock speed/ $(m ⋅ s − 1)$	Experimental wave head/ $(m ⋅ s − 1)$	Experimental wave tail/ $(m ⋅ s − 1)$
2	P-R	17 479	$1.058 6 × 104$	$9.747 4 × 104$	410.1	$− 328.1$	$− 156.2$
	Ideal gas	17 394	$1.082 4 × 104$	$9.264 7 × 104$	405.9	$− 303.0$	$− 148.9$
	Improved	16 635	$1.043 8 × 104$	$9.494 5 × 104$	404.0	$− 324.9$	$− 175.0$
3	P-R	0	$1.859 5 × 104$	0	309.5	–	–
	Ideal gas	0	$1.831 1 × 104$	0	281.1	–	–
	Improved	0	$1.838 7 × 104$	0	288.7	–	–
4	P-R	8 346.1	$1.750 1 × 104$	$3.819 5 × 106$	406.6	$− 318.8$	$− 278.3$
4	Improved	9 829.4	$1.717 5 × 104$	$4.552 2 × 106$	313.1	$− 278.3$	$− 267.4$

Case	EoS	$Δ (ρ u)$ /%	$Δ (ρ u 2 + p)$ /%	$Δ ((ρ e t + p) u)$ /%	Error (shock speed)/%	Error (experimental head)/%	Error (experimental tail)/%
2	Ideal gas	0.49	2.25	4.95	1.02	7.65	8.30
2	Improved	4.83	1.40	2.59	1.49	0.98	12.2
3	Ideal gas	0	1.53	0	9.18	–	–
3	Improved	0	1.12	0	6.72	–	–
4	Improved	15.09	1.86	19.18	23.0	12.7	3.92

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