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

doi:10.1007/s10483-025-3324-7

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

2010 MSC Number:

O351.2

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

TrendMD

Figures/Tables 16

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Table 1

Properties of real fluids"

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

Table 1

Table 2

Initial values of RP calculation"

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

Table 2

Fig. 5

Fig. 6

Fig. 7

Table 3

Initial values of RP calculation under different test cases"

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

Table 3

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Table 4

Comparison of RP results for different EoSs"

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	$− 278.3$	$− 267.4$

Table 4

Table 5

Relative error in calculation of ideal gas and improved ideal gas"

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

Table 5

References 17

[18]	KUILA, S., RAJA SEKHAR, T., and ZEIDAN, D. A robust and accurate Riemann solver for a compressible two-phase flow model. Applied Mathematics and Computation, 265(2), 681–695 (2015)
[19]	KUILA, S. and SEKHAR, T. R. Riemann solution for one dimensional non-ideal isentropic magnetogasdynamics. Computational and Applied Mathematics, 35(1), 119–133 (2014)
[20]	WANG, J. C. H. and HICKEY, J. P. Analytical solutions to shock and expansion waves for non-ideal equations of state. Physics of Fluids, 32(8), 086105 (2020)
[21]	WANG, J. C. H. and HICKEY, J. P. A class of structurally complete approximate Riemann solvers for trans- and supercritical flows with large gradients. Journal of Computational Physics, 468, 111521 (2022)
[22]	WANG, J. C. H. Riemann Solvers with Non-Ideal Thermodynamics: Exact, Approximate, and Machine Learning Solutions, Ph. D. dissertation, University of Waterloo (2022)
[23]	KOUREMENOS, D. A. The normal shock waves of real gases and the generalized isentropic exponents. Forschung Im Ingenieurwesen, 52(1), 23–31 (1986)
[24]	SHAO, Z. Q. The Riemann problem for a traffic flow model. Physics of Fluids, 35(3), 036104 (2023)
[25]	SHAO, Z. Q. Almost global existence of classical discontinuous solutions to genuinely nonlinear hyperbolic systems of conservation laws with small BV initial data. Journal of Differential Equations, 254(7), 2803–2833 (2013)
[26]	SHAO, Z. Q. Global weakly discontinuous solutions to the mixed initial-boundary value problem for quasilinear hyperbolic systems. Mathematical Models and Methods in Applied Sciences, 19(7), 1099–1138 (2009)
[27]	SEDOV, L. I. Mechanics of a Continuous Medium, Nauka, Moscow (1973)
[28]	PENG, D. and ROBINSON, D. A new two-constant equation of state. Industrial & Engineering Chemistry Fundamentals, 15(1), 59–64 (1976)
[29]	REDLICH, O. and KWONG, J. On the thermodynamics of solutions; an equation of state; fugacities of gaseous solutions. Chemical Reviews, 44(1), 233–237 (1949)
[30]	LIOU, M. S., VAN LEER, B., and SHUEN, J. S. Splitting of inviscid fluxes for real gases. Journal of Computational Physics, 87(1), 1–24 (1990)
[31]	LANDAU, L. D. and LIFSHITZ, E. M. Fluid Mechanics, Pergamon Press, Oxford, 310–346 (1959)
[32]	GODUNOV, S. K. A finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Matematicheskii Sbornik, 47(3), 271–306 (1959)
[33]	NIST. NIST Chemistry WebBook (2023) http://webbook.nist.gov
[34]	MOHAMED, K. and PARASCHIVOIU, M. Real gas numerical simulation of hydrogen flows. 2nd International Energy Conversion Engineering Conference, Providence, Rhode Island, 16–19 (2004)
[35]	TERASHIMA, H. and KOSHI, M. Approach for simulating gas-liquid-like flows under supercritical pressures using a high-order central differencing scheme. Journal of Computational Physics, 231(20), 6907–6923 (2012)
[36]	LACAZE, G., SCHMITT, T., RUIZ, A., and OEFELEIN, J. C. Comparison of energy-, pressure- and enthalpy-based approaches for modeling supercritical flows. Computers & Fluids, 181, 35–56 (2019)
[37]	ABGRALL, R. How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach. Journal of Computational Physics, 125(1), 150–160 (1996)
[38]	XU, B. N., JIN, H. H., GUO, Y., and FAN, J. R. An adaptive primitive-conservative scheme for high speed transcritical flow with an arbitrary equation of state. arXiv Preprint, arXiv: 2206.11639 (2022)
[39]	MA, P. C., LV, Y., and IHME, M. An entropy-stable hybrid scheme for simulations of transcritical real-fluid flows. Journal of Computational Physics, 340, 330–357 (2017)
[1]	RIEMANN, B. Ueber die fortpflanzung ebener luftwellen von endlicher schwingungsweite. Abhandlungen der Köeniglichen Gesellschaft der Wissenschaften zu Göettingen, 8, 43–65 (1860)
[2]	TORO, E. F. Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer, Heidelberg (2009)
[3]	HICKEY, J. P. and IHME, M. Supercritical mixing and combustion in rocket propulsion. Center for Turbulence Research Annual Research Briefs 2023, Stanford University, 21–36 (2013)
[4]	POPP, M., HULKA, J., YANG, V., and HABIBALLAH, M. Liquid rocket thrust chambers. Aspects of Modeling, Analysis, and Design, 13–109 (2004)
[5]	SUN, M. B., ZHONG, Z., LIANG, J. H., and WANG, Z. G. Experimental investigation of supersonic model combustor with distributed injection of supercritical kerosene. Journal of Propulsion and Power, 30(6), 1537–1542 (2014)
[6]	LUO, S., XU, D., SONG, J., and LIU, J. A review of regenerative cooling technologies for scramjets. Applied Thermal Engineering, 190, 116754 (2021)
[7]	SHEARER, M. Nonuniqueness of admissible solutions of Riemann initial value problems for a system of conservation laws of mixed type. Archive for Rational Mechanics and Analysis, 93, 45–59 (1986)
[8]	MENIKOFF, R. and PLOHR, B. J. The Riemann problem for fluid flow of real materials. Reviews of Modern Physics, 61, 75–130 (1989)
[9]	TRUSKINOVSKII, L. M. Dynamics of non-equilibrium phase boundaries in a heat conducting non-linearly elastic medium. Journal of Applied Mathematics and Mechanics, 51(6), 777–784 (1987)
[10]	LIU, T. P. The entropy condition and the admissibility of shocks. Journal of Mathematical Analysis and Applications, 53(1), 78–88 (1976)
[11]	LANDAU, L. D. and LIFSHITZ, E. M. Statistical Physics, Pergamon Press, Oxford (1980)
[12]	COLELLA, P. and GLAZ, H. M. Efficient solution algorithms for the Riemann problem for real gases. Journal of Computational Physics, 59(2), 264–289 (1985)
[13]	GLAISTER, P. An approximate linearised Riemann solver for the Euler equations for real gases. Journal of Computational Physics, 74(2), 382–408 (1988)
[14]	SAUREL, R., LARINI, M., and LORAUD, J. C. Exact and approximate Riemann solvers for real gases. Journal of Computational Physics, 112(1), 126–137 (1994)
[15]	IVINGS, M. J., CAUSON, D. M., and TORO, E. F. On Riemann solvers for compressible liquids. International Journal for Numerical Methods in Fluids, 28(3), 395–418 (1998)
[16]	QUARTAPELLE, L., CASTELLETTI, L., GUARDONE, A., and QUARANTA, G. Solution of the Riemann problem of classical gasdynamics. Journal of Computational Physics, 190(1), 118–140 (2003)
[17]	SEKHAR, T. R. and SHARMA, V. D. Solution to the Riemann problem in a one-dimensional magnetogasdynamic flow. International Journal of Computer Mathematics, 89(2), 200–216 (2012)