Elsevier

Journal of Computational Physics

Volume 340, 1 July 2017, Pages 330-357
Journal of Computational Physics

An entropy-stable hybrid scheme for simulations of transcritical real-fluid flows

https://doi.org/10.1016/j.jcp.2017.03.022Get rights and content

Abstract

A finite-volume method is developed for simulating the mixing of turbulent flows at transcritical conditions. Spurious pressure oscillations associated with fully conservative formulations are addressed by extending a double-flux model to real-fluid equations of state. An entropy-stable formulation that combines high-order non-dissipative and low-order dissipative finite-volume schemes is proposed to preserve the physical realizability of numerical solutions across large density gradients. Convexity conditions and constraints on the application of the cubic state equation to transcritical flows are investigated, and conservation properties relevant to the double-flux model are examined. The resulting method is applied to a series of test cases to demonstrate the capability in simulations of problems that are relevant for multi-species transcritical real-fluid flows.

Introduction

The accurate and robust simulation of transcritical real-fluid effects is crucial for many engineering applications, such as fuel injection in internal-combustion engines, rocket motors and gas turbines. For example, in diesel engines, the liquid fuel is injected into the ambient gas at a pressure that exceeds its critical value, and the fuel jet will be heated to a supercritical temperature before combustion takes place. This process is often referred to as a transcritical injection (see Fig. 1). The largest thermodynamic gradient in the transcritical regime occurs as the fluid undergoes a liquid-like to a gas-like transition when crossing the pseudo-boiling line [1], which is shown by the black dashed line in Fig. 1. At elevated pressures, the mixture properties exhibit liquid-like densities and gas-like diffusivities, and the surface tension and enthalpy of vaporization approach zero [2]. This phenomenon was shown by recent experimental studies [1], [3], [4]. However, these complex processes are still not well understood experimentally and numerically. Therefore, to provide insights into high-pressure combustion systems, reliable numerical simulation tools are required for the characterization of supercritical and transcritical flows.

Due to the unique thermodynamic behavior and large gradients of thermodynamic quantities in the transcritical regime, several challenges must be overcome to enable the numerical prediction of transcritical flows. One challenge is the accurate description of thermodynamic and transport properties across the transcritical regime. Cubic equations of state (EoSs), such as Peng–Robinson (PR) EoS [5] and Soave–Redlich–Kwong (SRK) EoS [6], have been used extensively in transcritical and supercritical simulations for their acceptable accuracy and computational efficiency. Volume-translation methods for cubic EoS were applied to supercritical simulations to further improve the accuracy of thermodynamic descriptions [7]. More complex state equations are also available, such as the Benedict–Webb–Rubin (BWR) EoS [8], and state relations that are explicit in Helmholtz energy [9], but they often involve complicated implementation and high computational cost. Tabulation methods have been developed and applied to single-species simulations [10]; however, these approaches are generally not extendable to mixtures of more than two species due to the prohibitive memory usage.

Another obstacle in studying transcritical flows with variable thermodynamic properties is the generation of spurious pressure oscillations when a fully conservative scheme is adopted. This is similar to that of multicomponent compressible ideal gas flow calculations [11]. Due to strong nonlinearities of the thermodynamic system, this issue is more severe in transcritical real-fluid flow predictions. The problem cannot be mitigated by introducing numerical dissipation from using lower-order schemes or artificial viscosity, as pointed out in previous works [10], [12], [13], [14], [15]. Terashima and Koshi [13] solved a transport equation for pressure instead of the total energy equation in their finite-difference solver. This approach was inspired by the work of Karni [16] on multi-species calorically perfect gas, by which the pressure equilibrium can be maintained since pressure is explicitly solved for instead of derived from flow variables. However, for finite-volume schemes, it is not straightforward to solve the pressure equation since pressure is not a conserved quantity. Schmitt et al. [12] derived a quasi-conservative scheme by relating the artificial dissipation terms in the mass, momentum, and energy conservation equations and setting the pressure differential to zero. This procedure was later adopted by Ruiz [14]. The performance of this method to maintain pressure equilibrium across contact discontinuities was not reported. For calorically perfect gas flows, Johnson and Ham [17] adopted a scheme in which an auxiliary advection equation for the specific heat ratio was solved. Although this method is suitable for well-defined interfacial flows with inert species, it is not applicable for thermodynamically complex transcritical and reacting flows in which the thermodynamic properties are dependent on temperature and species compositions. Saurel and Abgrall [18] and Saurel et al. [19], [20] developed several methods for interfacial flows where the two phases are solved separately. Recently, Pantano et al. [21] formulated a numerical scheme for transcritical contact and shock problems, which introduced an additive non-conservative transport equation to maintain the mechanical equilibrium of pressure. Another approach originally developed for calorically perfect gas is the double-flux model. This method was proposed by Abgrall and Karni [11], extended by Billet and Abgrall [22] for reacting flows, and later formulated for high-order schemes [23], [24], [25]. This quasi-conservative method has been reported to correctly predict shock speeds even for very strong shock waves [11]. In the present work, the double-flux method will be modified and extended to transcritical flows for general real-fluid state equations.

Capturing large density gradients that exist between liquid-like and gas-like fluid phases, as well as diffusion effects at the interfaces subject to strong turbulence, is another challenge that requires consideration. A contact interface with a density ratio on the order of O(100) can be obtained in the transcritical regime. One way to address these large density gradients is to introduce artificial viscosity [13], [14], [26]. Another approach is to consider hybrid schemes, which combine a high-order non-dissipative flux with an upwind-biased flux to minimize numerical dissipation. For ideal gas flows, multiple applications have shown the performance for predicting turbulent flows using hybrid schemes [27], [28], [29]. However, the performance of hybrid schemes on transcritical real-fluid flows has not been evaluated. Entropy-stable concepts were shown to be related to the dissipation in numerical schemes [30]. It was shown that entropy-stable schemes can dampen numerical oscillations across the contact interface [31]. For high-order unstructured finite-volume schemes, it is generally not easy to fulfill the entropy-stability condition, and these schemes have so far not been applied to transcritical flows. By addressing this issue, the objective of this work is to develop a numerical method that is capable of simulating transcritical real-fluid turbulent flows robustly, accurately, and efficiently. Specifically, the following aspects will be addressed:

  • Analysis of the generation mechanism of spurious pressure oscillations in the context of transcritical flows and the extension of a double-flux approach [11] to real-fluid flows;

  • Formulation of a hybrid scheme using an entropy-stable flux to accurately represent large density gradients and to ensure numerical stability in application to turbulent flows;

  • Efficient implementation of the numerical schemes for practical applications of transcritical flows.

The remainder of this paper is structured as follows: Section 2 introduces the governing equations, followed by Section 3, providing the description of thermodynamic relations and transport properties related to transcritical fluids. Section 4 discusses spurious pressure oscillations related to the solution of fully conservative formulations, the development of the double-flux method for real-fluid flows, and the entropy-stable hybrid scheme to represent large density gradients in transcritical flows. In Section 5, a series of test cases is considered and results are compared with available data in literature to examine the capability of this scheme for dealing with real-fluid thermodynamics while maintaining robustness and high-order accuracy. The paper finishes with conclusions in Section 6.

Section snippets

Governing equations

The governing equations are the conservation of mass, momentum, total energy, and species, which take the following form:ρt+(ρu)=0,(ρu)t+(ρuu+pI)=τ,(ρet)t+((ρet+p)u)=(τu)q,(ρYk)t+(ρuYk)=(ρDkYk),fork=1,,NS1, where ρ is the density, u=(u,v,w)T is the velocity vector, p is the pressure, et is the specific total energy, Yk is the mass fraction of species k, Dk is the diffusion coefficient for species k, and NS is the number of species. The viscous stress tensor and heat

Equation of state

The numerical framework developed in this work is not limited to a single type of EoS. For computational efficiency and the accurate representation of the thermodynamic state near the critical point [35], the PR cubic EoS [5], [36] is used in this study. This state equation can be written as:p=RTvbav2+2bvb2, where R is the gas constant, v is the specific volume, and the coefficients a and b are dependent on temperature and composition to account for effects of intermolecular forces. Extended

Spurious pressure oscillations

One major issue that exists in numerical simulations of transcritical flows is the occurrence of spurious pressure oscillations. This is related to nonlinearities introduced by the real-fluids EoS. To understand the cause of these spurious pressure oscillations and its sensitivity in transcritical simulations, in the following, we consider the relation between internal energy and pressure. This analysis extends the pioneering work by Abgrall and Karni [11]. For a general multicomponent fluid,

Numerical tests

A series of test cases are considered to demonstrate the capability of the developed numerical scheme. All test cases are performed using an unstructured finite-volume code, CharlesX, which has been developed at the Center for Turbulence Research at Stanford University. In these test cases, flows with different species are considered, which are relevant for practical applications. The critical properties of the species considered in this study are listed in Table 1. Table 2 summarizes the

Conclusions

A finite-volume algorithm for simulating transcritical real-fluid mixing with large density gradients is presented. The spurious pressure oscillations that are present in transcritical simulations are due to large jumps in thermodynamic properties that arise from nonlinearities in the real-fluid EoS. Pressure oscillations are more severe than for ideal gas conditions and could cause the divergence of the simulation. A double-flux model is formulated for transcritical simulations by introducing

Acknowledgements

Financial support through NASA with award number NNM13AA11G and Army Research Laboratory with award number W911NF-16-2-0170 are gratefully acknowledged. The authors would like to thank Prof. Jean-Pierre Hickey for the help on this paper, Dr. Luis Bravo for helpful discussions, and Dr. Guilhem Lacaze for sharing the data for comparison in Section 5.5.

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