Summary
This paper describes the implementation of a numerical solver that is capable of simulating compressible flows of nonideal single‐phase fluids. The proposed method can be applied to arbitrary equations of state and is suitable for all Mach numbers. The pressure‐based solver uses the operator‐splitting technique and is based on the PISO/SIMPLE algorithm: the density, velocity, and temperature fields are predicted by solving the linearized versions of the balance equations using the convective fluxes from the previous iteration or time step. The overall mass continuity is ensured by solving the pressure equation derived from the continuity equation, the momentum equation, and the equation of state. Nonphysical oscillations of the numerical solution near discontinuities are damped using the Kurganov‐Tadmor/Kurganov‐Noelle‐Petrova (KT/KNP) scheme for convective fluxes. The solver was validated using different test cases, where analytical and/or numerical solutions are present or can be derived: (1) A convergent‐divergent nozzle with three different operating conditions; (2) the Riemann problem for the Peng‐Robinson equation of state; (3) the Riemann problem for the covolume equation of state; (4) the development of a laminar velocity profile in a circular pipe (also known as Poiseuille flow); (5) a laminar flow over a circular cylinder; (6) a subsonic flow over a backward‐facing step at low Reynolds numbers; (7) a transonic flow over the RAE 2822 airfoil; and (8) a supersonic flow around a blunt cylinder‐flare model. The spatial approximation order of the scheme is second order. The mesh convergence of the numerical solution was achieved for all cases. The accuracy order for highly compressible flows with discontinuities is close to first order and, for incompressible viscous flows, it is close to second order. The proposed solver is named rhoPimpleCentralFoam and is implemented in the open‐source CFD library OpenFOAM®. For high speed flows, it shows a similar behavior as the KT/KNP schemes (implemented as rhoCentralFoam‐solver, Int. J. Numer. Meth. Fluids 2010), and for flows with small Mach numbers, it behaves like solvers that are based on the PISO/SIMPLE algorithm.
A numerical framework implemented in the open-source tool OpenFOAM is presented in this work combining a hybrid, pressure-based solver with a vapor-liquid equilibrium model based on the cubic equation of state. This framework is used in the present work to investigate underexpanded jets at engine-relevant conditions where real-gas effects and mixture induced phase separation are probable to occur. A thorough validation and discussion of the applied vapor-liquid equilibrium model is conducted by means of general thermodynamic relations and measurement data available in the literature. Enginerelevant simulation cases for two different fuels were defined. Analyses of the flow field show that the used fuel has a first order effect on the occurrence of phase separation. In the case of phase separation two different effects could be revealed causing the single-phase instability, namely the strong expansion and the mixing of the fuel with the chamber gas. A comparison of single-phase and two-phase jets disclosed that the phase separation leads to a completely different penetration depth in contrast to single-phase injection and therefore commonly used analytical approaches fail to predict the penetration depth.
In the present work, numerical simulations are carried out to investigate underexpanded methane jets with phase separation effects. In order to predict the fuel injection and the mixture formation in the constant volume chamber, a hybrid, pressure-based solver is combined with a vapor-liquid equilibrium model and a moving mesh methodology. The thermodynamic models are based on the cubic equation of state of Soave, Redlich, and Kwong. A compressibility correction for the widely known kωSST turbulence model is implemented additionally. Application-relevant simulations with a total fuel pressure of 300 bars and five different chamber pressures ranging from 12 to 60 bars were defined. Furthermore, the influence of two fuel and chamber temperatures, 294 and 363 K, is analyzed. Depending on the chamber pressure, two different flow structures of the potential core can be distinguished: (1) A series of typical shock barrels for small pressure ratios and moderately underexpanded jets and (2) a shear layer consisting of a two-phase mixture which enfolds the potential core for high pressure ratios and highly underexpanded jets. Increasing the fuel temperature leads to less significant phase separations, while an increase in the chamber pressure does not affect the structure of the potential core. A comparison with experimental measurements shows a very good agreement of the simulated structure of the potential core, providing evidence that the underlying phenomena are predicted correctly and suggesting that a moving mesh strategy and consistent two-phase thermodynamics implementation are necessary for a physical representation of high-pressure injections.
The choked mass flux density and the choked momentum flux density for the nonideal fluids methane and nitrogen have been calculated using the Soave–Redlich–Kwong equation of state (EoS). For the computation a steady, one-dimensional (1D), isenthalpic and isentropic flow is assumed. The developed algorithm for the calculation of the choked flow properties includes a bounded multidimensional Newton method. A possible second phase emerging in the critical nozzle area is excluded using the saturation properties of the considered fluids. The critical ratios of pressure, density, temperature, and speed of sound are discussed and compared to other publications. Formulations of the choked mass flux density and the choked momentum flux density explicit in Tr, pr, and Zr are given valid for different reduced pressures and temperatures depending on the fluid. Additional computational fluid dynamics (CFD) simulations are carried out in order to validate the findings of the algorithm and the proposed correlations.
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