Homogeneous Relaxation Model Research Articles

Validation of a three-phase Eulerian CFD model to account for cavitation and spray atomization phenomena

Cavitation phase change phenomenon appears in many engineering applications, often eroding and damaging surfaces, so deteriorating the performance of devices. Therefore, it is a phenomenon of great interest for the research and industry communities. In this work, three different cavitation models, the Homogeneous Relaxation Model (HRM), the Schnerr and Sauer, and the Kunz, are implemented in a Eulerian multiphase homogeneous flow Computational Fluid Dynamics (CFD) solver previously developed for simulating fully atomized sprays. The improved solver can be used then to study not only cases with cavitation, such a hydrofoil, but also situations where cavitation occurs together with liquid atomization, such as high pressure injection systems. Validation of this solver is carried out for three different cases under diverse operating conditions: a two-dimmensional throttle, a hydrofoil and a single-hole fuel injector. The Reynolds-Averaged Navier-Stokes (RANS) approach is employed for taking into account the turbulence effects. Simulation results are compared to experimental data available in the literature. Among the tested cavitation models, the HRM is the one that provides the best accuracy in the three validation cases. Nevertheless, the onset of cavitation and the area occupied by the vapor cavities are always underpredicted, by all cavitation models in all validation cases. This can be associated to the unsteady and turbulent nature of the cavitation phenomenon. Even so, the computational prediction of several parameters, such as mass flow rate through the nozzles or spray spreading angle, has an error below 5-10%, which proves the capabilities of the solver.

Experimental and numerical investigation of cavitation in marine Diesel injectors

To further increase the efficiency and decrease emissions of large two-stroke marine Diesel engines, the understanding of the fuel injection, spray breakup and the resulting combustion plays a vital role. Investigations have shown that the strongly asymmetrically and eccentrically arranged nozzle bores of the fuel injectors can lead to undesirable spray deflections that provoke increased component temperatures, emissions and fuel consumption. In order to investigate the origin of these spray deviations, transparent nozzles have been used to qualitatively visualize the in-nozzle flow under realistic geometrical and fuel pressure conditions. Three different, 0.75 mm diameter, single-hole nozzle geometries that represent typical geometrical characteristics have been used in cavitating nozzle flow experiments. The optical measurement technique Shadowgraphy has been applied to visualize the in-nozzle flow over the complete fuel injection process. The experiments have been performed with Diesel fuel at a rail pressure of 50 MPa with ambient back-pressure and temperature. Impingement measurements have been executed to compare the nozzle performance and validate CFD simulations using URANS with cavitation modeling in order to provide qualitative and quantitative support to the experimental results. The volume of fluid (VOF) method has been applied to simulate the multiphase flow with High Resolution Interface Capturing (HRIC). The cavitation model is based on a flash-boiling method with rapid heat transfer between the liquid and vapor phases. A Homogeneous Relaxation Model (HRM) has been utilized to describe the rate at which the instantaneous quality, the mass fraction of vapor in a two-phase mixture, will approach its equilibrium value.The numerical modeling of the cavitation inside the nozzle bore and the evaluated momentum flux have been compared to the experimental findings and show good agreement for the qualitative comparison of the cavitation patterns and differences of less than 6% for the quantitative momentum flux comparison.

CFD modelling of R410A flow through an expansion valve using equilibrium and modified relaxation models

The two-phase flow of R410A refrigerant through an electronic expansion valve (EEV) is modelled using two models, the Homogeneous Equilibrium Model (HEM) and Homogeneous Relaxation Model (HRM) in a CFD numerical analysis. The reason for this is to take a step towards developing a more generic description of two-phase flow calculations. The EEV, applied in air-conditioning and refrigeration systems, was examined by measuring the mass flow rate of the refrigerant, the pressure and the temperature at the valve inlet, and the pressure at the outlet. The EEV was regulated using pulse width modulation; however, during the experiments the EEV was fully opened. Operation of the valve was carried out over a sub-cooling range of 3.8 to 7.7 K. The expansion process started from around 20 bar and ended within the pressure range of 6.8 to 11.5 bar. The mass flow rates of the HEM are inaccurate when compared to the experimental results with a relative difference of approximately 42%. This outcome was expected because the theoretical background of the model assumes that there would be choking at saturation pressure. The modification of the HRM approach was performed with the use of a genetic algorithm (GA) in order to adapt the model’s constants, originally defined for water, to R410A. The HRM approach from literature using constants for water yielded average differences of 17 and 26% using two different relaxation time correlations. The modified HRM approach produced an average relative difference of around 5% for optimised relaxation time correlation constants. The study proved the feasibility of the adaptation of the HRM approach to various working fluids, both synthetic and natural.

Development and validation of the cavitation-induced erosion risk assessment tool

This work presents the development of a cavitation-induced erosion risk assessment (CIERA) tool that links multiphase flow simulation predictions with the progress towards material erosion. To develop a robust erosion modeling tool, the cavitation and erosion predictions for pressurized diesel fuel flow within a channel geometry were validated over a range of Reynolds and cavitation number conditions in two different aluminum channel geometries, one featuring a rounded inlet corner and the other with a sharp inlet corner. The multiphase flow development within the channel was modeled using a compressible mixture model, where phase change was represented with the homogeneous relaxation model and the turbulent flow evolution was modeled using a dynamic structure approach for Large Eddy Simulations. To improve representation of the incubation period before material rupture over existing approaches, a physics-based metric was derived based on the cumulative energy absorbed by the solid material from repeated hydrodynamic impacts. When the average peak pressure was related to the incubation period, the incubation period and its sensitivity to changes in flow conditions were found to be overpredicted. In contrast, predictions from CIERA provided a more accurate means to qualitatively and quantitatively predict the influence of flow conditions on the incubation period before material erosion. When the predicted stored energy was related to the solid material properties to estimate the incubation period, multiphase flow simulations demonstrated accurate representation of the sensitivity of erosion severity to changes in flow conditions. The use of CIERA led to quantitative agreement of the predicted incubation period within 5% of the experimentally measured incubation period.

Coupled in-nozzle flow and spray simulation of Engine Combustion Network Spray-G injector

A coupled Eulerian-Lagrangian approach was employed to Engine Combustion Network (ECN) Spray-G simulations. The Eulerian in-nozzle flow simulation was conducted with a small plenum attached to the nozzles, and the results were fed to the Lagrangian spray simulation. For Eulerian simulation, the homogeneous relaxation model (HRM) coupled with the volume of fluid (VOF) method was used. HRM proved to be good at predicting the phase change phenomena due to vaporization mechanisms, that is, both cavitation and flash boiling. As a one-way coupling, quantities such as rate of injection (ROI), mass injected through each hole, discharge coefficient, spray plume angle and half cone angle predicted from the Eulerian simulations were used as the initial and boundary conditions for the subsequent Lagrangian spray simulations using the blob injection model. Non-flashing (Spray-G1) and flashing (Spray-G2) spray was simulated, and the results were validated quantitatively against the published data in terms of the liquid and vapor penetration lengths, and good agreements were obtained. Furthermore, the simulation predicted the liquid and gas axial velocity and sauter mean diameter for Spray-G1 condition in agreement with the droplet size and particle image velocimetry (PIV) measurements from literature.

A homogeneous relaxation model for multi-phase CO2 jets following the release of supercritical CO2 pipelines

The pressurized CO2 pipeline is a necessary component of Carbon Capture and Storage (CCS) infrastructures. However, the potential risk caused by CO2 pipeline is still not very clear. Especially the scientific community is focusing on the understanding of multi-phase (gas-liquid-solid) flow of CO2 jet in the near-field of dispersion. In this paper, a compressible multi-phase Computational Fluid Dynamics (CFD) model considering real gas behavior is developed to predict the strength of source terms of CO2 release. The model is based on a Homogeneous Relaxation Model (HRM) and accounts for non-equilibrium phase transition. A CO2 release experiment was carried out to obtain near-field data, and it was used to validate the CFD model. The results show that the predictions are in good agreement with the experimental data in terms of velocity and jet structure. The relaxation time has a significant impact on the jet temperature and the condensed phase fraction.

A thermally-limited bubble growth model for the relaxation time of superheated fuels

We propose a novel approach to evaluate the relaxation time of vapor bubble growth in the context of the flash boiling of a superheated liquid. In alternative to the empirical correlation derived from superheated water experiments almost fifty years ago, the new model describes the thermally-dominated growth of vapor bubbles in terms that are dependent on the local Jakob number (the ratio of sensible heat to latent heat during phase change) and the number density of vapor bubbles. The model is tested by plugging the resulting relaxation time into the Homogenous Relaxation Model (HRM). Flash-boiling simulations carried out with HRM are compared with n-pentane (C5H12) injection and boil-off experiments conducted with a real-size, axial-hole, transparent gasoline injector discharging into a constant-pressure vessel. The long-distance microscopy images from the experiments, processed to derive the projected liquid volume (PLV) of the spray, provide a unique set of time-resolved validation data for direct fuel injection simulations. At conditions ranging from flaring to mild and minimal flash boiling, we show that switching to the new relaxation time improves the agreement with the measured PLV profiles with respect to the standard empirical model. Particularly at flaring conditions, the predicted increase in gas cooling caused by rapid vapor production is shown to be more consistent with the observed boil-off.

Numerical and experimental analysis of flashing cryogenic nitrogen

The development of new upper orbit thrusters using cryogenic propellants requires an improved understanding of the dynamics of oxidizer and fuel injection at near vacuum conditions before ignition. Due to the low ambient pressure, the propellants enter a superheated state and flash evaporation occurs. Flash boiling of cryogenic liquid nitrogen is studied experimentally on the newly developed test bench at DLR Lampoldshausen and numerically with a newly developed OpenFOAM© solver. Here, a one-fluid approach is selected where phase properties, such as density, enthalpy and saturation conditions are determined with the thermodynamic library CoolProp and tabulated before runtime. The phase change is modeled by the homogeneous relaxation model. For highly superheated jets the flow becomes supersonic and forms a shock after the injector outlet. The solver is validated with the aid of flashing acetone spray experiments where the shock structures are more clearly visible. The results show that the developed solver is capable to predict the all important gas dynamics by matching shock structure and spray angle to the experiment. The experiments using cryogenic liquid, however, do not reveal any shock structures but regions with low negative axial velocities can be identified on the jet centerline. A comparison with the simulations now demonstrates that shocks continue to persist but the respective shadowgraph signals may be obscured in these flows. The joint experimental and numerical study thus provides a consistent understanding of the observed flow features that govern the fluid dynamics and jet breakup of cryogenic flashing flows.

Numerical analysis of spray characteristics with liquid nitrogen

Spray of cryogenic fluids, which involves complicated aerodynamic instability and superheat flashing, is receiving increasing attention in cryogenic cooling applications and cryogenic propellant ignition. In the present study, a two-fluid numerical model coupling the Algebraic Interfacial Area Density (AIAD) model and Homogeneous Relaxation Model (HRM) framework was built to investigate the underlying physics of liquid nitrogen (LN2) spray. The proposed approach aims at describing the underlying physics affected by both aerodynamic and flashing dynamics. Behaviors of spray characteristics, including flow regime, pressure distribution, velocity distribution and intact core length, are analyzed subsequently. Accordingly, effects of aerodynamic instability and flashing on the LN2 spray were analyzed. Finally, a modified correlation of liquid-intact-core length which involves both of the aerodynamic and thermodynamic effects was proposed and the weight of two effects were evaluated accordingly.

Computational study on the influence of nozzle eccentricity in spray formation by means of Eulerian Σ - Y coupled simulations in diesel injection nozzles

The present work analyses the effect of the eccentricity of diesel nozzle orifices over the spray behaviour by means of CFD simulations. Several orifice geometries with varying horizontal eccentricity (from 0.50 to 0.94) are selected. Their performance is assessed at a high injection pressure of 200 MPa, a 3 MPa back-pressure and non-evaporative conditions. The nozzle flow characteristics, including cavitation modelled by a Homogeneous Relaxation Model (HRM), are accounted for in the spray performance by means of a Σ - Y model. The code is validated via two reference nozzles, the so called ”Spray A” of the Engine Combustion Network plus a second nozzle from a production injector, and then extended to the eccentric geometries. The results and discussions include spray angle and penetration, air entrainment and flow parameters of the nozzle inner conditions versus the eccentricity value.

Transient nozzle flow simulations of gasoline direct fuel injectors

In the field of Internal Combustion Engines (ICE) the usage of Gasoline Direct fuel injectors (GDi) with gasoline, iso-octane, ethanol (or other alternative fuels) has gained relevance in the past years with the goal of reducing fuel consumption and thus emissions. In this type of direct injections, the injector plays a major role in defining the air-fuel mixture quality. Nevertheless, the study of the phenomena inside the nozzle becomes a challenge due to its reduced size, high flow velocities and multiphase flow nature. Computational Fluid Dynamics (CFD) tools allow gaining valuable insight and understanding into such complex flow physics. Therefore, the objective of this work is the development of a predictive methodology for simulating two GDi nozzles. Unsteady Reynolds-Averaged Navier Stokes (URANS) is chosen for modeling the turbulence. The Homogeneous Relaxation Model (HRM) is used to investigate the possible phase change of the fuel through cavitation or flash boiling. Different injection conditions are simulated and results are compared against experimental data of mass flow and momentum rate for validation. CFD is able to accurately predict steady state values, but transients are very dependent on the initial and boundary conditions imposed on the model. A methodology for their definition is proposed and tested, and with it the accuracy in the prediction of the opening transient is improved.

Modelling start-up injection of CO2 into highly-depleted gas fields

The development and verification of a homogeneous relaxation model for simulating the highly transient flow phenomena taking place during the start-up injection of CO2 into deep highly depleted gas fields is presented. The constituent mass, momentum, and energy conservation equations, incorporating a relaxation time to account for non-equilibrium effects, are solved numerically for single and two-phase flows along the steel lined injection well leading to the storage reservoir. Wall friction, gravitational field effects and heat transfer between the expanding fluid and the outer well layers are taken into account as source terms in the conservation equations. At the well inlet, the opening of the upstream flow regulator valve is modelled as an isenthalpic expansion process; whilst at the well outlet, a formation-specific pressure-mass flow rate correlation is adopted to characterise the storage site injectivity. The testing of the model is based on its application to CO2 injection into the depleted Golden Eye Reservoir in the North Sea for which the required design and operational data are publically available. Three injection scenarios involving a rapid, medium and slow linear ramping up of the injected CO2 flow rate to the peak nominal value of 33.5 kg/s are simulated. In each case, the predicted pressure and temperature transients at the top and bottom of the well are employed to ascertain the risks of well-bore thermal shocking, and interstitial ice or CO2 hydrate formation leading to its blockage due to the rapid expansion cooling of the CO2. The results demonstrate the efficacy of the proposed model as a tool for the development of optimal injection strategies and best-practice guidelines for the minimisation of the risks associated with the start-up injection of CO2 into depleted gas fields.

Modelling phase change in a novel turbo expander for application to heat pumps and refrigeration cycles

A novel turbo expander based on the Tesla turbine is proposed to be applied to a heat pump or refrigeration cycle to improve the overall cycle efficiency. Initial numerical modelling of this turbo expander at representative conditions was carried out using the homogeneous relaxation model (HRM) to assess the influence of phase change on performance. The presence of a dense cloud of liquid droplets within the rotor was predicted to produce a significant back pressure on the turbine nozzle postponing the phase change. This was expected to occur in the vicinity at the outlet of the nozzle, but high volume fractions of liquid was predicted to penetrate deeper inside the rotor, especially at higher RPM. The resulting lower velocities of the liquid flow at the inlet of the rotor was predicted to significantly degrades the performance of the turbine. It is thus important for a successful implementation of this concept to remove as much liquid droplets as possible before the flow enters the rotor in order to minimise the back pressure.

Numerical investigation on flashing jet behaviors of single-hole GDI injector

In this paper, n-hexane flashing jets discharged from a single-hole gasoline direct injector (GDI) were studied numerically with the adoption of diffuse Eulerian framework and the homogeneous relaxation model (HRM). The fuel temperature ranged from 30 to 130 °C, and the ambient pressure varied from 20 to 101 kPa. The results showed that considerable vaporization started at the counter bore and a liquid core existed near the nozzle exit. Due to drastic vaporization, the pressure within the liquid core increased so the two-phase flow became under-expanded. Violent expansion then occurred and a low-pressure region was formed, which is believed as the origin of the spray collapse under flashing conditions for multi-hole GDI injectors. At high superheat levels, shock wave structures similar to those in highly under-expanded gaseous jets were identified. However, the transonic position located at some distances from the nozzle rather than at the throttle. Besides, vapor fraction played the dominant role in the onset of expansion, while the expansion was ended by the pressure difference between the two sides of the Mach disk.

A computational approach to predict external spray characteristics for flashing and cavitating nozzles

A computational approach to predict external spray characteristics for flashing and cavitating nozzles was presented and validated. It is developed as a fully Eulerian and compressible two-phase flow solver that simulates vaporization and condensation of the fuel using a Homogeneous Relaxation Model (HRM). The flow solver together with the interface area density model was applied to predict spray spreading angle. A method to identify spray plume boundary from the predicted flow field that offers a meaningful comparison to experimental definition was discussed in detail. Using the experimental data available in literature, a comparison between axi-symmetric and asymmetric nozzles was made to assess the nature of the influence of the nozzle geometry and the operating conditions on the ensuing spray. On the basis of this comparison, it was inferred that the spray plume angle of asymmetric nozzles is largely geometry dependent for a wide range of pressure ratios.

Numerical simulation of subcooled and superheated jets under thermodynamic non-equilibrium

Modelling and simulating the rapid pressure drop inside nozzles is a significant challenge because of the complexity of the multiple associated phenomena. In the present study, FlashFOAM a compressible solver for calculating the phase change within various nozzle geometries undergoing rapid pressure drops has been developed in the frame of the open source Computational Fluid Dynamics (CFD) code OpenFOAM. FlashFOAM accounts for the inter-phase heat transfer with the Homogeneous Relaxation Model (HRM). The work describes the development of a pressure equation within a different formulation than in other studies. The surface forces due to liquid-gas interfacial instabilities are modelled here in a novel coupling of HRM with the volume of fluid method giving rise to a conservative method for modelling primary atomisation. This new pressure equation is validated with published experimental measurements. A validation series dedicated to long nozzles is included for the first time. Novel additional tests for the flow characteristics and vapour generation in cryogenic liquid cases are included showing that the solver can be employed to gain some new insights into the physics of the flow regimes of sudden depressurising cryogenic liquids. The dependency of the geometry of the nozzles, pressure and subcooled degree on the vapour generation has been analysed including the effect of turbulence on the nozzle flow avoiding the laminar flow scenarios of previous validation studies. The validation study has demonstrated that FlashFOAM can be used to simulate flash boiling scenarios accurately and predict the properties of flash atomisation.

Investigations of effect of phase change mass transfer rate on cavitation process with homogeneous relaxation model

Abstracts The super high fuel injection pressure and micro size of nozzle orifice has been an important development trend for the fuel injection system. Accordingly, cavitation transient process, fuel compressibility, amount of non-condensable gas in the fuel and cavitation erosion have attracted more attention. Based on the fact of cavitation in itself is a kind of thermodynamic phase change process, this paper takes the perspective of the cavitation phase change mass transfer process to analyze above mentioned phenomenon. The two-phase cavitating turbulent flow simulations with VOF approach coupled with HRM cavitation model and U-RANS of standard k-e turbulence model were performed for investigations of cavitation phase change mass transfer process. It is concluded the mass transfer time scale coefficient in the Homogenous Relaxation Model (HRM) representing mass transfer rate should tend to be as small as possible in a condition that ensured the solver stable. At very fast mass transfer rate, the phase change occurs at very thin interface between liquid and vapor phase and condensation occurs more focused and then will contribute predictably to a more serious cavitation erosion. Both the initial non-condensable gas in fuel and the fuel compressibility can accelerate the cavitation mass transfer process.

Modified homogeneous relaxation model for the R744 trans-critical flow in a two-phase ejector

The proposed modified homogeneous relaxation model (HRM) applied to the numerical model of a CO2 two-phase ejector was numerically investigated and developed based on the optimisation of the relaxation time (RT) correlation. The optimisation procedure was performed using a genetic algorithm. The study of the RT definition on model accuracy was carried out using literature correlations and constant relaxation time, and by comparing the developed modified HRM with experimental results. The modified HRM showed the higher accuracy of the motive nozzle mass flow rate (MFR) than that of the other available numerical models for the subcritical operating regimes and similar high accuracy as the homogeneous equilibrium model (HEM) for the trans-critical operating regimes. The application range of the modified HRM was defined for the motive nozzle pressure above 59 bar to predict the motive nozzle MFR with the relative error below 15%. For the motive nozzle pressure level below 59 bar, the modified HRM improved the accuracy of the motive MFR prediction by 5%–10% compared to the HEM formulation.

Homogeneous two-phase flow models and accurate steam-water table look-up method for fast transient simulations

The accurate simulation of fast steam-water transients requires precise algorithms for calculating fluid properties. The system of the governing flow equations must be closed with an Equation of State (EoS) to calculate the pressure as a function of the system conservative variables. For water, accurate analytical EoS for this purpose are not available yet. The aim of this paper is to show an efficient and very accurate algorithm to calculate water properties when the independent variables of the EoS are the density and the specific internal energy. Our algorithm uses a new table look-up method with bicubic interpolation based on the IAPWS-IF97 EoS formulation, and it is able to account for metastable states. The liquid metastability domain is extended until the spinodal curve, here determined and compared with other formulations.The EoS algorithm is coupled to two classical homogeneous two-phase flow models, namely the Homogeneous Equilibrium Model (HEM) and the Homogeneous Relaxation Model (HRM). HEM and HRM are used to simulate fast depressurization, waterhammer and steam explosion problems. Comparison of the numerical results with available experimental data shows the good performance of the proposed algorithms.

Influence of fuel properties on internal nozzle flow development in a multi-hole diesel injector

Fuel physical properties are known to influence in-nozzle flow behavior, in turn affecting spray formation in internal combustion engines. A series of 3D simulations was performed to model the internal nozzle flow in a five-hole mini-sac diesel injector. The goal of the study was to evaluate the behavior of two gasoline-like fuels (full-range naphtha and light naphtha) and compare them against n-Dodecane, selected from a palette used as a diesel surrogate. Simulations were carried out using a multi-phase flow representation based on the mixture model assumption with the Volume of Fluid (VOF) method, and including cavitation effects by means of the Homogeneous Relaxation Model (HRM). Validated methodologies from our previous studies were employed to account for full needle motion. Detailed simulations revealed the influence of the fuel properties on injector performance, injected fuel energy and propensity to cavitation. The three fuels were compared with respect to global parameters such as mass flow rate and area contraction coefficients, and local parameters such as pressure and velocity distribution inside the sac and orifices. Parametric investigations were also performed to understand the fuel response to changes in the fuel injection temperature, injection pressure, and geometry details. Cavitation magnitude was observed to be strongly associated with the values of saturation pressure. Owing to their higher volatility, the two gasoline-like fuels were observed to cavitate more than n-Dodecane across all the investigated conditions. While at full needle lift cavitation was reduced for all fuels, during the injection transients the gasoline-like fuels showed more propensity to cavitate inside the orifice and seat regions. This is expected to have a profound influence on nozzle erosion. Although full-range and light naphtha have lower densities compared to n-Dodecane, owing to their lower viscosity, the mass flow rate differences between the naphtha fuels and n-Dodecane were small. The analysis of fuel energy content showed that the higher lower heating value (LHV) of light naphtha helped compensate for the slightly lower total delivered mass.