![Axial density distributions of nitrogen injection experiments by Mayer et al. [109] and Branam and Mayer [23]. Case numbering follows Mayer.](https://www.researchgate.net/profile/Daniel-Banuti/publication/279381888/figure/fig1/AS:614282488209409@1523467766200/Axial-density-distributions-of-nitrogen-injection-experiments-by-Mayer-et-al-109-and_Q320.jpg)
![Visualizations of injection regimes from Fig. 2.6(b). Conditions in Tab. 2.1. A and B modified from Chehroudi et al. [34], C modified from Lamanna et al. [85], D from Branam and Mayer [23], E modified from Stotz et al. [154].](https://www.researchgate.net/profile/Daniel-Banuti/publication/279381888/figure/fig4/AS:614282492391456@1523467767363/sualizations-of-injection-regimes-from-Fig-26b-Conditions-in-Tab-21-A-and-B_Q320.jpg)
![Shadowgraph images of transcritical nitrogen injection, injector diameter 0.25 mm. (a) and (b) into pure nitrogen at p r = 0.83 and p r = 2.03; (c) into mixture of nitrogen and helium at p r,N2 = 2.03. Taken at axial locations downstream of injector exit (1) 0.0 to 1.9 mm; (2) 7.4 to 9.5 mm; (3) 15.0 to 17.1 mm. Modified from Woodward and Talley [170].](https://www.researchgate.net/profile/Daniel-Banuti/publication/279381888/figure/fig7/AS:614282492399638@1523467767541/Shadowgraph-images-of-transcritical-nitrogen-injection-injector-diameter-025-mm-a_Q320.jpg)
![Shadowgraphs of cryogenic nitrogen surrounded by gaseous helium stream, from Mayer et al. [105].](https://www.researchgate.net/profile/Daniel-Banuti/publication/279381888/figure/fig3/AS:614282492391454@1523467767317/Shadowgraphs-of-cryogenic-nitrogen-surrounded-by-gaseous-helium-stream-from-Mayer-et-al_Q320.jpg)
![High pressure H 2 /O 2 flame near coaxial injector, from Mayer and Tamura [107].](https://www.researchgate.net/profile/Daniel-Banuti/publication/279381888/figure/fig6/AS:614282492407819@1523467767505/High-pressure-H-2-O-2-flame-near-coaxial-injector-from-Mayer-and-Tamura-107_Q320.jpg)
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Thermodynamic Analysis and Numerical Modeling of Supercritical Injection
Abstract and Figures
Although liquid propellant rocket engines are operational and have been studied for decades, cryogenic injection at supercritical pressures is still considered essentially not understood. This thesis intends to approach this problem in three steps: by developing a numerical model for real gas thermodynamics, by extending the present thermodynamic view of supercritical injection, and finally by applying these methods to the analysis of injection.
A new numerical real gas thermodynamics model is developed as an extension of the DLR TAU code. Its main differences to state-of-the-art methods are the use of a precomputed library for fluid properties and an innovative multi-fluid-mixing approach. This results in a number of advantages: There is effectively no runtime penalty of using a real gas model compared to perfect gas formulations, even for high fidelity equations of state (EOS) with associated high computational cost. A dedicated EOS may be used for each species. The model covers all fluid states of the real gas component, including liquid, gaseous, and supercritical states, as well as liquid-vapor mixtures. Numerical behavior is not affected by local fluid properties, such as diverging heat capacities at the critical point. The new method implicitly contains a vaporization and condensation model. In this thesis, oxygen is modeled using a modified Benedict-Webb-Rubin equation of state, all other involved species are treated as perfect gases.
A quantitative analysis of the supercritical pseudo-boiling phenomenon is given. The transition between supercritical liquid-like and gas-like states resembles subcritical vaporization and is thus called pseudo-boiling in the literature. In this work it is shown that pseudo-boiling differs from its subcritical counterpart in that heating occurs simultaneously to overcoming molecular attraction. In this process, the dividing line between liquid-like and gas-like, the so called Widom line, is crossed. This demarcation is characterized by the set of states with maximum specific heat capacity. An equation is introduced for this line which is more accurate than previous equations. By analyzing the Clausius-Clapeyron equation towards the critical limit, an expression is derived for its sole parameter. A new nondimensional parameter evaluates the ratio of overcoming molecular attraction to heating: It diverges towards the critical point but shows a significant pseudo-boiling effect for up to reduced pressures of 2.5 for various fluids.
It appears reasonable to interpret the Widom-line, which divides liquid-like from gas-like supercritical states, as a definition of the boundary of a dense supercritical fluid. This may be used to uniquely determine the radius of a droplet or the dense core length of a jet. Then, a quantitative thermodynamic analysis is possible. Furthermore, as the pseudo-boiling process may occur during moderate heat addition, this allows for a previously undescribed thermal jet disintegration mechanism which may take place within the injector.
This thermal jet break-up hypothesis is then applied to an analysis of Mayers and Branams nitrogen injection experiments. Instead of the constant density cores as predicted by theory, the majority of their cases show an immediate drop in density upon entering the chamber. Here, three different axial density modes are identified. The analysis showed that heat transfer did in fact take place in the injector. The two cases exhibiting a dense core are the cases which require the largest amount of power to reach the pseudo-boiling temperature. After this promising application of pseudo-boiling analysis, thermal break-up is tested numerically. By accounting for heat transfer inside the injector, a non dense-core injection can indeed be simulated for the first time with CFD.
Finally, the CFD model is applied to the A60 Mascotte test case, a reactive GH2/LOX single injector operating at supercritical pressure. The results are compared with experimental and other researchers numerical data. The flame shape lies well within the margins of other CFD results. Maximum OH* concentration is found in the shear layer close to the oxygen core and not in the shoulder, in agreement with experimental data. The axial temperature distribution is matched very well, particularly concerning position and value of the maximum temperature.
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... While the non-linear continuous transition of fluid and its thermophysical properties around the critical point at supercritical pressures were suspected by the pioneering experimentalists themselves (Mayer et al., 3,4,8,12 Oschwald et al., 5,6,14 and Chehroudi et al. 7,9,10,21,22 ) to be involved in the absence of traditional dense core and associated with the unique characteristics of supercritical cryogenic jets, Banuti et al.'s [23][24][25] pioneering work has revealed in detail the thermodynamic phenomena and mechanisms that causes it. Particularly, Banuti et al. proposed that the effect of the pseudoboiling phenomena and the associated distributed latent heat results in a predominantly thermaldriven jet decay. ...
... As a consequence, the surface tension values (at injection) are greatly diminished to play a significant role, in contrast to low-temperature liquid injection. In the previous work, 2 numerically simulated supercritical jets that develop due to nearcritical injection were examined extensively based on the thermal disintegration of supercritical jets and classification of supercritical fluids proposed by Banuti et al. 23,25,30 This work aims to review the supercritical and/or cryogenic fluid numerical simulations, particularly including those solver approaches where surface tension has been modeled. Following this, a new inhouse compressible volume of fluid (VoF) numerical solver with inbuilt adaptive surface tension as well as molecular and thermo(Soret)-diffusion will be presented. ...
The injection of cryogenic fluids into environments where the prevailing conditions are supercritical in comparison to the critical point of the injected cryogenic fluid is encountered in cryogenic rocket engines, and novel engine architectures such as the recuperated split cycle engine. The physical characteristics of cryogens injected into supercritical environment are rather unclear. While surface tension is usually assumed to be absent/negligible for supercritical fluids, recent experimental research has identified the existence of surface tension and its effects on liquid hydrocarbons in supercritical environment. This research work proposes an alternative computationally simple adaptive surface tension algorithm for the simulation of a liquid injected into supercritical environment. The numerical simulations presented here correspond to single- and binary-specie cases of iquid nitrogen and liquid methane respectively, undergoing phase transition post their injection into supercritical conditions. Following a critical review of related numerical works, this paper begins with a brief explanation of the physics behind the surface tension effect in a binary-fluid interface in which a supercritical fluid is involved and we present why this effect is of relevance to supercritical cryogenic jets? Then, the rationale and specifics of the the new modelling framework based on adaptive surface tension is discussed along with its implications. The results of the numerical simulations of low-temperature vs near-critical temperature iquid nitrogen and liquid methane injection dynamics revealed the drastically different fluid- and thermo-dynamics at play in these two cases. The role of surface tension at these conditions is also explored.
... Based on the ideas of supercritical fluid behaviour [3], Banuti developed a two-phase real fluid model which is capable of calculating phase changes and mixtures of a real fluid with ideal gases [4]. This real-gas extensions consist mainly of two parts: A separate preprocessing addon, which generates a database with fluid properties over ρ-T and a thermodynamic library which solves the dominating ρ-e-problem and some auxiliary thermodynamic problems like ρ-p and p-T . ...
... Peng-Robinson (PR), European Gas Research Group (GERG) and Modified Benedict-Webb-Rubin (MBWR). For the present investigations the MBWR EOS is chosen because of its accuracy compared to the experimental database of 1 NIST, see [4] and [3]. Detailed information about the equation of state can be found in [6]. ...
... High pressures are popular in a wide range of transport applications for road, air, and space. It can improve the efficiency of burning fuel and increase the thrust produced by engines, which is crucial for designing and improving propulsion systems [57,58,59,60,61,62,63,64,65,66,67,68,69,70,71]. ...
Driven by fundamental thermodynamic efficiency considerations, an emerging trend in the energy and propulsion systems is that the working fluid operates at a pressure above the critical pressure. Energy transport is thus accompanied by dramatic and strongly nonlinear variations of thermophysical properties, which cause abnormal heat transfer behavior and non-ideal gas effects. This situation raises a crucial challenge for the heat exchanger and turbomachinery design, overall energy and exergy efficiency. We aim to provide a multi-scale overview of the flow and thermal behavior of fluid above the critical point: microscopic physics and macroscopic transport. Microscopic physics, i.e. near-critical thermophysical properties, phase transition and fluid dynamics, are introduced. A particular focus will be on the supercritical pseudo boiling process, which is a generalization of classical liquid-vapor phase transitions to non-equilibrium supercritical states. These new perspectives lead to a revised view of the state space. Further, recent results demonstrated the possibility of stable supercritical fluid interfaces without surface tension. On the macroscale, recent progress on the modeling of turbulent flow and convective heat transfer of supercritical fluids are summarized. Direct numerical simulation is able to offer insights into the physics of thermal fluids. We start with a description of fundamental fluid mechanics problems related to supercritical fluids. In addition, the heat transfer deterioration in supercritical fluids is found to be closely connected to the flow relaminarization by the non-uniform body-force. Finally, various modeling approaches such as recently developed advanced Reynolds-averaged turbulence modeling as well as machine-learning methods are summarized.
... There is no fundamental theoretical conflict between pseudo-boiling and single-phase heat transfer, because so far there is no evidence to deny the first-order continuity of the thermophysical properties of supercritical fluids. Therefore, the numerical simulation based on the single-phase transport equations, and the pseudo-boiling theory can be combined [31]. In fact, a lot of research on this combination have been made in multiple fields [32][33][34]. ...
Large eddy simulations have been conducted to study upward transitional flows and heat transfer characteristics of supercritical CO2 in a vertical mini-channel. The numerical simulation was carried out on a modified buoyantPimpleFOAM solver in OpenFOAM 7, and was verified using experiment data. Numerical results indicate that increasing the Grashof numbers can reduce the flow stability and make the flow transition earlier. There are four stages of heat transfer in the transition process, i.e., weakened, improved, recovered and normal heat transfer. These heat transfer phenomena in the transition process were explained from three perspectives: thermal boundary layer theory, turbulent transport and pseudo-boiling theory. Heat transfer enhancement during transition is related to the transport of supercritical molecular clusters, and these molecular clusters are regarded as pseudo-bubbles in pseudo-boiling theory. The flow pattern of the pseudo-phases in the dia-Widom process contains single-phase flow, steady pseudo-film flow, unsteady pseudo-film flow, partial pseudo-bubbles flow and flocculent pseudo-film flow. Pseudo-bubbles have similar behaviors to subcritical bubbles, i.e., break-up, deformation, condensation and coalescence. Relevant researches in this work are favorable for understanding the heat transfer mechanism of supercritical fluids during flow transition.
... As reviewed by Banuti [42], the seemingly overwhelming variety of computational fluid dynamics (CFD) codes is misleading, because successful approaches and methods are used as almost canonical procedures. Given the amount of experimental evidence, the variable-density behavior of supercritical jets can then be modeled through the usage of real gas relationships for density (in the form of an EoS) and transport properties such as dynamic viscosity and thermal conductivity, detailing ideal gas behavior and departure functions to account for high-pressure effects. ...
Supercritical nitrogen jet behavior is modeled using an incompressible but variable density approach developed for variable density jets. Following mechanical and thermal breakup concepts, several injection conditions relevant to liquid rocket propulsion are analyzed, considering heat transfer in the injector. Regarding axial density distributions, different levels of agreement with experimental data are encountered for potential core, subsided core, and plateau formations. Further comparisons with compressible formulations from the literature are a good indicator of the proposed methodology's suitability for the simulation of supercritical injection behavior.
... [4] examined and analyzed the theory, and the results were encouraging enough to warrant further testing in other, more general contexts. The seemingly vast range of CFD (computational fluid dynamics) codes, as detailed by [6], is misleading, since successful approaches and methodologies are utilized as nearly standard procedures. ...
The simplification of the incompressible but variable density jet is based on the visualization of data similarity. Given the initial encouraging results obtained in the past, we extend, in the present paper, this hypothesis to a broader range of conditions to ascertain its applicability and its role as an alternative to the more commonly fully-compressible formulations encountered in the literature. Transcritical and supercritical injection conditions are considered for nitrogen, which works as a surrogate for the oxygen-hydrogen mixture combination, characteristic of liquid rocket propulsion. A close agreement is found between experiments and numerical results in terms of axial profiles and jet spreading rates.
... In other words, the evolution of such injected cryogenic fluids/jets is controlled by both thermal and inertia phenomena. The thermal influence on the cryogenic jet evolution known as 'thermal disintegration mechanism' has been first analysed in significant detail by Banuti [26], who focused on the liquid nitrogen injection into supercritical pressure and temperature environment corresponding to the experiments of Mayer et al. [10]. He demonstrated that the decay of such nitrogen jets were strongly influenced by the thermodynamic state (and properties) of the fluid. ...
Computational Fluid Dynamics (CFD) frameworks of supercritical cryogenic fluids need to employ Real Fluid models such as cubic Equations of State (EoS) to account for thermal and inertial driven mechanisms of fluid evolution and disintegration. Accurate estimation of the non-linear variation in density, thermodynamic and transport properties is required to computationally replicate the relevant thermo and fluid dynamics involved. This article reviews the availability, performance and the implementation of common Real Fluid EoS and data-based models in CFD studies of supercritical cryogenic fluids. A systematic analysis of supercritical cryogenic fluid (N2, O2 and CH4) thermophysical property predictions by cubic (PR and SRK) and non-cubic (SBWR) Real Fluid EoS, along with Chung’s model, reveal that: (a) SRK EoS is much more accurate than PR at low temperatures of liquid phase, whereas PR is more accurate at the pseudoboiling region and (b) SBWR EoS is more accurate than PR and SRK despite requiring the same input parameters; however, it is limited by the complexity in thermodynamic property estimation. Alternative data-based models, such as tabulation and polynomial methods, have also been shown to be reliably employed in CFD. At the end, a brief discussion on the thermophysical modelling of cryogenic fluids affected by quantum effects is included, in which the unsuitability of the common real fluid EoS models for the liquid phase of such fluids is presented.
... With a given thermal equation of state, the deviations can be calculated. The specific relations for ψ res are derived in standard textbooks such as Baehr and Kabalac [3] and a summary is given in Banuti [4]. ...
Bei der Beschreibung von Strömungen wird klassischerweise zwischen inkompressiblen und kompressiblen Bereichen unterschieden. Während inkompressible Strömungen durch ein divergenzfreies Geschwindigkeitsfeld charakterisiert werden, sind kompressible Strömungsfelder durch Expansionsfächer, Kontaktunstetigkeiten und Stoßwellen gekennzeichnet. Die beiden Bereiche werden damit durch stark unterschiedliche Systeme partieller Differentialgleichungen beschrieben. Diese Unterscheidung zeigt sich auch in der numerischen Strömungsmechanik durch die Entwicklung separater Ansätze für die beiden Bereiche. Dichtebasierte Verfahren eignen sich dabei zur Simulation von kompressiblen Strömungen höherer Mach-Zahlen, während druckbasierte Verfahren für inkompressible Strömungen bei kleinen Mach-Zahlen geeignet sind. Beide Verfahren sind ohne Anpassungen nicht für den jeweils anderen Bereich verwendbar. In vielen praktischen Anwendungen treten jedoch Strömungen mit einer großen Variation der lokalen Mach-Zahl auf, weshalb seit einigen Jahrzehnten intensiv an numerischen Verfahren für die Strömungssimulation in allen Mach-Zahlbereichen gearbeitet wird. Für dichtebasierte Verfahren wird meist die sogenannte Präkonditionierung angewendet. Die durchgeführten Arbeiten beschränken sich allerdings meist auf Strömungen idealer Gase, womit Realgaseffekte nicht berücksichtigt werden, die jedoch für viele praktische Anwendungen wichtig sind. Die vorliegende Arbeit beschäftigt sich in diesem Zusammenhang mit der Analyse numerischer Verfahren zur Simulation von Strömungen im Grenzfall einer verschwindenden Mach-Zahl für reale Gase. Als Modell eines realen Gases wird die Van der Waals-Zustandsgleichung verwendet.
... Summarized [17], "The acceptance of this change in understanding is perhaps best reflected in the shift of boundary conditions posed by the Rocket Combustion Modeling (RCM) workshops 2001 [18] and 2006 [19]. For the same configuration of a single injector combustion chamber (but at a different oxidizer/fuel ratio) at supercritical pressures, the 2001 workshop specified a spectrum of oxygen droplets to be prescribed in the CFD calculation. ...
Flows in liquid propellant rocket engines (LRE) are characterized by high pressures and extreme temperature ranges, resulting in complex fluid behavior that requires elaborate thermo-physical models. In particular, cubic equations of state and dedicated models for transport properties are firmly established for LRE simulations as a way to account for the non-idealities of the high-pressure fluids. In this paper, we review some shortcomings of the current modeling paradigm. We build on the common study of property errors, as a direct measure of the density or heat capacity accuracy, to evaluate the quality of cubic equations of state with respect to pseudo boiling of rocket-relevant fluids. More importantly, we introduce the sampling error as a new category, measuring how likely a numerical scheme is to capture real fluid properties during a simulation, and show how even reference quality property models may lead to errors in simulations because of the failure of our numerical schemes to capture them. Ultimately, a further evolution of our non-ideal fluid models is needed, based on the gained insight over the last two decades.
Verbrennungsinstabilitäten in Raketenbrennkammern stellen seit Beginn des Raumfahrtzeitalters eine große Herausforderung bei der Entwicklung neuer Antriebe dar, weil der genaue Mechanismus ihrer Entstehung bis heute nicht vollständig verstanden ist. Einen wichtigen Teilaspekt bei der Entstehung von Verbrennungsinstabilitäten stellt die Wechselwirkung zwischen der Flamme und den akustischen Brennkammereigenmoden dar, die in dieser Dissertation für eine Experimentalbrennkammer numerisch untersucht wird. In dieser Arbeit wird der DLR Strömungslöser TAU zur skalenauflösenden Simulation eines Lastpunkts von Brennkammer H (BKH) des DLR Lampoldshausen verwendet, bei dem sowohl Treibstoff als auch Oxidator unter kryogenen Bedingungen eingespritzt werden. Zu diesem Zweck wurde das TAU Verfahren um ein Realgas Flameletverbrennungsmodell erweitert, das eine effiziente Behandlung der chemischen Reaktionen erlaubt. Weiterhin wurden die Dissipationseigenschaften des numerischen Verfahrens eingehend untersucht und die Anwendbarkeit von Upwind-Flusslösern für skalenauflösende Simulationen diskutiert. Ein weiterer Aspekt ist die Analyse der akustischen Eigenschaften des TAU-Codes und der Randbedingungen, die dann zur gezielten Untersuchung der Kopplung von Brennkammer- und Injektoreigenmoden verwendet wurden. Diese Arbeit präsentiert Resultate mehrerer Detached-Eddy Simulationen (DES) der Strömung in BKH bei verschiedenen Anregungszuständen und vergleicht sie mit experimentellen Daten. Die Simulationsergebnisse bei nicht-resonanter Anregung durch eine Sirene weichen weniger als 2.1 % von den experimentellen Brennkammereigenfrequenzen ab. Allerdings wird in der Simulation die instationäre Anregung durch die Sirene überschätzt, was auf Modellannahmen der numerischen Konfiguration zurückgeführt wird. Die Ergebnisse einer resonant angeregten Simulation und eines Impulsantwort-Tests zeigen eine Verschiebung der Brennkammereigenfrequenzen, die bereits in früheren experimentellen Studien von BKH beobachtet wurde. Die Positionen der verschobenen Frequenzen stimmen sehr gut mit den experimentellen Ergebnissen überein. Darüber hinaus zeigen die Simulationsergebnisse der resonanten Anregungen eine starke Schwankung des mittleren Brennkammerdrucks, die im Experiment nicht zu beobachten war. Dieser Unterschied wird durch das abrupte Anschalten der Sirene in der Simulation erklärt, was zu einer niederfrequenten Modulation der Einströmbedingungen führt. Zum Schluss dieser Arbeit wird durch gezielte Modifikation des Sauerstoffinjektors eine Kopplung zwischen Injektor- und Brennkammereigenmoden künstlich herbeigeführt, die sich in früheren Untersuchungen als notwendige Bedingung für die Entstehung von Verbrennungsinstabilitäten herausgestellt hatte. Anhand der instationären Druckdaten wird gezeigt, dass zwar eine erfolgreiche Modenkopplung erreicht werden konnte, darüber hinaus aber kein signifikanter Einfluss auf die Brennkammermoden beobachtbar war.
The thermophysical properties of argon, ethylene, parahydrogen, nitrogen, nitrogen trifluoride and oxygen are presented. Properties are given in tables and a standard set of equations is described. The tables list pressure, density, temperature, internal energy, enthalpy, entropy, heat capacity at constant volume, heat capacity at constant pressure, and sound velocity. Also included are viscosity, thermal conductivity, and dielectric constant, for some of the fluids. The equation and related properties of this report represent a compilation from the cooperative efforts of two research groups: the NBS Thermophysical Properties Division and the Center for Applied Thermodynamics Studies of the University of Idaho.
This project looks at injection processes of a dense jet simulating oxygen core flow with nitrogen of a coaxial injector used in cryogenic rocket engines. The rocket engine performance is highly dependent on the injection processes such as mixing and jet dissipation of propellants in the supercritical regime. Experimental data at various temperatures and injection velocities taken by Raman imaging and Shadowgraphy were compared to computational models allowing comparisons of density, length scales and jet spreading angles providing insight into mass mixing and jet dissipation.
Contenido: Vectores y tensores; Propiedades de equilibrio; Mecánica estadística; La ecuación del estado de los gases a densidad baja y moderada; La ecuación del estado de gases densos y líquidos; Equilibrio del líquido-vapor y fenómenos críticos; Teoría cuántica y la ecuación de estado; La teoría cinética de los gases; Fenómenos de transporte y gases; Propiedades de transporte de gases densos y líquidos; Teoría cuántica y fenómenos de transporte; Aplicaciones hidrodinámicas de las ecuaciones de cambio; Bases electromagnéticas de las fuerzas intermoleculares; La teoría de las fuerzas intermoleculares; Cálculos de mecánica cuántica de fuerzas intermoleculares.
The mechanism of atomization, which is still unknown, was investigated by the simultaneous use of two photographic techniques. The initial transient was observed with a 10**6 frames/sec camera and the steady state by a technique similar to spark photography. It was found, for example, that: jet divergence begins progressively closer to the nozzle exit as the gas density increases until it reaches the exit with no evidence of abrupt change; the divergence angle (spray angle) increases with increasing gas density, and sharpness of nozzle inlet and with decreasing liquid viscosity and nozzle length; divergence angle and jet intact length are quasi-steady with respect to upstream pressure changes which occur on time scales greater than 10 to 30 mu s; aerodynamic effects, liquid turbulence, jet velocity profile rearrangements, and liquid pressure oscillations, each could not alone be the mechanism of atomization.