Compressible Flow Model Research Articles

Key features and highly effective prediction of complete startup from frozen state for high-temperature heat pipe in heat pipe reactor

High-temperature heat pipe is an efficient phase change heat transfer device, widely applied in the space nuclear reactors, the China Fusion Engineering Test Reactor, solar collectors, and the thermal protection of hypersonic vehicles. In order to understand the startup characteristics of the heat pipe and to predict its transient behavior with high accuracy, this paper first experimentally investigated the startup performance of a sodium heat pipe from the frozen state under different heating powers from 534 W to 1897 W. Then, a new compressible flow model for startup from the frozen state was developed and verified, solved the independence problems of mesh number and time-step used in the numerical treatments of most traditional heat pipe codes. Finally, based on the assessed model and experimental data, the startup procedure and the influences of heating power on the working temperature, kick-in time, and temperature difference were analyzed respectively. The comparisons indicated that the new model is extremely informative and effective, and the results provided knowledge for understanding the safe and optimum working regime of a high-temperature heat pipe using in heat pipe reactor.

Analysis of compressible bubbly flows. Part II: Derivation of a macroscopic model

This paper is the second of the series of two papers, which focuses on the derivation of an averaged 1D model for compressible bubbly flows. For this, we start from a microscopic description of the interactions between a large but finite number of small bubbles with a surrounding compressible fluid. This microscopic model has been derived and analysed in the first paper. In the present one, provided physical parameters scale according to the number of bubbles, we prove that solutions to the microscopic model exist on a timespan independent of the number of bubbles. Considering then that we have a large number of bubbles, we propose a construction of the macroscopic variables and derive the averaged system satisfied by these quantities. Our method is based on a compactness approach in a strong-solution setting. In the last section, we propose the derivation of the Williams–Boltzmann equation corresponding to our setting.

An Integrated Modeling Approach for Vertical Gas Migration Along Leaking Wells Using a Compressible Two-Fluid Flow Model

Gas migration behind casings can occur in wells where the annular cement barrier fails to provide adequate zonal isolation. A direct consequence of gas migration is annular pressure build-up at wellhead, referred to as sustained casing pressure (SCP). Current mathematical models for analyzing SCP normally assume gas migration along the cemented interval to be single-phase steady-state Darcy flow in the absence of gravity and use a drift-flux model for two-phase transport through the mud column above the cement. By design, such models do not account for the possible simultaneous flow of gas and liquid along the annulus cement or the impact of liquid saturation within the cemented intervals on the surface pressure build-up. We introduce a general compressible two-fluid model which is solved over the entire well using a newly developed numerical scheme. The model is first validated against field observations and used for a parametric study. Next, detailed studies are performed, and the results demonstrate that the surface pressure build-up depends on the location of cement intervals with low permeability, and the significance of two-phase co-current or counter-current flow of liquid and gas occurs along cement barriers that have an initial liquid saturation. As the magnitude of the frictional pressure gradient associated with counter-current of liquid and gas can be comparable to the relevant hydrostatic pressure gradient, two-phase flow effects can significantly impact the interpretation of the wellhead pressure build-up.

High-order modeling of multiphase flows: Based on discrete Boltzmann method

The high-order kinetic model for compressible multiphase flow is presented within the framework of the discrete Boltzmann method (DBM). Based on the Carnahan–Starling state equation, this model can describe the phase transition by introducing a source term of molecular interaction on the right-hand side of the Boltzmann equation. Meanwhile, the force term is incorporated to describe the external force. Through Hermite polynomial expansion, the equilibrium distribution function is expressed. Compared to the Navier–Stokes equations, the DBM provides more detailed and accurate information on both hydrodynamic and thermodynamic non-equilibrium effects. Finally, the model is verified through several typical benchmarks, including the liquid–vapor coexistence curve, free-falling process, shock wave, sound wave, thermal phase separation, and two-bubble oblique collision.

Numerical simulations of cavitating water jet by an improved cavitation model of compressible mixture flow with an emphasis on phase change effects

Focusing on cavitation phenomena caused by high-speed submerged water jets, this paper presents an improved cavitation model for a compressible fluid mixture based on a concise estimation of fluid compressibility that considers phase change effects. The homogeneous two-phase flow assumption is adopted, and the gas phase is assumed to consist of vapor and non-condensable components. Equations of state for a pure liquid and an ideal gas are employed to evaluate the compressibility of the liquid and non-condensable components, and the compressibility of the vapor is treated semi-empirically as a constant. The model is embedded in an unsteady Reynolds-averaged Navier–Stokes solver, with the realizable k-ε model employed to evaluate the eddy viscosity. The turbulent cavitating flow caused by an impulsively started submerged water jet is treated. The pattern of periodic cavitation cloud shedding is acceptably captured, and the mass flow rate coefficient and its fluctuation frequency evaluated by simulations agree with the experimental results well. The validity of the proposed method is confirmed. The results reveal that cavitation occurs when pin/Pin reaches 0.65 and fluid flow begins to pulsate. In the well-developed stage, the leading cavitation cloud and a subsequent cloud are successively shed downstream, and this process is repeated. The subsequent cloud catches the leading cloud, and they coalesce in the range x/d≈ 2–3. The pressure fluctuations concentrate in the range of x/d≈2–5 corresponding to the periodic shedding of cavitation clouds. The mass flow rate coefficient pulsates from 0.59–0.66 under the effect of cavitation.

Scaling Approaches to Quasigeostrophic Theory for Moist, Precipitating Air

Abstract Quasigeostrophic (QG) theory is of fundamental importance in the study of large-scale atmospheric flows. In recent years, there has been growing interest in extending the classical QG plus Ekman friction layer model (QG–Ekman) to systematically include additional physical processes known to significantly contribute to real-life weather phenomena. This paper lays the foundation for combining two of these developments, namely, Smith and Stechmann’s family of precipitating quasigeostrophic (PQG) models on the one hand, and the extension of QG–Ekman for dry air by a strongly diabatic layer (DL) of intermediate height (QG–DL–Ekman) on the other hand. To this end, Smith and Stechmann’s PQG equations for soundproof motions are first corroborated within a general asymptotic modeling framework starting from a full compressible flow model. The derivations show that the PQG model family is naturally embedded in the asymptotic hierarchy of scale-dependent atmospheric flow models introduced by one of the present authors. Particular emphasis is then placed on an asymptotic scaling regime for PQG that accounts for a generic Kessler-type bulk microphysics closure and is compatible with QG–DL–Ekman theory. The detailed derivation of a moist QG–DL–Ekman model is deferred to a future publication.

An experimental facility for detailed studies on energy absorbing components subjected to blast loading

AbstractDeformable components such as sandwich structures possess promising properties for use in protection systems. Detailed studies on energy absorption and fluid–structure interaction effects are necessary for the application of deformable sandwich structures in blast resistant design. In this paper, an existing shock tube facility has been extended with a transparent section to observe and measure fluid flow and the structural response of deformable components during transient dynamic loading. The extension was instrumented with pressure sensors and load cells to measure the pressure and force transmitted through the component during testing. The transparent design allows the use of optical measurement techniques. Here, high‐speed cameras were used both for digital image correlation and background‐oriented schlieren imaging. Tests with free‐standing plates and sandwich components were performed. A strong dependency was observed between the plate mass, and thus the velocity of the plates, and the pressure measured upstream and downstream of the components. The tests were simulated with a one‐dimensional numerical model for compressible shock flow with fluid–structure interaction. The numerical model accurately reproduced the shock flow and component displacements measured experimentally. Overall, the experimental set‐up presented in this study proved to be suitable for the detailed examination of deformable components subjected to airblast loading.

On the computation of compressible multiphase flows with heat and mass transfer in elastic pipelines

The present work is motivated by the engineering need to simulate the multiphase flows in the course of the start-up of a long-distance crude oil pipeline. We propose a model for compressible multiphase flows with heat and mass transfer and diffusion processes in elastic pipelines. The model is derived with two approaches, i.e., (a) the spatial averaging procedure of a single phase model and (b) the Arbitrary Lagrangian Eulerian (ALE) formulation of the Baer-Nunziato model with quasi-1D approximation. The diffusion processes include the viscous dissipation, wall friction, heat conduction, and heat exchange with external environment. In particular, the wall friction consists of steady friction term calculated by Darcy-Weisbach formula, and the unsteady friction determined by the instantaneous-acceleration-based (IAB) model. The unsteady friction modifies the characteristic structure of the hyperbolic part of the model. For the solution of the hyperbolic part, we propose a three-wave approximate Riemann solver incorporating the unsteady friction term. Mass and heat transfer are realized via instantaneous relaxations of the chemical potential and temperature, respectively. Efficient iterative relaxation procedures for N-phase flows have been proposed. We have validated the proposed model and numerical methods against some benchmark multiphase problems and applied the model to calculate the start-up of a realistic liquid pipeline with intermediate pump station boundary condition.

Shock location effect on direct contact condensation stability

Direct contact condensation (DCC) is an efficient way of transferring heat between fluids’ vapor and liquid phases by mixing them directly rather than using a device like a heat exchanger. However, in the field, DCC may produce destructive and unwanted vibrations due to the instability of the gaseous plume condensing in the process fluid under certain conditions. Previous research on this topic has focused on correlating the flow conditions of the liquid and vapor phases to engineering quantities, such as the heat transfer coefficient, plume length, stability, and the dominant frequency exhibited by the DCC process. The present study investigates the impact of nozzle geometry on plume stability. It was motivated by the drastic difference observed experimentally between the stability of two identically designed nozzles with slightly different microfeatures. Tests were performed with several nozzle geometries, including converging/diverging, straight, and converging nozzles. The results from these experiments show that a nozzle with a supersonic flow at the exit plane is more likely to be stable than one with a subsonic exit flow. It is, therefore, possible to correlate the pressure ratio across the nozzle and nozzle geometry to the stability of the plume. This understanding allows a DCC nozzle geometry to be analyzed and designed for high stability with relatively simple one- and two-dimensional compressible flow modeling tools capable of resolving only the single-phase, compressible flow in the nozzle itself

An entropy consistent and symmetric seven-equation model for compressible two-phase flows

We propose new seven-equation model and related solution methods for compressible two-phase flows. The model is built on new closure laws for the interaction between the coexisting two phases. It takes a symmetric form for the two phases and shares some favorable properties with the well-known Baer-Nunziato model [4]: the equations system is unconditionally hyperbolic and is conservative for the total momentum and energy; the interfacial interaction between phases is an isentropic process for each phase. Moreover, it permits to obtain a linear degenerate field associated with the wave which separates the mixture, and the full set of the associated Riemann invariants can be derived explicitly. The pressure and velocity relaxation terms are included for the additional interaction between phases. Numerical approximation of the entire model is performed in an operator splitting manner – two Godunov-type schemes utilizing respectively a HLL solver and a composite approximate Riemann solver are used to discretize the homogeneous system; stable implicit algorithms are explored for the non-instantaneous pressure and velocity relaxations. Several examples are provided to illustrate the property of the new model and the capability of the solution methods.

A consistent and conservative Phase-Field method for compressible multiphase flows with shocks

A key challenge with interface-capturing methods for compressible multiphase flows is that the numerical diffusion of upwind schemes causes material interfaces to continuously diffuse over time, possibly in a non-uniform fashion. In the present study, a consistent and conservative Phase-Field model for compressible multiphase flows is derived to control mixing between different fluids at material interfaces due to numerical diffusion. The model is general as it admits an arbitrary number of phases and has no prior assumption of the formulation of the Phase-Field mechanism. Necessary physical constraints to formulate the Phase-Field mechanism are obtained by analyzing physical principles, including the second law of thermodynamics. To numerically solve the model, two consistency requirements and a general numerical strategy, called reduction consistent formulation, are proposed. The proposed approach is consistent, conservative, bound/positivity preserving for the volume fraction/mass, and prevents spurious errors at interfaces. Various two-phase compressible flow problems including shocks and interfaces are performed to verify the analysis and demonstrate the efficacy of the method. Finally, temperature equilibrium and the incompressible limit are also discussed.

A unified consistent source term computational algorithm for the [formula omitted]-based compressible multi-fluid flow model

Using the criterion of one-fluid preservation, we developed a consistent algorithm for the source term to solve the γ-based compressible multi-fluid flow model with three approximate Riemann solvers, namely the Lax–Friedrichs (LxF), Kurganov, and Harten–Lax–van Leer contact (HLLC) solvers. The consistent algorithm comprises a standard Godunov solver with a high-order reconstruction and a consistent source term integration part. We prove that the present algorithm is consistent with Abgrall's criterion of the moving-material-interface property in the finite volume method framework. The cell boundary velocity for the source term discretization is found to be the same as HLLC solvers from five-equation model, but for the first time for LxF and Kurganov Riemann solvers. We develop 12 compressible multi-fluid solvers by combining four reconstruction schemes and three approximate Riemann solvers together with their consistent-source-term integration algorithms. All 12 solvers can maintain the one-fluid preservation and moving-material-interface properties numerically in several one- and two-dimensional example cases. The simulation of an underwater explosion demonstrates that a boundary-variation-diminishing reconstruction predicts an interface with width controlled within approximately three cells which is independent of the Riemann solver. The simulation of an explosion near a free surface demonstrates that the proposed model can simulate a severe compressible multi-fluid flow involving an interface across which there are large differences in density, pressure and parameters in the equation of state. In conclusion, the proposed consistent algorithm provides a unified framework for one kind of non-conservative hyperbolic system into a conservative hyperbolic system and a source term with velocity divergence, where the former can be computed by classical Godunov-type algorithms and the latter can be solved by the proposed consistent algorithm.

Generic five-equation model for compressible multi-material flows and its corresponding high-fidelity numerical algorithms

The five-equation model is widely used to simulate compressible multi-material flows. However, disunity and anomalous phenomena exist in both the model and corresponding numerical simulations. In this study, we first derived a unified formulation for the five-equation model, establishing a generic five-equation model. This model can not only recover existing typical five-equation models and generate new models but also provide a flexible framework for numerically simulating compressible multi-material flow problems in a consistent manner. Based on these properties, we constructed a suite of high-fidelity numerical algorithms, including a consistent algorithm for equations of volume fractions, technique for avoiding potential Mach number oscillation, and technique for restoring the non-monotonic sound speed if it is indeed physically possible. The proposed model and its corresponding high-fidelity numerical algorithms are verified on one-, two-, and three-dimensional problems, including artificial mixtures for separating pure or nearly pure fluids and physical fluid mixtures, and materials modeled as an ideal gas, stiffened gas, and Mie-Grüneisen-type equation of state. The numerical results validate our conclusions and confirm the broad applicability of the proposed model and its corresponding algorithm.

Numerical simulation and analysis of the chain-reaction implosions of multi-spherical hollow ceramic pressure hulls in deep-sea environment

Thin-walled hollow ceramic pressure hulls on deep-sea underwater vehicles are at risk for highly destructive chain-reaction implosions. A numerical method of simulating the chain-reaction implosions of multiple ceramic pressure hulls in deep-sea environment was developed. The fluid solver used for this method adopted the compressible multiphase flow model and adaptive mesh refinement, combining the finite element method and the failure criteria of brittle materials to determine the conditions that trigger an implosion. An implosion experiment was conducted for a single ceramic pressure hull, and the experimental results verified the accuracy of the fluid solver based on compressible multiphase flow theory. The chain-reaction implosions of two ceramic pressure hulls were also computed. The computation results showed that the air cavity in a spherical pressure hull diffused the expansion wave during the compression stage. The pressure drop in the flow field initiated by the expansion wave caused the ceramic spherical shell to reach its ultimate strength, thus triggering chain-reaction implosions. The two implosion shockwaves were superimposed during the diffusion process. The peak pressure at the superposition position of the two shockwaves is related to the spacing between the two pressure hulls.

URANOS: A GPU accelerated Navier-Stokes solver for compressible wall-bounded flows

URANOS, a massively parallel GPU-accelerated compressible flow solver for high-fidelity modeling of compressible wall flows, is presented. URANOS is based on modern high-fidelity and high-resolution discretization strategies for time-accurate compressible flow predictions. The solver provides six different convective scheme implementations, a cutting-edge method for viscous terms treatment, and three different frameworks for turbulence modeling (DNS, LES, and innovative WMLES), while utilizing a high-order finite-difference approach, ranging from second to sixth order spatial accuracy. This paper details all of these numerical models and implementation issues. Computationally, URANOS combines multiple three-dimensional MPI parallelization strategies with the open standard, OpenACC, for machine wide, on node, and on GPU parallelism, tailoring the software to match the state-of-the-art HPC computing facilities. Special attention is given to the GPU porting and data management, detailing the solver scaling properties and performance for multi-node/-GPU systems on three distinct architectures. In these experiments the different MPI strategies are compared for both GPU-aware MPI/GPU (Host-based MPI) and results are provided proving, consistently, that the GPU-aware MPI outperforms the GPU (Host-based MPI) approach. Furthermore, the GPU version is compared to the CPU only demonstrating over three times speed-up for a node-to-node comparison, even at large scales, with an efficiency of about 80% while using 1024 GPUs. Finally, the manuscript presents several validation benchmarks, from simple academic comparisons to turbulent channel and boundary layer configurations. URANOS is compared with the best DNS data available in the literature across various Mach numbers, ranging from almost incompressible conditions to hypersonic regimes, illustrating the solver's capability in treating a wide range of complex wall-flows problems using DNS, wall-resolved, and wall-modeled LES approaches. Thus, using OpenACC as a paradigm, URANOS provides a straightforward and efficient platform that can efficiently exploit the most modern GPU-accelerated computing architectures in a totally open-source and non-vendor-specific way. The solver represents a framework where the CFD community can explore the limits of modeling, test new models/methods all across a broad range of Mach and Reynolds numbers flows. The solver is released under a BSD license. Program summaryProgram title: Unsteady Robust All-around Navier-StOkes Solver (URANOS)CPC Library link to program files:https://doi.org/10.17632/pw5hshn9k6.1Developer's repository link:https://gitlab.com/fralusa/uranos_gpuLicensing provisions: BSD License 2.0Programming language: Fortran 90, OpenACC, MPINature of problem: Solving the compressible Navier-Stokes system of equations in a three-dimensional Cartesian framework from low to high Mach and Reynolds numbers conditions.Solution method: Convective terms are treated with several high-order high-resolution shock-capturing schemes (WENO/TENO) in hybrid and non-hybrid versions. The system dynamics is advanced in time with a three-stage, third-order total variation diminishing Runge-Kutta method. Three-dimensional MPI parallelization is implemented in several optimized arrangements and coupling OpenACC directives in a genuinely multi-GPU logic.

One-dimensional turbulence modeling of compressible flows. I. Conservative Eulerian formulation and application to supersonic channel flow

Accurate but economical modeling of supersonic turbulent boundary layers is a standing challenge due to the intricate entanglement of temperature, density, and velocity fluctuations on top of the mean-field variation. Application of the van Driest transformation may describe well the mean state but cannot provide detailed flow information. This lack-in modeling coarse and fine-scale variability is addressed by the present study using a stochastic one-dimensional turbulence (ODT) model. ODT is a simulation methodology that represents the evolution of turbulent flow in a low-dimensional stochastic way. In this study, ODT is extended to fully compressible flows. An Eulerian framework and a conservative form of the governing equations serve as the basis of the compressible ODT model. Computational methods for statistical properties based on ODT realizations are also extended to compressible flows, and a comprehensive way of turbulent kinetic energy budget calculation based on compressible ODT is put forward for the first time. Two canonical direct numerical simulation cases of supersonic isothermal-wall channel flow at Mach numbers 1.5 and 3.0 with bulk Reynolds numbers 3000 and 4880, respectively, are used to validate the extended model. A rigorous numerical validation is presented, including the first-order mean statistics, the second-order root mean square statistics, and higher-order turbulent fluctuation statistics. In ODT results, both mean and root mean square profiles are accurately captured in the near-wall region. Near-wall temperature spectra reveal that temperature fluctuations are amplified at all turbulent scales as the effects of compressibility increase. This phenomenon is caused by intensified viscous heating at a higher Mach number, which is indicated by the steeper profiles of viscous turbulent kinetic energy budget terms in the very near-wall region. The low computational cost and predictive capabilities of ODT suggest that it is a promising approach for detailed modeling of highly turbulent compressible boundary layers. Furthermore, it is found that the ODT model requires a Mach-number-dependent increase in a viscous penalty parameter Z in wall-bounded turbulent flows to enable accurate capture of the buffer layer.

One-dimensional turbulence modeling of compressible flows: II. Full compressible modification and application to shock–turbulence interaction

One-dimensional turbulence (ODT) is a simulation methodology that represents the essential physics of three-dimensional turbulence through stochastic resolution of the full range of length and time scales on a one-dimensional domain. In the present study, full compressible modifications are incorporated into ODT methodology, based on an Eulerian framework and a conservative form of the governing equations. In the deterministic part of this approach, a shock capturing scheme is introduced for the first time. In the stochastic part, one-dimensional eddy events are modeled and sampled according to standard methods for compressible flow simulation. Time advancement adjustments are made to balance comparable time steps between the deterministic and stochastic parts in compressible flows. Canonical shock–turbulence interaction cases involving Richtmyer–Meshkov instability at Mach numbers 1.24, 1.5, and 1.98 are simulated to validate the extended model. The ODT results are compared with available reference data from large eddy simulations and laboratory experiments. The introduction of a shock capturing scheme significantly improves the performance of the ODT method, and the results for turbulent kinetic energy are qualitatively improved compared with those of a previous compressible Lagrangian ODT method [Jozefik et al., “Simulation of shock–turbulence interaction in non-reactive flow and in turbulent deflagration and detonation regimes using one-dimensional turbulence,” Combust. Flame 164, 53 (2016)]. For the time evolution of profiles of the turbulent mixing zone width, ensemble-averaged density, and specific heat ratio, the new model also yields good to reasonable results. Furthermore, it is found that the viscous penalty parameter Z of the ODT model is insensitive to compressibility effects in turbulent flows without wall effects. A small value of Z is appropriate for turbulent flows with weak wall effects, and the parameter Z serves to suppress extremely small eddy events that would be dissipated instantly by viscosity.

On generalized compressible fluid systems on an evolving surface with a boundary

We consider compressible fluid flow on an evolving surface with a smooth boundary from an energetic point of view. We employ both an energetic variational approach and the first law of thermodynamics to make mathematical models for compressible fluid flow on the evolving surface. Moreover, we investigate the boundary conditions in co-normal direction for our fluid systems to study the conservation and energy laws of the systems.

Mathematical model of turbulent compressible flow based on two-fluid approach

BackgroundThe paper presents a two-fluid approach to modeling turbulent compressible flow. This study is an expansion of the two-fluid turbulence model. MethodsA mathematical model of turbulent compressible heat is constructed. The mathematical model of turbulence obtained is used to study the flow around a flat plate, the propagation of a hot and cold supersonic jets and high-speed mixing layer. To validate the model, the numerical results were compared with known experimental data. Significant findingsThe results obtained showed that the developed two-fluid model describes a turbulent compressible flow with good accuracy and versatility.

On the discrete equation model for compressible multiphase fluid flows

The modeling of multi-phase flow is very challenging, given the range of scales as well as the diversity of flow regimes that one encounters in this context. We revisit the discrete equation method (DEM) for two-phase flow in the absence of heat conduction and mass transfer. We analyze the resulting probability coefficients and prove their local convexity, rigorously establishing that our version of DEM can model different flow regimes ranging from the disperse to stratified (or separated) flow. Moreover, we reformulate the underlying mesoscopic model in terms of an one-parameter family of PDEs that interpolates between different flow regimes. We also propose two sets of procedures to enforce relaxation to equilibrium. We perform several numerical tests to show the flexibility of the proposed formulation, as well as to interpret different model components. The one-parameter family of PDEs provides an unified framework for modeling mean quantities for a multiphase flow, while at the same time identifying two key parameters that model the inherent uncertainty in terms of the underlying microstructure.