Compressible Flow Research Articles

Finite element methods for multicomponent convection-diffusion

Abstract We develop finite element methods for coupling the steady-state Onsager–Stefan–Maxwell (OSM) equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical species within a common thermodynamic phase is transported by convection and molecular diffusion. Developing a variational formulation for discretizing these equations is challenging: the formulation must balance physical relevance of the variables and boundary data, regularity assumptions, tractability of the analysis, enforcement of thermodynamic constraints, ease of discretization and extensibility to the transient, anisothermal and nonideal settings. To resolve these competing goals, we employ two augmentations: the first enforces the definition of mass-average velocity in the OSM equations, while its dual modifies the Stokes momentum equation to enforce symmetry. Remarkably, with these augmentations we achieve a Picard linearization of symmetric saddle point type, despite the equations not possessing a Lagrangian structure. Exploiting structure mandated by linear irreversible thermodynamics, we prove the inf-sup condition for this linearization, and identify finite element function spaces that automatically inherit well-posedness. We verify our error estimates with a numerical example, and illustrate the application of the method to nonideal fluids with a simulation of the microfluidic mixing of hydrocarbons.

Numerical simulation of vapor explosion bubbles in the presence of a non-condensable gas and a phase change

The complex dynamics of two flow problems of vapor explosion bubbles near a free surface and between two parallel plates are simulated by a compressible three-phase flow code. Firstly, a coupled high-order monotonicity-preserving (MP) interface capturing scheme is developed for the numerical computation of compressible three-phase flows, which is followed by validation with previous numerical and experimental data. Phase changes are taken into consideration and the empirical condensation coefficient is calibrated with the experimental data. For both two flow problems, good agreement between the numerical results and reference data is obtained. Most physical phenomena, including the free surface spike, the liquid jet before the impact and the complex interface deformations during multiple expanding and collapsing oscillations, are reasonably well reproduced. Afterwards, the effects of the initial vapor pressure and the non-condensable gas volume fraction in the surrounding water on the behavior of vapor bubbles are analyzed and discussed. It is found that both the initial vapor pressure and the non-condensable gas volume fraction have a substantial influence on the behavior of the bubble. Higher initial vapor pressure does indeed tend to result in larger bubbles. The presence of non-condensable gases tends to hinder the transfer of heat, slowing down condensation and potentially resulting in larger bubbles. When the initial vapor pressure is in the range of 10 to 20 atmospheres and the gas volume fraction is in the range of 0 to 0.2, the maximum vapor bubble volume is found to increase linearly with an increase of the amount of non-condensable gas.

Exploring nanoscale metallic multilayer Ta/Cu films: Structure and some insights on deformation and strengthening mechanisms

Nanoscale metallic multilayer (NMM) films are systems offering insight into the role of interfaces in metal plasticity, deformation, and strengthening mechanisms. Magnetron sputtering was used to fabricate the Ta/Cu NMM films with a periodicity (equal Ta and Cu layer thickness) from 6 to 80 nm, and with the structure exhibiting immiscible tetragonal β-Ta and face-centred cubic Cu phases. Transmission electron microscopy and X-ray diffraction analyses revealed that, irrespective of the period, all films manifested a polycrystalline structure. The growth direction of both Cu and Ta layers was found to be along 〈001〉 β-Ta || 〈111〉 Cu directions, with the crystallite size constrained by the layer thickness. The studies showed that the Ta/Cu NMMs exhibited compressive residual macro-stress and flow strength, and enhanced elastic recovery at the periodicity of ≤12 nm. Activation volume V⁎ value of 11 to 20 b3 as determined from the indentation creep test under a constant load, may indicate a mixed deformation mechanism. This mechanism likely involves the emission of dislocations from the incoherent Ta/Cu interfaces, as well as the formation of screw dislocations within Cu grains. The high-load indentation test, TEM studies, and the rCLS model collectively demonstrate that all NMM films predominantly undergo plastic deformation. This plastic deformation primarily occurs within the soft Cu layer, while the propagation of dislocations across the incoherent interface is largely excluded.

A viscoelastic flow model of Maxwell-type with a symmetric-hyperbolic formulation

Maxwell models for viscoelastic flows are famous for their potential to unify elastic motions of solids with viscous motions of liquids in the continuum mechanics perspective. But the usual Maxwell models allow one to define well motions mostly for one-dimensional flows only. To define unequivocal multi-dimensional viscoelastic flows (as solutions to well-posed initial-value problems) we advocated in [ESAIM:M2AN 55 (2021), p. 807-831] an upper-convected Maxwell model for compressible flows with a symmetric-hyperbolic formulation. Here, that model is derived again, with new details.

Convergence of a TPFA finite volume scheme for nonisothermal immiscible compressible two-phase flow in porous media

This paper presents and analyzes a finite volume scheme for a fully degenerate parabolic system modeling the compressible two-phase flow, such as water-gas, problem in porous media considering the effects of temperature. The problem is written in terms of the phase formulation, i.e. the saturation of one phase, the pressure of the second phase and the temperature are primary unknowns. We use a first-order time implicit Euler scheme. The spatial discretization uses a fully coupled fully implicit cell-centered two-point flux approximation scheme where upwinding of the flux across an interface between two control volumes is performed for each fluid phase. Under physically relevant assumptions on the data, we prove the maximum principle for both saturation and temperature. We obtain a priori estimates and we show that the discrete problem is well posed. Also, we exploit various a priori estimates as well as compactness arguments to establish the convergence of the numerical scheme toward the weak solution. The open-source platform DuMuX was employed for the implementation of the developed algorithm based on the above scheme. Numerical result are presented to demonstrate the efficiency of this scheme. The test addresses the evolution in 2D of CO2 injection into a layered aquifer. The algorithm's convergence with first order is validated by our numerical results, which support the theoretical results. To our best knowledge, this is the first convergence result of a finite volume scheme in the case of nonisothermal immiscible compressible two-phase flow in porous media.

The stability of simple plane-symmetric shock formation for three-dimensional compressible Euler flow with vorticity and entropy

The stability of simple plane-symmetric shock formation for three-dimensional compressible Euler flow with vorticity and entropy

A tensor basis neural network-based turbulence model for transonic axial compressor flows

Traditional turbulence models encounter limitations when simulating intricate flows within transonic axial compressors. In contrast, recent advancements in machine learning turbulence models have demonstrated enhanced potential in refining the precision of turbulence modeling. Notably, the tensor basis neural network (TBNN) methodology has successfully developed non-linear eddy viscosity turbulence models. These models possess the capability to capture the anisotropic characteristic of Reynolds stress. However, applying machine learning non-linear eddy viscosity models to aircraft turbomachinery remains relatively infrequent. Current research mainly focuses on linear eddy viscosity turbulence models based on the Boussinesq hypothesis that presupposes isotropy in Reynolds stress. In this work, we introduce a non-linear eddy viscosity turbulence model, denoted as the k-ω-SST-TBNN model based on the TBNN framework. This model has been employed to simulate the transonic axial compressor NASA Rotor 37. The TBNN is trained using large eddy simulation (LES) results datasets. The importance of input scalar features is analyzed using the random forest method, from which significant and low-degree variables are selected as inputs for the TBNN. Furthermore, this study proposes including the turbulent Mach number as one of the extra features, representing fluid compressibility, thereby extending the computational mass flow rate range of compressor. Additionally, this paper proposes a method involving the weighted average of Reynolds stress, combining the high-precision but less robust TBNN predictions with results based on the Boussinesq assumption to enhance the turbulence model's robustness. The trained k-ω-SST-TBNN model undergoes validation on Rotor 37, where it exhibits a marked improvement in the prediction of overall performance, tip-gap vortex, and radial distribution of flow parameters. The model also displays a commendable capacity for generalization.

Compressible FSI of elastic spikes for drag reduction under hypersonic flow

In this study, a new fluid structure interaction (FSI) algorithm is developed for the aerodynamic-elasticity problem, aiming to address the coupling problem between compressible fluids and deformable solid structures. To verify the feasibility of the proposed model, a two-dimensional cantilever panel case under shock waves was conducted. Our numerical results present good consistency with the data from literatures. Furthermore, the validated FSI algorithm is applied on an elastic spike under compressible hypersonic flow. The complicated interaction between the fluid and solid structure under various spike geometry sizes and materials were revealed in respect of the spike deformation, pressure, temperature, and flow fields distribution. The main findings show that the spike deformation largely affects pressure and temperature on the blunt, the aerodynamic drag and lift force, indicating the necessity of FSI. For the spike diameters and lengths, it is found that larger spike diameters and shorter spikes benefits for deformation reduction, thereby minimizing fluctuations in aerodynamic drag force. For spike materials, spikes with larger elastic moduli and lower densities are preferred for the drag reduction. Thus, the proposed FSI model can accurately predict flow fields around and deformations for elastic spikes and offer guidance for drag reduction design in aerospace engineering.

Accelerated construction of projection-based reduced-order models via incremental approaches

We present an accelerated greedy strategy for training of projection-based reduced-order models for parametric steady and unsteady partial differential equations. Our approach exploits hierarchical approximate proper orthogonal decomposition to speed up the construction of the empirical test space for least-square Petrov–Galerkin formulations, a progressive construction of the empirical quadrature rule based on a warm start of the non-negative least-square algorithm, and a two-fidelity sampling strategy to reduce the number of expensive greedy iterations. We illustrate the performance of our method for two test cases: a two-dimensional compressible inviscid flow past a LS89 blade at moderate Mach number, and a three-dimensional nonlinear mechanics problem to predict the long-time structural response of the standard section of a nuclear containment building under external loading.

A COMPARATIVE ANALYSIS OF HYPOTHESIS IN THE MODELING OF A HYBRID COMPRESSION REFRIGERATION CYCLE

This paper presents a comprehensive thermodynamic analysis of a hybrid refrigeration system that integrates an evacuated tube solar thermal collector into a vapor compression refrigeration cycle. The proposed analysis comprises two distinct approaches: an isovolumetric model and a compressible flow model. The former assumes a constant specific volume within the solar heat exchanger, while the latter applies principles of compressible fluid dynamics. The study compares these models with a reference work, emphasizing similarities and disparities. The investigation systematically varies key parameters, including evaporator and condenser temperatures, inlet and outlet temperatures of the heat exchanger, and heat load. By varying these parameters, the coefficient of performance (COP) and compressor work are evaluated, elucidating the impact of heat addition on the system's performance. Additionally, the influence of different working refrigerant fluids, such as R22 and R410A, is examined under various design point conditions. The results demonstrate the potential energy savings achievable by the hybrid system, with reductions in electrical power compared to the conventional compressor, as solar heat is introduced. While the isovolumetric analysis closely aligns with the reference work, the compressible flow modeling highlights sensitivity to inlet conditions. Although not directly comparable, the latter approach presents promising trends within specific design point ranges. Comparisons of performance curves for different refrigerant fluids further validate the models and assist in potential fluid selection. Overall, the outcomes indicate a promising avenue for energy-efficient refrigeration systems and provide valuable insights for engineering design and optimization.

Stability analysis of a uniform viscous transonic flow past a rotating circular cylinder

This work analyzes the transonic flow around a rotating circular cylinder. The combination of a uniform freestream flow and the flow due to the rotation of the cylinder (assuming clockwise rotation) generates a lift force (Robins-Magnus effect) that, for transonic flows, is conditioned by the presence of a shock wave on the upper side of the cylinder.Previous studies have focused primarily on incompressible flows, compressible flows at low Mach numbers, and hypersonic flows with detached shock waves. These works have barely considered the effect of the presence of the shock wave, its interaction with the wake of the cylinder and its impact in the aerodynamic coefficients.Two-dimensional numerical simulations have been carried out using the SU2 code. From these simulations, the value of the aerodynamic coefficients as a function of time and a set of snapshots of the fluid field have been obtained and analyzed using the POD and DMD techniques. This has allowed the identification of the dominant modes that characterize the time variations in the lift and drag coefficients acting on the circular cylinder for different values of the characteristic parameters of the problem.

Numerical investigation on the use of various blade tips for the helicopter's main rotor blade in forward flight regarding aerodynamic performance and HSI noise considerations

The goal of this research is to investigate aerodynamic performance and high-speed impulsive (HSI) noise of various blade tips for the helicopter's main rotor in forward flight. For this purpose, three-dimensional compressible flow field around the rotor blades is numerically simulated by the solution of the unsteady Reynolds averaged Navier-Stokes (URANS) equations with the "SST k-ω" turbulence model. Solution is quantified by the calculation of torque, drag, and thrust coefficients of CQ, CD, CT, and L/D ratio. HSI noise on the rotor plane is assessed by the calculation of sound pressure level at certain points of the flow field using the Ffowcs Williams-Hawking (FW-H) equation. Blade tips are eleven and include anhedral, dihedral, Eagle and combinations of them, and also Blue Edge and rectangular. The combinations are inspired by the tip configurations of the fixed wings. In this regard effects of the tip geometry parameters including curvature and height of its parts have been considered as well. All of the tips are examined for two flight conditions of Caradonna and Tung two-blade rotor with fixed pitch angle and time-varying pitch angles. Based on the present results, it is found that blade tips of dihedral family do not improve the aerodynamic performance of the helicopter blade. Also, addition of anhedral part to the blade tip improves the L/D ratio. According to the present acoustic analysis, it was seen that after the Blue Edge, the Eagle tip stands in order of priority. For the condition of Caradonna and Tung two-blade rotor with tip Mach number of 0.8 and advance ratio of 0.2, HSI noise of the Eagle-tip blade decreases by 2.39 % compared to that of the two-blade rotor with rectangular tip. However, the aerodynamics performance of the Eagle tip is much better than that of the Blue Edge.

Mean-line analysis and optimal design of turbines using a proper algorithm for choked conditions

Compressible flow turbines usually work under choked conditions, requiring an analysis for the performance estimation. The Mean-line modeling is a fast and accurate method extensively used for the analysis and design purposes. However, it has not already been well-investigated for axial flow turbines specially in choked conditions. The present study investigates the issue from three novel aspects. First, a detailed study is accomplished on the inconsistencies of Mean-line method respect to the physics of real flows. It is discussed that the conditions related to maximum mass flow rate through a nozzle and the Mach number one at the throat are different. In fact, the latter condition is not quite physical. Second, a complete algorithm is utilized to implement the aforementioned considerations for the analysis of turbines. This algorithm includes an innovative Blade Solution Algorithm (BSA) and Turbine Solution Algorithm (TSA). The BSA solves the blade rows of turbine using the basic laws of Physics and real-gas properties. In the TSA, a robust and efficient procedure is presented so the solution of turbine converges in wide ranges of boundary conditions with minimum iterations. The results demonstrate that the algorithm generate close maps to the previous studies. Eventually in the third part, since a fast and robust Mean-line algorithm is crucial in optimization, the code is utilized for a design based on optimization using the Genetic Algorithm. In this procedure, all geometrical parameters are optimized as decision parameters rather than the use of old and approximate correlations. This is done along with novel constraints, which not only maximizes efficiency, but also puts the output power, mass flow rate and turbine size in appropriate ranges. The optimization results in a 3.5% rise in efficiency.

A Compressible Formulation of the One-Fluid Model for Two-Phase Flows

In this paper, we introduce a compressible formulation for dealing with 2D/3D compressible interfacial flows. It integrates a monolithic solver to achieve robust velocity–pressure coupling, ensuring precision and stability across diverse fluid flow conditions, including incompressible and compressible single-phase and two-phase flows. Validation of the model is conducted through various test scenarios, including Sod’s shock tube problem, isothermal viscous two-phase flows without capillary effects, and the impact of drops on viscous liquid films. The results highlight the ability of the scheme to handle compressible flow situations with capillary effects, which are important in computational fluid dynamics (CFD).

A thermo-mechanical, viscoelasto-plastic model for semi-crystalline polymers exhibiting one-way and two-way shape memory effects under phase change

A finite strain phenomenological model is developed to simulate the shape memory behavior of semi-crystalline polymers under thermo-mechanical loading. The polymer is considered to be a composite of crystalline and amorphous phases with constant volume fractions. While the amorphous phase is stable, the crystalline one is considered to change phase with temperature. Therefore, the crystalline phase is considered further to be composed of two phases, whose volume fractions are controlled by a temperature and strain dependent function: the melted phase which is soft, and the crystallized phase which is stiff.A pressure dependent viscoelasto-plastic behavior is considered for the constitutive model of the different phases. In addition to pressure dependent plasticity, additional deformation measures are applied to the crystalline phase to model temporary (imperfect shape fixity) and permanent (imperfect shape recovery) deformations in a thermo-mechanical loading cycle. Formulating a compressible plastic flow during the phase changes yields the possibility to capture both one-way and two-way shape memory effects. As a consequence, the load-dependent and anisotropic thermal expansion observed experimentally in semi-crystalline polymers during phase change is naturally captured.The model is validated within a test campaign performed on nano-composite having a semi-crystalline polymer as a base material. It is shown that the model gives close results with the tests and it is able to capture the shape fixity and shape recovery behaviors of the polymer, for both one-way and two-way shape memory effects.

Urban Air Pollution by Laser Photoacoustic Spectroscopy and Simplified Numerical Modeling of Gas Pollution in Urban Canyon

With rapid urbanization and industrialization, atmospheric pollution has emerged as a significant environmental challenge in Romania. Employing a laser photoacoustic spectroscopy detector, researchers analyzed ethylene, benzene, and toluene simultaneously across three distinct environmental settings in the country's southern region. This investigation spanned from March to August 2021, covering both spring and summer seasons. Measurements were taken at a breathing height of 1.5 meters above ground level. The highest concentrations of ethylene (116.82 ± 82.37 ppb), benzene (1.13 ± 0.32 ppb), and toluene (5.48 ± 3.27 ppb) were recorded at measurement point P1, situated within the city amidst residential buildings during the summer season. Additionally, the highest ozone levels (154.75 ± 68.02 ppb) were observed at point P3, located in an industrial area, during the summer. The behavior of gas concentrations is influenced by meteorological factors such as temperature, wind speed, and direction. The high toluene/benzene ratio suggests that traffic and industrial emissions are the primary sources of these pollutants. Notably, benzene and ozone concentrations exceeded prescribed limit values based on the measurements. Concurrently, a numerical model was employed to assess the impact of greenery on mitigating pollution in urban canyons. Specifically, the study focused on how wind velocity affects the dispersion of benzene pollutants in a street canyon. This study's governing equations utilized for air pollutant flow were the Reynolds-averaged Navier–Stokes (RANS) equations for compressible turbulent flow and moisture transport in air, implemented through Comsol software.

Nektar++: Development of the compressible flow solver for jet aeroacoustics

A recently developed computational framework for jet noise predictions is presented. The framework consists of two main components, focusing on source prediction and noise propagation. To compute the noise sources, the turbulent jet is simulated using the compressible flow solver implemented in the open-source spectral/hp element framework Nektar++, which solves the unfiltered Navier-Stokes equations on unstructured grids using the high-order discontinuous Galerkin method. This allows high-order accuracy to be achieved on unstructured grids, which in turn is important in order to accurately simulate industrially relevant geometries. For noise propagation, the Ffowcs Williams - Hawkings method is used to propagate the noise between the jet and the far-field. The paper provides a detailed description of the computational framework, including how the different components fit together and how to use them. To demonstrate the framework, two configurations of a single stream subsonic jet are considered. In the first configuration, the jet is treated in isolation, whereas in the second configuration, it is installed under a wing. The aerodynamic results for these two jets show strong agreement with experimental data, while some discrepancies are observed in the acoustic results, which are discussed. In addition to this, we demonstrate close to linear scaling beyond 100,000 processors on the ARCHER2 supercomputer.

Rectangular-to-elliptical shape transition intakes with various leading-edge for performance enhancement

This study explores the efficiency of streamline-tracing intake in scramjets within the hypersonic domain, where the Mach number (Ma) exceeds five. Focusing on a rectangular-to-elliptical shape transition intake following Busemann streamlines with a contraction ratio of 4.67, tailored for Ma 7.1, this study innovates beyond traditional design method, employing numerical simulations to interpolate iso-Ma points and shape the leading edge of the intake to enhance the performance. Three distinct leading-edge designs—using a sinusoidal function to closely track all iso-Ma points, a convex second-order polynomial to bolster compression and mass flow, and a concave polynomial to widen the starting envelope—were conceptualized. The research included experiments with 0.06 scaled models for validation and performance assessments of full-scale intakes using three-dimensional viscous simulations. Key performance metrics, such as the pressure ratio, total pressure recovery, mass flow rate, and exit Ma, were evaluated to reveal the capability of all three intake designs to operate effectively over a broad Ma range (3–7). A key finding of this study is the ability to selectively enhance either the compression capability or the total pressure recovery by strategically designing the leading edge of the intake through interpolation methods. This study underscores the importance of using numerical simulation results in the notching process as an effective design strategy to handle a wide range of off-design conditions with fixed geometry and achieve performance improvements in scramjet intakes.

Structural stability of transonic shock flows with an external force

This paper is devoted to the structural stability of a transonic shock passing through a flat nozzle for two-dimensional steady compressible flows with an external force. We first establish the existence and uniqueness of one-dimensional transonic shock solutions to the steady Euler system with an external force by prescribing suitable pressure at the exit of the nozzle when the upstream flow is a uniform supersonic flow. It is shown that the external force helps to stabilize the transonic shock in flat nozzles and the shock position is uniquely determined. Then we are concerned with the structural stability of these transonic shock solutions when the exit pressure is suitably perturbed. One of the new ingredients in our analysis is to use the deformation-curl decomposition to the steady Euler system developed by Weng and Xin [Sci. Sinica Math., 49 (2019), pp. 307–320] to deal with the transonic shock problem.

Unified approach to artificial compressibility and local low Mach number preconditioning

This paper presents a unified approach to artificial compressibility and local low Mach number preconditioning methods. The numerical formulation is presented, which provides a seamless transition between incompressible and compressible systems. A single preconditioner that can act as the artificial compressibility or the local preconditioning under a given Mach number condition is derived. An artificial compressibility method is proposed for the unified system. The unified preconditioner is applied to the upwind approach, Roe's approximate Riemann solver, and the JST artificial dissipation approach. A Roe average, which can be applied to the unified systems in a single form is derived. The preconditioning matrix is applied to a single solver to analyze problems ranging from incompressible fluid flows to compressible flows with shocks.