Compressible Flow Equations Research Articles

Rigorous justification of hydrostatic approximation of compressible fluid flow equations

In this paper, we obtain the 3D compressible primitive equations approximation with gravity by taking the small aspect ratio limit to the Navier-Stokes equations. We use the versatile relative entropy inequality to prove rigorously the limit from the compressible Navier-Stokes equations with a pressure law of the form p(ρ) = ρ2 to the compressible primitive equations.

Efficient spectral element method for the Euler equations on unbounded domains

Mitigating the impact of waves leaving a numerical domain has been a persistent challenge in numerical modeling. Reducing wave reflection at the domain boundary is crucial for accurate simulations. Absorbing layers, while common, often incur significant computational costs. This paper introduces an efficient application of a Legendre-Laguerre basis for absorbing layers for two-dimensional non-linear compressible Euler equations. The method couples a spectral-element bounded domain with a semi-infinite region, employing a tensor product of Lagrange and scaled Laguerre basis functions. Semi-infinite elements are used in the absorbing layer with Rayleigh damping. In comparison to existing methods with similar absorbing layer extensions, this approach, a pioneering application to the Euler equations of compressible and stratified flows, demonstrates substantial computational savings. The study marks the first application of semi-infinite elements to mitigate wave reflection in the solution of the Euler equations, particularly in nonhydrostatic atmospheric modeling. A comprehensive set of tests demonstrates the method's versatility for general systems of conservation laws, with a focus on its effectiveness in damping vertically propagating mountain gravity waves, a benchmark for atmospheric models. Across all tests, the model presented in this paper consistently exhibits notable performance improvements compared to a traditional Rayleigh damping approach.

Influence of thermal buoyancy on the wake dynamics of a heated square cylinder

Direct numerical simulation of the three-dimensional (3-D) wake transition of a heated square cylinder subjected to horizontal cross-flow is performed in the presence of buoyancy. In order to capture the effects of large-scale heating, a non-Oberbeck–Boussinesq model is utilized, which includes the governing equations for compressible gas flow. All computations are performed at low free stream Mach number $M=0.1$ using air (free stream Prandtl number, $Pr=0.71$ ) as the working fluid. The 3-D instability modes A and B, which correspond to free stream Reynolds numbers of 180 and 250, are observed with longer and shorter spanwise wavelengths, respectively, and the onset of three-dimensionality is triggered at a Reynolds number of 173. In the presence of buoyancy, baroclinic vorticity production in the near-wake plays an important role for streamwise vorticity generation. The chaotic wake of the Mode-A instability bifurcates into periodic and quasiperiodic wakes at various heating levels, expressed by the overheat ratio, $\varepsilon =(T_w-T_\infty )/T_\infty$ , where $T_w$ and $T_\infty$ are the temperature of the cylinder surface and the ambient air, respectively. At low heating ( $\varepsilon =0.2$ ), the 3-D Mode-A instability is suppressed leading to a two-dimensional wake flow. Further increase in heating, again brings back the three-dimensionality in the wake through Mode-E instability. The variation of thermophysical properties and the effective Reynolds number with increase in heating level around the cylinder is examined. It is shown that the effect of thermophysical properties competes with the baroclinic streamwise vorticity generation at higher levels of heating ( $\varepsilon \geqslant 0.4$ ) to control the 3-D modes and wake dynamics.

Turbulence effects in the topology optimization of compressible subsonic flow

AbstractTurbulence significantly influences fluid flow topology optimization, and this has already been verified under the incompressible flow regime. However, the same cannot be said about the compressible flow regime, in which the density field now affects and couples all of the fluid flow and turbulence equations and makes obtaining the adjoint model, which is necessary for topology optimization, extremely difficult. Up to now, the turbulence phenomenon has still not been considered in compressible flow topology optimization, which is what is being proposed and analyzed here. Rather than being based in the Reynolds‐Averaged Navier–Stokes (RANS) equations which are defined only for incompressible flow, the equations are now based on the Favre‐Averaged Navier–Stokes (FANS) equations, which are the counterpart of the RANS equations for compressible flow and feature different dependencies and terms. The compressible turbulence model being considered is the compressible version of the Spalart–Allmaras model, which differs from the usual Spalart–Allmaras model, since now there are some new spatially varying density and specific heat terms that depend on the primal variables and that act over some of the turbulence terms of the overall model. The adjoint equations are obtained by using an automatic differentiation scheme through a coupled software platform. The optimization algorithm is IPOPT, and some examples are presented to show the effect of turbulence in the compressible flow topology optimization.

Global existence of entropy solutions for euler equations of compressible fluid flow

Global existence of entropy solutions for euler equations of compressible fluid flow

On steady state of viscous compressible heat conducting full magnetohydrodynamic equations

This paper is concerned with the study of equations of viscous compressible and heat-conducting full magnetohydrodynamic (MHD) steady flows in a horizontal layer under the gravitational force and a large temperature gradient across the layer. We assume as boundary conditions, periodic conditions in the horizontal directions, while in the vertical directions, slip-boundary is assumed for the velocity, vertical conditions for the magnetic field, and fixed temperature or fixed heat flux are prescribed for the temperature. The existence of stationary solution in a small neighborhood of a stationary profile close to hydrostatic state is obtained in Sobolev spaces as a fixed point of some nonlinear operator.

Vortex breakdown in the small Mach number regimes

The effect of compressibility for flow inside the cylinder with the top rotating lid (Vogel-Escudier flow) is examined. Three-dimensional Navier–Stokes equations for compressible flow in Cartesian coordinates are used to simulate the flow using open-source OpenFOAM software. The Mach number (Ma) of the flow is varied from 0.1 to 0.3, and the Reynolds number (Re) is varied from 1000 to 5000 for a fixed aspect ratio (Γ = 2.5) of the cylinder. The flow is found to have a transition from a steady axisymmetric state to a non-axisymmetric state exhibiting multiple azimuthal waves as the Mach number and the Reynolds number are varied. The flow field changes significantly with an increase in Ma for unsteady flow at higher Re. An increase in Ma increases the side wall azimuthal instability, as found in the perturbation contour plots and time series analysis. Further, we reconstruct phase portraits to show the dynamics of the flow finally becoming chaotic as the Reynolds number is increased to 5000. Finally, we support the argument with Lyapunov exponents for the higher Re samples. The Lyapunov Exponent is found to increase with Ma.

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.

Wall-Modeled Large-Eddy Simulation of Transonic Buffet over NASA-CRM Using FFVHC-ACE

This study investigates the capability of wall-modeled large-eddy simulation (WMLES) in predicting the transonic buffet phenomenon over three-dimensional aircraft configurations at high Reynolds numbers. The WMLES is conducted using the Cartesian-grid-based flow solver FrontFlow/Violet Hierarchical Cartesian for Aeronautics Based on Compressible-Flow Equations (FFVHC-ACE). To extend the capability of FFVHC-ACE to transonic flow simulations with shock waves, a hybrid kinetic energy and entropy preserving/upwind scheme is implemented in FFVHC-ACE. To validate the employed simulation framework, the flow around the NASA Common Research Model in the near-cruise condition is simulated. In this case, the surface pressure distributions and force coefficients are reasonably predicted and show the trend of grid convergence. Furthermore, the WMLES at the buffet conditions reproduces the wavy structures of the shock wave (i.e., buffet cells) propagating outboard. The obtained propagation speed of the buffet cells and surface pressure spectra show reasonable agreement with the wind tunnel experiments.

КОМП'ЮТЕРНЕ МОДЕЛЮВАННЯ НЕСТАЦІОНАРНОГО ПРОЦЕСУ РОЗВИТКУ НАДЗВУКОВОГО ПЕРЕРОЗШИРЕНОГО СТРУМЕНЮ

Questions of numerical simulation of non-stationary process of the development of a supersonic overexpanded jet are considered. Unsteady Reynolds-averaged Navier-Stokes equations (URANS) of compressible flows, written in an arbitrary coordinate system, along with the Spalart-Allmaras one-equation differential turbulence model are applied. The calculations of non-stationary formation of an air cold supersonic jet during the blowing through a conical Laval nozzle (simulation of engine start-up) were made. The physical features of the shock and acoustic waves generation and propagation are discussed. In the stationary phase an overexpanded free supersonic turbulent jet is formed, which includes a Mach disc in its structure. The obtained results were compared with experimental data and calculations by other authors based on the ANSYS Fluent and FloEFD CFD tools.

Numerical solution of steady nonlinear differential equations for compressible flow through a spinning convergent divergent nozzle

Many projectiles tend to spin about their longitudinal axis while progressing in the forward direction. It helps in providing stability and a reference direction for guidance during their run. Many different projectiles employ a supersonic convergent-divergent nozzle to produce thrust for their forward motion; hence, the nozzle and overall whole propulsion system tend to spin about its axis of rotation. The main aim of this study is to observe the effect of spin on the nozzle. In this research, a converging bell-shaped diverging nozzle is numerically designed using a method of characteristics (MOC) for exit Mach number 3.21. Viscous simulations are performed for both two- and three-dimensional cases. The analysis is then performed with nozzle spinning about its axis of symmetry with a constant angular velocity of 10 revolutions per second. The analysis is repeated for the value of constant angular velocities to be 15 and 20 revolutions per second, and the behavior of flow with increasing angular velocity is examined. It has been observed that the exit Mach number and velocity decrease due to the radial protrusion of the boundary layer, and it has a negative impact on the performance of the nozzle. Moreover, the decrease of exit Mach number is in direct relation to increasing angular velocity.

The limiting behavior of Riemann solutions to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force

In this paper, we investigate the limiting behavior of Riemann solutions to the Euler equations of compressible fluid flow for modified Chaplygin gas with the body force as the two parameters tend to zero. The formation of delta shock waves and the vacuum states is identified and analyzed during the process of vanishing pressure in the Riemann solutions. The concentration and cavitation are fundamental and physical phenomena in fluid dynamics, which can be mathematically described by delta shock waves and vacuums, respectively. In this paper, our main objective is to rigorously investigate the formation of delta shock waves and vacuums and observe the concentration and cavitation phenomena. First, the Riemann problem of the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force is solved. Second, we rigorously confirm that, as the pressure vanishes, any two shock Riemann solution to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force tends to a δ-shock solution to the pressureless gas dynamics model with a body force, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force tends to a solution consisting of four contact discontinuities together with vacuum states with three different virtual velocities in the limiting situation.

Modeling and simulation of the cavitation phenomenon in turbopumps

This paper presents a model for simulating a two-phase flow involving cavitation and condensation, focusing on its application to turbopump flows. The model is derived by extending Kapila's approach, which incorporates thermal and free energy relaxation terms, to compressible multiphase flow equations in a rotating reference frame. The numerical method employed is a finite volume approach based on the Godunov scheme, ensuring pressure equilibrium and total energy conservation through a relaxation step. The model utilizes separate Equations of State (EOS) for the liquid and vapor phases, with the experimental saturation curve, expressed as psat(T), being the only parameter needed for cavitation modeling. To broaden its applicability beyond axial pumps, a specific Riemann solver is developed to couple rotating and non-rotating regions within the flow domain. The thermodynamic phase change process is validated using simple cases, and the model is further utilized to simulate water flows in a turbopump inducer. The characteristic curve of the pump is obtained in the non-cavitating regime for several mass flow rates conditions, and the model is successfully assessed in severe cavitation conditions, demonstrating the performance breakdown caused by vapor pockets. The modeling also provides detailed flow descriptions, including analysis of vapor pockets and their impact on blade pressure loading.

A Spectral/hp-Based Stabilized Solver with Emphasis on the Euler Equations

The solution of compressible flow equations is of interest with many aerospace engineering applications. Past literature has focused primarily on the solution of Computational Fluid Dynamics (CFD) problems with low-order finite element and finite volume methods. High-order methods are more the norm nowadays, in both a finite element and a finite volume setting. In this paper, inviscid compressible flow of an ideal gas is solved with high-order spectral/hp stabilized formulations using uniform high-order spectral element methods. The Euler equations are solved with high-order spectral element methods. Traditional definitions of stabilization parameters used in conjunction with traditional low-order bilinear Lagrange-based polynomials provide diffused results when applied to the high-order context. Thus, a revision of the definitions of the stabilization parameters was needed in a high-order spectral/hp framework. We introduce revised stabilization parameters, τsupg, with low-order finite element solutions. We also reexamine two standard definitions of the shock-capturing parameter, δ: the first is described with entropy variables, and the other is the YZβ parameter. We focus on applications with the above introduced stabilization parameters and analyze an array of problems in the high-speed flow regime. We demonstrate spectral convergence for the Kovasznay flow problem in both L1 and L2 norms. We numerically validate the revised definitions of the stabilization parameter with Sod’s shock and the oblique shock problems and compare the solutions with the exact solutions available in the literature. The high-order formulation is further extended to solve shock reflection and two-dimensional explosion problems. Following, we solve flow past a two-dimensional step at a Mach number of 3.0 and numerically validate the shock standoff distance with results obtained from NASA Overflow 2.2 code. Compressible flow computations with high-order spectral methods are found to perform satisfactorily for this supersonic inflow problem configuration. We extend the formulation to solve the implosion problem. Furthermore, we test the stabilization parameters on a complex flow configuration of AS-202 capsule analyzing the flight envelope. The proposed stabilization parameters have shown robustness, providing excellent results for both simple and complex geometries.

A compressible solver for two phase-flows with phase change for bubble cavitation

The present work describes the development of a novel fully compressible solver for two-phase flows with liquid-vapour phase change. Phase change phenomena in compressible two-phase flows are driving mechanisms for a multitude of phenomena including cavitation, droplet evaporation in acoustic fields, phase change in closed environments. Those are fundamental for a large number of technological applications including combustion chambers, propellant management in space and submarine engine design. The accurate simulation of phase change phenomena in compressible flows requires a direct numerical solver thermodynamically consistent and able to describe the smallest driving scales and to handle capillary effects. The present solver has these characteristics. It solves the full system of conservation equations for real fluid compressible flows of single species written in primitive variables using a semi-implicit pressure based solver. It leads to the energy equation being a Screened Poisson equation for the pressure that can be solved as a linear system. For the sake of thermodynamic consistency, several cubic equations of state are proposed to describe liquid and vapour phases as well as evolving saturation conditions at the interface. The proposed solver can be considered as an extension of classical incompressible solvers for nucleate boiling based on sharp interface approaches towards configurations with temporally and spatially evolving saturation conditions at the interface. It enables a complete coupling between local liquid-vapour phase change at the interface and pressure variations due to local flow features or time evolving thermodynamic conditions. Its interest is demonstrated with several benchmarks of increasing complexity with an ultimate configuration involving bubble cavitation due to the pressure drop in a convergent pipe.

A COMBINED DESALINATION AND REFRIGERATION SYSTEM BASED ON SOLAR EJECTOR TECHNOLOGY

A solar ejector technology-based system that combines refrigeration and desalination was investigated for the present study. The proposed model combined a conventional ejector refrigeration system with a desalination unit to examine its ability to achieve cooling as well as produce clean water. An analytical model of the ejector was developed using 1D compressible flow equations based on mass, momentum, and energy conservation. The output from the ejector was then fed to a 1D heat exchanger model to compute the clean water production. The analytical model was implemented using the Matlab platform. A 2D axisymmetric numerical simulation of the ejector system was also performed to comprehend the internal flow structures. It has been observed that the entrainment ratio, which is the ratio of the vapor refrigerant's mass flow rate to the motive steam's mass flow rate, falls as the stagnation temperature of the motive steam increases. It was noted that the coefficient of performance (COP) rises as the evaporator temperature rises, but it is seen to decline with the rise in generator temperature. The amount of desalinated water that can be produced with the system was also explored. It was observed that the production of desalinated water increased proportionally with the rise in generator temperature. At a generator temperature of 140°C, the system obtained clean water at a rate of about 2.9 g/s, which corresponds to a 24.5% mass flow rate of the input steam.

An improved artificial compressibility method for aeroacoustics at low Mach numbers

This work presents an improved artificial compressibility method that enables correct sound propagation behaviour in low Mach number flows. As an extension of incompressible approaches, it offers the advantage of low computational costs since compressible flow equations are simplified under the isentropic assumption, and the coupling of flow and sound is preserved. The non-uniform speed of sound is considered in this method, and eigenvalue analysis of the modified governing equations reveals that the propagation speed of pseudo waves restores to that of physical acoustic waves. The effect of Mach number on the dissipation and dispersion error is then studied using a two-dimentional monopole example with uniform background flow. The results suggest that the proposed method can provide a satisfactory prediction of sound propagation when the Mach number is below 0.3. Lastly, a direct noise computation of the aerofoil trailing-edge noise problem is performed to assess its capability to deal with the interaction between flow and sound. It shows that the present method can well capture the multiple-tone phenomenon.

A Computational Fluid Dynamics Analysis of Hydrogen Leakage and Nitrogen Purging of a Solid Oxide Fuel Cell Stack

A computational study of the nitrogen purging of a solid oxide fuel cell stack enclosed in a hot box is presented. The stack operates on ammonia as a fuel, and in the case of a hydrogen leakage, the entire compartment is immediately purged with nitrogen to ensure that there are no regions with high oxygen concentrations. In addition to this, the speed at which a hydrogen leak can be detected is determined. The results are then compared to a case with a relocated nitrogen inlet. A computational fluid dynamics (CFD) model is developed using the Reynolds-averaged Navier–Stokes equations for compressible flow in combination with conservation of energy and species equations in OpenFOAM. The results suggest that for the maximum concentration of oxygen to be below 5%, the hot box should be purged for 35 s, corresponding to 1.1 kg of nitrogen, if the hot box was already heated. If the hot box was at T = 300 K, it should be purged for 95 s, corresponding to 3.0 kg of nitrogen. The purge of the heated hot box results in a heat loss of 18 kW on average. A leak could be detected in 3.2 s during open circuit voltage tests. Changing the location of the outlet does not affect the cold purge, but results in a minimum purge period of 48 s during the hot purge, and the leak could be detected in 2 s. This paper demonstrates how CFD methods can be employed in order to address questions related to hydrogen safety.

Flux approximation to the Euler equations of compressible fluid flow for extended Chaplygin gas

The concentration and cavitation are fundamental and physical phenomena in fluid dynamics, which can be mathematically described by delta shock waves and vacuums, respectively. In this paper, we are concerned with the Euler equations of compressible fluid flow when the state equation is governed by the extended Chaplygin gas, an important candidate for describing dark matter and dark energy. Our main objective is to apply the flux-approximation method to rigorously investigate the formation of delta shock waves and vacuums and observe the concentration and cavitation phenomena. First, the Riemann problem of the compressible fluid flow with n + 2 parameters including flux and pressure is solved. Then, two kinds of flux approximation are discussed, that is, both the flux perturbation and pressure tend to zero, or only the flux perturbation vanishes while the extended Chaplygin gas pressure partly tends to the Chaplygin gas pressure. The results indicate that different manners of flux approximation have their respective effects on the formation of delta shock waves. Finally, several numerical results are presented to confirm the theoretical analysis.

Sonic-supersonic solutions for the two-dimensional steady compressible multiphase flow equations

This paper is concerned with the sonic-supersonic structures for the two-dimensional steady compressible inviscid multiphase flow equations. We construct a local classical supersonic solution near a given smooth sonic curve. This problem is originated from the transonic channel multiphase flows, which are one kind of the most important problems in mathematical fluid dynamics. In order to overcome the difficulties caused by the parabolic degeneracy near sonic and the multivariable dependence of pressure, we adopt the mixed variables of the pressure and angle functions and derive the characteristic decompositions of these quantities. In terms of the angle coordinate system, the multiphase flow equations can be transformed into a new degenerate hyperbolic system with an explicit singularity-regularity structure. We verify the convergence of the iterative sequence generated by the new system and then return the solution to the original physical variables. As a by-product, we obtain the existence of sonic-supersonic solutions for the steady full Euler equations with non-polytropic gases.