Compressible Flow Model Research Articles

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.

Global existence and optimal time decay for the viscous liquid-gas two-phase flow model in the <inline-formula><tex-math id="M1">$ L^p $</tex-math></inline-formula> critical Besov space

This paper is dedicated to the study of viscous compressible liquid-gas two-phase flow model in multi-dimensional spaces $ d\geq2 $. We investigate the global existence of strong solutions to the Cauchy problem in the $ L^p $ critical regularity framework. In comparison with previous results, the high-frequency of dissipative variable $ (\widetilde{P},\widetilde{u}) $ may be bounded in the critical space $ \dot{B}_{p,1}^{d/p}\times\dot{B}_{p,1}^{d/p-1}(p\geq2) $ and the high-frequency of non-dissipative variable $ \widetilde{\rho} $ to be bounded in the $ L^2 $-type Besov space $ \dot{B}_{2,1}^{d/2} $. Moreover, a Lyapunov-type energy argument can be developed, which leads to the time-decay estimates of solutions without additional smallness assumptions.

Numerical study of dynamics of cavitation bubble collapse near oscillating walls

This paper presents a numerical study of the dynamics of an initially spherical bubble collapse near an oscillating rigid wall with a large amplitude; the wall oscillating amplitude is greater than 1% of the initial maximum bubble radius. Numerical simulations were conducted using a compressible two-phase flow model and the volume of fluid (VOF) interphase-sharpening technique on a general curvilinear moving grid. The numerical results for bubbles in the free field and near a wall were computed and compared with published experimental data. To study the effects of the oscillating wall on bubble collapse, a sinusoidal function was used for wall oscillation. The initial bubble conditions were set as a Rayleigh bubble located above the rigid wall at a dimensionless bubble-boundary distance with initial phases of 0° and 180°. During bubble collapse, the interface deformation, jetting behavior, bubble collapse time, and bubble migration were determined. Violent collapse of the bubble, jetting behavior, and shock propagation from the significant effects of the oscillating wall were observed in simulation cases with different wall motions. The effects of the non-dimensional amplitude scale and non-dimensional period timescale were considered with the initial phases in the problem. The trend lines of typical characteristics and critical points of bubble collapse were determined.

Empirijska jednačina mahovog broja kao funkcija odnosa stagnacionog pritiska za kvazijednodimenzionalni stišljivi protok

In the present work for a quasi-one-dimensional isentropic compressible flow model, an empirical equation of the Mach number is constructed as a function of the stagnation pressure ratio for an analytical equation that algebraic procedures cannot invert. The Excel 2019 Solver tool was applied to calibrate the coefficients and exponents of the empirical equation during its construction for the Mach number range from 1 to 10 and 1 to 5. A specific heat ratio from 1.1 to 1.67 and the generalized reduced gradient iterative method were used to minimize the sum of squared error, which was set as the objective function. The results show that for Mach 1 to 10, an error of less than 0.063% is obtained, and for Mach 1 to 5, an error of less than 0.00988% is obtained. It is concluded that the empirical equation obtained is a mathematical model that reproduces the trajectories of the inverted curves of the analytical equation studied.

A new compressible flow model for pneumatic directional control valves

In this study, a new compressible flow model for small orifice openings in pneumatic proportional directional control valves has been proposed. It is crucial to precisely control pneumatic valves over all control ranges; yet, conventional flow models fail around the closed position of the valve. The main deficit of the existing studies in the literature is to assume constant values for the parameters of the flow model over changing operating conditions. It has been demonstrated that these rough assumptions are insufficient in precisely predicting the mass flow rate, particularly for small orifice openings. An enhanced experimental setup has been introduced to improve the effectiveness of the proposed model. The cracking pressure ratio and parameters of the model have been identified with experimental method. In the proposed model, new empirical coefficients have been established after a thorough investigation of the impact of supply pressure on the flow behavior of the valve. Validation studies of the model in both the filling and exhausting states of the valve have been carried out at various supply pressures and orifice openings, yielding rather promising modeling performances. In validation tests, the real pressure and the pressure produced by new model have been compared, and good agreement has been achieved with 0.0039% absolute error. According to the findings, the proposed improved flow model can be selected in precision pneumatic control applications.

Convolutional neural network modeling and response surface analysis of compressible flow at sonic and supersonic Mach numbers

Ribs as passive control in suddenly expanded flow is exciting to enhance the base pressure around nozzle exit. Furthermore, predictions of compressible flow characteristics in duct expansion using machine learning are very rare. In this work, the evolution of compressible flow through a nozzle regulated by semi-circular rib passive control is analyzed experimentally using the data acquisition technique; to control the base pressure for sonic and four supersonic Mach numbers. Results shows nozzles flowing under favorable pressure becomes effective and significantly increases the base pressure. Artificial neural networks (ANN), deep ANN (DNN), convolutional NN (CNN), and deep CNN (DCNN) were used for the modeling of compressible flow data that are highly sensitive and non-linear. For the first time, CNN used to model high-speed data predicted accurately the flow characteristics. The base pressure continuously decreases, irrespective of Mach number and Nozzle Pressure Ratio, even when the flow is under-expanded due to a higher area ratio. DNN was found to be most effective, with R-squared above 0.9 for the base pressure and pressure loss, while for wall pressure, it is above 0.8. The accuracy of base pressure predictions is between 84% and 99% during the training and testing of all NN models.

A modified wall-adapting local eddy-viscosity model for large-eddy simulation of compressible wall-bounded flow

The wall-adapting local eddy-viscosity (WALE) model in large-eddy simulation can well predict wall-bounded flows but it is also well known for excessive dissipation. In this study, we apply the minimum-dissipation model to constrain the WALE model in compressible flows and obtain the coefficient of the WALE model. Through this process, the dissipation of WALE model can be lower while it still maintains strong stability. In the modified WALE model, the isotropic part of the subgrid-scale (SGS) stress is also reconstructed. In the filtered total energy equation, all of the extra SGS unclosed terms (besides SGS stress and SGS heat flux) are modeled instead of neglecting some SGS terms, such as the SGS viscous diffusion. The modified WALE model is tested in a compressible turbulent channel flow and a supersonic turbulent boundary layer over a compression corner. The new model can well predict the mean velocity, the mean temperature, the Reynolds stress, and the separation bubble.

Determination of fan design parameters for light-sport aircraft

This paper is focused on the preliminary design of an electric fan for light-sport aircraft. Usage of electric motors brings some advantages compared to piston engines, especially small size and the independence of power on shaft RPM. A 1D compressible fluid flow model is used for the determination of the performace. The influence of various system parameters is analysed. Results for the case of the UL-39 ultralight aircraft are presented. Finally, input parameters for the fan design are determined according to this analysis. This can be then used as input data for the standard fan (axial compressor) design procedure.

A discontinuous Galerkin spectral element method for a nonconservative compressible multicomponent flow model

In this work, we propose an accurate, robust (the solution remains in the set of states), and stable discretization of a nonconservative model for the simulation of compressible multicomponent flows with shocks and material interfaces. We consider the gamma-based model by Shyue (1998) [58] where each component follows a stiffened gas equation of state (EOS). We here extend the framework proposed in Renac (2019) [53] and Coquel et al. (2021) [15] for the discretization of hyperbolic systems, with both fluxes and nonconservative products, to unstructured meshes with curved elements in multiple space dimensions. The framework relies on a high-order discontinuous Galerkin spectral element method (DGSEM) using collocation of quadrature and interpolation points as proposed by Gassner (2013) [27] in the case of hyperbolic conservation laws. We modify the integrals over discretization elements where we replace the physical fluxes and nonconservative products by two-point numerical fluctuations. The contributions of this work are threefold. First, we analyze the semi-discrete DGSEM discretization of general hyperbolic systems with conservative and nonconservative terms and derive the conditions to obtain a scheme that is high-order accurate, free-stream preserving, and entropy stable when excluding material interfaces. Second, we design a three-point scheme with a HLLC solver for the gamma-based model that does not require a root-finding algorithm for the approximation of the nonconservative products. The scheme is proved to be robust and entropy stable for convex entropies, to preserve uniform profiles of pressure and velocity across material interfaces (material interface preservation), and to satisfy a discrete minimum principle on the specific entropy and maximum principles on the parameters of the EOS. Third, the HLLC solver is applied at interfaces in the DGSEM, while we consider two kinds of fluctuations in the integrals over discretization elements: the former is entropy conservative (EC), while the latter preserves material interfaces (CP). Time integration is performed using high-order strong-stability preserving Runge-Kutta schemes. The fully discrete scheme is shown to preserve material interfaces with CP fluctuations. Under a given condition on the time step, both EC and CP fluctuations ensure that the cell-averaged solution remains in the set of states; satisfy a minimum principle on any convex entropy and maximum principles on the EOS parameters. These results have allowed us to use existing limiters in order to restore positivity, and discrete maximum principles of degrees-of-freedom within elements. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and accuracy of the proposed DGSEM with either CP, or EC fluctuations, while the scheme with CP fluctuations is shown to offer better resolution capabilities.

Mathematical modeling of transport phenomena in compressible multicomponent flows

The present article proposes a diffuse interface model for compressible multicomponent flows with transport phenomena of mass, momentum and energy (i.e., mass diffusion, viscous dissipation and heat conduction). The model is reduced from the seven-equation Baer-Nuziato type model with asymptotic analysis in the limit of instantaneous mechanical relaxations. The main difference between the present model and the Kapila's five-equation model consists in that different time scales for pressure and velocity relaxations are assumed, the former being much smaller than the latter. Thanks to this assumption, the velocity disequilibrium is retained to model the mass diffusion process. Aided by the diffusion laws, the final model still formally consists of five equations. The proposed model satisfy two desirable properties: (1) it respects the laws of thermodynamics, (2) it is free of the spurious oscillation problem in the vicinity of the diffused interface zone. For solution of the model governing equations, we implement the fractional step method to split the model into five physical steps: the hydrodynamic step, the viscous step, the heat transfer step, the heat conduction step and the mass diffusion step. The split governing equations for the hydrodynamic step formally coincide with the Kapila's five-equation model and are solved with the Godunov finite volume method. The mass diffusion, viscous dissipation and heat conduction processes contribute parabolic partial differential equations that are solved with the Chebyshev method of local iterations. Numerical results show that the proposed model maintains pressure, velocity and temperature equilibrium near the diffused interface. Convergence tests demonstrate that the numerical methods achieve second order in space and time. The proposed model and numerical methods are applied to simulate the hydrodynamic instability problems in the inertial confinement fusion. Good agreement with experimental results is observed.

A finite particle method based on a Riemann solver for modeling incompressible flows

The weakly compressible SPH (WCSPH) method has been widely used for solving hydrodynamic problems, in which the fluid is assumed to be weakly compressible through limiting the density deviation to a small value. Despite its simplicity and robustness in implementation, the WCSPH method is generally hindered by its poor accuracy and spurious oscillations in the pressure field which may lead to numerical instability. In order to dampen the spurious oscillations without introducing excessive dissipation, a Roe-type approximate Riemann solver with a low-dissipation limiter was proposed and applied to the WCSPH method by Zhang et al. (2017). The present paper aims to extend this low-dissipation Riemann solver to the finite particle method (FPM), which can be referred to an enhanced or modified version of the conventional SPH method, for reducing spurious pressure oscillations and meanwhile achieving higher order of accuracy. Numerical results demonstrate that the Riemann solver based FPM is effective in modeling incompressible flows and it produces a more accurate flow field than the conventional WCSPH and Riemann solver based WCSPH methods.

Modeling and numerical analysis of compressible Darcy–Brinkman fluid flow in fractured media with finite volume method on non-matching grids

In this paper, a hybrid-dimensional coupled model of compressible fluid flow in fractured porous media is studied from theoretical and numerical points of view, in which the flow is governed by Brinkman model in fracture and Darcy model in surrounding porous media. A finite volume scheme with the fewest nodal points is constructed for solving the hybrid-dimensional coupled model on non-matching grids. The second-order error estimates with respect to different meshsizes are derived for both pressure and velocity in corresponding discrete norms. Numerical experiments are performed to verify the efficiency and accuracy of the proposed method. The effects of some parameters, such as the permeability, width and effective viscosity in fracture, together with the integration parameter introduced by the mathematical model, are investigated on the property of fluid flow in fractured system.

Discrete Boltzmann modeling of high-speed compressible flows with various depths of non-equilibrium

The non-equilibrium high-speed compressible flows present wealthy applications in engineering and science. With the deepening of Thermodynamic Non-Equilibrium (TNE), higher-order non-conserved kinetic moments of the distribution function are needed to capture the main feature of the flow state and the evolution process. Based on the ellipsoidal statistical Bhatnagar–Gross–Krook model, Discrete Boltzmann Models (DBMs) that consider various orders of TNE effects are developed to study flows in various depths of TNE. In numerical tests, DBMs including the first up to the sixth order TNE effects are demonstrated. Specifically, at first, the model's capability to capture large flow structures with zeroth-order TNE effects in two types of one-dimensional Riemann problems is demonstrated. The ability to capture large flow structures with first-order TNE effects is also shown in the Couette flow. Then, a shock wave structure given by Direct simulation Monte Carlo is used to verify the model's capability to capture fine structures at the level of the mean free path of gas molecules. Furthermore, we focus on the TNE degree of two colliding fluids mainly decided by two parameters: the relaxation time τ and relative speeds Δu of two fluids. Consequently, three numerical tests for flows with various depths of TNE are constructed. Due to any definition of the TNE strength is dependent on the perspective of investigation, we propose to use a N-component vector STNE to describe the TNE system from N perspectives. As specific applications, we use a three-component vector STNE=(τ,Δu,Δ2*) to roughly characterize three cases for numerical tests in this work. Then, we check the system TNE behavior from the perspective of the xx component of the TNE quantity, viscous stress Δ2*. It is found that, for the first two cases, at least up to the second-order TNE effects, i.e., the second-order terms in Knudsen number in the CE expansion, should be included in the model construction, while for the third case, at least up to the third-order TNE effects should be included. Similar to Δ2*, three numerical tests for flows in various depths of Δ3,1* are constructed. It is found that from the perspective of Δ3,1,x*, for case 1 and case 3, at least up to the second-order TNE effects should be required, while for case 2, the first-order TNE effects are enough. These findings demonstrate that the inadequacy of focusing only on the few kinetic moments appearing in Navier–Stokes increases with the degree of discreteness and deviation from thermodynamic equilibrium. Finally, a two-dimensional free jet is simulated to indicate that, to obtain satisfying hydrodynamic quantities, the DBM should include at least up to the third-order TNE effects. This study is meaningful for the understanding of the TNE behavior of complex fluid systems and the choice of an appropriate fluid model to handle desired TNE effects.

Performance parameters assessment of a pneumatic engine for low capacity commercial vehicles

In the present work, the performance evaluation of a low capacity pneumatic engine is shown, through modeling and simulation in order to identify the feasibility of its application in commercial vehicles for short distances. For the modeling of the pneumatic engine, a comprehensive approach has been used that includes the compressed air supply pipe, the cylinder compression - expansion chamber, and the intake and exhaust valves. The compressed air supply system is modeled using compressible fluid flow models. The fluid flow is considered adiabatic represented by the model known as Fanno flow (or non-isentropic) and it is assumed to be in a steady state. The flow in the opening and exhaust valves are evaluated as adiabatic and isentropic fluid, using a divergent - convergent compressible fluid model. The simulation results show that the feeding pressure and the pipe diameter have a great influence on the motor power. Also, the compression ratio has been found to have strong influence on the overall system efficiency. On the other hand, the simulations show that the initial pressure of the piston does not have a great influence on performance indicators, such as power or efficiency. However, from the results found in this work, it can be seen that the model and all the results obtained in this work can be used to design and generate an optimization model that describes a pneumatic engine proposed for commercial vehicles of low capacity and short distances.

Diffuse interface relaxation model for two-phase compressible flows with diffusion processes

The paper addresses a two-temperature model for simulating compressible two-phase flow taking into account diffusion processes related to the heat conduction and viscosity of the phases. This model is reduced from the two-phase Baer-Nunziato model in the limit of complete velocity relaxation and consists of the phase mass and energy balance equations, the mixture momentum equation, and a transport equation for the volume fraction. Terms describing effects of mechanical relaxation, temperature relaxation, and thermal conduction on volume fraction evolution are derived and demonstrated to be significant for heat conduction problems. The thermal conduction leads to instantaneous thermal relaxation so that the temperature equilibrium is always maintained in the interface region with meeting the entropy relations. A numerical method is developed to solve the model governing equations that ensures the pressure-velocity-temperature (PVT) equilibrium condition in its high-order extension. We solve the hyperbolic part of the governing equations with the Godunov method with the HLLC approximate Riemann solver. The non-linear parabolic part is solved with an efficient Chebyshev explicit iterative method without dealing with large sparse matrices. To verify the model and numerical methods proposed, we demonstrate numerical results of several numerical tests such as the multiphase shock tube problem, the multiphase impact problem, and the planar ablative Rayleigh–Taylor instability problem.

Numerical study of the thermodynamics and supercavitating flow around an underwater high-speed projectile using a fully compressible multiphase flow model

In this study, the thermodynamic behavior and supercavitating flow around projectiles of high-subsonic to supersonic speeds were numerically analyzed using a fully compressible multiphase flow with phase change. The mathematical model is formulated based on the typical conservation laws of mixtures, including mass, momentum, and energy balance, and by maximizing the thermodynamic properties of water and vapor to enable the analysis of the characteristics of high-speed flows. The model is solved on a body-fitted grid using a finite-volume Riemann solver combined with a monotonic upstream-centered scheme for conservation laws. The numerical method was validated by comparing the supercavitating flow around an underwater projectile with the experimental results available in the literature. The temperature, shock, velocity, and supercavity around the high-speed projectile and the effects of the moving velocities on the physical aspects were investigated. It was observed that the temperature around the stagnation point increased significantly with the increase in the stream velocity, and such high temperatures may soften the steel encasing of the projectile in a manner that the excessive pressure in the area could damage the projectile. The physical context and thermodynamics around the transonic projectile were further observed through numerical investigation.

Localized dynamic kinetic-energy model for compressible wavelet-based adaptive large-eddy simulation

Wavelet-based adaptive large-eddy simulation uses wavelet filtering to separate resolved from residual (more from less energetic) turbulent eddies, overcoming limitations of traditional lowpass filter-based methods that are not able to investigate the multi-resolution features and intermittency of turbulence. This is further developed by introducing a novel localized dynamic kinetic-energy-based model for wall-bounded turbulent compressible flows. Application to supersonic channel flow demonstrates the efficiency and accuracy of the method that is characterized by a high dynamic spatial-temporal adaptation, paving the way for modeling and simulation of complex engineering turbulent flows.

A kinetic energy–and entropy-preserving scheme for compressible two-phase flows

Accurate numerical modeling of compressible flows, particularly in the turbulent regime, requires a method that is non-dissipative and stable at high Reynolds (Re) numbers. For a compressible flow, it is known that discrete conservation of kinetic energy is not a sufficient condition for numerical stability, unlike in incompressible flows.In this study, we adopt the recently developed conservative diffuse-interface method (Jain et al., 2020 [30]) for the simulation of compressible two-phase flows. This method discretely conserves the mass of each phase, momentum, and total energy of the system and consistently reduces to the single-phase Navier-Stokes system when the properties of the two phases are identical. We here propose discrete consistency conditions between the numerical fluxes, such that any set of numerical fluxes that satisfy these conditions would not spuriously contribute to the kinetic energy and entropy of the system. We also present a set of numerical fluxes—which satisfies these consistency conditions—that results in an exact conservation of kinetic energy and approximate conservation of entropy in the absence of pressure work, viscosity, thermal diffusion effects, and time-discretization errors. Since the model consistently reduces to the single-phase Navier-Stokes system when the properties of the two phases are identical, the proposed consistency conditions and numerical fluxes are also applicable for a broader class of single-phase flows.To this end, we present coarse-grid numerical simulations of compressible single-phase and two-phase turbulent flows at infinite Re, to illustrate the stability of the proposed method in canonical test cases, such as an isotropic turbulence and Taylor-Green vortex flows. A higher-resolution simulation of a droplet-laden compressible decaying isotropic turbulence is also presented, and the effect of the presence of droplets on the flow is analyzed.

Arbitrary-rate relaxation techniques for the numerical modeling of compressible two-phase flows with heat and mass transfer

We describe compressible two-phase flows by a single-velocity six-equation flow model, which is composed of the phasic mass and total energy equations, one volume fraction equation, and the mixture momentum equation. The model contains relaxation source terms accounting for volume, heat and mass transfer. The equations are numerically solved via a fractional step algorithm, where we alternate between the solution of the homogeneous hyperbolic portion of the system via a HLLC-type wave propagation scheme, and the solution of a sequence of three systems of ordinary differential equations for the relaxation source terms driving the flow toward mechanical, thermal and chemical equilibrium. In the literature often numerical relaxation procedures are based on simplifying assumptions, namely simple equations of state, such as the stiffened gas one, and instantaneous relaxation processes. These simplifications of the flow physics might be inadequate for the description of the thermodynamical processes involved in various flow problems. In the present work we introduce new numerical relaxation techniques with two significant properties: the capability to describe heat and mass transfer processes of arbitrary relaxation time, and the applicability to a general equation of state. We show the effectiveness of the proposed methods by presenting several numerical experiments.

Experimental and numerical analysis of gas flow in nodeless antiresonant hollow-core fibers for optimization of laser gas spectroscopy sensors

In this paper, we present the pressure-driven gas flow studies in nodeless antiresonant hollow-core fibers (ARHCFs) for laser absorption spectroscopy (LAS) applications. The OpenFOAM® software was used for gas flow modeling for comparison with experimental data. The experimental LAS-based setup utilizing a 14.73 m long fiber, was used to gas exchange time measurements for various pressure differences (1.5–5 bar). The obtained results have shown that the application of a precise fiber geometrical representation and a compressible flow model improves the correlation between simulations and experiments for nodeless ARHCFs, in comparison to cylindrical channel approximation and an incompressible model. We believe that our work will help optimize the selection of the proper ARHCF during sensor design, where the fiber type and length can significantly determine the obtainable response time of the detector and its overall compactness.