Artificial Compressibility Method Research Articles

Boundary feedback control of Navier–Stokes equations around arbitrary bluff body with prescribed drag and lift coefficients

This paper presents a boundary feedback control method for the two- and three-dimensional Navier–Stokes equations around an arbitrary bluff body, with prescribed drag and lift coefficients. In order to determine the feedback control law, we consider an extended system coupling the equations governing the Navier–Stokes problem with an equation satisfied by the control on the bluff body, which is a part of the domain boundary. We utilize the artificial compressibility method, which approximates the Navier–Stokes equations in the limit, along with a priori estimation techniques, to establish a boundary control. Such a control law ensures that the prescriptions of the drag and lift coefficients are attained. Subsequently, a compactness result allows to let the compressibility parameter tend to zero. The application of the Faedo-Galerkin method is essential in reaching this result.

Online Bayesian Optimization of Polynomial-Multigrid Cycles for Flux Reconstruction

In this study, we present a novel strategy for dynamically optimizing polynomial multigrid cycles to accelerate convergence within the dual-time-stepping formulation of the artificial compressibility method. To accomplish this, a Gaussian process model is developed using Bayesian optimization to efficiently sample possible cycles to minimize run-time. To allow the use of conventional optimization methods, we developed fractional smoothing steps, moving the optimization from a discrete space to a continuous space. Initially, a static, offline, approach was developed, and optimal cycles were found for two flow past cylinder test cases with Re=200 and Re=500; however, when exchanging optimal cycles between the different test cases, there was significant degradation in speedup. Toward this, a dynamic, online, approach was developed where cycles are optimized during a simulation. The performance of the resulting optimal cycles gave a similar speedup to the offline approach while achieving a net reduction in run-time. Again testing the optimization strategy on the flow past a cylinder, this yielded candidates with mean speedups of ∼3.0× and ∼2.1×, respectively. Finally, testing online optimization on a turbulent flow past a cylinder at Re=3900 resulted in an overall speedup of ∼1.9×.

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.

NUMERICAL SIMULATION OF TURBULENT FLOW AROUND DARRIEUS AND SAVONIUS ROTORS

The study of unsteady processes during the flow around the rotors of vertical-axis wind turbines was carried out based on the Navier-Stokes equations. General definition of the coupled problem on vertical-axis wind turbines rotors aerodynamics and dynamics is formulated. The solution to the system of initial equations is obtained using an implicit finite-volume numerical algorithm based on the method of artificial compressibility and multi-block structured grids. A code for the calculation of vertical-axis wind turbines rotors aerodynamic and power characteristics is developed on the base of Reynolds averaged Navier-Stokes equations for incompressible fluids. Two types of Darrieus and Savonius rotors with different geometric parameters and number of blades were calculated. The influence of Reynolds numbers, tip-speed-ratios and solidity on the power characteristics of the rotors has been established.

Modal analysis for incompressible fluid flow

This paper presents a numerical method for incompressible fluid flow. A difficulty in analyzing incompressible fluid flow is that the continuity equation has no time evolution term. In the marker and cell (MAC) method, Poisson’s equation is solved iteratively, which takes most of the computation time, and in the artificial compressibility method (ACM), pseudo-time iteration is necessary to solve for unsteady solutions. Here, modal analysis that uses the velocity eigenvectors corresponding to zero eigenvalues is proposed for analyzing two-dimensional incompressible fluid flow. The proposed method involves only about one third of the number of variables needed in the MAC method and the ACM, and it does not require iterative calculation of Poisson’s equation or pseudo-time iteration. Numerical results for a simple flow system and a cavity flow obtained using the proposed method are compared with those obtained using the ACM and the simplified MAC method. The results agree well, thereby validating the proposed modal analysis.

Numerical investigation of cavitation vortex dynamics in different cavitation patterns coupled implicit large eddy simulation and boundary data immersion method

The objective of this paper is to investigate the flow characteristics of different cavitation flow patterns around a NACA (National Advisory Committee for Aeronautics) 66 hydrofoil by applying the BDIM (boundary data immersion method) and ILES (implicit large eddy simulation) with an artificial code. Meanwhile, an artificial compressibility method is also employed to consider the effects of compressibility on cavitating flow. The results present that the numerical method can effectively capture different cavitation patterns, which agrees well with the previous experimental data. Subsequently, the detailed analysis of vortex structures and dynamics for the non-cavitation (σ = 3.0), sheet cavitation (σ = 2.0), and cloud cavitation (σ = 1.6) cases with the Liutex method and the vortex enstrophy transport equation have been investigated. When cavitation occurs, the degree of turbulence and the enstrophy in the flow field have been enhanced, due to the disturbance of the velocity field. For sheet cavitation, complex vortex structures appear in the attached cavity region with high-intensity enstrophy causing by the highly intense velocity and density gradient. As the cavitation pattern transits from the sheet cavitation to the cloud cavitation, more complex vortex structures can be observed in the cavitation region. Furthermore, the value and the fluctuation amplitude of enstrophy intensity increase significantly under the effect of reentrant jet. Analysis of the enstrophy transport equation indicates that the vortex stretching term and dilatation term for cloud cavitation increase relatively significantly with the movement of the reentrant flow and are highly dependent on the cavitation evolution. In addition, the region affected by the baroclinic torque also increases.

Study on the steady-oscillatory transition of three-dimensional (3D) natural convection via Hopf bifurcation

The transition from steady-state flow to periodic oscillatory flow for the natural convection by Hopf bifurcation is investigated in a three-dimensional (3D) cavity. The spectral collocation method (SCM) in combination with the artificial compressibility method (ACM), which is developed by ourselves as a numerical method SCM-ACM with high accuracy, is employed to solve the governing equations directly instead of linear stability analysis method that is commonly used for the research on flow instability. The results show that the amplitude decays exponentially with time and the decay rate is linear with the Grashof number (Gr). The critical Grashof number for steady-oscillatory transition is obtained as Grcr = 3.423 × 106. The dimensionless angular frequency ωcr = 0.24 is also determined by Fourier analysis. In this work, we also examine the heat-momentum interactions within the boundary layers, visualize the periodic oscillations of temperature and velocity amplitudes, and analyze the origin of instability from multiple angles. The results show that large oscillations of velocity and temperature are observed near the isothermal walls. The oscillation is enhanced by the increase of thermal boundary layer thickness and flow velocity at both ends of isothermal walls. The maximum velocity and temperature amplitudes appear at the lower left and upper right corners of the mid-plane (Z = 0.5), where are the origin of instability, and the spanwise walls are almost independent of oscillations. The oscillatory flow of natural convection in three-dimensional cavity originates from the continuously increasing buoyancy force, and its transition occurs by Hopf bifurcation. Moreover, the temperature amplitude exhibits a wavy distribution on the mid-plane (X = 0.5) and strongly depends on the depth Z. These results provide benchmark data for future numerical studies and engineering application.

Artificial compressibility method for high-pressure transcritical fluids at low Mach numbers

Supercritical fluids possess unique properties that makes them relevant in various scientific and engineering applications. However, the experimental investigation of these fluids is challenging due to the high pressures involved and their complex thermophysical behavior. To overcome these limitations, computational researchers employ scale-resolving methods, such as direct numerical simulation and large-eddy simulation to study them. Nonetheless, these methods require substantial computational resources, especially in the case of low-Mach-number regimes due to the disparity between acoustic and hydrodynamic/thermal time scales. This work, therefore, addresses this problem by extending the artificial compressibility method to high-pressure transcritical fluids. This method is based on decoupling the thermodynamic and hydrodynamic parts of the pressure field, such that the acoustic time scales can be externally modified without severely impacting the flow physics of the problem. In addition, the method proposed has two key characteristics: (i) the splitting method presents low computational complexity, and (ii) an automatic strategy for selecting the speedup factor of the approach is introduced. The effectiveness of the resulting methodology is demonstrated through comprehensive numerical tests of increasing complexity, showcasing its ability to accurately simulate a wide range of high-pressure transcritical flows including turbulence. The results obtained indicate that the approach proposed can readily lead to computational speedups larger than 10× without significantly compromising the accuracy of the numerical solutions.

An investigation of the noise characteristics of a small-scale rotor in axial descent

In this work, the acoustic characteristics of a small-scale drone rotor in axial descent are numerically investigated. Various descent speeds covering different working states are tested, and the corresponding effects on the noise are analysed. The turbulent flow fields are obtained using delayed detached eddy simulations and an acoustic-wave preserved artificial compressibility method. The far-field noise is obtained by an integral solution of the Ffowcs-Williams and Hawkings equation. The aerodynamic force and the noise features are also evaluated experimentally in anechoic wind tunnel tests, and test data are used to validate the numerical simulations. The results show that the increase in descent speed leads to a reduction in the mean thrust. Significant thrust fluctuations are also observed when the rotor approaches the vortex ring state. Consequently, a substantial increase in sound pressure level in the low to medium frequency range is found in the far-field noise spectra, while the high-frequency sound pressure level shows a slight decrease. Furthermore, a noise source analysis is performed on the blade surface to help understand the corresponding noise mechanism.

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.

High-Order Implicit Large Eddy Simulation using Entropically Damped Artificial Compressibility

Performing industrial scale incompressible Large Eddy Simulation (LES) remains particularly challenging due to computational cost limitations. Recently, the Entropically Damped Artificial Compressibility (EDAC) method was introduced, which does not require coupled global solves or pseudo-time stepping. Parallel to this, high-order unstructured methods, such as the Flux Reconstruction (FR) approach, have been proposed for Implicit LES (ILES) of compressible turbulent flows. This work presents EDAC with FR as an ILES framework for incompressible flows. Results from a Taylor–Green vortex, turbulent channel flow, and transitional and turbulent flow over an SD7003 airfoil demonstrate the approach is accurate and stable for solutions polynomials of degree ps≤4, while some instabilities were observed beyond this. Higher-order solutions were found to be significantly more accurate than their lower-order counterparts compared to reference direct numerical simulation, and had lower divergence error even with less strict incompressibility factors. These results demonstrate that combining EDAC with FR provides a simple, accurate, efficient, and stable framework for performing ILES of incompressible flows. Additionally, this fully explicit framework does not require coupled solvers, pseudo-time stepping, or explicit subgrid scale models.

A second order ensemble algorithm for computing the Navier-Stokes equations

We present an efficient, second order ensemble algorithm for computing the Navier-Stokes equations multiple times with different model parameters. The algorithm is based on the Crank-Nicolson Leap-frog (CNLF) scheme and the ensemble timestepping incorporating the artificial compressibility method and the recent scalar auxiliary variable (SAV) idea for developing unconditionally stable schemes for nonlinear flows. The computation of the velocity and pressure is decoupled in the algorithm resulting in smaller linear systems to be solved at each time step. All realizations of the flow corresponding to different model parameters share the same coefficient matrix so that efficient block solvers can be used to reduce the computational cost. The proposed algorithm is efficient in terms of both computer storage and CPU time. We prove the algorithm is long time stable without any timestep conditions. Ample numerical experiments are performed for various flow problems to validate the efficiency and effectiveness of the algorithm.

Aerodynamic Performance of Vertical-Axis Wind Turbines

The nonstationary separated incompressible flows around Darrieus and Savonius rotors of vertical-axis wind turbines were investigated through computational simulation using the Reynolds averaged Navier–Stokes equations and Spalart–Allmaras turbulence model. The implicit finite-volume algorithm, the basis of which was artificial compressibility method, was chosen to obtain the numerical solution. The series of computational and physical experiments for Darrieus rotors with varied numbers and shapes of blades were performed. The detailed visualization of the flow was presented. The turbulent flows surrounding the Darrieus and Savonius rotors were studied, and as a part of these investigations, the major phases of vortex progress were identified. For this purpose, three series of computer tests on the aerodynamic and power properties of Savonius rotors with two and three buckets were performed, and their results are also presented. The influence of tip-speed ratio, solidity, and Reynolds numbers on the power coefficients of the Darrieus and Savonius rotors was investigated. It has been demonstrated that increasing Reynolds number from 104 to 106 causes a rise in Darrieus rotors power coefficient from 0.15 up to 0.5. The maximum values of power coefficient are moved away from higher values of tip-speed ratio from 2 to 5 as a result of a decrease in Darrieus rotor solidity from 1.0 to 0.33. The greatest power coefficient for a Savonius rotor with two blades is 0.23 and for a Savonius rotor with three blades is 0.19.

Computational Aeroacoustics Study of Propellers with Vibrational Motion

In this work, we conduct a numerical investigation of aerodynamics and aeroacoustics of propeller blades with vibrations, which could occur in practical unmanned aerial vehicles as the blades are often made of lightweight materials. The simulations resolve the sound generation from the unsteady turbulent flows using an acoustic-wave preserved artificial compressibility method. Then, the sound projection to the far-field observers is made using an integral solution of the Ffowcs-Williams and Hawkings equation. The study shows that periodic blade vibrations with small amplitudes can lead to aerodynamic thrust fluctuations. The blade vibration also affects the generation of tip vortices and the near-blade flow structures due to the periodic change of the effective angle of attack. Consequently, significant tonal noise at the harmonics of rotational frequency is produced, and the noise can propagate to both upstream and downstream directions of the rotor disc plane. A noise source analysis is performed to identify the contribution of different noise components. Results show that the extra tonal noise is mainly caused by the Doppler effect due to the blade axial motion and the influence of the thrust fluctuations. Moreover, the study also suggests that the high-frequency broadband noise seems to be insensitive to the blade vibration.

An unconditionally stable artificial compression method for the time‐dependent groundwater‐surface water flows

AbstractIn this article, we propose a second order, unconditionally stable artificial compression method for the fully evolutionary Stokes/Darcy and Navier‐Stokes/Darcy equations that model the coupling surface and groundwater flows. It uncouples the surface from the groundwater flow by the Crank‐Nicolson Leapfrog scheme for the discretization in time, and through the artificial compression method without artificial pressure boundary conditions to decouple the velocity and pressure of the incompressible flow. Finally, we have verified the stability and second‐order convergence of the algorithm from theoretical analysis and numerical experiments.

Artificial compressibility method using bulk viscosity term for an unsteady incompressible flow simulation

Artificial compressibility method using bulk viscosity term for an unsteady incompressible flow simulation

A Review of Preconditioning and Artificial Compressibility Dual-Time Navier–Stokes Solvers for Multiphase Flows

This review paper aims to summarize recent advancements in time-marching schemes for solving Navier–Stokes (NS) equations in multiphase flow simulations. The focus is on dual-time stepping, local preconditioning, and artificial compressibility methods. These methods have proven to be effective in achieving high time accuracy in simulations, as well as converting the incompressible NS equations into a hyperbolic form that can be solved using compact schemes, thereby accelerating the solution convergence and allowing for the simulation of compressible flows at all Mach numbers. The literature on these methods continues to grow, providing a deeper understanding of the underlying physical processes and supporting technological advancements. This paper also highlights the imposition of dual-time stepping on both incompressible and compressible NS equations. This paper provides an updated overview of advanced methods for the CFD community to continue developing methods and select the most suitable two-phase flow solver for their respective applications.

A high-order entropically-damped artificial compressibility approach on moving and deforming domains

A wide range of incompressible flow applications require accurate, efficient, and stable solutions on moving and deforming domains. The entropically damped artificial compressibility (EDAC) method offers a pressure-dampening alternative to conventional Poisson’s or artificial compressibility methods (ACM). EDAC uses an artificial compressibility parameter, similar to the ACM, but does not require pseudo-iterations to converge the pressure field. This paper explores the accuracy of the EDAC scheme on moving and deforming domains using a high-order flux reconstruction (FR) spatial discretization. The method of manufactured solutions is used to verify observed orders of accuracy for FR implementations of EDAC on moving and deforming domains, followed by confirmation of freestream preservation. Then, simulations of a double shear layer, flow over an oscillating cylinder, and dynamic stall of a 2D NACA 0012 airfoil undergoing heaving and pitching motions are used to assess the ability of the scheme to preserve physics and to determine the impacts of the artificial compressibility factor and solution polynomial degree. Results demonstrate that artificial compressibility factors on the order of 0.025–0.05, combined with high-order solution polynomials, recovered reference solutions. These results demonstrate the suitability of FR discretizations of EDAC for incompressible flows on moving and deforming domains.

Numerical Study of Compressible and Weak Compressible Flows in Channel Based on Artificial Compressibility Method and Fully Artificial Compressibility Method

In this article, a numerical study of compressible and weak compressible Newtonian flows is achieved for a time marching, Galerkin algorithm. A comparison between two numerical techniques for such flows, namely the artificial compressibility method (AC–method) and the fully artificial compressibility method (FAC–method) is performed. In the first artificial compressibility parameter ( is added to the continuity equation, while this parameter is added to both continuity and momentum equations in the second technique. This strategy is implemented to treat the governing equations of Newtonian flow in cylindrical coordinates (axisymmetric). Particularly, this study concerns with the effect of the artificial compressibility parameters on the convergence level of solutions components. To confirm the analysis of these approaches, Poiseuille flow along a circular channel under an isothermal state is used as a simple test problem. The results show that when the AC-method is used there is a significant reduction in the level of time convergence of pressure and axial velocity compared to that with FAC-method. Here, for compressible flow the Tail model of state is employed to relate the pressure to density. In this context, the effect of Tail parameters and Reynolds number on the time convergence of solution components is also investigated in the present study. The results indicate a significant reduction in the time-stepping convergence as increasing in the {B,m}-value. In contrast, more difficulties are faced in the convergence when the level of the Reynolds number is increasing.

Numerical investigation of distinct flow modes in a wide-gap spherical annulus using OpenMP

Spherical Couette flow (SCF) between two rotating concentric spheres has many applications in different fields of Mathematics, Physics, and Engineering. The transitions occurring in this flow phenomena have the relevancy with geophysical motions. In this paper, we use the time-dependent three-dimensional incompressible Navier–Stokes equations (3D-INSEs) to investigate distinct flow modes for a wide-gap clearance ratio (CR) . An Artificial Compressibility method has been used along with three implicit solvers, i.e. alternating direction implicit (ADI), line Gauss–Seidel (LGS), and point Gauss–Seidel (PGS). We apply the basic techniques of OpenMP to our Fortran code. First, we make a performance comparison of the three implicit solvers running on a single central processing unit (CPU) with that of the multi-core CPUs using Open Multi-Processing (OpenMP) to choose the best solver for the simulation of the SCF problem. The comparison shows that the ADI solver is the most efficient solver compared to LGS and PGS solvers. Using the most efficient parallel ADI solver, we compute distinct flow modes for this wide gap in a range of Reynolds number between . For this wide gap, first we obtain the basic 0-vortex flow (0-VF) at . By increasing the the flow becomes symmetric 1-vortex flow (1-VF) and asymmetric spiral 1-VF at and respectively. The flow returns back to super critical basic flow at . Further increasing the the flow becomes spiral 0-VF with m = 4 and m = 5 spiral waves at and respectively. Increasing the further the flow finally becomes turbulent. For this CR, we have first time numerically simulate these flow transitions from basic flow to super critical spiral basic flow with increasing . Abbreviations: ACM: artificial compressibility method; ADI: alternating directionimplicit; CPU: central processing unit; CR: clearance ratio; CVD: circumferentialvelocity distribution; INSEs: incompressible Navier–Stokes equations; LGS: lineGauss–Seidel; OpenMP: open multi-processing; PGS: point Gauss–Seidel; Re: Reynoldsnumber; SCF: spherical Couette flow; SG: spherical-gaps; TG: Taylor–Gortler; TVF:Taylor vortex flow; 0-VF: 0-vortex flow; 1-VF: 1-vortex flow; WENO: weightedessentially non-oscillatory.