Spurious Pressure Oscillations Research Articles

Pressure-equilibrium semi-implicit solver for real fluids

In this study, a semi-implicit pressure-based scheme which can suppress the spurious pressure oscillations is developed for real fluid flows. Conservation properties relevant to the proposed scheme are investigated through one-dimensional numerical simulations of the nitrogen interface advection. The efficiency and validity of this scheme are carefully examined with two- and three-dimensional numerical simulations of cryogenic nitrogen jets. From the one-dimensional simulation results, the conservation properties are well conserved as the past studies. The two-dimensional simulation results show that the developed scheme can reduce the computational cost by more than 92% (13–14 times faster) compared with the conventional density-based explicit solver employing the double flux model. It is also found that, through the three-dimensional simulation results, the developed scheme can predict the structure of the nitrogen jet under both transcritical and supercritical conditions, while taking into account the effects of real fluid properties on the mixing and spatiotemporal evolution of this jet, with excellent accuracy.

Adaptation of the Low Dissipation Low Dispersion Scheme for Reactive Multicomponent Flows on Unstructured Grids Using Density-Based Solvers

Abstract The low dissipation low dispersion (LD2) second-order accurate scheme is effective for scale-resolving simulations in finite volume computational fluid dynamics (CFD) solvers, thanks to its combination of a skew-symmetric split form for convective terms and matrix-valued artificial dissipation fluxes. However, reactive flow simulations face challenges due to steep gradients and varying gas properties. This study replaces the skew-symmetric scheme with the kinetic energy and entropy preserving (KEEP) scheme, utilizing quadratic and cubic split forms for convective terms, enhancing stability. The nonsmooth fluid interfaces in reactive flow simulations necessitate upwind fluxes for reactive scalars to limit total variation (TV), also requiring upwind fluxes for the mixture-dependent internal energy fluxes. Other convective terms use central discretizations from the KEEP scheme, leveraging LD2’s spatial reconstruction to minimize dispersive errors. Numerical assessments show this approach reduces spurious pressure oscillations in single and multicomponent flows. The absolute flux Jacobian for dissipation flux calculation is efficiently computed using an expanded Turkel’s approach for thermally perfect gas mixtures. Partial pressure derivatives are approximated when using the flamelet generated manifolds (FGM) combustion model. The proposed scheme is evaluated through scale-resolving simulations of the Cambridge burner flame SWB1 on an unstructured grid using the density-based solver TRACE, employing both finite rate chemistry (FRC) and FGM combustion models. Comparative analysis with the all-speed SLAU2 scheme shows the superior performance of the proposed scheme in handling turbulent reactive multicomponent flows.

المحاكاة العددية للتدفقات المضغوطة المكو نة من عنصرين مع المعادلة العامة للحالة

Numerical simulations of compressible two-component flows with a general equation of state (EOS), written in the form of the Mie-Gru¨neisen EOS, have been conducted using the diffuse interface method on a structured mesh. The unsteady and inviscid one-dimensional sixequation model of Kapila is employed to describe compressible two-component flows. The model is hyperbolic and non-conservative. The solution of the hyperbolic equations, including the non-conservative equation for the volume fraction evolution, is obtained using an extended Harten-Lax-Leer (HLL) approximate Riemann solver. A general formulation for various EOSs is proposed to enable simulations of a wide range of applications. In which the two constituents are either governed by the same EOS or different types of EOS. In this model, both fluids have the same velocity, but each fluid has its own pressure. Therefore, the pressure relaxation process is performed instantaneously to drive both constituents' pressures towards equilibrium. Several computational test problems were conducted in one and two dimensions to show the ability of the numerical method to simulate such problems. The computed results are presented without introducing spurious pressure oscillations in the method.

A coupled fully implicit method for solving drag-adjoint equations of steady incompressible flows

Numerical instability is one of the main concerns when applying adjoint-based gradient computation in aerodynamic design optimization for a variety of engineering problems. An adjoint pressure-adjoint velocity-coupled fully implicit method for solving the adjoint equations of steady incompressible flows is developed in the present work, taking the aerodynamic drag as the optimization target. The finite volume method for cell-centered, collocated variables on unstructured grids is used for spatial discretization. A stabilization term based on pressure-weighted interpolation (PWI) of the adjoint velocity is added to the adjoint continuity equation to avoid spurious oscillations of adjoint pressure. All the terms of the continuous adjoint equations are discretized implicitly, including the adjoint transpose convection (ATC) term, which is regarded as one of the major sources of numerical instability. The resulting discretized equations are integrated into one linear system, and the adjoint fields can be obtained simultaneously. To prevent the diagonal element of the coefficient matrix of the linear system from getting too small, a singular value decomposition is applied to the diagonal block of the coefficient matrix, and a limiter is applied to the diagonal elements. The condition number of the coefficient matrix may be quite large, so a parallel sparse linear solver that effectively combines direct and iterative methods through a Schur complement approach is applied to solve the linear system. Compared to the staggered iterative solver, no damping or masking is needed for the ATC term in the proposed fully implicit method. There is no need to sacrifice numerical accuracy for numerical stability. This method has been tested in several benchmark cases, and the numerical results show good numerical stability and accuracy. The present method provides a sound basis for industrial applications of adjoint-based aerodynamic optimization methods.

A weighted shifted boundary method for immersed moving boundary simulations of Stokes' flow

The Shifted Boundary Method (SBM) belongs to the class of unfitted (or immersed, or embedded) finite element methods, and relies on reformulating the original boundary value problem over a surrogate (approximate) computational domain. The surrogate domain is constructed so as to avoid cut cells and the associated problematic implementation and numerical integration issues. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions: hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we extend the SBM to the simulation of incompressible Stokes flow, by appropriately weighting its variational form with the elemental volume fraction of active fluid. This approach allows to drastically reduce spurious pressure oscillations in time, which are produced if the total volume of active fluid were to change abruptly over a time step. The proposed Weighted SBM (W-SBM) exactly preserves states of hydrostatic equilibrium, and induces small mass and momentum conservation errors, which converge as the grid is refined. This is in analogy to cutFEMs and related unfitted approaches, which rely on an affine representation of cut boundaries. We demonstrate the robustness and accuracy of the proposed method with an extensive suite of two-dimensional tests.

Performance Assessment of Pneumatic-Driven Automatic Valves to Improve Pipeline Fault Detection Procedure by Fast Transient Tests.

The use of fast transients for fault detection in long transmission networks makes the generation of controlled transients crucial. In order to maximise the information that can be extracted from the measured pressure time history (pressure signal), the transients must meet certain requirements. In particular, the manoeuvre that generates the transient must be fast and repeatable, and must produce a pressure wave that is as sharp as possible, without spurious pressure oscillations. This implies the use of small-diameter valves and often pneumatically operated automatic valves. In the present work, experimental transient tests are carried out at the Water Engineering Laboratory (WEL) of the University of Perugia using a butterfly valve and a ball pneumatic-driven valve to generate pressure waves in a pressurised copper pipe. A camera is used to monitor the valve displacement, while the pressure is measured by a pressure transducer close to the downstream end of the pipe where the pneumatic valve is installed. The experimental data are analysed to characterise the valve performance and to compare the two geometries in terms of valve closing dynamics, the sharpness of the generated pressure wave and the stability of the pressure time history. The present work demonstrates how the proposed approach can be very effective in easily characterising the transient dynamics.

Stabilized mixed material point method for incompressible fluid flow analysis

This paper proposes novel and robust stabilization strategies for accurately modeling incompressible fluid flow problems in the material point method (MPM). To address the modeling of Newtonian fluids with incompressibility constraints, a new mixed implicit MPM formulation is proposed. Here, instead of solving the velocity and pressure fields as the unknown variables like the typical Eulerian computational fluid dynamics (CFD) solver, the proposed approach adopts a monolithic displacement–pressure formulation inspired by the mixed-form updated-Lagrangian Finite Element Method (FEM). To satisfy the inf–sup stability condition, two stabilization strategies are integrated into the formulation: the variational multiscale method (VMS) and the pressure-stabilization Petrov–Galerkin method (PSPG). By concurrently solving the displacement and pressure fields, the developed monolithic solver obviates the need for free-surface detection as well as Dirichlet and Neumann pressure imposition, in contrast to the fractional-step method. This attribute mitigates spurious pressure and velocity oscillations in simulating dynamic and transient flow problems. This study also addresses other prevalent challenges in MPM simulations, such as the pressure oscillations triggered by cell-crossing errors, particle-distribution-induced quadrature errors, and particle-grid information transfer. To resolve these issues, the quadratic B-Spline basis function, the delta-correction method, and the Taylor particle-in-cell method are incorporated into the proposed mixed MPM formulation, thereby enhancing numerical stability. The efficacy of the proposed stabilized incompressible MPM framework is validated through several benchmark cases, comparing the obtained results with other numerical methods and analytical solutions. Furthermore, the method’s capability in simulating real-world problems involving violent free-surface fluid motion is demonstrated through comparisons with experimental results of water sloshing and dam break scenarios.

Kinetic-energy- and pressure-equilibrium-preserving schemes for real-gas turbulence in the transcritical regime

Numerical simulations of compressible turbulent flows governed by real-gas equations of state, such as high-pressure transcritical flows, are strongly susceptible to instabilities. In addition to the inherent multi-scale nature of the flow, the presence of a pseudo-interface can generate spurious pressure oscillations when conventional schemes are utilized. This study proposes a general framework to derive and analyze discretization methods that are able to preserve kinetic energy by convection, and simultaneously maintain pressure equilibrium in discontinuity-free compressible real-gas flows. The formal analysis reveals that the discrete pressure-equilibrium condition can be fulfilled at most to second-order accuracy, as it requires the spatial differential operator to satisfy a discrete chain rule when total or internal energy are directly discretized. A novel class of schemes based on the solution of a pressure equation is thus proposed, which preserves mass, momentum, kinetic energy and pressure equilibrium, but not total energy. Extensive numerical tests of increasing complexity confirm the theoretical predictions, and show that the proposed scheme is capable of providing non-dissipative, stable and oscillation-free simulations, unlike existing methods tailored for the transcritical regime.

Fully conservative and pressure-equilibrium preserving scheme for compressible multi-component flows

In compressible flows with a variable ratio of specific heats, such as multi-component flows, the conventional conservative schemes generate spurious pressure oscillations at multi-component interfaces even when the interfaces are smooth. In this study, we propose a novel spatial discretization scheme that maintains both the primary conservation and pressure equilibrium in discontinuity-free compressible multi-component flows. The key to this study is the compatibility condition, which is the discrete condition for implicitly satisfying the pressure equilibrium at the discrete level. The compatibility condition is derived based on the analytical relation between the mass of species and the ratio of specific heats in the governing equations. The proposed scheme is constructed by the spatial discretization that satisfies the compatibility condition and can be applied to the arbitrary number of species components. The inviscid smooth interfaces advection problems verify that the proposed scheme satisfies both the pressure equilibrium and primary conservation only by solving the conservation equations, unlike the existing schemes that solve non-conservative or overspecified equations.

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.

A Direct Ghost Fluid Method for Modeling Explosive Gas and Water Flows

This work presents a Direct Ghost Fluid Method (DGFM) as part of a two-fluid numerical framework suitable to model explosive gas and water flows resulting from underwater explosion (UNDEX). Due to the presence of explosive gas and water with shock waves in the modeling domain, classic Eulerian methods with inherent diffusion may not be effective. Numerical diffusion occurs due to nonphysical diffused density at material interfaces, which creates spurious pressure oscillations and significantly degrades the quality of the numerical results. To eliminate or minimize numerical diffusion, sharp interface methods having no mixed elements may be used in multifluid flow computations. The Direct Ghost Fluid Method (DGFM) described in this paper uses direct extrapolation of density (vice pressure) and tangential velocity from real to ghost fluid. The spurious pressure oscillations near the material interface are therefore minimized. One-, two-, and three-dimensional computational fluid dynamics (CFD) solvers that have DGFM as an essential part in their framework to model UNDEX interface conditions are developed, explored, and applied to the simulation of a series of benchmark problems. Excellent agreement is obtained among the simulations, the analytical solutions, and the experiments.

A conservative and consistent implicit Cartesian cut-cell method for moving geometries with reduced spurious pressure oscillations

A conservative and consistent three-dimensional Cartesian cut-cell method is presented for reducing the spurious pressure oscillations often observed in moving body simulations in sharp-interface Cartesian grid methods. By analysing the potential sources of the oscillation in the cut-cell framework, an improved moving body algorithm is proposed for the cut-cell method for the temporal discontinuity of the solid volume change. Strict conservation of mass and momentum for both fluid and cut cells is enforced through pressure-velocity coupling to reduce local mass conservation errors. A consistent mass and momentum flux computation is employed in the finite volume method. In contrary to the commonly cut-cell methods, an implicit time integration scheme is employed in the present method, which prevents numerical instability without any additional small cut-cell treatment. The effectiveness of the present cut-cell method for reducing spurious pressure oscillations is demonstrated by simulating various two- and three-dimensional benchmark cases (in-line and transversely oscillating cylinder, oscillating and free-falling sphere), with good agreement with previous experimental measurements and other numerical methods available in the literature.

Two-phase PFEM with stable nodal integration for large deformation hydromechanical coupled geotechnical problems

The node-based strain smoothed particle finite element method (NS-PFEM) has a high level of computational efficiency. However, it has not yet been extended to hydromechanical coupling. This study presents the development of a two-phase one-point stable NS-PFEM (SNS-PFEM) for solving large deformation hydromechanical coupled geotechnical problems. First, the soil–water coupling is achieved by the v–w formulation based on Biot’s mixture theory, which automatically ensures the consistency between water pressure and stress that the former should be interpolated one order lower than that of velocity. Secondly, the node-based strain smoothing method with linear strain fields over the smoothed domain on 3-node triangular elements is adopted and implemented to obtain a stable nodal integration. Thirdly, a novel efficient water pressure stabilisation scheme based on an inverse distance interpolation algorithm is implemented to cure spurious spatial water pressure oscillations. The performance of the hydromechanical SNS-PFEM is primarily validated by both static and dynamic 1D consolidation and 2D wave propagation, and then applied to analyse the progressive landslide and the Selborne cutting slope involving large deformation. All results demonstrate that the proposed method is powerful and easily extensible for analysing large deformation hydromechanical problems in geotechnical engineering.

A Framework of Runge–Kutta, Discontinuous Galerkin, Level Set and Direct Ghost Fluid Methods for the Multi-Dimensional Simulation of Underwater Explosions

This work describes the development of a hybrid framework of Runge–Kutta (RK), discontinuous Galerkin (DG), level set (LS) and direct ghost fluid (DGFM) methods for the simulation of near-field and early-time underwater explosions (UNDEX) in early-stage ship design. UNDEX problems provide a series of challenging issues to be solved. The multi-dimensional, multi-phase, compressible and inviscid fluid-governing equations must be solved numerically. The shock front in the solution field must be captured accurately while maintaining the total variation diminishing (TVD) properties. The interface between the explosive gas and water must be tracked without letting the numerical diffusion across the material interface lead to spurious pressure oscillations and thus the failure of the simulation. The non-reflecting boundary condition (NRBC) must effectively absorb the wave and prevent it from reflecting back into the fluid. Furthermore, the CFD solver must have the capability of dealing with fluid–structure interactions (FSI) where both the fluid and structural domains respond with significant deformation. These issues necessitate a hybrid model. In-house CFD solvers (UNDEXVT) are developed to test the applicability of this framework. In this development, code verification and validation are performed. Different methods of implementing non-reflecting boundary conditions (NRBCs) are compared. The simulation results of single and multi-dimensional cases that possess near-field and early-time UNDEX features—such as shock and rarefaction waves in the fluid, the explosion bubble, and the variation of its radius over time—are presented. Continuing research on two-way coupled FSI with large deformation is introduced, and together with a more complete description of the direct ghost fluid method (DGFM) in this framework will be described in subsequent papers.

Numerical method for reducing spurious pressure oscillations in transcritical flow simulations

Numerical method for reducing spurious pressure oscillations in transcritical flow simulations

A Local Semi-Fixed Ghost Particles Boundary Method for WCSPH

Due to the convenience and flexibility in modeling complex geometries and deformable objects, local ghost particles methods are becoming more and more popular. In the present study, a novel local semi-fixed ghost particles method is proposed for weakly compressible smoothed particle hydrodynamics (WCSPH). In comparison with the previous local ghost particles methods, the new boundary method can effectively reduce spurious pressure oscillations and smooth the flow field. Besides, the new generation mechanism of fictitious particles is simple and robust, which is suitable for all kinds of kernel functions with different sizes of the support domain. The numerical accuracy and stability of the new smoothed particle hydrodynamics (SPH) scheme are validated for several typical benchmark cases. A detailed investigation into the pressure on solid walls and the surface elevation in dynamic simulations is also conducted. A comparison of numerical results shows that the new boundary method helps reduce the oscillations in the numerical solutions and improves the numerical accuracy of the pressure field.

A diffuse-interface method for reducing spurious pressure oscillations in multicomponent transcritical flow simulations

Spurious pressure oscillations are a numerical instability that is commonly observed in multiphase and multicomponent flow problems near the critical point. A diffuse-interface model has been created to simulate transcritical mixing in multispecies and multiphase systems where spurious pressure oscillations pose a serious challenge in achieving convergence. To reduce the spurious pressure oscillations, three methods have been proposed and implemented in the present paper: (1) artificially thickening the interface between different species, (2) reconstruction of the primitive variables in the characteristics space, and (3) developing a hybrid method (HY) that switches between quasi-conservative (QC) double-flux (DF) and the classical fully-conservative numerical procedures based on the changes in the effective specific heat ratio and the effective reference internal energy. The cases considered include (1) a transcritical shock tube problem between nitrogen and n-dodecane, (2) a near-critical shock-droplet interaction, and (3) the transcritical Spray A benchmark case of Engine Combustion Network (ECN) that represents the injection of n-dodecane into a high-pressure and high-temperature nitrogen environment. Characteristic-wise reconstruction of the primitive variables in our proposed fully-conservative method tends to reduce the spurious pressure oscillations compared to primitive-wise reconstruction including the reconstruction of the speed of sound. The HY model is found to effectively reduce the magnitude of the spurious pressure oscillations, the loss of energy conservation, and provides more accurate results for all tested cases. The artificial thickening of the interface between different species is found to reduce spurious pressure oscillations and reduce energy conservation loss when used in conjunction with a QC method (HY or DF).

Improved predictions of thermal fluid pressurization in hydro-thermal models based on consistent incorporation of thermo-mechanical effects in anisotropic porous media

Coupled thermo-hydro-mechanical models are commonly used to model the evolution of temperature, pore pressure, and stress in a wide range of geotechnologies such as geothermal applications or around canisters of high-level radioactive waste in deep underground storage facilities. Their numerical modeling is often computationally highly demanding, especially if parameter identification, sensitivity analyses or uncertainty quantification require many model evaluations. Often, the thermally driven pore pressure evolution and the subsequently altered flow processes are the primary targets of an analysis. To benefit from the computational efficiency of hydro-thermal (HT) models while maintaining the accuracy of the thermo-hydro-mechanical (THM) model, we derived two cases of a simplified representation of mechanical deformations in a coupled hydro-thermal model. Deformations induced by pressure as well as temperature changes are consistently incorporated into the mass balance storage terms. We demonstrate the exact coincidence of THM and modified TH formulations in isotropic and orthotropic materials as long as the basic assumptions like constant hydrostatic stress conditions or uniaxial strain hold. By modeling of a point heat source in isotropic or anisotropic porous media it is shown that a good agreement between TH and THM models can be maintained even though the assumptions underlying the simplification are no longer valid exactly. On our test-machine, a significant speed-up could be achieved by the reduction of the problem size when transitioning from a THM to a TH model. The highest speed-ups were achieved when Taylor-Hood elements were employed in order to avoid the problem of spurious pressure oscillations in the fully coupled THM model.

On the structure preserving high-order approximation of quasistatic poroelasticity

We consider the systematic numerical approximation of Biot’s quasistatic model for the consolidation of a poroelastic medium. Various discretization schemes have been analysed for this problem and inf–sup stable finite elements have been found suitable to avoid spurious pressure oscillations in the initial phase of the evolution. In this paper, we first clarify the role of the inf–sup condition for the well-posedness of the continuous problem and discuss the choice of appropriate initial conditions. We then develop an abstract error analysis that allows us to analyse some approximation schemes discussed in the literature in a unified manner. In addition, we propose and analyse the high-order time discretization by a scheme that can be interpreted as a variant of continuous-Galerkin or particular Runge–Kutta methods applied to a modified system. The scheme is designed to preserve both, the underlying differential–algebraic structure and the energy-dissipation property of the problem. In summary, we obtain high-order Galerkin approximations with respect to space and time and derive order-optimal convergence rates. The numerical analysis is carried out in detail for the discretization of the two-field formulation by Taylor–Hood elements and a variant of a Runge–Kutta time discretization.

Preventing spurious pressure oscillations in split convective form discretization for compressible flows

This paper addresses issues in split convective form discretization in terms of the physical property of the pressure equilibrium to achieve physically-consistent, stable, and non-dissipative shock-free compressible flow simulations. Discrete pressure- and velocity-evolution equations are derived by using existing split convective form discretization, such as used in kinetic energy preserving (KEP) and kinetic energy and entropy preserving (KEEP) schemes. This analysis reveals that the existing KEP and KEEP schemes do not maintain the physical property of the pressure equilibrium due to the discretization of the internal energy convective term. The analysis also directly leads to the proposed split convective form discretization of the internal-energy convective term that strictly satisfies the pressure equilibrium at the discrete level. By applying the proposed discretization of the internal energy convective term to the existing KEEP scheme, this study shows that it is possible to satisfy the pressure equilibrium numerically with maintaining excellent kinetic energy and entropy preservation property. In numerical tests, the proposed scheme shows a superior numerical stability property without spurious pressure oscillations.