Treatment Of Boundary Conditions Research Articles

Effect of an incoming Gaussian wave packet on underlying turbulence

We present a simulation-based study of the effect of a passing wave packet on underlying fully developed turbulence. We propose a novel wave-phase-resolved simulation method inspired by Helmholtz decomposition to directly couple the turbulence simulation with instantaneous wave orbital motions without wave-phase averaging. We also introduce a boundary condition treatment for the turbulence at the wave surface, which allows the turbulence simulation to be conducted in a rectangular domain while retaining the wave-phase effect. The results obtained from the proposed method reveal considerable variations in turbulence statistics, including the enstrophy and Reynolds normal stresses, during wave packet passage. Most changes occur rapidly when the narrow bandwidth around the wave packet core passes. Further analyses of the energy spectra indicate that the enhancement of turbulence occurs across a wide range of scales, with the near-surface small-scale motions experiencing the most significant intensification. Meanwhile, large-scale motions with scales comparable to the boundary layer depth are also enhanced. The mechanisms underlying the Reynolds normal stress variation at different length scales are related to the energy transfer from the wave orbital straining to turbulence through production, the pressure–strain effect, the pressure diffusion and the wave advection. By assessing the turbulence statistics and dynamics impacted by a wave packet in detail, this study provides an improved understanding of the response of a developed turbulent flow to a transient wave field. The proposed simulation method also proves to be a promising phase-resolved approach for efficiently modelling the wave effect on turbulence.

On the use of fast projection methods with unsteady velocity boundary conditions

This article aims at clarifying and amending a previously published result on the use of pseudo-pressure approximations for fast incompressible Navier-Stokes solvers with time-dependent boundary conditions. Fast projection methods were proposed to speed up high-order incompressible Navier-Stokes solvers by replacing pressure projections with simple approximations. A generalized approximation theory was proposed in [1], in which the time-dependent boundary conditions were ignored. In this comment, we investigate the effect of unsteady boundary conditions on the order of accuracy and demonstrate that the pressure approximations derived in [1] hold for all RK2 integrators, while RK3 approximations are only third order for certain sets of coefficients. We show that third order is lost for other cases and shed light on why order is broken. This work serves as a reference for the treatment of unsteady boundary conditions for fast-projection methods. Numerical validation is performed on a well established channel flow problem with a variety of unsteady inlet conditions for a wide range of explicit RK2 and RK3 integrators.

An improved pressure gradient method for viscous incompressible flows

The pressure gradient method solves the viscous incompressible flow with the pressure gradients, rather than the pressure, as unknown variables. Two variants of the pressure gradient method have been developed in the past but have not received much attention due to their unsatisfactory performance or implementation complexity. Based on the artificial compressibility concept, this study proposes an improved pressure gradient method. One distinct feature of this method is that it requires no pressure/pressure gradient boundary condition or special treatment on wall boundaries. An auxiliary variable is introduced to represent the velocity dilatation, greatly simplifying the spatial discretization and computational procedure. The mathematical formulations are elaborated and compared with the previous pressure gradient methods, followed by discussions of compatibility relationships, boundary condition setup, and an extension to a pressure Poisson-like equation. Four validation examples are performed for various flow scenarios, and the solutions and domain solenoidity are examined for each case. The study also compares associated computational methods, different pressure boundary conditions, and flow characteristics, demonstrating the benefits of the present method.

A numerical approach to overcome the very-low Reynolds number limitation of the artificial compressibility for incompressible flows

We propose a numerical approach to solve a long-standing challenge which is the applicability of the artificial compressibility (AC) formulation for solving the incompressible Navier—Stokes equations at very-low Reynolds numbers. A wide range of engineering applications involves very-low Reynolds number flows in Micro-ElectroMechanical Systems (MEMS) and in the fields of chemical-, agricultural- and biomedical engineering. It is known that the already existing numerical methods using the AC approach fail to provide physically correct results at very-low Reynolds numbers (Re ≤ 1). To overcome the limitation of the AC method for these engineering applications, we propose a higher-order Neumann-type pressure outflow boundary condition treatment along with their up to fourth-order numerical approximations. We found that the numerical treatment of the pressure at the outlet boundary plays the main role in overcoming the limitation of the AC method at very-low Reynolds numbers (Re << 1). Therefore, we provide numerical evidence on the accuracy of the AC method beyond its previously reported limitations, e.g., the low Reynolds number Oseen flow (Re << 1) is first presented in this work. A third-order explicit total-variation diminishing (TVD) Runge–Kutta scheme has been employed with standard finite difference spatial discretisation schemes for improving the accuracy of the numerical solution. For modelling strongly viscous flows, the Reynolds number ranges from 10−1 to 10−4. Overall, we found that the accuracy limitation of the AC method below Re < 1 can be overcome with an accurate numerical treatment of the outlet pressure boundary condition instead of using high-order schemes in the governing equations. For the investigated Reynolds number range (10−1 ≤ Re ≤ 10−4), the obtained results show that the relative errors were smaller than 1% for the numerical simulations performed on the configurations of both the two- and three-dimensional, straight microfluidic channels. The imposition of high-order derivative Neumann-type pressure outflow boundary conditions reduced the maximum relative errors of the numerical solutions from 85% and 95% to below than 1% at the outlet section of the two- and three-dimensional, straight microfluidic channel flows, respectively. Taking the advantage of the numerical approach proposed here, two- and three-dimensional benchmark problems employed in the current investigation in comparison with analytical solutions available in the literature, clearly demonstrate that the artificial compressibility can be used beyond its previously known constraints for very-low Reynolds number incompressible flows.

The Riemann–Hilbert Approach to the Higher-Order Gerdjikov–Ivanov Equation on the Half Line

The Fokas method exhibits remarkable versatility in solving boundary value problems associated with both linear and nonlinear partial differential equations, particularly when conventional approaches encounter challenges in handling intricate boundary conditions. The existing literature often lacks the incorporation of unconventional boundary conditions, and this study addresses this issue by extending the application of the Fokas method to the higher-order Gerdjikov-Ivanov equation on the half line (−∞,0]. We have demonstrated the exclusive representation of the potential function u(z,t) in the higher-order Gerdjikov–Ivanov equation through the solution of a Riemann–Hilbert problem. The characteristic function is partitioned on the complex plane, and we obtain the jump matrix between each partition based on the positive and negative values of the partition as well as the spectral matrix determined by the initial data and boundary value data. The findings suggest that the spectral functions are not mutually independent; instead, they conform to a global relationship. The novel aspect of this study is the application of the Fokas method to a previously unexplored case, contributing to the theoretical and practical understanding of complex partial differential equations and filling a gap in the treatment of boundary conditions.

Boundary conditions in flutter simulations of subsonic, transonic and supersonic blade cascades

Simulations of blade flutter are highly sensitive to undesired wave reflections at inlet and outlet boundaries. A careful treatment of boundary conditions is required to prevent the generation of perturbations. This study is motivated by the need to perform flutter analysis of low-pressure steam turbine blades, for which supersonic inflow conditions may occur in the near-tip region. The exact steady non-reflecting boundary condition (NRBC), the spectral NRBC and a simple isentropic boundary condition are implemented in a time-marching flow solver and applied to turbomachinery flutter simulations covering a wide range of operating conditions. For the first time, the spectral NRBC is applied to a blade flutter simulation with a supersonic inlet and its performance is analysed and compared with other boundary condition formulations. It is shown that an effective non-reflective treatment in the design of the boundary condition is essential for an accurate aeroelastic prediction at all operating conditions, including the subsonic flow regime. The limitation of the exact steady NRBC to spatial modes causes it to perform poorly in some unsteady flow simulations, whereas the spectral NRBC achieves a satisfactory suppression of undesired wave reflections in all investigated cases.

Towards a momentum potential theory for reacting flows

Mutual coupling of (thermofluiddynamic) modes of perturbations can affect the thermo-acoustic stability of combustors and contribute to combustion noise. For example, vortical or entropic perturbations can be transferred to acoustic perturbations if accelerated by the mean flow. The decomposition of perturbation fields into the respective modes and a linear description of their interactions in terms of fluctuating primitive variables is challenging. In contrast, Doak’s momentum potential theory promises an unambiguous decomposition in terms of momentum fluctuations, which is not limited to the linear regime. Whereas classical momentum potential theory takes into account hydrodynamic, acoustic and entropic modes in unconfined flows, the investigation of noise generation in combustion chambers requires the extension of the momentum potential theory to capture modes linked to the fluctuation of species mass fractions (“species mode”) arising from the change in chemical composition due to the reaction. Furthermore, a rigorous treatment of boundary conditions due to the confinement of the flow inside the combustor is required. The herein presented extension to reactive flows consists of two steps, (i) the formulation of a potential for momentum fluctuations related to species modes and (ii) identification of the total fluctuating enthalpy related to species modes. The extended theory is applied to post-process computational fluid dynamic simulation data of the propagation of entropy and species perturbations through one-dimensional ducts, nozzles and premixed flames. We find that although momentum potential theory offers a complete decomposition of momentum perturbations for reactive flows, the meaningful interpretation of this decomposition is rather challenging, even for non-reactive flows.

Algorithm for the treatment of boundary conditions in NMM-SPH coupling models: Interface element-wise boundary particle scheme

Large deformation and complex fluid flow interactions, known as the fluid–structure interaction (FSI), are common in geotechnical and coastal projects. To address FSI problems involving crack development, block contact and large movement, a numerical model that utilises smoothed particle hydrodynamics for fluid field spatial discretisation and numerical manifold method for solid block domain solution is proposed in the present study. Specifically, to restore the particle inconsistency near the interface and the overflow issue around the sharp corner in traditional coupled methods, a recommended boundary particle approach is designed to provide full support for particles near the interface, regardless of the interface’s geometry. Moreover, this approach enables the choice of different spatial resolutions tailored to requirements of solid and fluid domains. Six numerical examples were conducted to verify the reliability, accuracy and robustness of the new method. Results showed that relative errors around the sharp corner are lower than those in previous studies. Furthermore, the new model performs well in dealing with the motion of blocks with arbitrary shapes, including large movements, block contacts, crack propagations, and large deformations. Additionally, various inner-variable field evolutions between fluids and solids were visualised, which is beneficial for the prediction of multi-hazard scenario outcomes.

On the calculation of irregular solutions of the Schrödinger equation for non-spherical potentials with applications to metallic alloys

The irregular solutions of the stationary Schrödinger equation are important for the fundamental formal development of scattering theory. They are also necessary for the analytical properties of the Green function, which in practice can greatly speed up calculations. Nevertheless, they are seldom considered in numerical treatments because of their divergent behavior at origin. This divergence demands high numerical precision that is difficult to achieve, particularly for non-spherical potentials which lead to different divergence rates in the coupled angular momentum channels. Based on an unconventional treatment of boundary conditions, an integral-equation method is here developed which is capable of dealing with this problem. The available precision is illustrated by electron-density calculations for NiTi in its monoclinic B19’ structure.

A fully coupled Pressure-Based method for compressible flows at all Mach numbers

When extending the fully coupled pressure-based method to resolve compressible flows, the effective treatment of the boundary conditions is crucial due to the inherent characteristics of compressible flows. Failure to properly account for this can result in numerical instability and divergence. In this article, a detailed treatment of boundary conditions for compressible flows has been formulated in terms of the boundary face’s contribution to the discretization equation of the boundary element. These boundary conditions involve subsonic inlet, supersonic inlet, subsonic outlet, and supersonic outlet boundary conditions. The analysis has been focused on test cases of flow through a bumped arc channel and a converging–diverging nozzle including quasi-incompressible, subsonic, transonic, supersonic and hypersonic speed regimes, where the Mach number ranges from 0.1 to 10. The results indicate the robust performance of the fully coupled method in resolving compressible flows. The computational results obtained from the fully coupled algorithm exhibit a high degree of consistency with available literature cases and analytical solutions, with a maximum error of less than 5%. Moreover, the findings also suggest that the coupled solver presents promising potential in handling hypersonic flows.

Very High-Order Accurate Discontinuous Galerkin Method for Curved Boundaries with Polygonal Meshes

Preserving the optimal convergence order of discontinuous Galerkin (DG) discretisations in curved domains is a critical and well-known issue. The proposed approach relies on the reconstruction for off-site data (ROD) method developed originally within the finite volume framework. The main advantages are simplicity, since the PDE solver only considers polygonal domains, and versatility, since any type of boundary condition can be imposed. The developed DG–ROD method consists in splitting the boundary conditions treatment and the leading discrete equations from a classical DG formulation into two independent solvers coupled in a simple and efficient iterative procedure. A numerical benchmark is provided to assess the capability of the method with Dirichlet and Neumann boundary conditions prescribed on curved boundaries, demonstrating that the optimal convergence order is effectively achieved.

Tobac v1.5: introducing fast 3D tracking, splits and mergers, and other enhancements for identifying and analysing meteorological phenomena

Abstract. There is a continuously increasing need for reliable feature detection and tracking tools based on objective analysis principles for use with meteorological data. Many tools have been developed over the previous 2 decades that attempt to address this need but most have limitations on the type of data they can be used with, feature computational and/or memory expenses that make them unwieldy with larger datasets, or require some form of data reduction prior to use that limits the tool's utility. The Tracking and Object-Based Analysis of Clouds (tobac) Python package is a modular, open-source tool that improves on the overall generality and utility of past tools. A number of scientific improvements (three spatial dimensions, splits and mergers of features, an internal spectral filtering tool) and procedural enhancements (increased computational efficiency, internal regridding of data, and treatments for periodic boundary conditions) have been included in tobac as a part of the tobac v1.5 update. These improvements have made tobac one of the most robust, powerful, and flexible identification and tracking tools in our field to date and expand its potential use in other fields. Future plans for tobac v2 are also discussed.

Multiple solutions for fractional elliptic systems

Abstract This paper investigates the existence and multiplicity of solutions to fractional elliptic systems on conical spaces. Specifically, we focus on the challenges posed by complex geometric configurations, including cones with rough bases, and their implications for the treatment of lateral boundary conditions. By utilizing the fibering map approach and iterative method, we aim to address these challenges and provide new insights into the field. Notably, these issues have not been previously explored in existing literature, highlighting the originality and significance of our study.

Multi-Passage harmonic balance method for flutter prediction in multi-row configurations

A multi-passage harmonic balance method is presented for flutter prediction in multi-row configurations, and particular focus is on simulation of more than two blade rows. The unsteady flows in such configurations contain two kinds of multiple perturbations: one refers to those who have different frequencies and the other one refers to those who have the same frequency but different inter-blade phase angles (IBPAs). Most of current harmonic balance methods only consider the former one and the applications have been restricted to two-blade-row configurations. In the few studies that consider more rows, a one-to-one correspondence assumption between frequency and IBPA is usually made putting extra limitation on the used modes. This motivates our work, and a simple but effective idea is proposed by adding passages adopted per row. The theoretical basis of this work is provided by the spinning mode analysis. The present method is developed based on the time-spectral and multi-frequential form of harmonic balance. The treatment of pitchwise and inter-row boundary conditions in the case of multi passages are particularly addressed. A recently-developed Jacobian-free Newton-Krylov method with a simplified LU-SGS preconditioner is employed to solve the resultant harmonic balance equations. For validation, two typical multi-row flutter cases are investigated. The numerical results show the validity and accuracy of the developed method.

A numerical B-spline Galerkin method with proper generalized decomposition for reduced order modeling of partial differential equations

A numerical study is presented on reduced order modeling of low and high-dimensional partial differential equations using a new B-spline Galerkin proper generalized decomposition (PGD) method. Galerkin-PGD schemes based on B-splines are numerically investigated for various benchmark problems, including the Poisson equation, linear/nonlinear advection–diffusion equations, and the Navier–Stokes equations. The treatment of inhomogeneous boundary conditions, non-separable terms and nonlinearities is numerically investigated. Numerical errors for high-fidelity B-spline solutions and reduced order solutions are provided to demonstrate the effectiveness and accuracy of the proposed method. Computational costs are reported to illustrate the efficiency of the developed approaches.

Cut-PFEM: a Particle Finite Element Method using unfitted boundary meshes

AbstractIn this work, we present a novel unfitted mesh boundary strategy in the context of the Particle Finite Flement Method (PFEM) aiming to improve endemic limitations of the PFEM relative to boundary conditions treatment and mass conservation. In this new methodology, which we called Cut-PFEM, the fluid–wall interaction is not performed by adding interface elements, as is done in the standard PFEM boundaries. Instead, we use an implicit representation of (all or some of) the boundaries by introducing the use of a level set function. Such distance function detects the elements trespassing the (virtual) contours of the domain to equip them with opportunely boundary conditions, which are variationally enforced using Nitsche’s method. The proposed Cut-PFEM circumvents important issues associated with the standard PFEM contact detection algorithm, such as the artificial addition of mass to the computational domain and the anticipation of contact time. Furthermore, the Cut-PFEM represents a natural ground for the imposition of alternative wall boundary conditions (e.g., pure slip) which pose significant difficulties in a standard PFEM framework. Several numerical examples, featuring both no-slip and slip boundary conditions, are presented to prove the accuracy and robustness of the method in two-dimensional and three-dimensional scenarios.

Perturbative RG flows in AdS. An étude

We discuss general properties of perturbative RG flows in AdS with a focus on the treatment of boundary conditions and infrared divergences. In contrast with flat-space boundary QFT, general covariance in AdS implies the absence of independent boundary flows. We illustrate how boundary correlation functions remain conformally covariant even if the bulk QFT has a scale. We apply our general discussion to the RG flow between consecutive unitary diagonal minimal models which is triggered by the ϕ(1,3) operator. For these theories we conjecture a flow diagram whose form is significantly simpler than that in flat-space boundary QFT. In several stand-alone appendices we discuss two-dimensional BCFTs in general and the minimal model BCFTs in particular. These include both an extensive review as well as the computation of several new BCFT correlation functions.

Grad's 13 moments-based gas kinetic flux solver with triangle unstructured meshes for simulating continuum and rarefied fluid flows

A numerical framework based on the gas kinetic flux solver (GKFS) with unstructured meshes of triangle cells has been proposed and assessed in this work. Like conventional GKFS, the present scheme only updates the cell-average macroscopic conservative variables in time, thereby saving computational costs. At each cell interface, the numerical normal flux is computed with the GKFS scheme proposed by Liu et al. [J. Comput. Phys. 415, 109548 (2020)], in which the distribution functions at the surrounding points are reconstructed with the Grad's 13 moments (G13). Additionally, a solid wall boundary condition treatment has been proposed in the present G13-GKFS framework, which allows one to model from purely specular to perfectly thermalized solid walls. The present framework has been validated with the steady cylindrical Couette flow, the lid-driven cavity flow, the unsteady Rayleigh flow, and the rarefied flow around the NACA0012 airfoil test-cases, where good agreements are found with references for a quite wide range of flow regimes, from continuum to transitional flow regimes.

Practical considerations for the adoption of Anderson acceleration in nonlinear diffusion accelerated transport

The use of the Anderson Acceleration (AA) method within the context of the coarse mesh finite difference (CMFD)-accelerated neutron transport k-eigenvalue calculation is explored for practical problems. Current application of CMFD only has a linear convergence rate. The AA method is similar to a quasi-Newton method and can improve the convergence rate of fixed-point iterations. We therefore implement AA for the 2D calculation in MPACT and study its performance in pressurized water reactor (PWR) problems. The initial results show that the traditional AA method can improve the convergence rate in a trivial problem and improve the stability of the CMFD method for some problems. However, the success of AA in the practical problems, where simulations with symmetry or domain decomposition, relies on one essential factor–the treatment of iterative angular flux boundary conditions in the AA. Additionally, the efficiency of AA is also affected by the computational cost for solving the optimization problem, and the computational cost of the transport solver and the CMFD solver. To address these issues, we propose a new implementation of AA that considers only the mutligroup scalar flux in the optimization problem, but the angular flux and the scalar flux in the mixing. The new approach achieves better stability and efficiency in practical simulations, and can be explained by the theory of Randomized Numerical Linear Algebra (RandNLA).

Implicit boundary conditions in partial differential equations discretizations: Identifying spurious modes and model reduction

We revisit the problem of spurious modes that are sometimes encountered in partial differential equations discretizations. It is generally suspected that one of the causes for spurious modes is due to how boundary conditions are treated, and we use this as the starting point of our investigations. By regarding boundary conditions as algebraic constraints on a differential equation, we point out that any differential equation with homogeneous boundary conditions also admits a typically infinite number of hidden or implicit boundary conditions. In most discretization schemes, these additional implicit boundary conditions are violated, and we argue that this is what leads to the emergence of spurious modes. These observations motivate two definitions of the quality of computed eigenvalues based on violations of derivatives of boundary conditions on the one hand, and on the Grassmann distance between subspaces associated with computed eigenspaces on the other. Both of these tests are based on a standardized treatment of boundary conditions and do not require a priori knowledge of eigenvalue locations. The effectiveness of these tests is demonstrated on several examples known to have spurious modes. In addition, these quality tests show that in most problems, about half of the computed spectrum of a differential operator is of low quality. The tests also specifically identify the low accuracy modes, which can then be projected out as a type of model reduction scheme.