Simulation Of Incompressible Flows Research Articles

A mixed virtual element method for the two-dimensional Navier-Stokes equations in stream-function formulation

This work presents the formulation and analysis of a H1− conforming mixed virtual element method (VEM) for the two-dimensional stationary incompressible Navier-Stokes (NS) equations in stream-function formulation. By representing the velocity field as the curl of a stream function, we recast the second-order NS system into a fourth-order nonlinear equation for the scalar stream function, inherently satisfying the incompressibility constraint. Introducing a vorticity variable enables construction of H1− conforming VEM spaces for both stream function and vorticity and circumventing stringent C1− continuity constraints. The proposed method provides an initial exploration of stream function-vorticity discretizations on general polygonal meshes using the flexible VEM of arbitrary order. Existence and uniqueness of discrete solutions are established theoretically under a small data assumption. Optimal error estimates are then derived in the energy norm for the stream function, H1− norm for the stream function and L2− norm for the vorticity, rigorously demonstrating convergence. Numerical results validate the error analysis and illustrate the accuracy and robustness of the mixed VEM for simulation of incompressible flows on complex geometries.

Subgrid-scale model for large eddy simulations of incompressible turbulent flows within the lattice Boltzmann framework

Subgrid-scale model for large eddy simulations of incompressible turbulent flows within the lattice Boltzmann framework

On the boundary conditions for GFMxP high-order schemes on staggered grids in the simulation of incompressible multiphase flows

The simulation of incompressible multiphase flows through the so-called fractional step method needs to solve a variable coefficient Poisson equation for discontinuous functions. Recently, it has been shown how the solution of this equation may be found out through a novel coding of the Ghost Fluid Method (named GFMxP), by avoiding any fit to evaluate the interface position and providing, anyhow, a perfect sharp modeling of the same interface. Furthermore, the accuracy order of the numerical solutions exactly corresponds to the order of the adopted finite difference scheme. The effectiveness and reliability of the new procedure were successfully checked by a lot of tests. However, the a-priori knowledge of the unknown function allowed to elude a fundamental aspect of the numerical approach: the appropriate encoding of the boundary conditions. This topic has often been debated in the past, especially from a theoretical viewpoint, and still represents a rather thorny point in the whole simulation process. The paper shows how to handle the problem in practice and in the context of the GFMxP approach, i.e. by accounting for the presence of the discontinuity and the possible use of high-order solving schemes on a staggered grid.

A fourth-order compact finite volume method on unstructured grids for simulation of two-dimensional incompressible flow

This paper introduces a novel and efficient compact reconstruction procedure for high-order finite volume methods applied to unstructured grids. In this procedure, we establish a set of constitutive relations that ensure the continuity of the reconstruction polynomial and its normal derivatives between adjacent elements at control points. The paper delves into the details of the fourth-order compact reconstruction method specifically designed for two-dimensional triangular grids. This method can be considered an extension of the one-dimensional compact scheme and cubic spline interpolation methods to two-dimensional triangular unstructured grids. In the cubic polynomial reconstruction, a two-dimensional cubic polynomial is reconstructed using the cell averages of elements in the standard stencil and the function values at control points. The determination of function values at control points is achieved by adhering to the constraint of normal derivative continuity. This reconstruction is solved using a relaxation iteration method and has been verified to be solvable for triangular grids. Compared to other fourth-order implicit compact polynomial reconstructions, our method requires solving a smaller number of unknowns, which means less computational cost in reconstruction. By applying this method to solve the Poisson equation, the linear convection equation, and incompressible flow benchmarks, it demonstrates that the proposed method exhibits the expected high-order accuracy and performs well in incompressible flow problems.

DualSPHysics+: An enhanced DualSPHysics with improvements in accuracy, energy conservation and resolution of the continuity equation

This paper presents an enhanced version of the well-known SPH (Smoothed Particle Hydrodynamics) open-source code DualSPHysics for the simulation of free-surface fluid flows, leading to the DualSPHysics+ code. The enhancements are made through incorporation of several schemes with respect to stability, accuracy and energy/volume conservation issues in simulating incompressible free-surface fluid flows within the weakly compressible SPH formalism. The Optimized Particle Shifting (OPS) scheme is implemented to improve the accuracy of particle shifting vectors in the free-surface region. To mitigate energy dissipation and maintain consistency, the artificial viscosity in δ-SPH is substituted with a Riemann stabilization term, leading to the δR-SPH. The Velocity divergence Error Mitigating (VEM) and Volume Conservation Shifting (VCS) schemes are adopted in DualSPHysics+ to mitigate the velocity divergence error and improve the volume conservation, and hence to enhance the resolution of the continuity equation. To further reduce both the instantaneous and accumulated errors in velocity divergence, a Hyperbolic/Parabolic Divergence Cleaning (HPDC) scheme is incorporated in addition to the VEM scheme. The implementations of the introduced schemes on both CPU and GPU-based versions of the DualSPHysics+ code along with details on the compilation, running and computational performance are presented. Validations in terms of accuracy, energy conservation and convergence of DualSPHysics+ are shown via several relevant benchmarks. It is demonstrated that a better velocity divergence error cleaning in both instantaneous and accumulated errors can be achieved by the combination of VEM and HPDC. Meanwhile, the excessive energy dissipation by the artificial viscosity is shown to be suppressed by adopting the Riemann stabilization term. Enhanced resolution of the continuity equation along with improved energy conservation of DualSPHysics+ advance the SPH-based simulation of incompressible free-surface fluid flows.

Adaptive refinement in incompressible fluid flow simulation based on THB-splines-powered isogeometric analysis

In this paper, we deal with adaptive refinement in numerical simulation of incompressible flow solved by isogeometric analysis. We study various error estimators and compare them with respect to convergence to the exact solution. Further, we propose a new class of error estimators based on stabilization methods for numerical solving of incompressible flow and we show that they provide viable option to standard error estimators. Moreover, we comment on different choices of marking strategies and their suitability to the case of incompressible flow and provide comparison of error estimators also with respect to selected marking strategies and selected representative pairs of discretization spaces in isogeometric analysis.

Numerical simulation of steady incompressible slip flow around a circular cylinder at low Reynolds numbers

Numerical simulation of steady incompressible slip flow around a circular cylinder at low Reynolds numbers

Modeling and simulation of simultaneous transport and incompressible flows on all level sets in a bulk domain

Modeling and simulation of simultaneous transport and incompressible flows on all level sets in a bulk domain

Bridging Large Eddy Simulation and Reduced-Order Modeling of Convection-Dominated Flows through Spatial Filtering: Review and Perspectives

Reduced-order models (ROMs) have achieved a lot of success in reducing the computational cost of traditional numerical methods across many disciplines. In fluid dynamics, ROMs have been successful in providing efficient and relatively accurate solutions for the numerical simulation of laminar flows. For convection-dominated (e.g., turbulent) flows, however, standard ROMs generally yield inaccurate results, usually affected by spurious oscillations. Thus, ROMs are usually equipped with numerical stabilization or closure models in order to account for the effect of the discarded modes. The literature on ROM closures and stabilizations is large and growing fast. In this paper, instead of reviewing all the ROM closures and stabilizations, we took a more modest step and focused on one particular type of ROM closure and stabilization that is inspired by large eddy simulation (LES), a classical strategy in computational fluid dynamics (CFD). These ROMs, which we call LES-ROMs, are extremely easy to implement, very efficient, and accurate. Indeed, LES-ROMs are modular and generally require minimal modifications to standard (“legacy”) ROM formulations. Furthermore, the computational overhead of these modifications is minimal. Finally, carefully tuned LES-ROMs can accurately capture the average physical quantities of interest in challenging convection-dominated flows in science and engineering applications. LES-ROMs are constructed by leveraging spatial filtering, which is the same principle used to build classical LES models. This ensures a modeling consistency between LES-ROMs and the approaches that generated the data used to train them. It also “bridges” two distinct research fields (LES and ROMs) that have been disconnected until now. This paper is a review of LES-ROMs, with a particular focus on the LES concepts and models that enable the construction of LES-inspired ROMs and the bridging of LES and reduced-order modeling. This paper starts with a description of a versatile LES strategy called evolve–filter–relax (EFR) that has been successfully used as a full-order method for both incompressible and compressible convection-dominated flows. We present evidence of this success. We then show how the EFR strategy, and spatial filtering in general, can be leveraged to construct LES-ROMs (e.g., EFR-ROM). Several applications of LES-ROMs to the numerical simulation of incompressible and compressible convection-dominated flows are presented. Finally, we draw conclusions and outline several research directions and open questions in LES-ROM development. While we do not claim this review to be comprehensive, we certainly hope it serves as a brief and friendly introduction to this exciting research area, which we believe has a lot of potential in the practical numerical simulation of convection-dominated flows in science, engineering, and medicine.

Conservative transport model for surfactant on the interface based on the phase-field method

We propose a new surfactant transport model on the interface based on the phase-field method. The new model has two features: (a) preventing surfactants on the interface from leaking out of the interface due to numerical diffusion even when the diffusion coefficient is extremely low, and (b) satisfying the conservation of surfactant mass. The concentration profile on the interface is matched to the surface delta function to prevent numerical diffusion, and the finite volume method is applied to the resulting model to satisfy the conservation of surfactant mass. The proposed model is incorporated in a weakly compressible scheme and applied in several numerical tests related to incompressible two-phase flows with surfactants. Numerical results show that the proposed model can be applied to various incompressible two-phase flow simulations with surfactant.

Evolution of underwater noise emission prediction technology for ship-propulsor combinations in an industry environment

ABSTRACT The prediction of underwater noise emissions by marine vessels has evolved significantly over the last decade as the interest shifted from exclusive military to civil use as a result of environmental concerns. Among different research groups, the best practices for the estimation with high-fidelity simulation methods based on hydrodynamic sound emissions are developed considering the complete vessel and propulsor. Here the current state-of-the-art computational fluid dynamics (CFD) approach, centered around propulsion units, is worked out in steps leading to the application for vessels. This consists of incompressible finite volume method flow simulations with large eddy turbulence modelling and volume-of-fluid phase capturing with a Schnerr-Sauer cavitation model in combination with an added permeable surface Ffowcs-Williams-Hawkings method for far-field acoustics. Many mechanistic arguments support it and the final validation cases lead to acceptable accuracy and enrich simulation data acquisition, however, few technical limitations reveal weaknesses compared to traditional approaches.

A computational study for simulating MHD duct flows at high Hartmann numbers using a stabilized finite element formulation with shock-capturing

The concern of this manuscript is stabilized finite element simulations of two-dimensional incompressible and viscous magnetohydrodynamic (MHD) duct flows governed by a coupled system of partial differential equations of the convection–diffusion type. In such flows, the high values of the Hartmann numbers (Ha) indicate that the convection process dominates the flow field. As is typical trouble encountered in convection-dominated flow simulations, classical discretization methods fail to function properly for MHD flows at high Hartmann numbers, yielding several numerical instabilities. In this study, the core computational tool we employ in order to overcome such challenges is the streamline-upwind/Petrov–Galerkin (SUPG) formulation. Beyond that, since the SUPG-stabilized formulation requires additional treatment where solutions experience rapid changes, we also enhance the SUPG-stabilized finite element formulation with a shock-capturing operator for achieving better solution profiles around sharp layers. The formulations are derived for both steady-state and time-dependent models. The proposed method, the so-called SUPG-YZβ formulation, techniques used, and in-house-developed finite element solvers are evaluated on a comprehensive set of numerical experiments. The test computations reveal that the SUPG-YZβ formulation yields quite good solution profiles near strong gradients without any significant numerical instabilities, even for quite challenging values of Ha, whereas the SUPG usually fails alone. Furthermore, by using only linear elements on relatively coarser meshes, the proposed formulation achieves this without the need for any adaptive mesh strategy, in contrast to most previously reported studies.

A face-centred finite volume method for laminar and turbulent incompressible flows

This work develops, for the first time, a face-centred finite volume (FCFV) solver for the simulation of laminar and turbulent viscous incompressible flows. The formulation relies on the Reynolds-averaged Navier–Stokes (RANS) equations coupled with the negative Spalart–Allmaras (SA) model and three novel convective stabilisations, inspired by Riemann solvers, are derived and compared numerically. The resulting method achieves first-order convergence of the velocity, the velocity-gradient tensor and the pressure. FCFV accurately predicts engineering quantities of interest, such as drag and lift, on unstructured meshes and, by avoiding gradient reconstruction, the method is less sensitive to mesh quality than other FV methods, even in the presence of highly distorted and stretched cells. A monolithic and a staggered solution strategies for the RANS-SA system are derived and compared numerically. Numerical benchmarks, involving laminar and turbulent, steady and transient cases are used to assess the performance, accuracy and robustness of the proposed FCFV method.

Large-eddy simulation of wall-bounded incompressible turbulent flows based on multi-moment finite volume formulation

The multi-moment based numerical model is proposed for performing large-eddy simulation (LES) of wall-bounded incompressible turbulent flows on the unstructured grids. In this model, velocity is defined as volume integral average (VIA) and point values (PVs) which are updated as prognostic variables simultaneously in time. Using VIA and PV in each cell, this method constructs higher-order polynomial on a compact stencil and more importantly, provides a new approach to design subgrid-scale (SGS) models for LES. Various SGS models are developed in this framework where the eddy-viscosity is computed separately at both VIA and PVs using the least-square method on different interpolation stencils. Compared with conventional finite volume schemes, the proposed method effectively reduces numerical dissipation and largely suppresses the sensitivity of numerical solution on SGS models and resolves more elaborated flow structures on the same mesh resolution. It also reduces solution dependence on the shape and quality of grid elements, as demonstrated by comparing numerical results on the different types of grids. The proposed method can attain high-fidelity simulation without loss of numerical efficiency and robustness, which is highly appealing for complex turbulent problems with high Reynolds numbers and complicated geometries.

Development of an accurate central finite-difference scheme with a compact stencil for the simulation of unsteady incompressible flows on staggered orthogonal grids

In scale-resolving simulations such as Large Eddy Simulations (LES), the spatial discretization scheme of the convective term plays a crucial role in avoiding interference between the numerical errors and the subgrid-scale model. Accurate schemes lead to lower truncation errors and better predictions of turbulent flows without the need for an excessively refined grid. To this end, a new second-order finite-difference scheme (HCDS6) has been developed for incompressible flows and orthogonal staggered grids. Compared to the standard second-order scheme, the new scheme has significantly lower dispersion errors. Compared to existing high-order schemes, the numerical stencil of HCDS6 is more compact, which makes it easier to implement, especially considering boundary conditions around complex geometries using Immersed Boundary Methods (IBM). The HCDS6 scheme conserves the discrete momentum with limited production or dissipation of discrete kinetic energy, which guarantees its numerical stability. Its performance is evaluated using an open-source CFD package called REEF3D. Three benchmarks demonstrate the key properties and performance of the scheme: the convection of an isentropic vortex, the Taylor-Green vortex flow, and turbulent channel flow. Its relatively low dispersion errors, combined with ease of implementation, make the HCDS6 scheme a promising candidate for efficient scale-resolving simulations of turbulent flows.

Numerical investigation of the effect of perforated almond, tunnel, and bishop shaped pin fin heat sinks on the hydrothermal performance enhancement

The pin-fin design of the heat sink optimizes thermal dissipation by enhancing surface area and enabling efficient heat transfer in high-performance devices. The primary objective of the highly functionalized and convenient heat sink is to provide a competitive advantage in the field of efficient thermal management systems. The primary objective of this study is to investigate various designs of pin-fin heat sinks with unique shapes. These heat sinks are intended to operate in varying dynamic situations and necessitate an efficient convective heat transfer. The current technique has been thoroughly confirmed using earlier research, and computational investigations were conducted to improve the hydrothermal performance of staggered organized pin-fin heat sinks compared to the baseline case adopted from previous studies. Research is underway to examine the collective impact of the fin's unique structure, ideal aerodynamic form, and quality of perforations. A simulation of three-dimensional incompressible flow was conducted using the realizable k- ԑ model to generate turbulence flow that satisfies the relevant boundary conditions for solving the governing equations. When comparing fins shaped like a bishop to the standard hemispherical instance mentioned in recent work, the Nusselt number is 33 % higher. In addition, when the Reynolds number reaches 21,367, the presence of almond-shaped fins with circular perforations leads to a significant reduction in pressure drop, namely by 53.24 %. The almond-shaped hexagonal perforated fins, formed of a copper-diamond composite material, achieved the maximum hydrothermal performance factor (HTPF) of around 1.52 at a Reynolds number of 21,367. These layouts can be considered in various industrial applications.

A hierarchical level-set numerical approach for immiscible incompressible n-phase flows ([formula omitted])

In this article, we propose an approach for interface-resolved simulations of immiscible incompressible n-phase flows (n≥3). The standard level set (LS) method can accurately simulate the motion of two phases. However, when more than two phases are considered, the interfaces cannot intersect and numerical overlaps occur, leading to unphysical spurious velocities and unrealistic distortions of the interfaces. The proposed approach organizes a group of n−1 LS functions into a hierarchy to resolve the issue of numerical overlaps. The n-phase flow field is discretized using the extended finite element method (XFEM), which adequately addresses the discontinuities occurring in the pressure and velocity fields due to distinct fluid properties and surface tension effects at the interface between the fluids. This approach is applied to three-phase and four-phase fluid dynamics, involving gas and liquids under various conditions of surface tension. The 2D numerical tests prove that the proposed numerical methods can effectively model interactions between multi-fluid interfaces, avoiding numerical overlaps.

A Pressure Projection Scheme With Near‐Spectral Accuracy for Nonhydrostatic Flow in Domains With Open Boundaries

AbstractWe describe a pressure projection scheme for the simulation of incompressible flow in cubic domains with open boundaries based on fast Fourier transforms. The scheme is implemented in flow_solve, a numerical code designed for process studies of rotating, density‐stratified flow. The main algorithmic features of the open‐boundary code are the near‐spectral accuracy of the discrete differentiation and a dynamic two‐dimensional domain decomposition that scales efficiently to large numbers of processors. The simulated flows are not required to be periodic or to satisfy symmetry conditions at the open boundaries owing to the use of mixed series expansions combining cosine and singular Bernoulli polynomial basis functions. These expansions facilitate the imposition of inhomogeneous boundary conditions and allow the code to be used for offline, one‐way nesting within an arbitrarily embedded subdomain of a larger scale simulation. The projection scheme is designed to exploit a simple and powerful numerical engine: inversion of Poisson's equation with homogeneous Neumann boundary conditions using fast cosine transforms. Here, we describe the mathematical transformations used to accommodate the imposition of space‐ and time‐varying boundary conditions. The utility of the approach for process studies and for nesting within submesoscale‐resolving ocean models is demonstrated with simulations of wind‐driven near‐inertial waves in the upper ocean.

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.

Lighter and faster simulations on domains with symmetries

A strategy to improve the performance and reduce the memory footprint of simulations on meshes with spatial reflection symmetries is presented in this work. By using an appropriate mirrored ordering of the unknowns, discrete partial differential operators are represented by matrices with a regular block structure that allows replacing the standard sparse matrix–vector product with a specialised version of the sparse matrix-matrix product, which has a significantly higher arithmetic intensity. Consequently, matrix multiplications are accelerated, whereas their memory footprint is reduced, making massive simulations more affordable. As an example of practical application, we consider the numerical simulation of turbulent incompressible flows using a low-dissipation discretisation on unstructured collocated grids. All the required matrices are classified into three sparsity patterns that correspond to the discrete Laplacian, gradient, and divergence operators. Therefore, the above-mentioned benefits of exploiting spatial reflection symmetries are tested for these three matrices on both CPU and GPU, showing up to 5.0x speed-ups and 8.0x memory savings. Finally, a roofline performance analysis of the symmetry-aware sparse matrix–vector product is presented.