Second-order Spatial Convergence Research Articles

Large–Eddy Simulation of the Flow Inside a Scour Hole Around a Circular Cylinder Using a Cut Cell Immersed Boundary Method

We present a novel symmetry-preserving cut cell finite volume method which is a three-dimensional generalisation of the method by Dröge and Verstappen (Int J Numer Method Fluids 47:979–985, 2005). A colour-coding scheme for the three-dimensional cut momentum cell faces reduces the number of possible cut cell configurations. A cell merging strategy is employed to alleviate time step constraints. We demonstrate the energy conservation property of the convective and pressure gradient terms, and the second-order spatial convergence with suitable benchmark cases. We used the scheme to perform highly resolved large–eddy simulations of the flow inside a scour hole around a circular cylinder mounted vertically in a flume. The simulation results are extensively compared to a stereoscopic particle image velocimetry experiment of the same configuration performed by Jenssen and Manhart (Exp Fluids 61:217, 2020). We demonstrate that for the investigated Reynolds numbers (20,000 and 40,000) nearly converged solutions are obtained; however at large computational efforts (up to 2.35 billion cells for the higher Reynolds number). It turns out that the flow topology of the horseshoe vortex system is strongly dependent on the grid resolution. For simulation results obtained on the finest grid, the mean flow and turbulence quantities agree well with the experiment. We investigate the shape and turbulence structure of the horseshoe vortex based on three-dimensional fields, and discuss the distribution of the mean and standard deviation of the wall shear stress in the scour hole and the implications for the physics of the scouring process over a sand bed.

Numerical Schemes for Solving the Time-Fractional Dual-Phase-Lagging Heat Conduction Model in a Double-Layered Nanoscale Thin Film

This article proposes a time fractional dual-phase-lagging (DPL) heat conduction model in a double-layered nanoscale thin film with the temperature-jump boundary condition and a thermal lagging effect interfacial condition between layers. The model is proved to be well-posed. A finite difference scheme with second-order spatial convergence accuracy in maximum norm is then presented for solving the fractional DPL model. Unconditional stability and convergence of the scheme are proved by using the discrete energy method. A numerical example without exact solution is given to verify the accuracy of the scheme. Finally, we show the applicability of the time fractional DPL model by predicting the temperature rise in a double-layered nanoscale thin film, where a gold layer is on a chromium padding layer exposed to an ultrashort-pulsed laser heating.

Dynamics of dusty vortices – I. Extensions and limitations of the terminal velocity approximation

ABSTRACT Motivated by the stability of dust laden vortices, in this paper we study the terminal velocity approximation equations for a gas coupled to a pressureless dust fluid and present a numerical solver for the equations embedded in the FARGO3D hydrodynamics code. We show that for protoplanetary discs it is possible to use the barycentre velocity in the viscous stress tensor, making it trivial to simulate viscous dusty protoplanetary discs with this model. We also show that the terminal velocity model breaks down around shocks, becoming incompatible with the two-fluid model it is derived from. Finally we produce a set of test cases for numerical schemes and demonstrate the performance of our code on these tests. Our implementation embedded in FARGO3D using an unconditionally stable explicit integrator is fast, and exhibits the desired second-order spatial convergence for smooth problems.

An hr-adaptive method for the cubic nonlinear Schrödinger equation

The nonlinear Schrödinger equation (NLSE) is one of the most important equations in quantum mechanics, and appears in a wide range of applications including optical fibre communications, plasma physics and biomolecule dynamics. It is a notoriously difficult problem to solve numerically as solutions have very steep temporal and spatial gradients. Adaptive moving mesh methods ( r -adaptive) attempt to optimise the accuracy obtained using a fixed number of nodes by moving them to regions of steep solution features. This approach on its own is however limited if the solution becomes more or less difficult to resolve over the period of interest. Adaptive mesh refinement ( h -adaptive), where the mesh is locally coarsened or refined, is an alternative adaptive strategy which is popular for time-independent problems. In this paper, we consider the effectiveness of a combined method ( h r -adaptive) to solve the NLSE in one space dimension. Simulations are presented indicating excellent solution accuracy compared to other moving mesh approaches. The method is also shown to control the spatial error based on the user’s input error tolerance. Evidence is also presented indicating second-order spatial convergence using a novel monitor function to generate the adaptive moving mesh.

Lattice Boltzmann methods in porous media simulations: From laminar to turbulent flow

The lattice Boltzmann method has become a popular tool for determining the correlations for drag force and permeability in porous media for a wide range of Reynolds numbers and solid volume fractions. In order to achieve accurate and relevant results, it is important not only to implement very efficient code, but also to choose the most appropriate simulation setup. Moreover, it is essential to accurately evaluate the boundary conditions and collision models that are effective from the Stokes regime to the inertial and turbulent flow regimes. In this paper, we compare various no-slip boundary schemes and collision operators to assess their efficiency and accuracy. Instead of assuming a constant volume force driving the flow, a periodic pressure drop boundary condition is employed to mimic the pressure-driven flow through the simple sphere pack in a periodic domain. We first consider the convergence rates of various boundary conditions with different collision operators in the Stokes flow regime. Additionally, we choose different boundary conditions that are representatives of first- to third-order schemes at curved boundaries in order to evaluate their convergence rates numerically for both inertial and turbulent flow. We find that the multi-reflection boundary condition yields second order for inertial flow while it converges with third order in the Stokes regime. Taking into account the both computational cost and accuracy requirements, we choose the central linear interpolation bounce-back scheme in combination with the two-relaxation-time collision model. This combination is characterized by providing viscosity independent results and second-order spatial convergence. This method is applied to perform simulations of touching spheres arranged in a simple cubic array. Full- and reduced-stencil lattice models, i.e., the D3Q27 and D3Q19, respectively, are compared and the drag force and friction factor results are presented for Reynolds numbers in the range of 0.001 to 2,477. The drag forces computed using these two different lattice models have a relative difference below 3% for the highest Reynolds number considered in this study. Using the evaluation results, we demonstrate the flexibility of the models and software in two large scale computations, first a flow through an unstructured packing of spherical particles, and second for the turbulent flow over a permeable bed.

A New Efficient and Stable 3D Conformal FDTD

A novel conformal technique for the FDTD method, here referred to as Conformal Relaxed Dey-Mittra method, is proposed and assessed in this letter. This technique helps avoid local time-step restrictions caused by irregular cells, thereby improving the global stability criterion of the original Dey-Mittra method. The approach retains a second-order spatial convergence. A numerical experiment based on the NASA almond has been chosen to show the improvement in accuracy and computational performance of the proposed method.

A hybrid incremental projection method for thermal-hydraulics applications

A new second-order accurate, hybrid, incremental projection method for time-dependent incompressible viscous flow is introduced in this paper. The hybrid finite-element/finite-volume discretization circumvents the well-known Ladyzhenskaya–Babuška–Brezzi conditions for stability, and does not require special treatment to filter pressure modes by either Rhie–Chow interpolation or by using a Petrov–Galerkin finite element formulation. The use of a co-velocity with a high-resolution advection method and a linearly consistent edge-based treatment of viscous/diffusive terms yields a robust algorithm for a broad spectrum of incompressible flows. The high-resolution advection method is shown to deliver second-order spatial convergence on mixed element topology meshes, and the implicit advective treatment significantly increases the stable time-step size. The algorithm is robust and extensible, permitting the incorporation of features such as porous media flow, RANS and LES turbulence models, and semi-/fully-implicit time stepping. A series of verification and validation problems are used to illustrate the convergence properties of the algorithm. The temporal stability properties are demonstrated on a range of problems with 2 ≤ C F L ≤ 100 . The new flow solver is built using the Hydra multiphysics toolkit. The Hydra toolkit is written in C++ and provides a rich suite of extensible and fully-parallel components that permit rapid application development, supports multiple discretization techniques, provides I/O interfaces, dynamic run-time load balancing and data migration, and interfaces to scalable popular linear solvers, e.g., in open-source packages such as HYPRE, PETSc, and Trilinos. • A new second-order hybrid finite-element/finite-volume projection algorithm for transient viscous flow has been introduced. • The hybrid discretization prevents pressure modes without using Rhie–Chow interpolation or a Petrov–Galerkin formulation. • A monotonicity-preserving advection method shown to deliver second-order accuracy on mixed element topology meshes. • Verification studies demonstrate hybrid projection solver accuracy and temporal stability for super-CFL conditions.

A second-order Cartesian method for the simulation of electropermeabilization cell models

In this paper, we present a new finite differences method to simulate electropermeabilization models, like the model of Neu and Krassowska or the recent model of Kavian et al. These models are based on the evolution of the electric potential in a cell embedded in a conducting medium. The main feature lies in the transmission of the voltage potential across the cell membrane: the jump of the potential is proportional to the normal flux thanks to the well-known Kirchoff law. An adapted scheme is thus necessary to accurately simulate the voltage potential in the whole cell, notably at the membrane separating the cell from the outer medium. We present a second-order finite differences scheme in the spirit of the method introduced by Cisternino and Weynans for elliptic problems with immersed interfaces. This is a Cartesian grid method based on the accurate discretization of the fluxes at the interface, through the use of additional interface unknowns. The main novelty of our present work lies in the fact that the jump of the potential is proportional to the flux, and therefore is not explicitly known. The original use of interface unknowns makes it possible to discretize the transmission conditions with enough accuracy to obtain a second-order spatial convergence. We prove the second-order spatial convergence in the stationary linear one-dimensional case, and the first-order temporal convergence for the dynamical non-linear model in one dimension. We then perform numerical experiments in two dimensions that corroborate these results.

A technique implemented in the VOLNA code package to simulate the natural gas flux in pipelines

The paper presents an explicit Lagrangian-Eulerian technique implemented in the VOLNA code package for simulating the transient fluxes of natural gas in pipelines. The test calculations of stationary problems show that the technique has attained the second-order spatial convergence for smooth fluxes and provides acceptable accuracy in practical calculations.

Immersed Boundary-Lattice Boltzmann Coupling Scheme for Fluid-Structure Interaction with Flexible Boundary

AbstractCoupling the immersed boundary (IB) method and the lattice Boltzmann (LB) method might be a promising approach to simulate fluid-structure interaction (FSI) problems with flexible structures and complex boundaries, because the former is a general simulation method for FSIs in biological systems, the latter is an efficient scheme for fluid flow simulations, and both of them work on regular Cartesian grids. In this paper an IB-LB coupling scheme is proposed and its feasibility is verified. The scheme is suitable for FSI problems concerning rapid flexible boundary motion and a large pressure gradient across the boundary. We first analyze the respective concepts, formulae and advantages of the IB and LB methods, and then explain the coupling strategy and detailed implementation procedures. To verify the effectiveness and accuracy, FSI problems arising from the relaxation of a distorted balloon immersed in a viscous fluid, an unsteady wake flow caused by an impulsively started circular cylinder at Reynolds number 9500, and an unsteady vortex shedding flow past a suddenly started rotating circular cylinder at Reynolds number 1000 are simulated. The first example is a benchmark case for flexible boundary FSI with a large pressure gradient across the boundary, the second is a fixed complex boundary problem, and the third is a typical moving boundary example. The results are in good agreement with the analytical and existing numerical data. It is shown that the proposed scheme is capable of modeling flexible boundary and complex boundary problems at a second-order spatial convergence; the volume leakage defect of the conventional IB method has been remedied by using a new method of introducing the unsteady and non-uniform external force; and the LB method makes the IB method simulation simpler and more efficient.

Convergence analysis of moving Godunov methods for dynamical boundary layers

In this paper, a Godunov scheme on moving meshes is studied for kinds of time-dependent convection-dominated equations with dynamical boundary layers. The stability and a second-order spatial convergence are proved. A numerical example is provided to confirm the theoretical results.

Modeling Arteriolar Flow and Mass Transport Using the Immersed Boundary Method

Flow in arterioles is determined by a number of interacting factors, including perfusion pressure, neural stimulation, vasoactive substances, the intrinsic contractility of arteriolar walls, and wall shear stress. We have developed a two-dimensional model of arteriolar fluid flow and mass transport. The model includes a phenomenological representation of the myogenic response of the arteriolar wall, in which an increase in perfusion pressure stimulates vasoconstriction. The model also includes the release, advection, diffusion, degradation, and dilatory action of nitric oxide (NO), a potent, but short-lived, vasodilatory agent. Parameters for the model were taken primarily from the experimental literature of the rat renal afferent arteriole. Solutions to the incompressible Navier-Stokes equations were approximated by means of a splitting that used upwind differencing for the inertial term and a spectral method for the viscous term and incompressibility condition. The immersed boundary method was used to include the forces arising from the arteriolar walls. The advection of NO was computed by means of a high-order flux-corrected transport scheme; the diffusion of NO was computed by a spectral solver. Simulations demonstrated the efficacy of the numerical methods employed, and grid refinement studies confirmed anticipated first-order temporal convergence and demonstrated second-order spatial convergence in key quantities. By providing information about the effective width of the immersed boundary and sheer stress magnitude near that boundary, the grid refinement studies indicate the degree of spatial refinement required for quantitatively reliable simulations. Owing to the dominating effect of NO advection, relative to degradation and diffusion, simulations indicate that NO has the capacity to produce dilation along the entire length of the arteriole.