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This paper proposes a hybrid adaptive mesh refinement (AMR) algorithm for the immersed boundary method (IBM) combination, and the AMR code is split into mesh and physics codes to optimize each part individually. The uniform parent grid solver is used for AMR grids by constructing hanging cells for the high-order extension. The restrictive spatial refinement in a fully threaded tree (FTT) data structure is explored, and a simplified stencil search algorithm for hanging cells construction is introduced, including current and following child AMR level cells, if any. The proposed AMR method was applied to IBM, which offers flexibility in treating complex geometries in the Cartesian grid, leading to algorithmic simplicity and computational efficiency. Local near immersed boundary refinement is proposed to avoid complex and computationally expensive IBM and AMR algorithms near the solid bodies. Finally, a high-order flux scheme extension at AMR level transition cells and the proposed method's applicability for steady and transient flows are demonstrated. Besides state and flux variables storage, the proposed hybrid AMR method's additional cost is 4.5 words/cell instead of 3.5 words/cell for 2D and 3.75 words/cell instead of 2.75 words/cell for 3D compared to the conventional FTT-based AMR method. By comparing the uniform grid, the AMR final grid distribution shows that the optimal cell distribution based on flow physics reduces the cells required by more than 40% to resolve the complex flow features in less computational time. The enhanced performance and accuracy of the proposed AMR method in resolving different scale flow features are validated through benchmark problems.

► On the accuracy, robustness, and performance of high order interpolation schemes for the overset method on unstructured grids

18 Oct, 2021

For a second-order finite volume code, on diffusion dominant flows, interpolation schemes higher than second order are needed not to increase the total discretization errors. For convection dominated flows, the results suggest that second order schemes are sufficient to prevent overset schemes tainting discretization errors. Considering performance, the total overset-grid interpolation over-head (including donor search) may be less than 4% of the total run time for second-order schemes and 8% for third-order ones, therefore empowering higher-order schemes and more accurate solutions.

Abstract

A comprehensive study on interpolation schemes used in overset grid techniques is here presented. Based on a literature review, numerous schemes are implemented, and their robustness, accuracy, and performance are assessed. Two code verification exercises are performed for this purpose: a 2D analytical solution of a laminar Poiseuille steady flow; and an intricate manufactured solution of a turbulent flow case, characteristic of a boundary layer flow combined with an unsteady separation bubble. For both cases, the influence of grid layouts, grid refinement, and time-step is investigated. Local and global errors, convergence orders, and mass imbalance are quantified. In terms of computational performance, strong scalability, cpu timings, load imbalancing, and domain connectivity information (DCI) overhead are reported. The effect of the overset-grid interpolation schemes on the numerical performance of the solver, that is, number of nonlinear iterations, is also scrutinized. The results show that, for a second order finite volume code, once diffusion is dominant (low Reynolds number), interpolation schemes higher than second order, for example, least squares of degree 2, are needed not to increase the total discretization errors. For convection dominated flows (high Reynolds numbers), the results suggest that second order schemes, for example, nearest cell gradient, are sufficient to prevent overset grid schemes to taint the underlying discretization errors. In terms of performance, by single-process and parallel communication optimization, the total overset-grid overhead (with DCI done externally to the CFD code) may be less than 4% of the total run time for second-order schemes and 8% for third-order ones, therefore empowering higher-order schemes and more accurate solutions.

► High‐order implicit time‐stepping with high‐order central essentially‐non‐oscillatory methods for unsteady three‐dimensional computational fluid dynamics simulations

15 Oct, 2021

Parallel high-order implicit time-stepping methods coupled with a fourth-order central essentially-non-oscillatory (CENO) finite volume scheme are developed to handle stiff three-dimensional inviscid, viscous, and magnetohydrodynamic flows in a Newton–Krylov–Schwarz framework. A fourth-order accurate explicitly singly diagonally implicit Runge–Kutta (ES-DIRK4) method is shown to be robust and efficient in providing speedups of up to 150 compared with explicit RK methods for stiff large-scale problems. Backward differentiation formulas (BDF) and Rosenbrock-type methods, however, lack stability, robustness, or accuracy on fine grids.

Abstract

This article develops high-order implicit time-stepping methods combined with the fourth-order central essentially-non-oscillatory (CENO) scheme for stiff three-dimensional computational fluid dynamics problems having disparate characteristic time scales. Both aerodynamic and magnetohydrodynamic problems are considered on three-dimensional multiblock body-fitted grids with hexahedral cells. Several implicit time integration methods of third- and fourth-order accuracy are considered, including the multistep backward differentiation formulas (BDF4), multistage explicitly singly diagonally implicit Runge-Kutta (ESDIRK4), and Rosenbrock-type methods (ROS34POW2). The resulting nonlinear algebraic system of equations is solved via a preconditioned Jacobian-free inexact Newton–Krylov method with additive Schwarz preconditioning using block-based incomplete LU decomposition. The performance of the high-order implicit time-stepping methods on smooth and stiff problems is compared with a standard fourth-order explicit Runge-Kutta (RK4) method. It is shown that the Rosenbrock methods, despite their advantage of only requiring the solution of linear systems, have significant drawbacks in terms of robustness issues for highly nonlinear compressible flows. The implicit BDF4 and ESDIRK4 methods are found to be much more efficient than the explicit fourth-order RK4 method for a stiff resistive magnetohydrodynamic (MHD) problem discretized with the fourth-order CENO method. When applied to the problem of vortex shedding governed by the Navier–Stokes equations, an A-stable ESDIRK4 scheme proved to be the more robust and accurate implicit time-marching scheme and was able to offer significant speedup compared with the RK4 method. Initial results are also shown for high-order implicit time integration applied to two problems with discontinuities. The current study represents the first to achieve high-order implicit time integration for MHD, enabling large time steps and substantial speedups for stiff MHD problems with high-order accuracy, and it also represents the first to establish high-order implicit time integration for high-order CENO in space.

► An explicit finite volume scheme on staggered grids for the Euler equations: Unstructured meshes, stability analysis, and energy conservation

14 Oct, 2021

We set up a finite volume scheme for the Euler equations on staggered unstructured grids with numerical densities, energies, and velocities stored on different locations. This time-explicit method is strongly inspired, on the one hand, from the kinetic framework for the definition of the numerical fluxes, and, on the other hand, from the discrete duality finite volume (DDFV) framework. We exhibit stability conditions that guaranty the positivity of the discrete densities and internal energies; and, remarkably, while the scheme works on the internal energy equation, we can define a discrete total energy which satisfies a local conservation equation.

Abstract

We set up a numerical strategy for the simulation of the Euler equations, in the framework of finite volume staggered discretizations where numerical densities, energies, and velocities are stored on different locations. The main difficulty relies on the treatment of the total energy, which mixes quantities stored on different grids. The proposed method is strongly inspired, on the one hand, from the kinetic framework for the definition of the numerical fluxes, and, on the other hand, from the discrete duality finite volume (DDFV) framework, which has been designed for the simulation of elliptic equations on complex meshes. The time discretization is explicit and we exhibit stability conditions that guaranty the positivity of the discrete densities and internal energies. Moreover, while the scheme works on the internal energy equation, we can define a discrete total energy which satisfies a local conservation equation. We provide a set of numerical simulations to illustrate the behavior of the scheme.

► Development and assessment of an intrusive polynomial chaos expansion‐based continuous adjoint method for shape optimization under uncertainties

12 Oct, 2021

The intrusive Polynomial Chaos Expansion (iPCE) to the Navier–Stokes equations for laminar and incompressible flows is mathematically developed, along with its continuous adjoint counterpart, to support aerodynamic shape optimization under uncertainties. Two variants of the adjoint equations are developed, which prove to be analytically and numerically equivalent. The primal and adjoint iPCE solvers, developed in OpenFOAM, are validated against Monte Carlo and finite differences, respectively, and are, then, used for the shape optimization of two airfoils with uncertain farfield conditions.

Abstract

This article contributes to the development of methods for shape optimization under uncertainties, associated with the flow conditions, based on intrusive Polynomial Chaos Expansion (iPCE) and continuous adjoint. The iPCE to the Navier–Stokes equations for laminar flows of incompressible fluids is developed to compute statistical moments of the Quantity of Interest which are, then, compared with those obtained through the Monte Carlo method. The optimization is carried out using a continuous adjoint-enabled, gradient-based loop. Two different formulations for the continuous adjoint to the iPCE PDEs are derived, programmed, and verified. Intrusive PCE methods for the computation of the statistical moments require mathematical development, derivation of a new system of governing equations and their numerical solution. The development is presented for a chaos order of two and two uncertain variables and can be used as a guide to those willing to extend this development to a different set of uncertain variables or chaos order. The developed method and software, programmed in OpenFOAM, is applied to two optimization problems pertaining to the flow around isolated airfoils with uncertain farfield conditions.

► Multiscale evaluation method of the drag effect on shallow water flow through coastal forests based on 3D numerical simulations

11 Oct, 2021

A method for determining the drag parameter in the 2D shallow water (SW) equation for flows through a coastal forest by conducting 3D numerical simulations (3D NSs) is presented. Following the theory of multiscale modeling, we prepare a local test domain that contains a sufficient number of trees to constitute part of a coastal forest. With various inflow conditions, 3D NSs are conducted for constructing the response surfaces that surrogates the drag parameter in the global SW flow simulations.

Abstract

This study presents a method for determining the drag parameter in the 2D shallow water (SW) equation for flows through a coastal forest by conducting a series of 3D numerical simulations (3D NSs). Following the theory of multiscale modeling, an evaluation method procedure is proposed. We first prepare a local test domain that contains a sufficient number of trees to constitute part of a coastal forest. Then, 3D NSs are conducted in this test domain with various inflow conditions. Based on the corresponding results, the momentum losses over the test domain are converted into the drag parameter of the global SW equation. A response surface of the drag parameter is constructed as a function of the flow conditions. The stabilized finite element method is employed for both the local and the global NSs, and the phase-field method is utilized to represent 3D free surfaces. Comparisons between the 2D SW calculation results and the 3D NS results are also performed to verify the validity of the proposed method.

► Lattice Boltzmann modeling of the capillary rise of non‐Newtonian power‐law fluids

7 Oct, 2021

We proposed a lattice Boltzmann (LB) model to study the capillary rise phenomena of two-phase immiscible non-Newtonian power-law fluids. The present LB model is more stable for the problems with large density difference. The capillary rising process under the influence of gravity, viscosity, contact angle, power-law index and displaced fluids are analyzed in detail.

Abstract

In this work, a lattice Boltzmann (LB) model is proposed to study the capillary rise phenomena of two-phase immiscible non-Newtonian power-law fluids. The main advantage of the LB model is that it not only allows us to solve the problems of two-phase flow with a large density difference, but also can overcome the numerical instability caused by the change of relaxation time in the collision operator. In this model, two LB evolution equations are used to solve the conservative Allen–Cahn equation for capturing phase interface and incompressible Navier–Stokes equations for non-Newtonian power-law fluid dynamics. Some benchmark examples, including a droplet spreading on a smooth wall and two-phase power-law fluid flows between parallel plates, are used to test present LB model, and the results show that the present model is efficient and accurate in the study of two-phase power-law fluid flows. Furthermore, we pay attention to the phenomena of capillary rise of non-Newtonian fluids, and carry out a parametric study on the effects of the gravity, viscosity, contact angle, power-law index, and displaced fluids. The results show that the factors mentioned above have some significant influences on the rising process of the absorbed fluids.

► An iterative method to compute conformal mappings and their inverses in the context of water waves over topographies

6 Oct, 2021

In this article we present a numerical approach to compute conformal mappings that requires lower computational time than the Schwarz–Christoffel mapping. The method can be easily extended to solve time-dependent water wave problems with large topographic obstacles. Besides, we present an alternative approach to compute the inverse conformal mapping.

Abstract

An iterative numerical method to compute the conformal mapping in the context of propagating water waves over uneven topographies is investigated. The map flattens the fluid domain onto a canonical strip in which computations are performed. The accuracy of the method is tested by using the MATLAB Schwarz–Christoffel toolbox mapping as a benchmark. Besides, we give a numerical alternative to compute the inverse of the conformal map.

► Economically high‐order unstructured‐grid methods: Clarification and efficient FSR schemes

6 Oct, 2021

We clarify reconstruction-based discretization schemes for unstructured grids and discuss their economically high-order versions that achieve high-order accuracy under certain conditions at little extra cost. The clarification leads to the most economical approaches: the flux-and-solution- reconstruction (FSR) approach with an extended κ-scheme and cheap flux reconstruction formulas, achieving up to 5th-order accuracy (6th-order with zero dissipation) when a grid is regular.

Abstract

In this article, we clarify reconstruction-based discretization schemes for unstructured grids and discuss their economically high-order versions, which can achieve high-order accuracy under certain conditions at little extra cost. The clarification leads to one of the most economical approaches: the flux-and-solution-reconstruction (FSR) approach, where highly economical schemes can be constructed based on an extended κ-scheme combined with economical flux reconstruction formulas, achieving up to fifth-order accuracy (sixth-order with zero dissipation) when a grid is regular. Various economical FSR schemes are presented and their formal orders of accuracy are verified by numerical experiments.

► An efficient and simplified Gay‐Lussac approach in secondary variables form for the non‐Boussinesq simulation of free convection problems

6 Oct, 2021

An efficient form of the Gay-Lussac approach in the context of the secondary variables, that is, vorticity stream-function is presented for buoyancy driven flows in which the density variations are extended to the advection and convection terms of the momentum and energy equations, respectively. In the proposed formulation, the density ratio is removed from the continuity equation under the traditional Gay-Lussac approach. It is shown that, the proposed simplified Gay-Lussac approach gives identical results to the traditional one with a reduced computational cost. The figures show the difference for different parameters between the traditional and simplified Gay-Lussac approaches in a steady state at Ra = 10⁵ that ranges within 1% of the considered parameters values.

Abstract

The Gay-Lussac (GL) approach is an incompressible-based strategy for non-Boussinesq treatment of the governing equations for free convection problems that is established based on extending the density variations beyond the gravity term. Such a strategy leads to emerging the GL parameter as a non-Boussinesq prefactor of different terms in the governing equations. In this article, the GL approach is expressed/discussed in terms of the secondary variables, that is, vorticity and stream-function, for the first time and a simplified version of this approach is proposed by removing density variations from the continuity equation. The difference of results under the simplified and traditional GL approach ranges within a maximum of 1% for different parameters. The lower computational cost of numerical solution of governing equations in the secondary variables formula and the corresponding convergence rate is scrutinized for the simplified GL approach showing around 25% lower computational cost. The performance of this approach is evaluated at high relative temperature differences against the low Mach number scheme and the Boussinesq approximations. In this respect, natural convection in an annulus cavity is numerically simulated using a CVFEM solver under the aforementioned approximations up to Rayleigh number Ra=105 at Prandtl number Pr=1 and high relative temperature differences (ϵ=0.15 and 0.3). The largest deviations found for either the simplified GL or Boussinesq methods from the low Mach number scheme solution are less than 20% for velocity magnitude, 14% for stream function, 6% for vorticity, and 5% for temperature. Results under the three approximations are also analyzed in terms of the skin friction and local and average Nusselt number, indicating that the Gay-Lussac approach requires some revisions to act more accurately than the classical Boussinesq approximation at high relative temperature differences in natural convection problems, especially within the convection dominated regime.