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We introduce a parallel cell‐centered finite volume lattice Boltzmann method on 3D unstructured grids for incompressible flow simulations. The proposed method is validated by three benchmark problems and the results agree well with the published data. The parallel algorithm scales well on a supercomputer with a large number of processors.

Abstract

The standard lattice Boltzmann method, which employs certain regular lattices coupled with discrete velocities as the computational grid, is limited in its flexibility to simulate flows in irregular geometries. To simulate large‐scale complex flows, we present a cell‐centered finite volume lattice Boltzmann method for incompressible flows on three‐dimensional (3D) unstructured grids and its corresponding parallel algorithm. The advective fluxes are calculated by the low‐diffusion Roe scheme, and the gradients of the particle distribution functions are computed with a least squares method. The presented scheme is validated by three benchmark flows: (a) a 3D Poiseuille flow, (b) cubic cavity flows with Reynolds numbers Re = 100 and 400, and (c) flows past a sphere with Re = 50, 100, 150, 200, and 250. Some parallel performance results are presented to show the scalability of the method, which reveal that the proposed parallel algorithm has considerable scalability and that the parallel efficiency is higher than 87% on 3840 processor cores. It can be seen that the presented parallel solver has significant potential for the accurate simulation of flows in complex 3D geometries.

► Phase‐field formulation of a fictitious domain method for particulate flows interacting with complex and evolving geometries

4 May, 2021

We introduce a phase‐field formulation of an existing fictitious domain method and a corresponding collision model in order to simulate rigid particulate flows interacting with arbitrary solid structures. The phase‐field framework enables the consideration of both the particulate flow and the structure on a fix Cartesian grid. It also enables the consideration of phase transition within the boundary structure, for example, due to grain growth, making it change its shape during runtime.

Abstract

A distributed Lagrange multiplier/fictitious domain method in a phase‐field formulation for the simulation of rigid bodies in incompressible fluid flow is presented. The phase‐field method yields an implicit representation of geometries and thus rigid body particulate flows within arbitrary geometries can be simulated based on a fixed Cartesian grid. Therefore, a phase‐field based collision model is introduced in order to address contact of particles with arbitrary solid structures as boundaries. In addition, grain growth within the boundary geometry can be considered leading to changes in its shape during the simulation. The method is validated on benchmark problems and a convergence study is performed. Multiple numerical experiments are carried out in order to show the methods' capability to simulate problems with differently shaped rigid bodies and particulate flows involving complex boundary geometries like foam structures.

► Variationally derived closure models for large eddy simulation of incompressible turbulent flows

4 May, 2021

A variationally approach for deriving residual‐based closure models for incompressible Navier–Stokes equations is presented. Variational consistency of the model lends itself to rigorous linearization that results in quadratic rate of convergence of the method in the nonlinear solution strategy. The method is shown to work for a family of linear and quadratic hexahedral and tetrahedral elements as well as composite discretizations that are comprised of hexahedral and tetrahedral elements in the same computational domain.

Abstract

We present a variationally consistent method for deriving residual‐based closure models for incompressible Navier–Stokes equations. The method is based on the fine‐scale variational structure facilitated by the variational multiscale framework where fine scales are driven by the residuals of the Euler–Lagrange equations of the resolved scales in the balance of momentum and conservation of mass equations. A bubble‐functions based approach is applied directly to the fine‐scale variational equation to derive analytical expressions for the closure model. Variational consistency of the model lends itself to rigorous linearization that results in quadratic rate of convergence of the method in the iterative solution strategy for the nonlinear equations. The method is shown to work for a family of linear and quadratic hexahedral and tetrahedral elements as well as composite discretizations that are comprised of hexahedral and tetrahedral elements in the same computational domain. Numerical tests with the proposed model are presented for various classes of turbulent flow problems to show its generality and range of applicability. The test cases investigated include Taylor–Green vortex stretching, statistically stationary wall‐bounded channel flows, and modeling the effects of the geometry of the leading edge of the plate on the instability of the boundary layer that leads to flow separation and flow reversal over flat plates of finite thickness.

► Lattice Boltzmann method for the simulation of White–Metzner fluid flows

3 May, 2021

A coupled lattice Boltzmann method with two types of distribution functions is used to simulate non‐Newtonian White‐Metzner fluid flows inside a lid driven square cavity. A very detailed investigation of the effect of each parameter on the behavior of the WM fluid flows is presented. The method proves to be good in simulating WM fluids and encounter the instabilities that arise in such flows.

Abstract

The simulation of non‐Newtonian fluids is a challenging task from theoretical and numerical points of view. Different numerical methods has been used to study this class of fluids. In this article, a novel numerical scheme based on lattice Boltzmann method is presented for the simulation of White–Metzner (WM) fluid flows, where two types of distribution functions are defined for the evolution of momentum and stress, respectively. We study the accuracy, and the influence of different parameters on the flow on different benchmarks: we validate our model first for a two‐dimensional planar channel, and then we investigate in details the behavior of WM fluid flows in a square lid driven cavity. In the numerical results, we give a very detailed investigation to elaborate the effect over a large range of each parameter on the flow field.

► Front capturing by level set method for the reactive Euler equations

28 Apr, 2021

Abstract

In this paper, we present a front capturing method for compressible reactive flows, where a shock tracking technique is applied with level set method. In stiff reaction problems, the difficulty arises when the time scale of the source term is significantly shorter than the time scale of the homogeneous conservation law. The spurious numerical phenomenon may occur due to the smeared out shock profiles. In order to overcome the difficulty, the detonation front is captured sharply by level set technique, and a modified finite volume scheme is developed at the computational cells cut by detonation front. The mass, momentum and energy transitions are occurred on shock front for chemical reactive flows. In our method, the interface exchanges are calculated by considering a Riemann problem. Unlike the standard level set/ghost fluid method, our method can maintain the conservation on the shock discontinuity. One‐ and two‐dimensional numerical examples including stable and unstable detonation problems are illustrated to verify the good reliability and robustness of our method.

► Reduced order models for the incompressible Navier‐Stokes equations on collocated grids using a ‘discretize‐then‐project’ approach

28 Apr, 2021

Summary

A novel reduced order model (ROM) for incompressible flows is developed by performing a Galerkin projection based on a fully (space and time) discrete full order model (FOM) formulation. This ‘discretize‐then‐project’ approach requires no pressure stabilization technique (even though the pressure term is present in the ROM) nor a boundary control technique (to impose the boundary conditions at the ROM level). These are two main advantages compared to existing approaches. The fully discrete FOM is obtained by a finite volume discretization of the incompressible Navier‐Stokes equations on a collocated grid, with a forward Euler time discretization. Two variants of the time discretization method, the inconsistent and consistent flux method, have been investigated. The latter leads to divergence‐free velocity fields, also on the ROM level, whereas the velocity fields are only approximately divergence‐free in the former method. For both methods, accurate results have been obtained for test cases with different types of boundary conditions: a lid‐driven cavity and an open‐cavity (with an inlet and outlet). The ROM obtained with the consistent flux method, having divergence‐free velocity fields, is slightly more accurate but also slightly more expensive to solve compared to the inconsistent flux method. The speedup ratio of the ROM and FOM computation times is the highest for the open cavity test case with the inconsistent flux method.

► A stable discontinuous Galerkin method based on high‐order dual splitting scheme without additional stabilization term for incompressible flows

27 Apr, 2021

The DG formulation of the velocity divergence term and the pressure gradient term is required in order to improve the stability of the scheme at small time steps. Moreover, an appropriate penalty term for pressure Poisson equation still provides inf‐sup stabilization for equal‐order polynomials. The combination of these methods is a relatively simple and effective way to improve the stability and robustness of the solver for incompressible flows with high‐Reynolds numbers.

Abstract

In this work, we consider equal‐order discontinuous Galerkin (DG) solver for incompressible Navier–Stokes equations based on high‐order dual splitting scheme. In order to stay stable, the time step size of this method has been reported that is strictly limited. The upper bound of time step size is restricted by Courant–Friedrichs–Lewy (CFL) condition (Hesthaven and Warburton, 2007) and lower bound is required to be larger than the critical value which depends on Reynolds number and spatial resolution (Ferrer et al., 2014). For high‐Reynolds‐number flow problems, if the spatial resolution is low, the critical value may be larger than CFL condition, then instability will occur for any time step size. Therefore, sufficiently high spatial resolution is indispensable in order to maintain stability, which increases the computational cost. To overcome these difficulties and develop a robust solver for high‐Reynolds‐number flow problem, it is necessary to further study the instability problem at small time steps. We numerically investigate the effect of the pressure gradient term in projection step and the velocity divergence term in pressure Poisson equation on the stability for small time step size, respectively, and conclude that the DG formulation of the pressure gradient term has a more significant effect on the stability of the scheme than that of the velocity divergence term. Integration by parts of these terms is essential in order to improve the stability of the scheme. Based on this discretization format, an appropriate penalty parameter for pressure Poisson equation is utilized so as to provide the scheme with an inf‐sup stabilization. Moreover, the lid‐driven cavity flow is considered to verify that this numerical algorithm enhances the stability without additional stabilization term at small time step size and high‐Reynolds number for equal‐order polynomial approximations.

► Imposing C0 and C1 continuity constraints during CAD‐based adjoint optimization

27 Apr, 2021

This article presents a method for performing CAD‐based aerodynamic shape optimization by manipulating the standard geometry. The method addresses the continuity issues that emerge during the update of the boundary representation (BRep) of the shape by imposing up to C1 constraints between neighboring patches and defines a parameterization scheme which inherently satisfies them. The proposed method is demonstrated in an academic case and in industrial‐like cases such as the S‐section of a duct and the tail surface of a car.

Abstract

This article presents a method for performing adjoint‐based aerodynamic shape optimization by manipulating the standard CAD geometry of the shape to be optimized. A standard CAD file gives access to the boundary representation (BRep) of the shape and consequently its boundary surfaces which are usually trimmed patches. This is a sensible choice as the open format of such files is a requirement for the computation of the shape derivatives. The method addresses the continuity issues that emerge during the update of the shape by imposing up to C ₁ constraints between different CAD patches. A parameterization scheme based on NURBS surfaces is then defined, in which the aforementioned constraints are inherently satisfied. The proposed method is firstly demonstrated in a simple geometric case and then in the full‐scale optimization of industrial‐like cases such as the S‐section of a cooling duct and the tail surface of a passenger car.

► Computational aeroacoustic modeling using hybrid Reynolds averaged Navier–Stokes/large‐eddy simulations methods with modified acoustic analogies

26 Apr, 2021

We develop and implement a hybrid Reynolds averaged Navier–Stokes/large‐eddy simulations improved delayed detached eddy simulation model for flow and noise predictions of tandem cylinder experiment. The aeroacoustic simulations were conducted via modified acoustic analogies, namely the Curle's and Ffowcs‐Williams and Hawkings analogies. The results from the implemented computational methodologies were in good agreement with experimental acoustic power spectral density data as measured at NASA's Quiet Flow Facility.

Abstract

This study employs a numerical approach to investigate noise mechanisms in tandem cylinders. The tandem cylinder configuration is an excellent model for aircraft landing gear. The study is therefore primarily motivated by the need to understand the aircraft landing gear as a primary contributor to airframe noise during approach and landing. Fluctuations in the flow properties induced by turbulent flow around the tandem cylinders are computed and analyzed. A hybrid improved delayed detached eddy simulation turbulence model is employed to compute the boundary layer flow as well as to compute the fluctuations in the flow properties. The numerical methodologies for the turbulent flow computations are implemented on the OpenFOAM software platform. The study also considers the impact of neglecting the volume sources and employs two modified versions of Curle's and Ffowcs‐Williams and Hawkings analogies. The results are in good agreement with published numerical and experimental data in the existing literature.

► On the Stabilization of Steady Continuous Adjoint Solvers in the Presence of Unsteadiness, in Shape Optimization

25 Apr, 2021

Abstract

Adjoint‐based shape optimization using unsteady solvers is costly and/or memory demanding. When mild unsteadiness is present or the flow in/around the optimized shape is not expected to be time‐varying, steady primal and adjoint solvers can be used instead. However, in such a case, convergence difficulties caused by flow unsteadiness must properly be treated. In this article, the steady primal and the corresponding (continuous) adjoint solvers are both stabilized by implementing the Recursive Projection Method (RPM). This is carried out in the adjointOptimisation library of OpenFOAM, developed and made publicly available by the group of authors. Upon completion of the optimization using steady solvers, unsteady reevaluations of the optimized solutions confirm a reduction in the time‐averaged objective function. In complex cases, in which the RPM may not necessarily ensure convergence of the adjoint solver on its own, the controlled damping of the adjoint transposed‐convection (ATC) term is additionally implemented. This is demonstrated in the shape optimization of a motorbike fairing where averaged primal fields over a number of iterations of the steady flow solver are used for the solution of the adjoint equations. Cases in which the RPM is, on its own, sufficient in ensuring convergence of the adjoint solver are additionally studied by using a controlled ATC damping, to assess its impact on the computed sensitivity derivatives. Comparisons show that controlled/mild ATC damping is harmless and greatly contributes to robustness.