Lid-driven Cavity Flow Research Articles

A simple one-step index algorithm for implementation of lattice Boltzmann method on GPU

We proposed a simple one-step index (OSI) algorithm for solving the lattice Boltzmann equation, particularly the streaming of particle distribution functions (PDFs) on a single grid system. The OSI algorithm is derived from the conventional A-B pattern. The memory addresses of the PDFs are fixed in this algorithm and consistent with collision principles. The streaming process is implicitly computed by reassigning their indexes corresponding to the time steps, spatial coordinates, and directions of the PDFs. The algorithm is simple to program because it reads and writes the PDFs only once per time step and does not require the synchronization of odd and even time steps. In this implementation, the data layout of the PDFs is the structure of arrays (SoA), suitable for the memory access pattern of graphics processing units (GPUs). The accuracy and single-precision performance of the proposed algorithm for the three-dimensional lid-driven cavity flow simulation with the D3Q19 model were validated and tested on an NVIDIA A100 having a 40 GB PCIe using CUDA and OpenACC. Performances of 8.4 and 8.1 giga lattice updates per second were obtained for CUDA and OpenACC, respectively. OpenACC can outperform CUDA by up to 95% with significantly less programming work. The bandwidth usage rates on a single GPU were 96% and 94% for CUDA and OpenACC, respectively, close to the theoretical values. Lattice Boltzmann method parallelism is implemented using CUDA and MPI for multi-GPU usage. Finally, computation and communication overlaps were implemented to optimize the parallel efficiency, where the weak scaling parallel efficiency exceeded 0.98 on up to 512 GPUs.

Numerical Study of Lid-Driven Square Cavity Flow with Embedded Circular Obstacles Using Spectral/hp Element Methods

We show a successful numerical study of lid-driven square cavity flow with embedded circular obstacles based on the spectral/hp element methods. Various diameters of embedded two-dimensional circular obstacles inside the cavity and Reynolds numbers Re (from 100 to 5000) are considered. In order to verify the effectiveness and accuracy of the current methods, numerical results are investigated by comparing with those available in the literature obtained by the moving immersed boundary method (MIBM) and the lattice Boltzmann method (LBM). The present spectral/hp element methods have been not only successfully applied to study and visualize the primary and induced vortices but also capture new vortices on the lower right, upper left and upper right positions of the circular obstacle when Reynolds number Re = 100 and Re = 5000, which is not observed in the lattice Boltzmann method. The current data and figures are in good agreement with the published results. The results of the present study show that the spectral/hp element methods are effective and accurate in simulation of lid-driven cavity flow with embedded circular obstacles, and the present methods have the following advantages: less preprocesses required and high-resolution characteristics.

A less time-consuming upwind compact difference method with adjusted dissipation property for solving the unsteady incompressible Navier-Stokes equations

In this paper, a new strategy to establish less time-consuming upwind compact difference method with adjusted dissipation is introduced for solving the incompressible Navier-Stokes (N-S) equations in the streamfunction-velocity form efficiently. By weighted combination of the numerical solutions calculated using the upwind term and the downwind term of the general third-order upwind compact scheme (UCD3), a new fourth-order compact formulation and a third-order upwind compact formulation with adjusted dissipation nature are proposed for computing the first derivatives. Further, they are used to approximate the biharmonic term and the convective terms in the streamfunction-velocity formulation of the N-S equations, respectively. Meanwhile, the first derivatives of the streamfunction (velocities) in the coefficients in the convective terms are solved by the newly proposed fourth-order compact formulation. Temporal discretization for the streamfunction-velocity formulation is addressed with the help of the second-order Crank-Nicolson scheme. Moreover, the newly proposed scheme for the linear models is proved to be unconditionally stable by virtue of the discrete Fourier analysis. Finally, five numerical problems, viz. the analytic solution, Taylor-Green vortex problem, doubly periodic double shear layer flow problem, lid-driven square cavity flow problem and two-sided square cavity flow problem are solved numerically to verify the efficiency and accuracy of the present method. Results solved by the present method match well with the analytic solutions and the existing results proving the accuracy of it. What is more, it is less time-consuming and has lower dissipation than the existing method [20] .

Third-order less oscillatory and less diffusive compact stencil-based upwind schemes, and their applications to incompressible flows and free surface flows

We propose novel third-order less oscillatory and less diffusive compact stencil-based upwind schemes for the approximation of the continuity equation. The proposed schemes are based on the constrained interpolation profile-conservative semi-Lagrangian schemes. An important feature of the proposed schemes is that the interpolation functions are constructed using only variables within one upwind cell (a cell average and two boundary values). Furthermore, the proposed schemes have third-order accuracy and are also less oscillatory, less diffusive, and fully conservative. The proposed schemes are validated through various benchmark problems and comparisons with experiments of two droplets collision/separation and droplet splashing. The numerical results have shown that the proposed schemes have third-order accuracy for smooth solution, and capture discontinuities and smooth solutions simultaneously without numerical oscillations. The proposed schemes can capture the secondary vorticity of lid-driven cavity flow of Re = 7500 with a Cartesian grid of 64 × 64. The numerical results of two droplets collision/separation of We = 40 show that the proposed schemes can reproduce droplets collision/separation with quite coarse grids. These numerical results of droplet splashing have demonstrated that proposed schemes can reduce numerical diffusions well against existing schemes and robust.

Error analysis of a residual-based stabilization-motivated POD-ROM for incompressible flows

This article presents error bounds for a velocity–pressure segregated POD reduced order model discretization of the Navier–Stokes equations. The stability is proven in L∞(L2) and energy norms for velocity, with bounds that do not depend on the viscosity, while for pressure it is proven in a semi-norm of the same asymptotic order as the L2 norm with respect to the mesh size. The proposed estimates are calculated for the two flow problems, the flow past a cylinder and the lid-driven cavity flow. Their quality is then assessed in terms of the predicted logarithmic slope with respect to the velocity POD contribution ratio. We show that the proposed error estimates allow a good approximation of the real errors slopes and thus a good prediction of their rate of convergence.

A novel high-order solver for simulation of incompressible flows using the flux reconstruction method and lattice Boltzmann flux solver

In this paper, a novel high-order solver using the flux reconstruction (FR) method and the lattice Boltzmann flux solver (LBFS) is proposed for accurately and efficiently simulating nearly incompressible flows. The existing LBFSs are all based on the finite volume (FV) scheme. The present work extends the application of the LBFS for the first time by combining the high-order FR scheme. Compared with the traditional high-order incompressible Navier-stokes (N-S) solvers, no particular techniques are needed for the current FR-LBFS to overcome the difficulty of the pressure-velocity coupling since the present method is a weakly compressible model. Moreover, unlike the traditional N-S solvers where the inviscid and viscous terms are treated separately, the inviscid and viscous fluxes in the current scheme are coupled and computed uniformly in FR-LBFS. In the present method, the common inviscid and viscous fluxes at the cell interface of the FR scheme are evaluated simultaneously by the local reconstruction of lattice Boltzmann equation (LBE) solution from macroscopic flow variables at solution points. Thus the discretization of the second-order partial derivative term is avoided. Compared with the high-order off-lattice Boltzmann methods (OLBM), the FR-LBFS is more stable, efficient and low-storage. No special techniques are needed to remove the stiffness of the governing equations which existing in the discrete velocity Boltzmann equation (DVBE). Compared with the high-order FV-LBFS, the FR-LBFS is compact for parallel computing by avoiding wide stencils on meshes. In addition, the present scheme has lower dissipation and can be more flexible to increase accuracy order. Numerical validations of the proposed method are implemented by simulating (a) Taylor-Green vortex problem, (b) steady plane Poiseuille flow, (c) lid-driven cavity flow, (d) laminar boundary layer and (e) flow past a square cylinder.

Local mesh refinement sensor for the lattice Boltzmann method

A novel mesh refinement sensor is proposed for lattice Boltzmann methods (LBMs) applicable to either static or dynamic mesh refinement algorithms. The sensor exploits the kinetic nature of LBMs by evaluating the departure of distribution functions from their local equilibrium state. This sensor is first compared, in a qualitative manner, to three state-of-the-art sensors: (1) the vorticity norm, (2) the Q-criterion, and (3) spatial derivatives of the vorticity. This comparison shows that our kinetic sensor is the most adequate candidate to propose tailored mesh structures across a wide range of physical phenomena: incompressible, compressible subsonic/supersonic single phase, and weakly compressible multiphase flows. As a more quantitative validation, the sensor is then used to produce the computational mesh for two existing open-source LB solvers based on inhomogeneous, block-structured meshes with static and dynamic refinement algorithms, implemented in the Palabos and AMROC-LBM software, respectively. The sensor is first used to generate a static mesh to simulate the turbulent 3D lid-driven cavity flow using Palabos. AMROC-LBM is then adopted to confirm the ability of our sensor to dynamically adapt the mesh to reach the steady state of the 2D lid-driven cavity flow. Both configurations show that our sensor successfully produces meshes of high quality and allows to save computational time.

High-order flux reconstruction thermal lattice Boltzmann flux solver for simulation of incompressible thermal flows.

In this paper, a high-order solver combining the flux reconstruction (FR) method and the thermal lattice Boltzmann flux solver (FRTLBFS) is developed for accurately and efficiently simulating incompressible thermal flows. The conservative differential equations recovered from Chapman-Enskog analysis of the thermal lattice Boltzmann equation are solved by the high-order FR method. The thermal lattice Boltzmann method is only applied to reconstruct the local solution used for evaluating fluxes at the solution and flux points. Unlike the traditional Navier-Stokes-Boussinesq (NSB) solvers where the inviscid and viscous terms are treated separately, the inviscid and viscous fluxes in the current FRTLBFS are coupled and computed uniformly. In comparison with the recently developed high-order flux reconstruction thermal lattice Boltzmann method, the FRTLBFS holds advantages such as high-order accuracy, good stability, and compactness but is more efficient and low storage, since only macroscopic flow variables including density, velocity, and temperature are stored and evolved. In addition, the physical boundary conditions in FRTLBFS can be directly implemented by using the same method as in conventional NSB solvers. Numerical validations of the proposed method are implemented by simulating (a) the porous plate problem, (b) natural convection in a square cavity, (c) unsteady natural convection in a tall cavity, and (d) thermal lid-driven cavity flow. Numerical results demonstrate that the present solver is an attractive tool to simulate incompressible thermal flows due to its high-order accuracy, stability, and low memory cost.

A low-storage adjoint lattice Boltzmann method for the control of incompressible flows

In this paper, we present a low-storage adjoint lattice Boltzmann method (LSALBM) for the control of unsteady incompressible flows. The core of the method is to adopt a velocity-independent approximate equilibrium for the adjoint variable. Thus, the space–time history of the flow field is not needed to be stored for the adjoint variable and the well-known drawback of the existing adjoint methods is overcome. The new method is validated through both initial-value and boundary control problems. Numerical examples demonstrate that the LSALBM agrees well with the standard adjoint lattice Boltzmann method when the terminal time of the problem is not too long. For long-time unsteady flow problems, the standard method loses its effectiveness, while the LSALBM still produces ideal results. Moreover, the efficiency of our method for steady-state problems is shown by simulating a lid-driven grooved cavity flow.

A fully-coupled framework for solving Cahn-Hilliard Navier-Stokes equations: Second-order, energy-stable numerical methods on adaptive octree based meshes

We present a fully-coupled, implicit-in-time framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes system that models two-phase flows. In this work, we extend the block iterative method presented in Khanwale et al. [Simulating two-phase flows with thermodynamically consistent energy stable Cahn-Hilliard Navier-Stokes equations on parallel adaptive octree based meshes, J. Comput. Phys. (2020)], to a fully-coupled, provably second-order accurate scheme in time, while maintaining energy-stability. The new method requires fewer matrix assemblies in each Newton iteration resulting in faster solution time. The method is based on a fully-implicit Crank-Nicolson scheme in time and a pressure stabilization for an equal order Galerkin formulation. That is, we use a conforming continuous Galerkin (cG) finite element method in space equipped with a residual-based variational multiscale (RBVMS) procedure to stabilize the pressure. We deploy this approach on a massively parallel numerical implementation using parallel octree-based adaptive meshes. We present comprehensive numerical experiments showing detailed comparisons with results from the literature for canonical cases, including the single bubble rise, Rayleigh-Taylor instability, and lid-driven cavity flow problems. We analyze in detail the scaling of our numerical implementation.

Velocity discretization for lattice Boltzmann method for noncontinuum bounded gas flows at the micro- and nanoscale

The lattice Boltzmann (LB) method intrinsically links to the Boltzmann equation with the Bhatnagar–Gross–Krook collision operator; however, it has been questioned to be able to simulate noncontinuum bounded gas flows at the micro- and nanoscale, where gas moves at a low speed but has a large Knudsen number. In this article, this point has been verified by simulating Couette flows at large Knudsen numbers (e.g., Kn=10 and Kn=100) through use of the linearized LB models based on the popular half-range Gauss–Hermite quadrature. The underlying cause for the poor accuracy of these conventional models is analyzed in the light of the numerical evaluation of the involved Abramowitz functions. A different thought on velocity discretization is then proposed using the Gauss–Legendre (GL) quadrature. Strikingly, the resulting GL-based LB models have achieved high accuracy in simulating Couette flows, Poiseuille flows, and lid-driven cavity flows in the strong transition and even free molecular flow regimes. The numerical study in this article reveals an essentially distinct but workable way in constructing the LB models for simulating micro- and nanoscale low-speed gas flows with strong noncontinuum effects.

Stabilized reduced-order models for unsteady incompressible flows in three-dimensional parametrized domains

In this work we derive a parametric reduced-order model (ROM) for the unsteady three-dimensional incompressible Navier–Stokes equations without additional pre-processing on the reduced-order subspaces. Concerning the high-fidelity, full-order model, we start from a streamline-upwind Petrov–Galerkin stabilized finite element discretization of the equations using P1−P1 elements for velocity and pressure, respectively. We rely on Galerkin projection of the discretized equations onto reduced basis subspaces for the velocity and the pressure, respectively, obtained through Proper Orthogonal Decomposition on a dataset of snapshots of the full-order model. Both nonlinear and nonaffinely parametrized algebraic operators of the reduced-order system of nonlinear equations, including the projection of the stabilization terms, are efficiently assembled exploiting the Discrete Empirical Interpolation Method (DEIM), and its matrix version (MDEIM), thus obtaining an efficient offline–online computational splitting. We apply the proposed method to (i) a two-dimensional lid-driven cavity flow problem, considering the Reynolds number as parameter, and (ii) a three-dimensional pulsatile flow in stenotic vessels characterized by geometric and physiological parameter variations. We numerically show that the projection of the stabilization terms on the reduced basis subspace and their reconstruction using (M)DEIM allows to obtain a stable ROM with coupled velocity and pressure solutions, without any need for enriching the reduced velocity space, or further stabilizing the ROM. Additionally, we demonstrate that our implementation allows to compute the ROM solution about 20 times faster than the full order model.

Periodic Unsteady Natural Convection on CNT Nano-powder Liquid in a Triangular Shaped Mechanical Chamber

Nanoparticle is highly used in enhancing thermal performance, especially in lid-driven cavity flow in powder-related applications. Many researchers considered different conditions, such as MHD, radiation, and mixed convection incorporating nano liquid in such flow. However, no study was carried out with CNT-water nano liquid in a triangular cavity, heated sinusoidally under natural convection to determine the Rayleigh number (Ra) and nano-powder liquid concentration effect on heat transfer. As a result, this study attempts to find the Rayleigh number and particle concentration effects in such transient cavity flow. The governing equations were employed with the Galerkin residual method. While calculating the thermal conductivity and dynamic viscosity of the nano liquid, Brownian motion was taken into consideration. The four solid volume fractions (0.01, 0.05, 0.1, 0.15) and the Ra (104 ≤ Ra ≤ 106) were chosen to evaluate the impact within dimensionless time (0.1 ≤ τ ≤ 1). The findings have been shown by plotting the streamlines, isotherms, heat transfer variation, and pressure gradient. It is found that when the Rayleigh number goes up, the velocity, vorticity, and pressure gradient magnitude become high. But, the fluid flow vortices decrease with dimensionless time. Due to the sinusoidal heat flux conditions, the variation of the result was found to be sinusoidal as well. When the nanoparticle concentration rises, the heat transfer rate and average fluid temperature also increase. Therefore, the highest heat transfer rate is found using the nano liquid with a15% concentration.

Lattice Boltzmann model for simulation of flow in intracranial aneurysms considering non-Newtonian effects

We propose a robust modified central Hermite polynomial-based multiple relaxation time lattice Boltzmann model with independent control over relaxation of acoustic modes for non-Newtonian fluids, more specifically in the context of blood flow in intracranial aneurysms. The use of the robust collision operator along with the implicit computation of the non-linear stress allows for a very wide operation domain in terms of time step and grid-size. The solver is first validated via well-documented configurations such as the 2D Poiseuille–Hagen and lid-driven cavity flows with a power-law fluid. The results clearly show second-order convergence of the scheme. The model is then used to simulate pulsating flow in an ideal aneurysm geometry with four different viscosity laws, namely, Newtonian, power-law, Carreau–Yasuda, and Cross. The results show that the assumption of high shear rates does not necessarily hold within the aneurysm sac. Finally, the solver is used to simulate pulsating blood flow in a patient-specific configuration.

Three-Dimensional Simulations of Anisotropic Slip Microflows Using the Discrete Unified Gas Kinetic Scheme.

The specific objective of the present work study is to propose an anisotropic slip boundary condition for three-dimensional (3D) simulations with adjustable streamwise and spanwise slip length by the discrete unified gas kinetic scheme (DUGKS). The present boundary condition is proposed based on the assumption of nonlinear velocity profiles near the wall instead of linear velocity profiles in a unidirectional steady flow. Moreover, a 3D corner boundary condition is introduced to the DUGKS to reduce the singularities. Numerical tests validate the effectiveness of the present method, which is more accurate than the bounce-back and specular reflection slip boundary condition in the lattice Boltzmann method. It is of significance to study the lid-driven cavity flow due to its applications and its capability in exhibiting important phenomena. Then, the present work explores, for the first time, the effects of anisotropic slip on the two-sided orthogonal oscillating micro-lid-driven cavity flow by adopting the present method. This work will generate fresh insight into the effects of anisotropic slip on the 3D flow in a two-sided orthogonal oscillating micro-lid-driven cavity. Some findings are obtained: The oscillating velocity of the wall has a weaker influence on the normal velocity component than on the tangential velocity component. In most cases, large slip length has a more significant influence on velocity profiles than small slip length. Compared with pure slip in both top and bottom walls, anisotropic slip on the top wall has a greater influence on flow, increasing the 3D mixing of flow. In short, the influence of slip on the flow field depends not only on slip length but also on the relative direction of the wall motion and the slip velocity. The findings can help in better understanding the anisotropic slip effect on the unsteady microflow and the design of microdevices.

A Convex Relaxation Approach for Learning the Robust Koopman Operator

Although data-driven models, especially deep learning, have achieved astonishing results on many prediction tasks for nonlinear sequences, challenges still remain in finding an appropriate way to embed prior knowledge of physical dynamics in these models. In this work, we introduce a convex relaxation approach for learning robust Koopman operators of nonlinear dynamical systems, which are intended to construct approximate space spanned by eigenfunctions of the Koopman operator. Different from the classical dynamic mode decomposition, we use the layer weights of neural networks as eigenfunctions of the Koopman operator, providing intrinsic coordinates that globally linearize the dynamics. We find that the approximation of space can be regarded as an orthogonal Procrustes problem on the Stiefel manifold, which is highly sensitive to noise. The key contribution of this paper is to demonstrate that strict orthogonal constraint can be replaced by its convex relaxation, and the performance of the model can be improved without increasing the complexity when dealing with both clean and noisy data. After that, the overall model can be optimized via backpropagation in an end-to-end manner. The comparisons of the proposed method against several state-of-the-art competitors are shown on nonlinear oscillators and the lid-driven cavity flow.

A high-order finite volume method solving viscous incompressible flows using general pressure equation

A high-order accurate finite volume method on Cartesian meshes has been developed to solve viscous incompressible flows using the general pressure equation (GPE). The GPE is a pressure evolution equation that replaces the divergence-free constraint equation when computing incompressible flows. The high-order spatial accuracy of the current method is deduced from the high-order flow reconstruction model using the conserved volume integrals, and the high-order accurate computations of the flux integrals. The Roe’s flux difference scheme was adapted to compute the inviscid fluxes. The grid convergence tests using the solution of Taylor-Green decaying vortices showed that the current method is spatially fifth order accurate for both velocity and pressure. The problem of doubly periodic shear layers and the lid-driven cavity flow were computed to demonstrate the capability of the current method in transient and steady state computations.

Unified X-space parallelization algorithm for conserved discrete unified gas kinetic scheme

In this paper, the open source multiscale flow solver dugksFoam is optimized with a newly proposed parallelization strategy and conserved algorithm. A novel X-space parallel framework is introduced. In contrast to traditional discrete physical/velocity space decomposition parallelization methods, it can decompose each space in a hybrid manner and efficiently parallelize the computation. For better conservation of the macroscopic physical variables, flow properties in each cell are updated by moments of microscopic variable fluxes, which preserves particle multiscale information under sparse velocity space mesh. Several test cases, which include 2D lid-driven cavity flow in the transition flow regime, hypersonic rarefied flow past a 3D sphere, are carried out to verify the accuracy of the conserved discrete unified gas kinetic scheme to simulate the flows in all flow regimes. A large-scale parallel computational test case of 3D sphere with 101,600 physical mesh cells and 62,744 velocity mesh cells is conducted to measure the parallel efficiency of the conserved dugksFoam code. A significant superlinear speedup of 1631.87 has been achieved on 1120 cores.

A priorisparsification of Galerkin models

A methodology to generate sparse Galerkin models of chaotic/unsteady fluid flows containing a minimal number of active triadic interactions is proposed. The key idea is to find an appropriate set of basis functions for the projection representing elementary flow structures that interact minimally one with the other, resulting in a triadic interaction coefficient tensor with sparse structure. Interpretable and computationally efficient Galerkin models are obtained, since a reduced number of triadic interactions are computed to evaluate the right-hand side of the model. To find the basis functions, a subspace rotation technique is used, whereby a set of proper orthogonal decomposition (POD) modes is rotated into a POD subspace of larger dimension using coordinates associated with low-energy dissipative scales to alter energy paths and the structure of the triadic interaction coefficient tensor. This rotation is obtained as the solution of a non-convex optimisation problem that maximises the energy captured by the new basis, promotes sparsity and ensures long-term temporal stability of the sparse Galerkin system. We demonstrate the approach on two-dimensional lid-driven cavity flow at$Re = 2 \times 10^4$where the motion is chaotic and energy interactions are scattered in modal space. We show that the procedure generates Galerkin models with a reduced set of active triadic interactions, distributed in modal space according to established knowledge of scale interactions in two-dimensional flows. This property, however, is observed only if long-term temporal stability is included explicitly in the formulation, indicating that a dynamical constraint is necessary to obtain a physically consistent sparsification.

Application of 2D adaptive mesh refinement method to estimation of the center of vortices for flow over a wall-mounted plate

This paper describes the application of an adaptive mesh refinement (AMR) method to estimate centers of vortices of two-dimensional (2D) incompressible fluid flow over a wall-mounted plate. Following the accuracy verification of the AMR method using the benchmarks of 2D lid-driven cavity flows and backward-facing step flows, this study considers the application of the AMR method to the flow over a wall-mounted plate. The AMR method refines a mesh using numerical solutions of the Navier–Stokes equations computed using an open-source flow solver, Navier2D. The AMR is applied to seven test cases considering flows with different Reynolds numbers ([Formula: see text]) and the estimated centers of vortices after the plate are reported. The results show that AMR can capture the location of the center of vortices within the once refined cells. Furthermore, improved estimation of vortex centers is obtained using twice refined meshes. The AMR aims to get a refined mesh which captures the characteristics of flow with required accuracy at a lower computational cost.