2D Lid-driven Cavity Flow Research Articles

Physics-informed neural networks with domain decomposition for the incompressible Navier–Stokes equations

Physics-informed neural network (PINN) has emerged as a promising approach for solving differential equations in recent years. However, their application to large-scale complex problems has faced challenges regarding accuracy and efficiency. To address these limitations, domain decomposition has gained popularity as an effective strategy. This paper studies a domain decomposition PINN method for solving incompressible Navier–Stokes equations. We assess the method's predicted accuracy, convergence, and the impact of different strategies on performance. In the domain decomposition PINN method, individual PINN is employed for each subdomain to compute local solutions, which are seamlessly connected by enforcing additional continuity conditions at the interfaces. To improve the method's performance, we investigate various continuity conditions at the interfaces and analyze their influence on the predictive accuracy and interface continuity. Furthermore, we introduce two approaches: the dynamic weight method and a novel neural network architecture incorporating attention mechanisms, both aimed at mitigating gradient pathologies commonly encountered in PINN methods. To demonstrate the effectiveness of the proposed method, we apply it to a range of forward and inverse problems involving diverse incompressible Navier–Stokes flow scenarios. This includes solving benchmark problems such as the two-dimensional (2D) Kovasznay flow, the three-dimensional (3D) Beltrami flow, the 2D lid-driven cavity flow, and the 2D cylinder wake. Additionally, we conduct 3D blood flow simulations for synthetic flow geometries and real blood vessels. The experimental results demonstrate the capability and versatility of the domain decomposition PINN method in accurately solving incompressible Navier–Stokes flow problems.

Three-dimensional laminar flow using physics informed deep neural networks

Physics informed neural networks (PINNs) have demonstrated their effectiveness in solving partial differential equations (PDEs). By incorporating the governing equations and boundary conditions directly into the neural network architecture with the help of automatic differentiation, PINNs can approximate the solution of a system of PDEs with good accuracy. Here, an application of PINNs in solving three-dimensional (3D) Navier–Stokes equations for laminar, steady, and incompressible flow is presented. Notably, our approach involves deploying PINNs using feed-forward deep neural networks (DNNs) without depending on any simulation or experimental data. This investigation focuses on 3D square channel flow and 3D lid-driven cavity flow. For each case, one deep neural network was trained using only the governing equations and boundary conditions. Finally, the PINNs' results were compared with the computational fluid dynamics results. The goal was to assess the ability of PINNs (with DNN architectures) to predict the solution of Navier–Stokes equations in the 3D domain without any simulation or experimental data (unsupervised learning).

Large-Scale Cluster Parallel Strategy for Regularized Lattice Boltzmann Method with Sub-Grid Scale Model in Large Eddy Simulation

As an improved method of the lattice Boltzmann method (LBM), the regularized lattice Boltzmann method (RLBM) has been widely used to simulate fluid flow. For solving high Reynolds number problems, large eddy simulation (LES) and RLBM can be combined. The computation of fluid flow problems often requires a large number of computational grids and large-scale parallel clusters. Therefore, the high scalability parallel algorithm of RLBM with LES on a large-scale cluster has been proposed in this paper. The proposed parallel algorithm can solve complex flow problems with large-scale Cartesian grids and high Reynolds numbers. In order to achieve computational load balancing, the domain decomposition method (DDM) has been used in large-scale mesh generation. Three mesh generation strategies are adopted, namely 1D, 2D and 3D. Then, the buffer on the grid interface is introduced and the corresponding 1D, 2D and 3D parallel data exchange strategies are proposed. For the 3D lid-driven cavity flow and incompressible flow around a sphere under a high Reynolds number, the given parallel algorithm is analyzed in detail. Experimental results show that the proposed parallel algorithm has a high scalability and accuracy on hundreds of thousands of cores.

An improved MRT-LBM and investigation to the transition and periodicity of 2D lid-driven cavity flow with high Reynolds numbers

An improved MRT-LBM and investigation to the transition and periodicity of 2D lid-driven cavity flow with high Reynolds numbers

Deep Neural Network Modeling for CFD Simulations: Benchmarking the Fourier Neural Operator on the Lid-Driven Cavity Case

In this work we present the development, testing and comparison of three different physics-informed deep learning paradigms, namely the ConvLSTM, CNN-LSTM and a novel Fourier Neural Operator (FNO), for solving the partial differential equations of the RANS turbulence model. The 2D lid-driven cavity flow was chosen as our system of interest, and a dataset was generated using OpenFOAM. For this task, the models underwent hyperparameter optimization, prior to testing the effects of embedding physical information on performance. We used the mass conservation of the model solution, embedded as a term in our loss penalty, as our physical information. This approach has been shown to give physical coherence to the model results. Based on the performance, the ConvLSTM and FNO models were assessed in forecasting the flow for various combinations of input and output timestep sizes. The FNO model trained to forecast one timestep from one input timestep performed the best, with an RMSE for the overall x and y velocity components of 0.0060743 m·s−1.

Numerical simulation of the 2D lid-driven cavity flow of chiral liquid crystals

ABSTRACT In this study, the two-dimensional (2D), lid-driven cavity flow of chiral liquid crystals (LCs) was modelled using the Landau de-Gennes (LdG) theory. Parametric studies investigating the effect of Er (Ericksen number) and Θ (chiral strength) on the microstructure of chiral LCs were performed. In this study, we observed that an increase in Θ caused the chiral texture to have more striations and shorter pitch lengths as causes for an increased number of defects. When Θ was held constant, an increase in Er disrupted the chiral structure and even broke it at very high Er. Interestingly, a transition from low Er (10) to moderate Er (1,000) increased the number of defects; however, further increases in Er reduced the number of defects since much of the chiral structure was destroyed by the high viscous flow effects. In particular, even at very high Er, the chiral structure and defects were still present as a vortex was always present with lower velocities, where the viscous flow effect was smaller. We also found that a hexagonal structure with penta-hepta defects formed at high chiral strengths.

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.

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.

Turbulent kinetic dissipation analysis for residual-based large eddy simulation of incompressible turbulent flow by variational multiscale method

The underlying physical mechanism of the residual-based large eddy simulation (LES) based on the variational multiscale (VMS) method is clarified. Resolved large-scale energy transportation equation is mathematically derived for turbulent kinetic energy budget analysis. Firstly, statistical results of benchmark turbulent channel flow at Reτ=180 obtained using a coarse mesh are compared with the results obtained by the classical LES with the Smagorinsky and dynamic subgrid stress (SGS) model. The present LES shows an advantage in predicting the statistical results of the incompressible turbulent flows. Secondly, the contributions of the unresolved small-scale presentation terms (Term I-IV in Eq. (10)) to the turbulent kinetic dissipation are analysed for the VMS method. The results show that the turbulent kinetic dissipation provided by the numerical diffusion in the VMS method is smaller in the inner layer, larger in the outer layer of the channel flow than those by the Smagorinsky and dynamic SGS model. The turbulent kinetic dissipation in the VMS method is mainly given by the numerical diffusion provided by one of the “cross-stress” terms (Term I, same as the stabilization term in the SUPG method) and LSIC term (Term IV). The other one of the “cross-stress” terms (Term II) gives rise to the positive turbulent kinetic energy budget, and does not dissipate the turbulent kinetic energy. The so-called “Reynolds stress” term (Term III) dissipates the turbulent energy but provides a very small numerical diffusion. Finally, on the basis of the turbulent kinetic energy dissipation analysis, a new residual-based stabilized finite element formulation is proposed by modifying the large-scale equation in the VMS method. Numerical experiments of 2D lid-driven cavity flow and 3D incompressible turbulent channel flow are tested to validate the proposed formulation. It is shown that all the stabilization terms in the proposed formulation produce additional numerical diffusions and physically increase the total turbulent kinetic dissipation. Consequently, an apparent improvement in both the first-order and second-order statistical quantities are pursued by the new stabilized finite element formulation.

Numerical simulation of non-Newtonian Carreau fluid in a lid-driven cavity

In this work, the 2D lid-driven cavity flow of non-Newtonian Carreau fluids has been studied by finite difference method on a staggered grid. A finite-difference algorithm on staggered grid based on projection method is adopted to solve the lid-driven cavity flow, which includes a second-order central difference scheme for the non-Newtonian viscous stress term. This study has been conducted for the certain pertinent parameters of Reynolds number (Re=100-1000), power-law index (n=0.6-1.4). The results show that as the Reynolds number increases, the influence of the power-law index on the flow increases. As the power-law index decreases, the flow field becomes more complicated.

ADAPTIVE GRID REFINEMENT USING THE GENERALISED FINITE-DIFFERENCE METHOD

The combination of the Generalised Finite Difference Method (GFDM), and adaptive grid refinement is applied to solve 2D fluid flow problems. The accuracy of this combination is demonstrated by solving the 2D lid-driven cavity flow, and 2D backward-facing step flow problems, and comparing the results against the benchmarks. This new Computational Fluid Dynamics (CFD) formulation is applied to solve a 2D meter flow application to determine the velocity profiles through the centre of the meter for higher Reynolds numbers. To verify the accuracy of this combnation, analytical 2D and 3D Laplace partial differential equations (PDE's) are solved by two methods. The first method uses the Finite Difference Method (FDM) over a uniform grid of nodes, and the second method uses the GFDM over a non-uniform grid of nodes. Computational cost and accuracy comparisons are made for both methods.

Accuracy Verification of a 2D Adaptive Mesh Refinement Method Using Backward-Facing Step Flow of Low Reynolds Numbers

Identifying centers of vortices of fluid flow accurately is one of the accuracy measures for computational methods. After verifying the accuracy of the 2D adaptive mesh refinement (AMR) method in the benchmarks of 2D lid-driven cavity flow, this paper shows the accuracy verification by the benchmarks of 2D backward-facing step flow. The AMR method refines a mesh using the numerical solution of the Navier–Stokes equations computed on the mesh by an open source software Navier2D which implemented a vertex centered finite volume method (FVM) using the median dual mesh to form control volumes about each vertex. The accuracy is shown by the comparison between vortex center locations calculated from the linearly interpolated numerical solutions and those obtained in the benchmark. The AMR method is proposed based on the qualitative theory of differential equations, and it can be applied to refine a mesh as many times as required and used to seek accurate numerical solutions of the mathematical models including the continuity equation for incompressible fluid or steady-state compressible flow with low computational cost.

Numerical verification of a non-residual orthogonal term-by-term stabilized finite element formulation for incompressible convective flow problems

The modeling of turbulent flows is relevant in many engineering applications and, is an active field of research on numerical methods. Convergence and stability of proposed formulations are crucial to predict transitional flows from laminar to turbulent flows. In this work, a recently developed stabilized finite element formulation is used as a powerful tool to describe such kind of problems. The essential point of the formulation is the time dependent nature of the subscales and, contrary to residual based formulations, the introduction of two velocity subscale components. The theoretical rate of convergence of the method is verified numerically using linear and quadratic equal-order finite element discretizations. To this end, a standard convergence test of L2-norm is presented where the computed solutions are compared when manufactured solutions are imposed at Gauss-point level. Moreover, the Hopf bifurcation is studied for two well-known benchmark problems: flow past a cylinder and the three-dimensional lid-driven cavity flow. For the flow past a cylinder case, the Hopf bifurcation is verified using dynamic subscales and is assessed so that they do not disturb the solution. In particular, a dominant convective problem (Re=4000) is solved using both the quasistatic and dynamic versions of the method, evaluating the performance of each one in the quality of the solution and in the CPU time needed to obtain a converged solution. For the 3D lid-driven cavity flow problem, the Hopf bifurcation is determined using two different boundary conditions, analyzing their effect on the dynamics of the problem and on the thickness and shape of the boundary layer. The final test case is the turbulent 3D lid-driven cavity problem (Re=12000), where velocity profiles are compared with experimental, LES and DNS reference solutions. Additionally, pressure and velocity spectra are shown at certain representative points of the domain, as well as phase diagrams, correlation function graphs, the Poincaré map, and the Lyapunov exponent, typical mathematical tools used in dynamical system analysis. From the results, the method is robust and accurate for all the numerical tests both in viscous dominant problems as in dominant convective ones.

GPU acceleration of an iterative scheme for gas-kinetic model equations with memory reduction techniques

This paper presents a Graphics Processing Unit (GPU) acceleration of an iteration-based discrete velocity method (DVM) for gas-kinetic model equations. Unlike the previous GPU parallelization of explicit kinetic schemes, this work is based on a fast converging iterative scheme. The memory reduction techniques previously proposed for DVM are applied for GPU computing, enabling full three-dimensional (3D) solutions of kinetic model equations in the contemporary GPUs usually with a limited memory capacity that otherwise would need terabytes of memory. The GPU algorithm is validated against the direct simulation Monte Carlo (DSMC) simulation of the 3D lid-driven cavity flow and the supersonic rarefied gas flow past a cube with the phase-space grid points up to 0.7 trillion. The computing performance profiling on three models of GPUs shows that the two main kernel functions can utilize 56%∼79% of the GPU computing and memory resources. The performance of the GPU algorithm is compared with a typical parallel CPU implementation of the same algorithm using the Message Passing Interface (MPI). The comparison shows that the GPU program on K40 and K80 achieves 1.2∼2.8 and 1.2∼2.4 speedups for the 3D lid-driven cavity flow, respectively, compared with the MPI parallelized CPU program running on 96 CPU cores.

An implicit simplified sphere function-based gas kinetic scheme for simulation of 3D incompressible isothermal flows

In this work, an implicit simplified sphere function-based gas kinetic scheme (SGKS) is presented for simulation of 3D incompressible isothermal flows. At first, the numerical fluxes of governing equations are reconstructed by the local solution of Boltzmann equation with sphere function distribution. Due to incompressible limit, the sphere at cell interface can be approximately considered to be symmetric as shown in the work. Besides that, the energy equation is usually not needed for simulation of incompressible isothermal flows. With all these simplifications, the formulations of the simplified SGKS can be expressed concisely and explicitly. Secondly, the commonly-used implicit Lower-Upper Symmetric Gauss-Seidel (LU-SGS) method is adopted to further improve the computational efficiency and numerical stability of present scheme. In LU-SGS method, only a forward and a backward sweep are needed for marching the conservative variables in time. As a result, the simplified SGKS with the LU-SGS method can be implemented easily. Numerical experiments, including the 3D lid-driven cavity flow and flow over a backward-facing step, showed that the incompressible isothermal flows can be well simulated by the developed scheme and its computational efficiency is significantly higher than that of the original SGKS and the lattice Boltzmann flux solver (LBFS). In addition, it was found that the present scheme with the LU-SGS method is more efficient than that with the explicit Euler method, and the speedup ratio is about 2 to 5.

Lagrangian transport in a class of three-dimensional buoyancy-driven flows

The present study concerns the Lagrangian dynamics of three-dimensional (3D) buoyancy-driven cavity flows under steady and laminar conditions due to a global temperature gradient imposed via an opposite hot and cold sidewall. This serves as the archetypal configuration for natural-convection flows in which (contrary to the well-known Rayleigh–Bénard flow) gravity is perpendicular (instead of parallel) to the global temperature gradient. Limited insight into the Lagrangian properties of this class of flows, despite its relevance to observed flow phenomena as well as scalar transport, motivates this study. The 3D Lagrangian dynamics are investigated in terms of the generic structure and associated transport properties of the global streamline pattern (‘Lagrangian flow topology’) by both theoretical and computational analyses. The Grashof number$Gr$is the principal control parameter for the flow topology: limit$Gr=0$yields a trivial state of closed streamlines;$Gr>0$induces symmetry breaking by fluid inertia and buoyancy and thus causes formation of toroidal coherent structures (‘primary tori’) embedded in chaotic streamlines governed by Hamiltonian mechanisms. Fluid inertia prevails for ‘smaller’$Gr$and gives behaviour that is dynamically entirely analogous to 3D lid-driven cavity flows. Buoyancy-induced bifurcation of the flow topology occurs for ‘larger’$Gr$and underlies the emergence of ‘secondary rolls’ observed in the literature and to date unreported secondary tori for ‘larger’ Prandtl numbers$Pr$. Key to these dynamics are stagnation points and corresponding heteroclinic manifold interactions.

Three-dimensional simplified and unconditionally stable lattice Boltzmann method for incompressible isothermal and thermal flows

In this paper, a three-dimensional simplified and unconditionally stable lattice Boltzmann method (3D-USLBM) is proposed for simulating incompressible isothermal/thermal flows. This method is developed by reconstructing solutions to the macroscopic governing equations recovered from the lattice Boltzmann equation and resolved in a predictor-corrector scheme. The final formulations of 3D-USLBM only involve the equilibrium and the non-equilibrium distribution functions. Among them, the former is calculated from the macroscopic variables and the latter is evaluated from the difference between two equilibrium distribution functions at different locations and time levels. Thus, 3D-USLBM directly tracks the evolution of macroscopic variables, which yields lower cost in virtual memory and facilitates the implementation of physical boundary conditions. A von Neumann stability analysis was performed on the present method to theoretically prove its unconditional stability. By imposing a regular Lagrange interpolation algorithm, this method can be flexibly extended to a non-uniform Cartesian mesh or body-fitted mesh with curved boundaries. Four numerical tests, that is, plane Poiseuille flow, 3D lid-driven cavity flow and 3D natural convection in a cubic cavity, and concentric annulus, were conducted to verify the stability, accuracy, and flexibility of the presented method.

GPU Acceleration of CFD Algorithm: HSMAC and SIMPLE

CFD (Computational Fluids Dynamic) is an important branch of fluid dynamics. It applies various kinds of discrete mathematical method to analyze and simulate problems in fluid mechanics with the use of computer. During the computation, huge computational tasks on a single CPU often makes it very inefficient to get the result, so there is an increasing number of application of parallel computation in CFD. With more powerful computing capability and lower price, GPU (Graphic Processing Unit) has become a better solution for parallel computing than CPU in some cases in recent years. In this paper, we implemented the HSMAC and SIMPLE algorithms on GPU. For the simulation of 2D lid-driven cavity flow, the GPU version could get a speedup up to 58x and 21x respectively with double precision, and 78x and 32x with single precision, compared to the sequential CPU version. It demonstrates a good prospects of GPU acceleration of CFD algorithms.

Compressibility effects on the first global instability mode of the vortex formed in a regularized lid-driven cavity flow

Abstract The effect of compressibility on the stability of a two-dimensional (2D) lid-driven cavity flow was investigated using a global linear stability analysis in this study. In a previous study, Bergamo et al. (2015) [12] revealed that compressibility has a stabilizing effect on dominant instability modes of this flow. However, the detailed mechanism of this stabilization effect has not been elucidated. The present study focused on the mechanism of this stabilizing effect on the first instability mode which causes a periodic oscillation to a primary vortex formed in the cavity. Our results show that the compressibility has a stabilizing effect on this flow due to the baroclinic torque and vorticity-dilatation term which appear in the vorticity transport equation. The distribution of these terms show that the vorticity-dilatation effect suppresses the changes in the perturbed vorticity (i.e., stabilizes the flow), and the baroclinic torque deforms the perturbed vorticity distribution. Furthermore, the effect of baroclinic torque was quantitatively estimated by solving the forced compressible stability problem in which the corresponding linearized equations are artificially forced to cancel the baroclinic torque. The results clearly show that the baroclinic torque decreases the growth rate of the first instability mode. However, the quantitatively estimated variation in the critical Reynolds number caused by the baroclinic torque indicates this term is not the most dominant mechanism of the stabilization effect and dilatation effects would be more significant.

A Simplified Lattice Boltzmann Method without Evolution of Distribution Function

AbstractIn this paper, a simplified lattice Boltzmann method (SLBM) without evolution of the distribution function is developed for simulating incompressible viscous flows. This method is developed from the application of fractional step technique to the macroscopic Navier-Stokes (N-S) equations recovered from lattice Boltzmann equation by using Chapman-Enskog expansion analysis. In SLBM, the equilibrium distribution function is calculated from the macroscopic variables, while the non-equilibrium distribution function is simply evaluated from the difference of two equilibrium distribution functions. Therefore, SLBM tracks the evolution of the macroscopic variables rather than the distribution function. As a result, lower virtual memories are required and physical boundary conditions could be directly implemented. Through numerical test at high Reynolds number, the method shows very nice performance in numerical stability. An accuracy test for the 2D Taylor-Green flow shows that SLBM has the second-order of accuracy in space. More benchmark tests, including the Couette flow, the Poiseuille flow as well as the 2D lid-driven cavity flow, are conducted to further validate the present method; and the simulation results are in good agreement with available data in literatures.