Two-dimensional Lid-driven Cavity Flow Research Articles

An efficient framework for solving forward and inverse problems of nonlinear partial differential equations via enhanced physics-informed neural network based on adaptive learning

In recent years, the advancement of deep learning has led to the utilization of related technologies to enhance the efficiency and accuracy of scientific computing. Physics-Informed Neural Networks (PINNs) are a type of deep learning method applied to scientific computing, widely used to solve various partial differential equations (PDEs), demonstrating tremendous potential. This study improved upon original PINNs and applied them to forward and inverse problems in the nonlinear science field. In addition to incorporating the constraints of PDEs, the improved PINNs added constraints on gradient information, which further enhanced the physical constraints. Moreover, an adaptive learning method was used to update the weight coefficients of the loss function and dynamically adjust the weight proportion of each constraint term. In the experiment, the improved PINNs were used to numerically simulate localized waves and two-dimensional lid-driven cavity flow described by partial differential equations. Meanwhile, we critically evaluate the accuracy of the prediction results. Furthermore, the improved PINNs were utilized to solve the inverse problems of nonlinear PDEs, where the results showed that even with noisy data, the unknown parameters could be discovered satisfactorily. The study results indicated that the improved PINNs were significantly superior to original PINNs, with shorter training time, increased accuracy in prediction results, and greater potential for application.

Numerical Simulation and Computational Fluid Dynamics Analysis of Two-Dimensional Lid-Driven Cavity Flow Within the Weapon Bay of an Autonomous Fighter Drone

The efficient deployment of weapons in military operations is critical for mission success, and the flow of air within the weapon bay of an autonomous fighter drone plays a vital role in achieving this objective. In this paper, we present a comprehensive numerical simulation and computational fluid dynamics (CFD) analysis of the three-dimensional lid-driven cavity flow within the weapon bay of an autonomous fighter drone. To address this challenging problem, we employ CFD analysis and a multigrid approach to solve the Navier-Stokes equations for the aerodynamic problem. Our simulations include high Reynolds numbers of up to 10,000, which demonstrates the potential of CFD analysis in optimizing the design of autonomous fighter drones for military operations. We evaluate the effectiveness of the linked strongly implicit multigrid technique in estimating high-Re fine-mesh flow solutions using the vorticity-stream function formulation of the two-dimensional incompressible Navier-Stokes equations. The model issue is the driven flow in a square cavity, and we consider meshes up to 1024 x 1024 points and combinations with Reynolds numbers as high as 1000. To further improve the accuracy of our simulations, we employ one-dimensional grid clustering coordinate transformations instead of uniform mesh refinement, as the flow field exhibits one or more secondary vortices. Our findings demonstrate that CFD analysis can provide valuable insights into the complex flow dynamics within the weapon bay of autonomous fighter drones, which can lead to the optimization of their design for enhanced mission capabilities. Overall, our study highlights the significance of numerical simulations and CFD analysis in the design and optimization of autonomous fighter drones for military applications. Our results can serve as a basis for future research in the field of UAV aerodynamics and contribute to the development of more efficient and effective military operations.

GPU implementation of the discrete unified gas kinetic scheme for low-speed isothermal flows

In this paper, two GPU parallel algorithms are proposed for the discrete unified gas kinetic scheme (DUGKS) for simulating low-speed isothermal flows. Algorithm-I uses a two-level fine-grain technique for the parallelization of physical spatial space, while Algorithm-II adopts this technique for both physical spatial and particle velocity spaces. To evaluate the performance of the proposed algorithms, several typical benchmark problems are simulated, including the two-dimensional (2D) and three-dimensional (3D) lid-driven cavity flows, the micro channel and cavity flows. Numerical results show that our GPU algorithms can achieve satisfactory computational efficiency. For Algorithm-I, the speedup can reach 250 and 338 on a Tesla V100 GPU card for the 2D and 3D continuum cavity flows, respectively, and a hundredfold acceleration can be obtained for the rarefied cases. While for Algorithm-II, a speedup of about 70 can be attained for rarefied cases. However, it is not applied to continuum problems that only require a small number of velocity points. Moreover, comparisons between the two GPU algorithms are also conducted for the rarefied flows with various grid meshes and velocity directions. The results show that Algorithm-I performs better when physical mesh size is large, while Algorithm-II can provide higher efficiency for a coarser mesh with medium number of discrete velocities. Special attention is also paid to comparisons between Algorithm-I and MPI parallelization with 128 CPU cores based on physical space discretization approach, and it is found that Algorithm-I has a clear advantage on V100 GPU when dealing with sparse physical grids in both continuum and rarefied cases.

An incompressible flow solver on a GPU/CPU heterogeneous architecture parallel computing platform

A computational fluid dynamics (CFD) solver for a GPU/CPU heterogeneous architecture parallel computing platform is developed to simulate incompressible flows on billion-level grid points. To solve the Poisson equation, the conjugate gradient method is used as a basic solver, and a Chebyshev method in combination with a Jacobi sub-preconditioner is used as a preconditioner. The developed CFD solver shows good performance on parallel efficiency, which exceeds 90% in the weak-scalability test when the number of grid points allocated to each GPU card is greater than 2083. In the acceleration test, it is found that running a simulation with 10403 grid points on 125 GPU cards accelerates by 203.6x over the same number of CPU cores. The developed solver is then tested in the context of a two-dimensional lid-driven cavity flow and three-dimensional Taylor-Green vortex flow. The results are consistent with previous results in the literature.

A multiscale discrete velocity method for diatomic molecular gas

In the previous study, the multiscale discrete velocity method (MDVM) has been developed for monatomic gas with particle translational motion only. Unlike the unified gas-kinetic scheme (UGKS) and discrete unified gas-kinetic scheme, which are the typical representative of multiscale kinetic methods, MDVM achieves multiscale property by mixing the solution of macroscopic control equations and the Boltzmann equation, without the need to calculate complex interface flux. Therefore, MDVM has a higher computational efficiency. To broaden the application scope of MDVM, the Rykov model, which elucidates the exchange of energy between molecular translational and rotational energies, is introduced into MDVM in this paper. Numerical simulations are conducted for various cases, including one-dimensional shock tube, one-dimensional nitrogen shock structure, two-dimensional lid-driven cavity flow, and two-dimensional hypersonic flows around a flat plate and a blunt circular cylinder. The present results agree well with those from the diatomic UGKS method, demonstrating the developed diatomic MDVM can simulate multi-scale, strongly non-equilibrium, diatomic molecular gas flow while exhibiting certain efficiency improvements compared to the diatomic UGKS.

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.

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.

Chaotic advection in a recirculating flow: Effect of a fluid-flexible-solid interaction.

The present work is on laminar recirculating flow-induced deformation as well as motion of a neutrally buoyant flexible elliptical solid, resulting in Lagrangian chaos in a two-dimensional lid-driven cavity flow. Using a fully Eulerian and monolithic approach-based single-solver for the fluid flow and flexible-solid deformation, a chaotic advection study is presented for various aspect ratios β ( =0.5-1.0) and a constant volume fraction Φ=10% of an elliptical solid at a constant Ericksen number Er=0.05 and Reynolds number Re=100. Our initial analysis reveals maximum chaotic advection at β=0.5 for which a comprehensive nonlinear dynamical analysis is presented. The Poincaré map revealed elliptic islands and chaotic sea in the fluid flow. Three large elliptic islands, apart from certain smaller islands, were identified near the solid. Periodic point analysis revealed the lowest order hyperbolic/elliptic periodic points to be three. Adaptive material tracking gave a physical picture of a deforming material blob revealing its exponential stretch along with steep folds and demonstrated unstable/stable manifolds corresponding to lowest order hyperbolic points. Furthermore, adaptive material tracking demonstrates heteroclinic connections and tangles in the system that confirm the existence of chaos. For the transient as compared to the periodic flow, adaptive material tracking demonstrates a larger exponential increase of the blob's interfacial area. The finite-time Lyapunov exponent field revealed attracting/repelling Lagrangian coherent structures and entrapped fluid zones. Our work demonstrates an immersed deformable solid-based onset of chaotic advection, for the first time in the literature, which is relevant to a wide range of applications.

Polygonal Finite Element for Two-Dimensional Lid-Driven Cavity Flow

This paper investigates a polygonal finite element (PFE) to solve a two-dimensional (2D) incompressible steady fluid problem in a cavity square. It is a well-known standard benchmark (i.e., lid-driven cavity flow)-to evaluate the numerical methods in solving fluid problems controlled by the Navier–Stokes (N–S) equation system. The approximation solutions provided in this research are based on our developed equal-order mixed PFE, called Pe1Pe1. It is an exciting development based on constructing the mixed scheme method of two equal-order discretisation spaces for both fluid pressure and velocity fields of flows and our proposed stabilisation technique. In this research, to handle the nonlinear problem of N-S, the Picard iteration scheme is applied. Our proposed method’s performance and convergence are validated by several simulations coded by commercial software, i.e., MATLAB. For this research, the benchmark is executed with various Reynolds numbers up to the maximum . All results then numerously compared to available sources in the literature.

Stability of obliquely driven cavity flow

The linear stability of the incompressible flow in an infinitely extended cavity with rectangular cross-section is investigated numerically. The basic flow is driven by a lid which moves tangentially, but at yaw with respect to the edges of the cavity. As a result, the basic flow is a superposition of the classical recirculating two-dimensional lid-driven cavity flow orthogonal to a wall-bounded Couette flow. Critical Reynolds numbers computed by linear stability analysis are found to be significantly smaller than data previously reported in the literature. This finding is confirmed by independent nonlinear three-dimensional simulations. The critical Reynolds number as a function of the yaw angle is discussed for representative aspect ratios. Different instability modes are found. Independent of the yaw angle, the dominant instability mechanism is based on the local lift-up process, i.e. by the amplification of streamwise perturbations by advection of basic flow momentum perpendicular to the sheared basic flow. For small yaw angles, the instability is centrifugal, similar as for the classical lid-driven cavity. As the spanwise component of the lid velocity becomes dominant, the vortex structures of the critical mode become elongated in the direction of the bounded Couette flow with the lift-up process becoming even more important. In this case the instability is made possible by the residual recirculating part of the basic flow providing a feedback mechanism between the streamwise vortices and the streamwise velocity perturbations (streaks) they promote. In the limit when the basic flow approaches bounded Couette flow the critical Reynolds number increases very strongly.

Solving steady-state lid-driven square cavity flows at high Reynolds numbers via a coupled improved element-free Galerkin–reduced integration penalty method

Steady-state two-dimensional lid-driven square cavity flows at high Reynolds numbers are solved in this communication using a velocity-based formulation developed in the context of the improved element-free Galerkin–reduced integration penalty method (IEFG–RIPM). The analyses based on the IEFG–RIPM are performed under a standard Galerkin weak-formulation, i.e. without the need of introducing streamline-upwind or pressure stabilizing terms in order to suppress the appearance of non-physical oscillations. Solutions in a wide range of high Reynolds numbers are successfully achieved, and the results obtained have exhibited an excellent agreement with mesh-based solutions reported in the literature. The numerical performance of the proposed IEFG–RIPM in the solution of such benchmark problem has been analysed in terms of velocity distribution, streamline patterns, vorticity and pressure contours, and also in terms of the properties of primary and secondary vortices. The outcomes of this study demonstrate the potential of the IEFG–RIPM as a feasible and reliable numerical technique for the analysis of lid-driven square cavity flows at high Reynolds numbers, allowing the fulfilment of accuracy and stability numerical requirements demanded in the solution of this benchmark problem in a remarkably simple manner.

Discrete unified gas kinetic scheme for incompressible Navier-Stokes equations

The discrete unified gas kinetic scheme (DUGKS) combines the advantages of both the unified gas kinetic scheme (UGKS) and the lattice Boltzmann method. It can adopt the flexible meshes, meanwhile, the flux calculation is simple. However, the original DUGKS is proposed for the compressible flows. When we try to solve a problem governed by the incompressible Navier-Stokes (N-S) equations, the original DUGKS may bring some undesirable errors because of the compressible effect. To eliminate the compressible effect, the DUGKS for incompressible N-S equations is developed in this work. In addition, the Chapman-Enskog analysis ensures that the present DUGKS can solve the incompressible N-S equations exactly, meanwhile, a new non-extrapolation scheme is adopted to treat the Dirichlet boundary conditions. To test the present DUGKS for incompressible N-S equations, four problems are adopted. The first one is a periodic problem driven by an external force, which is used to test the influences of Courant–Friedrichs–Lewy condition number and the Mach number (Ma). Besides, some comparisons between the present DUGKS and some available results are also conducted. The second problem is Womersley flow, it is also used to test the influence of Ma, and the results show that the compressible effect is reduced obviously. Then, the two-dimensional lid-driven cavity flow is considered. In these simulations, the Reynolds number is varied from 400 to 1000000 to illustrate the accuracy, stability and efficiency of the present DUGKS. Finally, the numerical solutions of the three-dimensional lid-driven cavity flow suggest that the present DUGKS is suitable for the three-dimensional problems.

A data-driven, physics-informed framework for forecasting the spatiotemporal evolution of chaotic dynamics with nonlinearities modeled as exogenous forcings

We introduce a data-driven, linear method for the spatiotemporal prediction of high-dimensional and chaotic dynamical systems. In this method, the observables are vector-valued and delay-embedded, and the nonlinearities are treated as external forcings. The proper representation of the nonlinear terms is found in a physics-informed way or a purely data-driven fashion, depending on whether any knowledge of governing dynamics is available or not. The unknown matrices appearing in the method are found using the dynamic mode decomposition with control technique. The distinctive features of the method enable it to accurately capture the system's dynamically important unstable modes while suppressing their unbounded growth via the included nonlinearities. The predictive capabilities of the method are demonstrated for well-known prototypes of chaotic dynamics such as the Kuramoto-Sivashinsky equation and Lorenz-96 system, for which the method predictions are accurate for several Lyapunov timescales. Similar performance is shown for two-dimensional lid-driven cavity flows at high Reynolds numbers.

Application of fully convolutional neural networks for feature extraction in fluid flow

Accurate extraction of features in fluid flows is of importance due to the presence in many natural and technological systems. Recently, methods based on machine learning have emerged as an alternative to traditional Eulerian-based methods to extract features in fluid flows. One broad category in ML is the convolution operation-based methods. The precision of feature extraction in convolution operation-based methods increases by constraining the measurement box, such as dividing the input data into small patches and using them as input boxes in convolutional neural networks. In this work, we propose a method that transforms each cell of the computational domain into a detection pixel (measurement box) to perform the task of feature extraction at the smallest possible computational level. To demonstrate the performance, we extract the vortical structures in a benchmark two-dimensional lid-driven cavity flow employing a symmetric, fully convolutional network. The number of convolution and deconvolution blocks in the network’s structure is studied to obtain the highest accuracy and yet to avoid the degradation problem. Different parameters, such as the Reynolds number and velocity boundary values, are considered in the complete and clipped cavity cases to create the training and test datasets. The semantic segmentation metrics, including Jaccard and Dice, yield values close to 1 for the test set on complete and clipped cavity cases with varying the Reynolds number or the velocity boundary values.

Transitions of power-law fluids in two-dimensional lid-driven cavity flow using lattice Boltzmann method

Studying transitions from laminar to turbulence of non-Newtonian fluids can provide a theoretical basis to further mediate their dynamic properties. Compared with Newtonian fluids, transitions of non-Newtonian fluids turning are less focused, thus being lack of good predictions of the critical Reynolds number (<i>Re</i>) corresponding to the first Hopf bifurcation. In this study, we employ the lattice Boltzmann method as the core solver to simulate two-dimensional lid-driven flows of a typical non-Newtonian fluid modeled by the power rheology law. Results show that the critical <i>Re</i> of shear-thinning (5496) and shear-thickening fluids (11546) are distinct from that of Newtonian fluids (7835). Moreover, when <i>Re</i> is slightly larger than the critical one, temporal variations of velocity components at the monitor point all show a periodic trend. Before transition of the flow filed, the velocity components show a horizontal straight line, and after transition , the velocity components fluctuate greatly and irregularly. Through fast Fourier transform for the velocity components, it is noted that the velocity has a dominant frequency and a harmonic frequency when <i>Re</i> is marginally larger than the critical one. Besides, the velocity is steady before transition of flow filed, so it appears as a point on the frequency spectrum. As the flow filed turns to be turbulent, the frequency spectrum of the velocity component appears multispectral. Different from a single point in the velocity phase diagram before transition, the velocity phase diagram after transition forms a smooth and closed curve, whose area is also increasing as <i>Re</i> increases. The center point of the curve moves along a certain direction, while movement directions of different center points are different. Proper orthogonal decompositions for the velocity and vorticity field reveal that the first two modes, in all types of fluids, are the dominant modes when <i>Re</i> is close to the critical one, with energy, occupying more than 95% the whole energy. In addition, for one type of fluid, the dominant modes at different <i>Re</i> values have similar structures. Results of the first and second modes of velocity field show that the modal peak is mainly distributed in vicinity of the cavity wall.

Applying physics informed neural network for flow data assimilation

Data assimilation (DA) refers to methodologies which combine data and underlying governing equations to provide an estimation of a complex system. Physics informed neural network (PINN) provides an innovative machine learning technique for solving and discovering the physics in nature. By encoding general nonlinear partial differential equations, which govern different physical systems such as fluid flows, to the deep neural network, PINN can be used as a tool for DA. Due to its nature that neither numerical differential operation nor temporal and spatial discretization is needed, PINN is straightforward for implementation and getting more and more attention in the academia. In this paper, we apply the PINN to several flow problems and explore its potential in fluid physics. Both the mesoscopic Boltzmann equation and the macroscopic Navier-Stokes are considered as physics constraints. We first introduce a discrete Boltzmann equation informed neural network and evaluate it with a one-dimensional propagating wave and two-dimensional lid-driven cavity flow. Such laminar cavity flow is also considered as an example in an incompressible Navier-Stokes equation informed neural network. With parameterized Navier-Stokes, two turbulent flows, one within a C-shape duct and one passing a bump, are studied and accompanying pressure field is obtained. Those examples end with a flow passing through a porous media. Applications in this paper show that PINN provides a new way for intelligent flow inference and identification, ranging from mesoscopic scale to macroscopic scale, and from laminar flow to turbulent flow.

L1-based calibration of POD-Galerkin models of two-dimensional unsteady flows

This paper discusses a physics-informed methodology aimed at reconstructing efficiently the fluid state of a system. Herein, the generation of an accurate reduced order model of two-dimensional unsteady flows from data leverages on sparsity-promoting statistical learning techniques. The cornerstone of the approach is l1 regularised regression, resulting in sparsely-connected models where only the important quadratic interactions between modes are retained. The original dynamical behaviour is reproduced at low computational costs, as few quadratic interactions need to be evaluated. The approach has two key features. First, interactions are selected systematically as a solution of a convex optimisation problem and no a priori assumptions on the physics of the flow are required. Second, the presence of a regularisation term improves the predictive performance of the original model, generally affected by noise and poor data quality. Test cases are for two-dimensional lid-driven cavity flows, at three values of the Reynolds number for which the motion is chaotic and energy interactions are scattered across the spectrum. It is found that: (A) the sparsification generates models maintaining the original accuracy level but with a lower number of active coefficients; this becomes more pronounced for increasing Reynolds numbers suggesting that extension of these techniques to real-life flow configurations is possible; (B) sparse models maintain a good temporal stability for predictions. The methodology is ready for more complex applications without modifications of the underlying theory, and the integration into a cyber-physical model is feasible.

Thel1-based sparsification of energy interactions in unsteady lid-driven cavity flow

Abstract

A study of the regularized lid-driven cavity’s progression to chaos

Computational simulations of a two-dimensional incompressible regularized lid-driven cavity flow were performed and analyzed to identify the dynamic behavior of the flow through multiple bifurcations which ultimately result in Eulerian chaotic flow. Semi-implicit, pseudo-spectral numerical simulations were performed at Reynolds numbers from 1000 to 25,000. Poincaré maps were used to identify transitions as the Reynolds number increased from stable-laminar flow to periodic flow, periodic flow to quasi-periodic flow, quasi-periodic flow to chaotic flow, and a sudden, brief return from chaotic flow to periodic flow. The first critical Reynolds number, near 10,250, is found in agreement with existing literature. An additional bifurcation is observed near a Reynolds number of 15,500. A power spectrum analysis, in which the novel concepts of frequency shredding and power capacity are introduced, was performed with the conclusion that no further bifurcations occurred at Reynolds numbers above 15,500 even though Eulerian chaos was not formally observed until Reynolds numbers above 18,000. While qualitative changes in the fluid flow’s attractor were apparent from trajectories in the phase space, mechanisms by which such changes can occur were apparent from power spectra of flow time histories. The novel power spectrum analysis may also serve as a new approach for characterizing multi-scale nonlinear dynamical systems.

The barely implicit correction algorithm for low-Mach-Number flows

A new Barely Implicit Correction (BIC) algorithm is presented for the simulation of low-Mach-number flows. This new algorithm is based on the original, introduced by G. Patnaik et al. [G. Patnaik, R. H. Guirguis, J. P. Boris and E. S. Oran, A barely implicit correction for flux-corrected transport. In: Journal of Computational Physics 71.1 (1987), pp. 120], which was a solution procedure including an explicit predictor step to solve the convective portion of the Navier–Stokes equations and an implicit corrector step to remove the acoustic limit on the integration time-step. The explicit predictor uses a high-order monotone algorithm while the implicit corrector solves an elliptic equation for a pressure correction to equilibrate acoustic waves. In this paper, we develop and extend BIC for multidimensional viscous flows. We introduce a new filter to further stabilize the algorithm and clarify the solution procedure for the inclusion of the viscous fluxes. The new algorithm is examined in three test problems with successively increased difficulty. First, a two-dimensional lid-driven cavity flow is simulated to demonstrate the ability of BIC on solving steady-state swirling flows. Using time steps at least 100 times larger than the explicit limit, good agreements are obtained for solutions when compared with an incompressible calculation by a prior work. A two-dimensional (2D) doubly periodic shear layer flow is simulated to examine the algorithm on solving a transient flow with strong vorticity gradients. Finally, vortex breakdown in three-dimensional (3D) swirling flows are used to further test the stability and performance of the new BIC algorithm. Comparisons of explicit and implicit BIC calculations of both the 2D doubly periodic shear layer and 3D vortex breakdown are presented side by side. They demonstrate that the new BIC algorithm is able to predict accurate and robust solutions using time steps varying from near the explicit stability limit to tens and hundreds of times larger. Excellent agreement is also obtained when compared with results from other algorithms. We discuss our observations of these computations and features which were found to be critical for robustly simulating low-speed, highly dynamic flows.