Three-dimensional Lid-driven Cavity Research Articles

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.

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.

Combined effects of velocity ratio and radii sizes of internal cylinder on the flow instability in a two-sided lid-driven cubical cavity

This paper computes the laminar flow in a three-dimensional (2D) two-sided lid-driven cubical cavity induced by a centrally located circular cylinder by Finite Volume Method and multigrid acceleration. The cavity has left and right parallel lid-driven walls and all the other walls completing the domain are motionless. Furthermore, different wake parameters are considered throughout this study; velocity ratios [[Formula: see text] and 0.75] and radius size of the inner cylinder [[Formula: see text] and 0.2]. Thereafter, the study has been extended to the unsteady state by further increasing Reynolds number till reaching up to 2000. It is noticed that the highest critical Reynolds number corresponds to the lowest radius size and conversely. By further increasing the velocity ratio till reaching 0.25, 0.5 and 0.75, where the left moving lid began to have an effect, the opposite phenomenon occurs and we note that the highest critical Reynolds number corresponds to the highest radius. This can be explained by the fact that when the cavity is two-sided lid-driven, the velocity of the fluid particles accelerates. Besides, it has been proved that both parameters [Formula: see text] and a radius size [Formula: see text] promote faster transition to the unsteadiness rather than the remaining velocity ratios and radii sizes.

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.

Global transition dynamics of flow in a lid-driven cubical cavity

The dynamics of a fully three-dimensional lid-driven cubical cavity (3D-LDC) flow at several post-critical conditions, i.e., beyond the first bifurcation, are elucidated using both linear and nonlinear analyses. When the Reynolds number is increased beyond the critical value, symmetry breaks down intermittently with subsequent gradual growth in spanwise inhomogeneity. This results in crossflow as well as pronounced secondary flow due to enhanced imbalance between centrifugal and pressure forces. Thus, while a stable solution is obtained at \(\hbox {Re}=1900\) (Reynolds number based on lid velocity and cavity side length), nonlinear analysis identifies intermittent and nearly saturated regimes at \(\hbox {Re}=2100\) and \(\hbox {Re}=3000\), respectively. These changes in the regime are examined by considering five basic states at different Reynolds numbers starting from \(\hbox {Re}=1900\). At the lowest Reynolds number, linear analysis yields only symmetric modes, characterized by Taylor–Görtler-like (TGL) vortices. However, at \(\hbox {Re}=2100\), the intermittent breakdown of symmetry results in the emergence of an antisymmetric low-frequency mode apart from primary high-frequency mode. The frequencies of both these modes are numerically close to those obtained from corresponding nonlinear simulations. When the Reynolds number is increased even further, the TGL structures drift under the influence of the crossflow to occupy the previously structureless region near the wall. The frequency of each mode decreases with increase in \(\hbox {Re}\); between 1900 and 3000, the frequency of the primary mode changes by more than 20%. Furthermore, the spatial support of each mode becomes larger within the cavity. Both primary and secondary modes are increasingly destabilized with \(\hbox {Re}\); however, the secondary antisymmetric mode is destabilized at a higher rate. The current study thus provides a comprehensive picture of the overall dynamics of 3D-LDC flows in pre- and post-bifurcation regimes in an extended \(\hbox {Re}\) range not considered hitherto.

Large-eddy simulation of wall-bounded turbulent flow with high-order discrete unified gas-kinetic scheme

In this paper, we introduce the discrete Maxwellian equilibrium distribution function for incompressible flow and force term into the two-stage third-order Discrete Unified Gas-Kinetic Scheme (DUGKS) for simulating low-speed turbulent flows. The Wall-Adapting Local Eddy-viscosity (WALE) and Vreman sub-grid models for Large-Eddy Simulations (LES) of turbulent flows are coupled within the present framework. Meanwhile, the implicit LES are also presented to verify the effect of LES models. A parallel implementation strategy for the present framework is developed, and three canonical wall-bounded turbulent flow cases are investigated, including the fully developed turbulent channel flow at a friction Reynolds number (Re) about 180, the turbulent plane Couette flow at a friction Re number about 93 and lid-driven cubical cavity flow at a Re number of 12000. The turbulence statistics, including mean velocity, the r.m.s. fluctuations velocity, Reynolds stress, etc. are computed by the present approach. Their predictions match precisely with each other, and they are both in reasonable agreement with the benchmark data of DNS. Especially, the predicted flow physics of three-dimensional lid-driven cavity flow are consistent with the description from abundant literature. The present numerical results verify that the present two-stage third-order DUGKS-based LES method is capable for simulating inhomogeneous wall-bounded turbulent flows and getting reliable results with relatively coarse grids.

Lagrangian chaos in steady three-dimensional lid-driven cavity flow.

Steady three-dimensional flows in lid-driven cavities are investigated numerically using a high-order spectral-element solver for the incompressible Navier-Stokes equations. The focus is placed on critical points in the flow field, critical limit cycles, their heteroclinic connections, and on the existence, shape, and dependence on the Reynolds number of Kolmogorov-Arnold-Moser (KAM) tori. In finite-length cuboidal cavities at small Reynolds numbers, a thin layer of chaotic streamlines covers all walls. As the Reynolds number is increased, the chaotic layer widens and the complementary KAM tori shrink, eventually undergoing resonances, until they vanish. Accurate data for the location of closed streamlines and of KAM tori are provided, both of which reach very close to the moving lid. For steady periodic Taylor-Görtler vortices in spanwise infinitely extended cavities with a square cross section, chaotic streamlines occupy a large part of the flow domain immediately after the onset of Taylor-Görtler vortices. As the Reynolds number increases, the remaining KAM tori vanish from the Taylor-Görtler vortices, while KAM tori grow in the central region further away from the solid walls.

Energy transport of two-phase nanofluid approach inside a three-dimensional lid-driven cubic cavity containing solid cylinder and heat source

The energy transport intensification can be achieved by an addition of nanoparticles to the host liquid. Simultaneously, a location of partial heating and the velocity of moving cavity surface can be control parameters for the thermal transmission. The present research deals with a computational investigation of mixed convective energy transport within a cubical chamber saturated by nanofluids, including the impacts of cold surface motion, local heating, and central heat-conducting cylinder. The governing equations are written using the Buongiorno's nanoliquid approach with the Boussinesq approximation in primitive dimensionless variables and are solved numerically by the finite-element technique. An investigation of the nanoliquid circulation and energy transport has resulted in an extensive range of control characteristics, namely, Reynolds number, Richardson number, nanoparticle volume fraction, dimensionless radius of solid cylinder, dimensionless heat source length and the dimensionless heat source position. It has been observed that a nonlinear impact of nanoparticles volume fraction toward the energy transport strength for high Reynolds number exists. Whilst for low Reynolds number, a growth of the nano-sized particles concentration leads to the energy transference strengthening. At the same time, a rise of the inner cylinder size characterizes a diminution of the heat transport strength.

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.

Turbulence characteristics of high Reynolds number flow inside a three-dimensional cubic lid-driven cavity

Large eddy simulation (LES) with dynamic Smagorinsky model (DSM) has been implemented to elicit the turbulence characteristic of the flow inside a cubic lid-driven cavity at Reynolds number of 10000. The coherent organized structures tear, pair and disintegrate leading to the evolution of turbulence. They are reflected in the RMS (root mean square) plots of the turbulent fluctuations in different planes. The flow has a near zero helicity value; unlike the other two velocity component, the probability density function of span wise velocity component shows the flow symmetry on the both side of the mid plane in span wise direction; kurtosis being largely leptokurtic except near wall locations indicates their peakedness. The quadrant analysis shows presence of more coherent structures near the down stream wall and the joint probability distribution function shows that turbulence is anisotropic. Power spectra is observed to follow the −5∕3 law. Both Taylor microscale λ and Kolmogorov length scale η show greater near-wall dissipation

Adaptive mesh refinement for steady flows in Nek5000

Adaptive mesh refinement is performed in the framework of the spectral element method augmented by approaches to error estimation and control. The h-refinement technique is used for adapting the mesh, where selected grid elements are split by a quadtree (2D) or octree (3D) structure. Continuity between parent–child elements is enforced by high-order interpolation of the solution across the common faces. Parallel mesh partitioning and grid management respectively, are taken care of by the external libraries ParMETIS and p4est. Two methods are considered for estimating and controlling the error of the solution. The first error estimate is local and based on the spectral properties of the solution on each element. This method gives a local measure of the L2-norm of the solution over the entire computational domain. The second error estimate uses the dual-weighted residuals method — it is based on and takes into account both the local properties of the solution and the global dependence of the error in the solution via an adjoint problem. The objective of this second approach is to optimize the computation of a given functional of physical interest. The simulations are performed by using the code Nek5000 and three steady-state test cases are studied: a two-dimensional lid-driven cavity at Re=7,500, a two-dimensional flow past a cylinder at Re=40, and a three-dimensional lid-driven cavity at Re=2,000 with a moving lid tilted by an angle of 30∘. The efficiency of both error estimators is compared in terms of refinement patterns and accuracy on the functional of interest. In the case of the adjoint error estimators, the trend on the error of the functional is shown to be correctly represented up to a multiplicative constant.

Application of a Projection Method for Simulating Flow of a Shear-Thinning Fluid

In this paper, a first-order projection method is used to solve the Navier–Stokes equations numerically for a time-dependent incompressible fluid inside a three-dimensional (3-D) lid-driven cavity. The flow structure in a cavity of aspect ratio δ = 1 and Reynolds numbers ( 100 , 400 , 1000 ) is compared with existing results to validate the code. We then apply the developed code to flow of a generalised Newtonian fluid with the well-known Ostwald–de Waele power-law model. Results show that, by decreasing n (further deviation from Newtonian behaviour) from 1 to 0.9, the peak values of the velocity decrease while the centre of the main vortex moves towards the upper right corner of the cavity. However, for n = 0.5 , the behaviour is reversed and the main vortex shifts back towards the centre of the cavity. We moreover demonstrate that, for the deeper cavities, δ = 2 , 4 , as the shear-thinning parameter n decreased the top-main vortex expands towards the bottom surface, and correspondingly the secondary flow becomes less pronounced in the plane perpendicular to the cavity lid.

Development of a three dimensional meshless numerical method for the solution of the Boltzmann equation on complex geometries

We propose a three dimensional meshless numerical scheme based on the Boltzmann equation for the simulation of fluid flows in domains of complex geometries. The velocity space discretization in our method is the same as the standard lattice Boltzmann method, i.e. the same lattices are used to discretize microscopic velocities. The velocity-discrete Boltzmann equation with the BGK collision approximation is discretized in time using the Lax–Wendroff scheme, and subsequently in space using the meshless local Petrov–Galerkin method based on local radial basis functions augmented with polynomials. The method is first validated and tuned by solving two benchmark problems: The pressure-driven flow in an elliptical tube, and the three-dimensional lid-driven cavity flow. The error analysis for the first case shows the second order accuracy of the proposed method. The flow in a packed bed is then solved in order to illustrate the efficiency of the proposed method in the simulation of geometrically complex flows. The results show that both the computational time and the memory usage of our method are lower than those of the standard lattice Boltzmann method for the geometrically complex flows.

An Augmented Lagrangian Preconditioner for the 3D Stationary Incompressible Navier--Stokes Equations at High Reynolds Number

In Benzi & Olshanskii (SIAM J.~Sci.~Comput., 28(6) (2006)) a preconditioner of augmented Lagrangian type was presented for the two-dimensional stationary incompressible Navier--Stokes equations that exhibits convergence almost independent of Reynolds number. The algorithm relies on a highly specialized multigrid method involving a custom prolongation operator and is tightly coupled to the use of piecewise constant finite elements for the pressure. However, the prolongation operator and velocity element used do not directly extend to three dimensions: the local solves necessary in the prolongation operator do not satisfy the inf-sup condition. In this work we generalize the preconditioner to three dimensions, proposing alternative finite elements for the velocity and prolongation operators for which the preconditioner works robustly. The solver is effective at high Reynolds number: on a three-dimensional lid-driven cavity problem with approximately one billion degrees of freedom, the average number of Krylov iterations per Newton step varies from $7$ at $\mathrm{Re} = 4.5$ to $3$ at $\mathrm{Re} = 1000$ and $5$ at $\mathrm{Re} = 5000$.

A Hybrid High-Order method for the incompressible Navier–Stokes equations based on Temam's device

In this work we propose a novel Hybrid High-Order method for the incompressible Navier–Stokes equations based on a formulation of the convective term including Temam's device for stability. The proposed method has several advantageous features: it supports arbitrary approximation orders on general meshes including polyhedral elements and non-matching interfaces; it is inf-sup stable; it is locally conservative; it supports both the weak and strong enforcement of velocity boundary conditions; it is amenable to efficient computer implementations where a large subset of the unknowns is eliminated by solving local problems inside each element. Particular care is devoted to the design of the convective trilinear form, which mimicks at the discrete level the non-dissipation property of the continuous one. The possibility to add a convective stabilisation term is also contemplated, and a formulation covering various classical options is discussed. The proposed method is theoretically analysed, and an energy error estimate in hk+1 (with h denoting the meshsize) is proved under the usual data smallness assumption. A thorough numerical validation on two and three-dimensional test cases is provided both to confirm the theoretical convergence rates and to assess the method in more physical configurations (including, in particular, the well-known two- and three-dimensional lid-driven cavity problems).

High-Reynolds-number turbulent cavity flow using the lattice Boltzmann method

We present a boundary condition scheme for the lattice Boltzmann method that has significantly improved stability for modeling turbulent flows while maintaining excellent parallel scalability. Simulations of a three-dimensional lid-driven cavity flow are found to be stable up to the unprecedented Reynolds number $\mathrm{Re}=5\ifmmode\times\else\texttimes\fi{}{10}^{4}$ for this setup. Excellent agreement with energy balance equations, computational and experimental results are shown. We quantify rises in the production of turbulence and turbulent drag, and determine peak locations of turbulent production.

Mixed convection heat transfer of turbulent flow in a three-dimensional lid-driven cavity with a rotating cylinder

A numerical study has been carried out to investigate the combined forced and natural convection heat transfer in a differentially heated 3D obstructed cavity with a thermally insulated rotating circular cylinder. The cavity has a hot stationary bottom wall and a cold top lid-driven wall, and all the other walls completing the domain are motionless and adiabatic. The simulations are performed for different Reynolds numbers, Re=5000, 10,000, 15,000 and 30,000, and for dimensionless rotational speeds of the cylinder, 0≤Ω≤10. The performance of two turbulence methods, Large Eddy Simulation (LES) and Unsteady Reynolds-Averaged Navier-Stokes (URANS), has been evaluated in this research. The flow and thermal fields are studied through flow vectors, isotherm contours and iso-surfaces temperature, as well as through the average Nusselt number (Nuav) and velocity components. The results demonstrate clearly that the flow patterns and the thermal fields are influenced strongly by increasing either the rotating cylinder speed or the Reynolds number. Furthermore, both LES and URANS solutions can capture the essential feature of the primary eddies in the cavity. But this study has shown convincing evidence that only the LES method can predict the structure details of the secondary eddies that have profound effects on the heat transfer behaviour within the enclosure.

A theoretical model for the porosity–permeability relationship

An analytical permeability model is proposed for incompressible flow in porous media. Based on a general non-equilibrium thermodynamic stability theory, the model is derived from maximizing the entropy production rate to achieve a stable system. The resulting formula is in a form similar to previous empirical equations, but with calculable coefficients. Both three-dimensional lid-driven cavity and cubicle natural convection problems have been computed, via a modern computational fluid dynamics (CFD) solver, for numerical investigations of its performance.

Numerical Modeling of 2-D and 3-D Flows using Artificial Compressibility Method and Collocated Mesh

In this paper, applications of a numerical model on simulation of two and three-dimensional flows are presented. This model solves Navier-Stokes equations using finite volume method and large eddy simulation (LES) in a collocated mesh. Artificial compressibility method with dual t ime stepping is used to solve the time dependent equations. Also a modified m omentum i nterpolation method (MIM) based on the unsteady flows i s deployed t o overcome t he non-physical pressure oscillation. Capability of the presented numerical code for flow s imulation, i s a ssessed by a pplication f or twodimensional square and three-dimensional lid-driven cavity flows. Numerical r esults of cavity flow presents very good agreement with the numerical and experimental data of other existent researches.