Lid-driven Cavity Research Articles

Optimization of heat transfer in a double lid-driven cavity with isoperimetric heated blocks using GFEM.

This article is concerned with the examination of flow dynamics and heat transfer characteristics in a 1:4 double lid driven cavity in presence of isoperimetric heated blocks of various shapes. The focus is to identify the optimal shape that enhances the heat transfer in a tall cavity. The parametric settings are chosen in such a way that all the convection regimes including natural, forced and mixed convection could be generated. This cavity has lids positioned at the top and bottom, moving in opposite directions along the x-axis. The physical system is represented as a set of coupled partial differential equations incorporating the rheological properties of the power-law fluids (PL). The governing equations in conjunction with various non-dimensional physical parameters are simulated via Galerkin's Finite Element Method (GFEM) on a very fine hybrid grid. The study includes the computation of the Kinetic Energy and Average Nusselt number to determine the optimal shape. It is concluded that the circular block is superior to the other two in terms of heat transmission efficiency.

Differential quadrature lattice Boltzmann method for simulating fluid flows

In this study, a novel method for simulating incompressible flows using the lattice Boltzmann method is presented, aiming to improve robustness. This method is enhanced by the Qadyan numerical method. In the Qadyan method, Partial differential equations are solved using semi-discrete schemes combined with the differential quadrature method. To discretize the spatial derivatives of the lattice Boltzmann equation, the upwind differential quadrature method is employed, while the first-order forward difference scheme and/or fourth-order Runge-Kutta method handle the temporal term. The decision to exclude more complex methods stems from their requirement for refined computational grids. This work focuses on achievable results with similar types of uniform grids. The proposed scheme introduces and evaluates a novel method for solving both steady and unsteady problems. The present numerical method is validated by solving five benchmark problems, including heat diffusion on a slab, Stokes’ first problem, Taylor-Green vortex flow, flow around a flat plate, and flow in a lid-driven cavity. Reasonable agreements are obtained between the solutions obtained using this method and those from analytical/other numerical approaches. The Qadyan formulation delivers more accurate results compared to the finite-difference lattice Boltzmann method, (FDLBM), without requiring an increase in the number of grids. Notably, the novel Qadyan method achieves this increased accuracy without introducing additional complexity to the standard lattice Boltzmann method or resorting to non-uniform grids. This is possible due to the nature of the differential quadrature method, which utilizes more combinations of candidate stencils. The Qadyan method has a higher computational cost in comparison with the FDLBM.

Machine Learning-Based Predictions of Flow and Heat Transfer Characteristics in a Lid-Driven Cavity with a Rotating Cylinder

Machine learning-based predictions of heat transfer characteristics in lid-driven cavities are transforming the field of computational fluid dynamics (CFD). Lid-driven cavities are a fundamental problem in fluid mechanics, characterized by the motion of a fluid inside a square cavity driven by the motion of one of its walls. The goal of this study was to develop multiple machine-learning regression models and highlight the discrepancies between the predicted and actual average Nusselt numbers. Additionally, the study utilized physics-informed neural networks (PINNs) to model the flow and thermal behavior at both low and high Reynolds numbers. The results were compared among actual data from computational fluid dynamics (CFD) simulations, PINN models trained with CFD data, and purely PINN models created without any prior data input. The findings of this study showed that the random forest model exhibited an exceptional stability in its predictions, consistently maintaining low errors even as the Reynolds number increased compared with other machine-learning regression models. Further, the results of this study in terms of flow and thermal behavior within the cavity were found to depend significantly on the PINN method. The data-driven PINN exhibited a much lower mean average errors at both Reynolds numbers, while the physics-based PINN showed lower physics loss.

Heat transfer augmentation in a double lid-driven trapezoidal porous cavity with heated block using hybrid nanofluid

This study intends to understand the fundamental characteristics of hybrid nanofluids (HNF) in a porous cavity using computational modeling, which will aid in developing HNF within hydrodynamics and enable adaptability. The present research investigates the enhancement of the transmission of heat in a double lid-driven trapezoidal porous cavity containing a heated block, utilizing a HNF composed of aluminum oxide/copper-water (Al2O3/Cu-H2O) employing the Finite Element Method (FEM) based on Galerkin weighted residual (GWR). Additionally, the Darcy-Brinkman-Forchheimer comprehensive approach has been used to demonstrate fluid flow in the porous media. The governing parameters such as the Darcy number (10−4 ≤ Da ≤ 10−2), the porosity of the porous medium (0.6 ≤ ε ≤ 0.9), inclined angle (π/16 ≤ ϕ ≤ π/3), and solid volume fraction (0.001 ≤ δ ≤ 0.05) have been chosen to evaluate the impact within dimensionless time (0.1 ≤ τ ≤ 1). The simulation outcomes of flow and thermal fields are shown through streamlines, isotherms, average heat transfer rate at the hot surfaces, and average temperature inside the cavity. The outcomes of this study show that the HT rate rises by 41.02% when increasing the solid volume fraction (svf) from 0.1% to 5%. Also, it has been found that if the inclined angle is enhanced from π/16 to π/3 at non-dimensional time τ = 0.5, the average HT augmented by 63.16%.

Assessing physics-informed neural network performance with sparse noisy velocity data

The utilization of data in physics-informed neural network (PINN) may be considered as a necessity as it allows the simulation of more complex cases with a significantly lower computational cost. However, doing so would also make it prone to any issue with the data quality, including its noise. This study would primarily focus on developing a special loss function in the PINN to allow an effective utilization of noisy data. However, a study regarding the data location and amount was also conducted in order to allow a better data utilization in PINN. This study was conducted on a lid-driven cavity flow at Re = 200, 1000, and 5000 with a dataset of less than 100 velocity data and a maximum noise of 10% of the maximum velocity. The results show that by ensuring the data are distributed in a certain configuration, it has zero noise, and by using as much data as possible, the computational cost of PINN can be significantly reduced compared to without using any data at all. For Re = 200, it is 7.4 faster by using data, and this speedup is potentially higher for higher Re cases. For the noise in particular, it does not only make the PINN more inaccurate but also necessitate the usage of more data as this is the only way to make it more accurate. This issue though is capable to be solved with our new method, which only uses the data as an approximate solution, and the governing equation would figure out the details. This method was also shown to be capable to improve the PINN accuracy with the potential to almost completely eliminating the noise effect.

A new higher-order super compact finite difference scheme to study three-dimensional non-Newtonian flows

This work introduces a new higher-order accurate super compact (HOSC) finite difference scheme for solving complex unsteady three-dimensional (3D) non-Newtonian fluid flow problems. As per the author's knowledge, the proposed scheme is the first ever developed higher-order compact finite difference scheme to solve 3D non-Newtonian flow problems. The proposed scheme is fourth-order accurate in space and second-order accurate in time, utilizing only seven adjacent grid points at the (n+1)th time level for the finite difference discretization. A time-marching methodology is employed with pressure calculated via a pressure-correction strategy based on the modified artificial compressibility method. Using the power-law viscosity model, we tackle the benchmark problem of a 3D lid-driven cavity, systematically analyzing the varied rheological behavior of shear-thinning (n = 0.5), shear-thickening (n = 1.5), and Newtonian (n = 1.0) fluids across different Reynolds numbers (Re=1,50,100,200). Both Newtonian and non-Newtonian results are carefully investigated in terms of streamlines, velocity variation, pressure distributions, and viscosity contours, and the computed results are validated with the existing benchmark results. The findings demonstrate excellent agreement with the existing results. It is found that for shear-thinning fluid (n = 0.5), u velocity is higher near the top moving wall than the case of Newtonian (n = 1.0) and shear-thickening fluid (n = 1.5) for all Re values. This extensive analysis, using the new HOSC scheme, not only increases our understanding of non-Newtonian fluid behavior but also provides a robust foundation for future research and practical applications.

A multiscale stochastic particle method based on the Fokker-Planck model for nonequilibrium gas flows

The Fokker-Planck (FP) model characterizes the intermolecular collisions as continuous stochastic processes in velocity space. Compared to the Direct Simulation Monte Carlo (DSMC) method, which is grounded in the Boltzmann equation, the stochastic particle method based on the FP model, referred to as the SP-FP method, demonstrates a relative improvement in computational efficiency. However, the traditional SP-FP method, utilizing operator splitting scheme, encounters analogous limitations to DSMC, thereby limiting the application of large temporal-spatial discretization for simulating near-continuum and continuum flows. In this study, two critical advancements are introduced to improve the traditional SP-FP method. First, the exact integration scheme of the ellipsoidal-statistical FP (ESFP) model is derived, enabling the replication of the theoretical solutions for homogeneous relaxation with any time step. Furthermore, leveraging the integration scheme of the ESFP model, we propose a Multiscale Stochastic Particle (MSP) method. The MSP method quantifies the numerical errors stemming from the operator splitting scheme and corrects them in the relaxation step. This capability allows the MSP method to be employed with larger temporal-spatial discretization for inhomogeneous problems, achieving second-order accuracy while retaining implementation simplicity. Application of the MSP method to various benchmark problems, including Couette flow, Fourier flow, Poiseuille flow, Sod tube flow, Lid-driven cavity flow, and flow through a slit, demonstrates its promise for simulating multiscale nonequilibrium gas flows with high computational efficiency and accuracy.

Simulation of Bingham–Papanastasiou model within trapezoidal cavity with mixed convection effects

The study of Bingham–Papanastasiou fluids is conducted in lid-driven cavity with consideration of viscous dissipation. The upper and left wall of the cavity is cold while other walls are insulated. Numerical simulations are conducted to study the isotherms, temperature profile, local and average Nusselt number. The main focus of work is to analyse the behaviour of heat transfer within trapezoidal cavity. The governing system of nonlinear dimensionless partial differential equations is analysed by using PDE solver of finite element method in COMSOL. The analysis is carried out for different parameters like Reynolds number, Bingham parameter, stress growth parameter, Eckert number and Prandtl number. It is observed that impact of Bingham parameter on temperature variation is negligible while the impact of stress parameter leads to the reduction in temperature within cavity. The novelty of this work is that no work is done for the case of trapezoidal cavity where Bingham–Papanastasiou fluid behaviour is observed under the consideration of viscous dissipation and mixed convection.

Least-squares pressure recovery in reduced order methods for incompressible flows

In this work, we introduce a method to recover the reduced pressure for Reduced Order Models (ROMs) of incompressible flows. The pressure is obtained as the least-squares minimum of the residual of the reduced velocity with respect to a dual norm. We prove that this procedure provides a unique solution whenever the full-order pair of velocity-pressure spaces is inf-sup stable. We also prove that the proposed method is equivalent to solving the reduced mixed problem with reduced velocity basis enriched with the supremizers of the reduced pressure gradients. Optimal error estimates for the reduced pressure are obtained for general incompressible flow equations and specifically, for the transient Navier-Stokes equations. We also perform some numerical tests for the flow past a cylinder and the lid-driven cavity flow which confirm the theoretical expectations, and show an improved convergence with respect to other pressure recovery methods.

Efficient finite element strategy using enhanced high-order and second-derivative-free variants of Newton's method

In this work, we propose a stable finite element approximation by extending higher-order Newton's method to the multidimensional case for solving nonlinear systems of partial differential equations. This approach relies solely on the evaluation of Jacobian matrices and residuals, eliminating the need for computing higher-order derivatives. Achieving third and fifth-order convergence, it ensures stability and allows for significantly larger time steps compared to explicit methods. We thoroughly address accuracy and convergence, focusing on the singular p-Laplacian problem and the time-dependent lid-driven cavity benchmark. A globalized variant incorporating a continuation technique is employed to effectively handle high Reynolds number regimes. Through two-dimensional and three-dimensional numerical experiments, we demonstrate that the improved cubically convergent variant outperforms others, leading to substantial computational savings, notably halving the computational cost for the lid-driven cavity test at large Reynolds numbers.

Accuracy Verification of a 2D Adaptive Mesh Refinement Method by the Benchmarks of Lid-Driven Cavity Flows with an Arbitrary Number of Refinements

Accuracy Verification of a 2D Adaptive Mesh Refinement Method by the Benchmarks of Lid-Driven Cavity Flows with an Arbitrary Number of Refinements

Unified Gas Kinetic Simulations of Lid-Driven Cavity Flows: Effect of Compressibility and Rarefaction on Vortex Structures

Unified Gas Kinetic Simulations of Lid-Driven Cavity Flows: Effect of Compressibility and Rarefaction on Vortex Structures

Numerical analysis of magnetohydrodynamic mixed convection and entropy generation in a curvelinear lid-driven cavity with carbon nanotubes and an adiabatic cylinder

Mixed convection convection is a vital subject and it is beneficial in many engineering applications. The current paper addresses this subject with a novel geometry and very vital variables including magnetohydrodynamic influences on the forced/free convection as well as the reproduction of irreversibilities in an enclosure filled with water/carbon nanotubes (CNT) and a nonadiabatic cylinder. The top wall is split from the middle and moves in different directions to drive the isotherms which are generated from the bottom wall and cold from the vertical surfaces. The numerical analysis was carried out using finite element method; the variables are Reynolds number (40–200), Richardson number (0.01–10), Hartmann number (0–62), inclined magnetohydrodynamic angle (0–60), volume concentration (0–0.08) while Prandtl number has kept constant at 6.2. The results show that the transformation of heat, as well as the fluid flow, are largely influenced by the change of variables, where increasing Reynolds number, Richardson number enhances heat and increases the flow circulation. Furthermore, heat transfer enhances by 57 % when increasing Ri from 0.1 to 10 at Re=41 and this enhancement increases to 62.5 % at Re = 200. Furthermore, increasing the concentration of the carbon nanotube can cause heat transfer but decrease the circulation of the fluid. In contrast, the transfer of heat as well as the flow streams are remarkably decreased with the increase of the Hartmann at zero inclination angle; however, the value of the Nusselt average increases with the increase of the inclination angle. Moreover, the value of Nusselt average decreses by 34.7 % when increasing Ha from 0 to 62 at Re = 200. Furthermore, the total entropy generation increases as Richardson number, Reynolds number, and volume concentration increase; in contrast, detraction with the rise of the MHD.

Sensitivity study on convective heat transfer in a driven cavity with star-shaped obstacle and hybrid nanofluid using response surface methodology

Sensitivity analysis is significant for understanding and measuring the impact of various parameters and input variables on heat transfer phenomena. The main objective of the current work is to examine the sensitivity of a numerical analysis of mixed convection in a lid-driven square cavity with a magnetic field. The cavity also contains a heated, star-shaped obstacle and is filled with a hybrid nanofluid. The sensitivity analysis was conducted employing the statistical response surface methodology (RSM), while the numerical simulations used the Galerkin weighted residual finite element approach to solve the governing PDEs. The study investigates the impacts of four dimensionless factors: Ri, Re, Ha, and ϕ. The numerical observation was made that there exists an upward trend between the average heat transfer rate with Ri, Re, and ϕ, while there exists a downward trend with Ha. Furthermore, the average heat transfer rate increases by almost half (49.54 %) when ϕ increases from 1 % to 10 % and decreases by 5.97 % when the Ha increases from 0 to 60. Finally, the statistical investigation of the current model and testing techniques imply that R2 values for the response function are high (98.72 %), suggesting that this model is appropriate for estimating Nu.

A parameter-free particle relaxation technique for smoothed particle hydrodynamics

In this paper, we present a parameter-free particle relaxation technique to improve the accuracy and stability of smoothed particle hydrodynamics (SPH). Instead of imposing a background pressure, particles are regularized following the criteria of 0th-order consistency, i.e., the gradient of a constant to be zero. Specifically, the modifications of particles' position are solved by a gradient decent method according to the error between zero value and the gradient of a constant. This modification decreases the integration error and leads a more uniform particles distribution. A set of challenging benchmarks including lid-driven cavity flow, Taylor-Green vortex, FSI (fluid-solid interaction) problem, 2D (two-dimensional) dam-break case, and water exit of a cylinder are investigated to validate the effectiveness of the present technique for addressing the well-known tensile instability and particle clumping problems. Finally, the study of 3D (three-dimensional) dam-break against an obstacle demonstrates the stability and versatility of the present method.

Greedy identification of latent dynamics from parametric flow data

Projection-based reduced-order models (ROMs) play a crucial role in simplifying the complex dynamics of fluid systems. Such models are achieved by projecting the Navier-Stokes equations onto a lower-dimensional subspace while preserving essential dynamics. However, this approach requires prior knowledge of the underlying high-fidelity model, limiting its effectiveness when applied to black-box data. This article introduces a novel, non-intrusive, data-driven method–Greedy Identification of Latent Dynamics (GILD)–for constructing parametric fluid ROMs. Unlike traditional methods, GILD constructs models directly from data, without relying on specific high-fidelity model information. It also employs interpolation within the manifold R∗N×q/Oq to accommodate parameter variability. Numerical experiments on various fluid dynamics scenarios, including lid-driven cavity flow, flow past a cylinder with varying Reynolds number, and Ahmed body flow with variable geometry, demonstrate GILD’s robust performance across both training and unseen parameter values. GILD’s ability to accurately capture system dynamics and its adaptability to diverse data sources highlight its potential as a powerful tool for constructing parametric reduced-order models in an easy and general way for complex fluid dynamics and beyond.

Machine learning in viscoelastic fluids via energy-based kernel embedding

The ability to measure differences in collected data is of fundamental importance for quantitative science and machine learning, motivating the establishment of metrics grounded in physical principles. In this study, we focus on the development of such metrics for viscoelastic fluid flows governed by a large class of linear and nonlinear stress models. To do this, we introduce energy-compatible families of kernel functions corresponding to a given viscoelastic stress model. Each kernel implicitly embeds flowfield snapshots into a Reproducing Kernel Hilbert Space (RKHS) in which distances and angles are computed and whose squared norm equals the total mechanical energy. Additionally, we present a solution to the preimage problem for these kernels, enabling accurate reconstruction of flowfields from their RKHS representations. Through numerical experiments on an unsteady viscoelastic lid-driven cavity flow, we demonstrate the utility of energy-compatible kernels for extracting energetically-dominant coherent structures in viscoelastic flows across a range of Reynolds and Weissenberg numbers. Specifically, the features extracted by Kernel Principal Component Analysis (KPCA) of flowfield snapshots using energy-compatible kernel functions yield reconstructions with superior accuracy in terms of mechanical energy compared to conventional methods such as ordinary Principal Component Analysis (PCA) with naïvely-defined state vectors or KPCA with ad-hoc choices of kernel functions. Our findings underscore the importance of principled choices of metrics in both scientific and machine learning investigations of complex fluid systems.

Lid-Driven Cavity Flow Containing a Nanofluid

In this paper, we consider the flow of a nanofluid in an enclosed lid-driven cavity using a single-phase model. Two cases are considered: one in which the top and bottom walls are kept at adiabatic conditions, and a second case in which the left- and right-side walls are kept in adiabatic conditions. The impact of different viscosity models on the mixed convection heat transfer is examined, and numerical methods are used to obtain solutions for the Navier–Stokes equations for various parameter ranges. Using our robust methods, we are able to obtain novel solutions for large Reynolds numbers and very small Richardson numbers. Using water as the base fluid and aluminium oxide nanoparticles, our results suggest that heat transfer enhancement occurs with increasing particle concentration and decreasing Richardson numbers. There are also significant differences depending on the viscosity model used in terms of the impact of reducing corner recirculation regions in the cavity.

A stochastic Galerkin lattice Boltzmann method for incompressible fluid flows with uncertainties

Efficient modeling and simulation of uncertainties in computational fluid dynamics (CFD) remains a crucial challenge. In this paper, we present the first stochastic Galerkin (SG) lattice Boltzmann method (LBM) built upon the generalized polynomial chaos (gPC). The proposed method offers an efficient and accurate approach that depicts the propagation of uncertainties in stochastic incompressible flows. Formal analysis shows that the SG LBM preserves the correct Chapman–Enskog asymptotics and recovers the corresponding macroscopic fluid equations. Numerical experiments, including the Taylor–Green vortex flow, lid-driven cavity flow, and isentropic vortex convection, are presented to validate the solution algorithm. The results demonstrate that the SG LBM achieves the expected spectral convergence and the computational cost is significantly reduced compared to the sampling-based non-intrusive approaches, e.g., the routinely used Monte Carlo method. We obtain a speedup factor of 5.72 compared to Monte Carlo sampling in a randomized two-dimensional Taylor–Green vortex flow test case. By leveraging the accuracy and flexibility of LBM and the efficiency of gPC-based SG, the proposed SG LBM provides a powerful framework for uncertainty quantification in CFD practice.

Lagrangian Split-Step Method for Viscoelastic Flows.

This research addresses and resolves current challenges in meshless Lagrangian methods for simulating viscoelastic materials. A split-step scheme, or pressure Poisson reformulation of the Navier-Stokes equations, is introduced for incompressible viscoelastic flows in a Lagrangian context. The Lagrangian differencing dynamics (LDD) method, which is a thoroughly validated Lagrangian method for Newtonian and non-Newtonian incompressible flows, is extended to solve the introduced split-step scheme to simulate viscoelastic flows based on the Oldroyd-B constitutive model. To validate and evaluate the new method's capabilities, the following benchmarks were used: lid-driven cavity flow, droplet impact response, 4:1 planar sudden contraction, and die swelling. These findings highlight the LDD method's effectiveness in accurately simulating viscoelastic flows and capturing large deformations and memory effects. Even though the extra stress was directly modeled without any regularization approach, the method produced stable simulations for high Weissenberg numbers. The stability and performance of the the Lagrangian numerics for complex temporal evolution of material properties and stress responses encourage its use for industrial problems dealing with polymers.