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Highly-accurate solutions for the lid-driven cavity flow are computed by a Chebyshev collocation method. Accuracy of the solution is achieved by using a substraction method of the leading terms of the asymptotic expansion of the solution of the Navier–Stokes equations in the vicinity of the corners, where the velocity is discontinuous. Critical comparison with former numerical experiments confirms the high-accuracy of the method, and extensive results for the flow at Reynolds number Re=1000 are presented.

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... The steady lid-driven cavity flow serves as a popular computational benchmark, due to the combination of simple geometry and numerical challenges due to the discontinuous boundary conditions (Botella and Peyret, 1998;Gelfgat, 2006;Hansen and Kelmanson, 1994;Gupta et al., 1981). At the same time, lid-driven cavity flows exhibit fundamental fluid-dynamical phenomena such as an infinite sequence of viscous corner eddies (Moffatt, 2019(Moffatt, , 1964Kalita et al., 2018), hydrodynamic instabilities (Albensoeder et al., 2001;Kuhlmann and Albensoeder, 2014, and others), and finite-size Lagrangian coherent structures of advected particles (Romanò et al., 2019a;Wu et al., 2021Wu et al., , 2023. ...

... The solver was verified and the grid convergence was tested by computing a steady twodimensional lid-driven cavity flow (S −1 = 0) starting from rest with u(t = 0) = 0 and Table 1: Extrema [u min (x = 0), v max (y = 0), v min (y = 0)] of the velocity components and their respective locations (y min , x max , x min ) along the centerlines for the steady flow in a lid-driven square cavity at Re = 100 computed without (NR) and with (R) regularization of the lid velocity (see text) according to (8). The reference data of Botella and Peyret (1998) ...

... where j enumerates all nodal points, the solution was compared to the benchmark spectral data of Botella and Peyret (1998). As shown in Table 1 the values of the velocity extrema along the centerlines of the cavity agree with the benchmark data up to O(10 −6 ) for Re = 100. ...

The Lagrangian transport in the laminar incompressible flow in a two-dimensional square cavity driven by a harmonic tangential oscillation of the lid is investigated numerically for a wide range of Reynolds and Strouhal numbers. The topology of fluid trajectories is analyzed by stroboscopic projections revealing the co-existence of chaotic trajectories and regular Kolmogorov--Arnold--Moser tori. The pathline structure strongly depends on the Reynolds number and the oscillation frequency of the lid. Typically, most pathlines are chaotic when the oscillation frequency is small, with few KAM tori being strongly stretched along instantaneous streamlines of the flow. As the frequency is increased the fluid motion becomes more regular and the size of the KAM tori grows until, at high frequencies, they resemble streamlines of a mean flow.

... Our results, obtained with various B-spline space-based discretizations for three consecutive mesh refinement levels h ∈ [1/32, 1/64, 1/128] of unstretched meshes, are compared to classical reference results from the literature such as those of Ghia [65] using a second-order upwind finite difference method on a stretched mesh with 192 2 grid points. Moreover, additional comparisons are done with highly accurate, spectral method-based (Chebyshev Collocation) solutions of Botella [20] that show convergence up to seven digits. Furthermore, whenever comparable data is provided, results of two recently published articles [54,144], both applying IGA to the cavity flow problem, are addressed. ...

... Starting off with the lid-driven cavity flow problem including its regularized version, we have shown that the approximated flow attributes are very well comparable with reference results partially obtained with a highly accurate spectral (Chebyshev Collocation) method [20]. Moreover, we have extended our view to global quantities such as kinetic energy and enstrophy, and have provided results which are in very good agreement with reference results obtained with other approaches such as a Q 2 P 1 finite element discretization [112] and a high order finite difference scheme utilized in [25]. ...

... The FSI 1 problem features a very low Reynolds number (20). This yields a laminar flow pattern and the solution tends to a "steady state" as t → ∞. ...

... Both as a numerical benchmark problem and as a test bed for the investigation of specific physical phenomena, it has been used. There are more than 1800 results returned when you search the term "lid-driven" on the Web of Science [7] [8]. A review on lid-driven cavity flows seems to be warranted for the aforementioned reasons, in addition to the rapid development of this area of research. ...

... Considering that approximately 20 years have elapsed since Shankar and Deshpande published an introduction of the topic, this seems to be the case. Following the first numerical investigations conducted by Kawaguti and Burggraf, the pursuit of efficiency and accuracy got underway with the work of Ghia [8] [9] [10], and Schreiber, who computed the steady two-dimensional flow for Reynolds numbers up to 10 4 in a square cavity that was surrounded by three rigid walls and a lid that moved at a constant velocity. This was the beginning of the quest for efficiency and accuracy. ...

... (11) in term of variables (u,v & p) [8] [30] [31]. The discretized equations with non-dimensional form of variables are shown below (1). ...

... In the literature, there has been a considerable body on numerical calculations of twodimensional flows in a single-lid-driven cavity treated square and some rectangular geometry configurations over a wide range of Reynolds numbers. Their efficient analyses emphasize multiple aspects where their topics can be titled as: the analysis of steady flow with the evolving streamline topology [10][11][12][13][14][15][16][17], the analysis of the flow instabilities [18][19][20][21] and the study of the onset of the flow unsteadiness (Hopf bifurcation) and transition to chaos [22][23][24][25][26]. A series of practical experiments have also been performed [27][28][29] for cavities with different third lengths dimensions. ...

... To increase the reliability of the present code, brief literature set from the three-driving mode is considered. The well-known calculations by Ghia et al. [10], Botella and Peyret [12], Erturk [14], and Abdelmigid et al. [16] are addressed for a single-sided lid-driven cavity. Next, Lemée et al. [39] and Perumal [41] are taken for a two-sided lid-driven cavity and Kamel et al. [50] for a three-sided lid-driven cavity. ...

... Vortices Properties and location Present Ghia et al. [10] Botella and Peyret [12] E r t u r k [ 14] Abdelmigid et al. [16] PV give respectively a maximum relative deviation of stream-function and vorticity of 0.56% and 0.55% for primary vortices and 3.20% and 4.54% for secondary vortices. Consistent with the previously published works, present results are in good agreement except for errors resulting from mesh nonuniformity and the variety of numerical techniques. ...

This paper reports a numerical investigation of the steady two-dimensional incompressible flow in a three-sided lid-driven cavity of unit aspect ratios (Г = 1). The two opposite horizontal walls move in parallel and antiparallel motions, while the left vertical sidewall moves upwards and downwards. The right vertical sidewall is stationary. A detailed analysis of the fluid flow has been carried out with the finite volume method for a Reynolds number up to 5000 using a fine mesh resolution, whereas the coupled algorithm has been employed to handle the pressure–velocity coupling. The results are displayed in terms of stream-function contours, fluid properties, and velocity profiles and have indicated a good agreement with the set of literature. Among the three driving processes considered, a most complex topological flow pattern has been shown with the antiparallel-downwards case. This is embodied by a significant amount of robustness induced in three opposing directions leading to multiple changes of streamline patterns. This is accompanied by high rotation rates of secondary vortices in the near-wall regions.

... Results for final vorticity ω along moving boundary are shown in Fig. 7. Good agreement with the results of [65] can be found. For further verification, we compare the final pressure distribution of our results with those presented in [66,67]. Noted that their lid moving direction was the opposite of ours. ...

... The final distributions of the pressure are shown in Fig. 9. The pattern of our pressure distribution of Re 1000 is quite similar to those presented in [66,67]. Noted they Analysis of Navier-Stokes equations did not present the pressure distribution of Re 400. ...

A meshless numerical model is developed for the simulation of two-dimensional incompressible viscous flows. We directly deal with the pressure–velocity coupling system of the Navier–Stokes equations. With an efficient time marching scheme, the flow problem is separated into a series of time-independent boundary value problems (BVPs) in which we seek the pressure distribution at discretized time instants. Unlike in conventional works that need iterative time marching processes, numerical results of the present model are obtained straightforwardly. Iteration is implemented only when dealing with the linear simultaneous equations while solving the BVPs. The method for solving these BVPs is a strong form meshless method which employs the local polynomial collocation with the weighted-least-squares (WLS) approach. By embedding all the constraints into the local approximation, i.e. ensuring the satisfaction of governing equation at both the internal and boundary nodes and the satisfaction of the boundary conditions (BCs) at boundary points, this strong form method is more stable and robust than those just collocate one boundary condition at one boundary node. We innovatively use this concept to embed the satisfaction of the continuity equation into the local approximation of the velocity components. Consequently, their spatial derivatives can be accurately calculated. The nodal arrangement is quite flexible in this method. One can set the nodal resolution finer in areas where the flow pattern is complicated and coarser in other regions. Three benchmark problems are chosen to test the performance of the present novel model. Numerical results are well compared with data found in reference papers.

... This particular problem has been studied by numerous researchers over the years, and a variety of different approaches have been employed. [43][44][45][46][47][48] In the present article, the work of Ghia et al. 43 will be used as a benchmark for the velocity profiles and the streamline plots, while the produced pressure field will be compared with the results of Hou et al. 44 and Botella and Peyret. 45 The square cavity initially contains idle fluid, while the upper surface of the fluid has a constant horizontal velocity. ...

... [43][44][45][46][47][48] In the present article, the work of Ghia et al. 43 will be used as a benchmark for the velocity profiles and the streamline plots, while the produced pressure field will be compared with the results of Hou et al. 44 and Botella and Peyret. 45 The square cavity initially contains idle fluid, while the upper surface of the fluid has a constant horizontal velocity. The size of the cavity's sides is L, and the top boundary moves with a constant velocity U 0 . ...

In the present work, we present a new version of the pressure‐based Implicit Potential (IPOT) method for incompressible flows, which can be applied on a fully collocated mesh. The new version combines the IPOT algorithm with the Rhie and Chow (RC) technique, to produce solutions on collocated grids that are free of spurious pressure modes. The IPOT‐RC method retains all the benefits of the original algorithm, i.e. explicit velocity‐pressure coupling, easy implementation and reduced iteration time, without requiring a special grid topology. The presentation of the IPOT‐RC method, is accompanied by an extensive discussion on the cause of the spurious oscillations in zero‐div problems in general, and a possible cure that is linked to the Rhie and Chow technique. The IPOT‐RC method is validated through several benchmark problems including the lid‐driven cavity flow, flow over a backward facing step and Direct Numerical Simulation (DNS) of turbulent channel flow.

... Since the LDC is a canonical problem with well-established and easily implementable boundary conditions, there are numerous benchmark results present in the literature. 4,[7][8][9]15 The present work aims to provide a benchmark set of results for a shallow rectangular LDC subjected to a dual forced convection. ...

Numerical investigation of a compressible fluid in a two-dimensional rectangular lid-driven cavity (LDC) with a vertical temperature gradient is performed by solving the compressible Navier-Stokes equation. Here, we explore the role of aspect ratio (AR) (width/height) on the vorticity dynamics and redistribution by considering three ARs of 1:1, 2:1, and 3:1. The onset and propagation of the instability are explored via time-resolved and instantaneous distributions of vorticity, time-series of streamwise velocity, and its associated spectra. The flow physics reveal that the precessing vortical structures in certain square sub-cells of the rectangular LDC resemble that of orbital motion with a primary core eddy surrounded by gyrating satellite vortices, typical of a supercritical flow in a square LDC. Upon increasing the AR, there is a major shift in the vorticity transfer from the top right corner (acting as the source of maximum vorticity generation) toward the left square sub-cells in the domain. This is further aided by the convective motion due to the imposed destabilizing vertical thermal gradient. The spectra demonstrate that a multi-periodic, chaotic flow is the consistent flow feature for the rectangular LDC for Re ¼ 5500, irrespective of the AR. The compressible enstrophy budget of the rectangular LDC with varying AR is computed for the first time. This shows the dominance of the baroclinic vorticity over the viscous diffusion terms, which was conceived of as the major contributor to the creation of rotational flow structures.

... The lid-driven cavity flow as a benchmark problem has attracted much attention in PINNs [13] . The Reynolds number Re is the characteristic parameter of the cavity flow, and it is widely accepted that the cavity flow is steady when Re < 5 000 [14] . ...

Physics-informed neural networks (PINNs) are proved methods that are effective in solving some strongly nonlinear partial differential equations (PDEs), e.g., Navier-Stokes equations, with a small amount of boundary or interior data. However, the feasibility of applying PINNs to the flow at moderate or high Reynolds numbers has rarely been reported. The present paper proposes an artificial viscosity (AV)-based PINN for solving the forward and inverse flow problems. Specifically, the AV used in PINNs is inspired by the entropy viscosity method developed in conventional computational fluid dynamics (CFD) to stabilize the simulation of flow at high Reynolds numbers. The newly developed PINN is used to solve the forward problem of the two-dimensional steady cavity flow at Re = 1 000 and the inverse problem derived from two-dimensional film boiling. The results show that the AV augmented PINN can solve both problems with good accuracy and substantially reduce the inference errors in the forward problem.

... Again, it can be noted that all the produced profiles are in well agreement with their correspondents from the references. These works also compare their results with those from Ghia et al. [1], Bruneau and Saad [33], and Botella and Peyret [34]. This way, the present results are also in well agreement with those. ...

An alternative approach to solve the steady-state incompressible Navier-Stokes equations using the multigrid (MG) method is presented. The mathematical model is discretized using the finite volume method with second-order approximation schemes in a uniform collocated (nonstaggered) grid. MG is employed through a full approximation scheme-full MG algorithm based on V-cycles. Pressure-velocity coupling is ensured by means of a developed modified SIMPLEC algorithm which uses independent V-cycles for relaxing the pressure-correction and momentum equations. The coarser grids are used only internally in these cycles. All other original SIMPLEC steps can be performed only on the finest grid of the current full MG level. The model problem of the lid-driven flow in the unitary square cavity is used for the tests of the numerical model. Computational performance is measured through error and residual decays and execution times. Good performances were obtained for a wide range of Reynolds numbers, with speedups of orders as high as Oð10 3 Þ: Linear relationships between execution times and grid sizes were observed for low and high Re values (Re ¼ 0:1,

... The final PDE used to test the method is the incompressible Navier-Stokes non-stationary lid-driven cavity flow. This is a setup for studying fundamental aspects of confined fluid-flows [52]. The solution field is a 3-dim. ...

We introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning probability measures to the spatial domain, which allows us to treat collocation grids probabilistically as random variables to be marginalised out. Adapting this spatial statistics view, we solve forward and inverse problems for parametric PDEs in a way that leads to the construction of Gaussian process models of solution fields. The implementation of these random grids poses a unique set of challenges for inverse physics informed deep learning frameworks and we propose a new architecture called Grid Invariant Convolutional Networks (GICNets) to overcome these challenges. We further show how to incorporate noisy data in a principled manner into our physics informed model to improve predictions for problems where data may be available but whose measurement location does not coincide with any fixed mesh or grid. The proposed method is tested on a nonlinear Poisson problem, Burgers equation, and Navier-Stokes equations, and we provide extensive numerical comparisons. We demonstrate significant computational advantages over current physics informed neural learning methods for parametric PDEs while improving the predictive capabilities and flexibility of these models.

... This test is classical, and numerous computations are available (see e.g. [4,5,20]); the reference used in this paper is a converged-in-space computation that can be found in [5]. ...

... This test is classical, and numerous computations are available (see e.g. [4,5,20]); the reference used in this paper is a converged-in-space computation that can be found in [5]. ...

We propose in this paper a discretization of the momentum convection operator for fluid flow simulations on quadrangular or hexahedral meshes. The space discretization is performed by the loworder nonconforming Rannacher-Turek finite element: the scalar unknowns are associated to the cells of the mesh, while the velocities unknowns are associated to the edges or faces. The momentum convection operator is of finite volume type, and its almost second order expression is derived by a MUSCL-like technique. The latter is of algebraic type, in the sense that the limitation procedure does not invoke any slope reconstruction, and is independent from the geometry of the cells. The derived discrete convection operator applies both to constant or variable density flows, and may thus be implemented in a scheme for incompressible or compressible flows. To achieve this goal, we derive a discrete analogue of the computation ui ($\partial$t($\rho$ui)+div($\rho$uiu) = 1 2 $\partial$t($\rho$u 2 i)+ 1 2 div($\rho$u 2 i u) (with u the velocity, ui one of its component, $\rho$ the density, and assuming that the mass balance holds) and discuss two applications of this result: firstly, we obtain stability results for a semi-implicit in time scheme for incompressible and barotropic compressible flows; secondly, we build a consistent, semi-implicit in time scheme that is based on the discretization of the internal energy balance rather than the total energy. The performance of the proposed discrete convection operator is assessed by numerical tests on the incompressible Navier-Stokes equations, the barotropic and the full compressible Navier-Stokes and the compressible Euler equations.

... In the past several decades, there were a lot of studies about the lid-driven square cavity flow using different numerical approaches, for example, the implicit multigrid method (BIMM) [1,2], the p-type finite element method [3], Chebyshev collocation method [4], stream function-velocity formulation [5,6], the moving immersed boundary method (MIBM) [7][8][9] and the lattice Boltzmann method [10][11][12][13], etc. Among the methods mentioned above, the most commonly used method is the lattice Boltzmann method. ...

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

... Although the geometry of the problem is very simple, the lid-driven cavity a primary vortex in the center and some secondary vortices at the corners will form with the increase of the Reynolds number (Re = LU 1 /ν). Actually, the lid-driven cavity flow, as a classic benchmark problem, has also been widely used to test the capacity of numerical methods [45,[68][69][70][71]. In this part, we conduct some numerical simulations of lid-driven cavity flows at different Reynolds numbers, and to ensure the incompressible condition is valid, the discrete velocity c = 10 is adopted to give a small Mach number (Ma = U 1 / √ ηc). ...

In this paper, a multiple-distribution-function lattice Boltzmann method (MDF-LBM) with a multiple-relaxation-time model is proposed for incompressible Navier-Stokes equations which are considered as coupled convection-diffusion equations. Through direct Taylor expansion analysis, we show that the Navier-Stokes equations can be recovered correctly from the present MDF-LBM, and additionally, it is also found that the velocity and pressure can be directly computed through the zero and first-order moments of the distribution function. Then in the framework of the present MDF-LBM, we develop a locally computational scheme for the velocity gradient in which the first-order moment of the nonequilibrium distribution is used; this scheme is also extended to calculate the velocity divergence, strain rate tensor, shear stress, and vorticity. Finally, we also conduct some simulations to test the MDF-LBM and find that the numerical results not only agree with some available analytical and numerical solutions but also have a second-order convergence rate in space.

... Finally, the velocity profiles across the cavity center and the pressure profiles along the central vertical line at Re ¼ 1000 for various K are compared in Fig. 14, together with benchmark data. 59,60 It can be observed that the velocity profiles for different K are in a good agreement with the reference data and have no evident difference except at the peaks. Meanwhile, an obvious difference of the pressure profiles can be found in the region near the bottom wall, especially when K > 3=16. ...

Bounce-back schemes represent the most popular boundary treatments in the lattice Boltzmann method (LBM) when reproducing the no-slip condition at a solid boundary. While the lattice Boltzmann equation used in LBM for interior nodes is known to reproduce the Navier-Stokes (N-S) equations under the Chapman-Enskog (CE) approximation, the unknown distribution functions reconstructed from a bounce-back scheme at boundary nodes may not be consistent with the CE approximation. This problem could lead to undesirable effects such as non-physical slip velocity, grid-scale velocity and pressure noises, the local inconsistency with the N-S equations, and sometimes even a reduction of the overall numerical-accuracy order of LBM. Here we provide a systematic study of these undesirable effects. We first derive the explicit structure of the mesoscopic distribution function for interior nodes. Then the bounce-back distribution function is examined to identify the hidden errors. It is shown that the relaxation parameters in the collision models play a key role in determining the magnitude of the hidden error terms, and there exists an optimal setting which can suppress or eliminate most of these undesirable effects. While the existence of this optimal setting is derived previously for unidirectional flows, here we show that this optimal setting can be extended to non-uniform flows under certain conditions. Finally, a systematic numerical benchmark study is carried out, including non-uniform and unsteady flows. It is shown that, in all these flows, our theoretical analyses of the hidden errors can guide us to significantly improve the quality of the simulation results.

... 3.1 Test Case 1: Lid-Driven Cavity. Test case 1 is based on the well-known stationary lid-driven cavity problem (see, e.g., Ref. [40]), for which we consider an extension with variable geometry. The challenge of this test case is to capture the different vortex topologies formed for different Reynolds numbers and different aspect ratios of the geometry. ...

We introduce Universal Solution Manifold Network (USM-Net), a surrogate model, based on Artificial Neural Networks (ANNs), which applies to differential problems whose solution depends on physical and geometrical parameters. We employ a mesh-less architecture, thus overcoming the limitations associated with image segmentation and mesh generation required by traditional discretization methods. Our method encodes geometrical variability through scalar landmarks, such as coordinates of points of interest. In biomedical applications, these landmarks can be inexpensively processed from clinical images. We present proof-of-concept results obtained with a data-driven loss function based on simulation data. Nonetheless, our framework is non-intrusive and modular, as we can modify the loss by considering additional constraints, thus leveraging available physical knowledge. Our approach also accommodates a universal coordinate system, which supports the USM-Net in learning the correspondence between points belonging to different geometries, boosting prediction accuracy on unobserved geometries. Finally, we present two numerical test cases in computational fluid dynamics involving variable Reynolds numbers as well as computational domains of variable shape. The results show that our method allows for inexpensive but accurate approximations of velocity and pressure, avoiding computationally expensive image segmentation, mesh generation, or re-training for every new instance of physical parameters and shape of the domain.

... = 1000 have been generated by Botella & Peyret [368]. The first results on the linear stability of a square lid-driven cavity flow were obtained by Poliashenko et al. [369] in 1995, followed by Fortin et al. [370] and Gervais et al. [371] in 1997. ...

This work focuses on the computation and stability analysis of both steady-state and time-periodic solutions with an emphasis on very high-dimensional systems such as the discrete Navier-Stokes equations. Our results are obtained with nekStab, a user-friendly open-source toolbox for global stability analysis based on Krylov methods and a time-stepper formulation. Our package nekStab inherits the flexibility and all the capabilities of the highly parallel spectral element-based open-source solver Nek5000, enabling the characterization of the stability of complex flow configurations and several post-processing options. The performances and accuracy of our toolbox are first illustrated using standard benchmarks from the literature before turning our attention to the persistent sequence of bifurcations in the wake of bluff bodies. Using a Newton-Krylov algorithm, unstable periodic orbits are computed and fully three-dimensional Floquet modes obtained, highlighting a sequence of bifurcations leading to the onset of quasi-periodic dynamics as well as the existence of subharmonic cascade before the onset of temporal chaos. The stability of a jet in crossflow is also investigated for a range of jet to crossflow velocity ratios. After the first bifurcation, we note a surprising change in the nature of the perturbations before the onset of quasi-periodic dynamics and chaos. Finally, we present a parametric study of the influence of the aspect ratio on the first bifurcation taking place in lid-driven cavity flows. We find that very large spanwise aspect ratios need to be considered in order to tend toward the results obtained for cavities homogeneous in the spanwise direction.

... The Vakilha kernel function with the EISPH method gives satisfactory results. To verify the approximated pressure field, we compare the pressure results with results of a finite volume-based software (STAR-CD) data [43] and Bottela's results [44]. The pressure profile is very close to the FV approach in Fig. 11(a) and, as well the resemblance of pressure contours to the contours of the Chebyshev collocation method is acceptable (Fig. 11(c)) ...

Multiphase flow is a challenging area of computational fluid dynamics (CFD) due to their potential large topological change and close coupling between the interface and fluid flow solvers. As such, Lagrangian meshless methods are very well suited for solving such problems. In this paper, we present a new fully explicit incompressible Smoothed Particle Hydrodynamics approach (EISPH) for solving multiphase flow problems. Assuming that the change in pressure between consecutive time-steps is small, due to small time steps in explicit solvers, an approximation of the pressure for following time-steps is derived. To verify the proposed method, several test cases including both single-phase and multi-phase flows are solved and compared with either analytical solutions or available literature. Additionally, we introduce a novel kernel function, which improves accuracy and stability of the solutions, and the comparison with a well-established quintic spline kernel function is discussed. For the presented benchmark problems, results show very good agreements in velocity and pressure fields and the interface-capturing with those in the literature. To the best knowledge of the authors, the EISPH method is presented for the first time for multiphase flow simulations.

... The Vakilha kernel function with the EISPH method gives satisfactory results. To verify the approximated pressure field, we compare the pressure results with results of a finite volume-based software (STAR-CD) data [43] and Bottela's results [44]. The pressure profile is very close to the FV approach in Fig. 11(a) and, as well the resemblance of pressure contours to the contours of the Chebyshev collocation method is acceptable (Fig. 11(c)) ...

... Being the benchmark model in incompressible fluid mechanics systems, researchers sought advantage on the lid-driven cavity problem mainly to compare and test the accuracy of favored numerical methods. Several researchers revisited the lid-driven model to validate and improve the novel numerical techniques [6][7][8] and to check the proposed error estimator for the numerical method [9]. Al-Amiri et al. [10] adopted Galerkin weighted residual method to resolve the conventional 2-D cavity model with the top horizontal well as the moving lid. ...

The main emphasis of the current work is to deal with the forced convection of magnetohydrodynamics (MHD) flow and its heat transfer due to force convection generated by a moving lid in a trapezoidal enclosure. Various cases of temperature at the surface of the circular obstacle inside the cavity are determined. The trapezoidal cavity has a moveable and partially heated top lid while the bottom wall is kept at a low temperature. The linearly inclined walls on the left and right sides of the cavity are both adiabatic. The finite element method is applied for computations validated with existing work. Numerical simulations are conducted to analyze this trapezoidal cavity model for various thermal conditions of the inner circular obstacle, various heated lengths (0 ≤ LH ≤ 1), various Reynolds number (100 ≤ Re ≤ 700), Richardson number (0.001 ≤ Ri ≤ 10) and Hartmann number (0 ≤ Ha ≤ 100). The entire analysis describes that high Reynolds numbers improve the thermal performance of liquid. However, moving lid generates fluid molecules specifically directed according to wall movement. The force convection phenomenon becomes more dominant as the Reynolds number and heated length increase. A cold circular obstacle resists the circulation of heat in the cavity whereas the local Nusselt number drops when the simultaneous effects of the moving lid force and heated sources move away from the surface.

... Test Case 1 is based on the well-known stationary liddriven cavity problem (see, e.g., [32]), for which we consider an extension with variable geometry. The challenge of this test case is to capture the different vortex topologies formed for different Reynolds numbers and different aspect ratios of the geometry. ...

We introduce Universal Solution Manifold Network (USM-Net), a novel surrogate model, based on Artificial Neural Networks (ANNs), which applies to differential problems whose solution depends on physical and geometrical parameters. Our method employs a mesh-less architecture, thus overcoming the limitations associated with image segmentation and mesh generation required by traditional discretization methods. Indeed, we encode geometrical variability through scalar landmarks, such as coordinates of points of interest. In biomedical applications, these landmarks can be inexpensively processed from clinical images. Our approach is non-intrusive and modular, as we select a data-driven loss function. The latter can also be modified by considering additional constraints, thus leveraging available physical knowledge. Our approach can also accommodate a universal coordinate system, which supports the USM-Net in learning the correspondence between points belonging to different geometries, boosting prediction accuracy on unobserved geometries. Finally, we present two numerical test cases in computational fluid dynamics involving variable Reynolds numbers as well as computational domains of variable shape. The results show that our method allows for inexpensive but accurate approximations of velocity and pressure, avoiding computationally expensive image segmentation, mesh generation, or re-training for every new instance of physical parameters and shape of the domain.

... Depending on the Reynolds number (based on lid speed and box dimension), this exhibits a range of laminar and turbulent flow, for which there is experimental data available. Given the relative simplicity of the domain, this has also been used extensively for DNS simulation, and this also provides significant validation resources [9]. ...

The standard lid-driven cavity test case is one of the most used validation cases in CFD. Whilst comparisons with experimental and particularly DNS simulations are possible, there is no analytical solution, and the case is ill-posed when considering the boundary conditions. A modified lid driven cavity (MLDC) case exists in the literature in which the lid velocity is non-uniform and which introduces a spatially varying body force, and for which there is a closed-form analytical solution to the Navier-Stokes equations which is a function of the Reynolds number. In this paper I present an implementation of the MLDC as a modification of the standard OpenFOAM case, using run time coding for the boundary conditions and fvOptions, and show how convergence to the solution is affected by numerical parameters ofsimpleFoam such as choice of matrix inversion. The existance of an analytical solution also allows the investigation of the relation between the solver residual and the true solution error.

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relationship between the energy density and material properties for different applications which will help researchers working in this
area to get an insight into this new emerging technology.

Hybrid joints provide flexibility in design for joining dissimilar materials i.e., metals and composites. Hybrid joints are increasingly being used in automobile, aircraft, marine, and construction applications because of their high strength to weight ratio. Many applications use adhesive bonding and mechanical fastening to join dissimilar adherends. The present work deals with the optimization of bond strength between dissimilar adherends such as aluminum and carbon fiber-reinforced polymer(CFRP). The better bond strength can be achieved by creating mechanical interlocking on the surfaces on adherends. Surface-treatment techniques such as phosphoric acid anodizing(PAA), NaOH etching, and resin pre-coating(RPC) were applied to aluminum adherend to create better
mechanical interlocking. Non-destructive testing (NDT) was used to ensure the uniformity of bond quality before mechanical testing. A numerical simulation model was developed to validate the experimental results.
Keywords: Hybrid joint, Carbon fiber-reinforced polymer(CFRP), Surface-treatments, Non-destructive testing (NDT).

Complex Concentrated Alloys (CCAs) are one of the most studied group of advanced materials. They provide immense scope for compositional design and microstructural tailoring to obtain superior properties that may emerge as promising solutions for variety of functional and structural requirements in the aerospace and automotive industries. In the present work, novel Fe20Cr20Mo20Nb30Ti10 CCA is presented as a potential material for elevated temperature structural applications as well as for marine environment. Synthesis of the CCA was carried out by the vacuum arc melting technique. The microstructural characterization, microhardness, and corrosion behavior of as-cast Fe20Cr20Mo20Nb30Ti10 CCA were studied. XRD analysis shows the alloy has a disordered BCC phase with a small amount of Laves phase. The measured density, lattice parameter, and microhardness of CCA are 7.82 g/cm3, 3.169 Å, and 825.31 Hv, respectively. A potentiodynamic polarization experiment in a 3.5 wt% NaCl solution and electrochemical impedance spectroscopy was used to investigate the corrosion behavior at room temperature. The Fe20Cr20Mo20Nb30Ti10 CCA exhibits superior corrosion resistance compared to 316L stainless steel.

Aluminium and its composites are notable materials that may be utilized to replace conventional steel materials while maintaining strength and safety. Because of its high strength-to-weight ratio, stiffness, and capacity to absorb energy, high entropy alloy reinforced composites are an emerging and novel class of materials. Stir squeeze casting with an ultrasonic transducer is used to develop the CoCrFeMnNi High entropy alloy reinforced Al6082 metal matrix composite. FESEM is used to examine the morphology and microstructure of composite and fractured surfaces. It is revealed from the FESEM micrograph that there is a uniform distribution of HEA particles in the metal matrix however, the fracture surface micrograph showed the governing mode of failure. Impact strength is highly reduced with the inclusion of HEA particles.

This article proposes a meshless, semi-explicit solution technique for time-dependent transient laminar mixed convection problems. The approach approximates both spatial and temporal derivatives using high-order differential quadrature rules, with spatial derivatives approximated using the radial basis function-based differential quadrature method (RBF-DQM) and temporal derivatives using the conventional global differential quadrature method (DQM). The method offers the unique ability to utilize DQM for both spatial and temporal derivatives of partial differential equations simultaneously. The RBF-DQM's meshless nature makes it suitable for irregular spatial domains, allowing for the analysis of problems with irregular geometries. The proposed algorithm was evaluated using transient laminar mixed convection simulations within both rectangular and irregular cavities, with results validated against benchmark solutions. The study concludes that the proposed algorithm is a reliable and accurate tool for the analysis of transient laminar mixed convection problems, with the added advantages of being easy to program and mesh-free, albeit with a need for suitable shape parameter selection.

The lattice Boltzmann method (LBM) is known for its capability to model complex fluid systems. Python is becoming one of the most valuable tools for scientific programming and prototyping of commercial software. In parallel, there is nearly no information on LBM simulation libraries in Python. The current study addresses the issue of Pythonbased LBM simulations. We have developed and tested the serial and two concurrent approaches (multiprocessing and multithreading) through 7 different versions of the LBM solvers for a range of the various numbers of nodes. Parallelization is also defined for three individual levels of domain decomposition, and RunTime profile data for entire solvers have been discussed comparatively in detail. Apart from the implementation strategy, a preliminary comparison is made between two data storage styles to investigate the effect of data shape on RunTime. In contrast with the literature, serial processing provides the fastest results for task execution of LBM simulation in Python due to the strengths of Python in data manipulation. One version of multithreaded solver also provides the same results with serial implementation; however, in conclusion, and due to the ease of serial programming, serial implementation of LBM in Python is defined as the most practical candidate.

The two-phase Navier-Stokes equations are involved in several industrial and environmental applications. However, their resolution poses many problems and becomes a major issue for mathematicians and engineers. In the context of incompressible two-phase flows, the challenges are two-fold : on the one hand, the velocity-pressure coupling induced by the incompressibility constraint and the high density and viscosity ratios lead to very ill-conditioned linear systems and, on the other hand, an inconsistent discretization between momentum transport and mass
transport lead to a divergence of the computational code in some situations. This work is devoted to the development and implementation of a direct simulation tool for two-phase flows taking into account all interfacial scales in order to meet our needs. In this context, an original 3D parallel fully-coupled solver allowing the direct coupling between velocity and pressure has been developed. This solver has been validated on problems using several analytical solutions, numerical benchmarks and experiments, regardless of the ratios of density and viscosity discontinuities across the interface. Good scalability associated with the fully-coupled solver has been
observed, for a number of processors ranging from 40 to 12800. A conservative scheme AMP (Algebraic Momentum Preserving) based on a consistent discretization of density and momentum has been constructed and validated. Quantitative and qualitative comparisons have shown the robustness of our fully-coupled solver combined with the new AMP scheme when facing complex problems.

Purpose
The purpose of this paper is to present a new discretisation scheme, based on equation-coupled approach and high-order five-point integrated radial basis function (IRBF) approximations, for solving the first biharmonic equation, and its applications in fluid dynamics.
Design/methodology/approach
The first biharmonic equation, which can be defined in a rectangular or non-rectangular domain, is replaced by two Poisson equations. The field variables are approximated on overlapping local regions of only five grid points, where the IRBF approximations are constructed to include nodal values of not only the field variables but also their second-order derivatives and higher-order ones along the grid lines. In computing the Dirichlet boundary condition for an intermediate variable, the integration constants are used to incorporate the boundary values of the first-order derivative into the boundary IRBF approximation.
Findings
These proposed IRBF approximations on the stencil and on the boundary enable the boundary values of the derivative to be exactly imposed, and the IRBF solution to be much more accurate and not influenced much by the RBF width. The error is reduced at a rate that is much greater than four. In fluid dynamics applications, the method is able to capture well the structure of steady highly non-linear fluid flows using relatively coarse grids.
Originality/value
The main contribution of this study lies in the development of an effective high-order five-point stencil based on IRBFs for solving the first biharmonic equation in a coupled set of two Poisson equations. A fast rate of convergence (up to 11) is achieved.

Cryoballoon ablation (CBA) is a cryo-energy based minimally invasive treatment procedure for patients suffering from left atrial (LA) fibrillation. Although this technique has proved to be effective, it is prone to reoccurrences and some serious thermal complications. Also, the factors affecting thermal distribution at the pulmonary vein-antrum junction that are critical to the treatment success is poorly understood. Computer modeling of CBA can resolve this issue and help understand the factors affecting this treatment. To do so, however, numerical challenges associated with the simulation of advection-dominant transport process must be resolved. Here, we describe the development of a thermal-hemodynamics computational framework to simulate incomplete occlusion in a patient-specific LA geometry during CBA. The modeling framework uses the finite element method to predict hemodynamics, thermal distribution, and lesion formation during CBA. An incremental pressure correction scheme is used to decouple velocity and pressure in the Navier-Stokes equation, whereas several stabilization techniques are also applied to overcome numerical instabilities. The framework was implemented using an open-source FE library (FEniCS). We show that model predictions of the hemodynamics in a realistic human LA geometry match well with measurements. The effects of cryoballoon position, pulmonary vein blood velocity and mitral regurgitation on lesion formation during CBA was investigated. For a -700C cryoballoon temperature, the model predicts lesion formation for gaps less than 2.5 mm and increasing efficiency of CBA for higher balloon tissue contact areas. The simulations also predict that lesion formation is not sensitive to variation in pulmonary vein blood velocity and mitral regurgitation. The framework can be applied to optimize CBA in patients for future clinical studies.

In this paper, we have developed new hybrid compact schemes for the simulation of stream function-velocity formulation of the two-dimensional incompressible Navier-Stokes equation. The first-order spatial derivatives are approximated by the optimized upwind compact scheme, and the Laplacian and biharmonic operators are discretized using fourth-order hybrid compact schemes. Moreover, we have also performed Fourier analysis to assess the resolution and added numerical diffusion properties of numerical schemes for stream function-velocity formulation of the linear Navier-Stokes equation. For time discretization, we have used explicit fourth-stage fourth-order Runge-Kutta method and hybrid filters. Furthermore, to validate the accuracy and efficiency of the schemes, several fluid flow problems, including a test problem with a non-homogeneous source term and a lid-driven cavity problem, are considered. Numerical results exhibit a great match to the results reported in the literature at lower computational cost with hybrid filters.

This paper deals with the formulation of the tridiagonal compact difference schemes for derivatives up to second‐order with boundary stencils on non‐uniform grids. A compact scheme for the first derivative with interpolation is also devised on staggered non‐uniform grids. The developed schemes with non‐uniform spacing transform to respective classical compact schemes for the case of uniform mesh spacing. The resolution, numerical diffusion, and anti‐diffusion features of the devised schemes are evaluated using global‐spectral analysis. Applications to the direct numerical simulation (DNS) of two‐dimensional lid‐driven cavity (LDC) flow governed by Navier‐Stokes equations and wave‐propagation following linear rotating shallow water (LRSWE) equations with variable grid‐spacing are discussed at different choices of parameters. Computed results are also compared with solutions available in the literature.

This article presents a novel stabilized finite element analysis for the transient Stokes model. The algebraic subgrid variational multiscale finite element scheme with dynamic subscales approach has been employed to arrive at the stabilized formulation. Both the coarse and the fine scale solutions are of time dependent nature and the unknown fine scale solution is completely eliminated in terms of the coarse scale solution during the derivation. This elimination results into the emergence of a new subgrid multiscale stabilized formulation in the transient framework. This formulation facilitates the theoretical derivations of the robustness properties of the scheme. The fully implicit backward Euler scheme has been applied for the time discretization. Here we have analyzed the stability property of the approximate solution. As well as a detailed derivation of the a posteriori error estimate has been presented. The scheme is numerically validated for a benchmark problem and appropriate numerical experiments have been carried out to verify the theoretically established order of convergence results.

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

The complexity of scale interactions, arising from the increasing number of dynamically active flow structures, is a well-known problem for the numerical modelling of high Reynolds number flows. Without doubts, this complexity is the main obstacle to the development of computationally affordable and physically interpretable models of complex flows. This research focuses on the nonlinear energy interactions across modes in reduced order Galerkin models of turbulent flows demonstrating a novel approach to automatically identify relevant interactions. This work is motivated by the key observation that, in the dynamics of high Reynolds number flows, not all the interactions have the same contribution to the energy transfer between flow structures. With the proposed work, we aim to develop a set of techniques to systematically select the dominant interactions in Galerkin models of turbulent flows, therefore identifying dominant triadic interactions. In the present work, two different approaches have been developed. First, a regression-based approach where the relevant interactions are identified a posteriori according to their relative strength. Second, an a priori approach, where a new set of basis functions, encoding the sparsity features of the flow, is generated. The key aspect of the latter approach is that the reduced-order model obtained by Galerkin projection onto the subspace spanned by the basis has sparse matrix coefficients without the need for any a posteriori evaluation. Both approaches have been tested on a set of flow configurations of increasing complexity. Results show that both approaches can identify the subset of dominant interactions preserving their physics throughout the sparsification process. In addition, further analysis showed that the a priori sparsification method preserves better the physics of triadic interactions, resulting in a better long term time stability and, therefore, should be preferred. Looking into the future, to scale up the a priori methodology to a more complex configuration some aspects need to be further investigated such as the role of the initial guess on the uniqueness of the result and its properties.

We propose an efficient, accurate and robust implicit solver for the incompressible Navier‐Stokes equations, based on a DG spatial discretization and on the TR‐BDF2 method for time discretization. The effectiveness of the method is demonstrated in a number of classical benchmarks, which highlight its superior efficiency with respect to other widely used implicit approaches. The parallel implementation of the proposed method in the framework of the deal.II software package allows for accurate and efficient adaptive simulations in complex geometries, which makes the proposed solver attractive for large scale industrial applications.

Des schémas de projection de type Chorin-Temam approximant les équations de Navier Stokes pour un fluide incompressible sont présentés. %La discrétisation spatiale utilisée est de type Tchebychev Collocation à grille unique, où la pression est approximée par des polynômes de degré moins élevé de deux unités que ceux approchant la vitesse (méthode $\PP_N-\PP_N-2$). De manière à obtenir un champ de pression non pollué par des modes parasites, deux discrétisations spatiales de collocation Tchebychev à grille unique, où la pression est approximée par des polynômes de degré moins élevé de deux unités que ceux approchant la vitesse, sont comparées. La résolution du problème de Navier-Stokes se réduit à la résolution successive d' équations de Helmholtz pour la vitesse et d'équations de type pseudo-Poisson, sans condition de Neumann, pour la pression. En utilisant des solution exactes, on vérifie que la précision spatiale des schémas construits est de type spectral, et que la précision temporelle est d'ordre deux ou trois, pour la vitesse ainsi que la pression.

Explicit solutions of two-dimensional, steady-state Navier-Stokes equations are derived in the neighborhood of sharp corners where a sliding wall meets a stationary wall and causes a mathematical singularity. These solutions are valid for small Reynolds numbers. A semi-analytic technique is used to derive these solutions. Some comparisons with numerical solutions are also carried out.

The computation of incompressible three-dimensional viscous flow is discussed. A new physically consistent method is presented for the reconstruction for velocity fluxes which arise from the mass and momentum balance discrete equations. This closure method for fluxes allows the use of a cell-centered grid in which velocity and pressure unknowns share the same location, while circumventing the occurrence of spurious pressure modes. The method is validated on several benchmark problems which include steady laminar flow predictions on a two-dimensional cartesian (lid driven 2D cavity) or curvilinear grid (circular cylinder problem at Re = 40), unsteady three-dimensional laminar flow predictions on a cartesian grid (parallelopipedic lid driven cavity) and unsteady two-dimensional turbulent flow predictions on a curvilinear grid (vortex shedding past a square cylinder at Re = 22,000).

Some simple similarity solutions are presented for the flow of a viscous fluid near a sharp corner between two planes on which a variety of boundary conditions may be imposed. The general flow near a corner between plane boundaries at rest is then considered, and it is shown that when either or both of the boundaries is a rigid wall and when the angle between the planes is less than a certain critical angle, any flow sufficiently near the corner must consist of a sequence of eddies of decreasing size and rapidly decreasing intensity. The ratios of dimensions and intensities of successive eddies are determined for the full range of angles for which the eddies exist. The limiting case of zero angle corresponds to the flow at some distance from a two-dimensional disturbance in a fluid between parallel boundaries. The general flow near a corner between two plane free surfaces is also determined; eddies do not appear in this case. The asymptotic flow at a large distance from a corner due to an arbitrary disturbance near the corner is mathematically similar to the above, and has comparable properties. When the fluid is electrically conducting, similarity solutions may be obtained when the only applied magnetic field is that due to a line current along the intersection of the two planes; it is shown that the effect of such a current is to widen the range of corner angles for which eddies must appear.

Chebyshev pseudospectral solutions of the biharmonic equation governing two-dimensional Stokes flow within a driven cavity converge poorly in the presence of corner singularities. Subtracting the strongest corner singularity greatly improves the rate of convergence. Compared to the usual stream function/ vorticity formulation, the single fourth-order equation for stream function used here has half the number of coefficients for equivalent spatial resolution and uses a simpler treatment of the boundary conditions. We extend these techniques to small and moderate Reynolds numbers. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/44983/1/10915_2005_Article_BF01061264.pdf

We introduce a new filter or sum acceleration method which is the complementary error function with a logarithmic argument. It was inspired by the large order asymptotics of the Euler and Vandeven accelerations, which we show are both proportional to the erfc function also. We also show the relationship between Vandeven's filter, the Erfc-Log filter and the "lagged-Euler" method. The theory for the last of these is used to predict a spatially-varying optimal order for filtering of a Fourier or Chebyshev series for a function with a discontinuity, front or shock.

In developing this book, we decided to emphasize applications and to provide methods for solving problems. As a result, we limited the mathematical devel opments and we tried as far as possible to get insight into the behavior of numerical methods by considering simple mathematical models. The text contains three sections. The first is intended to give the fundamen tals of most types of numerical approaches employed to solve fluid-mechanics problems. The topics of finite differences, finite elements, and spectral meth ods are included, as well as a number of special techniques. The second section is devoted to the solution of incompressible flows by the various numerical approaches. We have included solutions of laminar and turbulent-flow prob lems using finite difference, finite element, and spectral methods. The third section of the book is concerned with compressible flows. We divided this last section into inviscid and viscous flows and attempted to outline the methods for each area and give examples.

The steady laminar two-dimensional flow of a viscous incompressible fluid within a rectangular corner between two plane walls is studied. It is assumed that one of the walls is moving with uniform velocity in its plane and the other is at rest. In the case of very large Reynolds number, the stream function is obtained by the use of the boundary-layer successive approximation. The possibility of appearance of the eigensolutions which satisfy the homogeneous boundary conditions and cause the nonuniqueness of the approximation solution is also examined. In the case of very small Reynolds number, the stream function is determined to the term of second order.

The development of an efficient general purpose calculation procedure
for turbulent chemically reacting internal flows is discussed. The
approach is to solve the time averaged Navier-Stokes equations governing
the conservation of mass, momentum, and chemical species and two
turbulence quantities such as the kinetic energy of turbulence and its
rate of dissipation. The equations in such a set are highly nonlinear
and strongly coupled. Of particular mention is the coupling between the
pressure and velocity fields implicitly through the momentum and
continuity equations.

The driven-cavity problem, a renowned bench-mark problem of computational, incompressible fluid dynamics, is physically unrealistic insofar as the inherent boundary singularities (where the moving lid meets the stationary walls) imply the necessity of an infinite force to drive the flow: this follows from G.I. Taylor's analysis of the so-called scaper problem. Using a boundary integral equation (BIE) formulation employing a suitable Green's function, we investigate herein, in the Strokes approximation, the effect of introducing small “leaks” to replace the singularities, thus rendering the problem physically realizable, The BIE approach used here incorporates functional forms of both the asymptotic far-field and singular near-field solution behaviours, in order to improve the accuracy of the numerical solution. Surprisingly, we find that the introduction of the leaks effects notably the global flow field a distance of the order of 100 leak widths away from the leaks. However, we observe that, as the leak width tends to zero, there is exellent agreement between our results and Taylor's thus justifying the use of the seemingly unrealizble boundary conditions in the driven-cavity problem. We also discover that the far-field, asymptotic, closed-form solution mentioned above is a remarkably accurate representation of the flow even in the near-field. Several streamline plots, over a range of spatial scales, are presented.

First published in 1967, Professor Batchelor's classic work is still one
of the foremost texts on fluid dynamics. His careful presentation of the
underlying theories of fluids is still timely and applicable, even in
these days of almost limitless computer power. This reissue ensures that
a new generation of graduate students experiences the elegance of
Professor Batchelor's writing.

Numerical approaches are discussed, taking into account general
equations, finite-difference methods, integral and spectral methods, the
relationship between numerical approaches, and specialized methods. A
description of incompressible flows is provided, giving attention to
finite-difference solutions of the Navier-Stokes equations,
finite-element methods applied to incompressible flows, spectral method
solutions for incompressible flows, and turbulent-flow models and
calculations. In a discussion of compressible flows, inviscid
compressible flows are considered along with viscous compressible flows.
Attention is given to the potential flow solution technique, Green's
functions and stream-function vorticity formulation, the discrete vortex
method, the cloud-in-cell method, the method of characteristics,
turbulence closure equations, a large-eddy simulation model,
turbulent-flow calculations with a closure model, and direct simulations
of turbulence.

The viscous structure of a separated eddy is investigated for two cases of simplified geometry. In § 1, an analytical solution, based on a linearized model, is obtained for an eddy bounded by a circular streamline. This solution reveals the flow development from a completely viscous eddy at low Reynolds number to an inviscid rotational core at high Reynolds number, in the manner envisaged by Batchelor. Quantitatively, the solution shows that a significant inviscid core exists for a Reynolds number greater than 100. At low Reynolds number the vortex centre shifts in the direction of the boundary velocity until the inviscid core develops; at large Reynolds number, the inviscid vortex core is symmetric about the centre of the circle, except for the effect of the boundary-layer displacement-thickness. Special results are obtained for velocity profiles, skin-friction distribution, and total power dissipation in the eddy. In addition, results of the method of inner and outer expansions are compared with the complete solution, indicating that expansions of this type give valid results for separated eddies at Reynolds numbers greater than about 25 to 50. The validity of the linear analysis as a description of separated eddies is confirmed to a surprising degree by numerical solutions of the full Navier–Stokes equations for an eddy in a square cavity driven by a moving boundary at the top. These solutions were carried out by a relaxation procedure on a high-speed digital computer, and are described in § 2. Results are presented for Reynolds numbers from 0 to 400 in the form of contour plots of stream function, vorticity, and total pressure. At the higher values of Reynolds number, an inviscid core develops, but secondary eddies are present in the bottom corners of the square at all Reynolds numbers. Solutions of the energy equation were obtained also, and isotherms and wall heat-flux distributions are presented graphically.

A discretization scheme is presented which, unlike the standard higher-order finite difference and spline methods, does not give rise to unphysical solution modes and boundary conditions. Practical application of this scheme is achieved via the DCMG algorithm recently developed by the same author, which turns out to be able to find a converged solution of the ψ-ζ Navier-Stokes equations in about the same time for highorder as for low-order discretization schemes. Examples are presented for the driven cavity problem to explore the accuracy of the new method. Finally, a local analysis is performed of the corner singularities which exist in driven cavity flow, and their effect on the overall accuracy of the solutions obtained by polynomial interpolation methods is investigated.

It is well known that spectral approximations of hyperbolic time-dependent equations can lead to incoherent results in the case where the solution is discontinuous. However, it has been proved that the spectral coefficients of the approximation are computed precisely. In this article we present and analyze a class of filters that allows the recovering of the solution with an exponential accuracy.

The unsteady 2D Navier-Stokes equations on an unregularized driven cavity are solved in vorticity-streamfunction variables using Incremental Unknowns. Periodic asymptotic solutions have been found for Re = 10000 and Re = 12500.

Calculations for the two-dimensional driven cavity incompressible flow problem are presented. A p-type finite element scheme for the fully coupled stream function-vorticity formulation of the Navier-Stokes equations is used. Graded meshes are used to resolve vortex flow features and minimize the impact of corner singularities. Incremental continuation in the Reynolds number allows solutions to be computed for Re = 12 500. A significant feature of the work is that new tertiary and quaternary corner vortex features are observed in the flow field. Comparisons are made with other solutions in the literature.

The Stokes equations are solved using spectral methods with staggered and nonstaggered grids. Numerous ways to avoid the problem of spurious pressure modes are presented, including new techniques using the pseudospectral method and a method solving the weak form of the governing equations (a variation on the “spectral element” method developed by Patera). The pseudospectral methods using nonstaggered grids are simpler to implement and have comparable or better accuracy than the staggered grid formulations. Three test cases are presented: a formulation with an exact solution, a formulation with homogeneous boundary conditions, and the driven cavity problem. The solution accuracy is shown to be greatly improved for the driven cavity problem when the analytical solution of the singular flow behavior in the upper corners is separated from the computational solution.

The steady incompressible Navier-Stokes equations in a 2D driven cavity are solved in primitive variables by means of the multigrid method. The pressure and the components of the velocity are discretized on staggered grids, a block-implicit relaxation technique is used to achieve a good convergence and a simplified FMG-FAS algorithm is proposed. Special focus on the finite differences scheme used to approach the convection terms is made and a large discussion with other schemes is given. Results in a square driven cavity are obtained for Reynolds numbers as high as 15,000 on fine uniform meshes and the solution is in good agreement with other studies. For Re = 5000 the secondary vortices are very well represented showing the robustness of the method. For Reynolds numbers higher than 5000 the loss of stability for the steady solution is discussed. Moreover, some computations on a rectangular cavity of aspect ratio equal to two are presented. In addition the method is very efficient as far as CPU time is concerned; for instance, the solution for Re = 1000 on a 128 × 128 grid is obtained within 24 s on a SIEMENS VP 200.

Efficient and reliable numerical techniques of high-order accuracy are presented for solving problems of steady viscous incompressible flow in the plane, and are used to obtain accurate solutions for the driven cavity. A solution is obtained at Reynolds number 10,000 on a 180 × 180 grid. The numerical methods combine an efficient linear system solver, an adaptive Newton-like method for nonlinear systems, and a continuation procedure for following a branch of solutions over a range of Reynolds numbers.

The vorticity-stream function formulation of the two-dimensional incompressible Navier-Stokes equations is used to study the effectiveness of the coupled strongly implicit multigrid (CSI-MG) method in the determination of high-Re fine-mesh flow solutions. The driven flow in a square cavity is used as the model problem. Solutions are obtained for configurations with Reynolds number as high as 10,000 and meshes consisting of as many as 257 × 257 points. For Re = 1000, the (129 × 129) grid solution required 1.5 minutes of CPU time on the AMDAHL 470 V/6 computer. Because of the appearance of one or more secondary vortices in the flow field, uniform mesh refinement was preferred to the use of one-dimensional grid-clustering coordinate transformations.

A calculation procedure for rapid computation of steady multidimensional viscous flows is presented. The method solves the Navier-Stokes equations in primitive variables using a coupled block-implicit multigrid procedure. The procedure is applicable to finite-difference formulations using staggered locations of the flow variables. A smoothing technique called symmetrical coupled Gauss-Seidel (SCGS) is proposed and is empirically observed to provide good smoothing rates. The viscous flow in a square cavity with a moving top wall is calculated for a range of Reynolds numbers. Calculations with finite difference grids as large as 321 × 321 nodes have been made to test the accuracy and efficiency of the calculation scheme. The CPU times for these calculations are observed to be significantly smaller than other solution algorithms with primitive variable formulation. The calculated flow fields in the cavity are in good agreement with earlier studies of the same flow situation.

A third-order time-accurate projection method for approximating the Navier-Stokes equations for incompressible flow is presented. In order to compute a pressure unpolluted by spurious modes, two Chebyshev collocation spatial discretizations, where the pressure is approximated by lower-order polynomials than for the velocity, are compared. Only one collocation grid is used, and no pressure boundary condition is needed. The Navier-Stokes problem is reduced to the successive solution of Helmholtz problems for the velocity and pseudo-Poisson problems for the pressure. These problems are solved by direct methods. Using an exact solution, spectral spatial accuracy, and third-order time accuracy, for both the velocity and the pressure, are checked. The stability properties are discussed by considering the regularized cavity flow at various Reynolds numbers.

The Chebyshev approximation for the solution of singular Navier-Stokes problems

- O Botella
- R Peyret

Botella, O. and Peyret, R., The Chebyshev approximation for the solution of singular Navier±Stokes problems. To
be published in, Proceedings of the Third Summer Conference on Numerical Modelling in Continuum Mechanics,
Prague, September 8±11, 1997.