Parameter-free numerical method for modeling thermal convection in square cavities in a wide range of Rayleigh numbers

Some numerical results for the two- and three-dimensional de Vahl Davis benchmark are presented. This benchmark describes thermal convection in a square (cubic) cavity with vertical heated walls in a wide range of Rayleigh numbers (10⁴ to 10¹⁴), which covers both laminar and highly turbulent f lows. Turbulent f lows are usually described using a turbulence model with parameters that depend on the Rayleigh number and require adjustment. An alternative is Direct Numerical Simulation (DNS) methods, but they demand extremely large computational grids. Recently, there has been an increasing interest in DNS methods with an incomplete resolution, which, in some cases, are able to provide acceptable results without resolving Kolmogorov scales. On the basis of this approach, the so-called parameter-free computational techniques have been developed. These methods cover a wide range of Rayleigh numbers and allow computing various integral properties of heat transport on relatively coarse computational grids. In this paper, a new numerical method based on the CABARET scheme is proposed for solving the Navier–Stokes equations in the Boussinesq approximation. This technique does not involve a turbulence model or any tuning parameters and has a second-order approximation scheme in time and space on uniform and nonuniform grids with a minimal computational stencil. Testing the technique on the de Vahl Davis benchmark and a sequence of refined grids shows that the method yields integral heat f luxes with a high degree of accuracy for both laminar and highly turbulent f lows. For Rayleigh numbers up to 10¹⁴, a several percent accuracy is achieved on an extremely coarse grid consisting of 20 × 20 cells refined toward the boundary. No definite or comprehensive explanation of this computational phenomenon has been given. Cautious optimism is expressed regarding the perspectives of using the new method for thermal convection computations at low Prandtl numbers typical of liquid metals.     

Finite volume multigrid prediction of laminar natural convection: Bench-mark solutions

A finite volume multigrid procedure for the prediction of laminar natural convection flows is presented, enabling efficient and accurate calculations on very fine grids. The method is fully conservative and uses second-order central differencing for convection and diffusion fluxes. The calculations start on a coarse (typically 10 × 10 control volumes) grid and proceed to finer grids until the desired accuracy or maximum affordable storage is reached. The computing times increase thereby linearly with the number of control volumes. Solutions are presented for the flow in a closed cavity with side walls at different temperatures and insulated top and bottom walls. Rayleigh numbers of 10⁴, 10⁵ and 10⁶ are considered. Grids as fine as 640 × 640 control volumes are used and the results are believed to be accurate to within 0–01%. Second-order monotonic convergence to grid-independent values is observed for all predicted quantities.     

An aspect ratio agglomeration multigrid for unstructured grids

A robust aspect ratio‐based agglomeration algorithm to generate high quality of coarse grids for unstructured and hybrid grids is proposed in this paper. The algorithm focuses on multigrid techniques for the numerical solution of Euler and Navier–Stokes equations, which conform to cell‐centered finite volume special discretization scheme, combines vertex‐based isotropic agglomeration and cell‐based directional agglomeration to yield large increases in convergence rates. Aspect ratio is used as fusing weight to capture the degree of cell convexity and give an indication of cell stretching. Agglomeration front queue is established to propagate inward from the boundaries, which stores isotropic vertex and also high‐stretched cell marked with different flag according to aspect ratio. We conduct the present method to solve Euler and Navier–Stokes equations on unstructured and hybrid grids and compare the results with single grid as well as MGridGen, which shows that the present method is efficient in reducing computational time for large‐scale system equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Developing a physical influence upwind scheme for pressure-based cell-centered finite volume methods

In the framework of a cell-centered finite volume method (FVM), the advection scheme plays the most important role in developing FVMs to solve complicated fluid flow problems for a wide range of Reynolds numbers. Advection schemes have been widely developed for FVMs employing pressure-velocity coupling methodology in the incompressible flow limit. In this regard, the physical influence upwind scheme (PIS) is developed for a cell-centered finite volume coupled solver (FVCS) using a pressure-weighted interpolation method for linking the pressure and velocity fields. The well-known exponential differencing scheme and skew upwind differencing scheme are also deployed in the current FVCS and their numerical results are presented. The accuracy and convergence of the present PIS are evaluated solving flow in a lid-driven square cavity, a lid-driven skewed cavity, and over a backward-facing step (BFS). The flow within the lid-driven square cavity is numerically solved at Reynolds numbers from 400 to 10 000 on a relatively coarse mesh with respect to other reported solutions. The lid-driven skewed cavity test case at Reynolds number of 1000 demonstrates the numerical performance of the present PIS on nonorthogonal grids. The flow over a BFS at Reynolds number of 800 is numerically solved to examine capabilities of current FVCS employing the current PIS in inlet-outlet flow computations. The numerical results obtained by the current PIS are in excellent agreement with those of benchmark solutions of corresponding test cases. Incorporating implicit role of pressure terms in a pressure-weighted interpolation method and development of PIS provides satisfactory solution convergence alongside the numerical accuracy for the current FVCS. A particular numerical verification is performed for the V velocity calculation within the BFS flow field, which confirms the reliability of present PIS.     

Global flow instability in a lid‐driven cavity

The stability of flow in a lid‐driven cavity is investigated using an accurate numerical technique based on a hybrid scheme with spectral collocation and high‐order finite differences. A global stability analysis is carried out and critical parameters are identified for various aspect ratios. It is found that while there is reasonable agreement with the literature for the critical parameters leading to loss of stability for the square cavity, there are significant discrepancies for cavities of aspect ratios 1.5 and 2. Simulations of the linearized unsteady equations confirm the results from the global stability analysis for aspect ratios A = 1, 1.5 and A = 2. Copyright © 2009 John Wiley & Sons, Ltd.     

A numerical investigation of the influence of the aspect ratio in three‐dimensional separated flows

The influence of aspect ratio in three‐dimensional, numerical experiments of separated flows is studied in the case of the backward‐facing step at Reynolds numbers 600, 800, and 950. The computational domain is designed as an actual laboratory experiment. The governing equations are the steady state, isothermal, and incompressible Navier–Stokes equations. The expansion ratio of the computational domain is 1:2. The aspect ratio varies from 1:10 to 1:40. The results of the computations focus on the spanwise variations of the length and the strength of the two eddies along the lower and upper wall. It is concluded that both numerical and laboratory experiments should be designed with an aspect ratio of at least 1:20, if only the accuracy of the position of the detachment and the re‐attachment points matters. If the accuracy of the shear‐stress distributions is also taken into account, then an aspect ratio of at least 1:30 should be chosen. Finally, if the magnitudes of the vortex centers are also considered, then only the aspect ratio of 1:40 qualifies for a realization of two‐dimensional flow conditions in the plane of symmetry. This is contrary to the common practice in the field, at least from the side of laboratory experiments, where an aspect ratio of 1:10 is still considered adequate for this purpose. Copyright © 2011 John Wiley & Sons, Ltd.     

Start‐up flow in a three‐dimensional lid‐driven cavity by means of a massively parallel direction splitting algorithm

The purpose of this paper is to validate a new highly parallelizable direction splitting algorithm. The parallelization capabilities of this algorithm are illustrated by providing a highly accurate solution for the start‐up flow in a three‐dimensional impulsively started lid‐driven cavity of aspect ratio 1 × 1 × 2 at Reynolds numbers 1000 and 5000. The computations are done in parallel (up to 1024 processors) on adapted grids of up to 2 billion nodes in three space dimensions. Velocity profiles are given at dimensionless times t = 4, 8, and 12; at least four digits are expected to be correct at Re = 1000. Copyright © 2011 John Wiley & Sons, Ltd.     

2D thermal/isothermal incompressible viscous flows

2D thermal and isothermal time‐dependent incompressible viscous flows are presented in rectangular domains governed by the Boussinesq approximation and Navier–Stokes equations in the stream function–vorticity formulation. The results are obtained with a simple numerical scheme based on a fixed point iterative process applied to the non‐linear elliptic systems that result after a second‐order time discretization. The iterative process leads to the solution of uncoupled, well‐conditioned, symmetric linear elliptic problems. Thermal and isothermal examples are associated with the unregularized, driven cavity problem and correspond to several aspect ratios of the cavity. Some results are presented as validation examples and others, to the best of our knowledge, are reported for the first time. The parameters involved in the numerical experiments are the Reynolds number Re, the Grashof number Gr and the aspect ratio. All the results shown correspond to steady state flows obtained from the unsteady problem. Copyright © 2005 John Wiley & Sons, Ltd.     

Level set方法在自适应Cartesian网格上的应用

采用基于自适应Cartesian网格的level set方法对多介质流动问题进行数值模拟。采用基于四叉树的方法来生成自适应Cartesian网格。采用有限体积法求解Euler方程,控制面通量的计算采用HLLC（Hartern, Lax, van Leer, Contact）近似黎曼解方法。level set方程也采用有限体积法求解,采用Lax-Friedchs方法计算通量,通过窄带方法来减少计算量,界面的处理采用ghost fluid方法。Runge-Kutta显式时间推进,时间、空间都是二阶精度。对两种不同比热比介质激波管问题进行数值模拟,其结果和精确解吻合;对空气/氦气泡相互作用等问题进行模拟,取得令人满意的结果。     

Chebyshev finite spectral method with extended moving grids

A Chebyshev finite spectral method on non-uniform meshes is proposed. An equidistribution scheme for two types of extended moving grids is used to generate grids. One type is designed to provide better resolution for the wave surface, and the other type is for highly variable gradients. The method has high-order accuracy because of the use of the Chebyshev polynomial as the basis function. The polynomial is used to interpolate the values between the two non-uniform meshes from a previous time step to the current time step. To attain high accuracy in the time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme is used. To avoid numerical oscillations caused by the dispersion term in the Korteweg-de Vries (KdV) equation, a numerical technique on non-uniform meshes is introduced. The proposed numerical scheme is validated by the applications to the Burgers equation (nonlinear convectiondiffusion problems) and the KdV equation (single solitary and 2-solitary wave problems), where analytical solutions are available for comparisons. Numerical results agree very well with the corresponding analytical solutions in all cases.     

Natural convection in partially cooled tilted cavities

Two-dimensional numerical simulations of laminar natural convection in a partially cooled, differentially heated inclined cavities are performed. One of the cavity walls is entirely heated to a uniformly high temperature (heat source) while the opposite wall is partially cooled to a lower temperature (heat sink). The remaining walls are adiabatic. The tilt angle of the cavity is varied from 0° (heated from left) to −90° (heated from top). The fast false implicit transient scheme (FITS) algorithm, developed earlier by the same authors, is modified to solve the derived variables vorticity-streamfunction formulation. The effects of aspect ratio (AR), sink–source ratio and tilt angle on the average Nusselt number are examined through a parametric study; solutions are obtained for two Grashof numbers, 10⁵ and 10⁷. Flow patterns and isotherms are used to investigate the heat transfer and fluid flow mechanisms inside the cavity. © 1998 John Wiley & Sons, Ltd.     

Temperature and velocity field characteristics of turbulent natural convection in a vertical parallel-plate channel with asymmetric heating

Turbulent natural convection in an asymmetrically heated vertical parallel-plate channel has been studied experimentally and numerically using LDA and CFD. Simultaneous velocity and temperature measurements across the channel at different elevations have been carried out. Three different Ra(b/h) values of 1.91 × 10⁷, 2.74 × 10⁷ and 3.19 × 10⁷ are considered with the channel aspect ratio of b/h = 1/20. Experimental and numerical data are presented in the form of streamwise direction heated wall surface temperature, mean velocity, mean temperature, Reynolds shear stress and turbulent kinetic energy profiles along the channel for one case. These profiles exhibit the flow field development along the channel emphatically. The numerical technique used predicts temperature field fairly well, considerably over-estimating velocity field in the core region.     

Three‐dimensional numerical simulation of Marangoni instabilities in liquid bridges: influence of geometrical aspect ratio

Oscillatory Marangoni convection in silicone oil–liquid bridges with different geometrical aspect ratios is investigated by three‐dimensional and time‐dependent numerical simulations, based on control volume methods in staggered cylindrical non‐uniform grids. The three‐dimensional oscillatory flow regimes are studied and compared with previous experimental and theoretical results. The results show that the critical wavenumber (m), related to the azimuthal spatio‐temporal flow structure, is a monotonically decreasing function of the geometrical aspect ratio of the liquid bridge (defined as the ratio of length to diameter). For this function, a general correlation formula is found, which is in agreement with the previous experimental findings. The critical Marangoni number and the oscillation frequency are decreasing functions of the aspect ratio; however, the critical Marangoni number, based on the axial length of the bridge, does not change much with the aspect ratio. For each aspect ratio investigated, the onset of the instability from the axisymmetric steady state to the three‐dimensional oscillatory one is characterized by the appearance of a standing wave regime that exhibits, after a certain time, a second transition to a travelling wave regime. The standing wave regime is more stable for lower aspect ratios since it lasts for a long time. This behaviour is explained on the basis of the propagation velocity of the disturbances in the liquid phase. For this velocity, a general correlation law is found as a function of the aspect ratio and of the Marangoni number. Copyright © 2001 John Wiley & Sons, Ltd.     

Multiplicity of steady solutions in a two-sided lid-driven cavity with different aspect ratios

This study presents a continuation method to calculate flow bifurcation in a two-sided lid-driven cavity with different aspect ratios for anti-parallel motion. In anti-parallel motion, the top and bottom walls of the cavity move in opposite directions simultaneously, while the two walls both moving to the right give parallel motion at the same speed. Comprehensive bifurcation diagrams of the cavity flows with different aspect ratios of the cavities are derived via Keller’s continuation method, and linear- stability analysis is used to identify the nature of the various flow solutions. The Reynolds number (1 ≤ Re ≤ 1,200) is used as the continuation parameter to trace the solution curves. In anti-parallel motion, the evolution of the bifurcation diagrams in cases with different aspect ratios (1 ≤ AR ≤ 2.5) is illustrated. Two stable symmetric flows and one stable asymmetric flow are identified, and the existent regions of the stable flows in the aspect ratios and Reynolds numbers are distinguished. The newly found asymmetric flow state can be obtained at a high aspect ratio and a low Reynolds number.     

A convective weakly viscoelastic rotating flow with pressure Neumann condition

The objective of this work is to investigate through the numeric simulation, the effects of the weakly viscoelastic flow within a rotating rectangular duct subject to a buoyancy force due to the heating of one of the walls of the duct. A direct velocity–pressure algorithm in primitive variables with a Neumann condition for the pressure is employed. The spatial discretization is made with finite central differences on a staggered grid. The pressure field is directly updated without any iteration. Numerical simulations were done for several Weissemberg numbers (We) and Grashof numbers (Gr) . The numerical results show that for high Weissemberg numbers (We>7.4 × 10^?5) and for ducts with aspect ratio 2:1 and 8:1, the secondary flow is restabilized with a stretched double vortex configuration. It is also observed that when the Grashof number is increased (Gr>17 × 10^?4) , the buoyancy force neutralizes the effects of the Coriolis force for ducts with aspect ratio 8:1. Copyright © 2008 John Wiley & Sons, Ltd.     

Modeling of Two-Phase Flow in Fractured Porous Media on Unstructured Non-Uniformly Coarsened Grids

In this article, we investigate two strategies for coarsening fractured geological models. The first approach, which generates grids that resolve the fractures, is referred to as explicit fracture-matrix separation (EFMS). The second approach is based on a non-uniform coarsening strategy introduced in Aarnes et al. (Adv Water Resour 30(11):2177–2193, 2007a). A series of two-phase flow simulations where the saturation is modeled on the respective coarse grids are performed. The accuracy of the resulting solutions is examined, and the robustness of the two strategies is assessed with respect to number of fractures, degree of coarsening, well locations, phase viscosities, and fracture permeability. The numerical results show that saturation solutions obtained on the non-uniform coarse grids are consistently more accurate than the corresponding saturation solutions obtained on the EFMS grids. The numerical results also reveal that it is much easier to tune the upscaling factor with the non-uniform coarsening approach.     

Implementation of some higher-order convection schemes on non-uniform grids

A generalized formulation is applied to implement the quadratic upstream interpolation (QUICK) scheme, the second-order upwind (SOU) scheme and the second-order hybrid scheme (SHYBRID) on non-uniform grids. The implementation method is simple. The accuracy and efficiency of these higher-order schemes on non-uniform grids are assessed. Three well-known bench mark convection-diffusion problems and a fluid flow problem are revisited using non-uniform grids. These are: (1) transport of a scalar tracer by a uniform velocity field; (2) heat transport in a recirculating flow; (3) two-dimensional non-linear Burgers equations; and (4) a two-dimensional incompressible Navier-Stokes flow which is similar to the classical lid-driven cavity flow. The known exact solutions of the last three problems make it possible to thoroughly evaluate accuracies of various uniform and non-uniform grids. Higher accuracy is obtained for fewer grid points on non-uniform grids. The order of accuracy of the examined schemes is maintained for some tested problems if the distribution of non-uniform grid points is properly chosen.     

An immersed boundary method for the incompressible Navier–Stokes equations in complex geometry

A nodally exact convection–diffusion–reaction scheme developed in Cartesian grids is applied to solve the flow equations in irregular domains within the framework of immersed boundary (IB) method. The artificial momentum forcing term applied at certain points in the flow and inside the body of any shape allows the imposition of no‐slip velocity condition to account for the body of complex boundary. Development of an interpolation scheme that can accurately lead to no‐slip velocity condition along the IB is essential since Cartesian grid lines generally do not coincide with the IB. The results simulated from the proposed IB method agree well with other numerical and experimental results for several chosen benchmark problems. The accuracy and fidelity of the IB flow solver to predict flows with irregular IBs are therefore demonstrated. Copyright © 2007 John Wiley & Sons, Ltd.     

A collocated grid,projection method for time‐accurate calculation of low‐Mach number variable density flows in general curvilinear coordinates

A time‐accurate algorithm is proposed for low‐Mach number, variable density flows on curvilinear grids. Spatial discretization is performed on collocated grid that offers computational simplicity in curvilinear coordinates. The flux interpolation technique is used to avoid the pressure odd–even decoupling of the collocated grid arrangement. To increase the stability of the method, a two‐step predictor–corrector time integration scheme is employed. At each step, the projection method is used to calculate the hydrodynamic pressure and to satisfy the continuity equation. The robustness and accuracy of the method is illustrated with a series of numerical experiments including thermally driven cavity, polar cavity, three‐dimensional cavity, and direct numerical simulation of non‐isothermal turbulent channel flow. Copyright © 2012 John Wiley & Sons, Ltd.