Conjugate natural convection from an array of discrete heat sources: part 2 — a numerical parametric study

Coupled conduction and natural convection transport within a discretely heated cavity have been investigated numerically. One vertical wall of the cavity is composed of discrete, isoflux heat sources mounted in a substrate of finite thermal conductivity. The opposite vertical wall and the horizontal walls are assumed to be isothermal and adiabatic, respectively. The governing steady-state partial differential equations for the fluid and solid region are solved simultaneously using a control volume formulation, coupled with an additive correction multigrid procedure that increases the convergence rate of the solution. The fluid Prandtl number and heater/fluid thermal conductivity ratio are fixed at 25 and 2350, respectively, corresponding to a dielectric fluid (FC-77) and heaters manufactured from silicon. With increasing modified Rayleigh number (10⁴ < Ra_Lz^* < 109), the cavity flow becomes more boundary layer-like along the vertical walls, and multiple fluid cells develop in the central region. Thermal spreading in the substrate increases with decreasing modified Rayleigh number and with increasing values of the substrate/fluid thermal conductivity ratio (10⁻¹ <- R_s ≤ 10³). For large R_s, the discrete heat sources lose their thermal identity, and the streamlines and isotherms resemble those associated with a differentially heated cavity. Thermal spreading in the substrate also has a significant effect on circulation in the cavity and on maximum surface temperatures.     

Direct numerical simulation of low Reynolds number flows in an open‐channel with sidewalls

A direct numerical simulation of low Reynolds number turbulent flows in an open‐channel with sidewalls is presented. Mean flow and turbulence structures are described and compared with both simulated and measured data available from the literature. The simulation results show that secondary flows are generated near the walls and free surface. In particular, at the upper corner of the channel, a small vortex called inner secondary flows is simulated. The results show that the inner secondary flows, counter‐rotating to outer secondary flows away from the sidewall, increase the shear velocity near the free surface. The secondary flows observed in turbulent open‐channel flows are related to the production of Reynolds shear stress. A quadrant analysis shows that sweeps and ejections are dominant in the regions where secondary flows rush in toward the wall and eject from the wall, respectively. A conditional quadrant analysis also reveals that the production of Reynolds shear stress and the secondary flow patterns are determined by the directional tendency of the dominant coherent structures. Copyright © 2009 John Wiley & Sons, Ltd.     

A FULLY NONLINEAR BOUSSINESQ MODELFOR WAVE PROPAGATION BASED ON MUSTA SCHEME

建立了求解二维全非线性布氏（Boussinesq）水波方程的有限差分/有限体积混合数值格式. 针对守恒形式的控制方程,采用有限体积方法并结合 MUSTA格式计算数值通量, 剩余项则采用有限差分方法求解, 采用具有总变差减小（totalvariation diminishing, TVD）性质的三阶龙格-库塔法进行时间积分.该格式具备间断捕捉、程序实现简单、数值稳定性强、海岸动边界以及波浪破碎处理方便和可调参数少等优点.利用典型算例对数值模型进行了验证,计算结果与实验数据吻合较好.     

Simulation of 2D external viscous flows by means of a domain decomposition method using an influence matrix technique

Two-dimensional external viscous flows are numerically approximated by means of a domain decomposition method which combines a vortex method and a finite difference method. The vortex method is used in the flow region which is dominated by convective effects, whereas the finite difference method is used in the flow region where viscous diffusion effects are dominant. An influence matrix technique combined with the uniformity condition of the pressure is used to enforce the tangential velocity boundary condition. Comparisons between numerical and experimental data show that the method is well adapted for simulating two-dimensional flows.     

A direct boundary-layer stability analysis of steady-state cavity convection flow

Natural convection flow in cavities with insulated top and bottom and heated and cooled walls is known to exhibit travelling wave instabilities in the thermal boundary layers that form on the walls. In water (Pr = 7.5) at Rayleigh number Ra = 6 × 10⁸, these waves have been observed at start-up. However no such waves have been observed for the fully developed flow, although it may be assumed that the stability character of the boundary layers is at least approximately the same. The start-up waves are generated by perturbations to the system. In the present paper, an artificial perturbation is applied to the system to determine the stability character of the boundary layers in fully developed flow. It is shown that the thermal boundary layers in the fully developed flow have approximately the same stability character as the start-up flow.     

A numerical continuous model for the hydrodynamics of fluid particle systems

In order to understand the hydrodynamic interactions that can appear in a fluid particle motion, an original method based on the equations governing the motion of two immiscible fluids has been developed. These momentum equations are solved for both the fluid and solid phases. The solid phase is assumed to be a fluid phase with physical properties, such as its behaviour can be assimilated to that of pseudo‐rigid particles. The only unknowns are the velocity and the pressure defined in both phases. The unsteady two‐dimensional momentum equations are solved by using a staggered finite volume formulation and a projection method. The transport of each particle is solved by using a second‐order explicit scheme. The physical model and the numerical method are presented, and the method is validated through experimental measurements and numerical results concerning the flow around a circular cylinder. Good agreement is observed in most cases. The method is then applied to study the trajectory of one settling particle initially off‐centred between two parallel walls and the corresponding wake effects. Different particle trajectories related to particulate Reynolds numbers are presented and commented. A two‐body interaction problem is investigated too. This method allows the simulation of the transport of particles in a dilute suspension in reasonable time. One of the important features of this method is the computational cost that scales linearly with the number of particles. Copyright © 1999 John Wiley & Sons, Ltd.     

A numerical study of three-dimensional laminar natural convection in a vented enclosure

A three-dimensional numerical investigation of steady laminar natural convection in vented enclosures is carried out. A discrete flush-type heat source mounted on the substrate is used to simulate an electronic component. Four different vent locations are investigated. Combined natural convection in the air and conduction in the heat source, the substrate, and the enclosure walls are solved. Solutions are obtained for Rayleigh numbers ranging from 10⁴ to 10⁶, different substrate thermal conductivity ratios, and varied vent sizes. The calculation domain is extended beyond the cubic enclosure in x-, y-, and z-directions. Appropriate boundary conditions are prescribed on the extended computational domain. The resulting flow and temperature patterns are discussed. Also, the local and overall heat transfer from the heat source and the substrate, in terms of Nusselt numbers and the surface temperatures, are presented to illustrate the vent effects.     

Investigation of low-intensity convection regimes in a rectangular cavity with a heat flux on the boundary

The characteristics of heat transfer during natural thermogravitational fluid convection of low intensity in a rectangular cavity heated from below (cooled from above) are investigated. Local convection effects are studied. The dependence of local superheating (supercooling) on the Grashof number and the cavity side ratio is found for single-, two-and three-vortex steady motions. The limits of the convection regimes are estimated.     

A 3D second‐order accurate projection‐based Finite Volume code on non‐staggered,non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows

It is well known that exact projection methods (EPM) on non‐staggered grids suffer for the presence of non‐solenoidal spurious modes. Hence, a formulation for simulating time‐dependent incompressible flows while allowing the discrete continuity equation to be satisfied up to machine‐accuracy, by using a Finite Volume‐based second‐order accurate projection method on non‐staggered and non‐uniform 3D grids, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by first solving an elliptic equation on a compact stencil that is by performing a standard approximate projection method (APM). In such a way, three sets of divergence‐free normal‐to‐face velocities can be computed. Then, a second elliptic equation for a scalar field is derived by prescribing that its additional discrete gradient ensures the continuity constraint based on the adopted linear interpolation of the velocity. Characteristics of the double projection method (DPM) are illustrated in details and stability and accuracy of the method are addressed. The resulting numerical scheme is then applied to laminar buoyancy‐driven flows and is proved to be stable and efficient. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical simulation of thermally and chemically nonequilibrium flows and heat transfer in underexpanded induction plasmatron jets

The results of numerical simulation are presented for thermally and chemically nonequilibrium air plasma flows in a plasmatron discharge channel and underexpanded dissociated and partially ionized air jets flowing past a cylindrical model with a blunt leading edge and cooled copper surface under the experimental conditions realized in a VGU-4 100 kW induction plasmatron (Institute for Problems in Mechanics of the Russian Academy of Sciences) (see, for example, [1, 2]). The nonequilibrium excitation of the vibrational degrees of freedom of the molecules in the modal approximation and the difference between the electron and translational heavy-particle temperatures are taken into account in the calculations. The calculated data on the heat transfer and pressure at the stagnation point are compared with the results obtained within the framework of the thermally equilibrium model. Comparison with the experimental data obtained in the Institute for Problem in Mechanics of the Russian Academy of Sciences (Laboratory for interaction between plasma and radiation and materials) and kindly provided for comparison purposes gives satisfactory agreement.     

High-order DG solvers for underresolved turbulent incompressible flows: A comparison of L2 and H(div) methods

The accurate numerical simulation of turbulent incompressible flows is a challenging topic in computational fluid dynamics. For discretisation methods to be robust in the underresolved regime, mass conservation and energy stability are key ingredients to obtain robust and accurate discretisations. Recently, two approaches have been proposed in the context of high-order discontinuous Galerkin (DG) discretisations that address these aspects differently. On the one hand, standard L²-based DG discretisations enforce mass conservation and energy stability weakly by the use of additional stabilisation terms. On the other hand, pointwise divergence-free H(div)-conforming approaches ensure exact mass conservation and energy stability by the use of tailored finite element function spaces. This work raises the question whether and to which extent these two approaches are equivalent when applied to underresolved turbulent flows. This comparative study highlights similarities and differences of these two approaches. The numerical results emphasise that both discretisation strategies are promising for underresolved simulations of turbulent flows due to their inherent dissipation mechanisms.     

Magnetohydrodynamic flow of a Sisko fluid in annular pipe: A numerical study

This paper presents a numerical study for the unsteady flow of a magnetohydrodynamic (MHD) Sisko fluid in annular pipe. The fluid is assumed to be electrically conducting in the presence of a uniform magnetic field. Based on the constitutive relationship of a Sisko fluid, the non‐linear equation governing the flow is first modelled and then numerically solved. The effects of the various parameters especially the power index n, the material parameter of the non‐Newtonian fluid b and the magnetic parameter B on the flow characteristics are explored numerically and presented through several graphs. Moreover, the shear‐thinning and shear‐thickening characteristics of the non‐Newtonian Sisko fluid are investigated and a comparison is also made with the Newtonian fluid. Copyright © 2009 John Wiley & Sons, Ltd.     

An attempt to improve accuracy of higher‐order statistics and spectra in direct numerical simulation of incompressible wall turbulence by using the compact schemes for viscous terms

We attempt to improve accuracy in the high‐wavenumber region in DNS of incompressible wall turbulence such as found in fully developed turbulent channel flow. In particular, it is shown that the improvement of accuracy of viscous terms in the Navier–Stokes equations leads to the improvement of accuracy of higher‐order statistics and various spectra. It is emphasized that increase in required computational cost will not be crucial when incompressible flow is simulated, because the introduction of a higher‐order scheme into the viscous terms does not increase computational cost for solving the Poisson equation. We introduced fourth‐order and eighth‐order central compact schemes for discretizing the viscous terms in DNS of a fully developed turbulent channel flow. The results are compared with those using second‐order and fourth‐order central‐difference schemes applied to the viscous terms and those obtained by the spectral method. The results show that accuracy improvement of the viscous terms improve accuracy of higher‐order statistics (i.e., skewness and flatness factors of streamwise velocity fluctuation) and various spectra of velocity and pressure fluctuations in the high‐wavenumber region. Copyright © 2013 John Wiley & Sons, Ltd.     

A wavelet-based variational multiscale method for the LES of incompressible flows in a high-order DG-FEM framework

This work focuses upon the development of a wavelet-based variant of the variational multiscale method (VMS) for accurate and efficient large eddy simulation (LES) called wavelet-based VMS-LES (WMS-LES). This approach has been incorporated within the framework of a high-order incompressible flow solver based upon the pressure-stabilized discontinuous Galerkin finite element method (DG-FEM). The VMS approach is designed to produce an a priori scale separation of the governing equations, in a manner which makes no assumptions on either the boundary conditions or the mesh uniformity. Using second-generation wavelets (SGWs) elementwise for scale separation ensures, on one hand, the preservation of the computational compactness of the DG-FEM scheme and, on the other hand, the ability to achieve scale separation in wavenumber space. The optimal space-frequency localization property of the SGW provides an improvement over the commonly used Legendre polynomials. The suitability of the elementwise SGW scale-separation operation as a tool for error indication has been demonstrated in an h-adaptive computation of the reentrant corner test case. Finally, the DG-FEM solver and the WMS-LES method have been assessed through simulations upon the three-dimensional Taylor-Green vortex test case. Our results indicate that the WMS-LES approach exhibits a distinct improvement over the monolevel LES approach. This effect is not produced by a change in the magnitude of the subgrid dissipation but rather by the redistribution of the subgrid dissipation in wavenumber space.     

Analysis of the local truncation error in the pressure‐free projection method for incompressible flows: a new accurate expression of the intermediate boundary conditions

The numerical integration of the Navier–Stokes equations for incompressible flows demands efficient and accurate solution algorithms for pressure–velocity splitting. Such decoupling was traditionally performed by adopting the Fractional Time‐Step Method that is based on a formal separation between convective–diffusive momentum terms from the pressure gradient term. This idea is strictly related to the fundamental theorem on the Helmholtz–Hodge orthogonal decomposition of a vector field in a finite domain, from which the name projection methods originates. The aim of this paper is to provide an original evaluation of the local truncation error (LTE) for analysing the actual accuracy achieved by solving the de‐coupled system. The LTE sources are formally subdivided in two categories: errors intrinsically due to the splitting of the original system and errors due to the assignment of the boundary conditions. The main goal of the present paper consists in both providing the LTE analysis and proposing a remedy for the inaccuracy of some types of intermediate boundary conditions associated with the prediction equation. Such evaluations will be directly performed in the physical space for both the time continuous formulation and the finite volume discretization along with the discrete Adams–Bashforth/Crank–Nicolson time integration. A new proposal for a boundary condition expression, congruent with the discrete prediction equation is herein derived, fulfilling the goal of accomplishing the closure of the problem with fully second order accuracy. In our knowledge, this procedure is new in the literature and can be easily implemented for confined flows. The LTE is clearly highlighted and many computations demonstrate that our proposal is efficient and accurate and the goal of adopting the pressure‐free method in a finite domain with fully second order accuracy is reached. Copyright © 2003 John Wiley & Sons, Ltd.     

On the distributed Lagrange multiplier/fictitious domain method for rigid‐particle‐laden flows: a proposition for an alternative formulation of the Lagrange multipliers

The distributed Lagrange multiplier/fictitious domain method proposed for the direct numerical simulation of particle‐laden flows is considered in this work. First, it is demonstrated that improved accuracy is obtained with a coupled numerical scheme, whereby the pressure and the Lagrange multiplier fields enforcing incompressibility and rigid body motion, respectively, are calculated and applied together. However, the convergence characteristics of the iterative solution of the coupled scheme are poor because symmetric but indefinite and poorly conditioned matrices are produced. An analysis is then presented, which suggests that the cause for the matrix pathologies lies in the interaction of the respective matrix operators enforcing incompressibility and rigid body motion. On the basis of this analysis, an alternative formulation is developed for the Lagrange multipliers, being now composed of a set of forces distributed only on the particle boundary together with a set of couples distributed within the particle core. The new formulation is tested with several types of flows with stationary or moving particles under creeping or finite Reynolds number conditions and it is demonstrated that it produces correct results and better conditioned matrices, thus enabling faster and more reliable convergence of the conjugate gradient method. The analysis and tests, therefore, support the expectation that the proposed formulation is promising and worthy of further study and improvement. Copyright © 2011 John Wiley & Sons, Ltd.     

A combined vortex and panel method for numerical simulations of viscous flows: a comparative study of a vortex particle method and a finite volume method

This paper describes and compares two vorticity‐based integral approaches for the solution of the incompressible Navier–Stokes equations. Either a Lagrangian vortex particle method or an Eulerian finite volume scheme is implemented to solve the vorticity transport equation with a vorticity boundary condition. The Biot–Savart integral is used to compute the velocity field from a vorticity distribution over a fluid domain. The vorticity boundary condition is improved by the use of an iteration scheme connected with the well‐established panel method. In the early stages of development of flows around an impulsively started circular cylinder, and past an impulsively started foil with varying angles of attack, the computational results obtained by the Lagrangian vortex method are compared with those obtained by the Eulerian finite volume method. The comparison is performed separately for the pressure fields as well. The results obtained by the two methods are in good agreement, and give a better understanding of the vorticity‐based methods. Copyright © 2005 John Wiley & Sons, Ltd.     

A comparative investigation of hydrofoil at angles of attack

In this paper, a numerical investigation of incompressible flow around a hydrofoil is presented. The laminar flow was modeled at different angles of attack. Momentum and continuity equations were coupled by the artificial compressibility scheme. In finite‐volume method, convective fluxes were calculated and compared by four schemes. Flux averaging with pressure correction was used. The other characteristic‐based (CB) methods consisted of Roe scheme and original CB scheme. A revised CB scheme was implemented in this research, which demonstrated very accurate solutions with respect to others. The results confirmed the superiority of the revised upwind scheme regarding accuracy and convergence without any requirement to artificial viscosity. The problem was studied at high Reynolds numbers at the onset of turbulence. For time discretization, the fifth‐order Runge–Kutta scheme was used. Results were compared with those of others in which good agreement was observed. Numerical experiments were performed on the NACA0012 hydrofoil. Copyright © 2011 John Wiley & Sons, Ltd.