A meshless local Petrov–Galerkin approach for solving the convection-dominated problems based on the streamline upwind idea and the variational multiscale concept

A meshless local Petrov–Galerkin (MLPG) approach based on the streamline upwind (SU) idea and the variational multiscale (VMS) concept, called as VMS-SUMLPG method, is herein proposed to solve the convection-dominated problems. In the present VMS-SUMLPG method, the streamline upwind is constructed in the test function to solve the non-self-adjoint matrix. Meanwhile, the VMS concept as a stability term is adopted to alleviate the numerical instability such as spurious oscillations, overshoots, and undershoots. Its numerical accuracy and stability are validated by comparing with the streamline upwind Petrov–Galerkin (SUMLPG) method and the finite volume method with high-order difference schemes for two classical convection-dominated problems at the Peclet number ranging from 10⁶ to 10⁸. It is shown that the numerical solutions of the present VMS-SUMLPG method are accuracy, smoothness, and stability.     

Variational formulation of hyperbolic heat conduction problems applying Laplace transform technique

In this paper, a non-Fourier heat conduction problem is analyzed by employing newly developed theory. Application of conventional numerical schemes leads to strong oscillations of the results around discontinuities in solution domain. To overcome this difficulty the variational formulation of the Laplace-transformed hyperbolic heat conduction equation is developed. The results were used for evaluation of parameters used in approximate transformed temperature profiles. To validate the approach the results were compared with the exact analytical solution solved at special case and with an approach previously reported in the literature. Both showed a close agreement with the proposed approach.     

Numerical investigation of the thermal separation in a Ranque–Hilsch vortex tube

The application of a mathematical model for the simulation of thermal separation in a Ranque–Hilsch vortex tube is presented in this paper. The modelling of turbulence for compressible, swirling flows used in the simulation is discussed. The work has been carried out in order to provide an understanding of the physical behaviors of the flow, pressure, temperature in a vortex tube. A staggered finite volume approach with the standard k–ε turbulence model and an algebraic stress model (ASM) is used to carry out all the computations. To investigate the effects of numerical diffusion on the predicted results, the second-order upwind (SOU) and the QUICK numerical schemes are used and compared with the first-order upwind and the hybrid schemes. The computations show that the differences of results obtained from using the various schemes are marginal. In addition, results predicted by both turbulence models generally are in good agreement with measurements but the ASM performs better agreement between the numerical results and experimental data. The computations with selective source terms of the energy equation suppressed show that the diffusive transport of mean kinetic energy has a substantial influence on the maximum temperature separation occurring near the inlet region. In the downstream region far from the inlet, expansion effects and the stress generation with its gradient transport are also significant.     

Transient analytical solution to heat conduction in composite circular cylinder

An analytical method leading to the solution of transient temperature filed in multi-dimensional composite circular cylinder is presented. The boundary condition is described as time-dependent temperature change. For such heat conduction problem, nearly all the published works need numerical schemes in computing eigenvalues or residues. In this paper, the proposed method involves no such numerical work. Application of ‘separation of variables’ is novel. The developed method represents an extension of the analytical approach derived for solving heat conduction in composite slab in Cartesian coordinates. Close-formed solution is provided and its agreement with numerical result is good which demonstrates a good accuracy of the developed solution form.     

Exact factorization technique for numerical simulations of incompressible Navier–Stokes flows

Splitting techniques break an ill-conditioned indefinite system resulting from incompressible Navier–Stokes equations into well-conditioned subsystems, which can be solved reliably and efficiently. Apart from the ambiguity regarding numerical boundary conditions for the pressure (and for intermediate velocities, whenever introduced), splitting techniques usually incur splitting errors which reduce time accuracy. The discrete approach of approximate factorization techniques eliminates the need of numerical boundary conditions and restores time accuracy by an approximate inversion of some matrix in the case of semi-implicit time schemes. For linear implicit, non-linear implicit, and higher-order semi-implicit time schemes, however, approximate factorization techniques are laborious. In this paper, we systematically present a new and straightforward exact factorization technique. The main contributions of this work include: (1) the idea of removing the splitting error or the idea of restoring time accuracy for fully discrete systems, (2) the introduction of the pressure-update type and the pressure-correction type of exact factorization techniques for any time schemes, and (3) an analysis of several established techniques and their relations to the exact factorization technique. The exact factorization technique is implemented with a standard second-order finite volume method and is verified numerically.     

Transient analytical solution to heat conduction in multi-dimensional composite cylinder slab

The analytical solution for the problem of transient heat conduction in multi-dimensional composite cylinder slab is developed for a time-dependent boundary condition. For such problems, numerical programs are needed to obtain eigenvalues and residues in most of the published papers. The numerical schemes may become unstable due to the existence of imaginary eigenvalues in multi-dimensional cases. In this paper, the proposed analytical method involves no numerical complications. By a novel application of the methods of the Laplace transform and separation of variables together with variable transformations, the residue calculation is avoided. The developed analytical method is powerful which represents extension of the analytical approach derived for the heat conduction problem in Cartesian coordinates. A closed form solution is provided. Calculation examples show that the analytical solutions predict good agreement with the numerical results.     

Modeling studies for production potential of Chingshui geothermal reservoir

The Chingshui geothermal power plant was decommissioned in 1993 due to a continued decline in production. Although some geothermal exploration and field investigation had been exercised, the production potential of the reservoir is still not well understood. In this paper, numerical modeling approaches for characterization of the geothermal reservoir, investigation of reservoir production performance, and evaluation of exploitation scheme design are presented. At first, a site-scale refined grid numerical model was developed for simulating the natural state of Chingshui geothermal reservoir. Through the model, the production potential of the geothermal reservoir was estimated and the availability of water resources was assessed. We further built production model to simulate the production history during 1981–1993. From the production model, we can conclude that the abnormal drop of the reservoir production capacity is mainly caused by carbonate scaling. Potential production schemes with different reinjection designs were evaluated through the model. Simulation results indicated that a sustainable hot water production capacity of Chingshui geothermal reservoir is about 200 t/h without reinjection, and 300 t/h or even higher with reinjection which is enough for a 3 MWe power plant. The simulation results indicate that reinjection provides an effective approach for maintaining reservoir pressure during hot water/steam production.     

Conservative and nondispersive schemes for diffusion terms with strong property variations

The spatial discretization of diffusion terms leads to a theoretical order loss, specially in the presence of strong property and variable gradients. New conservative and nondispersive numerical schemes are proposed to circumvent this issue. Their spectral resolution, numerical order, and cost are evaluated and compared to traditional finite volume and difference schemes. Cost is measured through the computer time and memory it takes these schemes to reach a user-prescribed error tolerance. A manufactured test case is constructed to evaluate these characteristics. Its analysis shows that the new schemes are more cost-effective than traditional schemes under the conditions analyzed.     

FLOW AND HEAT TRANSFER IN RECTANGULAR ENCLOSURES USING A NEW BLOCK-IMPLICIT NUMERICAL METHOD

This work reports a numerical investigation on buoyancy-induced flows occurring in enclosures of small aspect ratio and inclined with respect to the horizontal direction. The numerical method used consists of the control-volume approach and a new block-implicit error-smoothing operator. Governing equations are written in terms of primitive variables and are recast into a general form. In the proposed method, all governing equation are relaxed locally, in contrast with commonly used segregated schemes. The effects of Rayleigh number, aspect ratio, and cavity inclination on temperature and velocity patterns are discussed. It is expected that more advanced parallel computer architectures can benefit from the error-smoothing operator described here.     

Effect of the heating rate on crystallization behavior of mechanically alloyed amorphous alloy

The mechanical alloying is the most convenient method to produce Mg–Ni alloys. In this study, the effect of ball-to-powder weight ratios and the mechanical alloying time on amorphization of Mg₅₀Ni₅₀ alloy and its thermal stabilities were investigated. Mg₅₀Ni₅₀ alloy has been produced by using Spex 8000 D mixer/mill with different ball-to-powder weight ratios (5:1, 10:1, 20:1). Amorphization times by XRD analysis are found to be 60 h for 5:1 ball-to-powder weight ratio, 10 h for 10:1 ball-to-powder weight ratio and 5 h for 20:1 ball-to-powder weight ratio. The thermal stabilities of amorphous Mg₅₀Ni₅₀ alloys, obtained by different ball-to-powder weight ratios, have been determined and the effect of heating rates on the crystallization temperatures have also been investigated by DSC. The heating rates employed were 5, 10, 15, 20 °C/min. During the first crystallization reaction, the amorphous and Mg₂Ni intermetallic phases occurred. DSC studies show that increase in heating rates increased the crystallization temperatures for all samples. The apparent activation energies were determined by means of the Kissinger method.     

GDQ METHOD FOR NATURAL CONVECTION IN A SQUARE CAVITY USING VELOCITY–VORTICITY FORMULATION

ABSTRACT

This article describes a compact numerical algorithm based on the generalized differential quadrature (GDQ) method for the numerical analysis of natural convection in a differentially heated square cavity. The velocity–vorticity form of the Navier–Stokes equations and energy equation are used to represent the mass, momentum, and energy conservations of the fluid medium in the cavity. The GDQ form of the governing equations and the vorticity definition at the boundaries are solved by a coupled solution algorithm using a global matrix scheme for all the field variables. The vorticity values at the boundary are correctly imposed using the GDQ method, which approximates a given space derivative with higher-order accuracy compared to the existing schemes based on Taylor's series expansion. This has assured a divergence-free solution for the flow field by satisfying the continuity constraint, though the pressure term is not used directly in the present formulation. The proposed algorithm is validated for a lid-driven cavity flow for Reynolds number Re = 100, 400, and 1,000, and the predicted velocity profiles are in excellent agreement with the benchmark solutions. The algorithm is then used to compute the average Nusselt number and flow parameters for natural convection in a square cavity for Rayleigh number Ra = 10³, 10⁴, 10⁵, and 10⁶. These results are in better agreement with the benchmark solutions than the results obtained by other numerical schemes, which used much finer grids compared to the present scheme.     

A COMPARISON OF TIME DISCRETIZATION SCHEMES FOR THE GREEN ELEMENT SOLUTION OF THE TRANSIENT HEAT CONDUCTION EQUATION

We present three time discretization schemes for the Green element solution of the linear conduction equation. Numerical results obtained from the three methods are assessed by their convergence and stability-related properties, namely, numerical amplitude and amplification factor through Fourier analysis. The main emphasis is the ability of any of the schemes to handle conduction problems typified by those properties that are known to be taxing to most numerical schemes. This was found in some cases to be related to how accurately the harmonics of the Fourier waves are propagated.     

LES of Impingement Heat Transfer on a Concave Surface

Wall-resolved and zonal numerical large eddy type simulations are performed for a round jet impinging on a concave hemisphere at Re = 23,000. The zonal method uses a near-wall k–l model and a Hamilton-Jacobi equation to match this to the large eddy simulation zone. To minimize numerical dissipation, a self-adaptive discretization (SDS) scheme is examined. Both second- (n = 2) and sixth- (n = 6) order-based central discretization schemes are tested. The characteristics of the schemes is assessed using two test cases: the development of a subcritical Tollmien-Schlichting (T-S) stability wave in a plane channel and the decay of homogenous, isotropic turbulence (DHIT). It is found, that Smagorinsky LES simulations tend to be too dissipative in the high wave-number region, even with the SDS scheme; hence, the SGS model is omitted. Significant flow feedback is observed for the hemisphere case. Both shear-layer excitation and stabilization is observed. Computed wall pressure coefficients for the zonal NLES method are encouraging; for the wall-resolved case the stagnation region value is overpredicted. Heat transfer for the wall-resolved and zonal large eddy simulations are encouraging. For both quantities the difference between the n = 2 and n = 6 schemes is small, and the modeling approach used appears to be more influential. It is concluded that the presence of feedback mechanisms should be considered when designing experiments and/or numerical simulations for this case, and that the importance of boundary conditions for LES should not be neglected.     

GDQ Method for Natural Convection in a Cubic Cavity Using Velocity–Vorticity Formulation

ABSTRACT

Natural convection in a differentially heated cubic enclosure is studied by solving the velocity–vorticity form of the Navier–Stokes equations by a generalized differential quadrature (GDQ) method. The governing equations in the form of velocity Poisson equations, vorticity transport equations, and energy equation are solved using a coupled numerical scheme via a single global matrix for velocities, vorticities, and temperature. Vorticity and velocity coupling at the solid boundaries is enforced through a higher-order approximation by the GDQ method, thus assuring accurate satisfaction of the continuity equation. Nusselt numbers computed for Ra = 10³, 10⁴, 10⁵, and 10⁶ show good agreement with the benchmark results. A mesh independence study indicates that the present numerical procedure requires much coarse mesh compared to other numerical schemes to produce the benchmark solutions of the flow and heat transfer problems.     

Simulation of propagating fronts in geothermal reservoirs with the implicit Leonard total variation diminishing scheme

Geothermal reservoir engineering requires accurate numerical solution of the advective–diffusive transport equations for strong advective flows of multiphase nonisothermal fluids. Conventional interface weighting schemes such as upstream weighting cause numerical dispersion. Numerical dispersion can be reduced by grid refinement, but this increases execution times and computer memory requirements. As an alternative, higher-order differencing schemes can be used to reduce numerical dispersion, but they often lead to spurious oscillations. These limitations have led to the development of higher-order schemes called total variation diminishing (TVD) schemes. For geothermal reservoir engineering, these schemes must be capable of handling flows that may not be physically total variation diminishing. We have implemented TVD schemes into the implicit geothermal reservoir simulator TOUGH2. We verify the Leonard TVD (LTVD) scheme by comparison to an analytical solution for two-dimensional flow and transport. The LTVD scheme reduces numerical dispersion for tracer transport in a two-phase geothermal reinjection problem. One-dimensional simulations show that the LTVD scheme works well even if the saturation variation increases with time. Because the location of the phase front is strongly coupled to temperature, phase front propagation is sensitive to grid resolution insofar as it affects the temperature field. Phase front propagation in a composite porous medium Buckley–Leverett flow problem, where phase saturations increase upon encountering a second medium, are slightly more accurate for the LTVD scheme as compared to upstream weighting. We find that the LTVD scheme only performs well if the weighting and limiter are applied to saturation rather than to relative permeability. While there is some increased computational cost with the LTVD scheme due to increased linear equation solution time and smaller time-step size, the LTVD scheme is a practical and robust method for reducing numerical dispersion in complex flow problems relevant to geothermal reservoir engineering.     

Heat transfer through periodic macro-contact with constriction

A quadrupole method is developed to solve heat transfer through a periodic macro-contact with time varying constriction. The solution is based on Fourier developments of time periodic variables. The result shows that it is necessary to introduce the concept of “building-up” of constriction to explain the thermal behavior for short and moderate periods. It is demonstrated that three characteristic times govern the problem: contact period, characteristic time of the rod and “building-up” time of constriction. Simplified schemes of the apparent resistance are presented corresponding to three limiting states. The analytical approach is validated by a numerical solution.     

Solution of the Initial Inverse Problems in the Heat Equation Using the Finite Difference Method with Positivity-Preserving Padé Schemes

The classical inverse problem of recovering the initial temperature distribution from the final temperature distribution is extremely ill-posed. We propose a class of numerical schemes based on positivity-preserving Padé approximations to solve initial inverse problems in the heat equation. We also utilize a partial fraction decomposition technique to solve the problem more efficiently when higher order Padé approximations are used. We apply the proposed numerical schemes on the parabolic heat equation. Our aim is to model the problem as a direct problem and use our numerical schemes to recover the initial profile in a stable and efficient way.     

Mean-field/PDF numerical approach for polydispersed turbulent two-phase flows

The purpose of this paper is to give an overview in the realm of numerical computations of polydispersed turbulent two-phase flows, using a mean-field/PDF approach. In this approach, the numerical solution is obtained by resorting to a hybrid method, where the mean fluid properties are computed by solving mean-field (RANS) equations with a classical finite volume procedure whereas the local instantaneous properties of the particles are determined by solving stochastic differential equations (SDEs). The fundamentals of the general formalism are recalled and particular attention is focused on a specific theoretical issue: the treatment of the multiscale character of the dynamics of the discrete particles, i.e. the consistency of the system of SDEs in asymptotic cases. Then, the main lines of the particle/mesh algorithm are given and some specific problems, related to the integration of the SDEs, are discussed, for example, issues related to the specificity of the treatment of the averaging and projection operators, the time integration of the SDEs (weak numerical schemes consistent with all asymptotic cases), and the computation of the source terms. Practical simulations, for three different flows, are performed in order to demonstrate the ability of both the models and the numericals to cope with the stringent specificities of polydispersed turbulent two-phase flows.