First-order Upwind Difference Scheme Research Articles

An adaptive grid method for a two-parameter singularly perturbed problem with non-smooth data

In this paper, a two-parameter singularly perturbed problem with discontinuous source and convection coefficient is studied. This problem is discretized by using a first-order upwind finite difference scheme for which a posteriori error analysis in the maximum norm is derived. Then, based on this a posteriori error estimation, a grid iteration algorithm is designed to generate an adaptive nonuniform mesh. Finally, Numerical experiments are proposed to confirm that our proposed method is first-order uniformly convergent.

Analysis of internal flow field for a three‐stage centrifugal fan under various operating conditions

Centrifugal fan in series is the core facility for pneumatic transportation. Its operation is required to be adjusted real time as the flow rate of transported materials changes. For optimisation performance, internal flow of a type 700 three-stage fan at rated rotation speed is analysed with FLUENT6.3 for various total pressure. Multiple reference frame (MRF) model is used for the impellers rotation in simulation, and Re-Normalisation group (RNG) κ-ɛ turbulence model and Roe-FDS flux type with first-order upwind difference scheme are used for calculation of compressible flow under various total pressure. Flow characteristics are obtained and differences of total pressure and velocity distribution in every impeller are analysed in different conditions, and velocity distribution on meridian plane as well as section of wind guide plates and volute are compared. Results show that there are many vortexes and backflows in fan with higher total pressure, and the lower pressure, the more smoothly flows, besides flow at inlet and outlet of impellers and in guide plates is seriously influenced by total pressure. Curves of P-Q and P-η at 4600 r/min are predicted based on simulation, which is helpful reducing the impact caused by off-design operation and improving efficiency.

Large Eddy Simulation Analysis of Fluid Temperature Fluctuations at a T-junction for Prediction of Thermal Loading

T-junctions are widely used for fluid mixing in power and process plants. Temperature fluctuations generated by the mixing of hot and cold fluids at a T-junction can cause high cycle thermal fatigue (HCTF) failure. The existing Japanese guideline for evaluating HCTF provides margin that varies greatly depending on the case for the evaluation result. Computational fluid dynamics (CFD)/finite element analysis (FEA) coupling analysis is expected to be a useful tool for the more accurate evaluation of HCTF. Precise temperature fluctuation histories are necessary to determine the thermal loads because fatigue damage prediction requires temperature fluctuation amplitudes and their cycle numbers. The present investigation was intended to discover the accurate prediction methods of fluid temperature fluctuations, prior to performing CFD/FEA coupling analysis. Large eddy simulation (LES) turbulence models suitable for the simulation of unsteady phenomena were investigated. The LES subgrid scale (SGS) models used included the standard Smagorinsky model (SSM) and the dynamic Smagorinsky model (DSM). The effects of numerical schemes for calculating the convective term in the energy equation on the simulation results were also investigated. LES analyses of the flow and temperature fields at a T-junction were carried out using these numerical methods. For comparison, the simulation conditions were the same as the experiment in literature. All of the simulation results show the flow pattern of a wall jet with strong flow and temperature fluctuations, as observed in the experiment. The simulation results indicate the numerical schemes have a great effect on the temperature distribution and the temperature fluctuation intensity (TFI). The first-order upwind difference scheme (1UD) significantly underestimates the TFI for each LES SGS model, although it exhibits good numerical stability. However, the hybrid scheme (HS), which is mainly the second-order central difference scheme (2CD) blended with a small fraction of 1UD, can better predict the TFI for each LES SGS model. Furthermore, the DSM model gives a prediction closer to the experimental results than the SSM model, while using the same numerical scheme. As a result, it was found through the systematic investigations of various turbulence models and numerical schemes that the approach using the DSM model and the HS with a large blending factor could provide accurate predictions of the fluid temperature fluctuations. Furthermore, it is considered that this approach is also applicable to the accurate prediction of any other scalar (e.g., concentration), based on the analogy of scalar transport phenomena.

Unsteady Non-Newtonian Solver on Unstructured Grid for the Simulation of Blood Flow

Blood is in fact a suspension of different cells with yield stress, shear thinning, and viscoelastic properties, which can be represented by different non-Newtonian models. Taking Casson fluid as an example, an unsteady solver on unstructured grid for non-Newtonian fluid is developed to simulate transient blood flow in complex flow region. In this paper, a steady solver for Newtonian fluid is firstly developed with the discretization of convective flux, diffusion flux, and source term on unstructured grid. For the non-Newtonian characteristics of blood, the Casson fluid is approximated by the Papanastasiou's model and treated as Newtonian fluid with variable viscosity. Then considering the transient property of blood flow, an unsteady non-Newtonian solver based on unstructured grid is developed by introducing the temporal term by first-order upwind difference scheme. Using the proposed solver, the blood flows in carotid bifurcation of hypertensive patients and healthy people are simulated. The result shows that the possibility of the genesis and development of atherosclerosis is increased, because of the increase in incoming flow shock and backflow areas of the hypertensive patients, whose WSS was 20~87.1% lower in outer vascular wall near the bifurcation than that of the normal persons and 3.7~5.5% lower in inner vascular wall downstream the bifurcation.

Development of a Numerical Method for Analyzing Fire Plume Equations Using an Improved Version of a Quasi-Third-Order Accurate CIP Method That Eliminates Numerical Oscillation

This study is concerned with simulations of fire phenomena using field equation models. When performing numerical computation of a fire plume accompanied by gas and smoke, the phenomena are expressed through simultaneous non-stationary nonlinear second-order partial differential equations containing advective terms. Those advective terms are then subjected to difference approximations. The ordinary CIP method, which allows us to find an approximate solution using a third-order interpolation function, is often influenced by numerical oscillations emerging specifically with the third-order accuracy, thereby drastically reducing the numerical stability. We developed a new computational algorithm called mCIP method to eliminate these numerical oscillations generated in the numerical computation of partial differential equations when a third-order accurate method is used. A characteristic feature of this mCIP is that it is based on the first-order upwind difference scheme, which does not generate numerical oscillations in those regions where such oscillations may normally occur. We successfully evaluated the performance of this mCIP method and verified its effectiveness by comparing the results of computations with other representative methods, such as CIP method, TVD method and first-order accurate difference method.

Revaluation of the first-order upwind difference scheme to solve coarse-grained master equations

This study addresses the initial-boundary value problem of coarse-grained probability measure on the state space in which a differentiable vector field v is given and, as a consequence, the differenced continuity equation using the first-order upwind difference scheme (UDS) based on the finite volume method appears as the physical substance on the coarse-grained dynamics. At first, the UDS is theoretically shown to be equivalent to a class of coarse-grained master equations (CGME), brought by a principle that we cannot distinguish state points in the same partition with each other. The principle is based on the formulation of non-equilibrium statistical mechanics to resolve the macroscopic irreversibility. Moreover the entropy production evaluated by the UDS is also shown to be in accord with the average volume contraction rate in the steady state. This is essential for the non-equilibrium statistical dynamics and was numerically confirmed. Under the principle of coarse graining the UDS is very superior to the conventional Monte-Carlo method in computer time and storage and is very useful to solve the CGME.

DIFFUSIVITY OF TWO-COMPONENT ISOTHERMAL FINITE DIFFERENCE LATTICE BOLTZMANN MODELS

Diffusion equations are derived for an isothermal lattice Boltzmann model with two components. The first-order upwind finite difference scheme is used to solve the evolution equations for the distribution functions. When using this scheme, the numerical diffusivity, which is a spurious diffusivity in addition to the physical diffusivity, is proportional to the lattice spacing and significantly exceeds the physical value of the diffusivity if the number of lattice nodes per unit length is too small. Flux limiter schemes are introduced to overcome this problem. Empirical analysis of the results of flux limiter schemes shows that the numerical diffusivity is very small and depends quadratically on the lattice spacing.

On a set of singularly perturbed convection–diffusion equations

We study convergence properties of a first-order upwind difference scheme applied to a weakly coupled system of singularly perturbed convection–diffusion equations. We derive a priori and a posteriori error estimates that are robust with respect to the perturbation parameters. Thereby strengthening and generalising recent results (Appl. Numer. Math. 51 (2004) 171; in: A. Ansari, A Hegarty, G.I. Shishkin, Numerical Methods for Problems with Layer Phenomena, Limerick, 2004, pp. 33–39). The key ingredient of our analysis are strong negative-norm stability results obtained earlier by Andreev (Differential Equations 37(7) (2001) 923) and by Andreev and Kopteva (Differential Equations 34(7) (1998) 921)).

Error expansion for a first-order upwind difference scheme applied to a model convection-diffusion problem

A singularly perturbed convection–diffusion problem is considered. The problem is discretized using a simple first‐order upwind difference scheme on general meshes. We derive an expansion of the error of the scheme that enables uniform error bounds with respect to the perturbation parameter in the discrete maximum norm for both a defect correction method and the Richardson extrapolation technique. This generalizes and simplifies results obtained in earlier publications by Fröhner et al.(2001, Numer. Algorithms, 26, 281–299) and by Natividad & Stynes (2003, Appl. Numer. Math., 45, 315–329). Numerical experiments complement our theoretical results.

Analysis of Thin Film Flows Using a Flux Vector Splitting1

We propose a flux vector splitting (FVS) for the solution of film flows radially spreading on a flat surface created by an impinging jet using the shallow-water approximation. The governing equations along with the boundary conditions are transformed from the physical to the computational domain and solved in a rectangular grid. A first-order upwind finite difference scheme is used at the point of the shock while a second-order upwind differentiation is applied elsewhere. Higher-order spatial accuracy is achieved by introducing a MUSCL approach. Three thin film flow problems (1) one-dimensional dam break problem, (2) radial flow without jump, and (3) radial flow with jump, are investigated with emphasis in the prediction of hydraulic jumps. Results demonstrate that the method is useful and accurate in solving the shallow water equations for several flow conditions.

Effect of spatial discretization schemes on numerical solutions of viscoelastic fluid flows

This study examines the effect of discretization schemes for the convection term in the constitutive equation on numerical solutions of viscoelastic fluid flows. For this purpose, a temporally evolving mixing layer, a two-dimensional vortex pair interacting with a wall, and a fully developed turbulent channel flow are selected as test cases, and eight different discretization schemes are considered. Among them, the first-order upwind difference scheme (UD) and artificial diffusion scheme (AD), which are commonly used in the literature, show most stable and smooth solutions even for highly extensional flows. However, the stress fields are smeared too much by these schemes and the corresponding flow fields are quite different from those obtained by higher-order upwind difference schemes. Among higher-order upwind difference schemes investigated in this study, a third-order compact upwind difference scheme (CUD3) with locally added AD shows stable and most accurate solutions for highly extensional flows even at relatively high Weissenberg numbers.

Numerical Modeling of Flow in Chlorine Disinfection Tanks

The reliability of simulations of flow and disinfection processes in disinfection contact tanks is determined by the accuracy of computations in terms of the hydrodynamic characteristics in the tanks, particularly the accuracy of the calculations of advection and shear stress, which are two problematic terms currently under intensive research. As part of the development of a simulation module for a software package of flow and disinfection process predictions in chlorine contact tanks, various modeling methods and difference schemes have been tested for typical tank configurations, and the results are compared and analyzed. In the software the computation of shear stress terms was originally represented by a depth mean viscosity model, and its results have been compared with that of calculations using the k¯-ε¯ and the Smagorinsky subgrid scale stress models in an attempt to improve upon the model predictions. The discretization of advection terms in the software module was a first-order upwind difference scheme, the results of which also have been compared with that of its two alternatives, the QUICK scheme and a third-order upwind difference scheme. Predictions made by using several combinations of these models and schemes were tested against measurements from a physical model study.

Simulating disinfection processes in chlorine contact tanks using various turbulence models and high-order accurate difference schemes

The numerical simulation of the disinfection processes in a contact tank involves determining both the flow and solute transport distributions. In this study it has been established that the reliability of the solute transport predictions depends strongly upon the accuracy of the hydrodynamic predictions. In the hydrodynamic computations, it is common practice to combine various turbulence models and numerical schemes to approximate the shear stress and advection terms, e.g. the depth mean eddy viscosity model with either a first-order or third-order upwind difference scheme; the k- ε model or the Smagorinsky model in Large Eddy Simulation (LES) with a first-order upwind scheme, etc. A question which this study also sought to answer, is to what extent the accuracy of the discretization of the advection terms in the solute transport equations affects the discrepancy between the concentration prediction and the physical model, or field measured results, whilst the flow is simulated using various difference and turbulence models. In this study the advection terms in the solute transport equation were evaluated using the QUICK and the third-order upwind difference schemes in separate simulation runs. The numerical predictions of the dye transport within a contact tank for a pulse injection, were compared with experimental measurements carried out in a physical model. It has been demonstrated that the reliability of some predicted bulk characteristics of the flow in a contact tank, such as the depth of flow and the cross-sectional averaged velocity, are of crucial importance in the computation of solute transport processes.

Higher-order numerical model for simulation of time-dependent variable-density flows

A numerical model, first-order accurate in time discretization and second-order accurate in space discretization, is presented for the simulation of time-dependent laminar diffusion flames. The numerical model uses a semi-implicit scheme for time marching and incorporates flux-corrected transport (FCT) for convection terms and projection method for velocity-pressure coupling. A direct solver is used for solving the pressure Poisson equation. A time-splitting method is adopted to split the transport equation into a convection equation in explicit form and a diffusion equation in implicit form. The explicit convection equation allows for implementation of the FCT without introducing an artificial viscosity. The implicit diffusion equation removes the time step size restriction of the von Neumann stability criterion. Theoretical assessments are derived for the order of accuracy of time difference by viewing the projection method as a lower-upper decomposition, and formulation is also given to establish the second-order accuracy. The current numerical model is used to solve the Smith-Hutton problem and the Burke-Schumann diffusion flame with or without external forcing of the fuel jet. The results are compared with solutions using the schemes of lower-order accuracy in the space discretization. The comparison shows that significant numerical diffusion error exists in the solution when a first-order upwind difference scheme or power-law approximation is used. The present study suggests that a higher-order space difference scheme should be used for the simulation of time-dependent laminar jet diffusion flames, although a first-order time difference scheme would be adequate.

Error in the upwind differencing of the convection-diffusion equation

In convection-diffusion problems a first-order upwind difference approximation is usually used on a locally refined mesh to get a wiggle free solution, which is then corrected using some form of deferred correction to get second-order accuracy. Pereyra (1966) has shown that improving accuracy, of a low-order scheme using deferred correction, requires the existence of an asymptotic expansion for the discretization error of the scheme concerned. In this paper bounds on the discrezazation error are obtained when the first-order upwind difference scheme is applied to the convection-diffusion equation on a locally refined mesh, both in one and two dimensions. Existence of an asymptotic expansion for the discretization error is established. Different formulae, at the nodes separating subregions of differing mesh sizes, are analysed and numerical tests performed to illustrate validity of the resulting expressions.

A study of upstream‐weighted high‐order differencing for approximation to flow convection

AbstractThis paper is concerned with a number of upstream‐weighted second‐ and third‐order difference schemes. Also considered are the conventional upwind and central difference schemes for comparison. It commences with a general difference equation which unifies all the given first‐, second‐ and third‐order schemes. The various schemes are evaluated through the use of the general equation. The unboundedness and accuracy of the solutions by the difference schemes are assessed via various analyses: examination of the coefficients of the difference equation, Taylor series truncation error analysis, study of the upstream connection to numerical diffusion, single‐cell analysis. Finally, the difference schemes are tested on one‐ and two‐dimensional model problems. It is shown that the high‐order schemes suffer less from the problem of numerical diffusion than the first‐order upwind difference scheme. However, unboundedness cannot be avoided in the solutions by these schemes. Among them the linear upwind difference scheme presents the best compromise between numerical diffusion and solution unboundedness.