General Convection-diffusion Equation Research Articles

Enthalpy-based cascaded lattice Boltzmann method for conjugate heat transfer

Lattice Boltzmann (LB) methods for conjugate heat transfer are widely studied because of the simplicity and efficiency in dealing with complex boundary conditions. Recently, cascaded (or central-moments-based) LB (CLB) method has shown the potential to improve the stability of LB method. For conventional CLB method which is constructed through the continuous equilibrium distribution function, the temperature-based energy equation is recovered and solved. The continuity of temperature and heat flux at conjugate interface is unable to be accurately enforced without special treatment. To circumvent this deficiency, an enthalpy-based CLB method for conjugate heat transfer is developed in this work. In the proposed method, the enthalpy-based formulation of equilibrium moments is directly designed in the central-moment space, and the macroscopic quantity solved is converted from temperature to enthalpy. As demonstrated by the Chapman-Enskog multiscale analysis, the enthalpy-based energy equation can be recovered by the CLB equation. Compared with the conventional CLB method, the continuity condition for temperature and heat flux on the conjugate interface can be enforced automatically without special treatment. In addition, the proposed CLB method provides a new way to develop a CLB equation that can recover to a general convection-diffusion equation. Finally, the accuracy and stability of the temperature field are validated by benchmarks for the proposed enthalpy-based CLB method.

A high order moving boundary treatment for convection-diffusion equations

In this paper, a new type of inverse Lax-Wendroff boundary treatment is designed for high order finite difference schemes for solving general convection-diffusion equations on time-varying domain. This new method can achieve high order accuracy on Dirichlet boundary conditions with moving boundary. To ensure stability of the boundary treatment, we give a convex combination of the boundary treatments for the diffusion-dominated and the convection-dominated cases. A group convex combination of weights is carefully designed to avoid zero denominator, resulting in a unified algorithm for pure convection, convection-dominated, convection-diffusion, diffusion-dominated and pure diffusion cases. In order to match the time levels when constructing values of ghost points in the two intermediate stages of the third order Runge-Kutta method, we propose a new approximation to the mixed derivatives at the boundaries to ensure high order accuracy and to improve computational efficiency. In particular, we extend the boundary treatment to the compressible Navier-Stokes equations, which satisfies the isothermal no-slip wall boundary condition at any Reynolds number. We provide numerical tests for one- and two-dimensional problems involving both scalar equations and systems, demonstrating that our boundary treatment is high order accurate for problems with smooth solutions and also performs well for problems involving interactions between viscous shocks and moving rigid bodies.

Lattice Boltzmann method for general convection-diffusion equations: MRT model and boundary schemes

This paper is concerned with the lattice Boltzmann method for general convection-diffusion equations. For such equations, we develop a multiple-relaxation-time lattice Boltzmann model and show its consistency under the diffusive scaling. The second-order accuracy of the half-way anti-bounce-back scheme accompanying the present MRT model is justified based on an elegant relation of the collision matrix. Using the half-way anti-bounce-back scheme as a central step, we further construct some parameterized single-node second-order schemes for curved boundaries. The accuracy of the proposed model and boundary schemes are numerically validated with several nonlinear convection-diffusion equations.

Simulation of Sedimentation in Lake Taihu with Geostationary Satellite Ocean Color Data

In this study, the goal is to estimate the sedimentation on the bottom bed of Lake Taihu using numerical simulation combined with geostationary satellite ocean color data. A two-dimensional (2D) model that couples the dynamics of shallow water and sediment transport is presented. The shallow water equations are solved using a semi-implicit finite difference method with an Alternating Direction Implicit (ADI) method. Suspended sediment transport is simulated by solving the general convection-diffusion equation with resuspension and deposition terms using a second-order explicit central difference method in space and two-step Adams–Bashforth method in time. Moreover, the total suspended particulate matter (TSM) is retrieved by the world’s first geostationary satellite ocean color sensor Geostationary Ocean Color Imager (GOCI) using atmospheric correction algorithm for turbid waters using ultraviolet wavelengths (UV-AC) and regional empirical TSM algorithm. The 2D model and GOCI-retrieved TSM are applied to study the sediment transport and sedimentation in Lake Taihu. Validation results show rationale TSM concentration retrieved by GOCI, and the simulated TSM concentrations are consistent with GOCI observations. In addition, simulated sedimentation results reveal the dangerous locations that must be observed and desilted.

Analysis of melting with natural convection and volumetric radiation using lattice Boltzmann method

A unified computational framework based on the Lattice Boltzmann method (LBM) is applied to solving the coupled heat transfer during phase change process with natural convection and volumetric radiation. A general convection-diffusion equation is introduced for a unified description of coupled heat transfer including conduction, convection and radiation in phase change media, and a corresponding unified lattice Boltzmann equation is developed. The D2Q9 model is used to solve the LBE. In order to validate the LBM for the coupled heat transfer process, three 2-D test cases of uncoupled and coupled heat transfer are examined. The computational results by LBM agree well with the benchmark solutions or other numerical solutions, which shows that the present model is of high accuracy and good flexibility to simulate coupled heat transfer in multidimensional participating medium. Then, the model is expanded to examine the problems of melting with natural convection and thermal radiation.

Inverse Lax–Wendroff procedure for numerical boundary conditions of convection–diffusion equations

We consider numerical boundary conditions for high order finite difference schemes for solving convection–diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax–Wendroff procedure [28] , in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, has been quite successful. In this paper, we extend this methodology to convection–diffusion equations. It turns out that this extension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combination of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection–diffusion equations. We provide extensive numerical tests for one- and two-dimensional problems involving both scalar equations and systems, including the compressible Navier–Stokes equations, to demonstrate the good performance of our numerical boundary conditions.

Conservative and Finite Volume Methods for the Convection-Dominated Pricing Problem

AbstractThis work presents a comparison study of different numerical methods to solve Black-Scholes-type partial differential equations (PDE) in the convection-dominated case, i.e., for European options, if the ratio of the risk-free interest rate and the squared volatility-known in fluid dynamics as Péclet number-is high. For Asian options, additional similar problems arise when the “spatial” variable, the stock price, is close to zero.Here we focus on three methods: the exponentially fitted scheme, a modification of Wang’s finite volume method specially designed for the Black-Scholes equation, and the Kurganov-Tadmor scheme for a general convection-diffusion equation, that is applied for the first time to option pricing problems. Special emphasis is put in the Kurganov-Tadmor because its flexibility allows the simulation of a great variety of types of options and it exhibits quadratic convergence. For the reduction technique proposed by Wilmott, a put-call parity is presented based on the similarity reduction and the put-call parity expression for Asian options. Finally, we present experiments and comparisons with different (non)linear Black-Scholes PDEs.

Application of an Iterative High Order Difference Scheme Along With an Explicit System Solver for Solution of Stream Function-Vorticity Form of Navier–Stokes Equations

This paper describes the general convection-diffusion equation in 2D domain based on a particular fourth order finite difference method. The current fourth-order compact formulation is implemented for the first time, which offers a semi-explicit method of solution for the resulting equations. A nine point finite difference scheme with uniform grid spacing is also put into action for discretization purpose. The proposed numerical model is based on the Navier–Stokes equations in a stream function-vorticity formulation. The fast convergence characteristic can be mentioned as an advantage of this scheme. It combines the enhanced Fournié's fourth order scheme and the expanded fourth order boundary conditions, while offering a semi-explicit formulation. To accomplish this, some coefficients which do not influence the solutions are also omitted from Fournié's formulation. Consequently, very accurate results can be acquired with a relatively coarse mesh in a short time. The robustness and accuracy of the proposed scheme is proved using the benchmark problems of flow in a driven square cavity at medium and relatively high Reynolds numbers, flow over a backward-facing step, and flow in an L-shaped cavity.

An exact discretization for a transport equation with piecewise-constant coefficients and arbitrary source

The search for a robust and accurate discretization scheme for a general convection diffusion equation has been a driving force in the improvement of codes employed in the CFD community. As part of this avenue of inquiry the research in this paper was aimed at improving the accuracy of schemes implemented in an in-house RANS code by incorporating the source in the discretization stencil. To understand how to do this we started working with the equation for a transported scalar and we came up with a systematic way of transforming a source-dominant equation into a source-free equation that allowed us to obtain an exact solution, i.e., a solution whose error is within machine accuracy, for an arbitrary source and an arbitrary number of discretization nodes, as long as the coefficients are piecewise constant in the intervals. The final equation, being a source-free equation, allows the use of the exponential scheme for the coefficients which is known to be exact for the homogeneous 1D convection diffusion equation. This paper shows how to convert the exponential scheme into an exact scheme for a transport equation with piecewise constant coefficients and arbitrary source in a 1D domain.

Monte Carlo approximations of the Neumann problem

We introduce Monte Carlo methods to compute the solution of elliptic equations with pure Neumann boundary conditions. We first prove that the solution obtained by the stochastic representation has a zero mean value with respect to the invariant measure of the stochastic process associated to the equation. Pointwise approximations are computed by means of standard and new simulation schemes especially devised for local time approximation on the boundary of the domain. Global approximations are computed thanks to a stochastic spectral formulation taking into account the property of zero mean value of the solution. This stochastic formulation is asymptotically perfect in terms of conditioning. Numerical examples are given on the Laplace operator on a square domain with both pure Neumann and mixed Dirichlet-Neumann boundary conditions. A more general convection-diffusion equation is also numerically studied.

Magnetic resonance electrical impedance tomography (MREIT) based on the solution of the convection equation using FEM with stabilization

Most algorithms for magnetic resonance electrical impedance tomography (MREIT) concentrate on reconstructing the internal conductivity distribution of a conductive object from the Laplacian of only one component of the magnetic flux density (∇2Bz) generated by the internal current distribution. In this study, a new algorithm is proposed to solve this ∇2Bz-based MREIT problem which is mathematically formulated as the steady-state scalar pure convection equation. Numerical methods developed for the solution of the more general convection–diffusion equation are utilized. It is known that the solution of the pure convection equation is numerically unstable if sharp variations of the field variable (in this case conductivity) exist or if there are inconsistent boundary conditions. Various stabilization techniques, based on introducing artificial diffusion, are developed to handle such cases and in this study the streamline upwind Petrov–Galerkin (SUPG) stabilization method is incorporated into the Galerkin weighted residual finite element method (FEM) to numerically solve the MREIT problem. The proposed algorithm is tested with simulated and also experimental data from phantoms. Successful conductivity reconstructions are obtained by solving the related convection equation using the Galerkin weighted residual FEM when there are no sharp variations in the actual conductivity distribution. However, when there is noise in the magnetic flux density data or when there are sharp variations in conductivity, it is found that SUPG stabilization is beneficial.

Osmotic backwash mechanism of reverse osmosis membranes

A new osmotic backwash (BW) model for reverse osmosis (RO) membranes was developed for conditions of no applied pressures across the membrane. This analytical model has one adjustable parameter representing the coefficient of a linearized convection term in the general convection–diffusion equation. An experimental RO/BW system was used for 12 data sets to verify the proposed BW model and illustrate its predictability. Results show deviations of the model from the data within a range of 5–15%. The described dilution mechanism of the feed concentration polarization (CP) layer is based on RO originated concentrated layer detachment from the membrane surface followed by its gradual dilution. The understanding gained in this research may be applied to automatic RO/BW cleaning cycles. A dominant RO parameter of the BW process is the RO initial driving force—the concentration difference across the membrane. Other RO process parameters – applied pressure and feed flow rate – have lesser effects. Both theoretical and experimental methods provide quantitative relationships between RO and BW variables that enable an understanding and control of the BW process.

Simulation of nitrogen contaminant transportation by a compact difference scheme in the downstream Yellow River, China

The rapid agricultural development of the Yellow River basin in China has caused serious potential water pollution to the vicinal areas. In this paper, a compact scheme is developed and employed to simulate the unsteady transportation of three representative water quality variables including ammonia nitrogen, nitrite nitrogen and nitrate nitrogen in the downstream Yellow River. As an extension of Ref. [Yang ZF, Chen GQ. The compact fourth-order finite difference scheme for the unsteady convection–diffusion equation involving source term. Chin Sci Bull 1993;38(2):113–6 [in Chinese]], the scheme is applied to the more general convection–diffusion equation with complicated terms reflecting the mutual action of water and sediment, which conforms to the characteristic of high sediment concentration of the Yellow River. The simulation results indicate that the error yielded from the proposed method is smaller than that of the referenced scheme (five-point hybrid scheme) in solving the water quality equation.

On the interface diffusion coefficient for solving Reynolds equation by control volume method

When using the control volume method to solve the Reynolds equation, the mass flux crossing the control surface should be calculated properly. According to Pantakar’s formulation, which is commonly used in solving general convection–diffusion equations, the mass flux can be expressed as a function of the convection and diffusion coefficients. Consequently, the performance of the numerical algorithm depends strongly on the scheme employed for the calculation of the interface diffusion coefficient. Two diffusion schemes have been proposed in the literature. One scheme (referred to as scheme I) employs the arithmetic mean of the pressure values at the neighboring grid points to evaluate the interface diffusion coefficient, while the other (referred to as scheme II) uses the harmonic mean of the neighbor diffusion coefficients. Scheme I has been used for solving the Reynolds equation successfully. On the other hand, scheme II, while being popular for solving convection–diffusion types of equations when the diffusion coefficient is known, has not been implemented to solve the Reynolds equation for air bearings, in which the diffusion coefficient depends on the unknown dependent variable. In this paper, we implement scheme II for solving the Reynolds equation and compare its performance with that of scheme I. Both numerical and analytical results indicate that scheme II may yield unrealistic results for air bearings with large clearance discontinuities.

3D Numerical Modeling of Flow and Sediment Transport in Open Channels

A 3D numerical model for calculating flow and sediment transport in open channels is presented. The flow is calculated by solving the full Reynolds-averaged Navier-Stokes equations with the k − ε turbulence model. Special free-surface and roughness treatments are introduced for open-channel flow; in particular the water level is determined from a 2D Poisson equation derived from 2D depth-averaged momentum equations. Suspended-load transport is simulated through the general convection-diffusion equation with an empirical settling-velocity term. This equation and the flow equations are solved numerically with a finite-volume method on an adaptive, nonstaggered grid. Bed-load transport is simulated with a nonequilibrium method and the bed deformation is obtained from an overall mass-balance equation. The suspended-load model is tested for channel flow situations with net entrainment from a loose bed and with net deposition, and the full 3D total-load model is validated by calculating the flow and sediment transport in a 180° channel bend with movable bed. In all cases, the agreement with measurements is generally good.