A staggered-grid numerical algorithm for the extended Boussinesq equations

The two-dimensional Reynolds averaged compressible Navier-Stokes equations are solved using MacCormack's second-order accurate explicit finite difference method to simulate the separated transonic tur- bulent flowfield over an airfoil. Four different algebraic eddy viscoisity models are tested for viability to achieve turbulence closure for the class of flows considered. These models range from an unmodified boundary-layer mixing-length model to a relaxation model incorporating special considerations for the separation bubble region. Results of this study indicate the necessity for special attention to the separated flow region and suggest limits of applicability of algebraic turbulence models to these separated flowfield. each of these studies the time-dependent Reynolds averaged Navier-Stokes equations for two-dimensional compressive flow are used and tur- bulence closure is achieved by means of model equations for the Reynolds stresses. Wilcox1'2 used a first-order accurate numerical scheme and the two equation differential tur- bulence model of Saffman 12 to simulate the supersonic shock boundary-layer interaction experiment of Reda and Mur- phy 13 and the compression corner flow of Law.14 Good quan- titative agreement with the Reda and Murphy data was ob- tained, but only the qualitative features of the compression corner flow were well simulated. Using a more sophisticated second-order accurate numerical scheme, Baldwin3'4 con- sidered both the two equation differential model of Saffman and a simpler algebraic mixing-length model to simulate the hypersonic shock boundary-layer interaction experiment of Holden.15 He found the more elaborate model of Saffman to yield somewhat better results than the algebraic model, but at the cost of considerably more computing time. Good quan- titative agreement with experiment was not obtained with either model. Following Baldwin's approach all subsequent investigations have been performed using the more rigorous second-order accurate numerical scheme of Mac- Cormack.17'18 Deiwert5'6'11 considered an algebraic mixing- length model to simulate the transonic airfoil experiment of McDevitt et al. 16 while Horstman et al. 8 used a similar ap- proach to simulate their hypersonic shock boundary-layer ex- periment on an axisymmetric cylinder. In each of these studies, while qualitative features of the flows were described well, good quantitative agreement with experiment in the in- teraction regions was not obtained. Using a relaxing turbulence model Shang and Hankey7 simulated the compression corner flow of Law, and Baldwin and Rose10 simulated the flat plate flow of Reda and Murphy. In each of these studies the relaxing model was found to per- form significantly better than the simpler algebraic model and, according to Shang and Hankey, provided significantly better comparisons with measurements than were obtained by Wilcox using the two equation differential model of Saffman. In each of these studies it was essential that the full Navier- Stokes equations be considered to describe the viscous- inviscid interaction and the elliptic nature of separating-

In this paper, we propose and analyze a second-order accurate numerical scheme for the Poisson-Nernst-Planck equations. The proposed scheme combines a novel second-order temporal discretization with the centered finite difference method in space. By using a Taylor expansion for the logarithmic term in the chemical potential, this second-order accurate numerical scheme is able to preserve original energy dissipation. Based on the gradient flow formulation, the resulting scheme guarantees several crucial physical properties: mass conservation, positivity of ionic concentration, preservation of original energy dissipation, and steady states preservation. Remarkably, these properties are ensured without any restriction on the time step size. Furthermore, an optimal rate convergence estimate is provided for the proposed numerical scheme. Due to the non-constant mobility and the nonlinear and singular properties of the logarithm terms, a higher-order asymptotic expansion and a combination of rough and refined error estimation techniques are introduced to accomplish this analysis. A few numerical tests are provided to validate our theoretical claims.

A second-order accurate (in time) and linear numerical scheme is proposed and analyzed for the nonlocal Cahn–Hilliard equation. The backward differentiation formula is used as the temporal discretization, while an explicit extrapolation is applied to the nonlinear term and the concave expansive term. In addition, an $O (\varDelta {t}^2)$ artificial regularization term, in the form of $A \varDelta _N (\phi ^{n+1} - 2 \phi ^n + \phi ^{n-1})$, is added for the sake of numerical stability. The resulting constant-coefficient linear scheme brings great numerical convenience; however, its theoretical analysis turns out to be very challenging, due to the lack of higher-order diffusion in the nonlocal model. In fact, a rough energy stability analysis can be derived, where an assumption on the $\ell ^\infty $ bound of the numerical solution is required. To recover such an $\ell ^\infty $ bound, an optimal rate convergence analysis has to be conducted, which combines a high-order consistency analysis for the numerical system and the stability estimate for the error function. We adopt a novel test function for the error equation, so that a higher-order temporal truncation error is derived to match the accuracy for discretizing the temporal derivative. Under the view that the numerical solution is actually a small perturbation of the exact solution, a uniform $\ell ^\infty $ bound of the numerical solution can be obtained, by resorting to the error estimate under a moderate constraint of the time step size. Therefore, the result of the energy stability is restated with a new assumption on the stabilization parameter $A$. Some numerical experiments are carried out to display the behavior of the proposed second-order scheme, including the convergence tests and long-time coarsening dynamics.

Essentially Compact Schemes for Unsteady Viscous Incompressible Flows

A three-dimensional unsteady, viscous aerodynamic analysis has been developed for the flow inside a transonic, high-through-flow, single stage compressor. The compressor stage is comprised of a low-aspect-ratio rotor and a closely coupled stator. The analysis is based on a numerical method for solving the three-dimensional Navier-Stokes equation for unsteady viscous flow through multiple turbomachinery blade rows. The method solves the fully three-dimensional Navier-Stokes equation with an implicit scheme. A two-equation turbulence model with a low-Reynolds-number modification is applied for the turbulence closure. A third-order accurate upwinding scheme is used to approximate convection terms while a second-order accurate central difference scheme is used for the discretization of the viscous terms. A second-order accurate scheme is employed for the temporal discretization. The numerical method is applied to study the unsteady flow field inside a transonic, high-through-flow, axial compressor stage. The numerical results are compared with available experimental data.

Comparison of higher-order accurate schemes for solving the two-dimensional unsteady Burgers' equation

Efficient decoupled second-order numerical scheme for the flow-coupled Cahn–Hilliard phase-field model of two-phase flows

A high-order accurate scheme for Maxwell's equations with a Generalized Dispersive Material (GDM) model and material interfaces

Laminar flow control and drag reduction research requires accurate boundary layer solutions as input to the three-dimensional stability analysis procedures currently under development. In support of these major programs, a fourth-order accurate finite difference scheme for solving the three-dimensional, compressible boundary layer equations has been developed and is presented in this paper. The method employs a two-point scheme in the wall normal direction and second order zigzag scheme in the cross flow direction. Accurate procedures to interface with the inviscid results are also presented. The results of applying the procedure to laminar flow on wings and fuselages are presented.

Chapter 1 Boussinesq models and applications to nearshore wave propagation, surf zone processes and wave-induced currents

On a novel fully-decoupled, linear and second-order accurate numerical scheme for the Cahn–Hilliard–Darcy system of two-phase Hele–Shaw flow

Finite Difference Methods for 3D Viscous Incompressible Flows in the Vorticity–Vector Potential Formulation on Nonstaggered Grids

This paper develops a high-order accurate gas-kinetic scheme in the framework of the finite volume method for the one- and two-dimensional flow simulations, which is an extension of the third-order accurate gas-kinetic scheme [Q.B. Li, K. Xu, and S. Fu, J. Comput. Phys., 229(2010), 6715-6731] and the second-order accurate gas-kinetic scheme [K. Xu, J. Comput. Phys., 171(2001), 289-335]. It is formed by two parts: quartic polynomial reconstruction of the macroscopic variables and fourth-order accurate flux evolution. The first part reconstructs a piecewise cell-center based quartic polynomial and a cell-vertex based quartic polynomial according to the “initial” cell average approximation of macroscopic variables to recover locally the non-equilibrium and equilibrium single particle velocity distribution functions around the cell interface. It is in view of the fact that all macroscopic variables become moments of a single particle velocity distribution function in the gas-kinetic theory. The generalized moment limiter is employed there to suppress the possible numerical oscillation. In the second part, the macroscopic flux at the cell interface is evolved in fourth-order accuracy by means of the simple particle transport mechanism in the microscopic level, i.e. free transport and the Bhatnagar-Gross-Krook (BGK) collisions. In other words, the fourth-order flux evolution is based on the solution (i.e. the particle velocity distribution function) of the BGK model for the Boltzmann equation. Several 1D and 2D test problems are numerically solved by using the proposed high-order accurate gas-kinetic scheme. By comparing with the exact solutions or the numerical solutions obtained the second-order or third-order accurate gas-kinetic scheme, the computations demonstrate that our scheme is effective and accurate for simulating invisid and viscous fluid flows, and the accuracy of the high-order GKS depends on the choice of the (numerical) collision time.

SUMMARYThe accurate numerical modelling of 3-D seismic wave propagation is essential in understanding details to seismic wavefields which are, observed on regional and global scales on the Earth’s surface. The diffusive-viscous wave (DVW) equation was proposed to study the connection between fluid saturation and frequency dependence of reflections and to characterize the attenuation property of the seismic wave in a fluid-saturated medium. The attenuation of DVW is primarily described by the active attenuation parameters (AAP) in the equation. It is, therefore, imperative to acquire these parameters and to additionally specify the characteristics of the DVW. In this paper, quality factor, Q is used to obtain the AAP, and they are compared to those of the visco-acoustic wave. We further derive the 3-D numerical schemes based on a second order accurate finite-volume scheme with a second order Runge–Kutta approximation for the time discretization and a fourth order accurate finite-difference scheme with a fourth order Runge–Kutta approximation for the time discretization. We then simulate the propagation of seismic waves in a 3-D fluid-saturated medium based on the derived schemes. The numerical results indicate stronger attenuation when compared to the visco-acoustic case.

A second-order time accurate and fully-decoupled numerical scheme of the Darcy-Newtonian-Nematic model for two-phase complex fluids confined in the Hele-Shaw cell