Numerical solutions of fully non‐linear and highly dispersive Boussinesq equations in two horizontal dimensions

This paper investigates preconditioned iterative techniques for finite difference solutions of a high‐order Boussinesq method for modelling water waves in two horizontal dimensions. The Boussinesq method solves simultaneously for all three components of velocity at an arbitrary z‐level, removing any practical limitations based on the relative water depth. High‐order finite difference approximations are shown to be more efficient than low‐order approximations (for a given accuracy), despite the additional overhead. The resultant system of equations requires that a sparse, unsymmetric, and often ill‐conditioned matrix be solved at each stage evaluation within a simulation. Various preconditioning strategies are investigated, including full factorizations of the linearized matrix, ILU factorizations, a matrix‐free (Fourier space) method, and an approximate Schur complement approach. A detailed comparison of the methods is given for both rotational and irrotational formulations, and the strengths and limitations of each are discussed. Mesh‐independent convergence is demonstrated with many of the preconditioners for solutions of the irrotational formulation, and solutions using the Fourier space and approximate Schur complement preconditioners are shown to require an overall computational effort that scales linearly with problem size (for large problems). Calculations on a variable depth problem are also compared to experimental data, highlighting the accuracy of the model. Through combined physical and mathematical insight effective preconditioned iterative solutions are achieved for the full physical application range of the model. Copyright © 2004 John Wiley & Sons, Ltd.     

Implicit schemes with large time step for non‐linear equations: application to river flow hydraulics

In this work, first‐order upwind implicit schemes are considered. The traditional tridiagonal scheme is rewritten as a sum of two bidiagonal schemes in order to produce a simpler method better suited for unsteady transcritical flows. On the other hand, the origin of the instabilities associated to the use of upwind implicit methods for shock propagations is identified and a new stability condition for non‐linear problems is proposed. This modification produces a robust, simple and accurate upwind semi‐explicit scheme suitable for discontinuous flows with high Courant–Friedrichs–Lewy (CFL) numbers. The discretization at the boundaries is based on the condition of global mass conservation thus enabling a fully conservative solution for all kind of boundary conditions. The performance of the proposed technique will be shown in the solution of the inviscid Burgers' equation, in an ideal dambreak test case, in some steady open channel flow test cases with analytical solution and in a realistic flood routing problem, where stable and accurate solutions will be presented using CFL values up to 100. Copyright © 2004 John Wiley & Sons, Ltd.     

基于HLL-HLLC的高阶WENO格式及其应用研究

HLL-HLLC格式能够克服HLLC在强激波附近的激波不稳定现象,并且保持了HLLC的低耗散特性,是一种适合更大马赫数范围的近似黎曼求解器。本文从RANS方程出发,将HLL-HLLC近似黎曼求解器结合五阶WENO重构,实现了对无粘通量的高阶离散;同时,采用完全守恒形式的四阶中心差分格式处理粘性项,建立了RANS 方程的高阶数值求解格式。通过对四个经典算例,钝头体、 ONERA M6机翼、DLR F6-WB翼身组合体和DLR F6-WBNP复杂外形的数值模拟,考察了两种WENO改进格式在复杂流场中的表现,研究了高阶格式的收敛特性;给出了在复杂流动中 WENO自由参数的推荐值,以增强求解的收敛性。算例结果表明,本文构造的高阶格式鲁棒性好,能够显著改善激波位置和激波强度,捕获更丰富的流场细节,满足复杂工程应用需求。     

A fluid solver based on vorticity–helical density equations with application to a natural convection in a cubic cavity

We study numerically a recently introduced formulation of incompressible Newtonian fluid equations in vorticity–helical density and velocity–Bernoulli pressure variables. Unlike most numerical methods based on vorticity equations, the current approach provides discrete solutions with mass conservation, divergence‐free vorticity, and accurate kinetic energy balance in a simple and natural way. The method is applied to compute buoyancy‐driven flows in a differentially heated cubic enclosure in the Boussinesq approximation for Ra ∈ {10⁴,10⁵,10⁶}. The numerical solutions on a finer grid are of benchmark quality. The computed helical density allows quantification of the three‐dimensional nature of the flow. Copyright © 2011 John Wiley & Sons, Ltd.     

A linear solver based on algebraic multigrid and defect correction for the solution of adjoint RANS equations

A new solution approach for discrete adjoint Reynolds‐averaged Navier–Stokes equations is suggested. The approach is based on a combination of algebraic multigrid and defect correction. The efficient interplay of these two methods results in a fast and robust solution technique. Algebraic multigrid deals very well with unstructured grids, and defect correction completes it in such a way that a required second‐order accuracy of the solution is achieved. The performance of the suggested solution approach is demonstrated on a number of representative benchmarks.Copyright © 2014 John Wiley & Sons, Ltd.     

A generic framework for time-stepping partial differential equations (PDEs): general linear methods,object-oriented implementation and application to fluid problems

Time-stepping algorithms and their implementations are a critical component within the solution of time-dependent partial differential equations (PDEs). In this article, we present a generic framework – both in terms of algorithms and implementations – that allows an almost seamless switch between various explicit, implicit and implicit–explicit (IMEX) time-stepping methods. We put particular emphasis on how to incorporate time-dependent boundary conditions, an issue that goes beyond classical ODE theory but which plays an important role in the time-stepping of the PDEs arising in computational fluid dynamics. Our algorithm is based upon J.C. Butcher's unifying concept of general linear methods that we have extended to accommodate the family of IMEX schemes that are often used in engineering practice. In the article, we discuss design considerations and present an object-oriented implementation. Finally, we illustrate the use of the framework by applications to a model problem as well as to more complex fluid problems.