Abaqus/CFD求解不可压缩流体问题

Numerical implementation

The solution of the incompressible Navier-Stokes equations poses a number of algorithmic issues due to the divergence-free velocity condition and the concomitant spatial and temporal resolution required to achieve solutions in complex geometries for engineering applications. The Abaqus/CFD incompressible solver uses a hybrid discretization built on the integral conservation statements for an arbitrary deforming domain. For time-dependent problems, an advanced second-order projection method is used with a node-centered finite-element discretization for the pressure. This hybrid approach guarantees accurate solutions and eliminates the possibility of spurious pressure modes while retaining the local conservation properties associated with traditional finite volume methods. An edge-based implementation is used for all transport equations permitting a single implementation that spans a broad variety of element topologies ranging from simple tetrahedral and hexahedral elements to arbitrary polyhedral. In Abaqus/CFD only tetrahedral and hexahedral elements are supported.

Abaqus/CFD不可压求解器采用混合离散方式。对于时变问题,采用一种先进的二阶投影法,压力采用基于节点的有限元离散。这种混合方法保证了精度,消除了伪压模式出现的可能性,保留了传统有限体积法的局部守恒特性。Abaqus/CFD目前仅支持四面体和六面体单元。

Projection method

The basic concept for projection methods is the legitimate segregation of pressure and velocity fields for efficient solution of the incompressible Navier-Stokes equations. Over the past decade, projection methods have found broad application for problems involving laminar and turbulent fluid dynamics, large density variations, chemical reactions, free surfaces, mold filling, and non-Newtonian behavior.

除了SIMPLE方法外,求解不可压缩流体NS方程的方法还有投影法。在过去的几十年里,投影法被广泛应用于层流动力学、湍流动力学、大密度变动、化学反应、自由表面、铸模及非牛顿行为等问题。

In practice, the projection is used to remove the part of the velocity field that is not divergence-free (“div-free”). The projection is achieved by splitting the velocity field into div-free and curl-free components using a Helmholtz decomposition. The projection operators are constructed so that they satisfy prescribed boundary conditions and are norm-reducing, resulting in a robust solution algorithm for incompressible flows.

(背景介绍:一般来说,由于显式格式不需要进行迭代求解,其计算量通常远远低于隐式格式.因此,在大多数的细观数值模拟中都采用显式格式.但对于不可压湍流燃烧过程的数值求解而言存在一个很大的困难,当Ma数很低时声速将趋向于无穷大,这就对显式格式的时间步长提出了严格的要求,为了保证数值计算稳定必须采用极小的时间步长.同时,由于压力波瞬间传遍全场,因此即使采用显式格式求解时,压力项也必须采用隐式格式.于是,如何将压力项从动量方程中解耦就成为求解方法的关键问题. Chorin在1968年提出了一种应用于常密度不可压湍流流动数值模拟的分步投影方法.该方法利用不可压流动速度散度为零的条件将动量方程分解为分别包含速度和压力的两组方程,对这两组方程分别求解.这就使得求解的动量方程中不包含压力项,于是时间步长不再受声速极大的限制,可以采用较大的时间步长进行计算.)

Least-squares gradient estimation

The solution methods in Abaqus/CFD use a linearly complete second-order accurate least-squares gradient estimation. This permits accurate evaluation of dual-edge fluxes for both advective and diffusive processes. All transport equations in Abaqus/CFD make use of the second-order least-squares operators.

Abaqus/CFD采用线性完全二阶最小二乘梯度法求解传递方程。

Advection methods

The implementation of advection in Abaqus/CFD is edge-based, monotonicity-preserving, and preserves smooth variations to second-order in space. The advection relies on a least-squares gradient estimation with unstructured-grid slope limiters that are topology independent. Sharp gradients are captured within approximately 2–3 elements; i.e., , and the use of slope limiting in conjunction with a local diffusive limiter precludes over-/under-shoots in advected fields. The advection is treated explicitly (see the stability discussion in“Time incrementation” below).

Abaqus/CFD中对流的实施是基于边缘、保单调性的,保留空间平滑二次项。高梯度能够在2到3个单元中捕捉到。

Energy equation

The energy transport equation is optionally activated in Abaqus/CFD for non-isothermal flows. For small density variations, the Boussinesq approximation provides the coupling between momentum and energy equations. In turbulent flows, the energy transport includes a turbulent heat flux based on the turbulent eddy viscosity and turbulent Prandtl number. Abaqus/CFD provides a temperature-based energy equation.

对于非等温流,Abaqus/CFD提供了能量传递方程。对于小的密度变动,布辛涅斯克近似提供了动量方程与能量方程的耦合。

The energy equation, in temperature form, can be obtained from the first law of thermodynamics and is given by由热力学第一定律得到

 

where is the specific enthalpy, is heat flux due to conduction defined by Fourier's law, and is the heat supplied externally into the body per unit volume. The energy equation is solved in terms of temperature in Abaqus/CFD.

其中h为比焓(读han),q为传导引起的热通量,r为单位体积的外部供热。

Deforming-mesh ALE

Many industrial CFD/FSI/CHT problems involve moving boundaries or deforming geometries. This class of problem includes prescribed boundary motion that induces fluid flow or where the boundary motion is relatively independent of the fluid flow. Abaqus/CFD uses an arbitrary Lagrangian Eulerian (ALE) formulation and automated mesh deformation method that preserves element size in boundary layers. The ALE and deforming-mesh algorithms are activated automatically for problems that involve a moving boundary prescribed by the user or identified as a moving boundary in an FSI co-simulation.

对于有移动边界或变形几何的情况,Abaqus/CFD提供了任何ALE法和自动网格变形法来保存边界层的网格尺寸。对于移动边界问题ALE和网格变形算法自动被激活。

To properly control the mesh motion during a simulation, it is the user’s responsibility to prescribe appropriate displacement boundary conditions on the computational mesh.

为合适控制网格运动,用户应该定义合适的位移边界条件。

Linear equation solvers

The solution methods for the momentum and auxiliary transport equations in Abaqus/CFD rely on scalable parallel preconditioned Krylov solvers. The pressure, pressure-increment, and distance function equations are solved with user-selectable Krylov solvers and a robust algebraic multigrid preconditioner. A set of preselected default convergence criteria and iteration limits are prescribed for all linear equation solvers. The default solver settings should provide computationally efficient and robust solutions across a spectrum of CFD problems. However, full access to diagnostic information, convergence criteria, and optional solvers is provided. In practice, the pressure-increment equation may be the most sensitive linear system and could require user intervention based on knowledge of the specific flow problem.