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不可壓求解器采用混合離散方式。對于時變問題,采用一種先進的二階投影法,壓力采用基于節(jié)點的有限元離散。這種混合方法保證了精度,消除了偽壓模式出現(xiàn)的可能性,保留了傳統(tǒng)有限體積法的局部守恒特性。Abaqus/CFD目前僅支持四面體和六面體單元。

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,求解不可壓縮流體NS方程的方法還有投影法。在過去的幾十年里,投影法被廣泛應(yīng)用于層流動力學(xué)、湍流動力學(xué)、大密度變動、化學(xué)反應(yīng)、自由表面、鑄模及非牛頓行為等問題。

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.

(背景介紹:一般來說,由于顯式格式不需要進行迭代求解,其計算量通常遠遠低于隱式格式.因此,在大多數(shù)的細觀數(shù)值模擬中都采用顯式格式.但對于不可壓湍流燃燒過程的數(shù)值求解而言存在一個很大的困難,當(dāng)Ma數(shù)很低時聲速將趨向于無窮大,這就對顯式格式的時間步長提出了嚴格的要求,為了保證數(shù)值計算穩(wěn)定必須采用極小的時間步長.同時,由于壓力波瞬間傳遍全場,因此即使采用顯式格式求解時,壓力項也必須采用隱式格式.于是,如何將壓力項從動量方程中解耦就成為求解方法的關(guān)鍵問題. Chorin在1968年提出了一種應(yīng)用于常密度不可壓湍流流動數(shù)值模擬的分步投影方法.該方法利用不可壓流動速度散度為零的條件將動量方程分解為分別包含速度和壓力的兩組方程,對這兩組方程分別求解.這就使得求解的動量方程中不包含壓力項,于是時間步長不再受聲速極大的限制,可以采用較大的時間步長進行計算.)

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采用線性完全二階最小二乘梯度法求解傳遞方程。

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中對流的實施是基于邊緣、保單調(diào)性的,保留空間平滑二次項。高梯度能夠在2到3個單元中捕捉到。

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.

對于非等溫流,Abaqus/CFD提供了能量傳遞方程。對于小的密度變動,布辛涅斯克近似提供了動量方程與能量方程的耦合。

The energy equation, in temperature form, can be obtained from the first law of thermodynamics and is given by由熱力學(xué)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為比焓(讀han),q為傳導(dǎo)引起的熱通量,r為單位體積的外部供熱。

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.

對于有移動邊界或變形幾何的情況,Abaqus/CFD提供了任何ALE法和自動網(wǎng)格變形法來保存邊界層的網(wǎng)格尺寸。對于移動邊界問題ALE和網(wǎng)格變形算法自動被激活。

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.

為合適控制網(wǎng)格運動,用戶應(yīng)該定義合適的位移邊界條件。

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.