Method For Unsteady Flows Research Articles

Accuracy and Efficiency Assessment of Harmonic Balance Method for Unsteady Flow in Multi-Stage Turbomachinery

With the relative movement of neighboring blade rows, flows in multi-stage turbomachinery are unsteady and periodic in time at the design condition. As an alternative to the widely used time domain time marching method, the harmonic balance (HB) method has been successfully applied to simulate the essentially unsteady flow of multi-stage turbomachinery. By modelling various number of harmonics, the accuracy of this method could be adjusted at different level of computational cost. In this article, accuracy of the harmonic balance method is not only validated against the time domain time marching method, as in most previous works on this topic, but also against the data from an experiment campaign of a two-stage high-pressure turbine where strong tip leakage flow exists. Efficiency of this method is also assessed in detail by adjusting the number of harmonics and comparing with time domain time marching solution results. Results show that the harmonic balance method is a flexible tool with adjustable accuracy for fast-turnaround unsteady flow simulation of multi-stage turbomachinery. Results from this work can provide a guidance in applying the harmonic balance method with balance between accuracy and computational cost.

Robust high‐order semi‐implicit backward difference formulas for Navier‐Stokes equations

SummaryTaylor‐Hood finite elements provide a robust numerical discretization of Navier‐Stokes equations (NSEs) with arbitrary high order of accuracy in space. To match the accuracy of the lowest degree Taylor‐Hood element, we propose a very efficient time‐stepping methods for unsteady flows, which are based on high‐order semi‐implicit backward difference formulas (SBDF) and the inclusion of grad‐div term in the NSE. To mitigate the impact on the numerical accuracy (in time) of the extrapolation of the nonlinear term in SBDF, several variants of nonlinear extrapolation formulas are investigated. The first approach is based on an extrapolation of the nonlinear advection term itself. The second formula uses the extrapolation of the velocity prior to the evaluation of the nonlinear advection term as a whole. The third variant is constructed such that it produces similar error on both velocity and pressure to that with fully implicit backward difference formulas methods at a given order of accuracy. This can be achieved by fixing one‐order higher than usually done in the extrapolation formula for the nonlinear advection term, while keeping the same extrapolation formula for the time derivative. The resulting truncation errors (in time) between these formulas are identified using Taylor expansions. These truncation error formulas are shown to properly represent the error seen in numerical tests using a 2D manufactured solution. Lastly, we show the robustness of the proposed semi‐implicit methods by solving test cases with high Reynolds numbers using one of the nonlinear extrapolation formulas, namely, the 2D flow past circular cylinder at Re=300 and Re = 1000 and the 2D lid‐driven cavity at Re = 50 000 and Re = 100 000. Our numerical solutions are found to be in a good agreement with those obtained in the literature, both qualitatively and quantitatively.

СУЧАСНІ ПІДХОДИ ДО ВИЗНАЧЕННЯ ПРОНИКНОСТІ ПОРІД-КОЛЕКТОРІВ ЗА ДАНИМИ ГЕОФІЗИЧНИХ ДОСЛІДЖЕНЬ

Permeability of rock is its physical property that describes its ability to conduct fluids under the pressure gradient. This paper presents short description and analysis of methods for determination of permeability of oil and gas reservoirs. Permeability is a function of different parameters that leads to difficulties during its estimation. Investigations of the void space structure of rocks, their anisotropy etc.were carried out in order to take into account all factors that have an influence on the permeability. Reservoir conditions could also be modeled for that purpose. Methods for determination of permeability of rocks can be divided into three groups: methods based on the laboratory studies of rocks; methods based on the well logging data; and methods based on the correlations between different parameters of rocks. The first two groups include methods for steady and unsteady fluid flow. Methods for the unsteady flow are usually more precise and rapid, thus prospects of extension of methods for permeability determination are mostly connected with them. Each of the presented methods to determine permeability is characterized by some pros and cons. The most appropriate method for the specific experiment is always chosen according to conditions and requirements and expected results. Further author's investigations will be related to the creation of petrophysical models of permeability of oil and gas reservoir rocks, including reservoirs of complex structure.

구조 변형이 있는 평면 위의 비정상 유동해석을 위한 준-정상 기법

구조 변형이 있는 평면 위의 비정상 유동해석을 위한 준-정상 기법

A Meshless Solution Method for Unsteady Flow with Moving Boundary

Using the concept of overlapping mesh method for reference, a new method called as Overlapping Clouds of Points Method (OCPM) is firstly proposed to simulate unsteady flow with moving boundary problems based on meshless method. Firstly, a set of static background discrete points is generated in the whole calculation zone. Secondly, moving discrete points are created around moving body. According to the initial position of moving object in the flow field, the two sets of discrete points can be overlapped. With the motion of moving objects in the calculation field, moving discrete points around the moving body will inherently move. The exchange of flow field information between static points and moving points is realized by the solution of the clouds of points made up of static and moving discrete points using weighted meshless method nearby overlapping boundary. Four cases including piston problem, NACA0012 airfoil vibration flow around a moving sphere in supersonic and multibody separation are given to verify accuracy and practicability of OCPM. The numerical results agree well with exact solution and experimental results, which shows that the proposed OCPM can be applied to the simulation of unsteady flow problem.

Spatial accuracy and performance of a mixed‐order, explicit multi‐stage method for unsteady flows

SUMMARYWe assess the spatial accuracy and performance of a mixed‐order, explicit multi‐stage method in which an inexpensive low‐order scheme is used for the initial stages, and a more expensive high‐order scheme is used for the final stage only. Compared with the use of a high‐order scheme for all stages, we observe that the mixed‐order scheme achieves comparable accuracy and convergence while providing a speed‐up of a factor of two on mesh sizes of O(106 − 107) tetrahedron. For calculations with significant adaptive mesh refinement, a more modest speed‐up of 30% is obtained. Published 2012. This article is a US Government work and is in the public domain in the USA.

REVIEWS OF MOVING GRID GENERATION TECHNIQUES AND NUMERICAL METHODS FOR UNSTEADY FLOW

CFD methods forunsteady flow simulations are reviewed in this paper, includingmoving grid generation techniques and unsteady numerical methods.Based on the topology of computational grids, two types ofunsteady numerical methods are discussed, e.g. the methods basedon structured grids and unstructured/hybrid grids, respectively.Their advantages and drawbacks are analyzed with regard toefficiency, robustness for complex geometries and other features.In the framework of structured grids, several moving gridtechniques are reviewed, including the rigid moving structuredgrid approach, the deforming structured grid technique withtrans-finite interpolation, overlapping moving structured gridtechnique, and patched moving structured grid technique. Then someunsteady numerical methods on structured grids are discussed inthe systems of inertial and non-inertial coordinates. In addition,the discretization schemes in temporal direction and the GeometricConservation Law (GCL) are emphasized for unsteady simulations onmoving grids. In the context of unstructured/hybrid grids,overlapping unstructured grid technique, re-meshing unstructuredgrid technique, node-relaxation approach based on `Spring analogy'and Delaunay mapping are reviewed and compared. And, a combineddynamic unstructured/hybrid grid method developed by the authorsis suggested, which is a kind of coupled method of `Spring'relaxation, Delaunay mapping and local re-meshing. The GCLapproach, the convergence acceleration methods, and numericalschemes for unsteady incompressible flows are discussed forunsteady flow solvers on unstructured/hybrid grids. Some examplesand applications performed by the authors are given at the end ofthe paper.

Gridless Method for Unsteady Flows Involving Moving Discrete Points and its Applications

A gridless method for dealing with unsteady flow involving moving discrete points is investigated. The moving principle of discrete points for gridless method is proposed. The spring analogy is modified to control the movements of points caused by the motion and torsion of boundaries, so the adaptation and precision of the analogy are enhanced. The algorithm for solving Arbitrary Lagrangian-Eulerian (ALE) equations based on gridless method is accomplished. On the base of clouds in the computational region, the spatial derivatives are approximated by local least-square curve fits, HLLC (Harten, Lax, van Leer, Contact) scheme is extended into the gridless method to calculate the numerical flux and a method of flux limiter is employed in order to improve the accuracy, a multistage Runge-Kutta algorithm is used to advance the equation in time. The oscillatory pitching movement of representative airfoils is simulated, the computational results agree well with the experimental data, and the plunging movement is simulated too. It indicates that the method is successful and good at catching shock waves.

Blending Dual Time Stepping and Newton‐Krylov Methods for Unsteady Flows

AbstractWe explore the idea of using nonlinear schemes as preconditioners in Newton‐Krylov schemes for unsteady flow computations. Analysis shows that left preconditioning changes the Newton scheme in a non equivalent way, leading to a stall in Newton convergence, whereas right preconditioning leads to a sound method. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)