Unsteady flow calculation in a radial flow centrifugal pump with spiral casing

The prediction of the two-dimensional unsteady flow established in a radial flow centrifugal pump is considered. Assuming the fluid incompressible and inviscid, the velocity field is represented by means of source and vorticity surface distributions as well as a set of point vortices. Using this representation, a grid-free (Lagrangian) numerical method is derived based on the coupling of the boundary element and vortex particle methods. In this context the source and vorticity surface distributions are determined through the non-entry boundary condition together with the unsteady Kutta condition. In order to satisfy Kelvin's theorem, vorticity is shed at the trailing edges of the impeller blades. Then the vortex particle method is used to approximate the convection of the free vorticity distribution. Results are given for a pump configuration experimentally tested by Centre Technique des Industries Mécaniques (CETIM). Comparisons between predictions and experimental data show the capability of the proposed method to reproduce the main features of the flow considered.     

Interaction of zones of flow separation in a centrifugal impeller-stationary vane system

In a radial flow pump operating in off-design conditions, regions of stall can exist on the rotating impeller blade and on the downstream diffuser blade, vane or tongue. Interaction of these stall zones can generate complex patterns of vorticity concentrations. In turn, these vorticity concentrations are related to sources of unsteady stagnation enthalpy. The form of these patterns is strongly dependent on the instantaneous location of the impeller trailing-edge relative to the leading-edge of the vane.Comparison of instantaneous with ensemble-averaged images shows that the flow structure in the gap region between the impeller and the vane is highly repetitive. Away from this region, in particular in the separated shear layer from the vane, the nonrepetitive nature of the vorticity field is manifested in substantial reduction of peak levels of vorticity in the ensemble-averaged image, relative to the instantaneous image.The three-dimensional flow structure resulting from these separation zone interactions was characterized via end views of the flow patterns. Particularly pronounced concentrations of vorticity can occur in this plane. They tend to be located in the shear layer at the outer edge of the large-scale separation zone. These vorticity concentrations are, however, highly non-stationary for successive passages of the impeller blade. Ensemble-averaging reveals that they persist primarily on the endwalls of the diffuser.The authors are grateful to the Office of Naval Research for support of this research program     

非定常流动的快速拉格朗日涡方法数值模拟

采用快速拉格朗日涡方法数值模拟有复杂旋涡运动的非定常流动. 利用离散涡元模拟旋涡的产生、聚集和输送过程. 拉格朗日描述法用来计算离散涡元的移动,而移动速度则利用广义毕奥-萨伐尔公式结合快速多极子展开法计算,修正的涡半径扩散模型用来模拟离散涡元的黏性扩散. 突然起动圆柱和大攻角下突然起动翼型的非定常有涡流动的数值模拟,及其与试验结果的对比验证了方法的有效性. 另外,大攻角下突然起动翼型的计算结果给出了翼型起动后吸力面旋涡的产生、发展,周期性非定常流动的形成,以及尾流旋涡结构等一些重要的流动特征.[关键词] 非定常流有涡流动快速涡方法      

A three‐dimensional vortex particle‐in‐cell method for vortex motions in the vicinity of a wall

A new vortex particle‐in‐cell method for the simulation of three‐dimensional unsteady incompressible viscous flow is presented. The projection of the vortex strengths onto the mesh is based on volume interpolation. The convection of vorticity is treated as a Lagrangian move operation but one where the velocity of each particle is interpolated from an Eulerian mesh solution of velocity–Poisson equations. The change in vorticity due to diffusion is also computed on the Eulerian mesh and projected back to the particles. Where diffusive fluxes cause vorticity to enter a cell not already containing any particles new particles are created. The surface vorticity and the cancellation of tangential velocity at the plate are related by the Neumann conditions. The basic framework for implementation of the procedure is also introduced where the solution update comprises a sequence of two fractional steps. The method is applied to a problem where an unsteady boundary layer develops under the impact of a vortex ring and comparison is made with the experimental and numerical literature. Copyright © 2001 John Wiley & Sons, Ltd.     

Research on the algorithm of binary image cross-correlation for unsteady flow field measurement

In the present paper, the algorithm of Binary Image Cross-Correlation (BICC) was developed to measure the unsteady flow field. A vortex flow field was used to test the algorithm by numerical simulation. The results show that BICC is an effective algorithm for particle identification from consecutive images, the accurate velocity vector field can be obtained. The real velocity field in a valve chamber was measured by BICC in this study. From the full-field velocity information, the pressure and vorticity fields were also extracted by post-processing. The project supported by the National Natural Science Foundation of China     

LES study on the influence of the diffuser inlet angle of a centrifugal pump on pressure fluctuations

Large-Eddy Simulations are conducted on a centrifugal pump at design and reduced flow-rates for three diffuser geometries, to investigate the effect of changing the diffuser inlet angle on the overall performance and the pressure fields. In particular, pressure fluctuations are investigated, which affect the unsteady loads acting on the pump, as well as vibrations, noise and cavitation phenomena. The considered modification of the diffuser geometry is targeted at decreasing the incidence angle at the off-design flow-rate by rotating the stationary blades of the pump around their leading edge. Results are compared against those of an earlier study, where the same modification of the diffuser inlet angle was achieved by increasing also the radial gap between impeller and diffuser, whose blades were rotated relative to their mid camber location. The comparisons across cases demonstrate that the radial gap between the trailing edge of the impeller blades and the leading edge of the diffuser blades has a more profound influence on pressure fluctuations, compared to the angle of incidence on the diffuser blades of the flow coming from the impeller.     

A combined vortex and panel method for numerical simulations of viscous flows: a comparative study of a vortex particle method and a finite volume method

This paper describes and compares two vorticity‐based integral approaches for the solution of the incompressible Navier–Stokes equations. Either a Lagrangian vortex particle method or an Eulerian finite volume scheme is implemented to solve the vorticity transport equation with a vorticity boundary condition. The Biot–Savart integral is used to compute the velocity field from a vorticity distribution over a fluid domain. The vorticity boundary condition is improved by the use of an iteration scheme connected with the well‐established panel method. In the early stages of development of flows around an impulsively started circular cylinder, and past an impulsively started foil with varying angles of attack, the computational results obtained by the Lagrangian vortex method are compared with those obtained by the Eulerian finite volume method. The comparison is performed separately for the pressure fields as well. The results obtained by the two methods are in good agreement, and give a better understanding of the vorticity‐based methods. Copyright © 2005 John Wiley & Sons, Ltd.     

Solution of the Navier–Stokes equations in velocity–vorticity form using a Eulerian–Lagrangian boundary element method

This paper describes the Eulerian–Lagrangian boundary element model for the solution of incompressible viscous flow problems using velocity–vorticity variables. A Eulerian–Lagrangian boundary element method (ELBEM) is proposed by the combination of the Eulerian–Lagrangian method and the boundary element method (BEM). ELBEM overcomes the limitation of the traditional BEM, which is incapable of dealing with the arbitrary velocity field in advection‐dominated flow problems. The present ELBEM model involves the solution of the vorticity transport equation for vorticity whose solenoidal vorticity components are obtained iteratively by solving velocity Poisson equations involving the velocity and vorticity components. The velocity Poisson equations are solved using a boundary integral scheme and the vorticity transport equation is solved using the ELBEM. Here the results of two‐dimensional Navier–Stokes problems with low–medium Reynolds numbers in a typical cavity flow are presented and compared with a series solution and other numerical models. The ELBEM model has been found to be feasible and satisfactory. Copyright © 2000 John Wiley & Sons, Ltd.     

Three‐dimensional vortex simulation of unsteady flow in hydraulic turbines

On the basis of the Helmholtz decomposition, a grid‐free numerical scheme is provided for the solution of unsteady flow in hydraulic turbines. The Lagrangian vortex method is utilized to evaluate the convection and stretch of the vorticity, and the BEM is used to solve the Neumann problem to define the potential flow. The no‐slip boundary condition is satisfied by generating vortex sticks at the solid surface. A semi‐analytical regularization technique is applied to evaluate the singular boundary surface integrals of the potential velocity and its gradients accurately. The fast multipole method was extended to evaluate the velocity and velocity gradients induced by the discretized vortex blobs in the Lagrangian vortex method. The successful simulation for the unsteady flow through a hydraulic turbine's runner has manifested the effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

Modeling the flow past an oscillating airfoil by the method of viscous vortex domains

The Lagrangian vortex method for solving the Navier-Stokes equations is applied for numerically modeling the unsteady flow past a wing airfoil executing angular oscillations in a viscous incompressible flow. Formulas relating the unsteady forces on the airfoil and the vorticity field are derived. The calculated results are compared with the experimental data for the NACA-0012 airfoil executing harmonic oscillations in an air flow at the Reynolds number Re = 4.4 × 10⁴.     

Diffusion velocity for a three-dimensional vorticity field

A discussion is presented on the existence of a diffusion velocity for the vorticity vector that satisfies extensions of the Helmholtz vortex laws in a three-dimensional, incompressible, viscous fluid flow. A general form for the diffusion velocity is derived for a complex-lamellar vorticity field that satisfies the property that circulation is invariant about a region that is advected with the sum of the fluid velocity and the diffusion velocity. A consequence of this property is that vortex lines will be material lines with respect to this combined velocity field. The question of existence of diffusion velocity for a general three-dimensional vorticity field is shown to be equivalent to the question of existence of solutions of a certain Fredholm equation of the first kind. An example is given for which it is shown that a diffusion velocity satisfying this property does not, in general, exist. Properties of the simple expression for diffusion velocity for a complex-lamellar vorticity field are examined when applied to the more general case of an arbitrary three-dimensional flow. It is found that this form of diffusion velocity, while not satisfying the condition of circulation invariance, nevertheless has certain desirable properties for computation of viscous flows using Lagrangian vortex methods. The significance and structure of the noncomplex-lamellar part of the viscous diffusion term is examined for the special case of decaying homogeneous turbulence.     

High-speed stereoscopic PIV study of rotating instabilities in a radial vaneless diffuser

This paper presents an experimental analysis of the unsteady phenomena developing in a vaneless diffuser of a radial flow pump. Partial flow operating conditions were investigated using 2D/3C high repetition rate PIV, coupled with unsteady pressure transducers. Pressure measurements were acquired on the shroud wall of the vaneless diffuser and on the suction pipe of the pump, whereas PIV flow fields were determined on three different heights in the hub to shroud direction, inside the diffuser. The classical Fourier analysis was applied to both pressure signals to identify the spectral characteristics of the developing instabilities, and the high-order spectral analysis was exploited to investigate possible non-linear interaction mechanisms between different unsteady structures. A dedicated PIV averaging procedure was developed and applied to the PIV flow fields so as to capture and visualize the topology of the spectrally identified phenomena. The influence of these phenomena on the diffuser efficiency was also investigated.     

Construction of three-dimensional images of flow structure via particle tracking techniques

Construction of three-dimensional images of flow structure, based on the quantitative velocity field, is assessed for cases where experimental data are obtained using particle tracking technique. The experimental data are in the form of contiguous planes of particle images. These contiguous data planes are assumed to correspond to successive spatial realizations in steady flow, or to phase-referenced realizations in an unsteady flow.Given the particle images on contiguous planes, the in-plane velocity fields are determined. Then, the out-of-plane velocity field is obtained using a spectral interpolation method. Application of this method allows, in principle, construction of the three-dimensional vorticity field and the streamline patterns.A critical assessment is made of the uncertainties arising from the in-plane interpolation of the velocity field obtained from particle tracking and the evaluation of the out-of-plane velocity component. The consequences of such uncertainties on the reconstructed vorticity distributions and streamline patterns are addressed for two basic types of vortex flows: a columnar vortex, for which the streamlines are not closed and are spatially periodic in the streamwise direction; and for a spherical (Hill's) vortex exhibiting closed streamline patterns, and no spatial periodicity.     

Vortex particle‐in‐cell method for three‐dimensional viscous unbounded flow computations

A new vortex particle‐in‐cell (PIC) method is developed for the computation of three‐dimensional unsteady, incompressible viscous flow in an unbounded domain. The method combines the advantages of the Lagrangian particle methods for convection and the use of an Eulerian grid to compute the diffusion and vortex stretching. The velocity boundary conditions used in the method are of Dirichlet‐type, and can be calculated using the vorticity field on the grid by the Biot–Savart equation. The present results for the propagation speed of the single vortex ring are in good agreement with the Saffman's model. The applications of the method to the head‐on and head‐off collisions of the two vortex rings show good agreement with the experimental and numerical literature. Copyright © 2000 John Wiley & Sons, Ltd.     

VISCOUS FLOWFIELD BASED ON DISCONTINUOUS BOUNDARY ELEMENT METHOD AND VORTEX METHOD

基于非协调边界元方法和涡方法的联合应用, 模拟了二维和三维黏性不可压缩流场. 计算中利用离散涡元对漩涡的产生、凝聚和输送过程进行模拟, 并将整体计算域分解为采用涡泡模拟的内部区域和用涡列模拟的数字边界层区域. 计算域中涡量场的拉伸和对流由Lagrangian涡方法模拟, 用随机走步模拟涡量场的扩散. 内部区域涡元涡量场速度由广义Biot-Savart公式计算, 势流场速度则采用非协调边界元方法计算. 非协调边界元将所有节点均取在光滑边界处, 从而避免了法向速度的不连续现象; 而对于系数矩阵不对称的大型边界元方程组,引入了非常高效的预处理循环型广义极小残余(the generalized minimum residual, GMRES)迭代算法, 使得边界元法的优势得到了充分发挥, 同时, 在内部涡元势流场计算中对近边界点采用了正则化算法, 该算法将奇异积分转化为沿单元围道上一系列线积分, 消除了势流计算中速度及速度梯度的奇异性. 二维、三维流场算例证明了所用方法的正确性, 也验证了该算法可以大幅度提高模拟精度和效率.     

二级轻气炮内弹道的数值模拟与性能分析

为了优化二级轻气炮的设计参数和工作参数,必须进行数值模拟。本文介绍了准一维不定常流的Lagrangian算法,讨论了数值模拟中的若干问题,分析了气动过程。根据我们的数值模拟,归纳了影响气炮性能的一些基本规律,可供选择工作参数时综合分析。     

Experimental investigation of the flow in the vaneless diffuser of a centrifugal pump by particle image displacement velocimetry

This work presents the application of particle image displacement velocimetry to the measurement of fluid velocities in a centrifugal pump diffuser. Measurements are taken at different operating points and allow to define the variation of radial and tangential velocity components along a pitch. They are further processed to determine the relative velocity and vorticity fields. Results are also compared with laser Doppler measurements taken in the same facility.     

Effects of modifying the blade trailing edge profile on unsteady pressure pulsations and flow structures in a centrifugal pump

In the present work, the effects of modifying the blade pressure side (EPS profile) on unsteady pressure pulsations and flow structures in a low specific speed centrifugal pump are carried out by experimental and numerical methods. Results are compared to the original trailing edge (OTE profile). Unsteady pressure signals are captured at twenty measuring points at flow rate of 0–1.6Q_d. It is observed that the pump head of the EPS profile is improved for all the concerned working conditions. Pressure amplitudes at the blade passing frequency are compared and discussed in detail. It is found that the EPS profile contributes to pressure pulsation reduction obviously. For all the measured flow rates, pressure amplitudes are attenuated evidently at major measuring positions, especially at high flow rates. As for the mean pressure amplitude of twenty measuring points, pressure amplitude is reduced more than 20% at the nominal flow rate using the EPS profile. From relative velocity distribution, it is found that the uniformity of flow field at the blade outlet region would be improved significantly by the EPS profile. Besides, the corresponding vorticity magnitude at the blade outlet would be reduced compared to the OTE profile. The combined effects contribute to the reduction of pressure amplitude using the EPS profile.     

Application of Particle Image Velocimetry and Reference Image Topography to jet shock cells using the hydraulic analogy

This paper examines the shock cell structure, vorticity and velocity field at the exit of an underexpanded jet nozzle using a hydraulic analogy and the Reference Image Topography technique. Understanding the flow in this region is important for the mitigation of screech, an aeroacoustic problem harmful to aircraft structures. Experiments are conducted on a water table, allowing detailed quantitative investigation of this important flow regime at a greatly reduced expense. Conventional Particle Image Velocimetry is employed to determine the velocity and vorticity fields of the nozzle exit region. Applying Reference Image Topography, the wavy water surface is reconstructed and when combined with the hydraulic analogy, provides a pressure map of the region. With this approach subtraction of surfaces is used to highlight the unsteady regions of the flow, which is not as convenient or quantitative with conventional Schlieren techniques. This allows a detailed analysis of the shock cell structures and their interaction with flow instabilities in the shear layer that are the underlying cause of jet screech.