High-order accurate simulations of unsteady flow past plunging and pitching airfoils

This paper presents simulations of unsteady flow past plunging and pitching airfoils using a high-order spectral difference (SD) method. Both third-order and fourth-order SD methods are employed on unstructured quadrilateral grids for the plunging airfoil at a low Reynolds number. The vortex shedding pattern of an airfoil in an oscillating plunge motion becomes asymmetric at a sufficiently high frequency. The SD method is able to capture this effect and reveal a fine structure that closely replicates the experimental photograph. Interestingly, our simulations also predict that the degree of this asymmetry increases with Reynolds number. Unsteady flow at a higher Reynolds number past a pitching airfoil is studied using the fifth-order SD method. Our predictions show very good agreements with the available experimental data. The developed high-order accurate SD algorithms could enable high-order accurate simulations of unsteady flow past flapping Micro-Air-Vehicles (MAVs).     

Investigation of flow around a pair of side-by-side square cylinders using the lattice Boltzmann method

The low-Reynolds number flow around two square cylinders placed side-by-side is investigated using the lattice Boltzmann method (LBM). The effects of the gap ratio s/d (s is the separation between the cylinders and d is the characteristic dimension) on the flow are studied. These simulations reveal the existence of regimes with either synchronized or non-synchronized vortex-shedding, with transition occurring at s/d ≈ 2, which is larger than for circular cylinders. Detailed results are presented at Re = 73 for s/d = 2.5 and 0.7 corresponding to the synchronized and flip-flop regimes, respectively. Vortex-shedding from the cylinder occurs either in-phase or in-antiphase in the synchronized regime. However, linear stochastic estimate (LSE) calculations show that in-phase locking is the predominant mode. LSE is also employed to educe the underlying modes in the flip-flop regime, where evidence for both in-phase and anti-phase locked vortices is found, indicating that this regime is in a quasi-stable state between these two modes. The merging of the wakes, which is gradual for the synchronized regime, occurs rapidly in the flip-flop regime. The mean pressure on the upstream surface is symmetric and asymmetric for the synchronized and flip-flop regimes, respectively. Differences in results between the two regimes are interpreted in terms of the interaction of the jet formed between the cylinders with the adjoining wakes, the strength of this interaction depending on the spacing.     

二维波动方程的一种高精度紧致差分方法

提出了一种求解二维波动方程的高精度紧致差分方法,该方法首先利用紧交替方向隐式差分格式,其截断误差为O(τ2+h4),分别在粗网格和细网格上对原方程进行求解,然后利用Richardson外推计算一次,进一步提高精度,得到了二维波动方程具有O(τ4+h6)精度的数值解。数值实验验证了该方法的可靠性、有效性和精确性。     

Spectral-element discontinuous Galerkin lattice Boltzmann simulation of flow past two cylinders in tandem with an exponential time integrator

In this paper, a spectral-element discontinuous Galerkin (SEDG) lattice Boltzmann discretization and an exponential time-marching scheme are used to study the flow field past two circular cylinders in tandem arrangement. The basic idea is to discretize the streaming step of the lattice Boltzmann equation by using the SEDG method to get a system of ordinary differential equations (ODEs) whose exact solutions are expressed by using a large matrix exponential. The approximate solution of the resulting ODEs are obtained from a projection method based on a Krylov subspace approximation. This approach allows us to approximate the matrix exponential of a very large and sparse matrix by using a matrix of much smaller dimension. The exponential time integration scheme is useful especially when computations are carried out at high Courant–Friedrichs–Lewy ($CFL$) numbers, where most explicit time-marching schemes are inaccurate. Simulations of flow were carried out for a circular cylinder at $Re=20$ and for two circular cylinders in tandem at $Re=40$ and a spacing of $2.5D$, where $D$ is the diameter of the cylinders. We compare our results with those from a fourth-order Runge–Kutta scheme that is restricted by the $CFL$ number. In addition, important flow parameters such as the drag coefficients of the two cylinders and the wake length behind the rear cylinder were calculated by using the exponential time integration scheme. These results are compared with results from our simulation using the RK scheme and with existing benchmark results.     

Numerical simulation of flow past row of square cylinders for various separation ratios

A systematic study for the flow around a row of five square cylinders placed in a side-by-side arrangement and normal to the oncoming flow at a Reynolds number of 150 is carried out through the numerical solution of the two-dimensional unsteady incompressible Navier-Stokes equations. Special attention is paid to investigate the effect of the spacing between the five cylinders on the wake structure and vortex shedding mechanism. The simulations are performed for the separation ratios (spacing to size ratio) of 1.2, 2, 3 and 4. Depending on the separation ratio the following flow patterns are observed: a flip-flopping pattern, in-phase and anti-phase synchronized pattern and non-synchronized pattern. These flow patterns are supposed to be a consequence of the interaction between two types of frequencies viz. the vortex shedding (primary) and the cylinder interaction (secondary) frequencies. At small separation ratio the flow is predominantly characterized by the jet in the gaps between successive cylinders and the secondary frequencies play a role in the resulting chaotic flow. On the contrary, at higher separation ratio the secondary frequencies almost disappear and the resulting flow becomes more synchronized dominated by the primary frequency.     

利用层流分布测量微流控芯片压力的方法研究

针对微流控芯片中压力相关测量的问题,提出了一种利用微尺度层流分布测量微流控芯片中的压力(流速)的方法。主要研究内容包括微汇合层流结构的设计(Y型结构),微尺度层流分布的压力(流速)关系的推导和利用显微图像处理进行测量的方法等,并通过计算机仿真和实际芯片实验对Y型测压结构进行了分析,结果证实了提出方法的有效性。     

Numerical simulation of the flow around rows of cylinders

Two-dimensional, laminar, unsteady, water flow around cylinder arrays of unequal sizes was simulated using FLUENT™ at Reynolds numbers below 150 (based on the free-stream velocity and first row cylinder diameter). The flow pattern through two rows of inline cylinders showed incomplete vortex shedding behind the first row at a separation distance less than 2d. Karman vortices were not formed and a near-stagnant separated flow region appeared between the aligned cylinders. Cylinders in staggered arrangements shed Karman vortices regardless of the separation between the two rows. This research has shed light on the detailed flow through paper machine forming fabrics.     

MHD flow past a circular cylinder using the immersed boundary method

The immersed boundary method (IB hereafter) is an efficient numerical methodology for treating purely hydrodynamic flows in geometrically complicated flow-domains. Recently Grigoriadis et als. [1] proposed an extension of the IB method that accounts for electromagnetic effects near non-conducting boundaries in magnetohydrodynamic (MHD) flows. The proposed extension (hereafter called MIB method) integrates naturally within the original IB concept and is suitable for magnetohydrodynamic (MHD) simulations of liquid metal flows. It is based on the proper definition of an externally applied current density field in order to satisfy the Maxwell equations in the presence of arbitrarily-shaped, non-conducting immersed boundaries. The efficiency of the proposed method is achieved by fast direct solutions of the two poisson equations for the hydrodynamic pressure and the electrostatic potential.The purpose of the present study is to establish the performance of the new MIB method in challenging configurations for which sufficient details are available in the literature. For this purpose, we have considered the classical MHD problem of a conducting fluid that is exposed to an external magnetic field while flowing across a circular cylinder with electrically insulated boundaries. Two- and three-dimensional, steady and unsteady, flow regimes were examined for Reynolds numbers Re_d ranging up to 200 based on the cylinder’s diameter. The intensity of the external magnetic field, as characterized by the magnetic interaction parameter N, varied from N=0 for the purely hydrodynamic cases up to N=5 for the MHD cases. For each simulation, a sufficiently fine Cartesian computational mesh was selected to ensure adequate resolution of the thin boundary layers developing due to the magnetic field, the so called Hartmann and sidewall layers. Results for a wide range of flow and magnetic field strength parameters show that the MIB method is capable of accurately reproducing integral parameters, such as the lift and drag coefficients, as well as the geometrical details of the recirculation zones. The results of the present study suggest that the proposed MIB methodology provides a powerful numerical tool for accurate MHD simulations, and that it can extend the applicability of existing Cartesian flow solvers as well as the range of computable MHD flows. Moreover, the new MIB method has been used to carrry out a series of accurate simulations allowing the determination of asymptotic laws for the lift and drag coefficients and the extent of the recirculation length as a function of the amplitude of the magnetic field. These results are reported herein.     

Molecular dynamics simulation of flow past a plate

Molecular dynamics simulations with a soft-sphere potential have been carried out to model two dimensional fluid flow obstructed by a plate. At fluid velocities large enough to obtain adequate signal to noise resolution, two counter-circulating vortices are observed behind the obstruction. The stationary state length scale of these vortices is found to be roughly proportional to the average velocity in the system, as predicted by the hydrodynamic theory.     

Numerical simulation of three-dimensional flow past bluff bodies

An improved understanding of the aerodynamics of bluff bodies, such as buildings and road vehicles, can lead to significant reductions in wind damage and gasoline consumption and to the increased safety and comfort of human occupants. To achieve this goal, improved theoretical and experimental techniques are urgently needed. This paper explores in a general way the potential of using numerical simulation methods for predicting and interpreting aerodynamic phenomena affecting bluff bodies. As a basis for discussion a prototype finite difference method is described and illustrated with sample calculations of air flow about simple bluff bodies. The limitations of this scheme are then discussed in detail, together with some suggestions for extensions that could be realized in the immediate future. The paper concludes with speculations on what could be achieved in the next five to ten years to produce a generally useful research tool for bluff body aerodynamics.     

The numerical solution of laminar flow in a re-entrant tube geometry by a Chebyshev spectral element collocation method

A Chebyshev spectral element collocation method is used for the numerical solution of laminar flow in a re-entrant tube geometry. The high accuracy of the method allows the study of the merging process of the vortices in the re-entrant region. Results are presented for values of the Reynolds number up to 400 and for different lengths of the re-entrant tube lip. For low Reynolds number, the observed behaviour of the flow in the re-entrant region may be interpreted by analytical results on Stokes flow from the literature. As the Reynolds number is increased, a vortex that grows in size with the Reynolds number, is detected below the tip of the re-entrant lip.     

A collocated finite volume embedding method for simulation of flow past stationary and moving body

A simple and conservative numerical scheme is introduced in this paper to simulate unsteady flow around stationary and moving body. Based on the embedding method (immersed boundary (IB) + volume of fluid (VOF)) implemented in the finite-volume framework, flow past the arbitrarily complex geometry can be readily computed on any existing mesh system. Flow variables stored at cell centers, including those residing within the immersed body, are computed where the induced effect on the flow due to the immersed body is realised via a simple acceleration term (forcing function) derived based on the VOF value. In the current work, an identical VOF value is used for all momentum equations, in contrast to that of the pre-existing method, whereby numerical interpolation is required. The method is verified with a number of flow cases, including flow in a 2D square cavity, flow past a stationary and oscillating cylinder and flow induced by a flapping ellipse in an enclosure.     

Numerical calculation of a laminar two dimensional straight cascade flow

The laminar and incompressible flow in a straight cascade is investigated. Numerical solutions of the full Navier-Stokes equations are obtained using the vorticity-stream function formulation and body fitted coordinate system. The numerical method includes a special force balance for the determination of the downstream boundary condition and a double sweep deferred correction which allows a second order accuracy but with the stability properties of an upwind first order scheme. Results for cylindrical, elliptical and NACA 0012 airfoils are presented including separated flow regions. Good agreement with experiments and previous computations is obtained.     

Aerodynamic noise simulation of the flow past an airfoil trailing-edge using a hybrid zonal RANS-LES

This paper document the evaluation of a zonal RANS-LES approach for the prediction of broadband and tonal noise generated by the flow past an airfoil trailing edge at a high Reynolds number. A multi-domain decomposition is considered, where the acoustic sources are resolved with a LES sub-domain embedded in the RANS domain. At the RANS-LES interface, a stochastic vortex method is used to generate synthetic turbulent perturbations. The simulations are performed with the general-purpose unstructured control-volume code FLUENT. The far-field noise is calculated using the aeroacoustic analogy of Ffowcs-Williams and Hawkings. The results of the simulation are compared with available acoustic and mean velocity measurements. The investigation demonstrates the ability of this approach to predict the aerodynamic and aeroacoustic properties of the flow. Two simulations are performed in order to address the sensitivity of the results to the perturbation model. The comparison clearly indicates the critical influence of the model.     

Examination of simulation output using spectral methods

Spectral procedures that have been developed to estimate the variance of the sample mean of simulation output are presented and evaluated. The evaluation is performed using a single server queueing situation with exponential interarrival and service times. This process is employed because theoretical values are available for use in the evaluation. In addition, the assumptions necessary to apply spectral methods are valid for this process. The results of the evaluation indicate that spectral methods produce unreliable estimates of the variance of the sample mean. Thus, it is concluded that spectral methods will not produce reliable estimates of the variance of the sample mean for general simulation output.     

CZ60/30型船用空压机缸内压差数值模拟

为了解压差式空压机二级缸内压力相互作用情况,提高工作效率,基于CZ60/30船用空压机气缸和活塞的结构以及该型空压机工作特点,建立高、低压级气缸内压力和压差关系的数学模型,通过MATLAB进行模拟.通过模拟结果分析气缸余隙高度对缸内压力和压差的影响情况,并得到高、低压级气缸压力随时间的变化曲线,为高效的空压机设计和故障分析提供参考.     

精确的互相关算法在超声波流速测量中的应用

互相关算法能够比较准确地计算出声波在流速正反2个方向上的传播时间,进而得出时间差,并根据时间差求出流体的流速。但离散的互相关函数峰值的精度取决于采样间隔的大小,间隔减小精度会相应提高,但在实际的应用中计算量也会大幅度提高。在以单片机为核心的测量仪表中,这种规模的计算通常是不会被采用的,找到一种更简便的计算方法是解决问题的关键。在离散的互相关函数峰值附近,利用抛物线算法可以得出更精确的互相关函数峰值。这种方法允许采样间隔较大,计算时间可大幅度缩短,使同一测量精度的计算时间从3.7 s缩短到0.9 s左右。     

Boundary conditions for shear flow past a permeable interface modeled as an array of cylinders

The boundary condition relating the macroscopic jump in the tangential velocity across a permeable interface consisting of a particulate lattice to the shear rate prevailing on either side of the interface is discussed. The computation of the velocity jump hinges on the realization that shear flow on one side of the interface induces a slip velocity on that side and a streaming drift velocity on the other side. The direction and magnitude of the slip and drift velocities depend on the interface constitution, solid fraction, and Reynolds number. Numerical computations are performed for a model two-dimensional interface consisting of a periodic array of cylinders. In the case of longitudinal unidirectional flow, the boundary conditions are defined in terms of previously computed drift and slip velocity coefficients for any ratio of the shear rates above and below the interface and any Reynolds number. To study the behavior in the complementary case of transverse flow, the Navier-Stokes equation is solved numerically using a finite-difference method on an orthogonal grid generated by conformal mapping, using the stream function/vorticity formulation. The results reveal that inertial effects promote the magnitude of the slip and drift velocity, and illustrate the streamline pattern near the interface.     

Numerical simulation of thrust generating flow past a pitching airfoil

Earlier analytical and experimental studies predict that pitching motions at high frequency can generate thrust on the airfoil. The present work is an effort towards a systematic understanding of the influence of various parameters on thrust generation from a harmonically pitching airfoil. Quantitative instantaneous force computations have been discussed together with qualitative vortex patterns using a 2-D discrete vortex simulation of incompressible viscous flow. In general, thrust force increases with the oscillation frequency. The trend is very similar to the inviscid theory prediction. Further, the thrust force is seen to decrease with the increase in mean angle of attack. However, in a clear deviation from inviscid theory trends, pitching at high amplitudes about high mean angle of attack, only drag is observed for high values of reduced frequency considered. The effect of location of the pitching axis is also found to be significant on the propulsive characteristics of the airfoil.     

A scalable interface-resolved simulation of particle-laden flow using the lattice Boltzmann method

We examine the scalable implementation of the lattice Boltzmann method (LBM) in the context of interface-resolved simulation of wall-bounded particle-laden flows. Three distinct aspects relevant to performance optimization of our lattice Boltzmann simulation are studied. First, we optimize the core sub-steps of LBM, the collision and the propagation (or streaming) sub-steps, by reviewing and implementing five different published algorithms to reduce memory loading and storing requirements to boost performance. For each, two different array storage formats are benchmarked to test effective cache utilization. Second, the vectorization of the multiple-relaxation-time collision model is discussed and our vectorized collision and propagation algorithm is presented. We find that careful use of Intel’s Advance Vector Extensions and appropriate array storage formats can significantly enhance performance. Third, in the presence of many finite-size, moving solid particles within the flow field, three different communication schemes are proposed and compared in order to optimize the treatment of fluid-solid interactions. These efforts together lead to a very efficient LBM simulation code for interface-resolved simulation of particle-laden flows. Overall, the optimized scalable code of particle-laden flow is a factor of 4.0-to-8.5 times faster than our previous implementation.