Denoising methods for time-resolved PIV measurements

The increasing capabilities of currently available high-speed cameras present several new opportunities for particle image velocimetry (PIV). In particular, temporal postprocessing methods can be used to remove spurious vectors but can also be applied to remove inherent noise. This paper explores this second possibility by estimating the error introduced by several denoising methods on manufactured velocity fields. It is found that PIV noise, while autocorrelated in space, is uncorrelated in time, which leads to a significant improvement in the efficiency of temporal denoising methods compared to their spatial counterparts. Among them, the optimal Wiener filter presents better results than convolution- or wavelet-based filters and has the valuable advantage that no adjustments are required, unlike other methods which generally involve the tuning of some parameters that depend on flow and measurement conditions and are not known a priori. Further refinements show that denoised data can be successfully deconvolved to increase the accuracy of remaining small-scale velocity fluctuations, leading in particular to the recovery of the true shape of turbulent spectra. In practice, the computation of the filter function is not always accurate and different procedures can be used to improve the method depending on the flow considered. Some of them are derived from the properties of the time-frequency spectrum provided by the wavelet transform.     

Superconvergence and H(div) projection for discontinuous Galerkin methods

We introduce and analyse a projection of the discontinuous Galerkin (DG) velocity approximations that preserve the local mass conservation property. The projected velocities have the additional property of continuous normal component. Both theoretical and numerical convergence rates are obtained which show that the accuracy of the DG velocity field is maintained. Superconvergence properties of the DG methods are shown. Finally, numerical simulations of complicated flow and transport problem illustrate the benefits of the projection. Copyright © 2003 John Wiley & Sons, Ltd.     

A 3D second‐order accurate projection‐based Finite Volume code on non‐staggered,non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows

It is well known that exact projection methods (EPM) on non‐staggered grids suffer for the presence of non‐solenoidal spurious modes. Hence, a formulation for simulating time‐dependent incompressible flows while allowing the discrete continuity equation to be satisfied up to machine‐accuracy, by using a Finite Volume‐based second‐order accurate projection method on non‐staggered and non‐uniform 3D grids, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by first solving an elliptic equation on a compact stencil that is by performing a standard approximate projection method (APM). In such a way, three sets of divergence‐free normal‐to‐face velocities can be computed. Then, a second elliptic equation for a scalar field is derived by prescribing that its additional discrete gradient ensures the continuity constraint based on the adopted linear interpolation of the velocity. Characteristics of the double projection method (DPM) are illustrated in details and stability and accuracy of the method are addressed. The resulting numerical scheme is then applied to laminar buoyancy‐driven flows and is proved to be stable and efficient. Copyright © 2006 John Wiley & Sons, Ltd.     

Discrete filter operators for large‐eddy simulation using high‐order spectral difference methods

The combination of a high‐order unstructured spectral difference (SD) spatial discretization scheme with sub‐grid scale (SGS) modeling for large‐eddy simulation is investigated with particular focus on the consistent implementation of a structural mixed model based on the scale similarity hypothesis. The difficult task of deriving a consistent formulation for the discrete filter within the SD element of arbitrary order led to the development of a new class of three‐dimensional constrained discrete filters. The discrete filters satisfy a set of selected criteria and are completely local within the SD element. Their weights can be automatically computed at run time from the number of solution points within each element and the expected filter cutoff length scale. The novel discrete filters can be applied to any SGS model involving explicit filtering and to a broad class of high‐order discontinuous finite element numerical schemes. The code is applied to the computation of turbulent channel flows at three Reynolds numbers, namely Re_τ = 180, 395, and 590 (based on the friction velocity u_τ and channel half‐width δ). Results from computations with and without the SGS model are compared against results from direct numerical simulation. The numerical experiments suggest that the results are sensitive to the use of the SGS model, even when a high‐order numerical scheme is used, especially when the grid resolution is kept relatively low and mostly in terms of resolved Reynolds stresses. Results obtained using existing filters based on the projection of the solution over lower‐order polynomial bases are also shown and demonstrate that these filters are inadequate for SGS modeling purposes, mostly because of their inability to enforce the selected cutoff length scale with sufficient accuracy. The use of the similarity mixed formulation proved to be particularly accurate in reproducing SGS interactions, confirming that its well‐known potential can be realized in conjunction with state‐of‐the‐art high‐order numerical schemes.Copyright © 2012 John Wiley & Sons, Ltd.     

Taylor and Lagrange correlations in a turbulent free shear layer

 The objective of this research was to study the effect of various Lagrange-tracking correlation methods in estimating the eddy lifetime for a two-stream, turbulent, planar free shear layer. Zeroth-, first- and second-order Lagrange correlation methods were applied to the time-evolving velocity field data collected from a cinematic particle image velocimetry technique. A time scale associated with the eddy lifetime was obtained based on a 2/e correlation of either vorticity or streamwise velocity fluctuations. When based on vorticity, this time scale significantly increased as expected when the tracking was computed with a second-order Lagrangian tracking technique as compared to a (zeroth-order) Taylor hypothesis approach. However when based on streamwise velocity fluctuations, this time scale did not increase significantly for the higher order projection methods. The latter result is attributed to occurrences of “reverse correlation” of the instantaneous streamwise velocity fluctuations caused by eddy rotation. Received: 2 April 1997/Accepted: 3 September 1997     

Velocity measurements on highly turbulent free surface flow using ADV

The 3D instantaneous velocity recorded with an acoustic Doppler velocimeter (ADV) in a highly turbulent free surface flow is analysed using several filters in order to eliminate the corrupted data from the sample. The filters used include the minimum/maximum threshold, the acceleration threshold, and the phase-space threshold. Following some ideas of the phase-space filter, a new method based on the 3D velocity cross-correlation is proposed and tested. A way of computing the constants of the acceleration threshold method is proposed, so no parameters need to be fixed by the user, which makes the filtering process simpler, more objective and more efficient. All the samples analysed are highly turbulent. Nevertheless, the turbulence intensity and the air entrainment vary widely in the flow under study, which produces data records of different quality depending on the measurement point. The performance of the filtering methods when applied to samples of different quality, and the effects of the filtering process in the mean velocity, turbulent kinetic energy and frequency spectra are discussed.     

Acceleration of conductors to supersonic speeds in a pulsed magnetic field

The effect of the shape of the current-density curve and the initial angle of acceleration on the velocity of projection of current-carrying conductors in a pulsed magnetic field produced in the discharge of a capacitive energy accumulator is investigated. A variational problem is solved to determine the optimum current-density function in the projected body for a given magnetic-induction function. It is shown that the relation j =KB, where K is a constant, gives the maximum velocity. For a uniform magnetic field varying as a damped sinusoid, expressions are obtained for the current density in the accelerated body, the velocity of projection, and the acceleration path. It is shown that there is an optimum initial angle of acceleration depending on the amplitude and frequency of the accelerating force and the acceleration path. A procedure is presented for the approximate design of a hypersonic electromagnetic accelerator. In accord with the conclusions of the theory an experimental arrangement is set up and a study is made of the projection of conductors in a pulsed magnetic field. A maximum velocity of 10.5 km/sec is obtained for 0.16-mm diameter aluminum wires.Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskoi Fiziki, No. 3, pp. 44–53, May–June, 1974.     

可解尺度各向同性湍流的标度律

针对大涡模拟, 首先利用 EDQNM 能谱和传输谱理论, 在3种不同的过滤器下分别计算可解尺度各向同性湍流的二阶、三阶结构函数标度律, 并通过速度差扭率验证了可解尺度湍流的 ESS 理论. 研究了该可解尺度湍流标度律及速度差扭率与多个因素的关系, 这些因素包括: 两点距离与过滤尺度的比、大涡模拟雷诺数和过滤器类型. 结果显示, 当两点距离位于过滤尺度量级时或大涡模拟雷诺数较小时, 可解尺度标度律与未过滤流场的标度律相差很大进而必须加以修正, 而且可解尺度流场也不再总是满足 ESS 理论进而对应的速度差扭率也需要修正. 然后通过3个例子介绍了这些结果在大涡模拟亚格子模型中的应用.      

On the application of the Helmholtz–Hodge decomposition in projection methods for incompressible flows with general boundary conditions

This paper is concerned with the analysis of the Helmholtz–Hodge decomposition theorem since it plays a fundamental role in the projection methods that are adopted in the numerical solution of the Navier–Stokes equations for incompressible flows. The paper highlights the role of the orthogonal decomposition of a vector field in a bounded domain when general boundary conditions are in effect. In fact, even if Fractional Time‐Step Methods are standard procedures for de‐coupling the pressure gradient and the velocity field, many problems are encountered in performing the decoupling with higher accuracy. Since the problem of determining a unique and orthogonal decomposition requires only one boundary condition to be well posed, thus either the normal or the tangential ones, result exactly imposed at the end of the projection. Numerical errors are introduced in terms of both the pressure and the velocity but the orthogonality of decomposition guarantees that the former does not contribute to affect the accuracy of the latter. Moreover, it is shown that depending on the meaning of the vector to be decomposed, i.e. acceleration or velocity, the true orthogonal projector can be defined only when suitable boundary conditions are verified. Conversely, it is shown that when the decomposition results non‐orthogonal, the velocity accuracy suffers of other errors. The issue on the resulting accuracy order of the procedure is clearly addressed by means of several accuracy studies and a strategy for improving it is proposed. This paper follows and integrates the issues reported in Iannelli and Denaro (Int. J. Numer. Meth. Fluids 2003; 42 : 399–437). Copyright © 2003 John Wiley & Sons, Ltd.     

Calculation of impulsively started incompressible viscous flows

The paper's focus is the calculation of unsteady incompressible 2D flows past airfoils. In the framework of the primitive variable Navier–Stokes equations, the initial and boundary conditions must be assigned so as to be compatible, to assure the correct prediction of the flow evolution. This requirement, typical of all incompressible flows, viscous or inviscid, is often violated when modelling the flow past immersed bodies impulsively started from rest. Its fulfillment can however be restored by means of a procedure enforcing compatibility, consisting in a pre‐processing of the initial velocity field, here described in detail. Numerical solutions for an impulsively started multiple airfoil have been obtained using a finite element incremental projection method. The spatial discretization chosen for the velocity and pressure are of different order to satisfy the inf–sup condition and obtain a smooth pressure field. Results are provided to illustrate the effect of employing or not the compatibility procedure, and are found in good agreement with those obtained with a non‐primitive variable solver. In addition, we introduce a post‐processing procedure to evaluate an alternative pressure field which is found to be more accurate than the one resulting from the projection method. This is achieved by considering an appropriate ‘unsplit’ version of the momentum equation, where the velocity solution of the projection method is substituted. Copyright © 2004 John Wiley & Sons, Ltd.     

Discrete filters for large eddy simulation

This paper summarizes several results relative to discrete filters for subgrid‐scale (SGS) models based on a multi‐level filtering procedure. First, a theoretical study of discrete filters in physical space is performed. The analysis is done in the uniform one‐dimensional case, and is then extended to the general multi‐dimensional case for arbitrary structured and unstructured meshes. Some equivalence classes for the discrete filters are defined, based either on a differential approximation or the associated transfer function. Methods for the definition of discrete filters are proposed in the general case, including the approximation of continuous convolution filters. Second, the sensitivity of several SGS models with respect to the test filter is investigated. The selected models are: the dynamic Smagorinsky model, the mixed scale model (MSM), the selective MSM and the Liu–Meneveau–Katz (LMK) similarity model. Improved versions, which explicitly account for the spectral width of the test filter of the MSM and the LMK similarity model are proposed. The analysis, which reveals a significant influence of the test filter, is done through a priori testing on a 128³ field issued from the large eddy simulation (LES) of freely decaying homogeneous isotropic turbulence. Copyright © 1999 John Wiley & Sons, Ltd.     

A posteriori analysis of low-pass spatial filters for approximate deconvolution large eddy simulations of homogeneous incompressible flows

The goal of this paper is twofold: first, it investigates the effect of low-pass spatial filters for approximate deconvolution large eddy simulation (AD-LES) of turbulent incompressible flows. Second, it proposes the hyper-differential filter as a means of increasing the accuracy of the AD-LES model without increasing the computational cost. Box filters, Padé filters, and differential filters with a wide range of parameters are studied in the AD-LES framework. The AD-LES model, in conjunction with these spatial filters, is tested in the numerical simulation of the two-dimensional and three-dimensional Taylor–Green vortex problems. We show that the most accurate results are obtained with the hyper-differential filter, followed by the differential filter. We also demonstrate that the results highly depend on the selection of the filtering procedure. It seems that filters whose transfer function resembles that of the Fourier cut-off filter (such as the hyper-differential filters) tend to perform the best.     

Hybrid finite element/volume method for 3D incompressible flows with heat transfer

An implicit hybrid finite element (FE)/volume solver has been extended to incompressible flows coupled with the energy equation. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centred finite volume (FV) method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centres and the auxiliary variable at vertices. The Generalized Minimal Residual (GMRES) matrix-free strategy is adapted to solve the governing equations in both FE and FV methods. The presented 2D and 3D numerical examples show the robustness and accuracy of the numerical method.     

A non‐diffusive,divergence‐free,finite volume‐based double projection method on non‐staggered grids

Second‐order accurate projection methods for simulating time‐dependent incompressible flows on cell‐centred grids substantially belong to the class either of exact or approximate projections. In the exact method, the continuity constraint can be satisfied to machine‐accuracy but the divergence and Laplacian operators show a four‐dimension nullspace therefore spurious oscillating solutions can be introduced. In the approximate method, the continuity constraint is relaxed, the continuity equation being satisfied up to the magnitude of the local truncation error, but the compact Laplacian operator has only the constant mode. An original formulation for allowing the discrete continuity equation to be satisfied to machine‐accuracy, while using a finite volume based projection method, is illustrated. The procedure exploits the Helmholtz–Hodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by solving a second elliptic field for a scalar field obtained by prescribing that its additional discrete gradients ensure discrete continuity based on the previously adopted linear interpolation of the velocity. The resulting numerical scheme is applied to several flow problems and is proved to be accurate, stable and efficient. This paper has to be considered as the companion of: 'F. M. Denaro, A 3D second‐order accurate projection‐based finite volume code on non‐staggered, non‐uniform structured grids with continuity preserving properties: application to buoyancy‐driven flows. IJNMF 2006; 52 (4):393–432. Now, we illustrate the details and the rigorous theoretical framework. Copyright © 2006 John Wiley & Sons, Ltd.     

广义联邦滤波器的全局最优性

基于分散化滤波算法和信息分配原理,建立了广义联邦滤波器设计理论。证明了联邦滤波器当其主滤波器和局部滤波器的维数都相同时,其全局滤波和集中卡尔曼滤波等价,是最优的;同时提出当主滤波器维数和局部滤波器维数不相同时,达到全局滤波最优的解析补偿方法,其附加计算量小,并可作为一种性能指标用于子系统的软故障检测。在组合导航系统中运用此方法对非公共状态信息进行补偿,仿真结果验证了该方法的有效性。     

Simultaneous velocity field measurements in two-phase flows for turbulent mixing of sprays by means of two-phase PIV

This paper describes a novel derivative of the PIV method for measuring the velocity fields of droplets and gas phases simultaneously using fluorescence images rather than Mie scattering images. Two-phase PIV allows the simultaneous and independent velocity field measurement of the liquid phase droplets and ambient gas in the case of two-phase flow mixing. For phase discrimination, each phase is labelled by a different fluorescent dye: the gas phase is seeded with small liquid droplets, tagged by an efficient fluorescent dye while the droplets of the liquid phases are tagged by a different fluorescent dye. For each phase, the wavelength shift of fluorescence is used to separate fluorescence from Mie scattering and to distinguish between the fluorescence of each phase. With the use of two cross-correlation PIV cameras and adequate optical filters, we obtain two double frame images, one for each phase. Thus standard PIV or PTV algorithms are used to obtain the simultaneous and independent velocity fields of the two phases. Because the two-phase PIV technique relies on the ability to produce two simultaneous and independent images of the two phases, the choice of the labelling dyes and of the associated optical filter sets is relevant for the image acquisition. Thus a spectroscopic study has been carried out to choose the optimal fluorescent dyes and the associated optical filters. The method has been evaluated in a simple two-phase flow: droplets of 30–40 μm diameter, produced by an ultrasonic nozzle are injected into a gas coflow seeded with small particles. Some initial results have been obtained which demonstrate the potential of the method.     

Optimal solenoidal interpolation of turbulent vector fields: application to PTV and super-resolution PIV

A new approach for the interpolation of a filtered turbulence velocity field given random point samples of unfiltered turbulence velocity data is described. In this optimal interpolation method, the best possible value of the interpolated filtered field is obtained as a stochastic estimate of a conditional average, which minimizes the mean square error between the interpolated filtered velocity field and the true filtered velocity field. Besides its origins in approximation theory, the optimal interpolation method also has other advantages over more commonly used ad hoc interpolation methods (like the adaptive Gaussian window). The optimal estimate of the filtered velocity field can be guaranteed to preserve the solenoidal nature of the filtered velocity field and also the underlying correlation structure of both the filtered and the unfiltered velocity fields. The a posteriori performance of the optimal interpolation method is evaluated using data obtained from high-resolution direct numerical simulation of isotropic turbulence. Our results show that for a given sample data density, there exists an optimal choice of the characteristic width of cut-off filter that gives the least possible relative mean square error between the true filtered velocity and the interpolated filtered velocity. The width of this optimal filter and the corresponding minimum relative error appear to decrease with increase in sample data density. Errors due to the optimal interpolation method are observed to be quite low for appropriate choices of the data density and the characteristic width of the filter. The optimal interpolation method is also seen to outperform the adaptive Gaussian window, in representing the interpolated field given the data at random sample locations. The overall a posteriori performance of the optimal interpolation method was found to be quite good and hence makes a potential candidate for use in interpolation of PTV and super-resolution PIV data.     

Motion and filling of cavities in a boundless liquid and close to a plane

The motion of bubbles in liquids has been studied in many earlier papers [1–8]. In this paper methods of the projection type are applied to the problem of a cavity in an ideal, incompressible liquid in the absence of vortices. The collapse of a bubble having a finite initial velocity in a boundless liquid is considered; also considered is the collapse of a stationary bubble close to a solid wall. Using the small-parameter method the generation of a jet is examined analytically. A numerical computing method not involving small parameters is developed; it is based on calculating the projection by numerical computation of the corresponding integrals. The method combines economy and simplicity of application with a high accuracy in the region in which the representation of the velocity potential by a series of spherical functions remains effective.     

数值求解不可压缩流动的投影方法研究进展

在过去的20多年中, 投影方法通过速度和压力的解耦计算, 获得了比全耦合方法更高的计算效率, 这个显著优点使之得以广泛应用. 目前, 在计算非定常不可压缩流动的原始变量形式的数值方法中, 投影方法得到了越来越广泛的应用. 本文根据投影方法的构造思路,将众多的投影方法分成了3类, 即: Helmholtz-Hodge分解类投影方法、算子分裂类投影方法和局部连续投影方法, 并详细的介绍了3类投影方法的发展历程和求解步骤. 从投影方法的求解过程不难发现, 通过速度和压力的解耦计算, 提高了投影方法的计算效率, 但同时也给投影方法的时间精度分析带来了困难, 并长期成为大家争论的焦点. 普遍认为, 速度的时间精度比较容易达到高阶, 但是压力一般来说只有一阶精度. 但通过对3类投影的对比分析后, 我们认为, 局部连续投影方法将有助于澄清目前投影方法存在的相关争议, 并使得发展高阶精度的投影方法在理论上和技术上成为可能.