Spectral proper orthogonal decomposition Sieber, Moritz; Paschereit, C. Oliver; Oberleithner, Kilian
Journal of fluid mechanics,
04/2016, Volume:
792
Journal Article
Peer reviewed
Open access
The identification of coherent structures from experimental or numerical data is an essential task when conducting research in fluid dynamics. This typically involves the construction of an empirical ...mode base that appropriately captures the dominant flow structures. The most prominent candidates are the energy-ranked proper orthogonal decomposition (POD) and the frequency-ranked Fourier decomposition and dynamic mode decomposition (DMD). However, these methods are not suitable when the relevant coherent structures occur at low energies or at multiple frequencies, which is often the case. To overcome the deficit of these ‘rigid’ approaches, we propose a new method termed spectral proper orthogonal decomposition (SPOD). It is based on classical POD and it can be applied to spatially and temporally resolved data. The new method involves an additional temporal constraint that enables a clear separation of phenomena that occur at multiple frequencies and energies. SPOD allows for a continuous shifting from the energetically optimal POD to the spectrally pure Fourier decomposition by changing a single parameter. In this article, SPOD is motivated from phenomenological considerations of the POD autocorrelation matrix and justified from dynamical systems theory. The new method is further applied to three sets of PIV measurements of flows from very different engineering problems. We consider the flow of a swirl-stabilized combustor, the wake of an airfoil with a Gurney flap and the flow field of the sweeping jet behind a fluidic oscillator. For these examples, the commonly used methods fail to assign the relevant coherent structures to single modes. The SPOD, however, achieves a proper separation of spatially and temporally coherent structures, which are either hidden in stochastic turbulent fluctuations or spread over a wide frequency range. The SPOD requires only one additional parameter, which can be estimated from the basic time scales of the flow. In spite of all these benefits, the algorithmic complexity and computational cost of the SPOD are only marginally greater than those of the snapshot POD.
Particulate flows have mainly been studied under the simplifying assumption of a one-way coupling regime where the disperse phase does not modify the carrier fluid. A more complete view of multiphase ...flows can be gained calling into play two-way coupling effects, i.e. by accounting for the inter-phase momentum exchange, which is certainly relevant at increasing mass loading. In this paper we present a new methodology rigorously designed to capture the inter-phase momentum exchange for particles smaller than the smallest hydrodynamical scale, e.g. the Kolmogorov scale in a turbulent flow. The momentum coupling mechanism exploits the unsteady Stokes flow around a small rigid sphere, where the transient disturbance produced by each particle is evaluated in a closed form. The particles are described as lumped point masses, which would lead to the appearance of singularities. A rigorous regularization procedure is conceived to extract the physically relevant interactions between the particles and the fluid which avoids any ‘ad hoc’ assumption. The approach is suited for high-efficiency implementation on massively parallel machines since the transient disturbance produced by the particles is strongly localized in space. We will show that hundreds of thousands of particles can be handled at an affordable computational cost, as demonstrated by a preliminary application to a particle-laden turbulent shear flow.
Research into high-Reynolds-number turbulent boundary layers in recent years has brought about a renewed interest in the larger-scale structures. It is now known that these structures emerge more ...prominently in the outer region not only due to increased Reynolds number (Metzger & Klewicki, Phys. Fluids, vol. 13(3), 2001, pp. 692–701; Hutchins & Marusic, J. Fluid Mech., vol. 579, 2007, pp. 1–28), but also when a boundary layer is exposed to an adverse pressure gradient (Bradshaw, J. Fluid Mech., vol. 29, 1967, pp. 625–645; Lee & Sung, J. Fluid Mech., vol. 639, 2009, pp. 101–131). The latter case has not received as much attention in the literature. As such, this work investigates the modification of the large-scale features of boundary layers subjected to zero, adverse and favourable pressure gradients. It is first shown that the mean velocities, turbulence intensities and turbulence production are significantly different in the outer region across the three cases. Spectral and scale decomposition analyses confirm that the large scales are more energized throughout the entire adverse pressure gradient boundary layer, especially in the outer region. Although more energetic, there is a similar spectral distribution of energy in the wake region, implying the geometrical structure of the outer layer remains universal in all cases. Comparisons are also made of the amplitude modulation of small scales by the large-scale motions for the three pressure gradient cases. The wall-normal location of the zero-crossing of small-scale amplitude modulation is found to increase with increasing pressure gradient, yet this location continues to coincide with the large-scale energetic peak wall-normal location (as has been observed in zero pressure gradient boundary layers). The amplitude modulation effect is found to increase as pressure gradient is increased from favourable to adverse.
The turbulent/non-turbulent interface in a zero-pressure-gradient turbulent boundary layer at high Reynolds number (
$\mathit{Re}_\tau =14\, 500$
) is examined using particle image velocimetry. An ...experimental set-up is utilized that employs multiple high-resolution cameras to capture a large field of view that extends
$2\delta \times 1.1\delta $
in the streamwise/wall-normal plane with an unprecedented dynamic range. The interface is detected using a criteria of local turbulent kinetic energy and proves to be an effective method for boundary layers. The presence of a turbulent/non-turbulent superlayer is corroborated by the presence of a jump for the conditionally averaged streamwise velocity across the interface. The steep change in velocity is accompanied by a discontinuity in vorticity and a sharp rise in the Reynolds shear stress. The conditional statistics at the interface are in quantitative agreement with the superlayer equations outlined by Reynolds (J. Fluid Mech., vol. 54, 1972, pp. 481–488). Further analysis introduces the mass flux as a physically relevant parameter that provides a direct quantitative insight into the entrainment. Consistency of this approach is first established via the equality of mean entrainment calculations obtained using three different methods, namely, conditional, instantaneous and mean equations of motion. By means of ‘mass-flux spectra’ it is shown that the boundary-layer entrainment is characterized by two distinctive length scales which appear to be associated with a two-stage entrainment process and have a substantial scale separation.
Considerable discussion over the past few years has been devoted to the question of whether the logarithmic region in wall turbulence is indeed universal. Here, we analyse recent experimental data in ...the Reynolds number range of nominally
$2\times 1{0}^{4} \lt {\mathit{Re}}_{\tau } \lt 6\times 1{0}^{5} $
for boundary layers, pipe flow and the atmospheric surface layer, and show that, within experimental uncertainty, the data support the existence of a universal logarithmic region. The results support the theory of Townsend (The Structure of Turbulent Shear Flow, Vol. 2, 1976) where, in the interior part of the inertial region, both the mean velocities and streamwise turbulence intensities follow logarithmic functions of distance from the wall.
Violent respiratory events such as coughs and sneezes play a key role in transferring respiratory diseases between infectious and susceptible individuals. We present the results of a combined ...experimental and theoretical investigation of the fluid dynamics of such violent expiratory events. Direct observation of sneezing and coughing events reveals that such flows are multiphase turbulent buoyant clouds with suspended droplets of various sizes. Our observations guide the development of an accompanying theoretical model of pathogen-bearing droplets interacting with a turbulent buoyant momentum puff. We develop in turn discrete and continuous models of droplet fallout from the cloud in order to predict the range of pathogens. According to the discrete fallout model droplets remain suspended in the cloud until their settling speed matches that of the decelerating cloud. A continuous fallout model is developed by adapting models of sedimentation from turbulent fluids. The predictions of our theoretical models are tested against data gathered from a series of analogue experiments in which a particle-laden cloud is ejected into a relatively dense ambient. Our study highlights the importance of the multiphase nature of respiratory clouds, specifically the suspension of the smallest drops by circulation within the cloud, in extending the range of respiratory pathogens.
This work is dedicated to systematically studying and predicting the wake characteristics of a yawed wind turbine immersed in a turbulent boundary layer. To achieve this goal, wind tunnel experiments ...were performed to characterize the wake of a horizontal-axis wind turbine model. A high-resolution stereoscopic particle image velocimetry system was used to measure the three velocity components in the turbine wake under different yaw angles and tip-speed ratios. Moreover, power and thrust measurements were carried out to analyse the performance of the wind turbine. These detailed wind tunnel measurements were then used to perform a budget study of the continuity and Reynolds-averaged Navier–Stokes equations for the wake of a yawed turbine. This theoretical analysis revealed some notable features of the wakes of yawed turbines, such as the asymmetric distribution of the wake skew angle with respect to the wake centre. Under highly yawed conditions, the formation of a counter-rotating vortex pair in the wake cross-section as well as the vertical displacement of the wake centre were shown and analysed. Finally, this study enabled us to develop general governing equations upon which a simple and computationally inexpensive analytical model was built. The proposed model aims at predicting the wake deflection and the far-wake velocity distribution for yawed turbines. Comparisons of model predictions with the wind tunnel measurements show that this simple model can acceptably predict the velocity distribution in the far wake of a yawed turbine. Apart from the ability of the model to predict wake flows in yawed conditions, it can provide valuable physical insight on the behaviour of turbine wakes in this complex situation.
The route to turbulence in pipe flow is a complex, nonlinear, spatiotemporal process for which an increasingly clear understanding has emerged in recent years. This paper presents a theoretical ...perspective on the problem, focusing on what can be understood from relatively few physical features and models that encompass these features. The paper proceeds step-by-step with increasing detail about the transition process, first discussing the relationship to phase transitions and then exploiting an even deeper connection between pipe flow and excitable and bistable media. In the end a picture emerges for all stages of the transition process, from transient turbulence, to the onset of sustained turbulence in a percolation transition, to the modest and then rapid expansion of turbulence, ultimately leading to fully turbulent pipe flow.
A model for the instantaneous wall-shear-stress distribution is presented for zero-pressure-gradient turbulent boundary layers. The model, based on empirical and theoretical considerations, is able ...to reconstruct a statistically representative fluctuating wall-shear-stress time-series, ${ \tau }_{w}^{\ensuremath{\prime} } (t)$, using only the low-frequency content of the streamwise velocity measured in the logarithmic region, away from the wall. Results, including spectra and second-order moments, show that the model is capable of successfully capturing Reynolds number trends as observed in other studies.
The logarithmic law for the mean velocity in turbulent boundary layers has long provided a valuable and robust reference for comparison with theories, models and large-eddy simulations (LES) of ...wall-bounded turbulence. More recently, analysis of high-Reynolds-number experimental boundary-layer data has shown that also the variance and higher-order moments of the streamwise velocity fluctuations
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}u^{\prime +}$
display logarithmic laws. Such experimental observations motivate the question whether LES can accurately reproduce the variance and the higher-order moments, in particular their logarithmic dependency on distance to the wall. In this study we perform LES of very high-Reynolds-number wall-modelled channel flow and focus on profiles of variance and higher-order moments of the streamwise velocity fluctuations. In agreement with the experimental data, we observe an approximately logarithmic law for the variance in the LES, with a ‘Townsend–Perry’ constant of
$A_1\approx 1.25$
. The LES also yields approximate logarithmic laws for the higher-order moments of the streamwise velocity. Good agreement is found between
$A_p$
, the generalized ‘Townsend–Perry’ constants for moments of order
$2p$
, from experiments and simulations. Both are indicative of sub-Gaussian behaviour of the streamwise velocity fluctuations. The near-wall behaviour of the variance, the ranges of validity of the logarithmic law and in particular possible dependencies on characteristic length scales such as the roughness length
$z_0$
, the LES grid scale
$\Delta $
, and subgrid scale mixing length
$C_s\Delta $
are examined. We also present LES results on moments of spanwise and wall-normal fluctuations of velocity.
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