Resolvent modelling of near-wall coherent structures in turbulent channel flow
Introduction
The subject of wall-bounded turbulence has been the topic of many studies in the past decades. Coherent turbulent structures have been studied in several frameworks in wall turbulence, and near-wall streaks are the most typically observed structures (Kline et al., 1967, Gupta et al., 1971). Streaks are elongated structures in the streamwise direction for low Reynolds numbers and close to the wall, composed by regions of alternating low and high momentum located in the viscous and buffer layers with a characteristic spanwise spacing of , where is the friction velocity and is the kinematic viscosity of the fluid (Marusic et al., 2017). Farther from the wall, larger structures with similar streaky shape are observed (Marusic et al., 2017). The study of a turbulent flow along the surface of a solid body and its interaction with the wall is one of the most fundamental problems in fluid mechanics. Thus, the pursuit for more effective methods to characterize and model near-wall coherent structures is a very relevant problem.
Statistical methods can be used to identify coherent structures present in turbulent flows. Certain turbulence statistics, taken in frequency and wavenumber domain whenever possible, can be optimally represented using a spectral proper orthogonal decomposition (SPOD), described in previous works (Abreu et al., 2017, Abreu et al., 2018, Abreu et al., 2019, Schmidt and Colonius, 2020). Essentially, SPOD extracts the most likely spatial structures that arise in turbulent-flow realisations, called SPOD modes, sorted in terms of their contribution to the overall energy; SPOD is thus a useful method to extract energetic coherent structures in turbulence.
A theoretical approach, involving the Navier–Stokes equations, called resolvent analysis, can be used to analyse the dynamics and to model coherent structures in turbulent flows. Resolvent analysis was introduced and explored in Jovanovic and Bamieh, 2005, Bagheri et al., 2009, McKeon and Sharma, 2010. The method consists in taking the singular value decomposition of the linearised Navier–Stokes operator, considering the mean field as base flow for linearisation, and treating non-linear terms, neglected in the linearisation, as an unknown external forcing. This provides two orthonormal bases, one for forcings and the other for associated flow responses, related by a gain. Response modes with high gains are expected to be dominant in the flow, and associated forcing shows non-linear terms that excite such amplified responses, leading to relevant information about how these are generated. Such linearised responses can often be related to results of hydrodynamic stability theory, with modes corresponding to linear stability eigenfunctions or to non-modal mechanisms such as lift-up (Jovanović and Bamieh, 2005), but in recent years resolvent analysis has been used to study coherent structures in turbulent flows (see the review of (McKeon, 2017) and references therein).
The connection between SPOD modes and the flow responses to stochastic forcing have been explored in recent works (Towne et al., 2018, Abreu et al., 2017, Abreu et al., 2018, Abreu et al., 2020). Indeed, resolvent modes have a direct relationship with energetic structures observed in experimental or numerical flow databases. An important result is that if the forcing can be modeled as spatial white noise, the response modes provided by resolvent analysis should be exactly equal to those obtained by SPOD (Towne et al., 2018). Moreover, for a flow with a dominant optimal forcing, leading to a gain much larger than that of suboptimal ones, the cross-spectral density matrix (CSD) will often be dominated by the leading response obtained in resolvent analysis (Cavalieri et al., 2019). However, despite such general trends, the lack of knowledge on non-linear forcing leads to no a priori guarantee of the accuracy of resolvent analysis in representing turbulent structures, and a quantitative comparison between modelled resolvent modes and educed SPOD modes is carried out on a case-by-case basis.
This paper focuses on the combined analysis of a turbulent channel flow, with SPOD on the one hand, serving to decompose turbulent flow fluctuations, and resolvent analysis on the other hand, used as a theoretical framework. This enables a reduced-order model of the dynamically-relevant flow features. The investigated flows have friction Reynolds numbers and 550 (note that is defined in terms of and the channel half-height h). Our previous studies (Abreu et al., 2020, Abreu et al., 2019) dealt with SPOD and resolvent analysis for turbulent pipe flow also at and 550, and in the present work we apply a similar approach to study turbulent channel flow. This allows a comparison between these two canonical flows for wall turbulence, with a quantitative examination of how resolvent analysis is capable to model coherent structures for both flows. We expect to find similar results for channel and pipe flows, since the same kind of turbulent coherent structures near the wall are found for channel, pipe and also for a zero pressure gradient turbulent boundary layer (ZPGTBL) (Monty et al., 2009), with differences only related to the amplitude of the forcing/pressure gradient (zero for the ZPGTBL, lower for pipes, higher for channels), the surface curvature and the potential interaction with the wake structures.
The remainder of the paper is organised as follows. Section §2 presents a brief description of the simulation databases, and some results of turbulence statistics and spectral analysis. Section §3 proceeds with a description of the mathematical models used to analyse the databases, with a description of the SPOD, used as a data-driven approach, and the mathematical formulation of the resolvent analysis, used as a theoretical framework. Section §4 shows the results of a detailed comparison between SPOD and resolvent modes. The paper is completed with the conclusions in §5.
Section snippets
Numerical simulation description
The direct-numerical-simulation (DNS) databases employed in this work were produced in Martini et al. (2020) () and Morra et al. (2020) () for a turbulend channel flow. The corresponding Reynolds numbers based on the bulk velocity are and 10000, respectively. The simulations were carried out using the code Channelflow (see Channelflow 2.0 (2018) for details), which is based on a pseudo-spectral spatial discretisation using Fourier modes in streamwise and spanwise
Data-driven approach
Proper orthogonal decomposition (POD) was first introduced by Lumley (Lumley, 1967) in the context of turbulence, and consists in obtaining, from an ensemble of realizations of the flow field, a basis functions that maximises the mean square energy. An approach, also referred to as space-only POD (Towne et al., 2018), is the most commonly employed form of POD to generates spatial modes. Alternatively, POD in frequency domain, or spectral proper orthogonal decomposition (SPOD) as introduced in
Projection coefficient
In order to perform detailed quantitative comparisons between the first SPOD mode and the optimal response from resolvent analysis, we adopted the metric used in our previous work (Abreu et al., 2020), which evaluates the agreement for several values of wavelengths and at a fixed frequency , defined as:where is the first SPOD mode; is the optimal response from resolvent
Conclusions
In the present study we used a data-driven approach based on SPOD, to identify space- and time-dependent coherent structures in a DNS database of turbulent channel flow for friction Reynolds numbers and 550. Resolvent analysis was used as a theoretical approach to model such structures, and the mean velocity was used to linearise the Navier–Stokes equations. The statistically stationary time, and the homogeneous directions in x and z, of this simulation allow the evaluation of SPOD
CRediT authorship contribution statement
Leandra I Abreu: Conceptualization, Methodology, Software, Validation, Investigation, Writing - original draft, Formal analysis. André V.G. Cavalieri: Conceptualization, Supervision, Investigation, Validation, Writing - review & editing, Project administration, Funding acquisition, Resources. Philipp Schlatter: Data curation, Investigation, Validation, Writing - review & editing. Ricardo Vinuesa: Data curation, Investigation, Validation, Writing - review & editing. Dan S. Henningson:
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors acknowledge the financial support received from Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, under Grant No. 310523/2017-6, and by CAPES through the PROEX program. We also acknowledge funding from Centro de Pesquisa e Inovação Sueco-Brasileiro (CISB).
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