We employ a resolvent-based methodology to estimate velocity and pressure fluctuations within turbulent channel flows at friction Reynolds numbers of approximately 180, 550 and 1000 using measurements of shear stress and pressure at the walls, taken from direct numerical simulation (DNS) databases. Martini et al. (J. Fluid Mech., vol. 900, 2021, A2) showed that the resolvent-based estimator is optimal when the true space-time forcing statistics are utilized, thus providing an upper bound for the accuracy of any linear estimator. We use this framework to determine the flow structures that can be linearly estimated from wall measurements, and we characterize these structures and the estimation errors in both physical and wavenumber space. We also compare these results to those obtained using approximate forcing models - an eddy-viscosity model and white-noise forcing - and demonstrate the significant benefit of using true forcing statistics. All models lead to accurate results up to the buffer-layer, but only using the true forcing statistics allows accurate estimation of large-scale log-layer structures, with significant correlation between the estimates and DNS results throughout the channel. The eddy-viscosity model displays an intermediate behaviour, which may be related to its ability to partially capture the forcing colour. Our results show that structures that leave a footprint on the channel walls can be accurately estimated using the linear resolvent-based methodology, and the presence of large-scale wall-attached structures enables accurate estimations through the logarithmic layer.
We combine resolvent-mode decomposition with techniques from convex optimization to optimally approximate velocity spectra in a turbulent channel. The velocity is expressed as a weighted sum of resolvent modes that are dynamically significant, non-empirical, and scalable with Reynolds number. To optimally represent DNS data at friction Reynolds number $2003$, we determine the weights of resolvent modes as the solution of a convex optimization problem. Using only $12$ modes per wall-parallel wavenumber pair and temporal frequency, we obtain close agreement with DNS-spectra, reducing the wall-no…
The ability of linear stochastic response analysis to estimate coherent motions is investigated in turbulent channel flow at friction Reynolds number Re$_τ$ = 1007. The analysis is performed for spatial scales characteristic of buffer-layer and large-scale motions by separating the contributions of different temporal frequencies. Good agreement between the measured spatio-temporal power spectral densities and those estimated by means of the resolvent is found when the effect of turbulent Reynolds stresses, modelled with an eddy-viscosity associated to the turbulent mean flow, is included in th…
The resolvent formulation of McKeon & Sharma (2010) is applied to supersonic turbulent boundary layers to study the validity of Morkovin's hypothesis, which postulates that high-speed turbulence structures in zero pressure-gradient turbulent boundary layers remain largely the same as its incompressible counterpart. Supersonic zero-pressure-gradient turbulent boundary layers with adiabatic wall boundary conditions at Mach numbers ranging from 2 to 4 are considered. Resolvent analysis highlights two distinct regions of the supersonic turbulent boundary layer in the wave parameter space: the rela…
This work introduces a formulation of resolvent analysis that uses wavelet transforms rather than Fourier transforms in time. This allows resolvent analysis to be extended to turbulent flows with non-stationary means in addition to statistically-stationary flows. The optimal resolvent modes for this formulation correspond to the potentially time-transient structures that are most amplified by the linearized Navier-Stokes operator. We validate this methodology for turbulent channel flow and show that the wavelet-based and Fourier-based resolvent analyses are equivalent for statistically-station…
Response modes computed via linear resolvent analysis of a turbulent mean-flow field have been shown to qualitatively capture characteristics of the observed turbulent coherent structures in both wall-bounded and free shear flows. To make such resolvent models predictive, the nonlinear forcing term must be closed. Strategies to do so include imposing self-consistent sets of triadic interactions, proposing various source models, or through turbulence modelling. For the latter, several investigators have proposed using the mean-field eddy viscosity acting linearly on the fluctuation field. In th…
Recent simulations indicate that streamwise-preferential porous materials have the potential to reduce drag in wall-bounded turbulent flows(Gomez-de-Segura & Garcia-Mayoral 2019). This paper extends the resolvent formulation to study the effect of such anisotropic permeable substrates on turbulent channel flow. Under the resolvent formulation, the Fourier-transformed Navier-Stokes equations are interpreted as a linear forcing-response system. The nonlinear terms are considered the endogenous forcing in the system that gives rise to a velocity and pressure response. A gain-based decomposition o…
The present work demonstrates the use of resolvent analysis to obtain physical insights for open-cavity flows. Resolvent analysis identifies the flow response to harmonic forcing, given a steady base state, in terms of the response and forcing modes and the amplification gain. The response and forcing modes reveal the spatial structures associated with this amplification process. In this study, we perform resolvent analysis on both laminar and turbulent flows over a rectangular cavity with length-to-depth ratio of $L/D=6$ at a free stream Mach number of $M_\infty=0.6$ in a spanwise periodic se…
This article accompanies a fluid dynamics video entered into the Gallery of Fluid Motion of the 66th Annual Meeting of the APS Division of Fluid Dynamics.
Coherent turbulent wave-packet structures in a jet at Reynolds number 460000 and Mach number 0.4 are extracted from experimental measurements and are modeled as linear fluctuations around the mean flow. The linear model is based on harmonic optimal forcing structures and their associated flow response at individual Strouhal numbers, obtained from analysis of the global linear resolvent operator. These forcing-response wave packets ("resolvent modes") are first discussed with regard to relevant physical mechanisms that provide energy gain of flow perturbations in the jet. Modal shear instabilit…
The predictions of resolvent analysis for turbulent channel flow are evaluated for a friction Reynolds number of Retau = 550. In addition to the standard resolvent operator with kinematic viscosity, a resolvent operator augmented with the Cess eddy viscosity is considered. Adding eddy viscosity significantly alters the low-rank behavior of the resolvent. Regardless of the wave speed selected, the eddy resolvent is low-rank for spanwise wavelengths of ly+ = 80 and ly/h = 3.5 in comparison to the standard resolvent whose low-rank behavior depends on the wave speed. The leading eddy modes have hi…
This work applies resolvent analysis to incompressible flow through a rectangular duct, in order to identify dominant linear energy-amplification mechanisms present in such flows. In particular, we formulate the resolvent operator from linearizing the Navier--Stokes equations about a two-dimensional base/mean flow. The laminar base flow only has a nonzero streamwise velocity component, while the turbulent case exhibits a secondary mean flow (Prandtl's secondary flow of the second kind). A singular value decomposition of the resolvent operator allows for the identification of structures corresp…
This paper extends the resolvent formalism for wall turbulence proposed by McKeon and Sharma(2010) to account for the effect of streamwise-constant riblets. Under the resolvent formulation, the Navier-Stokes equations are interpreted as a forcing-response system: the nonlinear convective term is interpreted as a feedback forcing on the remaining linear terms, which generates a velocity and pressure response. A gain-based decomposition of the linear forcing-response transfer function --- the resolvent operator --- yields highly amplified velocity and pressure modes, which can be considered key …
We analyse the inter-scale transfer of energy for two types of plane Poiseuille flow: the P4U exact coherent state of Park and Graham (2015) and turbulent flow in a minimal channel. For both flows, the dominant energy-producing modes are streamwise-constant streaks with a spanwise spacing of approximately 100 wall units. Since the viscous dissipation for these scales is not sufficient to balance production, the nonlinear terms redistribute the excess energy to other scales. Spanwise-constant scales (that is, Tollmien-Schlichting-like modes with zero spanwise wavenumber), in particular, account…
Self-similarity of wall-attached coherent structures in a turbulent channel at $Re_τ=543$ is explored by means of resolvent analysis. In this modelling framework, coherent structures are understood to arise as a response of the linearised mean-flow operator to generalised, frequency-dependent Reynolds stresses, considered to act as an endogenous forcing. We assess the self-similarity of both the wall-attached flow structures and the associated forcing. The former are educed from direct numerical simulation data by finding the flow field correlated with the wall shear, whereas the latter is ide…
We use resolvent analysis to determine an unsteady active control setup to attenuate pressure fluctuations in turbulent supersonic flow over a rectangular cavity with a length-to-depth ratio of 6 at a Mach number of 1.4 and a Reynolds number based on cavity depth of 10,000. Large-eddy simulations (LES) and dynamic modal decomposition (DMD) of the supersonic cavity flow reveal the dominance of two-dimensional Rossiter modes II and IV. These predominantly two-dimensional vortical structures generate high-amplitude unsteadiness over the cavity through trailing-edge impingement and create oblique …
Performing global resolvent analysis for high-Reynolds-number turbulent flow calls for the handling of a large discrete operator. Even though such large operator is required in the analysis, most applications of resolvent analysis extracts only a few dominant resolvent response and forcing modes. Here, we consider the use of randomized numerical linear algebra to reduce the dimension of the resolvent operator for achieving computational speed up and memory saving compared to the standard resolvent analysis. To accomplish this goal, we utilize sketching of the linear operator with random test m…
The cross-spectral density (CSD) of the non-linear forcing in resolvent analyses is here quantified for the first time for turbulent channel flows. Direct numerical simulations (DNS) at $Re_τ =179$ and $Re_τ =543$ are performed. The CSDs are computed for highly energetic structures typical of buffer-layer and large-scale motions, for different temporal frequencies. The CSD of the non-linear forcing is shown not to be uncorrelated (white) in space, which implies the forcing is structured. Since the non-linear forcing is non-solenoidal by construction and the velocity of an incompressible flow i…
The resolvent formulation of the Navier$\text{--}$Stokes equations gives a means for the characterization and prediction of features of turbulent flows$\text{---}$such as statistics, structures and their nonlinear interactions$\text{---}$using the singular value decomposition of the resolvent operator based on the appropriate turbulent mean, following the framework developed by McKeon & Sharma (2010). This work will describe a methodology for approximating leading resolvent (i.e., pseudospectral) modes for shear-driven turbulent flows using prescribed analytic functions. We will demonstrate th…
A resolvent-based reduced-order representation is used to capture time-averaged second-order statistics in turbulent channel flow. The recently-proposed decomposition of the resolvent operator into two distinct families related to the Orr-Sommerfeld and Squire operators [K. Rosenberg and B. J. McKeon, Efficient representation of exact coherent states of the Navier-Stokes equations using resolvent analysis, Fluid Dynamics Research 51, 011401 (2019)] results in dramatic improvement in the ability to match all components of the energy spectra and the $uv$ cospectrum. The success of the new repres…
An analysis of the statistics of the non-linear terms in resolvent analysis is performed in this work for turbulent Couette flow at low Reynolds number. Data from a direct numerical simulation of a minimal flow unit, at Reynolds number 400, is post-processed using Fourier analysis in both time and space, leading to the covariance matrix of the velocity. From the same data, we computed the non-linear terms of the Navier-Stokes equations (treated as forcing in the present formulation), which allowed us to compute the covariance matrix of the forcing for this case. The two covariances are related…
The interaction between shear driven turbulence and stratification is a key process in a wide array of geophysical flows with spatio-temporal scales that span many orders of magnitude. A quick numerical model prediction based on external parameters of stratified boundary layers could greatly benefit the understanding of the interaction between velocity and scalar flux at varying scales. For these reasons, here, we use the resolvent framework to investigate the effects of an active scalar on incompressible wall-bounded turbulence. We obtain the state of the flow system by applying the linear re…
Resolvent analysis is a powerful tool for modeling and analyzing turbulent flows and in particular provides an approximation of coherent flow structures. Despite recent algorithmic advances, computing resolvent modes for flows with more than one inhomogeneous spatial coordinate remains computationally expensive. In this two-part paper, we show how efficient and accurate approximations of resolvent modes can be obtained using a well-posed spatial marching method for flows that contain a slowly varying direction. In this first part of the paper, we derive a well-posed and convergent one-way equa…
We report that many exact invariant solutions of the Navier-Stokes equations for both pipe and channel flows are well represented by just few modes of the model of McKeon & Sharma J. Fl. Mech. 658, 356 (2010). This model provides modes that act as a basis to decompose the velocity field, ordered by their amplitude of response to forcing arising from the interaction between scales. The model was originally derived from the Navier-Stokes equations to represent turbulent flows and has been used to explain coherent structure and to predict turbulent statistics. This establishes a surprising new li…
The decomposition of hydrodynamic and acoustic components of cavity flows can aid the understanding of the flow-acoustic interaction, which produces a variety of adverse effects in applications. We apply the approach of combining the resolvent analysis and Doak's momentum potential theory to examine the input-output flow-acoustic characteristics of compressible flow over an open cavity at Re=10,000. The methodology can decompose hydrodynamic and acoustic components from an input-output framework. We further localize the forcing at the leading-edge and front wall of the cavity and filter with a…
We study the Reynolds number scaling and the geometric self-similarity of a gain-based, low-rank approximation to turbulent channel flows, determined by the resolvent formulation of McKeon & Sharma (2010), in order to obtain a description of the streamwise turbulence intensity from direct consideration of the Navier-Stokes equations. Under this formulation, the velocity field is decomposed into propagating waves (with single streamwise and spanwise wavelengths and wave speed) whose wall-normal shapes are determined from the principal singular function of the corresponding resolvent operator. U…
We develop a method to estimate space-time flow statistics from a limited set of known data. While previous work has focused on modeling spatial or temporal statistics independently, space-time statistics carry fundamental information about the physics and coherent motions of the flow and provide a starting point for low-order modeling and flow control efforts. The method is derived using a statistical interpretation of resolvent analysis. The central idea of our approach is to use known data to infer the statistics of the nonlinear terms that constitute a forcing on the linearized Navier-Stok…
We seek to quantify non-normality of the most amplified resolvent modes and predict their features based on the characteristics of the base or mean velocity profile. A 2-by-2 model linear Navier-Stokes (LNS) operator illustrates how non-normality from mean shear distributes perturbation energy in different velocity components of the forcing and response modes. The inverse of their inner product, which is unity for a purely normal mechanism, is proposed as a measure to quantify non-normality. In flows where there is downstream spatial dependence of the base/mean, mean flow advection separates t…
This paper presents a method for calculating the wall shear rate in pipe turbulent flow. It collapses adequately the data measured in laminar flow and turbulent flow into a single flow curve and gives the basis for the design of turbulent flow viscometers. Key words: non-Newtonian, wall shear rate, turbulent, rheometer
Previous work has established the usefulness of the resolvent operator that maps the terms nonlinear in the turbulent fluctuations to the fluctuations themselves. Further work has described the self-similarity of the resolvent arising from that of the mean velocity profile. The orthogonal modes provided by the resolvent analysis describe the wall-normal coherence of the motions and inherit that self-similarity. In this contribution, we present the implications of this similarity for the nonlinear interaction between modes with different scales and wall-normal locations. By considering the nonl…
Three amplification mechanisms present in turbulent jets, namely lift-up, Kelvin-Helmholtz, and Orr, are characterized via global resolvent analysis and spectral proper orthogonal decomposition (SPOD) over a range of Mach numbers. The lift-up mechanism was recently identified in turbulent jets via local analysis by Nogueira et al. (J. Fluid Mech., vol. 873, 2019, pp. 211-237) at low Strouhal number ($St$) and non-zero azimuthal wavenumbers ($m$). In these limits, a global SPOD analysis of data from high-fidelity simulations reveals streamwise vortices and streaks similar to those found in turb…
An extended version of the resolvent formulation is used to evaluate the use of anisotropic porous materials as passive flow control devices for turbulent channel flow. The effect of these porous substrates is introduced into the governing equations via a generalized version of Darcy's law. Model predictions show that materials with high streamwise permeability and low wall-normal permeability ($ϕ_{xy} = k_{xx}/k_{yy}\gg1$) can suppress resolvent modes resembling the energetic near-wall cycle. Based on these predictions, two anisotropic porous substrates with $ϕ_{xy} > 1$ and $ϕ_{xy} < 1$ were…
We study the laminar and turbulent channel flow over a viscous hyper-elastic wall and show that it is possible to sustain an unsteady chaotic turbulent-like flow at any Reynolds number by properly choosing the wall elastic modulus. We propose a physical explanation for this effect by evaluating the shear stress and the turbulent kinetic energy budget in the fluid and elastic layer. We vary the bulk Reynolds number from 2800 to 10 and identify two distinct mechanisms for turbulence production. At moderate and high Reynolds numbers, turbulent fluctuations activate the wall oscillations, which, i…
The turbulent flow around a squared-section cylinder at $Re=22000$ exhibits both a low-frequency Vortex-Shedding (VS), a common feature of all bluff-body flows, and high-frequency Kelvin-Helmholtz (KH) structures evolving on the shear layers formed at the top and bottom side of the cylinder. The VS motion triggers a periodic movement of these shear layers, inducing oscillations of the strength of their velocity gradients. The high-frequency KH structures respond to these low-frequency oscillations of the base-flow gradients in a quasi-steady manner. In this paper we will propose a general fram…
An analytical framework for studying the logarithmic region of turbulent channels is formulated. We build on recent findings (Moarref et al., J. Fluid Mech., 734, 2013) that the velocity fluctuations in the logarithmic region can be decomposed into a weighted sum of geometrically self-similar resolvent modes. The resolvent modes and the weights represent the linear amplification mechanisms and the scaling influence of the nonlinear interactions in the Navier-Stokes equations (NSE), respectively (McKeon & Sharma, J. Fluid Mech., 658, 2010). Originating from the NSE, this framework provides an a…
Scaling of turbulent wall-bounded flows is revealed in the gradient structures, for each of the Reynolds stress components. Within the dissipation structure, an asymmetrical order exists, that we can deploy to unify the scaling and transport dynamics within and across these flows. There are subtle differences in the outer boundary conditions between channel and flat-plate boundary-layer flows, which modify the turbulence structure far from the wall. The self-similarity exhibited in the dissipation structures and corresponding transport dynamics establish capabilities and encompassing knowledge…
The conceptual picture underlying resolvent analysis(RA) is that the nonlinear term in the Navier-Stokes(NS) equations provides an intrinsic forcing to the linear dynamics, a description inspired by control theory. The inverse of the linear operator, defined as the resolvent, is interpreted as a transfer function between the forcing and the velocity response. This inversion obscures the physical interpretation of the governing equations and is prohibitive to analytical manipulation, and for large systems leads to significant computational cost and memory requirements. In this work we suggest a…
We consider the frequency domain form of proper orthogonal decomposition (POD) called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space-time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic tools in turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently i…
Turbulent mixing is studied in the Lagrangian framework with an approach based on the complex network formalism. We consider the motion of passive, noninertial particles inside a turbulent channel simulated at $Re_τ = 950$. The time-dependent network is built to evaluate the transfer of tracers between thin wall-parallel layers which partition the channel in the wall-normal direction. By doing so, we are able to assess the spatial and temporal complexities arising from turbulence dynamics and their influence on the mixing process. This approach highlights the effects of small-scale features of…
