We identify forcing mechanisms that separately amplify subsonic and supersonic features obtained from a linearised Navier–Stokes based model for compressible parallel boundary layers. Resolvent analysis is used to analyse the linear model, where the nonlinear terms of the linearised equations act as a forcing to the linear terms. Considering subsonic modes, only the solenoidal component of the forcing to the momentum equations amplify these modes. When considering supersonic modes, we find that these are pressure fluctuations that radiate into the free stream. Within the free stream, these modes closely follow the trends of inviscid Mach waves. There are two distinct forcing mechanisms that amplify the supersonic modes: (i) the ‘direct route’, where the forcing to the continuity and energy equations and the dilatational component of the forcing to the momentum equations directly force the mode; and (ii) the ‘indirect route’, where the solenoidal component of the forcing to the momentum equations force a response in wall-normal velocity, and this wall-normal velocity in turn forces the supersonic mode. A majority of the supersonic modes considered are dominantly forced by the direct route. However, when considering Mach waves that are, like in direct numerical simulations, forced from the buffer layer of the flow, the indirect route of forcing becomes significant. We find that these observations are also valid for a streamwise developing boundary layer. These results are consistent with, and extend, the observations in the literature regarding the solenoidal and dilatational components of velocity in compressible turbulent wall-bounded flows.

Adverse pressure gradient (APG) turbulent boundary layers (TBL) require an understanding of the details of the pressure gradient, or history effect, to characterize the associated variation of spatiotemporal turbulent statistics. The streamwise-varying mean pressure gradient is reflected in the streamwise developing mean flow field and thus resolvent analysis, which captures the amplification of the Navier–Stokes equations linearized about the turbulent mean, can be used to understand linear amplification in APG TBLs. In particular, by using a biglobal approach in which the amplification is characterized by a temporal frequency and spanwise wavenumber, the streamwise and wall-normal inhomogeneities of the APG TBL can be resolved and related to the APG history. The linear response is able to identify multiscale phenomena, identifying a near-wall peak with $\lambda _{z}^+\approx 100$ for zero pressure gradient TBLs and mild to moderate APG TBLs as well as large-scale modes whose amplification increases with APG strength and Reynolds number. It is shown that the monotonic growth in the turbulent statistics with increasing APG is reflected in the linear growth in the associated resolvent amplification. Collapse in the Reynolds stresses is obtained through an augmented hybrid velocity scale, which replaces the local APG strength measure in the hybrid velocity scale presented in Romero et al. (Intl J. Heat Fluid Flow, vol. 93, 2022, 108885) with a velocity that encapsulates the pressure gradient history. While this resolvent approach is applicable to any APG TBL, it is shown from a scaling analysis of the linearized Navier–Stokes equations that the linear growth observed in the resolvent amplification with the history effect is limited to near-equilibrium APG TBLs.

Eigenvectors of the observability and controllability Gramians represent responsive and receptive flow structures that enjoy a well-established connection to resolvent forcing and response modes. However, whereas resolvent modes have demonstrated great potential to guide sensor and actuator placement, observability and controllability modes have been leveraged exclusively in the context of model reduction via input and output projections. In this work, we introduce interpolatory, rather than orthogonal, input and output projections, that can be leveraged for sensor and actuator placement and open-loop control design. An interpolatory projector is an oblique projector with the property of preserving certain entries in the vector being projected. We review the connection between the resolvent operator and the Gramians, and present several numerical examples where we perform both orthogonal and interpolatory input and output projections onto the dominant forcing and response subspaces. Input projections are used to identify dynamically relevant disturbances, place sensors to measure disturbances, and place actuators for feedforward control in the linearized Ginzburg–Landau equation. Output projections are used to identify coherent structures and place sensors aiming at state reconstruction in the turbulent flow in a minimal channel at $Re_{\tau }=185$. The framework does not require data snapshots and relies only on knowledge of the steady or mean flow.

The potential of frequency-tuned surfaces as a passive control strategy for reducing drag in wall-bounded turbulent flows is investigated using resolvent analysis. These surfaces are considered to have geometries with impedances that permit transpiration and/or slip at the wall in response to wall pressure and/or shear and are tuned to target the dynamically important structures of wall turbulence. It is shown that wall impedance can suppress the modes resembling the near-wall cycle and the very-large-scale motions and the Reynolds stress contribution of these modes. Suppression of the near-wall cycle requires a more reactive impedance. In addition to these dynamically important modes, the effect of wall impedance across the spectral space is analysed by considering varying mode speeds and wavelengths. It is shown that the materials designed for suppression of the near-wall modes lead to gain reduction over a wide range across the spectral space. Furthermore, a wall with only shear-driven impedance is found to suppress turbulent structures over a wider range in spectral space, leading to an overall turbulent drag reduction. Most importantly, the present analysis shows that the drag-reducing impedance is non-unique and the control performance is not sensitive to variations of the surface impedance within a favourable range. This implies that specific frequency bandwidths can be targeted with periodic material design.

Resolvent analysis identifies the most responsive forcings and most receptive states of a dynamical system, in an input–output sense, based on its governing equations. Interest in the method has continued to grow during the past decade due to its potential to reveal structures in turbulent flows, to guide sensor/actuator placement and for flow control applications. However, resolvent analysis requires access to high-fidelity numerical solvers to produce the linearized dynamics operator. In this work, we develop a purely data-driven algorithm to perform resolvent analysis to obtain the leading forcing and response modes, without recourse to the governing equations, but instead based on snapshots of the transient evolution of linearly stable flows. The formulation of our method follows from two established facts: (i) dynamic mode decomposition can approximate eigenvalues and eigenvectors of the underlying operator governing the evolution of a system from measurement data, and (ii) a projection of the resolvent operator onto an invariant subspace can be built from this learned eigendecomposition. We demonstrate the method on numerical data of the linearized complex Ginzburg–Landau equation and of three-dimensional transitional channel flow, and discuss data requirements. Presently, the method is suitable for the analysis of laminar equilibria, and its application to turbulent flows would require disambiguation between the linear and nonlinear dynamics driving the flow. The ability to perform resolvent analysis in a completely equation-free and adjoint-free manner will play a significant role in lowering the barrier of entry to resolvent research and applications.

The nonlinear mechanism in the self-sustaining process (SSP) of wall-bounded turbulence is investigated. Resolvent analysis is used to identify the principal forcing mode that produces the maximum amplification of the velocities in numerical simulations of the minimal channel for the buffer layer and a modified logarithmic (log) layer. The wavenumbers targeted in this study are those of the fundamental mode, which is infinitely long in the streamwise direction and once-periodic in the spanwise direction. The identified mode is then projected out from the nonlinear term of the Navier–Stokes equations at each time step from the simulation of the corresponding minimal channel. The results show that the removal of the principal forcing mode of the fundamental wavenumber can inhibit turbulence in both the buffer and log layer, with the effect being greater in the buffer layer. Removing other modes instead of the principal mode of the fundamental wavenumber only marginally affects the flow. Closer inspection of the dyadic interactions in the nonlinear term shows that contributions to the principal forcing mode come from a limited set of wavenumber interactions. Using conditional averaging, the flow structures that are responsible for generating the nonlinear interaction to self-sustain turbulence are identified as spanwise rolls interacting with oblique streaks. This method, based on the equations of motion, validates the similarities in the SSP of the buffer and log layer, and characterises the underlying quadratic interactions in the SSP of the minimal channel.

We present a framework for predicting the interactions between motion at a single scale and the underlying stress fluctuations in wall turbulence, derived from approximations to the Navier–Stokes equations. The dynamical equations for an isolated scale and stress fluctuations at the same scale are obtained from a decomposition of the governing equations and formulated in terms of a transfer function between them. This transfer function is closely related to the direct correlation coefficient of Duvvuri & McKeon (J. Fluid Mech., vol. 767, 2015, R4), and approximately to the amplitude modulation coefficient described in Mathis et al. (J. Fluid Mech., vol. 628, 2009, pp. 311–337), by consideration of interactions between triadically consistent scales. In light of the agreement between analysis and observations, the modelling approach is extended to make predictions concerning the relationship between very-large motions and small-scale stress in the logarithmic region of the mean velocity. Consistent with experiments, the model predicts that the zero-crossing height of the amplitude modulation statistic coincides with the wall-normal location of the very large-scale peak in the one-dimensional premultiplied spectrum of streamwise velocity fluctuations, the critical layer location for the very large-scale motion. Implications of fixed phase relationships between small-scale stresses and larger isolated scales for closure schemes are briefly discussed.

The probability density function (p.d.f.) of the streamwise velocity has been shown to indicate the presence of uniform momentum zones in wall-bounded turbulent flows. Most studies on the topic have focused on the instantaneous characteristics of this p.d.f. In this work, we show how the use of time-resolved particle image velocimetry data highlights robust features in the temporal behaviour of the p.d.f. and how these patterns are associated with the change of the number of zones present in the flow over time. The use of a limited resolvent model provides a clear link between this experimentally observed behaviour and the underlying velocity structures and their phase characteristics. This link is further supported by an extended resolvent model consisting of self-similar hierarchies centred in the logarithmic region, with triadically consistent members, yielding much more complex patterns in the p.d.f. Results indicate that the geometric similarity of these members instantaneously, as well as their relative evolution in time (dictated by their wall-normal varying wave speed), both inherent to the model, can reproduce many experimentally identified features.

Direct simulations of two-dimensional plane channel flow of a viscoelastic fluid at Reynolds number $Re=3000$ reveal the existence of a family of attractors whose structure closely resembles the linear Tollmien–Schlichting (TS) mode, and in particular exhibits strongly localized stress fluctuations at the critical layer position of the TS mode. At the parameter values chosen, this solution branch is not connected to the nonlinear TS solution branch found for Newtonian flow, and thus represents a solution family that is nonlinearly self-sustained by viscoelasticity. The ratio between stress and velocity fluctuations is in quantitative agreement for the attractor and the linear TS mode, and increases strongly with Weissenberg number, $\mathit{Wi}$. For the latter, there is a transition in the scaling of this ratio as $\mathit{Wi}$ increases, and the $\mathit{Wi}$ at which the nonlinear solution family comes into existence is just above this transition. Finally, evidence indicates that this branch is connected through an unstable solution branch to two-dimensional elastoinertial turbulence (EIT). These results suggest that, in the parameter range considered here, the bypass transition leading to EIT is mediated by nonlinear amplification and self-sustenance of perturbations that excite the TS mode.

The resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) 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 their incompressible counterparts. 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 relatively supersonic region and the relatively subsonic region. In the relatively supersonic region, where the flow is supersonic relative to the free-stream, resolvent modes display structures consistent with Mach wave radiation that are absent in the incompressible regime. In the relatively subsonic region, we show that the low-rank approximation of the resolvent operator is an effective approximation of the full system and that the response modes predicted by the model exhibit universal and geometrically self-similar behaviour via a transformation given by the semi-local scaling. Moreover, with the semi-local scaling, we show that the resolvent modes follow the same scaling law as their incompressible counterparts in this region, which has implications for modelling and the prediction of turbulent high-speed wall-bounded flows. We also show that the thermodynamic variables exhibit similar mode shapes to the streamwise velocity modes, supporting the strong Reynolds analogy. Finally, we demonstrate that the principal resolvent modes can be used to capture the energy distribution between momentum and thermodynamic fluctuations.

The flows around a NACA 0018 airfoil at a chord-based Reynolds number of $Re=10\,250$ and angles of attack of $\unicode[STIX]{x1D6FC}=0^{\circ }$ and $\unicode[STIX]{x1D6FC}=10^{\circ }$ are modelled using resolvent analysis and limited experimental measurements obtained from particle image velocimetry. The experimental mean velocity fields are data assimilated so that they are solutions of the incompressible Reynolds-averaged Navier–Stokes equations forced by Reynolds stress terms which are derived from experimental data. Resolvent analysis of the data-assimilated mean velocity fields reveals low-rank behaviour only in the vicinity of the shedding frequency for $\unicode[STIX]{x1D6FC}=0^{\circ }$ and none of its harmonics. The resolvent operator for the $\unicode[STIX]{x1D6FC}=10^{\circ }$ case, on the other hand, identifies two linear mechanisms whose frequencies are a close match with those identified by spectral proper orthogonal decomposition. It is also shown that the second linear mechanism, corresponding to the Kelvin–Helmholtz instability in the shear layer, cannot be identified just by considering the time-averaged experimental measurements as an input for resolvent analysis due to missing data near the leading edge. For both cases, resolvent modes resemble those from spectral proper orthogonal decomposition when the resolvent operator is low rank. The $\unicode[STIX]{x1D6FC}=0^{\circ }$ case is classified as an oscillator and its harmonics, where the resolvent operator is not low rank, are modelled using parasitic modes as opposed to classical resolvent modes which are the most amplified. The $\unicode[STIX]{x1D6FC}=10^{\circ }$ case behaves more like an amplifier and its nonlinear forcing is far less structured. The two cases suggest that resolvent-based modelling can be achieved for more complex flows with limited experimental measurements.

This work develops a methodology for approximating the shape of leading resolvent modes for incompressible, quasi-parallel, shear-driven turbulent flows using prescribed analytic functions. We demonstrate that these functions, which arise from the consideration of wavepacket pseudoeigenmodes of simplified linear operators (Trefethen, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 461, 2005, pp. 3099–3122. The Royal Society), give an accurate approximation for the energetically dominant wall-normal vorticity component of a class of nominally wall-detached modes that are centred about the critical layer. We validate our method on a model operator related to the Squire equation, and show for this simplified case how wavepacket pseudomodes relate to truncated asymptotic expansions of Airy functions. Following the framework developed in McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), we next apply a sequence of simplifications to the resolvent formulation of the Navier–Stokes equations to arrive at a scalar differential operator that is amenable to such analysis. The first simplification decomposes the resolvent operator into Orr–Sommerfeld and Squire suboperators, following Rosenberg & McKeon (Fluid Dyn. Res., vol. 51, 2019, 011401). The second simplification relates the leading resolvent response modes of the Orr–Sommerfeld suboperator to those of a simplified scalar differential operator – which is the Squire operator equipped with a non-standard inner product. This characterisation provides a mathematical framework for understanding the origin of leading resolvent mode shapes for the incompressible Navier–Stokes resolvent operator, and allows for rapid estimation of dominant resolvent mode characteristics without the need for operator discretisation or large numerical computations. We explore regions of validity for this method, and show that it can predict resolvent response mode shape (though not necessary the corresponding resolvent gain) over a wide range of spatial wavenumbers and temporal frequencies. In particular, we find that our method remains relatively accurate even when the modes have some amount of ‘attachment’ to the wall, and that that the region of validity contains the regions in parameter space where large-scale and very-large-scale motions typically reside. We relate these findings to classical lift-up and Orr amplification mechanisms in shear-driven flows.

Large-scale motions (LSMs) in wall-bounded turbulent flows have well-characterised instantaneous structural features (Kovasznay et al., J. Fluid Mech., vol. 41 (2), 1970, pp. 283–325; Meinhart & Adrian, Phys. Fluids, vol. 7 (4), 1995, pp. 694–696) and a known spectral signature (Monty et al., J. Fluid Mech., vol. 632, 2009, pp. 431–442). This work aims to connect these previous observations through the development and analysis of a representative model for LSMs. The model is constructed to be consistent with the streamwise energy spectrum (Monty et al. 2009) and amplification characteristics of the Navier–Stokes equations (McKeon & Sharma, J. Fluid Mech., vol. 658, 2010, pp. 336–382), and is found to naturally recreate characteristics of instantaneous turbulent structures, including a bulge shape (Kovasznay et al. 1970) and the presence of uniform momentum zones (Meinhart & Adrian 1995) in the streamwise velocity field. The observed structural similarity between the LSM representative model and instantaneous experimental data supports the use of travelling wave models to connect statistical and instantaneous descriptions of coherent structures, and clarifies a simple general equivalency between symmetry in a Reynolds-decomposed velocity field and asymmetry in the laboratory frame.

A formal relationship between the skewness and the correlation coefficient of large and small scales, termed the amplitude modulation coefficient, is established for a general statistically stationary signal and is analysed in the context of a turbulent velocity signal. Both the quantities are seen to be measures of phase in triadically consistent interactions between scales of turbulence. The naturally existing phase relationships between large and small scales in a turbulent boundary layer are then manipulated by exciting a synthetic large-scale motion in the flow using a spatially impulsive dynamic wall roughness perturbation. The synthetic scale is seen to alter the phase relationships, or the degree of modulation, in a quasi-deterministic manner by exhibiting a phase-organizing influence on the small scales. The results presented provide encouragement for the development of a practical framework for favourable manipulation of energetic small-scale turbulence through large-scale inputs in a wall-bounded turbulent flow.

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 (J. Fluid Mech., vol. 658, 2010, pp. 336–382), 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. Using the accepted scalings of the mean velocity in wall-bounded turbulent flows, we establish that the resolvent operator admits three classes of wave parameters that induce universal behaviour with Reynolds number in the low-rank model, and which are consistent with scalings proposed throughout the wall turbulence literature. In addition, it is shown that a necessary condition for geometrically self-similar resolvent modes is the presence of a logarithmic turbulent mean velocity. Under the practical assumption that the mean velocity consists of a logarithmic region, we identify the scalings that constitute hierarchies of self-similar modes that are parameterized by the critical wall-normal location where the speed of the mode equals the local turbulent mean velocity. For the rank-1 model subject to broadband forcing, the integrated streamwise energy density takes a universal form which is consistent with the dominant near-wall turbulent motions. When the shape of the forcing is optimized to enforce matching with results from direct numerical simulations at low turbulent Reynolds numbers, further similarity appears. Representation of these weight functions using similarity laws enables prediction of the Reynolds number and wall-normal variations of the streamwise energy intensity at high Reynolds numbers (${Re}_{\tau } \approx 1{0}^{3} {\unicode{x2013}} 1{0}^{10} $). Results from this low-rank model of the Navier–Stokes equations compare favourably with experimental results in the literature.