A new paradigm of rheological characterization, oscillatory simple shear with infinite forcing amplitudes, is introduced by Khair (J. Fluid Mech., vol. 791, 2016, R5). This pushes the technique of large-amplitude oscillatory shear (LAOS) to have two extremely large amplitudes (both strain-rate and strain), which we might call XXLAOS. Model-specific analytical predictions are derived for a suspension of nearly spherical rigid particles subject to Brownian rotational diffusion. The work illuminates a new regime of rheological characterization that may serve as a distinct proving ground for constitutive model selection and for probing the flow physics of rheologically complex fluids.

Three-dimensional direct numerical simulations (DNS) of a shock-induced laminar separation bubble are carried out to investigate the flow instability and origin of any low-frequency unsteadiness. A laminar boundary layer interacting with an oblique shock wave at $M=1.5$ is forced at the inlet with a pair of monochromatic oblique unstable modes, selected according to local linear stability theory (LST) performed within the separation bubble. Linear stability analysis is applied to cases with marginal and large separation, and compared to DNS. While the parabolized stability equations approach accurately reproduces the growth of unstable modes, LST performs less well for strong interactions. When the modes predicted by LST are used to force the separated boundary layer, transition to deterministic turbulence occurs near the reattachment point via an oblique-mode breakdown. Despite the clean upstream condition, broadband low-frequency unsteadiness is found near the separation point with a peak at a Strouhal number of $0.04$, based on the separation bubble length. The appearance of the low-frequency unsteadiness is found to be due to the breakdown of the deterministic turbulence, filling up the spectrum and leading to broadband disturbances that travel upstream in the subsonic region of the boundary layer, with a strong response near the separation point. The existence of the unsteadiness is supported by sensitivity studies on grid resolution and domain size that also identify the region of deterministic breakdown as the source of white noise disturbances. The present contribution confirms the presence of low-frequency response for laminar flows, similarly to that found in fully turbulent interactions.

The wakes and the drag forces of canopy patches with different densities, immersed in turbulent boundary layers, are investigated experimentally. The patches are circular (with outer diameter $D$) and are made of several identical circular cylinders (height, $H$, and diameter, $d$). The bulk aspect ratio of all of the patches ($AR=H/D$) was fixed at 1 and the patch density (${\it\phi}=N_{c}d^{2}/D^{2}$, also referred to as the solidity) is altered by varying the number of cylinders ($N_{c}$) in the patch. Drag measurements show that the patch drag coefficient increases with increasing density. However, the drag coefficient of the highest investigated density (${\it\phi}\approx 0.25$) is greater than the drag coefficient of a solid patch (i.e. ${\it\phi}=1$, which is a solid cylinder with $AR=1$). Particle image velocimetry (PIV) measurements were carried out along the streamwise–wall-normal ($x$–$y$) plane along the centreline of patch and in the streamwise–spanwise ($x$–$z$) plane at its mid height (i.e. $y=H/2$). Mean velocity fields show that the porosity of the patch promotes bleeding along all directions. It was observed that bleeding along the vertical/horizontal direction increases/decreases with increasing ${\it\phi}$. Furthermore, it was also observed that bleeding along the lateral direction dictates the point of flow separation along the sides of the patch and makes it independent of ${\it\phi}$. All of these aspects make wakes for porous patches markedly different from their solid counterpart and provide a rational basis to explain the observed trends in the drag coefficient.

A reduced model is developed for low-Rossby-number convection in a plane layer geometry with no-slip upper and lower boundaries held at fixed temperatures. A complete description of the dynamics requires the existence of three distinct regions within the fluid layer: a geostrophically balanced interior where fluid motions are predominantly aligned with the axis of rotation, Ekman boundary layers immediately adjacent to the bounding plates, and thermal wind layers driven by Ekman pumping in between. The reduced model uses a classical Ekman pumping parameterization to alleviate the need to resolve the Ekman boundary layers. Results are presented for both linear stability theory and a special class of nonlinear solutions described by a single horizontal spatial wavenumber. It is shown that Ekman pumping (which correlates positively with interior convection) allows for significant enhancement in the heat transport relative to that observed in simulations with stress-free boundaries. Without the intermediate thermal wind layer, the nonlinear feedback from Ekman pumping would be able to generate heat transport that diverges to infinity at finite Rayleigh number. This layer arrests this blowup, resulting in finite heat transport at a significantly enhanced value. With increasing buoyancy forcing, the heat transport transitions to a more efficient regime, a transition that is always achieved within the regime of asymptotic validity of the theory, suggesting that this behaviour may be prevalent in geophysical and astrophysical settings. As the rotation rate increases, the slope of the heat transport curve below this transition steepens, a result that is in agreement with observations from laboratory experiments and direct numerical simulations.

The method of matched asymptotic expansions is used to study the canonical problem of steady laminar flow through a narrow two-dimensional channel blocked by a tight-fitting finite-length highly permeable porous obstacle. We investigate the behaviour of the local flow close to the interface between the single-phase and porous regions (governed by the incompressible Navier–Stokes and Darcy flow equations, respectively). We solve for the flow in these inner regions in the limits of low and high Reynolds number, facilitating an understanding of the nature of the transition from Poiseuille to plug to Poiseuille flow in each of these limits. Significant analytical progress is made in the high Reynolds number limit, and we explore in detail the rich boundary layer structure that occurs. We derive general results for the interfacial stress and for the conditions that couple the flow in the outer regions away from the interface. We consider the three-dimensional generalization to unsteady laminar flow through and around a tight-fitting highly permeable cylindrical porous obstacle within a Hele-Shaw cell. For the high Reynolds number limit, we give the coupling conditions and interfacial stress in terms of the outer flow variables, allowing information from a nonlinear three-dimensional problem to be obtained by solving a linear two-dimensional problem. Finally, we illustrate the utility of our analysis by considering the specific example of time-dependent forced far-field flow in a Hele-Shaw cell containing a porous cylinder with a circular cross-section. We determine the internal stress within the porous obstacle, which is key for tissue engineering applications, and the interfacial stress on the boundary of the porous obstacle, which has applications to biofilm erosion. In the high Reynolds number limit, we demonstrate that the fluid inertia can result in the cylinder experiencing a time-independent net force, even when the far-field forcing is periodic with zero mean.

Integrals of the mean and turbulent energy dissipation rates are examined using direct numerical simulation (DNS) databases in a turbulent channel flow. Four values of the Kármán number ($h^{+}=180$, 395, 640 and 1020; $h$ is the channel half-width) are used. Particular attention is given to the functional $h^{+}$ dependence by comparing existing DNS and experimental data up to $h^{+}=10^{4}$. The logarithmic $h^{+}$ dependence of the integrated turbulent energy dissipation rate is established for $300\leqslant h^{+}\leqslant 10^{4}$, and is intimately linked to the logarithmic skin friction law, viz.$U_{b}^{+}=2.54\ln (h^{+})+2.41$ ($U_{b}$ is the bulk mean velocity). This latter relationship is established on the basis of energy balances for both the mean and turbulent kinetic energy. When $h^{+}$ is smaller than 300, viscosity affects the integrals of both the mean and turbulent energy dissipation rates significantly due to the lack of distinct separation between inner and outer regions. The logarithmic $h^{+}$ dependence of $U_{b}^{+}$ is clarified through the scaling behaviour of the turbulent energy dissipation rate $\overline{{\it\varepsilon}}$ in different parts of the flow. The overlap between inner and outer regions is readily established in the region $30/h^{+}\leqslant y/h\leqslant 0.2$ for $h^{+}\geqslant 300$. At large $h^{+}$ (${\geqslant}$5000) when the finite Reynolds number effect disappears, the magnitude of $\overline{{\it\varepsilon}}y/U_{{\it\tau}}^{3}$ approaches 2.54 near the lower bound of the overlap region. This value is identical between the channel, pipe and boundary layer as a result of similarity in the constant stress region. As $h^{+}$ becomes large, the overlap region tends to contribute exclusively to the $2.54\ln (h^{+})$ dependence of the integrated turbulent energy dissipation rate. The present logarithmic $h^{+}$ dependence of $U_{b}^{+}$ is essentially linked to the overlap region, even at small $h^{+}$.

Colonies of the green alga Volvox are spheres that swim through the beating of pairs of flagella on their surface somatic cells. The somatic cells themselves are mounted rigidly in a polymeric extracellular matrix, fixing the orientation of the flagella so that they beat approximately in a meridional plane, with axis of symmetry in the swimming direction, but with a roughly $20^{\circ }$ azimuthal offset which results in the eponymous rotation of the colonies about a body-fixed axis. Experiments on colonies of Volvox carteri held stationary on a micropipette show that the beating pattern takes the form of a symplectic metachronal wave (Brumley et al. Phys. Rev. Lett., vol. 109, 2012, 268102). Here we extend the Lighthill/Blake axisymmetric, Stokes-flow model of a free-swimming spherical squirmer (Lighthill Commun. Pure Appl. Maths, vol. 5, 1952, pp. 109–118; Blake J. Fluid Mech., vol. 46, 1971b, pp. 199–208) to include azimuthal swirl. The measured kinematics of the metachronal wave for 60 different colonies are used to calculate the coefficients in the eigenfunction expansions and hence predict the mean swimming speeds and rotation rates, proportional to the square of the beating amplitude, as functions of colony radius. As a test of the squirmer model, the results are compared with measurements (Drescher et al. Phys. Rev. Lett., vol. 102, 2009, 168101) of the mean swimming speeds and angular velocities of a different set of 220 colonies, also given as functions of colony radius. The predicted variation with radius is qualitatively correct, but the model underestimates both the mean swimming speed and the mean angular velocity unless the amplitude of the flagellar beat is taken to be larger than previously thought. The reasons for this discrepancy are discussed.

The distribution of particle accelerations in turbulence is intermittent, with non-Gaussian tails that are quite different for light and heavy particles. In this article we analyse a closure scheme for the acceleration fluctuations of light and heavy inertial particles in turbulence, formulated in terms of Lagrangian correlation functions of fluid tracers. We compute the variance and the flatness of inertial-particle accelerations and we discuss their dependency on the Stokes number. The closure incorporates effects induced by the Lagrangian correlations along the trajectories of fluid tracers, and its predictions agree well with results of direct numerical simulations of inertial particles in turbulence, provided that the effects induced by inertial preferential sampling of heavy/light particles outside/inside vortices are negligible. In particular, the scheme predicts the correct functional behaviour of the acceleration variance, as a function of $St$, as well as the presence of a minimum/maximum for the flatness of the acceleration of heavy/light particles, in good qualitative agreement with numerical data. We also show that the closure works well when applied to the Lagrangian evolution of particles using a stochastic surrogate for the underlying Eulerian velocity field. Our results support the conclusion that there exist important contributions to the statistics of the acceleration of inertial particles independent of the preferential sampling. For heavy particles we observe deviations between the predictions of the closure scheme and direct numerical simulations, at Stokes numbers of order unity. For light particles the deviation occurs for larger Stokes numbers.

The linearized water wave radiation problem for an oscillating submerged line source in an inviscid shear flow with a free surface is investigated analytically at finite, constant depth in the presence of a shear flow varying linearly with depth. The surface velocity is taken to be zero relative to the oscillating source, so that Doppler effects are absent. The radiated wave out from the source is calculated based on Euler’s equation of motion with the appropriate boundary and radiation conditions, and differs substantially from the solution obtained by assuming potential flow. To wit, an additional wave is found in the downstream direction in addition to the previously known dispersive wave solutions; this wave is non-dispersive and we show how it is the surface manifestation of a critical layer-like flow generated by the combination of shear and mass flux at the source, passively advected with the flow. As seen from a system moving at the fluid velocity at the source’s depth, streamlines form closed curves in a manner similar to Kelvin’s cat’s eye vortices. A resonant frequency exists at which the critical wave resonates with the downstream propagating wave, resulting in a downstream wave pattern diverging linearly in amplitude away from the source.

We investigate analytically the linearised water wave radiation problem for an oscillating submerged point source in an inviscid shear flow with a free surface. A constant depth is taken into account and the shear flow increases linearly with depth. The surface velocity relative to the source is taken to be zero, so that Doppler effects are absent. We solve the linearised Euler equations to calculate the resulting wave field as well as its far-field asymptotics. For values of the Froude number $F^{2}={\it\omega}^{2}D/g$ (where ${\it\omega}$ is the oscillation frequency, $D$ is the submergence depth and $g$ is the gravitational acceleration) below a resonant value $F_{res}^{2}$, the wave field splits cleanly into separate contributions from regular dispersive propagating waves and non-dispersive ‘critical waves’ resulting from a critical layer-like street of flow structures directly downstream of the source. In the subresonant regime, the regular waves behave like sheared ring waves, while the critical layer wave forms a street with a constant width of order $D\sqrt{S/{\it\omega}}$ (where $S$ is the shear flow vorticity) and is convected downstream at the fluid velocity at the depth of the source. When the Froude number approaches its resonant value, the downstream critical and regular waves resonate, producing a train of waves of linearly increasing amplitude contained within a downstream wedge.

In 1880, Stokes famously demonstrated that the singularity that occurs at the crest of the steepest possible water wave in infinite depth must correspond to a corner of $120^{\circ }$. Here, the complex velocity scales like $f^{1/3}$ where $f$ is the complex potential. Later in 1973, Grant showed that for any wave away from the steepest configuration, the singularity $f=f^{\ast }$ moves into the complex plane, and is of order $(f-f^{\ast })^{1/2}$ (Grant J. Fluid Mech., vol. 59, 1973, pp. 257–262). Grant conjectured that as the highest wave is approached, other singularities must coalesce at the crest so as to cancel the square-root behaviour. Despite recent advances, the complete singularity structure of the Stokes wave is still not well understood. In this work, we develop numerical methods for constructing the Riemann surface that represents the extension of the water wave into the complex plane. We show that a countably infinite number of distinct singularities exist on other branches of the solution, and that these singularities coalesce as Stokes’ highest wave is approached.

We investigate the convection and dissolution rate generated when a wall of ice dissolves into seawater under Antarctic Ocean conditions. In direct numerical simulations three coupled interface equations are used to solve for interface temperature, salinity and ablation velocity, along with the boundary layer flow and transport. The main focus is on ambient water temperatures between $-1\,^{\circ }\text{C}$ and $6\,^{\circ }\text{C}$ and salinities around 35 ‰, where diffusion of salt to the ice–water interface depresses the freezing point and enhances heat diffusion to the ice. We show that fluxes of both heat and salt to the interface are significant in governing the dissolution of ice, and the ablation velocity agrees well with experiments and a recent theoretical prediction. The same turbulent flow dynamics and ablation rate are expected to apply at any depth in a deeper ocean water column (after choosing the relevant pressure coefficient for the liquidus temperature). At Grashof numbers currently accessible by direct numerical simulation, turbulence is generated both directly from buoyancy flux and from shear production in the buoyancy-driven boundary layer flow, whereas shear production by the convective flow is expected to be more important at geophysical scales. The momentum balance in the boundary layer is dominated by buoyancy forcing and wall stress, with the latter characterised by a large drag coefficient.

Fluctuations in the gas-phase velocity can contribute significantly to the total gas-phase kinetic energy even in laminar gas–solid flows as shown by Mehrabadi et al. (J. Fluid Mech., vol. 770, 2015, pp. 210–246), and these pseudo-turbulent fluctuations can also enhance heat transfer in gas–solid flow. In this work, the pseudo-turbulent heat flux arising from temperature–velocity covariance, and average fluid-phase conduction during convective heat transfer in a gas–solid flow are quantified and modelled over a wide range of mean slip Reynolds number and solid volume fraction using particle-resolved direct numerical simulations (PR-DNS) of steady flow through a random assembly of fixed isothermal monodisperse spherical particles. A thermal self-similarity condition on the local excess temperature developed by Tenneti et al. (Intl J. Heat Mass Transfer, vol. 58, 2013, pp. 471–479) is used to guarantee thermally fully developed flow. The average gas–solid heat transfer rate for this flow has been reported elsewhere by Sun et al. (Intl J. Heat Mass Transfer, vol. 86, 2015, pp. 898–913). Although the mean velocity field is homogeneous, the mean temperature field in this thermally fully developed flow is inhomogeneous in the streamwise coordinate. An exponential decay model for the average bulk fluid temperature is proposed. The pseudo-turbulent heat flux that is usually neglected in two-fluid models of the average fluid temperature equation is computed using PR-DNS data. It is found that the transport term in the average fluid temperature equation corresponding to the pseudo-turbulent heat flux is significant when compared to the average gas–solid heat transfer over a significant range of solid volume fraction and mean slip Reynolds number that was simulated. For this flow set-up a gradient-diffusion model for the pseudo-turbulent heat flux is found to perform well. The Péclet number dependence of the effective thermal diffusivity implied by this model is explained using a scaling analysis. Axial conduction in the fluid phase, which is often neglected in existing one-dimensional models, is also quantified. As expected, it is found to be important only for low Péclet number flows. Using the exponential decay model for the average bulk fluid temperature, a model for average axial conduction is developed that verifies standard assumptions in the literature. These models can be used in two-fluid simulations of heat transfer in fixed beds. A budget analysis of the mean fluid temperature equation provides insight into the variation of the relative magnitude of the various terms over the parameter space.

In the present article, the cell model (or self-consistent scheme) is used to derive constitutive equations for rod suspensions in non-Newtonian viscous matrices such as power-law, Ellis and Carreau fluids. It is found that the shear-thinning character of the matrix influences considerably the rod contribution to the stress tensor, but has no impact on the rod orientation dynamics: the same microstructure evolution as the one encountered in Newtonian fluids is obtained. The rod suspension behaves differently than the unfilled matrix in the sense that, depending on rod orientation, the onset of shear thinning in the composite occurs at lower or higher shear rates. Our analysis also provides a semi-analytical model for rod suspensions in an Ellis fluid, which appears to be suitable for predicting a Newtonian plateau at low shear rates and a shear-thinning behaviour at high shear rates. In addition, the model predictions are in good agreement with the shear viscosity measurements of glass-fibre-filled polystyrene melts (Chan et al., J. Rheol., vol. 22 (5), 1978, pp. 507–524), demonstrating its ability to describe the rheological behaviour of such polymer composites. Finally, the proposed approach is extended to a Carreau fluid although its solution requires the numerical solution of a set of partial differential equations.

We investigate the early development of instabilities in the oscillatory viscous flow past cylinders with elliptic cross-sections using three-dimensional direct numerical simulations. This is a classical hydrodynamic problem for circular cylinders, but other configurations have received only marginal attention. Computed results for some different aspect ratios ${\it\Lambda}$ from 1 : 1 to 1 : 3, all with the major axis of the ellipse aligned in the main flow direction, show good qualitative agreement with Hall’s stability theory (J. Fluid Mech., vol. 146, 1984, pp. 347–367), which predicts a cusp-shaped curve for the onset of the primary instability. The three-dimensional flow structures for aspect ratios larger than 2 : 3 resemble those of a circular cylinder, whereas the elliptical cross-section with the lowest aspect ratio of 1 : 3 exhibits oblate rather than tubular three-dimensional flow structures as well as a pair of counter-rotating spanwise vortices which emerges near the tips of the ellipse. Contrary to a circular cylinder, instabilities for an elliptic cylinder with sufficiently high eccentricity emerge from four rather than two different locations in accordance with the Hall theory.

Previous experimental studies on turbulent square duct flow have focused mainly on high Reynolds numbers for which a turbulence-induced eight-vortex secondary flow pattern exists in the cross-sectional plane. More recently, direct numerical simulations (DNS) have revealed that the flow field at Reynolds numbers close to transition can be very different; the flow in this ‘marginally turbulent’ regime alternating between two states characterised by four vortices. In this study, we experimentally investigate the onset criteria for transition to turbulence in square ducts. In so doing, we highlight the potential importance of Coriolis effects on this process for low-Ekman-number flows. We also present experimental data on the mean flow properties and turbulence statistics in both marginally and fully turbulent flow at relatively low Reynolds numbers using laser Doppler velocimetry. Results for both flow categories show good agreement with DNS. The switching of the flow field between two flow states at marginally turbulent Reynolds numbers is confirmed by bimodal probability density functions of streamwise velocity at certain distances from the wall as well as joint probability density functions of streamwise and wall normal velocities which feature two peaks highlighting the two states.

Two-phase turbulent Taylor–Couette (TC) flow is simulated using an Euler–Lagrange approach to study the effects of a secondary phase dispersed into a turbulent carrier phase (here bubbles dispersed into water). The dynamics of the carrier phase is computed using direct numerical simulations (DNS) in an Eulerian framework, while the bubbles are tracked in a Lagrangian manner by modelling the effective drag, lift, added mass and buoyancy force acting on them. Two-way coupling is implemented between the dispersed phase and the carrier phase which allows for momentum exchange among both phases and to study the effect of the dispersed phase on the carrier phase dynamics. The radius ratio of the TC setup is fixed to ${\it\eta}=0.833$, and a maximum inner cylinder Reynolds number of $Re_{i}=8000$ is reached. We vary the Froude number ($Fr$), which is the ratio of the centripetal to the gravitational acceleration of the dispersed phase and study its effect on the net torque required to drive the TC system. For the two-phase TC system, we observe drag reduction, i.e. the torque required to drive the inner cylinder is lower compared with that of the single-phase system. The net drag reduction decreases with increasing Reynolds number $Re_{i}$, which is consistent with previous experimental findings (Murai et al., J. Phys.: Conf. Ser., vol. 14, 2005, pp. 143–156; Phys. Fluids, vol. 20(3), 2008, 034101). The drag reduction is strongly related to the Froude number: for fixed Reynolds number we observe higher drag reduction when $Fr<1$ than for with $Fr>1$. This buoyancy effect is more prominent in low $Re_{i}$ systems and decreases with increasing Reynolds number $Re_{i}$. We trace the drag reduction back to the weakening of the angular momentum carrying Taylor rolls by the rising bubbles. We also investigate how the motion of the dispersed phase depends on $Re_{i}$ and $Fr$, by studying the individual trajectories and mean dispersion of bubbles in the radial and axial directions. Indeed, the less buoyant bubbles (large $Fr$) tend to get trapped by the Taylor rolls, while the more buoyant bubbles (small $Fr$) rise through and weaken them.

An asymptotic model is constructed to analyse the interaction of turbulence generated far upstream with a thin elliptic-nosed solid body in uniform flow. The leading-edge stagnation point causes significant deformation of incident vorticity, and hence our analysis focuses on the region of size scaling with the nose radius close to the stagnation point. Rapid distortion theory is used to separate the flow field generated by a single unsteady gust perturbation into a convective non-acoustic part, containing the evolution of the upstream vortical disturbance, and an acoustic part generated by the interaction of the vorticity with the solid surface, as is typical in gust–aerofoil interaction theory. Using single-frequency gust response solutions, along with a von Kármán energy spectrum, we find the turbulent pressure spectrum generated by homogeneous isotropic turbulence incident from far upstream. Both high- and low-frequency gusts are considered to allow approximations to be found for the turbulent pressure spectra close to the leading edge, and far from the body close to the incident stagnation streamline. Good agreement is shown between the asymptotic results for the near- and far-field leading-edge turbulent pressure spectra and recent experimental findings.

Analytical solutions are derived to describe two-dimensional wave evolution in converging bays. Three bay types of different cross-sections are studied: U-shaped, V-shaped and cusped bays. For these bays, the two-dimensional linear shallow water equations can be reduced to one-dimensional linear dispersive wave equations if the transverse flow acceleration inside them is assumed to be small. The derived solutions are characterized as the leading-order plane-wave solutions with higher-order corrections for two-dimensionality due to wave refraction. Wave amplitude longitudinally increases with different rates for the three bay types, whereas it exhibits weak parabolic variations in the transverse direction. Wave refraction significantly affects relatively short waves, contributing to wave energy transfer to the inner bay in a different manner depending on the bay type. The perturbation analysis of very high-order wave celerity suggests that the solutions are valid only when the ratio of the bay width to the wavelength is smaller than a certain limit that differs with bay type. Beyond the limit, the higher-order effect is no longer a minor correction, implying that wave behaviours become highly two-dimensional and possibly cause total reflection. The higher-order effect on the run-up height at the bay head is found to be small within the applicable range of the solution, and thus, the run-up formula neglecting the transverse flows has a wide validity. We also discuss the limitation of run-up height by wave breaking on the basis of a breaking criterion from previous studies.

This article provides theoretical conditions for the use and meaning of a stability analysis around a mean flow. As such, it may be considered as an extension of the works by McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382) to non-parallel flows and by Turton et al. (Phys. Rev. E, vol. 91 (4), 2015, 043009) to broadband flows. Considering a Reynolds decomposition of the flow field, the spectral (or temporal Fourier) mode of the fluctuation field is found to be equal to the action on a turbulent forcing term by the resolvent operator arising from linearisation about the mean flow. The main result of the article states that if, at a particular frequency, the dominant singular value of the resolvent is much larger than all others and if the turbulent forcing at this frequency does not display any preferential direction toward one of the suboptimal forcings, then the spectral mode is directly proportional to the dominant optimal response mode of the resolvent at this frequency. Such conditions are generally met in the case of weakly non-parallel open flows exhibiting a convectively unstable mean flow. The spatial structure of the singular mode may in these cases be approximated by a local spatial stability analysis based on parabolised stability equations (PSE). We have also shown that the frequency spectrum of the flow field at any arbitrary location of the domain may be predicted from the frequency evolution of the dominant optimal response mode and the knowledge of the frequency spectrum at one or more points. Results are illustrated in the case of a high Reynolds number turbulent backward facing step flow.