Asymptotic homogenization is employed to formulate upscaled effective boundary conditions at a smooth virtual surface for a natural-convection flow over a periodically roughened vertical wall, to bypass the expensive numerical resolution of flow and temperature fields near and within wall corrugations. Microscale problems are found by expanding near-wall variables in terms of a small parameter $\epsilon$, the ratio between the microscopic and the macroscopic length scales. The expressions of the upscaled velocity and temperature boundary conditions are provided up to second-order accuracy in $\epsilon$. The case of transverse square ribs is considered as a representative example. The classical Navier-slip condition for the streamwise and the spanwise velocity components is modified at second order by the gradient of the normal stress and the time derivative of the shear stress. The streamwise slip velocity is additionally corrected by a buoyancy term at first order and a temperature-gradient term at second order. The normal velocity at the virtual surface appears only as a second-order transpiration condition. A Robin-like condition for the temperature is found, where the wall temperature is corrected with a temperature-gradient term representing thermal slip. The accuracy levels and the applicability range of the effective conditions to mimic the macroscopic flow behaviour are investigated under laminar flow conditions, in comparison with results of full feature-resolving simulations. A formal validity limit for the approximation is sought in terms of a single accuracy criterion ($C$), which combines the effects of the Grashof number and the ribs’ density. The introduced model is further tested on different rib geometries.

This work reports on modelling unsteady gas flow in porous media at the macroscopic scale in the slip regime, a topic of interest in a wide range of applications. The slip effect is modelled by means of a Navier-type boundary condition. A macroscopic model is derived from the initial-boundary-value problem governing unsteady, single-phase flow of a Newtonian fluid through homogeneous porous media in the creeping, isothermal and slightly compressible slip regime. For momentum transport, the macroscopic model involves two terms. The first consists of a convolution product between the macroscopic pressure gradient and the temporal derivative of an apparent dynamic permeability tensor; the second accounts for the memory of the initial condition. Both contributions are predicted from the solution of a unique closure problem that is independent of the initial flow configuration and of the macroscopic pressure gradient. The accuracy of the model is assessed by comparisons with direct numerical simulations performed at the pore-scale, which find excellent agreement. The simulations also show that a classical heuristic model, which is the consequence of assuming a separation of time scales between the pore-scale and the macroscale, is inadequate, in general, to correctly predict the macroscopic velocity. Results from this work provide a formal clear insight about unsteady flow in porous media in the slip regime, motivating further theoretical and experimental work.

Natural and engineered surfaces are never smooth, but irregular, rough at different scales, compliant, possibly porous, liquid impregnated or superhydrophobic. The correct numerical modelling of fluid flowing through and around them is important but poses problems. For media characterized by a periodic or quasi-periodic microstructure of characteristic dimensions smaller than the relevant scales of the flow, multiscale homogenization can be used to study the effect of the surface, avoiding the numerical resolution of small details. Here, we revisit the homogenization strategy using adjoint variables to model the interaction between a fluid in motion and regularly micro-textured, permeable or impermeable walls. The approach described allows for the easy derivation of auxiliary/adjoint systems of equations which, after averaging, yield macroscopic tensorial properties, such as permeability, elasticity, slip, transpiration, etc. When the fluid in the neighbourhood of the microstructure is in the Stokes regime, classical results are recovered. Adjoint homogenization, however, permits simple extension of the analysis to the case in which the flow displays nonlinear effects. Then, the properties extracted from the auxiliary systems take the name of effective properties and do not depend only on the geometrical details of the medium, but also on the microscopic characteristics of the fluid motion. Examples are shown to demonstrate the usefulness of adjoint homogenization to extract effective tensor properties without the need for ad hoc parameters. In particular, notable results reported herein include:

1. (i) an original formulation to describe filtration in porous media in the presence of inertial effects;

2. (ii) the microscopic and macroscopic equations needed to characterize flows through poroelastic media;

3. (iii) an extended Navier’s condition to be employed at the boundary between a fluid and an impermeable rough wall, with roughness elements which can be either rigid or linearly elastic;

4. (iv) the microscopic problems needed to define the relevant parameters for a Saffman-like condition at the interface between a fluid and a porous substrate; and

5. (v) the macroscopic equations which hold at the dividing surface between a free-fluid region and a fluid-saturated poroelastic domain.

A macroscopic boundary condition to be used when a fluid flows over a rough surface is derived. It provides the slip velocity $\boldsymbol{u}_{S}$ on an equivalent (smooth) surface in the form $\boldsymbol{u}_{S}=\unicode[STIX]{x1D716}{\mathcal{L}}\boldsymbol{ : }{\mathcal{E}}$, where the dimensionless parameter $\unicode[STIX]{x1D716}$ is a measure of the roughness amplitude, ${\mathcal{E}}$ denotes the strain-rate tensor associated with the outer flow in the vicinity of the surface and ${\mathcal{L}}$ is a third-order slip tensor arising from the microscopic geometry characterizing the rough surface. This boundary condition represents the tensorial generalization of the classical Navier slip condition. We derive this condition, in the limit of small microscopic Reynolds numbers, using a multi-scale technique that yields a closed system of equations, the solution of which allows the slip tensor to be univocally calculated, once the roughness geometry is specified. We validate this generalized slip condition by considering the flow about a rough sphere, the surface of which is covered with a hexagonal lattice of cylindrical protrusions. Comparisons with direct numerical simulations performed in both laminar and turbulent regimes allow us to assess the validity and limitations of this condition and of the mathematical model underlying the determination of the slip tensor ${\mathcal{L}}$.

The interaction between a fluid flow and a transversely isotropic porous medium is described. A homogenized model is used to treat the flow field in the porous region, and different interface conditions, needed to match solutions at the boundary between the pure fluid and the porous regions, are evaluated. Two problems in different flow regimes (laminar and turbulent) are considered to validate the system, which includes inertia in the leading-order equations for the permeability tensor through a Oseen approximation. The components of the permeability, which characterize microscopically the porous medium and determine the flow field at the macroscopic scale, are reasonably well estimated by the theory, both in the laminar and the turbulent case. This is demonstrated by comparing the model’s results to both experimental measurements and direct numerical simulations of the Navier–Stokes equations which resolve the flow also through the pores of the medium.

A simple approach is described for computing spatially extended, weakly nonlinear optimal disturbances, suitable for maintaining a disturbance-regeneration cycle in a simple shear flow. Weakly nonlinear optimals, computed over a short time interval for the expansion used to remain tenable, are oblique waves which display a shorter streamwise and a longer spanwise wavelength than their linear counterparts. Threshold values of the initial excitation energy, separating the region of damped waves from that where disturbances grow without bounds, are found. Weakly nonlinear optimal solutions of varying initial amplitudes are then fed as initial conditions into direct numerical simulations of the Navier–Stokes equations and it is shown that the weakly nonlinear model permits the identification of flow states which cause rapid breakdown to turbulence.

Receptivity is the process which describes how environmental disturbances (such as gusts, acoustic waves or wall roughness) are filtered by a boundary layer and turned into downstream-growing waves. It is closely related to the identification of initial conditions for the disturbances and requires knowledge of the characteristics of the specific external forcing field. Without such a knowledge, it makes sense to focus on worst case scenarios and search for those initial states which maximize the disturbance amplitude at a given downstream position, and hence to identify upper bounds on growth rates, which will be useful in predicting the transition to turbulence. This philosophical approach has been taken by Tempelmann, Hanifi & Henningson (J. Fluid Mech., 2010, vol. 646, pp. 5–37) in a remarkably complete parametric study of ‘optimal disturbances’ for a model of the flow over a swept wing; they pinpoint the crucial importance both of the spatial variation of the flow and of non-modal disturbances, even when the flow is ‘supercritical’ and hence subject to classical ‘normal mode’ instabilities.

The mechanisms leading to large transient growth of disturbances for the flow in a channel with compliant walls are investigated. The walls are modelled as thin spring-backed plates, and the flow dynamics is modelled using the Navier–Stokes equations linearized round the Poiseuille profile. Analysis for streamwise invariant perturbations show that this fluid-structure system can sustain oscillatory energy evolution of large amplitude, in the form of spanwise standing waves. Such waves are related to the travelling waves which a free wall can support, modified to account for an ‘added mass’ effect. Simple scaling arguments are found to provide results in excellent agreement with computations of optimal disturbances, for low-to-moderate values of the stiffness parameter characterizing the compliant surface.

It has been recently shown that the flow of a Bingham fluid in a channel is always linearly stable (Nouar et al., J. Fluid Mech., vol. 577, 2007, p. 211). To identify possible paths of transition we revisit the problem for the case in which the idealized base flow is slightly perturbed. No attempt is made to reproduce or model the perturbations arising in experimental environments – which may be due to the improper alignment of the channel walls or to imperfect inflow conditions – rather a general formulation is given which yields the transfer function (the sensitivity) for each eigenmode of the spectrum to arbitrary defects in the base flow. It is first established that such a function, for the case of the most sensitive eigenmode, displays a very weak selectivity to variations in the spanwise wavenumber of the disturbance mode. This justifies a further look into the class of spanwise homogeneous modes. A variational procedure is set up to identify the base flow defect of minimal norm capable of optimally destabilizing an otherwise stable flow; it is found that very weak defects are indeed capable to excite exponentially amplified streamwise travelling waves. The associated variations in viscosity are situated mostly near the critical layer of the inviscid problem. Neutrally stable conditions are found as function of the Reynolds number and the Bingham number, providing scalings of critical values with the amplitude of the defect consistent with previous experimental and numerical studies. Finally, a structured pseudospectrum analysis is performed; it is argued that such a class of pseudospectra provides information well suited to hydrodynamic stability purposes.

The stability of the laminar flow in the narrow gap between infinitely long concentric cylinders, the inner of which rotates, is examined for the case of compliant bounding walls, modelled as thin cylindrical shells supported by rigid frames through arrays of springs and dampers. Sufficiently soft walls have a destabilizing influence on the axisymmetric Taylor vortices produced by the centrifugal force, although the effect is limited to modes with large axial wavelengths. Due to the walls flexibility, hydroelastic modes are generated. Complex modal exchanges are observed, as function of the wall properties and the Reynolds number. For axisymmetric modes an asymptotic analysis is conducted in the limit of small axial wavenumber, to show the correspondence between such exchanges and singularities in the analytical solutions. While the axisymmetric modes dominate the spectrum when the walls are rigid or very mildly compliant, a critical non-zero azimuthal wavenumber exists for which the hydroelastic modes become more unstable. Shorter azimuthal waves are favoured by increasing spring stiffness.

A model of hairy medium is developed using a homogenized approach, and the fluid flow around a circular cylinder partially coated with hair is analysed by means of numerical simulations. The capability of this coating to adapt to the surrounding flow is investigated, and its benefits are discussed in the context of separation control. This fluid–structure interaction problem is solved with a partitioned approach, based on the direct resolution of the Navier–Stokes equations together with a nonlinear set of equations describing the dynamics of the coating. A volume force, arising from the presence of a cluster of hair, provides the link between the fluid and the structure problems. For the structure part, a subset of reference elements approximates the whole layer. The dynamics of these elements is governed by a set of equations based on the inertia, elasticity, interaction and losses effects of articulated rods. The configuration chosen is that of the two-dimensional flow past a circular cylinder at Re = 200, a simple and well-documented test case. Aerodynamics performances quantified by the Strouhal number, the drag and the maximum lift in the laminar unsteady regime are modified by the presence of the coating. A set of parameters corresponding to a realistic coating (length of elements, porosity, rigidity) is found, yielding an average drag reduction of 15% and a decrease of lift fluctuations by about 40%, associated to a stabilization of the wake.

The transition of the flow in a duct of square cross-section is studied. Like in the similar case of the pipe flow, the motion is linearly stable for all Reynolds numbers; this flow is thus a good candidate to investigate the ‘bypass’ path to turbulence. Initially the so-called ‘linear optimal perturbation problem’ is formulated and solved, yielding optimal disturbances in the form of longitudinal vortices. Such optimals, however, fail to elicit a significant response from the system in the nonlinear regime. Thus, streamwise-inhomogeneous sub-optimal disturbances are focused upon; nonlinear quadratic interactions are immediately caused by such initial perturbations and an unstable streamwise-homogeneous large-amplitude mode rapidly emerges. The subsequent evolution of the flow, at a value of the Reynolds number at the boundary between fully developed turbulence and relaminarization, shows the alternance of patterns with two pairs of large-scale vortices near opposing parallel walls. Such edge states bear a resemblance to optimal disturbances.

The laminar flow in a curved channel is studied numerically to analyse the initial formation, development and interaction phenomena of an array of centrifugally induced longitudinal vortices arranged across the span of the channel. Simulations employing streamwise periodic boundary conditions (temporal model) as well as inlet-outlet conditions (spatial model) are carried out. In the temporal approach the interactions (pairing of vortices and growth of new vortex pairs) of fully developed vortex pairs are time-dependent, whereas in the spatial approach these events are inherently steady and concern vortices not in their fully developed state. The initial spatial development of the vortices is in excellent agreement with results of a linear stability analysis up to fairly large disturbance amplitudes. In the nonlinear regime a good agreement with experimental results has also been found. The receptivity of the flow is very important in a convectively unstable situation such as the present one and different behaviour is found at fixed Reynolds number (equal to 2.43 times the critical value for the onset of Dean vortices): the flow can be either steady or undergo a continuous sequence of merging and splitting events, depending on the inlet conditions. In the latter situation decorrelated patterns of low- and high-speed streaks are produced in streamwise-spanwise planes and they bear several similarities to near-wall coherent structures of turbulent boundary layers.

Optimal and robust control for the three-dimensional algebraically growing instability of a Blasius boundary layer is studied in the nonlinear regime. First, adjoint-based optimization is used to determine an optimal control in the form of a spanwise-uniform wall suction that attenuates the transient growth of a given initial disturbance, chosen to be the optimal perturbation of the uncontrolled flow. Secondly, a robust control is sought and computed simultaneously with the most disrupting initial perturbation for the controlled flow itself. Results for both optimal and robust control show that the optimal suction velocity peaks near the leading edge. In the robust-control case, however, the peak value is smaller, located farther downstream from the leading edge, and the suction profile is much less dependent on the control energy than in the optimal-control case.

The Orr–Sommerfeld operator's eigenvalues determine the stability of exponentially growing disturbances in parallel and quasi-parallel flows. This work assesses the sensitivity of these eigenvalues to modifications of the base flow, which need not be infinitesimally small. Such base flow variations may represent differences between the laboratory flow and its ideal, theoretical counterpart. The worst case, i.e. the change in base flow with the most destabilizing effect on the eigenvalues, is found using variational techniques for the plane Couette flow. Relatively small changes in the base flow are shown to be destabilizing, although the ideal flow is unconditionally stable according to linear theory. These observations inspire a velocity-based definition of pseudospectra in the hydrodynamic stability context.

Optimal perturbations for the family of three-dimensional boundary layers described by the Falkner–Skan–Cooke similarity solution are obtained using a variational technique in the temporal framework. The disturbances experiencing the most growth take the form of vortices almost aligned with the external streamline at inception and evolve into streaks. In subcritical flows these can attain about twice the transient amplification observed in comparably forced two-dimensional flows. Possible connections between optimal perturbations and exponentially amplified crossflow vortices are explored.

A scenario of transition to turbulence likely to occur during the development of natural disturbances in a flat-plate boundary layer is studied. The perturbations at the leading edge of the flat plate that show the highest potential for transient energy amplification consist of streamwise aligned vortices. Due to the lift-up mechanism these optimal disturbances lead to elongated streamwise streaks downstream, with significant spanwise modulation. Direct numerical simulations are used to follow the nonlinear evolution of these streaks and to verify secondary instability calculations. The theory is based on a linear Floquet expansion and focuses on the temporal, inviscid instability of these flow structures. The procedure requires integration in the complex plane, in the coordinate direction normal to the wall, to properly identify neutral modes belonging to the discrete spectrum. The streak critical amplitude, beyond which streamwise travelling waves are excited, is about 26% of the free-stream velocity. The sinuous instability mode (either the fundamental or the subharmonic, depending on the streak amplitude) represents the most dangerous disturbance. Varicose waves are more stable, and are characterized by a critical amplitude of about 37%. Stability calculations of streamwise streaks employing the shape assumption, carried out in a parallel investigation, are compared to the results obtained here using the nonlinearly modified mean fields; the need to consider a base flow which includes mean flow modification and harmonics of the fundamental streak is clearly demonstrated.

A quantitative definition of the receptivity of the Görtler instability is given by the Green's functions that external disturbances must be scalarly multiplied by in order to yield the amplitude of the most amplified instability mode, defined sufficiently far downstream of the plate's leading edge. These Green's functions (one for each kind of external disturbance, either coming from the free stream or from the wall) are here displayed for the first time. Calculating such functions from a numerical solution of the instability equations would require repeating the calculation for each of a complete set of different initial and boundary conditions; although numerical simulations of the Görtler instability abound in the literature, such a systematic screening has never been attempted. Here, instead, we calculate the Green's functions directly from a numerical solution of the adjoint of the linearized boundary-layer equations, which exploits the fact that the direct and adjoint parabolic problems have opposite directions of stable time-like evolution. The Green's functions can thus be obtained by marching backward in time at the same computational cost as a single forward-in-time integration of the direct problem. The backward-in-time technique is not limited to the Görtler problem; quantitative receptivity calculations for other types of instability can easily be envisioned.