The concept of statistical stability is central to Malkus's 1956 attempt to predict the mean profile in shear flow turbulence. Here we discuss how his original attempt to assess this - an Orr-Sommerfeld analysis on the mean profile - can be improved by considering a cumulant expansion of the Navier-Stokes equations. Focusing on the simplest non-trivial closure (commonly referred to as CE2) which corresponds to the quasilinearized Navier-Stokes equations, we develop an extended Orr-Sommerfeld analysis (EOS) which also incorporates information about the fluctuation field. A more practical version of this - minimally extended Orr-Sommerfeld analysis (mEOS) - is identified and tested on a number of statistically-steady and therefore statistically stable turbulent channel flows. Beyond the concept of statistical stability, this extended stability analysis should also improve the popular approach of mean-flow linear analysis in time-dependent shear flows by including more information about the underlying flow in its predictions as well as for other flows with additional physics such as convection.
We introduce and validate an algorithm to compute transient amplication factors for the Orr-Sommerfeld-Squire linear theory for parallel two-phase flow. We further introduce direct numerical simulation as a way of comparing the linear theory with early-stage wave growth in simulations. The simulation results are drawn from a strongly supercritical parameter case wherein the modal growth rates are strong. In this case, the modal growth dominates the transient growth
This paper analyses the effects of small injection/suction Reynolds number, Hartmann number, permeability parameter and wave number on a viscous incompressilbe electrically conduction fluid flow in a parallel porous channel. The plates of the channel with small constant injection/suction, have constant temperature. The upper plate is allowed to mouve in flow direction and the lower plate is kept at rest. A magnetic field of uniform strength is also applied normally to the plates what are parallel. The originality of the paper is to study the effect of the above parameter in temporal linear sta…
Despite remarkable accomplishment, the classical hydrodynamic stability theory fails to predict transition in wall-bounded shear ow. The shortcoming of this modal approach was found 20 years ago and is linked to the non-orthogonality of the eigenmodes of the linearised problem, de noted by the Orr Sommerfeld and Squire equations. The associated eigenmodes of this linearised problem are the normal velocity and the normal vorticity eigenmodes, which are not dimensionally homogeneous quantities. Thus non-orthogonality condition between these two families of eigenmodes have not been clearly demons…
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…
Asymptotic eigenvalues and eigenfunctions for the Orr-Sommerfeld equation in two and three dimensional incompressible flows on an infinite domain and on a semi-infinite domain are obtained. Two configurations are considered, one in which a short-wave limit approximation is used, and another in which a long-wave limit approximation is used. In the short-wave limit, WKB methods are utilized to estimate the eigenvalues, and the eigenfunctions are approximated in terms Green's functions. The procedure consists of transforming the Orr-Sommerfeld equation into a system of two second order ordinary d…
For applications regarding transition prediction, wing design and control of boundary layers, the fundamental understanding of disturbance growth in the flat-plate boundary layer is an important issue. In the present work we investigate the stability of boundary layer in Poiseuille flow. We normalize pressure and time by inertial and viscous effects. The disturbances are taken to be periodic in the spanwise direction and time. We present a set of linear governing equations for the parabolic evolution of wavelike disturbances. Then, we derive modified Orr-Sommerfeld equations that can be applie…
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…
The classical Squire transformation is extended to the entire eigenfunction structure of both Orr-Sommerfeld and Squire modes. For arbitrary Reynolds numbers Re, this transformation allows the solution of the initial-value problem for an arbitrary three-dimensional (3D) disturbance via a two-dimensional (2D) initial-value problem at a smaller Reynolds number Re2D. Its implications for the transient growth of arbitrary 3D disturbances is studied. Using the Squire transformation, the general solution of the initial-value problem is shown to predict large-Reynolds-number scaling for the optimal g…
The linear amplification mechanisms leading to streamwise-constant large-scale structures in laminar and turbulent channel flows are considered. A key feature of the analysis is that the Orr--Sommerfeld and Squire operators are each considered separately. Physically this corresponds to considering two separate processes: (i) the response of wall-normal velocity fluctuations to external forcing; and (ii) the response of streamwise velocity fluctuations to wall-normal velocity fluctuations. In this way we exploit the fact that, for streamwise-constant fluctuations, the dynamics governing the wal…
The plane Poiseuille flow is one of the elementary flow configurations. Although its laminar-turbulent transition mechanism is investigated intensively in the last century, the significant difference in the critical Reynolds number between the experiments and theory lacks a clear explanation. In this paper, an attempt is made to reduce this gap by analysing the Reynolds-Orr equation solution. Recent literature results showed that the usage of enstrophy (the volume integral of the vorticity) instead of the kinetic energy as the norm of perturbations predicts higher Reynolds numbers in the two-d…
In this paper we derive consistent shallow water equations for thin films of power law fluids down an incline. These models account for the streamwise diffusion of momentum which is important to describe accurately the full dynamic of the thin film flows when instabilities like roll-waves arise. These models are validated through a comparison with Orr Sommerfeld equations for large scale perturbations. We only consider laminar flow for which the boundary layer issued from the interaction of the flow with the bottom surface has an influence all over the transverse direction to the flow. In this…
The linear stability of the laminar boundary layer flow of a Stokes wave in deep waters is investigated by means of a 'momentary' criterion of instability for unsteady flows (Blondeaux and Seminara, 1979). In the parameter range investigated, it is found that the flow is stable to 2-D perturbations. The least stable eigenmode of the resulting Orr-Sommerfield spectrum attains its maximum beneath the boundary layer of the Stokes wave. Moreover, an analysis of the associated pseudospectrum indicates that the laminar flow when modified by imperfections is unstable due to the non-normality of the O…
We investigate the effect of small suction Reynolds number and permeability parameter on the stability of Poiseuille fluid flow in a porous medium between two parallel horizontal stationary porous plates . We have shown that the perturbed flow is governed by an equation named modified Orr-Sommerfeld equation. We find also that the normalization of the wall-normal velocity with characteristic small suction (or small injection) velocity is important for a perfect command of porous medium fluid flow stability analysis. The stabilizing effect of the parameters in general and small suction Reynolds…
In this work, the linear stability of the viscous incompressible fluid flow between two parallel horizontal porous stationary plates with the assumption that there is a small constant suction at upper plate and a small constant injection at the lower plate is studied.The Navier-Stokes and continuous equations are reduced to an equation modified by the suction Reynolds number, which we call modified Orr-Sommerfeld equation. This equation is rewritten as an eigenvalue problem and is solved numerically using Matlab (Windows Version). The effect of small suction Reynolds number on the linear stabi…
We study two-phase stratified flow where the bottom layer is a thin laminar liquid and the upper layer is a fully-developed gas flow. The gas flow can be laminar or turbulent. To determine the boundary between convective and absolute instability, we use Orr--Sommerfeld stability theory, and a combination of linear modal analysis and ray analysis. For turbulent gas flow, and for the density ratio r=1000, we find large regions of parameter space that produce absolute instability. These parameter regimes involve viscosity ratios of direct relevance to oil/gas flows. If, instead, the gas layer is …
We present a robust and accurate discretization approach for incompressible turbulent flows based on high-order discontinuous Galerkin methods. The DG discretization of the incompressible Navier-Stokes equations uses the local Lax-Friedrichs flux for the convective term, the symmetric interior penalty method for the viscous term, and central fluxes for the velocity-pressure coupling terms. Stability of the discretization approach for under-resolved, turbulent flow problems is realized by a purely numerical stabilization approach. Consistent penalty terms that enforce the incompressibility cons…
A model-based description of the scaling and radial location of turbulent fluctuations in turbulent pipe flow is presented and used to illuminate the scaling behaviour of the very large scale motions. The model is derived by treating the nonlinearity in the perturbation equation (involving the Reynolds stress) as an unknown forcing, yielding a linear relationship between the velocity field response and this nonlinearity. We do not assume small perturbations. We examine propagating modes, permitting comparison of our results to experimental data, and identify the steady component of the velocit…
The presence and structure of an Orr-like inviscid mechanism is studied in fully developed, large-scale turbulent channel flow. Orr-like `bursts' are defined by the relation between the amplitude and local tilting angle of the wall-normal velocity perturbations, and extracted by means of wavelet-based filters. They span the shear-dominated region of the flow, and their sizes and lifespans are proportional to the distance from the wall in the logarithmic layer, forming a self-similar eddy hierarchy consistent with Townsend's attached-eddy model. Except for their amplitude, which has to be deter…
We study the nonlinear stability of plane Couette and Poiseuille flows with the Lyapunov second method by using the classical L2-energy. We prove that the streamwise perturbations are L2-energy stable for any Reynolds number. This contradicts the results of Joseph [10], Joseph and Carmi [12] and Busse [4], and allows us to prove that the critical nonlinear Reynolds numbers are obtained along two-dimensional perturbations, the spanwise perturbations, as Orr [16] had supposed. This conclusion combined with recent results by Falsaperla et al. [8] on the stability with respect to tilted rolls, pro…
Linear stability of stratified two-phase flows in horizontal channels to arbitrary wavenumber disturbances is studied. The problem is reduced to Orr-Sommerfeld equations for the stream function disturbances, defined in each sublayer and coupled via boundary conditions that account also for possible interface deformation and capillary forces. Applying the Chebyshev collocation method, the equations and interface boundary conditions are reduced to the generalized eigenvalue problems solved by standard means of numerical linear algebra for the entire spectrum of eigenvalues and the associated eig…
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…
In this paper, the physics of flow instability and turbulent transition in shear flows is studied by analyzing the energy variation of fluid particles under the interaction of base flow with a disturbance. For the first time, a model derived strictly from physics is proposed to show that the flow instability under finite amplitude disturbance leads to turbulent transition. The proposed model is named as "energy gradient method." It is demonstrated that it is the transverse energy gradient that leads to the disturbance amplification while the disturbance is damped by the energy loss due to visc…
A theoretical and numerical analysis of the linear stability of the boundary layer flow under a solitary wave is presented. In the present work, the nonlinear boundary layer equations are solved. The result is compared to the linear boundary layer solution in Liu et al. (2007) reveal- ing that both profiles are disagreeing more than has been found before. A change of frame of reference has been used to allow for a classical linear stability analysis without the need to redefine the notion of stability for this otherwise unsteady flow. For the linear stability the Orr-Sommerfeld equation and th…
We provide a detailed comparison of the two-dimensional, temporal and the spatiotemporal linearized analyses of the viscoelastic free shear flows in the limit of low to moderate Reynolds number and Elasticity number obeying four different types of stress-strain constitutive equations: Oldroyd-B, Upper Convected Maxwell, Johnson-Segalman (JS) and linear Phan-Thien Tanner (PTT). The resulting fourth-order Orr-Sommerfeld Equation is transformed into a set of six auxiliary equations that are numerically integrated via the Compound Matrix Method. The temporal stability analysis suggest (a) elastic …
We consider the evolution of arbitrarily large perturbations of a prescribed pure hydrodynamical flow of an electrically conducting fluid. We study whether the flow perturbations as well as the generated magnetic fields decay or grow with time and constitute a dynamo process. For that purpose we derive a generalized Reynolds-Orr equation for the sum of the kinetic energy of the hydrodynamic perturbation and the magnetic energy. The flow is confined in a finite volume so the normal component of the velocity at the boundary is zero. The tangential component is left arbitrary in contrast with pre…
In the dynamical systems approach to describing turbulent or otherwise chaotic flows, an important quantity is the Lyapunov exponents and vectors that characterize the strange attractor of the flow. In particular, knowledge of the Lyapunov exponents and vectors will help identify perturbations that the system is most sensitive to, and quantify the dimension of the attractor. However, reliably computing these Lyapunov quantities requires robust nu- merical algorithms. While several perturbation-based techniques are available in literature, their application to commonly used turbulent flow solve…
Recent experimental observations (Kuehnen et al., 2018) have shown that flattening a turbulent streamwise velocity profile in pipe flow destabilises the turbulence so that the flow relaminarises. We show that a similar phenomenon exists for laminar pipe flow profiles in the sense that the nonlinear stability of the laminar state is enhanced as the profile becomes more flattened. Significant drag reduction is also observed for the turbulent flow when triggered by sufficiently large disturbances. The flattening is produced by an artificial body force designed to mimick a baffle used in the exper…
We report the temporal and spatio-temporal stability analyses of anti-symmetric, free shear, viscoelastic flows obeying the Oldroyd-B constitutive equation in the limit of low to moderate Reynolds number and Weissenberg number. The resulting fourth order Orr-Sommerfeld equation is reduced to a set of six auxiliary equations which are numerically integrated starting from the rescaled far-field conditions, i.~e., via the Compound Matrix Method. Numerical results indicate that with increasing Weissenberg number: (a) the peak of the maximal growth rate (i. e., the maximum value of the imaginary co…
