Plane Parallel Shear Flows Research Articles

The nonlinear stability of plane parallel shear flows with respect to tilted perturbations

The nonlinear stability of plane parallel shear flows with respect to tilted perturbations

On the non-self-adjoint and multiscale character of passive scalar mixing under laminar advection

Except in the trivial case of spatially uniform flow, the advection–diffusion operator of a passive scalar tracer is linear and non-self-adjoint. In this study, we exploit the linearity of the governing equation and present an analytical eigenfunction approach for computing solutions to the advection–diffusion equation in two dimensions given arbitrary initial conditions, and when the advecting flow field at any given time is a plane parallel shear flow. Our analysis illuminates the specific role that the non-self-adjointness of the linear operator plays in the solution behaviour, and highlights the multiscale nature of the scalar mixing problem given the explicit dependence of the eigenvalue–eigenfunction pairs on a multiscale parameter $q=2{\rm i}k\,{\textit {Pe}}$ , where $k$ is the non-dimensional wavenumber of the tracer in the streamwise direction, and ${\textit {Pe}}$ is the Péclet number. We complement our theoretical discussion on the spectra of the operator by computing solutions and analysing the effect of shear flow width on the scale-dependent scalar decay of tracer variance, and characterize the distinct self-similar dispersive processes that arise from the shear flow dispersion of an arbitrarily compact tracer concentration. Finally, we discuss limitations of the present approach and future directions.

Global Stability of a Conduction-Diffusion System with Superimposed Plane Parallel Shear Flows

Global Stability of a Conduction-Diffusion System with Superimposed Plane Parallel Shear Flows

The miles theorem and new particular solutions to the Taylor–Goldstein equation

The direct Lyapunov method is used to prove the absolute linear instability of steadystate plane-parallel shear flows of an inviscid stratified incompressible fluid in the gravity field with respect to plane perturbations both in the Boussinesq and non-Boussinesq approximations. A strict description is given for the applicability of the known necessary condition for linear instability of steady-state plane-parallel shear flows of an ideal nonuniform (by density) incompressible fluid in the gravity field both in the Boussinesq and non-Boussinesq approximations (the Miles theorem). Analytical examples of illustrative character are constructed.

On the Nonlinear Stability of Plane Parallel Shear Flow in a Coplanar Magnetic Field

Lyapunov direct method has been used to study the nonlinear stability of laminar flow between two parallel planes in the presence of a coplanar magnetic field for streamwise perturbations with stress-free boundary planes. Two Lyapunov functions are defined. By means of the first, it is proved that the transverse components of the perturbations decay unconditionally and asymptotically to zero for all Reynolds numbers and magnetic Reynolds numbers. By means of the second, it is showed that the other components of the perturbations decay conditionally and exponentially to zero for all Reynolds numbers and the magnetic Reynolds numbers below \(\frac{\pi ^2}{2M}\), where M is the maximum of the absolute value of the velocity field of the laminar flow.

Reduced description of exact coherent states in parallel shear flows.

A reduced description of exact coherent structures in the transition regime of plane parallel shear flows is developed, based on the Reynolds number scaling of streamwise-averaged (mean) and streamwise-varying (fluctuation) velocities observed in numerical simulations. The resulting system is characterized by an effective unit Reynolds number mean equation coupled to linear equations for the fluctuations, regularized by formally higher-order diffusion. Stationary coherent states are computed by solving the resulting equations simultaneously using a robust numerical algorithm developed for this purpose. The algorithm determines self-consistently the amplitude of the fluctuations for which the associated mean flow is just such that the fluctuations neither grow nor decay. The procedure is used to compute exact coherent states of a flow introduced by Drazin and Reid [Hydrodynamic Stability (Cambridge University Press, Cambridge, UK, 1981)] and studied by Waleffe [Phys. Fluids 9, 883 (1997)]: a linearly stable, plane parallel shear flow confined between stationary stress-free walls and driven by a sinusoidal body force. Numerical continuation of the lower-branch states to lower Reynolds numbers reveals the presence of a saddle node; the saddle node allows access to upper-branch states that are, like the lower-branch states, self-consistently described by the reduced equations. Both lower- and upper-branch states are characterized in detail.

Eigenvalue problems modelling the stability of a plane-parallel shear in a two-layer viscous composite

We present the derivation of a precise statement of a linearized eigenvalue problem that models the stability of a plane-parallel shear flow in a composite consisting of two different layers of heavy Newtonian media. The peculiarities of the influence on the stability of five dimensionless criteria participating in the equations and boundary conditions are discussed. The specific feature of the power conditions of contact is that a spectral parameter α enters these conditions nonlinearly.

On the role of vortex stretching in energy optimal growth of three-dimensional perturbations on plane parallel shear flows

AbstractThe three-dimensional linearized optimal energy growth mechanism, in plane parallel shear flows, is re-examined in terms of the role of vortex stretching and the interplay between the spanwise vorticity and the planar divergent components. For high Reynolds numbers the structure of the optimal perturbations in Couette, Poiseuille and mixing-layer shear profiles is robust and resembles localized plane waves in regions where the background shear is large. The waves are tilted with the shear when the spanwise vorticity and the planar divergence fields are in (out of) phase when the background shear is positive (negative). A minimal model is derived to explain how this configuration enables simultaneous growth of the two fields, and how this mutual amplification affects the optimal energy growth. This perspective provides an understanding of the three-dimensional growth solely from the two-dimensional dynamics on the shear plane.

On the Nonlinear Stability of Inviscid Homogeneous Shear Flows in Sea Straits of Arbitrary Cross Sections

In this paper we study the nonlinear stability of steady flows of inviscid homogeneous fluids in sea straits of arbitrary cross sections. We use the method of Arnol'd [1] to obtain two general stability theorems for steady basic flows with respect to finite amplitude disturbances. For the special case of plane parallel shear flows we find a finite amplitude extension of the linear stability result of Deng et al [2]. We also present some examples of basic flows which are stable to finite amplitude disturbances.

The linear stability of inviscid shear flow of a vibrationally excited diatomic gas

The problem of the linear stability of plane-parallel shear flows of a vibrationally excited compressible diatomic gas is investigated using a two-temperature gas dynamics model. The necessary and sufficient conditions for stability of the flows considered are obtained using the energy integrals of the corresponding linearized system for the perturbations. It is proved that thermal relaxation produces an additional dissipation factor, which enhances the flow stability. A region of eigenvalues of unstable perturbations is distinguished in the upper complex half-plane. Numerical calculations of the eigenvalues and eigenfunctions of the unstable inviscid modes are carried out. The dependence on the Mach number of the carrier stream, the vibrational relaxation time τ and the degree of non-equilibrium of the vibrational mode is analysed. The most unstable modes with maximum growth rate are obtained. It is shown that in the limit there is a continuous transition to well-known results for an ideal fluid as the Mach number and τ approach zero and for an ideal gas when τ → 0.

Nonlinear stage of instability development in a stratified shear flow with an inflection-free velocity profile

A plane-parallel shear flow with an inflection-free velocity profile Vx=U(y), having an embedded thin layer of stable density stratification, is known to be unstable in ideal incompressible fluid, for an arbitrary bulk Richardson number J>0 [S. M. Churilov, J. Fluid Mech. 539, 25 (2005); S. M. Churilov, J. Fluid Mech.617, 301 (2008)], and it is the three- rather than two-dimensional (z-independent) disturbances that are most unstable within a wide range of parameters. We examine the weakly nonlinear evolution of a pair of unstable oblique waves in such a flow, in the unsteady critical layer regime. For this purpose, we derive the evolution equation which has the form of a nonlinear integral equation and is valid for both thin and thick critical layers, including the case where the critical layer width exceeds the stratification layer thickness. The solutions of this equation are inspected both analytically and numerically, and it is shown that during the nonlinear stage, the disturbance evolves, as a rule, explosively.

Stability analysis of stratified shear flows with a monotonic velocity profile without inflection points. Part 2. Continuous density variation

We investigate stability with respect to two-dimensional (independent of z) disturbances of plane-parallel shear flows with a velocity profile Vx=u(y) of a rather general form, monotonically growing upwards from zero at the bottom (y=0) to U0 as y → ∞ and having no inflection points, in an ideal incompressible fluid stably stratified in density in a layer of thickness ℓ, small as compared to the scale L of velocity variation. In terms of the ‘wavenumber k – bulk Richardson number J’ variables, the upper and lower (in J) boundaries of instability domains are found for each oscillation mode. It is shown that the total instability domain has a lower boundary which is convex downwards and is separated from the abscissa (k) axis by a strip of stability 0 < J < J0(−)(k) with minimum width J*=O(ℓ2/L2) at kL=O(1). In other words, the instability domain configuration is such that three-dimensional (oblique) disturbances are first to lose their stability when the density difference across the layer increases. Hence, in the class of flows under consideration, it is a three- not two-dimensional turbulence that develops as a result of primary instability.

Non-linear wave interactions from transient growth in plane-parallel shear flows

Based on the normal velocity–normal vorticity ( v – η ) formulation for the development of 3D disturbances in plane-parallel shear flows, the non-linear terms in the governing equations are derived as convolution integrals of the Fourier-transformed variables. They are grouped in three categories: v – v , v – η and η – η terms, and are expressed in a simple geometric form using the modulus of the two wave-vectors ( k ′ and k ″ ) appearing in the convolution integrals, and their intervening angle ( χ). The non-linear terms in the v-equation involving η are all weighted by sin χ (or sin 2 χ ). This confirms the known result that non-linear regeneration of normal velocity, necessary for a sustained driving of 3D disturbances, is not possible for stream-wise elongated structures ( α = 0 ), only. It is therefore suggested how transiently amplified η can interact with decaying 2D waves to activate (oblique) waves which may be less damped than the 2D wave. This is shown to be possible for Blasius flow. In the η-equation, non-linear effects are possible for elongated structures resulting in shorter spanwise scales appearing at a shorter time-scale than the (linear) transient growth. A numerical example shows the details of this process in plane Poiseuille flow. From an inspection of the y-dependency (wall-normal direction) of the non-linear terms it is suggested that higher y-derivatives may give rise to non-linear effects in the inviscid development of perturbations. Also, a result for the y-symmetry of the non-linear terms is derived, applicable to plane Poiseuille flow.

Interaction of shear flows of an ideal incompressible fluid in a channel

The problem of the decay of an arbitrary discontinuity for the equations describing plane-parallel shear flows of an ideal fluid in a narrow channel is considered. The class of particular solutions corresponding to fluid flows with piecewise constant vorticity is studied. In this class, the existence of self-similar solutions describing all possible unsteady wave configurations resulting from the nonlinear interaction of the specified shear flows is established.

Relating optimal growth to counterpropagating Rossby waves in shear instability

The optimal dynamics of conservative disturbances to plane parallel shear flows is interpreted in terms of the propagation and mutual interaction of components called counterpropagating Rossby waves (CRWs). Pairs of CRWs were originally used by Bretherton to provide a mechanistic explanation for unstable normal modes in the barotropic Rayleigh model and baroclinic two-layer model. One CRW has large amplitude in regions of positive mean cross-stream potential vorticity (PV) gradient, while the second CRW has large amplitude in regions of negative PV gradient. Each CRW propagates to the left of the mean PV gradient vector, parallel to the mean flow. If the mean flow is more positive where the PV gradient is positive, the intrinsic phase speeds of the two CRWs will be similar. The CRWs interact because the PV anomalies of one CRW induce cross-stream velocity at the location of the other CRW, thus advecting the mean PV. Although a single Rossby wave is neutral, their interaction can result in phase locking and mutual growth. Here the general initial value problem for disturbances to shear flow is analyzed in terms of CRWs. For the discrete spectrum (which could alternatively be described using normal modes), the singular value decomposition of the dynamical propagator can be obtained analytically in terms of the CRW interaction coefficient and the intrinsic CRW phase speeds. Using this formalism, optimal perturbations, the disturbances which grow fastest in a given norm over a specified time interval, can readily be found. The most natural norm for CRWs is related to air parcel displacements or enstrophy. However, if an energy norm is taken, it is shown to grow due to both mutual amplification of air parcel displacements and the untilting of PV structures (the Orr mechanism) associated with decreasing phase difference between the CRWs. A generalization of the CRW description to the optimal dynamics of the complete spectrum solution is outlined. Although the dynamics then involves the interaction between an infinite number of “CRW kernels,” the form of the simple interaction between any two CRW kernels is the same as in the discrete case.

Incompatibility of the exponential scaling law for a zero pressure gradient boundary layer flow with Reynolds averaged turbulence models

Analyzing the multipoint-correlation equations for parallel turbulent shear flows and zero pressure gradient turbulent boundary layer flows with the help of Lie symmetry methods, a new exponential scaling law has been derived in the work by Oberlack [“A unified approach for symmetries in plane parallel turbulent shear flows,” J. Fluid Mech. 427, 299 (2001)]. In the frame work of Lie group methods these solutions are called invariant solutions. From experiments [B. Lindgren, J. M. Österlund, and A. Johansson, “Evaluation of scaling laws derived from Lie group symmetry methods in zero-pressuere-gradient turbulent boundary layers,” J. Fluid Mech. 502, 127 (2004)] and direct numerical simulations [G. Khujadze and M. Oberlack, “DNS and scaling laws from new symmetry groups of ZPG turbulent boundary layer flow,” Theor. Comput. Fluid Dyn. 18, 5 (2004)] the exponential velocity profile was clearly validated in the mid-wake region of high Reynolds number flat-plate boundary layers. It was identified as an explicit analytic form of the velocity defect law. Implementing the latter invariant solution into various Reynolds stress models it was found that none of the investigated models are in accordance with the exponential velocity law.

A note on nonlinear stability of plane parallel shear flows

We present a generalized energy functional E for plane parallel shear flows which provides conditional nonlinear stability for Reynolds numbers Re below some value Re E depending on the shear profile. In the case of the experimentally important profiles, viz. combinations of laminar Couette and Poiseuille flow, Re E is shown to be at least 174.

Transition in plane parallel shear flows heated internally

The stability of internally heated inclined plane parallel shear flows is examined numerically for the case of finite value of the Prandtl number, Pr. The transition in a vertical channel has already been studied for 0⩽ Pr⩽100 with or without the application of an external pressure gradient, where the secondary flow takes the form of travelling waves (TWs) that are spanwise-independent (see works of Nagata and Generalis). In this work, in contrast to work already reported (J. Heat Trans. T. ASME 124 (2002) 635–642), we examine transition where the secondary flow takes the form of longitudinal rolls (LRs), which are independent of the steamwise direction, for Pr=7 and for a specific value of the angle of inclination of the fluid layer without the application of an external pressure gradient. We find possible bifurcation points of the secondary flow by performing a linear stability analysis that determines the neutral curve, where the basic flow, which can have two inflection points, loses stability. The linear stability of the secondary flow against three-dimensional perturbations is also examined numerically for the same value of the angle of inclination by employing Floquet theory. We identify possible bifurcation points for the tertiary flow and show that the bifurcation can be either monotone or oscillatory. To cite this article: M. Nagata, S. Generalis, C. R. Mecanique 332 (2004).

Transition in Homogeneously Heated Inclined Plane Parallel Shear Flows

The transition of internally heated inclined plane parallel shear flows is examined numerically for the case of finite values of the Prandtl number Pr. We show that as the strength of the homogeneously distributed heat source is increased the basic flow loses stability to two-dimensional perturbations of the transverse roll type in a Hopf bifurcation for the vertical orientation of the fluid layer, whereas perturbations of the longitudinal roll type are most dangerous for a wide range of the value of the angle of inclination. In the case of the horizontal inclination transverse roll and longitudinal roll perturbations share the responsibility for the prime instability. Following the linear stability analysis for the general inclination of the fluid layer our attention is focused on a numerical study of the finite amplitude secondary travelling-wave solutions (TW) that develop from the perturbations of the transverse roll type for the vertical inclination of the fluid layer. The stability of the secondary TW against three-dimensional perturbations is also examined and our study shows that for Pr=0.71 the secondary instability sets in as a quasi-periodic mode, while for Pr=7 it is phase-locked to the secondary TW. The present study complements and extends the recent study by Nagata and Generalis (2002) in the case of vertical inclination for Pr=0.

Non‐linear stability and convection for laminar flows in a porous medium with Brinkman law

AbstractThe non‐linear stability of plane parallel shear flows in an incompressible homogeneous fluid heated from below and saturating a porous medium is studied by the Lyapunov direct method.In the Oberbeck–Boussinesq–Brinkman (OBB) scheme, if the inertial terms are negligible, as it is widely assumed in literature, we find global non‐linear exponential stability (GNES) independent of the Reynolds number R. However, if these terms are retained, we find a restriction on R (depending on the inertial convective coefficient) both for a homogeneous fluid and a mixture heated and salted from below. In the case of a mixture, when the normalized porosity ε is equal to one, the laminar flows are GNES for small R and for heat Rayleigh numbers less than the critical Rayleigh numbers obtained for the motionless state. Copyright © 2003 John Wiley & Sons, Ltd.