Transient Growth Phenomenon Research Articles

Variational mode decomposition–based nonstationary coherent structure analysis for spatiotemporal data

The conventional modal analysis techniques face difficulties in handling nonstationary phenomena, such as transient, nonperiodic, or intermittent phenomena. This paper presents a variational mode decomposition–based nonstationary coherent structure (VMD-NCS) analysis that enables the extraction and analysis of coherent structures in the case of nonstationary phenomena from high-dimensional spatiotemporal data. The VMD-NCS analysis decomposes the input spatiotemporal data into intrinsic coherent structures (ICSs) that represent nonstationary spatiotemporal patterns and exhibit coherence in both spatial and temporal directions. Furthermore, unlike many conventional modal analysis techniques, the proposed method accounts for the temporal changes in the spatial distribution with time. The performance of the VMD-NCS analysis was validated based on the transient growth phenomena in the flow around a cylinder. It was confirmed that the temporal changes in the spatial distribution, depicting the transient growth of vortex shedding where fluctuations arising in the far-wake region gradually approach the near-wake region, were represented as a single ICS. Furthermore, in the analysis of the quasi-periodic flow field around a pitching airfoil, the temporal changes in the spatial distribution and the amplitude of vortex shedding behind the airfoil, influenced by the pitching motion of the airfoil, were captured as a single ICS. Additionally, the impact of two parameters that control the number of ICSs (K) and the penalty factor related to the temporal coherence (α), was investigated. The results revealed that K has a significant impact on the VMD-NCS analysis results. In the case of a relatively high K, the VMD-NCS analysis tends to extract more periodic spatiotemporal patterns resembling the results of dynamic mode decomposition. In the case of a small K, it tends to extract more nonstationary spatiotemporal patterns.

Transient growth in a flat plate boundary layer under a stream with uniform shear

One of the simplest problems involving external vorticity in boundary layer flows is the flow over a semi-infinite plate under a stream of uniform shear. We study the transient growth phenomenon in this flow to investigate the role of freestream shear on energy amplification and analyze the differences with the Blasius flow. The initial optimal disturbance that triggers the maximum growth is found to be streamwise vortices, as in other shear flows. Compared to the Blasius boundary layer, higher optimum energy and larger spanwise wavelength of streamwise vortices have been observed. We provide scaling laws for the maximum optimal amplification, which is found to increase exponentially with the freestream shear gradient.

Optimal disturbance growth in shear‐imposed falling film

AbstractA linear stability analysis of a three‐dimensional shear‐imposed fluid flowing down an inclined plane is studied based on the evolution equations for normal velocity and normal vorticity components. Both modal and nonmodal stability analyses are carried out. The modal stability analysis demonstrates that shear‐imposed flow is more unstable to solo streamwise perturbation. On the contrary, the nonmodal stability analysis demonstrates that shear‐imposed flow is more unstable to solo spanwise perturbation. There is evidence of existing transient energy growth in the wavenumber space that intensifies in the presence of imposed shear stress. Further, the boundary for the zone of transient growth appears far ahead of the boundary for the zone of exponential growth. This fact indicates that the onset of instability for the shear mode may occur before than that predicted from the eigenvalue analysis. A new critical surface parameter involving imposed shear stress is determined, which in fact reveals the existence criterion of instability for the surface mode. Moreover, we have found three different regimes of surface mode instability, shear mode instability, and transient growth phenomenon in the wavenumber space. The unstable region for the surface mode reduces while the unstable region for the shear mode enhances in the wavenumber space with the increasing value of surface parameter, or equivalently, with the increasing value of imposed shear stress. Finally, the unstable region for the surface mode disappears from the wavenumber space as soon as the surface parameter exceeds its critical value; instead, the wavenumber space is occupied by the regime of transient energy growth. In addition, the incident of pseudoresonance takes place on the response curve to external harmonic forcing.

Vortex pairing in jets as a global Floquet instability: modal and transient dynamics

The spontaneous pairing of rolled-up vortices in a laminar jet is investigated as a global secondary instability of a time-periodic spatially developing vortex street. The growth of subharmonic perturbations, associated with vortex pairing, is analysed both in terms of modal Floquet instability and in terms of transient growth dynamics. The article has the double objective to outline a toolset for the global analysis of time-periodic flows, and to leverage such an analysis for a fresh view on the vortex pairing phenomenon. Axisymmetric direct numerical simulations (DNS) of jets with single-frequency inflow forcing are performed, in order to identify combinations of the Reynolds and Strouhal numbers for which vortex pairing is naturally observed. The same DNS calculations are then repeated with an added time-delay control term, which artificially suppresses pairing, so as to obtain time-periodic unpaired base flows for linear stability analysis. It is demonstrated that the natural occurrence of vortex pairing in nonlinear DNS coincides with a linear subharmonic Floquet instability of the underlying unpaired vortex street. However, DNS results suggest that the onset of pairing involves much stronger temporal growth of subharmonic perturbations than that predicted by modal Floquet analysis, as well as a spatial distribution of these fast-growing perturbation structures that is inconsistent with the unstable Floquet mode. Singular value decomposition of the phase-shift operator (the operator that maps a given perturbation field to its state one flow period later) is performed for an analysis of optimal transient growth in the vortex street. Non-modal mechanisms near the jet inlet are thus found to provide a fast route towards the limit-cycle regime of established vortex pairing, in good agreement with DNS observations. It is concluded that modal Floquet analysis accurately predicts the parameter regime where sustained vortex pairing occurs, but that the bifurcation scenario under typical conditions is dominated by transient growth phenomena.

Floquet and transient growth stability analysis of a flow through a compressor passage

In this paper we investigate the instabilities arising in a flow through a compressor passage using BiGlobal stability analysis. The adopted geometry comes from the results of previous experimental and numerical investigations on a linear low-pressure (LP) compressor cascade [6,19,20]. Specifically, we address the role of laminar separation of the boundary layers at Re=138,500, where such separation effects are enhanced by the strong adverse pressure gradients that the flow experiences, in contrast to the more commonly studied low-pressure (LP) turbines. The vortical structures downstream the separation bubble on the suction surface were recognised to show a well-defined time periodicity, which could be precisely detected. Floquet stability analysis was then used to investigate the response of the flow to infinitesimal perturbations. To overcome the difficulty of performing a Floquet stability analysis when the periodicity is restricted just to a small region of the domain, a phase-averaged base flow was computed, such that only the organised motions are extracted, neglecting all the background unsteadiness. The same technique allowed us to confirm the presence of strong energy transient growth phenomena, which are directly associated with convective instabilities occurring in the region downstream from the separation bubble.

Revisiting the mixing-length hypothesis in the outer part of turbulent wall layers: mean flow and wall friction

AbstractWe reconsider foundations and implications of the mixing length theory as applied to wall-bounded turbulent flows in uniform pressure gradient. Based on recent channel-flow direct numerical simulation (DNS) data at sufficiently high Reynolds number, we find that Prandtl’s hypothesis of linear variation of the mixing length with the wall distance is rather inaccurate, hence overlap arguments are stronger in justifying the formation of a logarithmic layer in the mean velocity profile. Regarding the core region of the wall layer, we find that Clauser’s hypothesis of uniform eddy viscosity is strictly connected with the observed size of the eddy structures, and it delivers surprisingly good agreement with DNS and experiments for channels, pipes, and boundary layers. We show that the analytically derived composite mean velocity profiles can be used to accurately predict skin friction in canonical wall-bounded flows with a minimal number of adjustable parameters directly related to the mean velocity profile, and to obtain some insight into transient growth phenomena.

Transient growth in Rayleigh-Bénard-Poiseuille/Couette convection

An investigation of the effect of a destabilizing cross-stream temperature gradient on the transient growth phenomenon of plane Poiseuille flow and plane Couette flow is presented. Only the streamwise-uniform and nearly streamwise-uniform disturbances are highly influenced by the Rayleigh number Ra and Prandtl number Pr. The maximum optimal transient growth Gmax of streamwise-uniform disturbances increases slowly with increasing Ra and decreasing Pr. For all Ra and Pr, at moderately large Reynolds numbers Re, the supremum of Gmax is always attained for streamwise-uniform perturbations (or nearly streamwise-uniform perturbations, in the case of plane Couette flow) which produce large streamwise streaks and Rayleigh-Bénard convection rolls (RB). The optimal growth curves retain the same large-Reynolds-number scaling as in pure shear flow. A 3D vector model of the governing equations demonstrates that the short-time behavior is governed by the classical lift-up mechanism and that the influence of Ra on this mechanism is secondary and negligible. The optimal input for the largest long-time response is given by the adjoint of the dominant eigenmode with respect to the energy scalar product: the RB eigenmode without its streamwise velocity component. These short-time and long-time responses depict, to leading order, the optimal transient growth G(t). At moderately large Ra (or small Pr at a fixed Ra), the dominant adjoint mode is a good approximation to the optimal initial condition for all time. Over a general class of norms that can be considered as growth functions, the results remain qualitatively similar, for example, the dominant adjoint eigenmode still approximates the maximum optimal response.

Parabolic approach to optimal perturbations in compressible boundary layers

Optimal perturbations in compressible, non-parallel boundary layers are considered here. The flows past a flat plate and past a sphere are analysed. The governing equations are derived from the linearized Navier-Stokes equations by employing a scaling that relies on the presence of streamwise vortices, which are well-known for being responsible for the ‘lift-up’ effect. Consequently, the energy norm of the inlet perturbation encompasses the wall-normal and spanwise velocity components only. The effect of different choices of the energy norm at the outlet is studied, testing full (all velocity components and temperature) and partial (streamwise velocity and temperature only) norms. Optimal perturbations are computed via an iterative algorithm completely derived in the discrete framework. The latter simplifies the derivation of the adjoint equations and the coupling conditions at the inlet and outlet.Results for the flat plate show that when the Reynolds number is of the order of , a significant difference in the energy growth is found between the cases of full and partial energy norms at the outlet. The effect of the wall temperature is in agreement with previous parallel-flow results, with cooling being a destabilizing factor for both flat plate and sphere. Flow divergence, which characterizes the boundary layer past the sphere, has significant effects on the transient growth phenomenon. In particular, an increase of the sphere radius leads to a larger transient growth, with stronger effects in the vicinity of the stagnation point. In the range of interesting values of the Reynolds number that are typical of wind tunnel tests and flight conditions for a sphere, no significant role is played by the wall-normal and streamwise velocity components at the outlet.

A spatial analogue of transient growth in plane Couette flow

The linear response of an inviscid two-dimensional Couette flow disturbed by a time-periodic forcing is studied under the assumption that the forcing is distributed along a straight line. When the forcing is tilted against the shear, the disturbance streamfunction and energy are shown to be locally amplified downstream of the source before decaying at large distance. This spatially localized amplification is interpreted as an analogue of the transient growth phenomenon studied in the context of unforced intial-value problems. The self-consistency of the linear approximation and the instability of the disturbance are also examined.

Hydrodynamic Stability and Turbulence: Beyond Transients to a Self‐Sustaining Process

Transition from laminar to turbulent flows has generally been studied by considering the linear and weakly nonlinear evolution of small disturbances to the laminar flow. That approach has been fruitless for many shear flows, and a last hope for its success has been the existence of transient growth phenomena. The physical origin of those linear transient effects is elucidated, revealing serious limitations both of previous analyses and of the phenomena themselves, which preclude them from causing direct transition. Nonetheless, some transient effects are symptomatic of one element of a nonlinear process that becomes self‐sustaining at a small enough dissipation. The process is identified, and its description requires a departure from the traditional focus on the laminar flow. A theory is outlined in which the mean flow has an intrinsic spanwise variation. Evidence indicates this is also the central mechanism in the near wall‐region of fully turbulent shear flows.

Linear stability of planar solidification fronts

We review, extend and modify the classical linear stability theory of planar solidification fronts in Langer's moving symmetric model. Using a new integral equation for the front position, we compute an exact linear stability equation and solve it exactly for an important special case. The classical theory is seen to be the long-time limit of the exact solution. Finally, we show how to treat a general planar front, using a short-time approximation. Our conclusions generalize those of previous analyses, by treating initial temperature field perturbations and the resulting transient growth phenomena.