scholarly journals On the longitudinal optimal perturbations to inviscid plane shear flow: formal solution and asymptotic approximation

2013 ◽  
Vol 737 ◽  
pp. 387-411 ◽  
Author(s):  
C. Arratia ◽  
J.-M. Chomaz
Keyword(s):  
Shear Flow ◽  
Initial Conditions ◽  
Formal Solution ◽  
Base Flow ◽  
Flow Profile ◽  
Plane Shear ◽  
Optimal Perturbation ◽  
Energy Growth ◽  
Perturbation Problem ◽  
New Formulation

AbstractWe study the longitudinal linear optimal perturbations (which maximize the energy gain up to a prescribed time $T$) to inviscid parallel shear flow, which present unbounded energy growth due to the lift-up mechanism. Using the phase invariance with respect to time, we show that for an arbitrary base flow profile and optimization time, the computation of the optimal longitudinal perturbation reduces to the resolution of a single one-dimensional eigenvalue problem valid for all times. The optimal perturbation and its amplification are then derived from the lowest eigenvalue and its associated eigenfunction, while the remainder of the infinite set of eigenfunctions provides an orthogonal base for decomposing the evolution of arbitrary perturbations. With this new formulation we obtain, asymptotically for large spanwise wavenumber ${k}_{z} , $ a prediction of the optimal gain and the localization of inviscid optimal perturbations for the two main classes of parallel flows: free shear flow with an inflectional velocity profile, and wall-bounded flow with maximum shear at the wall. We show that the inviscid optimal perturbations are localized around the point of maximum shear in a region with a width scaling like ${ k}_{z}^{- 1/ 2} $ for free shear flow, and like ${ k}_{z}^{- 2/ 3} $ for wall-bounded shear flows. This new derivation uses the stationarity of the base flow to transform the optimization of initial conditions in phase space into the optimization of a temporal phase along each trajectory, and an optimization among all trajectories labelled by their intersection with a codimension-1 subspace. The optimization of the time phase directly imposes that the initial and final energy growth rates of the optimal perturbation should be equal. This result requires only time invariance of the base flow, and is therefore valid for any linear optimal perturbation problem with stationary base flow.

scholarly journals Multiscale Simulation of Non-Metallic Inclusion Aggregation in a Fully Resolved Bubble Swarm in Liquid Steel

Metals ◽  
2020 ◽  
Vol 10 (4) ◽  
pp. 517
Author(s):  
Jean-Sébastien Kroll-Rabotin ◽  
Matthieu Gisselbrecht ◽  
Bernhard Ott ◽  
Ronja May ◽  
Jochen Fröhlich ◽  
...  
Keyword(s):  
Particle Size ◽  
Shear Flow ◽  
Initial Conditions ◽  
Bubbly Flow ◽  
Multiscale Simulation ◽  
Particle Collision ◽  
Immersed Boundary ◽  
Collision Dynamics ◽  
Plane Shear ◽  
Boltzmann Method

Removing inclusions from the melt is an important task in metallurgy with critical impact on the quality of the final alloy. Processes employed with this purpose, such as flotation, crucially depend on the particle size. For small inclusions, the aggregation kinetics constitute the bottleneck and, hence, determine the efficiency of the entire process. If particles smaller than all flow scales are considered, the flow can locally be replaced by a plane shear flow. In this contribution, particle interactions in plane shear flow are investigated, computing the fully resolved hydrodynamics at finite Reynolds numbers, using a lattice Boltzmann method with an immersed boundary method. Investigations with various initial conditions, several shear values and several inclusion sizes are conducted to determine collision efficiencies. It is observed that although finite Reynolds hydrodynamics play a significant role in particle collision, statistical collision efficiency barely depends on the Reynolds number. Indeed, the particle size ratio is found to be the prevalent parameter. In a second step, modeled collision dynamics are applied to particles tracked in a fully resolved bubbly flow, and collision frequencies at larger flow scale are derived.

Thermoacoustic interplay between intrinsic thermoacoustic and acoustic modes: non-normality and high sensitivities

2019 ◽  
Vol 878 ◽  
pp. 190-220 ◽  
Author(s):  
Francesca M. Sogaro ◽  
Peter J. Schmid ◽  
Aimee S. Morgans
Keyword(s):  
Time Domain ◽  
Time Scale ◽  
Initial Conditions ◽  
Normal Modes ◽  
Time Domain Analysis ◽  
Normal Behaviour ◽  
Optimal Perturbation ◽  
Acoustic Modes ◽  
Energy Growth ◽  
The Time Domain

This study analyses the interplay between classical acoustic modes and intrinsic thermoacoustic (ITA) modes in a simple thermoacoustic system. The analysis is performed using a frequency-domain low-order network model as well as a time-domain spatially discretised model. Anti-correlated modal sensitivities are found to arise due to a pairwise interplay between acoustic and ITA modes. The magnitude of the sensitivities increases as the interplay between the modes grows stronger. The results show a global behaviour of the modes linked to the presence of exceptional points in the spectrum. The time-domain analysis results in a delay-differential equation and allows the investigation of non-normal behaviour and its consequences. Pseudospectral analysis reveals that energy amplification is crucially linked to an interplay between acoustic and ITA modes. While higher non-orthogonality between two modes is correlated with peaks in modal sensitivity, transient energy growth does not necessarily involve the most sensitive modes. In particular, growth estimates based on the Kreiss constant demonstrate that transient amplification relies critically on the proximity of the non-normal modes to the imaginary axis. The time scale for transient amplification is identified as the flame time delay, which is further corroborated by determining the optimal initial conditions responsible for the bulk of the non-modal energy growth. The flame is identified as an active and dominant contributor to energy gain. The frequency of the optimal perturbation matches the acoustic time scale, once more confirming an interplay between acoustic and ITA structures. Flame-based amplification factors of two to five are found, which are significant when feeding into the acoustic dynamics and eventually triggering nonlinear limit-cycle behaviour.

scholarly journals Transient energy growth of channel flow with cross-flow

2019 ◽  
Vol 286 ◽  
pp. 07008
Author(s):  
J. Benyza ◽  
M. Lamine ◽  
A. Hifdi
Keyword(s):  
Flow Rate ◽  
Initial Conditions ◽  
Cross Flow ◽  
High Sensitivity ◽  
Maximum Growth ◽  
Growth Function ◽  
High Energy ◽  
Optimal Perturbation ◽  
Energy Growth ◽  
Transient Energy

The effect of a uniform cross flow (injection/ suction) on the transient energy growth of a plane Poiseuille flow is investigated. Non-modal linear stability analysis is carried out to determine the two-dimensional optimal perturbations for maximum growth. The linearized Navier-Stockes equations are reduced to a modified Orr Sommerfeld equation that is solved numerically using a Chebychev collocation spectral method. Our study is focused on the response to external excitations and initial conditions by examining the energy growth function G(t) and the pseudo-spectrum. Results show that, the transient energy of the optimal perturbation grows rapidly at short times and decline slowly at long times when the cross-flow rate is low or strong. In addition, the maximum energy growth is very pronounced in low injection rate than that of the strong one. For the intermediate cross-flow rate, the transient energy growth of the perturbation, is only possible at the long times with a very high-energy gain. Analysis of the pseudo-spectrum show that the non-normal character of the modified Orr-Sommerfeld operator tends to a high sensitivity of pseudo-spectra structures.

Perturbation dynamics in unsteady pipe flows

2007 ◽  
Vol 570 ◽  
pp. 129-154 ◽  
Author(s):  
M. ZHAO ◽  
M. S. GHIDAOUI ◽  
A. A. KOLYSHKIN
Keyword(s):  
Growth Mechanism ◽  
Flow Instability ◽  
Initial Conditions ◽  
Laminar Flows ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Transient Growth ◽  
Energy Growth ◽  
Long Time ◽  
Short Time

This paper deals with perturbed unsteady laminar flows in a pipe. Three types of flows are considered: a flow accelerated from rest; a flow in a pipe generated by the controlled motion of a piston; and a water hammer flow where the transient is generated by the instantaneous closure of a valve. Methods of linear stability theory are used to analyse the behaviour of small perturbations in the flow. Since the base flow is unsteady, the linearized problem is formulated as an initial-value problem. This allows us to consider arbitrary initial conditions and describe both short-time and long-time evolution of the flow. The role of initial conditions on short-time transients is investigated. It is shown that the phenomenon of transient growth is not associated with a certain type of initial conditions. Perturbation dynamics is also studied for long times. In addition, optimal perturbations, i.e. initial perturbations that maximize the energy growth, are determined for all three types of flow discussed. Despite the fact that these optimal perturbations, most probably, will not occur in practice, they do provide an upper bound for energy growth and can be used as a point of reference. Results of numerical simulation are compared with previous experimental data. The comparison with data for accelerated flows shows that the instability cannot be explained by long-time asymptotics. In particular, the method of normal modes applied with the quasi-steady assumption will fail to predict the flow instability. In contrast, the transient growth mechanism may be used to explain transition since experimental transition time is found to be in the interval where the energy of perturbation experiences substantial growth. Instability of rapidly decelerated flows is found to be associated with asymptotic growth mechanism. Energy growth of perturbations is used in an attempt to explain previous experimental results. Numerical results show satisfactory agreement with the experimental features such as the wavelength of the most unstable mode and the structure of the most unstable disturbance. The validity of the quasi-steady assumption for stability studies of unsteady non-periodic laminar flows is discussed.

scholarly journals Optimal perturbation growth on a breaking internal gravity wave

2021 ◽  
Vol 925 ◽  
Author(s):  
J.P. Parker ◽  
C.J. Howland ◽  
C.P. Caulfield ◽  
R.R. Kerswell
Keyword(s):  
Shear Flow ◽  
Gravity Wave ◽  
Three Dimensional ◽  
Internal Gravity Wave ◽  
Internal Gravity Waves ◽  
Mixing Efficiency ◽  
Two Dimensional ◽  
Systematic Analysis ◽  
Optimal Perturbation ◽  
Energy Growth

The breaking of internal gravity waves in the abyssal ocean is thought to be responsible for much of the mixing necessary to close oceanic buoyancy budgets. The exact mechanism by which these waves break down into turbulence remains an active area of research and can have significant implications on the mixing efficiency. Recent evidence has suggested that both shear instabilities and convective instabilities play a significant role in the breaking of an internal gravity wave in a high Richardson number mean shear flow. We perform a systematic analysis of the stability of a configuration of an internal gravity wave superimposed on a background shear flow first considered by Howland et al. (J. Fluid Mech., vol. 921, 2021, A24), using direct–adjoint looping to find the perturbation giving maximal energy growth on this evolving flow. We find that three-dimensional, convective mechanisms produce greater energy growth than their two-dimensional counterparts. In particular, we find close agreement with the direct numerical simulations of Howland et al. (J. Fluid Mech., 2021, in press), which demonstrated a clear three-dimensional mechanism causing breakdown to turbulence. The results are shown to hold at realistic Prandtl numbers. At low mean Richardson numbers, two-dimensional, shear-driven mechanisms produce greater energy growth.

scholarly journals Localization of flow structures using -norm optimization

2013 ◽  
Vol 729 ◽  
pp. 672-701 ◽  
Author(s):  
D. P. G. Foures ◽  
C. P. Caulfield ◽  
P. J. Schmid
Keyword(s):  
Channel Flow ◽  
Energy Production ◽  
Worst Case Analysis ◽  
Two Dimensional ◽  
Time Intervals ◽  
Worst Case ◽  
Optimal Perturbation ◽  
Energy Growth ◽  
Perturbation Problem ◽  
Production Zone

AbstractStability theory based on a variational principle and finite-time direct-adjoint optimization commonly relies on the kinetic perturbation energy density ${E}_{1} (t)= (1/ {V}_{\Omega } )\int \nolimits _{\Omega } e(\boldsymbol{x}, t)\hspace{0.167em} \mathrm{d} \Omega $ (where $e(\boldsymbol{x}, t)= \vert \boldsymbol{u}{\vert }^{2} / 2$) as a measure of disturbance size. This type of optimization typically yields optimal perturbations that are global in the fluid domain $\Omega $ of volume ${V}_{\Omega } $. This paper explores the use of $p$-norms in determining optimal perturbations for ‘energy’ growth over prescribed time intervals of length $T$. For $p= 1$ the traditional energy-based stability analysis is recovered, while for large $p\gg 1$, localization of the optimal perturbations is observed which identifies confined regions, or ‘hotspots’, in the domain where significant energy growth can be expected. In addition, the $p$-norm optimization yields insight into the role and significance of various regions of the flow regarding the overall energy dynamics. As a canonical example, we choose to solve the $\infty $-norm optimal perturbation problem for the simple case of two-dimensional channel flow. For such a configuration, several solutions branches emerge, each of them identifying a different energy production zone in the flow: either the centre or the walls of the domain. We study several scenarios (involving centre or wall perturbations) leading to localized energy production for different optimization time intervals. Our investigation reveals that even for this simple two-dimensional channel flow, the mechanism for the production of a highly energetic and localized perturbation is not unique in time. We show that wall perturbations are optimal (with respect to the $\infty $-norm) for relatively short and long times, while the centre perturbations are preferred for very short and intermediate times. The developed $p$-norm framework is intended to facilitate worst-case analysis of shear flows and to identify localized regions supporting dominant energy growth.

Influence of Insulation Type on In-Plane Shear Behavior of Insulated Concrete Sandwich Panels (ICSP) with GFRP Grid Shear Connectors

2014 ◽  
Vol 525 ◽  
pp. 416-419 ◽  
Author(s):  
Hye Ran Kim ◽  
Dae Hyun Kang ◽  
Hyun Do Yun
Keyword(s):  
Shear Flow ◽  
Outer Wall ◽  
Expanded Polystyrene ◽  
Test Results ◽  
Flow Strength ◽  
Plane Shear ◽  
Shear Connectors ◽  
Reinforced Polymer ◽  
Glass Fiber Reinforced ◽  
Shear Performance

This paper reports the experimental results to evaluate in-plane shear performance of insulated concrete sandwich panel (ICSP) with glass fiber-reinforced polymer (GFRP) grid shear connectors. The variables considered in this study are the grid size (35 and 53mm) of GFRP shear connectors and the types of insulation (expanded polystyrene, EPS and extruded polystyrene with special slots, XPSS). For loading in-plane shear force to interface between inner and outer wall of ICSP system, the ICSP specimens were supported vertically at the bottom edge of the two concrete outer walls by steel blocks. The test results indicate that ICSP with XPSS developed higher shear flow strengths in ICSP with EPS when 35mm spacing of GFRP grid is used. Also, the test results indicated that as the grid spacing of GFRP shear connector decreases, the shear flow strength of ICSP with XPSS insulation was higher, but the shear flow strength of ICSP with EPS insulation was lower.

scholarly journals Structure of Plane Shear Flow in River with Vegetations.

1994 ◽  
pp. 41-50 ◽  
Author(s):  
Shoji Fukuoka ◽  
Akihide Watanabe ◽  
Takayuki Tsumori
Keyword(s):  
Shear Flow ◽  
Plane Shear

Growth of vortical disturbances entrained in the entrance region of a circular pipe

2021 ◽  
Vol 932 ◽  
Author(s):  
Pierre Ricco ◽  
Claudia Alvarenga
Keyword(s):  
Reynolds Number ◽  
Initial Conditions ◽  
Maximum Growth ◽  
Streamwise Velocity ◽  
Velocity Profiles ◽  
Boundary Region ◽  
Base Flow ◽  
Entrance Region ◽  
Circular Pipe ◽  
Large Reynolds Number

The development and growth of unsteady three-dimensional vortical disturbances entrained in the entry region of a circular pipe is investigated by asymptotic and numerical methods for Reynolds numbers between $1000$ and $10\,000$ , based on the pipe radius and the bulk velocity. Near the pipe mouth, composite asymptotic solutions describe the dynamics of the oncoming disturbances, revealing how these disturbances are altered by the viscous layer attached to the pipe wall. The perturbation velocity profiles near the pipe mouth are employed as rigorous initial conditions for the boundary-region equations, which describe the flow in the limit of low frequency and large Reynolds number. The disturbance flow is initially primarily present within the base-flow boundary layer in the form of streamwise-elongated vortical structures, i.e. the streamwise velocity component displays an intense algebraic growth, while the cross-flow velocity components decay. Farther downstream the disturbance flow occupies the whole pipe, although the base flow is mostly inviscid in the core. The transient growth and subsequent viscous decay are confined in the entrance region, i.e. where the base flow has not reached the fully developed Poiseuille profile. Increasing the Reynolds number and decreasing the frequency causes more intense perturbations, whereas small azimuthal wavelengths and radial characteristic length scales intensify the viscous dissipation of the disturbance. The azimuthal wavelength that causes the maximum growth is found. The velocity profiles are compared successfully with available experimental data and the theoretical results are helpful to interpret the only direct numerical dataset of a disturbed pipe-entry flow.

Assessment of Viscous Flows in High-Speed Micro Rotating Machinery for Energy Conversion Applications

2001 ◽  
Author(s):  
Luc G. Fréchette
Keyword(s):  
Shear Flow ◽  
Energy Conversion ◽  
Viscous Flow ◽  
High Speed ◽  
Applied Pressure ◽  
Secondary Flows ◽  
System Level ◽  
Flow Profile ◽  
System Level Design ◽  
Level Design

Abstract This paper investigates the characteristics of viscous flow in the micron-scale clearances surrounding high-speed micro-rotors currently being developed for miniature energy conversion applications. Analysis and experimental results from 4 mm diameter microfabricated rotors operated above 1 million rpm are used to describe the viscous flow characteristics, and provide guidelines for system-level design. To first order, the flow is characterized as fully developed shear flow (Couette flow) across the small gaps, induced by the rotor motion. However, secondary flows are induced perpendicular to the direction of rotor motion when externally applied pressure gradients exist along the small gaps. The developing flow in the entrance region of the small gaps in this secondary flow direction impacts the shear flow profile, hence affecting the drag on the disk. The effect of other inertial forces, such as Coriolis and centrifugal forces, are investigated analytically and numerically and found to affect the shear flow profile on the fluid in the motor gap at high rotational speeds. Since viscous losses are prevelant in microsystems, appropriate modeling is necessary for system-level design.

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