Fluid flows that are smooth at low speeds become unstable and then turbulent at higher speeds. This phenomenon has traditionally been investigated by linearizing the equations of flow and testing for unstable eigenvalues of the linearized problem, but the results of such investigations agree poorly in many cases with experiments. Nevertheless, linear effects play a central role in hydrodynamic instability. A reconciliation of these findings with the traditional analysis is presented based on the "pseudospectra" of the linearized problem, which imply that small perturbations to the smooth flow may be amplified by factors on the order of 105 by a linear mechanism even though all the eigenmodes decay monotonically. The methods suggested here apply also to other problems in the mathematical sciences that involve nonorthogonal eigenfunctions.

https://people.maths.ox.ac.uk/trefethen/publication/PDF/1993_57.pdf

Hydrodynamic Stability Without Eigenvalues

Direct numerical simulations of unsteady channel flow were performed at low to moderate Reynolds numbers on computational boxes chosen small enough so that the flow consists of a doubly periodic (in x and z) array of identical structures. The goal is to isolate the basic flow unit, to study its morphology and dynamics, and to evaluate its contribution to turbulence in fully developed channels. For boxes wider than approximately 100 wall units in the spanwise direction, the flow is turbulent and the low-order turbulence statistics are in good agreement with experiments in the near-wall region. For a narrow range of widths below that threshold, the flow near only one wall remains turbulent, but its statistics are still in fairly good agreement with experimental data when scaled with the local wall stress. For narrower boxes only laminar solutions are found. In all cases, the elementary box contains a single low-velocity streak, consisting of a longitudinal strip on which a thin layer of spanwise vorticity is lifted away from the wall. A fundamental period of intermittency for the regeneration of turbulence is identified, and that process is observed to consist of the wrapping of the wall-layer vorticity around a single inclined longitudinal vortex.

/pdf/the-minimal-flow-unit-in-near-wall-turbulence-21dsdqqfsk.pdf

The minimal flow unit in near-wall turbulence

Transition to turbulence in plane channel flow occurs even for conditions under which modes of the linearized dynamical system associated with the flow are stable. In this paper an attempt is made to understand this phenomena by finding the linear three‐dimensional perturbations that gain the most energy in a given time period. A complete set of perturbations, ordered by energy growth, is found using variational methods. The optimal perturbations are not of modal form, and those which grow the most resemble streamwise vortices, which divert the mean flow energy into streaks of streamwise velocity and enable the energy of the perturbation to grow by as much as three orders of magnitude. It is suggested that excitation of these perturbations facilitates transition from laminar to turbulent flow. The variational method used to find the optimal perturbations in a shear flow also allows construction of tight bounds on growth rate and determination of regions of absolute stability in which no perturbation growth is possible.

/pdf/three-dimensional-optimal-perturbations-in-viscous-shear-ttx3nmyjot.pdf

Three‐dimensional optimal perturbations in viscous shear flow

Numerical experiments on modified turbulent channels at moderate Reynolds numbers are used to differentiate between several possible regeneration cycles for the turbulent fluctuations in wall-bounded flows. It is shown that a cycle exists which is local to the near-wall region and does not depend on the outer flow. It involves the formation of velocity streaks from the advection of the mean profile by streamwise vortices, and the generation of the vortices from the instability of the streaks. Interrupting any of those processes leads to laminarization. The presence of the wall seems to be only necessary to maintain the mean shear. The generation of secondary vorticity at the wall is shown to be of little importance in turbulence generation under natural circumstances. Inhibiting its production increases turbulence intensity and drag.

/pdf/the-autonomous-cycle-of-near-wall-turbulence-tcvf904r54.pdf

The autonomous cycle of near-wall turbulence

The problem of transition from laminar to turbulent flow in viscous bound­ ary layers is of great practical interest. The low skin-friction coefficient of laminar boundary-layer flow is very attractive to those who lay out the engines or pay the fuel for high-speed vehicles such as airplanes. However, the low mixing of fluid properties such as chemical species, heat, or momen­ tum may be intolerable for others who design these engines or cope with the danger of separation in adverse pressure gradients; they may clearly prefer a turbulent state of the flow. Therefore, it would be highly desirable to at least predict, if not to control, whether the flow under consideration is laminar or turbulent. The tremendous efforts of decades of intense research, however, have been to little avail (Reshotko 1976). The empirical en-criterion is still the standard tool in engineering practice, although it is known to ignore essential ingredients of the physics of transition and therefore may dangerously mislead if used beyond the supporting data base. Numerical transition simulations have gained reliability in repro­ ducing the transition process in sufficient detail to extract information unobtainable from laboratory experiments. However, the inherent assumptions of stream wise periodicity and temporal growth of the bound­ ary layer, in addition to the uncertainty of initial conditions, prevent predicting transition in practice. Hence, theory still holds an important place in identifying inherent mechanisms and structures of the transition process and in explaining otherwise unintelligible observations. The past decade saw some important progress in stability theory, slow or fast, depending on the reader's judgment ..

Secondary Instability of Boundary Layers

The instability of the three-dimensional high-shear layer associated with a near-wall low-speed streak is investigated experimentally. A single low-speed streak, not unlike the near-wall low-speed streaks in transitional and turbulent flows, is produced in a laminar boundary layer by using a small piece of screen set normal to the wall. In order to excite symmetric and anti-symmetric modes separately, well-controlled external disturbances are introduced into the laminar low-speed streak through small holes drilled behind the screen. The growth of the excited symmetric varicose mode is essentially governed by the Kelvin–Helmholtz instability of the in ectional velocity profiles across the streak in the normal-to-wall direction and it can occur when the streak width is larger than the shear-layer thickness. The spatial growth rate of the symmetric mode is very sensitive to the streak width and is rapidly reduced as the velocity defect decreases flowing to the momentum transfer by viscous stresses. By contrast, the anti-symmetric sinuous mode that causes the streak meandering is dominated by the wake-type instability of spanwise velocity distributions across the streak. As far as the linear instability is concerned, the growth rate of the anti-symmetric mode is not so strongly affected by the decrease in the streak width, and its exponential growth may continue further downstream than that of the symmetric mode. As for the mode competition, it is important to note that when the streak width is narrow and comparable with the shear-layer thickness, the low-speed streak becomes more unstable to the anti-symmetric modes than to the symmetric modes. It is clearly demonstrated that the growth of the symmetric mode leads to the formation of hairpin vortices with a pair of counter-rotating streamwise vortices, while the anti-symmetric mode evolves into a train of quasi-streamwise vortices with vorticity of alternate sign.

The instability and breakdown of a near-wall low-speed streak

Velocity measurements were made in the flow field behind a circular cylinder at Reynolds numbers from 10 to 80 and results compared with existing numerical solutions. Takami & Keller's solution for the velocity distribution in the wake shows good agreement at low Reynolds numbers and fair agreement at high Reynolds numbers. The drag coefficient of the cylinder and the size of the standing eddies behind the cylinder were also determined. They are compatible with existing experimental and numerical results. Details of the velocity distribution in the standing eddies are clarified.

Measurements of velocity distributions in the wake of a circular cylinder at low Reynolds numbers

Two kinds of experiment were made in the wake of a cylinder at Reynolds numbers ranging between 20 and 150. One was a close look at the structure of the vortex street with a stationary cylinder at Reynolds numbers greater than 48. The other experiment was made at lower Reynolds numbers with a cylinder vibrating normal to the flow direction. In this case an artificially induced small-amplitude fluctuation grows exponentially with the rate predicted by the stability theory. Because of the similarity between the two kinds of wake, we postulate that the shedding of the vortex at low Reynolds numbers is initiated by the linear growth, namely, the fluctuation with the frequency of maximum linear growth rate develops into vortex streets. By using the measured width of the wake at the stagnation point in the wake and the result of the stability theory, we could calculate the Strouhal number for Reynolds numbers ranging from 48 to 120. The predicted Strouhal numbers agree well with the values from direct measurements.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112078002451

Mechanism of determination of the shedding frequency of vortices behind a cylinder at low Reynolds numbers

The subcritical transition in plane Poiseuille flow (generated in a long wind channel of rectangular cross-section) was studied experimentally. Cylinder-generated vortical disturbances were introduced into the parabolic flow (in the test section) or the inlet flow. The parabolic flow was also disturbed by a controlled periodic jet from a wall orifice. On the basis of the three kinds of observations, we come to the conclusion that the minimum transition Reynolds number is about 1000 and the related threshold intensity (of the external disturbance triggering the transition) is comparable to the maximum intensity of u-fluctuations in fully turbulent channel flows.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112085000210

Some observations of the subcritical transition in plane Poiseuille flow

In this paper, we present our experimental and numerical results for the supersonic mixing and combustion control. Our strategy of enhancing supersonic mixing is to use streamwise vortices. The reasons are (1) that in supersonic flows streamwise vortices can be generated quite easily and almost without additional losses in total pressure such as those due to shock waves, (2) that their breakdown into smaller scale, which is essential for mixing enhancement, can be controlled by their geometry in spanwise row configurations and by various combinations of their scales, intensity of circulation and rotational directions, and (3)that the hydrogen fuel can be injected into their core region. Several kind of alternating wedge struts are tested to examine the effects of the cited various combinations on the mixing enhancement. By presenting those results, we will discuss the formation mechanism of streamwise vortices and their downstream development, in particular the interaction between themselves in spanwise row configurations, their eventual breakdown into turbulent eddies. Firing tests are also carried out and the results for alternating wedge strut will be presented together with the results for a generic fuel injection strut for comparison. The results show the effectiveness of the present streamwise vortices for supersonic mixing enhancement and combustion control.

Michio Nishioka

Papers

The instability and breakdown of a near-wall low-speed streak

Measurements of velocity distributions in the wake of a circular cylinder at low Reynolds numbers

Mechanism of determination of the shedding frequency of vortices behind a cylinder at low Reynolds numbers

Some observations of the subcritical transition in plane Poiseuille flow

Supersonic mixing and combustion control using streamwise vortices