The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

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.

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The autonomous cycle of near-wall turbulence

We present a new mechanism for generation of near-wall streamwise vortices – which dominate turbulence phenomena in boundary layers – using linear perturbation analysis and direct numerical simulations of turbulent channel flow. The base flow, consisting of the mean velocity profile and low-speed streaks (free from any initial vortices), is shown to be linearly unstable to sinuous normal modes only for relatively strong streaks, i.e. for wall inclination angles of streak vortex lines exceeding 50°. Analysis of streaks extracted from fully developed near-wall turbulence indicates that about 20% of streak regions in the buffer layer exceed the strength threshold for instability. More importantly, these unstable streaks exhibit only moderate (twofold) normal-mode amplification, the growth being arrested by self-annihilation of streak-flank normal vorticity due to viscous cross-diffusion. We present here an alternative, streak transient growth (STG) mechanism, capable of producing much larger (tenfold) linear ampliflcation of x-dependent disturbances. Note the distinction of STG – responsible for perturbation growth on a streak velocity distribution U(y, z) – from prior transient growth analyses of the (streakless) mean velocity U(y). We reveal that streamwise vortices are generated from the more numerous normal-mode-stable streaks, via a new STG-based scenario: (i) transient growth of perturbations leading to formation of a sheet of streamwise vorticity ωx (by a ‘shearing’ mechanism of vorticity generation), (ii) growth of sinuous streak waviness and hence ∂u/∂x as STG reaches nonlinear amplitude, and (iii) the ωx sheet’s collapse via stretching by ∂u/∂x (rather than rollup) into streamwise vortices. Significantly, the three-dimensional features of the (instantaneous) streamwise vortices of x-alternating sign generated by STG agree well with the (ensemble-averaged) coherent structures educed from fully turbulent flow. The STG-induced formation of internal shear layers, along with quadrant Reynolds stresses and other turbulence measures, also agree well with fully developed turbulence. Results indicate the prominent – possibly dominant – role of this new, transient-growth-based vortex generation scenario, and suggest interesting possibilities for robust control of drag and heat transfer.

/pdf/coherent-structure-generation-in-near-wall-turbulence-36o0srxpc5.pdf

Coherent structure generation in near-wall turbulence

The structure of round jets in cross-flow was studied using flow visualization techniques and flying-hot-wire measurements. The study was restricted to jet to freestream velocity ratios ranging from 2.0 to 6.0 and Reynolds numbers based on the jet diameter and free-stream velocity in the range of 440 to 6200.Flow visualization studies, together with time-averaged flying-hot-wire measurements in both vertical and horizontal sectional planes, have allowed the mean topological features of the jet in cross-flow to be identified using critical point theory. These features include the horseshoe (or necklace) vortex system originating just upstream of the jet, a separation region inside the pipe upstream of the pipe exit, the roll-up of the jet shear layer which initiates the counter-rotating vortex pair and the separation of the flat-wall boundary layer leading to the formation of the wake vortex system beneath the downstream side of the jet.The topology of the vortex ring roll-up of the jet shear layer was studied in detail using phase-averaged flying-hot-wire measurements of the velocity field when the roll-up was forced. From these data it is possible to examine the evolution of the shear layer topology. These results are supported by the flow visualization studies which also aid in their interpretation.The study also shows that, for velocity ratios ranging from 4.0 to 6.0, the unsteady upright vortices in the wake may form by different mechanisms, depending on the Reynolds number. It is found that at high Reynolds numbers, the upright vortex orientation in the wake may change intermittently from one configuration of vortex street to another. Three mechanisms are proposed to explain these observations.

An experimental study of round jets in cross-flow

Bypass transition in an initially laminar boundary layer beneath free-stream turbulence is simulated numerically. New perspectives on this phenomenon are obtained from the numerical flow fields. Transition precursors consist of long backward jets contained in the fluctuating u-velocity field; they flow backwards relative to the local mean velocity. The jets extend into the upper portion of the boundary layer, where they interact with free-stream eddies. In some locations a free-stream perturbation to the jet shear layer develops into a patch of irregular motion – a sort of turbulent spot. The spot spreads longitudinally and laterally, and ultimately merges into the downstream turbulent boundary layer. Merging spots maintain the upstream edge of the turbulent region. The jets, themselves, are produced by low-frequency components of the free-stream turbulence that penetrate into the laminar boundary layer. Backward jets are a component of laminar region streaks.A method to construct turbulent inflow from Orr–Sommerfeld continuous modes is described. The free-stream turbulent intensity was chosen to correspond with the experiment by Roach & Brierly (1990). Ensemble-averaged numerical data are shown to be in good agreement with laboratory measurements.

Simulations of bypass transition

Situations where an effectively irrotational freest ream contains regions of concentrated vorticity are common in external aerodynamics, where vortices are known to play an important role in the flight of modern helicopters (Carr 1988) and aircraft (Cunningham 1989, Mabey 1989). Vortices may arise as a consequence of shedding from some upstream surface or near certain three-dimensional boundary geometries that act to promote vortex formation. Examples of the former type of vortex gen­ eration include: 1. vortices that trail from the tips of airfoils (Harvey & Perry 1971) and control surfaces on submarines (Lugt 1983), 2. transverse vortices shed from maneuvering airfoil surfaces and helicopter blades in a process known as dynamic stall (McCroskey 1982, Francis & Keesee 1985, Carr 1988), and 3. shedding from stationary obstacles. Geometry-induced creation can occur in any situation where a flow along a wall approaches a surface-mounted obstacle; examples include: 1. airframe features such as wing/body junctions, 2. conning towers on submarines, and 3 . computer chips mounted on electrical circuit boards. Similar geometries are en-

Vortex Interactions with Walls

The flow induced by a vortex ring approaching a plane wall on a trajectory normal to the wall is investigated for an incompressible fluid which is otherwise stagnant. The detailed characteristics of the interaction of the ring with the flow near the surface have been observed experimentally for a wide variety of laminar rings, using dye in water to visualize the flow in the ring as well as near the plane surface. Numerical solutions are obtained for the trajectory of the ring as well as for the unsteady boundary-layer flow that develops on the wall. The experimental and theoretical results show that an unsteady separation develops in the boundary-layer flow, in the form of a secondary ring attached to the wall. A period of explosive boundary-layer growth then ensues and a strong viscous-inviscid interaction occurs in the form of the ejection of the secondary vortex ring from the boundary layer. The primary ring then interacts with the secondary ring and in some cases was observed to induce the formation of a third, tertiary, ring near the wall. The details of this process are investigated over a wide Reynolds number range. The results clearly show how one vortex ring can produce another, through an unsteady interaction with a viscous flow near the wall.

The impact of a vortex ring on a wall

The response of a wall boundary layer to the motion of a convected vortex is investigated. The principal cases considered are for a rectilinear filament of strength –κ located a distance a above a plane wall and convected to the right in a uniform flow of speed U∞*. The inviscid solution predicts that such a vortex will remain at constant height a above the wall and be convected with constant speed αU∞*. Here α is termed the fractional convection rate of the vortex, and cases in the parameter range 0 [les ] α < 1 are considered. The motion is initiated at time t* = 0 and numerical calculations of the developing boundary-layer flow are carried out for α = 0, 0.2, 0.4, 0.55, 0.7, 0.75 and 0.8. For α < 0.75, a rapid lift-up of the boundary-layer streamlines and strong boundary-layer growth occurs in the region behind the vortex; in addition an unusual separation phenomenon is observed for α [les ] 0.55. For α [ges ] 0.75, the boundary-layer development is more gradual, but ultimately substantial localized boundary-layer growth also occurs. In all cases, it is argued that a strong inviscid–viscous interaction will take place in the form of an eruption of the boundary-layer flow. The generalization of these results to two-dimensional vortices with cores of finite dimension is discussed.

The boundary layer induced by a convected two-dimensional vortex

The unsteady boundary-layer flow produced by a two-dimensional vortex in motion above an infinite plane wall in an otherwise stagnant fluid is considered in the limit of infinite Reynolds number. This study is part of a continuing investigation into the nature of the physical processes that occur near the surface in transitional and fully turbulent boundary layers. The adverse pressure gradient due to the vortex leads to the development of a zone of recirculation in the viscous flow near the surface, and the boundary-layer flow then focuses rapidly toward an eruption along a band which is very narrow in the stream wise direction. The evolution of the unsteady boundary layer is posed in Lagrangian coordinates and computed using an efficient, factored ADI numerical method. The boundary-layer solution is found to develop a separation singularity and to evolve toward a terminal stage which is generic in two-dimensional unsteady flows. The computed results are compared with the results of asymptotic theory of two-dimensional boundary-layer separation and the agreement is found to be excellent.

Vortex-induced boundary-layer separation. Part 1. The unsteady limit problem Re [rightward arrow] [infty infinity]

The unsteady boundary layer induced by the motion of a rectilinear vortex above an infinite plane wall is calculated using interacting boundary-layer methods. The boundary-layer solution is computed in Lagrangian variables since it is possible to compute the flow evolution accurately in this formulation even when an eruption starts to evolve. Results are obtained over a range of Reynolds numbers, Re. For the limit problem Re - infinity (studied in Part 1), a singularity develops in the non-interacting boundary-layer solution at finite time. The present results show that the interacting boundary-layer calculations also terminate in a singularity at a time which is earlier than in the limit problem and which decreases with decreasing Reynolds number. The computed results are compared with the length-and timescales predicted by recent asymptotic theories and are found to be in excellent agreement. See also previous abstract.

J. D. A. Walker

Papers

Vortex Interactions with Walls

The impact of a vortex ring on a wall

The boundary layer induced by a convected two-dimensional vortex

Vortex-induced boundary-layer separation. Part 1. The unsteady limit problem Re [rightward arrow] [infty infinity]

Vortex-induced boundary-layer separation. part 2. unsteady interacting boundary-layer theory