On the nonlinear growth of single three-dimensional disturbances in boundary layers

Experiments indicate the importance of three-dimensional action during transition, while high-Reynolds-number-flow theory indicates a multi-structured type of analysis. In line with this, the three-dimensional nonlinear unsteady triple-deck problem is addressed here, for slower transition. High-amplitude/high-frequency properties show enhanced disturbance growth occurring downstream for single nonlinear oblique waves inclined at angles greater than tan−1 √2 (≈54.7°) to the free stream, in certain interesting special cases. The three-dimensional response there is very ‘spiky’ and possibly random, with sideband instabilities present. A second nonlinear stage, and then an Euler stage, are entered further downstream, although faster transition can go straight into these more nonlinear stages. More general cases are also considered. Sideband effects, sublayer bursting and secondary instabilities are discussed, along with the relation to experimental observations

Theoretical prediction and design for vortex generators in turbulent boundary layers

A theoretical study is presented of three-dimensional turbulent flow provoked in a boundary layer by an array of low-profile vortex generators (VGs) on the surface. The typical VG sits in the logarithmic region of the incident boundary layer, and the turbulence model used seems representative in this region. The governing equations yield a forward-marching three-dimensional vortex-type system, which is solved computationally and analytically for spanwise periodic VG arrays. Streamwise vortex patterns of various strengths are produced downstream, owing to three-dimensional distortion of the original logarithmic profile and to the turbulent stresses present. Predictions are given for certain basic VG shapes, e.g. triangular, with various spanwise spacings, and the predictions are found to agree favourably overall with recent experiments. In addition, the analytical formulae obtained prove useful in suggesting designs for favourable VG distributions, based on three factors: close spanwise packing, increased VG length, and suitably non-smooth spanwise shaping

PLATE-INJECTION INTO A SEPARATED SUPERSONIC BOUNDARY-LAYER

The structure of a supersonic laminar boundary layer near a flat plate is examined when fluid is injected into it with velocity of O(ε3U*[infty infinity]) over a distance of O(L). Here U*[infty infinity] is the undisturbed fluid velocity, L the length of the plate and ε−8 is a representative Reynolds number. An essential requirement of the theory is that separation must have occurred upstream of the blow through a free interaction. It is assumed that between separation and the blow the reversed flow region has a wedge-like shape, of semi-angle in which O(ε2), the fluid velocity has decayed to insignificant values at points just upstream of the blowing region. The blown fluid fills this wedge and the favourable pressure gradient necessary to drive this fluid downstream causes the boundary of the wedge to curve until at the end of the blow it is parallel to the plate. Explicit expressions for the pressure variation and boundary-layer thickness are worked out using a (crucially) modified form of the Cole-Aroesty theory. The relation. between the strong injection studied here and massive injection, when the blowing velocity is of O(U*[infty infinity]), is also discussed

ON INTERACTION BETWEEN FALLING BODIES AND THE SURROUNDING FLUID

Interactions between a finite number of bodies and the surrounding fluid, in a channel for instance, are investigated theoretically. In the planar model here the bodies or modelled grains are thin solid bodies free to move in a nearly parallel formation within a quasi-inviscid fluid. The investigation involves numerical and analytical studies and comparisons. The three main features that appear are a linear instability about a state of uniform motion, a clashing of the bodies (or of a body with a side wall) within a finite scaled time when nonlinear interaction takes effect, and a continuum-limit description of the body–fluid interaction holding for the case of many bodies

Planar flows past thin multi-blade configurations

Two-dimensional steady laminar flows past multiple thin blades positioned in near or exact sequence are examined for large Reynolds numbers. Symmetric configurations require solution of the boundary-layer equations alone, in parabolic fashion, over the successive blades. Non-symmetric configurations in contrast yield a new global inner-outer interaction in which the boundary layers, the wakes and the potential flow outside have to be determined together, to satisfy pressure-continuity conditions along each successive gap or wake. A robust computational scheme is used to obtain numerical solutions in direct or design mode, followed by analysis. Among other extremes, many-blade analysis shows a double viscous structure downstream with two streamwise length scales operating there. Lift and drag are also considered. Another new global interaction is found further downstream. All the interactions involved seem peculiar to multi-blade flows

Short-scale effects on model boundary-layer spots

This theoretical work, on the spot in an otherwise laminar boundary layer, concerns the initial-value problem for three-dimensional inviscid disturbances covering a wide range of scales. The study asks whether or not comparatively short scales can have a substantial impact on the spot spreading rate, as well as on other important features including the spot structure. It is found that such scales act to reduce the spread angle to approximately 11°, close to the experimental observations for transitional/turbulent spots, as opposed to the angle of 19.47° for waves near or behind the spot trailing edge. The scales emerge from coupling uniform shear flow directly with the local uniform stream and then analysing large-time features. The leading edge and trailing edge of the spot are also examined in detail, along with other structural properties. It is concluded that nonlinearity and short-scale effects probably combine to restrict the global spread angle as above, while viscous sublayer bursting among other things completes the spot structure. Related work on nonlinear, trailing-edge and leading-edge behaviours and comparisons with experiments are also discussed

On 'spot' evolution under an adverse pressure gradient

The unsteady travelling 'spots' or spot-like disturbances are produced, in an otherwise planar boundary layer, by an initial impulse/blip, from wall forcing or from nearby external forcing. Theory and computations are described for the evolving spot-like structure, yielding initial-value problems for inviscid spot-like disturbances, commencing near the onset of an adverse pressure gradient. A transient stage incorporates the initial conditions, following which adverse pressure gradient effects become significant. Leading and trailing critical layers then form, which confine and define the spot-like disturbance, and these depart from the wall downstream accompanied by disturbance amplification and mean flow distortion. The interplay of adverse pressure gradient effects with three-dimensionality, nonlinearity and non-parallelism is considered in turn.Three-dimensional effects provoke a universal closed planform of spot-like disturbance, which has a different side behaviour from the zero-gradient case. Nonlinear interactions eventually change the internal structure, particularly at the spot-like disturbance leading edge, while pointing to the mean-flow alteration underhanging the spot-like disturbance and to a pressure-feedback alteration for the region behind the spot-like disturbance. These two alterations offer complementary mechanisms for describing the calmed region trailing a spot-like disturbance, in which an attached thinned wall layer is identified. Non-parallel effects lead to enhanced spot-like disturbance growth and larger-scale/shorter-scale interactive behaviour downstream. The approach to separation is also considered, yielding maximal growth for small spot-like disturbances at 5/6 of the way from the minimum pressure position to the separation position. Links with recent experiments on adverse-gradient spot-like disturbances and with findings on calmed region properties are investigated, as well as the unsteady forcing effects from an incident relatively thick vortical wake outside the boundary layer

Singular modes in Rayleigh instability of three-dimensional streamwise-vortex flows

The upper-branch neutral modes of inviscid instability in a boundary-layer flow with significant longitudinal vortices present are shown to possess typically a logarithmically singular, non-inflectional, critical layer. This contrasts with previous linear and nonlinear suggestions implemented in vortex-wave interaction and secondary instability theories, which are re-examined. The analysis here is based first on perturbation techniques applied to a Rayleigh unstable planar motion supplemented by a vortex centred around the inflection level, followed by the extension to more general cases. Flows with order one and larger spanwise scales are considered. Multiple solutions, their limit properties and parametric continuations are illustrated with concrete examples

Direct simulations and modelling of basic three-dimensional bifurcating tube flows

Three-dimensional bifurcating internal flow is studied for a single mother tube branching into two equal but diverging daughter tubes. The mother tube is straight and of circular cross-section, containing a fully developed incident motion, while the diverging daughters are straight and of semi-circular cross-section. This basic configuration is treated first by direct numerical simulation and secondly by slenderflow modelling, for a variety of Reynolds numbers and angles of divergence. The direct simulations and modelling highlight different forms of three-dimensional separation or flow reversal as well as enhanced upstream and downstream influence and pressure loss induced by the bifurcations especially at increased divergence angles. Comparisons between the results from the simulations and those from the slender-flow modelling show relatively close agreement at medium values of Reynolds number. In particular, as the angle of divergence increases for a given Reynolds number, there is generally first an increase in flow attachment on to the inner divider wall(s) and then, at higher angles, an increasing trend to flow reversal at the corners formed by the junctions of the outer wall with the divider; longitudinal vortex motion is also enhanced then. The agreement persists over a surprisingly wide range of divergence angles

On vortex/wave interactions. Part 1. Non-symmetrical input and cross-flow in boundary layers

The paper studies certain effects of non-symmetry on vortex/wave interactions, for inviscid inflexional waves interacting nonlinearly with the vortex component of the mean flow in boundary-layer transition at large Reynolds number. Two types of non-symmetry are investigated, namely for unequal input wave amplitudes and for small cross-flows. These lead to coupled integro-differential equations for spatial development of the wave amplitudes, which are examined in an essentially equivalent differential form for various degrees of the non-symmetry present. Each type of non-symmetry can have a significant influence on the nonlinear interaction properties. Special emphasis is given to bounded solutions, and numerous interesting new flow responses are found analytically and computationally. The theory provides a basis for tackling enhanced non-symmetry in the input or stronger cross-flows