Asymptotic Suction Boundary Layer Research Articles

Turbulent spots in the asymptotic suction boundary layer

Amplitude thresholds for transition of localized disturbances, their breakdown to turbulence and the development of turbulent spots in the asymptotic suction boundary layer are studied using direct numerical simulations. A parametric study of the horizontal scales of the initial disturbance is performed and the disturbances that lead to the highest growth under the conditions investigated are used in the simulations. The Reynolds-number dependence of the threshold amplitude of a localized disturbance is investigated for 500≤ Re ≤ 1200, based on the free-stream velocity and the displacement thickness. It is found that the threshold amplitude scales as Re−1.5 for the considered Reynolds numbers. For Re ≤ 367, the localized disturbance does not lead to a turbulent spot and this provides an estimate of the critical Reynolds number for the onset of turbulence. When the localized disturbance breaks down to a turbulent spot, it happens through the development of hairpin and spiral vortices. The shape and spreading rate of the turbulent spot are determined for Re = 500, 800 and 1200. Flow visualizations reveal that the turbulent spot takes a bullet-shaped form that becomes more distinct for higher Reynolds numbers. Long streaks extend in front of the spot and in its wake a calm region exists. The spreading rate of the turbulent spot is found to increase with increasing Reynolds number.

Optimal disturbances in suction boundary layers

A well-known optimization procedure is used to find the optimal disturbances in two different suction boundary layers within the spatial framework. The maximum algebraic growth in the asymptotic suction boundary layer is presented and compared to previous temporal results. Furthermore, the spatial approach allows a study of a developing boundary layer in which a region at the leading edge is left free from suction. This new flow, which emulates the base flow of a recent wind-tunnel experiment, is herein denoted a semi-suction boundary layer. It is found that the optimal disturbances for these two suction boundary layers consist of streamwise vortices that develop into streamwise streaks, as previously found for a number of shear flows. It is shown that the maximum energy growth in the semi-suction boundary layer is obtained over the upstream region where no suction is applied. The result indicates that the spanwise scale of the streaks is set in this region, which is in agreement with previous experimental findings.

Transition thresholds in the asymptotic suction boundary layer

Energy thresholds for transition to turbulence in an asymptotic suction boundary layer is calculated by means of temporal direct numerical simulations. The temporal assumption limits the analysis to periodic disturbances with horizontal wave numbers determined by the computational box size. Three well known transition scenarios are investigated: oblique transition, the growth and breakdown of streaks triggered by streamwise vortices, and the development of random noise. Linear disturbance simulations and stability diagnostics are also performed for a base flow consisting of the suction boundary layer and a streak. The scenarios are found to trigger transition by similar mechanisms as obtained for other flows. Transition at the lowest initial energy is provided by the oblique wave scenario for the considered Reynolds numbers 500, 800, and 1200. The Reynolds number dependence on the energy thresholds are determined for each scenario. The threshold scales like Re−2.6 for oblique transition and like Re−2.1 for transition initiated by streamwise vortices and random noise, indicating that oblique transition has the lowest energy threshold also for larger Reynolds numbers.

Free stream turbulence induced disturbances in boundary layers with wall suction

An experimental investigation of free stream turbulence (FST) induced disturbances in asymptotic suction boundary layers (ASBL) has been performed. In the present study four different suction rates are used and the highest is 0.40% of the free stream velocity, together with three different FST levels (Tu=1.6, 2.0, and 2.3%). A turbulence generating grid of the active type is used and offers the possibility to vary the Tu-level while the scales of the turbulence remain almost constant. It is known that FST induces elongated disturbances consisting of high and low velocity regions, usually denoted streaky structures, into the boundary layer. The experiments show that wall suction suppresses the disturbance growth and may significantly delay or inhibit the break-down to turbulence. Two-point correlation measurements in the spanwise direction show that the averaged streak spacing decreases with increasing FST-level, whereas the spanwise scale in the ASBL is more or less constant if scaled with the free stream velocity and viscosity. This is in contrast to what is observed in a Blasius boundary layer where streaks develop and adapt their spanwise scale close to the boundary layer thickness.

On the disturbance growth in an asymptotic suction boundary layer

Both experimental and theoretical studies have beenconsidered on flat plate boundary layers as well as on wakesbehind porous cylinders. The main thread in this work iscontrol, which is applied passively and actively on boundarylayers in order to inhibit or postpone transition toturbulence; and actively through the cylinder surface in orderto effect the wakecharacteristics. An experimental set-up for the generation of the asymptoticsuction boundary layer (ASBL) has been constructed. This studyis the first, ever, that report a boundary layer flow ofconstant boundary layer thickness over a distance of 2 metres.Experimental measurements in the evolution region, from theBlasius boundary layer (BBL) to the ASBL, as well as in theASBL are in excellent agreement with boundary layer analysis.The stability of the ASBL has experimentally been tested, bothto Tollmien-Schlichting waves as well as to free streamturbulence (FST), for relatively low Reynolds numbers (Re). For the former disturbances good agreement is foundfor the streamwise amplitude profiles and the phase velocitywhen compared with linear spatial stability theory. However,the energy decay factor predicted by theory is slightlyoverestimated compared to the experimental findings. The latterdisturbances are known to engender streamwise elongated regionsof high and low speeds of fluid, denoted streaks, in a BBL.This type of spanwise structures have been shown to appear inthe ASBL as well, with the same spanwise wavelength as in theBBL, despite the fact that the boundary layer thickness issubstantially reduced in the ASBL case. The spanwise wavenumberof the optimal perturbation in the ASBL has been calculated andis β = 0.53, when normalized with the displacementthickness. The spanwise scale of the streaks decreases withincreasing turbulence intensity (Tu) and approaches the scale given by optimalperturbation theory. This has been shown for the BBL case aswell. The initial energy growth of FST induced disturbances hasexperimentally been found to grow linearly as Tu2Rexin the BBL, the transitional Reynolds numberto vary as Tu-2, and the intermittency function to have a relativelywell-defined distribution, valid for all Tu. The wake behind a porous cylinder subject to continuoussuction or blowing has been studied, where amongst other thingsthe Strouhal number (St) has been shown to increase strongly with suction,namely, up to 50% for a suction rate of 2.5% of the free streamvelocity. In contrast, blowing shows a decrease ofStof around 25% for a blowing rate of 5% of the freestream velocity in the considered Reynolds number range. Keywords:Laminar-turbulent transition, asymptoticsuction boundary layer, free stream turbulence,Tollmien-Schlichting wave, stability, flow control, cylinderwake.

Optimal linear growth in the asymptotic suction boundary layer

A variational technique in the temporal framework is used to study initial configurations of disturbance velocity which maximize perturbation kinetic energy in the asymptotic suction boundary layer (ASBL). These optimal perturbations (OP) excite significant and remarkably persistent transient growth, on the order of that which occurs in the Blasius boundary layer (BBL). In contrast, classical modal analysis of the ASBL predicts a critical Reynolds number two orders of magnitude larger than that for the BBL. As in other two-dimensional boundary layer flows, disturbances undergoing maximum amplification are infinitely elongated in the direction of the flow and take the form of streamwise-oriented vortices which induce strong variations in the streamwise perturbation velocity (streaks). The Reynolds number dependence of the maximum growth, and the best choice of scaling for the spanwise wavenumber of the perturbation causing it, are elucidated. There is good agreement between the streak resulting from OP and disturbances measured in experiments in which the asymptotic suction boundary layer is subject to free stream turbulence (FST). This agreement is shown to improve as the level of FST increases.

Primary and secondary instabilities of the asymptotic suction boundary layer on a curved plate

The temporal growth of Görtler vortices and the associated secondary instability mechanisms are investigated numerically in the case of an asymptotic suction boundary layer on a curved plate. Highly inflectional velocity profiles are generated in both the spanwise and vertical directions. The inflectional velocity profile develops earlier in the spanwise direction. There exist two distinct modes of instability that eventually lead to the breakdown of Görtler vortices: the sinuous mode and the varicose mode. The temporal secondary instability analysis of the three-dimensional inflectional velocity profile reveals that the sinuous mode becomes unstable earlier than the varicose mode. The sinuous mode is shown to be primarily related to shear in the spanwise direction, ∂U/∂z, and the varicose mode to shear in the vertical direction, ∂U/∂y.

Errata

Proc. R. Soc. Lond . A 399, 321–365 (1985) Uniform asymptotic approximations to solutions of the Orr–Sommerfeld equation for the asymptotic suction boundary layer profile By P. Baldwin Page 347, at the end of the second sentence in the statement of Theorem 1: insert the sentence ‘Suppose that f(z) may be expanded as a convergent series of the polynomials {P n (z)} for (z) ≼ p in R .

Finite-amplitude steady waves in plane viscous shear flows

Computations of two-dimensional solutions of the Navier–Stokes equations are carried out for finite-amplitude waves on steady unidirectional flow. Several cases are considered. The numerical method employs pseudospectral techniques in the streamwise direction and finite differences on a stretched grid in the transverse direction, with matching to asymptotic solutions when unbounded. Earlier results for Poiseuille flow in a channel are re-obtained, except that attention is drawn to the dependence of the minimum Reynolds number on the physical constraint of constant flux or constant pressure gradient. Attempts to calculate waves in Couette flow by continuation in the velocity of a channel wall fail. The asymptotic suction boundary layer is shown to possess finite-amplitude waves at Reynolds numbers orders of magnitude less than the critical Reynolds number for linear instability. Waves in the Blasius boundary layer and unsteady Rayleigh profile are calculated by employing the artifice of adding a body force to cancel the spatial or temporal growth. The results are verified by comparison with perturbation analysis in the vicinity of the linear-instability critical Reynolds numbers.

Uniform asymptotic approximations to solutions of the Orr—Sommerfeld equation for the asymptotic suction boundary layer profile

It is shown that Reid’s ( Stud. appl. Math . 53, 217-224 (1974)) proposed uniform asymptotic approximations to solutions of the Orr-Sommerfeld equation are valid for the asymptotic suction boundary layer profile. The dominant-recessive solutions are found to be free of logarithmic terms at all orders of approximation, and the partly balanced solutions to involve the generalized Airy functions B k ( z , p , q ) for q ═ 0 and 1 only. The analysis makes use of the theory of Chester, Friedman and Ursell ( Proc. Camb. phil. Soc . 53, 599-611 (1957)).

The effect of stable thermal stratification on the stability of viscous parallel flows

A unified linear viscous stability theory is developed for a certain class of stratified parallel channel and boundary-layer flows with Prandtl number equal to unity. Results are presented for plane Poiseuille flow and the asymptotic suction boundary-layer profile, which show that the asymptotic behaviour of both branches of the curve of neutral stability has a universal character. For velocity profiles without inflexion points it is found that a mode of instability disappears as η, the local Richardson number evaluated at the critical point, approaches 0.0554 from below. Calculations for Grohne's inflexion-point profile show both major and minor curves of neutral stability for 0 < η [les ] 0.0554; for\[ 0.0554 < \eta < 0.0773 \]there is only a single curve of neutral stability; and, for η > 0.0773, the curves of neutral stability become closed, with complete stabilization being achieved for a value of η of about 0·107.

The stability of the asymptotic suction boundary layer profile

The stability of the asymptotic suction boundary layer profile

Three-Dimensional Boundary Layer with Uniform Mass Transfer

The three-dimensional laminar boundary layer near the windward generator of a cone at an angle of attack subjected to uniform mass transfer, either suction or injection, is studied by means of a series solution. The important wall values which permit application of the results to the prediction of skin-friction, heat transfer, and surface streamline divergence are tabulated for a wide variety of Mach numbers, wall temperature ratios, and values of a parameter related to the angle of attack of the cone. For suction the results are shown to approach a three-dimensional, asymptotic suction boundary layer. For injection indications of “blow off,” and its behavior with the aforementioned parameters are found.

On the stability of the asymptotic suction boundary-layer profile

This paper presents a discussion of some aspects of the linear stability problem for the asymptotic suction profile. An exact solution of the inviscid equation is first obtained in terms of the usual hypergeometric function and its analytical continuation. This exact solution provides both a corrected version of an earlier treatment by Freeman and an independent check on the more general method suggested for solving the inviscid equation numerically. Various approximations to the characteristic equation, and hence to the curve of neutral stability, are then considered. In particular, it is found that, in a consistent asymptotic treatment of the related adjoint problem, at least one viscous correction to the singular inviscid solution must be considered. Based on the present results for the adjoint problem, it is suggested that Tollmien's original treatment of the viscous corrections must be slightly modified.