Attachment-line Boundary Layer Research Articles

Instability of the attachment line boundary layer in a supersonic swept flow

Theoretical analysis of attachment-line instabilities is performed for supersonic swept flows using the compressible Hiemenz approximation for the mean flow and the successive approximation procedures for disturbances. The theoretical model captures the dominant attachment-line modes in wide ranges of the sweep Mach number ${M_e}$ and the wall temperature ratio. It is shown that these modes behave similar to the first and second Mack modes in the boundary layer flow. This similarity allows us to extrapolate the knowledge gained for Mack modes to the attachment-line instabilities. In particular, we find that at sufficiently large ${M_e}$ , the dominant attachment-line instability is associated with the synchronisation of slow and fast modes of acoustic nature. Point-by-point comparisons of the theoretical predictions with the experiments of Gaillard et al. (Exp. Fluids, vol. 26, 1999, pp. 169–176) demonstrate that at ${M_e} > 4$ , the theory captures a significant drop of the transition onset Reynolds number, which is below the contamination criterion of Poll $({R_\mathrm{\ast }} = 250)$ at ${M_e} > 6$ . This contradicts the generally accepted assumption that the attachment-line flow is stable for ${R_\mathrm{\ast }} \le 250$ . The theoretical critical Reynolds numbers lie well below the experimental transition-onset Reynolds numbers. Stability computations using the Navier–Stokes mean flow and accounting for the leading-edge curvature effect do not eliminate this discrepancy. Most likely, in the experiments of Gaillard et al., we face with an unknown effect that does not fit to the concept of transition arising from linear instability.

Propagation of stationary and traveling waves in a leading-edge boundary layer of a swept wing

The transition characteristics around the leading edge of a swept-back wing shape were numerically investigated. We conducted direct numerical simulations (DNSs) of a swept-wing shape with a high Reynolds number Re=Rec/cos Λ=5.85×106 based on the chord length with a sweep angle Λ=70°. In the study, a randomly distributed impulsive local body force was applied at the wall to encourage a transition. Through impulsive local forcing, two coherent waves formed in both an attachment line and a three-dimensional boundary layer: A stationary elongated streak structure in the external flow direction and a traveling wave in the sweep direction. These characteristics in the attachment line were slightly different from those in the three-dimensional boundary layer. We computed the nonmodal transient energy growth for the present leading-edge boundary layer and compared the coherent waves observed in the DNSs. The stationary and traveling modes in the DNSs are found to be in a transient growth group; these modes temporally grow to the maximum in the short target time (τ<0.02). One of our conclusions is that both waves occurring in the present attachment line are strongly related to the short-term transient energy growth phenomena of the nonorthogonality of the flow field. When the roughness forcing was gradually increased, the traveling wave was not generated, whereas the stationary wave was. This was considered because the present attachment-line boundary layer was receptive to a small disturbance and more likely to generate a stationary wave than a traveling wave.

Attachment-line heating in a compressible flow

The attachment-line boundary layer on a swept wing can be subject to either an instability or contamination by wing-root turbulence. A model of the attachment-line boundary layer is first developed including compressibility and wall heating in a Falkner–Skan–Cooke class of three-dimensional boundary layers with a Hartree parameter of 1.0. This is verified with other formulations in the literature. For cases otherwise subcritical to either contamination or instability, the destabilizing effect of leading-edge heating under a variety of sweep angles and flight conditions is demonstrated. The results correlate with the attachment-line Reynolds number. Because the required heating levels are reasonable and achievable to trip the flow over the wing to turbulent, one possible application of this work is in the establishment of a baseline turbulent flow (on demand) for the calibration of a laminar-flow-control health-monitoring system.

Linear global instability of non-orthogonal incompressible swept attachment-line boundary-layer flow

AbstractFlow instability in the non-orthogonal swept attachment-line boundary layer is addressed in a linear analysis framework via solution of the pertinent global (BiGlobal) partial differential equation (PDE)-based eigenvalue problem. Subsequently, a simple extension of the extended Görtler–Hämmerlin ordinary differential equation (ODE)-based polynomial model proposed by Theofilis et al. (2003) for orthogonal flow, which includes previous models as special cases and recovers global instability analysis results, is presented for non-orthogonal flow. Direct numerical simulations have been used to verify the analysis results and unravel the limits of validity of the basic flow model analysed. The effect of the angle of attack, $\mathit{AoA}$, on the critical conditions of the non-orthogonal problem has been documented; an increase of the angle of attack, from $\mathit{AoA}= 0$ (orthogonal flow) up to values close to $\lrm{\pi} / 2$ which make the assumptions under which the basic flow is derived questionable, is found to systematically destabilize the flow. The critical conditions of non-orthogonal flows at $0\leq \mathit{AoA}\leq \lrm{\pi} / 2$ are shown to be recoverable from those of orthogonal flow, via a simple algebraic transformation involving $\mathit{AoA}$.

Algebraically decaying modes and wave packet pseudo-modes in swept Hiemenz flow

The modal structure of the swept Hiemenz flow, a model for the flow near the attachment line of a swept wing, consists of eigenfunctions which exhibit (super-)exponential or algebraic decay as the wall-normal coordinate tends to infinity. The subset of algebraically decaying modes corresponds to parts of the spectrum which are characterized by a significant sensitivity to numerical discretization. Numerical evidence further suggests that a continuous spectrum covering a two-dimensional range of the complex plane exists. We investigate the family of uniform swept Hiemenz modes using eigenvalue computations, numerical simulations and the concept of wave packet pseudo-modes. Three distinct branches of the family of algebraically decaying eigenmodes are identified. They can be superimposed to produce wavefronts propagating towards or away from the boundary layer and standing or travelling wave packets in the free stream. Their role in the exchange of information between the free stream and the attachment-line boundary layer for the swept Hiemenz flow is discussed. The concept of wave packet pseudo-modes has been critical in the analysis of this problem and is expected to lead to further insights into other shear flows in semi- or bi-infinite domains.

Experimental investigation of the flow instability near the attachment-line boundary layer on a yawed cylinder

The behavior of small disturbances in a 3-D laminar boundary layer on a yawed cylinder was experimentally investigated. This setup simulates the flow around the leading edge of swept wings. Since multiple instability modes appear near the attachment-line region, a point-source disturbance was artificially introduced to separate these modes. Amplitude and phase distributions of the disturbances originating from the point source were measured using a hotwire probe near the attachment-line flow to test existing theoretical predictions. Hotwire measurements show that two instability modes definitely coexist and overlap in the middle portion of the wedge-shaped region developing downstream of the point source. Decomposition by 2-D fast Fourier transform (FFT) analysis enables us to separate one oblique wave from the other. One of the oblique waves belongs to the cross-flow instability mode, which travels to the attachment line and grows even at Reynolds numbers that are slightly lower than the critical Reynolds number for the attachment-line instability. The origin of the other mode is not identifiable, because it has peculiar characteristics different from both the streamline-curvature instability mode and the cross-flow instability mode. This mode decays in the downstream direction for all frequencies examined. By investigating the spatial characteristics of the small disturbance, the critical Reynolds number for cross-flow instability was successfully determined in the off-attachment-line region. The value, Rc = 543, was lower than the critical Reynolds number of Rc = 583 for the attachment-line flow. Furthermore, the critical frequency and wavenumber were in good agreement with existing predictions based on linear stability theory.

Spatial optimal disturbances in swept attachment-line boundary layers

A formulation based on direct and adjoint parabolized equations is developed to account for the spatial evolution of perturbations in swept attachment-line boundary layers. For sweep Reynolds numbers larger than Re = 100 the dynamics is dominated by a lift-up mechanism which is responsible for large energy amplification by transforming spanwise vortices into spanwise streaks. This mechanism favours steady perturbations with a chordwise scale that quantitatively matches its counterpart for classical Blasius boundary layers.

Optimal energy growth and optimal control in swept Hiemenz flow

The objective of the study is first to examine the optimal transient growth of Görtler–Hämmerlin perturbations in swept Hiemenz flow. This configuration constitutes a model of the flow in the attachment-line boundary layer at the leading-edge of swept wings. The optimal blowing and suction at the wall which minimizes the energy of the optimal perturbations is then determined. An adjoint-based optimization procedure applicable to both problems is devised, which relies on the maximization or minimization of a suitable objective functional. The variational analysis is carried out in the framework of the set of linear partial differential equations governing the chordwise and wall-normal velocity fluctuations. Energy amplifications of up to three orders of magnitude are achieved at low spanwise wavenumbers (. The optimal wall blowing pattern consists of spanwise travelling waves which follow the naturally occurring vortices and qualitatively act in the same manner as a more simple constant gain feedback control strategy.

Instabilities in compressible attachment–line boundary layers

The hydrodynamic stability of the weakly compressible attachment–line boundary layer, with a sweep Mach number ranging from 0.1 to 1.3, is studied using a temporal compressible direct numerical simulation. A flow impinging non-normally onto an infinitely extended flat plate was computed. This complements the study of Hall et al. [Proc. R. Soc. London, Ser. A 395, 229 (1984)] who investigated the linear stability of an incompressible attachment–line boundary layer under the assumption of Görtler–Hämmerlin perturbation modes. In the present work, the base flow is modeled starting from the incompressible swept Hiemenz flow. Using Rayleigh-Jansen Mach number expansions, we obtain a family of base flows parameterized with the sweep Mach number ranging from 0.1 to 1.3. The Reynolds number of the simulation is higher than the incompressible critical Reynolds number, and the plate is adiabatic. Small purely vortical stochastic perturbations are inserted in the boundary layer and followed in time. For Mach numbers up to 0.3, developed velocity and pressure modes are similar to the ones assumed by Görtler and Hämmerlin. The chordwise dependencies of the temperature mode are presented. When increasing the Mach number, the structure of the modes changes; for high Mach numbers, a significantly slower decay of the eigenfunction with wall–normal distance is observed. Above M=0.5, the perturbations are exponentially decaying. This demonstrates the strong stabilizing effect of compressibility in the moderate Mach regime. Furthermore, for the same base flow, a higher exponential growth rate of the perturbation is obtained, if an isothermal wall boundary condition is applied instead of an adiabatic one.

Subcritical instability on the attachment-line of an infinite swept wing

The leading-edge contamination (LEC) problem of an infinite swept wing is shown here as vortex-induced instability. The governing equation for receptivity is presented for LEC in terms of disturbance energy based on the Navier-Stokes equation. The unperturbed shear layer given by the swept Hiemenz boundary-layer solution is two-dimensional and an exact solution of incompressible the Navier-Stokes equation. Thus, the LEC problem is solved numerically by solving the full two-dimensional Navier-Stokes equation. The contamination at the attachment-line is shown by solving a receptivity to a convecting vortex moving outside the attachment-line boundary layer, which triggers subcritical spatio-temporal instability

Receptivity of the inclined-cylinder attachment-line boundary layer to vortex perturbations

Receptivity of the inclined-cylinder attachment-line boundary layer to vortex perturbations

Receptivity of the inclined-cylinder attachment-line boundary layer to vortex perturbations

The receptivity of the boundary layer in the neighborhood of the attachment line of a cylinder inclined to the flow with respect to periodic vortex perturbations frozen into the stream is investigated. The problem considered simulates the interaction between external turbulence and the leading-edge swept wing boundary layer. It is shown that if the direction of the external perturbation vector is almost parallel to the leading edge, then the external perturbations are considerably strengthened at the outer boundary layer edge. This effect can cause laminar-turbulent transition on the attachment line at subcritical Reynolds numbers.

Computation of leading-edge contamination

The leading-edge contamination (LEC) problem of an infinite-swept wing is computed by solving the full two-dimensional Navier–Stokes equation using a very high spectral accuracy compact scheme. The initial condition is given by the steady attachment-line boundary layer forming over the leading-edge of an infinite swept wing or a yawed cylinder. The contamination is shown as due to either a convecting or a stationary vortex far outside the attachment-line boundary layer at the leading edge. This mechanism is proposed to account for the sub-critical instability onset observed for attachment-line boundary layer. (The criticality is for Reynolds number (based on momentum thickness) exceeding 100––an empirically suggested value based on earlier experiments.) The presented results show the problem to be essentially two-dimensional. Inflow and outflow of computational domain are chosen so that we consider cases of both sub- and super-critical instabilities. The receptivity mechanism of leading-edge contamination for both cases is investigated with respect to different parameters. Finally, we have also analyzed the computed solutions by proper orthogonal decomposition (POD) to highlight the low-dimensional nature and to provide a quantitative measure of leading-edge contamination events.

On the linear stability of swept attachment-line boundary layer flow. Part 1. Spectrum and asymptotic behaviour

The temporal stability of swept attachment-line boundary layer flow based on a swept Hiemenz flow model is studied. Starting from the global stability problem and motivated by analytical free-stream solutions, a Hermite expansion is employed in the chordwise coordinate direction which results in coupled local stability problems. A complete study of the temporal spectrum is presented and the discrete and continuous modes are classified according to their symmetry, chordwise polynomial order and asymptotic decay. Uniform, Gortler-Hammerlin and higher-order modes are described in detail. Estimates are given for the location of the continuous spectrum, and bounds are derived for the validity of the linear approximation

On the linear stability of swept attachment-line boundary layer flow. Part 2. Non-modal effects and receptivity

Following the study of the spectral properties of linearized swept Hiemenz flow (see Part 1, Obrist & Schmid 2003) we investigate the potential of swept Hiemenz flow to support transiently growing perturbations owing to the non-normal nature of the underlying linear stability operator. Transient amplification of perturbation energy is found for polynomial orders higher than zero, and a catalytic role of the continuous modes in increasing transient growth is demonstrated. The adjoint stability equations are derived and used in a numerical receptivity experiment to illustrate the scattering of vortical free-stream disturbances into the least stable boundary layer mode.

Large-eddy simulation of subcritical transition in an attachment-line boundary layer

The large-eddy simulation (LES) technique is applied to the subcritical transition to turbulence of a finite-amplitude instability in the attachment-line boundary layer of a swept wing. The three-dimensional swept Hiemenz solution is used to model the base laminar flow along the leading edge of a swept wing. The filtered Navier-Stokes equations are solved numerically using a localized dynamic eddy-viscosity model to parameterize the unresolved scales. Outflow conditions are imposed using the buffer-domain technique, and initial disturbances are introduced through a blowing/suction strip. The linear stage of transition is bypassed due to the finite-amplitude perturbation. The instability growth rate is found to be significantly higher than in previous two-dimensional results, due to the commonly-made Görtler-Hümmerlin assumption that the perturbation has the same spanwise structure as the base flow; this assumption is not supported by our results, most probably due to the finite spanwise length of the blowing/suction strip. As the flow propagates downstream, energy is fed into the spanwise modes, and turbulence generated on the attachment line is transported to other spanwise locations by the mean motion in this direction. The shape factor on the attachment line matches the turbulent boundary layer estimates, while the turbulent mean-velocity profiles on the attachment line deviate from the universal logarithmic behavior due to the appearance of inflectional points, which make the flow susceptible to inviscid instabilities.

Experimental Investigation of High-Enthalpy Effects on Attachment-Line Boundary-Layer Transition

A series of experiments was performed on high-enthalpy effects on hypersonic boundary-layer transition using nitrogen, air, and carbon dioxideas test gases in the T5 hypervelocity shock tunnel. Previous experiments on a sharp cone showed significant high enthalpy effects. This series concerns the attachment line boundary layer on swept cylinders with sweep angles of 60 and 45 deg. The observed trend of transition Reynolds number with enthalpy, which is found to be similar to that in the cone results, shows strong transition delay at the larger sweep angle for carbon dioxide, whereas no significant effect is observed in nitrogen and air. The acoustic wave absorption rate due to relaxation shows a quite similar trend in enthalpy dependence to transition Reynolds number for the carbon dioxide case. This suggests that the dominant effect in delaying transition in the case is vibrational relaxation. The comparisons between the magnitude of the strongest amplification rate from an inviscid linear stability analysis and the absorption rate due to relaxation show that they are of the same order of magnitude for carbon dioxide and that absorption is not significant for nitrogen or air, which supports the stated understanding of the effect of relaxation on transition.

The extended GrtlerHmmerlin model for linear instability of three-dimensional incompressible swept attachment-line boundary layer flow

A simple extension of the classic Gortler–Hammerlin (1955) (GH) model, essential for three-dimensional linear instability analysis, is presented. The extended Gortler–Hammerlin model classifies all three-dimensional disturbances in this flow by means of symmetric and antisymmetric polynomials of the chordwise coordinate. It results in one-dimensional linear eigenvalue problems, a temporal or spatial solution of which, presented herein, is demonstrated to recover results otherwise only accessible to the temporal or spatial partial-derivative eigenvalue problem (the former also solved here) or to spatial direct numerical simulation (DNS). From a numerical point of view, the significance of the extended GH model is that it delivers the three-dimensional linear instability characteristics of this flow, discovered by solution of the partial-derivative eigenvalue problem by Lin & Malik (1996a), at a negligible fraction of the computing effort required by either of the aforementioned alternative numerical methodologies. More significant, however, is the physical insight which the model offers into the stability of this technologically interesting flow. On the one hand, the dependence of three-dimensional linear disturbances on the chordwise spatial direction is unravelled analytically. On the other hand, numerical results obtained demonstrate that all linear three-dimensional instability modes possess the same (scaled) dependence on the wall-normal coordinate, that of the well-known GH mode. The latter result may explain why the three-dimensional linear modes have not been detected in past experiments; criteria for experimental identification of three-dimensional disturbances are discussed. Asymptotic analysis based on a multiple-scales method confirms the results of the extended GH model and provides an alternative algorithm for the recovery of three-dimensional linear instability characteristics, also based on solution of one-dimensional eigenvalue problems. Finally, the polynomial structure of individual three-dimensional extended GH eigenmodes is demonstrated using three-dimensional DNS, performed here under linear conditions.

Flow Past a Swept Wing with a Compliant Surface: Stabilizing the Attachment‐Line Boundary Layer

Many aquatic species such as dolphins and whales have fins, which can be modeled as swept wings. Some of these fins, such as the dorsal fin of a dolphin, are semi‐rigid and therefore can be modeled as a rigid swept wing with a compliant surface. An understanding of the hydrodynamics of the flow past swept compliant surfaces is of great interest for understanding potential drag reduction mechanisms, especially since swept wings are widely used in hydrodynamic and aerodynamic design. In this paper, the flow past a swept wing with a compliant surface is modeled by an attachment‐line boundary layer flow, which is an exact similarity solution of the Navier–Stokes equations, flowing past a compliant surface modeled as an elastic plate. The hydrodynamic stability of the coupled problem is studied using a new numerical framework based on exterior algebra. The basic instability of the attachment line boundary layer on a rigid surface is a traveling wave instability that propagates along the attachment line, and numerical results show that the compliance results in a substantial reduction in the instability region. Moreover, the results show that, although the flow‐field is three‐dimensional, the qualitative nature of the instability suppression is very similar to the qualitative reduction of instability of the two‐dimensional Tollmien–Schlichting modes in the classical boundary‐layer flow past a compliant surface.

The asymptotic structure of the turbulent attachment-line boundary layer

AbstractThe method of matched asymptotic expansions has been applied to analyse the asymptotic structure of the incompressible turbulent attachment-line boundary layer at large Reynolds numbers. Inner and outer solutions were obtained that give the ‘wall-unit’ and ‘velocity-defect’ represention of both the spanwise and chordwise velocity profiles. The predictions from the theory agree with experimental and DNS data in the range of fully developed turbulence.