Attachment-line Flow Research Articles

Boundary layer transition and linear modal instabilities of hypersonic flow over a lifting body

Boundary layer transition over a lifting body of 1.6 m length at $2^\circ$ angle of attack has been simulated at Mach 6 and a unit Reynolds number $1.0 \times 10^7$ m$^{-1}$. The model geometry is the same as the Hypersonic Transition Research Vehicle designed by the China Aerodynamics Research and Development Center. Four distinct transitional regions are identified, i.e. windward vortex region, shoulder vortex region, windward cross-flow region and shoulder cross-flow region. Multi-dimensional linear stability analyses by solving the two-dimensional eigenvalue problem (spatial BiGlobal approach) and the plane-marching parabolized stability equations (PSE3D approach) are further carried out to uncover the dominant instabilities in the last three regions as well as the shoulder attachment-line region. The shoulder vortex is conducive to both inner and outer modes of shear-layer instability, of which the latter most likely trigger the vortex breakdown. A novel method is presented to substantially reduce the computational cost of BiGlobal and PSE3D in resolving the cross-flow instabilities in cross-flow regions. The peak frequencies of cross-flow modes lie between 15 and 45 kHz. Whereas oblique second Mack modes are marginally unstable in the windward cross-flow region, they could be strong enough to compete with the cross-flow modes in the shoulder cross-flow region. In the shoulder attachment-line region, there exists only one unstable mode of Mack instability, differing from previous studies that show a hierarchy of modes in the context of symmetrical attachment-line flows. Results of the numerical simulation and multi-dimensional stability analyses are compared when possible, showing a fair agreement between the two approaches and highlighting the necessity of considering non-parallel effects.

Near-wall Lamb vector and its temporal–spatial evolution in the viscous sublayer of wall-bounded flows

This paper gives the relation between the Lamb vector and fundamental surface quantities in the vicinity of a no-slip flat wall using the Taylor-series expansion solution of the Navier–Stokes equations for incompressible viscous flow. The wall-normal component of the Lamb vector is dominated by the boundary enstrophy at the first order and the boundary enstrophy flux at the second order. The tangential Lamb vector is contributed by the terms related to both the boundary vorticity divergence and the skin friction divergence. Then, the derived relation is validated in the three-dimensional nonorthogonal Hiemenz flow (an attachment line flow) and a single-phase turbulent channel flow simulated using the lattice Boltzmann method. For this Hiemenz flow where the boundary vorticity divergence vanishes, the skin friction divergence-related term dominates the distribution of the near-wall tangential Lamb vector. In the turbulent channel flow, both the skin friction divergence and boundary vorticity divergence-related terms have significant contributions to the streamwise component of the Lamb vector, which are associated with the strong wall-normal velocity events (SWNVEs) in the viscous sublayer. In contrast, the boundary vorticity divergence-related terms have the main contribution to the spanwise component of the Lamb vector. Furthermore, the temporal–spatial evolution of the kinetic energy of the Lamb vector (half of the inner product of the Lamb vector denoted by J) is studied. In the turbulent channel flow, the SWNVEs are the main contributors to the temporal–spatial evolution of J in the viscous sublayer. This evolution is dominated by the viscous dissipation effect due to the gradient of the Lamb vector and the coupling effect between a viscous source term and the Lamb vector. The relations presented in this paper could be useful in understanding the physical mechanisms of the initial formation and evolution of the Lamb vector in the viscous sublayer of wall-bounded turbulence.

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.

Structures of Skin Friction, Surface Pressure, and Boundary Enstrophy Flux in Attachment-Line Flow

Abstract The on-wall structures in an attachment-line flow are discussed based on an exact solution of the Navier–Stokes (NS) equations, focusing on distinct features of skin friction, surface pressure, and boundary enstrophy flux (BEF) and intrinsic connections between them.

Measurement and modelling of the turbulent boundary layer near the attachment line of a swept wing

ABSTRACTThis work is motivated by the need for low-order aerodynamic models to predict accurately the effect on profile drag of controlling attachment line transition. Head's entrainment method(1), a rapid integral boundary layer technique used for design studies on swept wings, suffers from the governing swept-tapered turbulent integral boundary layer equations being ill-posed in the vicinity of the attachment line. This singularity has been treated using crude extrapolations of the attachment-line similarity solution for over half a century, but this approach is unlikely to deliver accurate predictions of the effect of changes in the attachment line flow on profile drag. An experimental study has been carried out to explore the nature of the turbulent flow in the vicinity of a highly swept swept attachment line and has revealed a quite complex, non-monotonic development of the momentum thickness in this region. It has also revealed lower levels of twist in the boundary layer velocity profiles than anticipated from the highly curved character of the inviscid flow streamlines. These observations have prompted an alternative approach to the modelling of the flow in this region which not only successfully eliminates the lack of robustness in the swept-tapered equations but which also matches the experimental results to within ±5%.

Spatially developing secondary instabilities in compressible swept airfoil boundary layers

Two-dimensional eigenvalue analysis is used on a massive scale to study the spatial instabilities of compressible shear flows with two inhomogeneous directions. The main focus of the study is crossflow dominated swept-wing boundary layers although the methodology can also be applied to study other types of flows, such as the attachment-line flow. Certain unique aspects of formulating a spatial, two-dimensional eigenvalue problem for the secondary instability of finite amplitude crossflow vortices are discussed, namely, fixing the spatial growth direction unambiguously through a non-orthogonal formulation of the linearized disturbance equations. A primary test case used for parameter study corresponds to Numerical results are presented for the low-speed, NLF-0415(b) airfoil configuration as tested in the ASU Unsteady Wind Tunnel, wherein a spanwise periodic array of roughness elements was placed near the leading edge in order to excite stationary crossflow modes with a specified fundamental wavelength. The two classes of flow conditions selected for this analysis include those for which the roughness array spacing corresponds to either the naturally dominant crossflow wavelength, or a subcritical wavelength that serves to reduce the growth of the naturally excited dominant crossflow modes. Numerical predictions are compared with the measured database, both as indirect validation for the spatial instability analysis and to provide a basis for comparison with a higher Reynolds number, supersonic swept-wing configuration. Application of the eigenvalue analysis to the supersonic configuration reveals that a broad spectrum of stationary crossflow modes can sustain sufficiently strong secondary instabilities as to potentially cause transition over this configuration. In particular, the control mode itself, if initiated with too large an amplitude, may lead to an earlier transition.

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 instabilities of the incompressible attachment-line flow using sparse matrix Jacobi-Davidson techniques

We consider the linear stability of incompressible attachment-line flow within the spatial framework. No similarity or symmetry assumptions for the instability modes are introduced and the full two-dimensional representation of the modes is used. The perturbation equations are discretized on a two-dimensional staggered grid. A high order finite difference scheme has been developed which gives rise to a large, sparse, quadratic, eigenvalue problem for the instability modes. The benefits of the Jacobi–Davidson method for the solution of this eigenvalue system are demonstrated and the approach is validated in some detail. Spatial stability results are presented subsequently. In particular, instability predictions at very high Reynolds numbers are obtained which show almost equally strong instabilities for symmetric and antisymmetric modes in this regime.

On the stability of attachment-line boundary layers. Part 1. The incompressible swept Hiemenz flow

The stability of the incompressible attachment-line boundary layer is studied by solving a partial-differential eigenvalue problem. The basic flow near the leading edge is taken to be the swept Hiemenz flow which represents an exact solution of the Navier-Stokes (N-S) equations. Previous theoretical investigations considered a special class of two-dimensional disturbances in which the chordwise variation of disturbance velocities mimics the basic flow and renders a system of ordinary-differential equations of the Orr-Sommerfeld type. The solution of this sixth-order system by Hall, Malik & Poll (1984) showed that the two-dimensional disturbance is stable provided that the Reynolds number . Both symmetric and antisymmetric two- and three-dimensional eigenmodes can be amplified. These unstable modes with the same spanwise wavenumber travel with almost identical phase speeds, but the eigenfunctions show very distinct features. Nevertheless, the symmetric two-dimensional mode always has the highest growth rate and dictates the instability. As far as the special two-dimensional mode is concerned, the present results are in complete agreement with previous investigations. One of the major advantages of the present approach is that it can be extended to study the stability of compressible attachment-line flows where no satisfactory simplified approaches are known to exist.

Spatial stability of incompressible attachment-line flow

Linear stability analysis of incompressible attachment-line flow is presented within the spatial framework. The system of perturbation equations is solved using spectral collocation. This system has been solved in the past using the temporal approach and the current results are shown to be in excellent agreement with neutral temporal calculations. Results amenable to direct comparison with experiments are then presented for the case of zero suction. The global solution method utilized for solving the eigenproblem yields, aside from the well-understood primary mode, the full spectrum of least-damped waves. Of those, a new mode, well separated from the continuous spectrum is singled out and discussed. Further, relaxation of the condition of decaying perturbations in the far-field results in the appearance of sinusoidal modes akin to those found in the classical Orr-Sommerfeld problem. Finally, the continuous spectrum is demonstrated to be amenable to asymptotic analysis. Expressions are derived for the location, in parameter space, of the continuous spectrum, as well as for the limiting cases of practical interest. In the large Reynolds number limit the continuous spectrum is demonstrated to be identical to that of the Orr-Sommerfeld equation.

Temperature and suction effects on the instability of an infinite swept attachment line

It is known that the incompressible, infinite swept attachment line flow is unstable to streamwise disturbances that originate in the boundary layer when the cross-flow exceeds a critical magnitude. Furthermore, a small degree of suction at the surface has a significant stabilizing influence while a small degree of blowing has a considerable destabilizing influence. This paper investigates the stabilizing and destabilizing effects of, respectively, cooling or heating the plate and the competing or enhancing effects of suction or blowing. A nonorthogonal flow with respect to the attachment line is also considered by adding a component of shear to the mean flow.

In-flight measurement of static pressures and boundary-layer state with integrated sensors

Advanced wing designs for transport aircraft using adaptive geometries for performance optimization and load control need reliable integrated sensors for feedback to the control system. More efficient air data sensing, eliminating the pitot-static piping, is also required, as in highly integrated modern flight management systems. Absolute pressure transducers from different manufacturers were tested and adapted to the flight-test environment. In laboratory and different flight tests, their applicability for steady measurements was proven in comparison to scanivalves. It was shown that sensing of pressure fluctuations in the buffet regime will improve the prediction of operational buffet limits. The pressure transducers were also used to investigate the laminar/ turbulent transition in the attachment-line flow of a swept wing in comparison to hot-film probes and were found to be an effective tool for such an experiment.

On the instability of a three-dimensional attachment-line boundary layer: weakly nonlinear theory and a numerical approach

The instability of a three-dimensional attachment-line boundary layer is considered in the nonlinear regime. Using weakly nonlinear theory, it is found that, apart from a small interval near the (linear) critical Reynolds number, finite-amplitude solutions bifurcate subcritically from the upper branch of the neutral curve. The time-dependent Navier–Stokes equations for the attachment-line flow have been solved using a Fourier–Chebyshev spectral method and the subcritical instability is found at wavenumbers that correspond to the upper branch. Both the theory and the numerical calculations show the existence of supercritical finite-amplitude (equilibrium) states near the lower branch which explains why the observed flow exhibits a preference for the lower branch modes. The effect of blowing and suction on nonlinear stability of the attachment-line boundary layer is also investigated.

On the effect of wing taper and sweep direction on leading edge transition

Summary An experimental study has been made of the conditions necessary to produce transition in a swept attachment-line boundary layer which is subjected to disturbances from trip wires of various sizes. The boundary layer considered is that formed on a long, untwisted, tapered wing which has been tested in both the backward swept and forward swept configurations. Results from these tests are compared with previous work on an untapered model. Some important similarities and some puzzling differences have been found. Finally consideration is given to the practical implications of this work. It is suggested that a forward swept wing may be capable of supporting a laminar attachment-line flow at a much higher free-stream Reynolds number than a corresponding swept-back wing.

Unsteady attachment line flow on a flat plate with attached cylinder

The unsteady laminar incompressible boundary-layer attachment-line flow on a flat plate with attached cylinder with heat and mass transfer has been studied when the free stream velocity, mass transfer and surface wall temperature vary arbitrarily with time. The governing partial differential equations with three independent variables have been solved numerically using an implicit finite-difference scheme. The heat transfer was found to be strongly dependent on the Prandtl number, variation of wall temperature with time and dissipation parameter (for large times). However, the free stream velocity distribution and mass transfer affect both the heat transfer and skin friction.

Attachment-Line Flow on an Infinite Swept Wing

A proposed method for calculating three-dimensional incompressible laminar and turbulent boundary layers is investigated with respect to its applicability to incompressible attachment line flow on an infinite swept wing. The calculation results obtained exhibit satisfactory agreement with experimental data.

The Calculation of the Three-Dimensional Turbulent Boundary Layer: Part III. Comparison of Attachment-Line Calculations with Experiment

SummaryWind tunnel measurements have been made of the attachment-line boundary layer on a swept wing. The combination of a 4-5 in (114 mm) nose radius and 60° angle of sweep enabled a wide range of values of the parameter V2/(v dU/dx) (denoted here by C*), to be covered, and gave a thick enough boundary layer for the velocity profiles to be accurately measured. Pitot traverses were performed for values of C* from 0·59 × 105 to 3·7 × 105. Without a trip wire fitted the attachment-line flow was laminar over the full range and the measured velocity profiles agreed closely with the theory for an infinite swept wing. With a large trip wire fitted, a transition régime was observed in the range 0·6 × 105 < C* < 1·4 × 105, and at higher values of C* the flow was fully turbulent and showed good agreement with the earlier calculations of Cumpsty and Head. Preston tubes were used for skin friction measurements in fully turbulent conditions and a hot wire was used to explore the development of turbulence in the transition régime.

The Calculation of Three-Dimensional Turbulent Boundary Layers: Part II: Attachment-Line Flow on an Infinite Swept Wing

SummaryAn earlier paper described a method of calculating the turbulent boundary layer flow over the rear of an infinite swept wing. It made use of an entrainment equation and momentum integral equations in streamwise and cross-flow directions, together with several auxiliary assumptions. Here the method is adapted to the calculation of the turbulent boundary layer flow along the attachment line of an infinite swept wing. In this case the cross-flow momentum integral equation reduces to the identity 0 = 0 and must be replaced by its differentiated form. Two alternative approaches are also adopted and give very similar results, in good agreement with the limited experimental data available. It is found that results can be expressed as functions of a single parameter C*, which is evidently the criterion of similarity for attachment-line flows.