Boundary-layer Stability Theory Research Articles

Toward Cost-Effective Boundary Layer Transition Computations With Large-Eddy Simulation

An efficient large-eddy simulation (LES) approach is investigated for laminar-to-turbulent transition in boundary layers. This approach incorporates the boundary-layer stability theory. Primary instability and subharmonic perturbations determined by the boundary-layer stability theory are assigned as forcing at the inlet of the LES computational domain. This LES approach reproduces the spatial development of instabilities in the boundary layer, as observed in wind tunnel experiments. Detailed linear growth and nonlinear interactions that lead to the H-type breakdown are well captured and compared well to previous direct numerical simulation (DNS). Requirements in the spatial resolution in the transition region are investigated with connections to the resolution in turbulent boundary layers. It is shown that the subgrid model used in this study is apparently dormant in the overall transitional region, allowing the right level of the growth of small-amplitude instabilities and their nonlinear interactions. The subgrid model becomes active near the end of the transition where the length scales of high-order instabilities become smaller in size compared to the given grid resolution. Current results demonstrate the benefit of the boundary-layer forcing method for the computational cost reduction.

Triple-deck theory in transonic flows and boundary layer stability

An analysis of the lower branch of the neutral curve for the Blasius boundary layer leads to a perturbed velocity field with a triple-deck structure, which is a rather unexpected result. It is the asymptotic treatment of the stability problem that has a rational basis, since it is in the limit of high Reynolds numbers that the basic flow has the form of a boundary layer. The principles for constructing a boundary layer stability theory based on the triple-deck theory are proposed. Although most attention is focused on transonic outer flows, a comparative analysis with the asymptotic theory of boundary layer stability in subsonic flows is given. The parameters of internal waves near the lower branch of the neutral curve are associated with a certain perturbation field pattern. These parameters satisfy dispersion relations derived by solving eigenvalue problems. The dispersion relations are investigated in complex planes.

Estimating amplitude ratios in boundary layer stability theory: a comparison between two approaches

We first demonstrate that, if the contributions of higher-order mean flow are ignored, the parabolized stability equations (Bertolotti et al. 1992) and the ‘full’ non-parallel equation of Govindarajan & Narasimha (1995, hereafter GN95) are both equivalent to order R−1 in the local Reynolds number R to Gaster's (1974) equation for the stability of spatially developing boundary layers. It is therefore of some concern that a detailed comparison between Gaster (1974) and GN95 reveals a small difference in the computed amplitude ratios. Although this difference is not significant in practical terms in Blasius flow, it is traced here to the approximation, in Gaster's method, of neglecting the change in eigenfunction shape due to flow non-parallelism. This approximation is not justified in the critical and the wall layers, where the neglected term is respectively O(R−2/3) and O(R−1) compared to the largest term. The excellent agreement of GN95 with exact numerical simulations, on the other hand, suggests that the effect of change in eigenfunction is accurately taken into account in that paper.

Analysis of ghost numerical solutions of differential equation caused by nonlinear instability

Jaffe, N. A., Okamura, T. T., and Smith, A. M. O., Determination of Spatial Amplification Factors and Their Application to Predicting Transition, AIAA Journal, Vol. 8, 1970, pp. 301-308. Malik, M. R., and Control of in Supersonic and Hypersonic Boundary Layers, AIAA Journal, Vol. 27, No. 11, 1989, pp. 1487-1493. Fisher, D. F., and Dougherty, N. S., Jr., In-Flight Measurement on a 10 deg Cone at Mach Numbers from 0.5 to 2.0, NASA TP 1971, June 1982. Masad, J. A., and Malik, M. R., Transition Correlation in Subsonic Flow over a Flat Plate, AIAA Journal, Vol. 31, No. 10, 1993, pp. 1953-1955. Masad, J. A., and lyer, V, Transition Prediction and Control in Subsonic Flow over a Hump, Physics of Fluids, Vol. 6, No. 1, 1994, pp. 313327. Mack, L. M., Boundary-Layer Stability Theory, Jet Propulsion Lab., Document 900-277, Rev. A., California Inst. of Technology, Pasadena, CA, May 1969. Chen, F.-J., Malik, M. R., and Beckwith, I. E., Boundary-Layer on a Cone and Flat Plate at Mach 3.5, AIAA Journal, Vol. 27, No. 6, 1989, pp. 687-693. difference schemes are applied

Remarks on disputed numerical results in compressible boundary-layer stability theory

Past work on the influence of Mach number on the viscous and inviscid instability of flat-plate boundary layers is reviewed, and new spatial calculations are presented. These calculations support the previous view that viscosity is only stabilizing for both two- and three-dimensional first-mode waves above M1=3.0, and for second-mode waves at all Mach numbers. It is concluded that the calculations of Wazzan, Taghavi, and Keltner that show viscous instability at M1=6.0 for first-mode 50° waves, and at M1=3.0 for two-dimensional second-mode waves, are not correct.

Higher eigenstates in boundary-layer stability theory

Using the Orr-Sommerfeld equation with the wavenumber as the eigenvalue, a search for higher eigenstates in the stability theory of the Blasius boundary layer has revealed the existence of a number of viscous states in addition to the long established fundamental state. The viscous states are discrete, belong to two series, and are all heavily damped in space. Within the limits of the investigation the number of viscous states existing in the layer increases as the Reynolds number and the angular frequency of the perturbation increase. It is suggested that the viscous eigenstates may be responsible for the excitation of some boundary-layer disturbances by disturbances in the free stream.

Non-local effects in the stability of flow between eccentric rotating cylinders

In this paper the linear stability of the flow between two long eccentric rotating circular cylinders is considered. The problem, which is of interest in lubrication technology, is an extension of the classical Taylor problem for concentric cylinders. The basic flow has components in the radial and azimuthal directions and depends on both of these co-ordinates. As a consequence the linearized stability equations arepartial differential equationsrather than ordinary differential equations. Thus standard methods of stability theory are not immediately useful. However, there are two small parameters in the problem, namely δ, the clearance ratio, and ε, the eccentricity. By letting these parameters tend to zero in such a way that δ½ is proportional to ε, a global solution to the stability problem is obtained without recourse to the concept of ‘local instability’, or ‘parallel-flow’ approximation, so commonly used in boundary-layer stability theory. It is found that the predictions of the present theory are at variance with what is given by a ‘local’ theory. First, the Taylor-vortex amplitude is found to be largest at about 90° downstream of the region of ‘maximum local instability’. This result is given support by some experimental observations of Vohr (1968) with δ = 0·1 and ε = 0·475, which yield a corresponding angle of about 50°. Second, the critical Taylor number rises with ε, rather than initially decreasing with ε as predicted by local stability theory using the criteria of maximum local instability. The present prediction of the critical Taylor number agrees well with Vohr's experiments for ε up to about 0·5 when δ = 0·01 and for ε up to about 0·2 when δ = 0.1.

Interaction of the wake from bluff bodies with an initially laminar boundary layer.

An experimental investigation was conducted to determine the mechanism and conditions governing transition induced by small bluff bodies suspended in a laminar boundary layer. The experiment was performed in the inlet section of a tube. The mechanism by which transition occurred appeared to be a local effect dependent upon the stability characteristics of the element wake rather than the stability characteristics of the boundary layer over the range of boundary-layer Reynolds numbers examined. That is, transition of the boundary layer appeared to involve the direct seeding of turbulence in the boundary layer from the wake of the element, rather than amplification of wavelike disturbances as considered in classical boundary-layer stability theory. In the central region of the boundary layer, the critical value of the element Reynolds number which resulted in transition of the boundary layer coincided, approximately, with the Reynolds number for incipient transition in the wake of the element. Laminar vortices shed by disturbance elements at subcritical values of element Reynolds numbers were found to decay in propagating downstream. For elements located outside the boundary layer, transverse turbulence contamination of the boundary layer was found to depend on diffusion of the turbulent wake below a critical (trigger) value of y/d. Turbulent patches were generated by elements traversing the boundary layer when the instantaneous slip velocity Reynolds number was slightly higher than the value of critical Reynolds number for elements fixed within the boundary layer.

Further study on flame stabilization in a boundary layer: A mechanism of flame oscillations

In this paper a mechanism of flame oscillation is proposed in an attempt to explain the often-sought flame oscillation phenomena in a boundary layer and also to bring together with a unified theory the two distinct types of flame stabilization in a boundary layer. These types are: (1) the steady flame observed by Hottel et al. and (2) the oscillating flame observed by Gross, Ziemer, and Turcotte. This proposed hypothesis postulates that the oscillation phenomenon of the flame in a boundary layer is caused by the instability of the boundary layer flow. The adverse pressure gradient immediately upstream of the flame nose in the boundary layer and the heat transfer to the unburned mixture from the solid boundary make the boundary-layer flow highly unstable. This instability, combined with the basic principles of flame stabilization and propagation, sets off the oscillation. A comparison of the experimental conditions with the existing boundary-layer stability theory indicates that the boundary layer in the experiments is in the unstable region. Evidences are provided to support the theory. A further study of the oscillation characteristics of the flame was made by means of high-speed motion pictures. It shows that the oscillating pattern of the flame repeats itself exactly and the advance speed and the oscillation frequency of the flame become higher when the plate temperature increases, while the retreat speed remains constant. The advance speed remains constant when the amplitude of oscillation is changed. Furthermore, the steady flame observed by Hottel et al. was also obtained by the authors at a very low-plate temperature and flow velocity. All these phenomena can be fully explained by the proposed hypothesis. As discussed, some of the present results agree with the observations of previous investigators, but some do not. The proposed hypothesis is further employed to examine and alleviate some of the doubts concerning the use of the equilibrium theory to correlate oscillating-flame data by a group of parameters. It is found that though the reason for choosing these parameters is different from that of the previous investigators, our results are quite similar. The correlation is readily explained qualitatively by the steady-flame theory.