Stability Of Boundary-layer Flow Research Articles

Stability of boundary-layer flow over a skin made of porous compliant wall and micro floating raft arrays

ABSTRACT To improve the drag reduction performance of underwater vehicles, a skin composed of a porous compliant wall and micro floating raft elements is proposed. A two-dimensional stability model of boundary-layer flow over the compliant skin is established, and the effect of structural parameters on the flow stability is investigated. It is found that enlarging the spacing l attenuates the Tollmien-Schlichting Instability (TSI) mode but promotes the occurrence of the Compliance-Induced Flow Instability (CIFI) mode, while the permeability coefficient a presents an opposite effect on the instability modes. Moreover, the CIFI mode is stabilised by increasing the stiffness ratio β, damping ratio γ and intermediate mass m, within limits. The skin proposed in this paper with specific parameters can better reduce the arising of TSI and CIFI modes in boundary-layer flow compared with the corresponding rigid and compliant walls, leading to a favourable performance for delaying boundary-layer transition.

On the stability of boundary-layer flow over a rotating cone in an enforced axial free stream

Studies into the boundary-layer stability of rotating cones in an axial free stream have considered both cases of broad-angled cones and slender-angled ones. This distinction is made due to the findings that the dominant instability in the quiescent-fluid case results from different physical mechanisms for broad and slender cones. These investigations made use of an Orr-Sommerfeld-type analysis derived from the assumption of a small perturbation to a similarity-type solution for the mean fluid flow around the cone. In this work, we investigate the efficacy of a combination of sophisticated numerical techniques to find the basic flow, using ideas both from automatic differentiation and spectral collocation methods, to maximise the accuracy of the obtained solution. We inspect the solutions obtained for the Orr-Sommerfeld system in this setting and compare them to the results from previous studies.

On the stability of boundary-layer flow over a rotating cone using new solution methods

In this study, a new solution is applied to the model problem of boundary-layer flow over a rotating cone in still fluid. The mean flow field is perturbed leading to disturbance equations that are solved via a more accurate spectral numerical method involving Chebyshev polynomials, both of which are compared with previous numerical and analytical approaches. Importantly, favourable comparisons are yielded with existing experiments [17] and theoretical investigations [6] in the literature. Meanwhile, further details will be provided of potential comparisons with new experiments currently in the pipeline.Physically, the problem represents a model of airflow over rotating machinery components at the leading edge of a turbofan. In such applications, laminar-turbulent transition within the boundary layer can lead to significant increases in drag, resulting in negative implications for fuel efficiency, energy consumption and noise generation. Consequently, delaying transition to turbulent flow is seen as beneficial, and controlling the primary instability may be one route to achieving this. Ultimately, control of the input parameters of such a problem may lead to future design modifications and potential cost savings.Our results are discussed in terms of existing experimental data and previous stability analyses on related bodies. Importantly, broad-angled rotating cones are susceptible to a crossflow instability [6], visualised in terms of co-rotating spiral vortices, whereas slender rotating cones have transition characteristics governed by a centrifugal instability [9], which is visualised by the appearance of counter-rotating Görtler vortices. We investigate both parameter regimes in this study and comment on the accuracy of the new solution method compared with previous methods of solving the stability equations.

Stability of boundary layer flow based on energy gradient theory

The flow of the laminar boundary layer on a flat plate is studied with the simulation of Navier–Stokes equations. The mechanisms of flow instability at external edge of the boundary layer and near the wall are analyzed using the energy gradient theory. The simulation results show that there is an overshoot on the velocity profile at the external edge of the boundary layer. At this overshoot, the energy gradient function is very large which results in instability according to the energy gradient theory. It is found that the transverse gradient of the total mechanical energy is responsible for the instability at the external edge of the boundary layer, which induces the entrainment of external flow into the boundary layer. Within the boundary layer, there is a maximum of the energy gradient function near the wall, which leads to intensive flow instability near the wall and contributes to the generation of turbulence.

On the stability of boundary-layer flows over rotating spheroids

We study linear convective instabilities within of the boundary-layer flows over spheroids rotating in otherwise still fluids. Particular spheroids within the prolate and oblate families are considered, each characterized by an eccentricity parameter, 0⩽e⩽0.7. Viscous and streamline-curvature effects are included and local analyses conducted at latitudes between 10°–70° from the axis of rotation. Both travelling and stationary convective modes of type I (crossflow) and type II (streamline curvature) are found at each latitude within specific parameter spaces. The results of existing rotating-sphere investigations are reproduced at all latitudes in the limit of zero eccentricity.In the prolate case, eccentricity is found to have a stabilizing effect on the type I mode at all latitudes and a destabilizing effect on the type II mode at latitudes above 50°. Eccentricity is therefore seen to be a destabilizing influence for low rotation rates (where instability occurs at high latitudes only and the type II mode dominates) and a stabilizing effect for high rotation rates (where instability occurs closer to the pole and the type I mode dominates). This effect is associated with the behavior of the local curvature of the prolate geometry as a function of eccentricity. In the oblate case, eccentricity is found to be universally stabilizing to both mode types at all latitudes. The oblate results demonstrate considerably lower sensitivity to eccentricity than the prolate results.The most amplified modes are found to travel at 75% of the surface speed at each latitude and eccentricity for both spheroid families. This is consistent with existing theoretical studies of boundary-layer flows over the related geometries of the rotating-disk, -sphere and -cone, and leads to the prediction of slow vortices over highly-polished bodies.

Stability analysis of a boundary layer over a hump using parabolized stability equations

Parabolized stability equations (PSEs) were used to investigate the stability of boundary layer flows over a small hump. The applicability of PSEs to flows with a small separation bubble was examined by comparing the result with DNS data. It was found that PSEs can efficiently track the disturbance waves with an acceptable accuracy in spite of a small separation bubble. A typical evolution scenario of Tollmien–Schlichting (TS) wave is presented. The adverse pressure gradient and the flow separation due to the hump have a strong effect on the amplification of the disturbances. The effect of hump width and height is also examined. When the width of the hump is reduced, the amplification factor is increased. The height of the hump is found to obviously influence the stability only when it is greater than the critical layer thickness.

Hermite–Padé approximation approach to steady flow of a liquid film with adiabatic free surface along an inclined heat plate

The thermal stability of boundary layer flows of a temperature-dependent viscosity liquid film with adiabatic free surface along an inclined heated plate is studied using a special type of Hermite–Padé approximation approach. This semi-numerical scheme offers some advantages over solutions obtained by using traditional methods such as finite differences, spectral method, shooting method, etc. It reveals the analytical structure of the solution function and the important properties of overall flow structure including velocity field, temperature field, thermal criticality and bifurcations are discussed.

The stability of boundary-layer flows under conditions of intense interfacial mass transfer: the effect of interfacial coupling

We consider the linear stability of a boundary-layer flow over a permeable surface under conditions of intense interfacial mass transfer. The stability of the flow is governed by an eigenvalue problem of Orr–Sommerfeld type coupled to a second-order differential equation for the concentration disturbance field through a flux boundary condition at the permeable surface. Previous studies on this problem have ignored the effect on the stability of the flow of this coupling. Curves of neutral stability and the critical Reynolds number for the flow are obtained. These show that the fully coupled system produce critical Reynolds numbers and wave-numbers that, in some cases, differ significantly from those obtained when the disturbance coupling is ignored.

Analysis of boundary layer control by heat transfer strips using an asymptotic approach to the PSE

The effect of heating strips on the stability of boundary layer flow over a flat plate is investigated. Heating strips alter the flow stability through the temperature dependence of the fluid properties. A stability study is carried out using the parabolized stability equations (PSE) that calculates the effects of temperature dependent fluid properties in terms of asymptotic expansions based on the total heat input. The leading order influence is obtained as a general result and, for the particular Prandtl number taken, is independent of any special set of property laws. In a fluid for which the intrinsic viscosity increases with temperature and the density decreases with temperature (such as air) the results show that the optimal location for a heating strip to stabilise the flow is upstream of the neutral point. The optimal location moves further upstream as the total heat input level is increased. For a given heat input widening the heating strip further stabilises the flow. Finally, the potential of the asymptotic method as a tool for further analysis of the flow is discussed.

The stability of boundary layers on curved heated plates

AbstractWe consider the effect the competing mechanisms of buoyancy-driven acceleration (arising from heating a surface) and streamline curvature (due to curvature of a surface) have on the stability of boundary-layer flows. We confine our attention to vortex type instabilities (commonly referred to as Görtler vortices) which have been identified as one of the dominant mechanisms of instability in both centrifugally and buoyancy driven boundary layers. The particular model we consider consists of the boundary-layer flow over a heated (or cooled) curved rigid body. In the absence of buoyancy forcing the flow is centrifugally unstable to counter-rotating vortices aligned with the direction of the flow when the curvature is concave (in the fluid domain) and stable otherwise. Heating the rigid plate to a level sufficiently above the fluid's ambient (free-stream) temperature can also serve to render the flow unstable. We determine the level of heating required to render an otherwise centrifugally stable flow unstable and likewise, the level of body cooling that is required to render a centrifugally unstableflow stable.

Stability of a spanwise nonuniform boundary layer flow

The linear stability of a boundary layer flow with a spanwise-periodic nonuniformity in the velocity profile is investigated. This flow can be considered as a model of a streaky structure occurring in the boundary layer at a high freestream turbulence level. It is shown that for a small nonuniformity amplitude symmetric modes similar to Tollmien-Schlichting waves are the most unstable. At higher nonuniformity amplitudes, antisymmetric modes, qualitatively different from Tollmien-Schlichting waves and having a larger phase velocity, are the most amplified.

The effects of displacement induced by thermal perturbations on the structure and stability of boundary-layer flows

The free-interaction influence of a thermal expansion process in boundary-layer gas flow is analyzed using the formalism of triple-deck theory. The physical model considered is the forced convection of a gas flowing over a flat plate subject to a heated slab. Both linearized and full nonlinear solutions are obtained using Fourier transform methods and spectral numerical techniques. The influence of monochromatic thermal perturbation on boundary-layer stability (lower branch) is studied and first-order correction of the lower branch neutral stability curve for the boundary-layer flow has been obtained. The shift of neutral stability is then computed for different values of the thermal perturbation wave number, making unstable some otherwise stable modes.

Some recent developments in the theory of boundary layer instability

The stability of boundary-layer flow over a flat-plate is investigated after taking into account the effects of boundary-layer growth. A critical review and analysis of earlier work is presented, mainly for those works that use an inhomogeneous Orr-Sommerfeld equation and a solvability condition to obtain corrections for the growth-rate based on the quasi-parallel (qp) approximation. During the course of this review and analysis an important result is proven; viz., for a ray, the basic spatial periodicity of the disturbance wave is indimensional space. Thereafter the energy integral equation is invoked, and an optimal monitorable property is found that has the same growth-rate as given by the eigensolution of the associated homogeneous problem. This also leads to the optimal normalisation of the eigenfunction at different downstream stations along the plate. A surprising result found is that the past non-parallel results can be virtually totally reproduced based on theqp-approximation and using the present methodology. And, by using the present methodology for the full non-parallel problem, the results obtained are in very much better agreement with past experimental results, and with the results of Smith based on the triple-deck theory.

The stability of boundary-layer flow over single-and multi-layer viscoelastic walls

In this paper, we are concerned with the linear stability of zero pressure-gradient laminar boundary-layer flow over compliant walls which are composed of one or more layers of isotropic viscoelastic materials and backed by a rigid base. Wall compliance supports a whole host of new instabilities in addition to the Tollmien-Schlichting mode of instability, which originally exists even when the wall is rigid. The perturbations in the flow and the compliant wall are coupled at their common interface through the kinematic condition of velocity continuity and the dynamical condition of stress continuity. The disturbance modes in the flow are governed by the Orr-Sommerfeld equation using the locally-parallel flow assumption, and the response of the compliant layers is described using a displacement-stress formalism. The theoretical treatment provides a unified formulation of the stability eigenvalue problem that is applicable to compliant walls having any finite number of uniform layers; inclusive of viscous sublayer. The formulation is well suited to systematic numerical implementation. Results for single- and multi-layer walls are presented. Analyses of the eigenfunctions give an insight into some of the physics involved. Multi-layering gives a measure of control over the stability characteristics of compliant walls not available to single-layer walls. The present study provides evidence which suggests that substantial suppression of disturbance growth may be possible for suitably tailored compliant walls.

Magnetohydrodynamic Stability of Boundary Layer Flow with Suction or Injection along a Flat Plate

AbstractThe flow of an incompressible electrically conducting fluid past, a semi‐infinite flat plate in the presence of a transverse magnetic field is studied considering the effect of uniform suction or injection. The transformed boundary layer equations are obtained, and the numerical calculations of resulting equations are performed for various values of magnetic parameter and suction or injection parameter. The neutral stability curves for wavelike disturbances of Tollmien‐Schlichting type are then presented for the velocity profiles obtained above. The results are given for the velocity profiles, coefficient of skin friction, displacement thickness and the critical Reynolds numbers.

Floquet theory for the stability of boundary layer flows

A Floquet theory is presented for the stability of boundary layer flows. The spectrum of the governing differential operator is partially discrete and partially continuous, which is different from confined flows. However, no instabilities appear from the continuous part of the spectrum.

Expansion problems in the linear stability of boundary layer flows

Several problems in the linearized stability of boundary layers are examined. They are all treated as perturbations of constant coefficient differential operators. Spectral theory and spectral expansions are developed. Possible anomalies, which might arise for nonparallel boundary layer flows with nonzero transverse component at infinity are also handled.

On the stability of boundary-layer flow over a spring-mounted piston

The stability of weakly compressible boundary-layer flow over a spring-mounted piston is examined theoretically by modelling the mean boundary layer at low Strouhal numbers by means of a step-function velocity profile. This constitutes a prototype problem for the treatment of the interaction of unsteady boundary-layer flow with a compliant surface, and the present discussion complements a recent analysis due to Ffowcs Williams and Purshouse by incorporating the influence of flow separation at the edges of the piston. This is effected analytically by application of the unsteady Kutta condition at both the leading and trailing edges of the piston. At high Reynolds numbers and in the case of light fluid loading i t is predicted that the separated flow can cause piston flutter. Stability criteria are derived for a rectangular piston of large aspect ratio.

Buried wire separation detector simulation in compressible flow

^aric, W, S. and Nayfeh, A. H., Nonparallel Stability of Boundary-Layer Flows, Physics of Fluids, Vol. 18, No. 8, 1975, pp. 945-950. Klebanoff, P. S. and Tidstrom, K. D., of Amplified Waves Leading to Transition of Boundary Layer with Zero Pressure Gradient, NASA Tech. Note DN-195, 1959. Kachanov, Y. S., Kozlov, V. V., and Levchenko, V. Y., Wave Evolution in the Boundary Layer, Izvestiya Akademii Nauk SSSR, Mechanika Zhidkosti i Gaza, No. 3, 1977, pp. 49-58. Caster, M., The Physical Processes Causing Breakdown to Turbulence, IUTAM Symposium on Laminar-Turbulent Transition, Abstracts, Stuttgart, Federal Republic of Germany, 1979. Zelman, M. B., On Nonlinear Evolutional Disturbances in Weakly Nonhomogeneous Flows of Unpressured Gas, Gasodinamika i Fizicheskaya Kinetika, ITPM SO AN SSSR, Novosibirsk, U.S.S.R., 1974. Craik, A. D. D., Resonant Instability in Boundary Layers, Journal of Fluid Mechanics, Vol. 50, No. 2, 1971, pp. 392413. Volodin, A. G. and Zelman, M. B., On Resonant TollmienSchlichting Wave Interaction in the Boundary Layer, Sb.Chislennye MetodyMech. Sploshnoy Sredy, Vol. 9, No. 1, 1978, pp. 31-38. Volodin, A. G. and Zelman, M. B., Three-Dimensional Resonant Interaction of Disturbances in the Boundary Layer, Izvestia Akademii Nauk. SSSR, Mechanika Zhidkosti i Gaza, No. 5, 1978, pp. 78-83. Volodin, A. G. and Zelman, M. B., Pairwise Nonlinear Interactions of Tollmien-Schlichting Waves in the Flows Like Boundary Layer, Izvestia Akademii Nauk SSSR, Mechanika Zhidkosti i Gaza, Vol. 2, 1977, pp. 33-37.

The Stability of Water Flow Over Heated and Cooled Flat Plates

The theory of two-dimensional instability of laminar flow of water over solid surfaces is extended to include the effects of heat transfer. The equation that governs the stability of these flows to Tollmien-Schlichting disturbances is the Orr-Sommerfeld equation “modified” to include the effect of viscosity variation with temperature. Numerical solutions to this equation at high Reynolds numbers are obtained using a new method of integration. The method makes use of the Gram-Schmidt orthogonalization technique to obtain linearly independent solutions upon numerically integrating the “modified Orr-Sommerfeld” equation using single precision arithmetic. The method leads to satisfactory answers for Reynolds numbers as high as Rδ* = 100,000. The analysis is applied to the case of flow over both heated and cooled flat plates. The results indicate that heating and cooling of the wall have a large influence on the stability of boundary-layer flow in water. At a free-stream temperature of 60 deg F and wall temperatures of 60, 90, 120, 135, 150, 200, and 300deg F, the critical Reynolds numbers Rδ* are 520, 7200, 15200, 15600, 14800, 10250, and 4600, respectively. At a free-stream temperature of 200F and wall temperature of 60 deg F (cooled case), the critical Reynolds number is 151. Therefore, it is evident that a heated wall has a stabilizing effect, whereas a cooled wall has a destabilizing effect. These stability calculations show that heating increases the critical Reynolds number to a maximum value (Rδ* max = 15,700 at a temperature of TW = 130 deg F) but that further heating decreases the critical Reynolds number. In order to determine the influence of the viscosity derivatives upon the results, the critical Reynolds number for the heated case of T∞ = 40 and TW = 130 deg F was determined using (a) the Orr-Sommerfeld equation and (b) the present governing equation. The resulting critical Reynolds numbers are Rδ* = 140,000 and 16,200, respectively. Therefore, it is concluded that the terms pertaining to the first and second derivatives of the viscosity have a considerable destabilizing influence.