Asymptotic description of incipient separation bubble bursting

The appearance of short laminar separation bubbles in high Reynolds number (Re) wall bounded flows due to appropriate adverse pressure gradient conditions is usually associated with minor effects on global flow properties (e.g. lift force). However, localized reverse flow regions are known to react very sensitively to perturbations and in further consequence may trigger the laminar-turbulent transition process or even cause global separation. The present investigation of marginally separated boundary layer flows is based on an asymptotic approach Re → ∞. Special emphasis is placed on solutions of the corresponding model equations which blow up within finite time indicating the ejection of a vortical structure and the emergence of shorter spatio-temporal scales reminiscent of the early transition scenario (‘ bubble bursting’ ). Within the framework of marginal separation theory, an alternative adjoint operator method is used to formulate evolution equations governing the viscous-inviscid interaction process in leading and higher order correction required for the study of later stages of the flow development. Their blow up structure specifies the initial condition of and the match to the subsequent triple deck stage. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Modulation Equations for the Asymptotic Suction Velocity Profile and the Ekman Boundary Layer

In this article two types of flows are considered, the asymptotic suction velocity profile, which is a nearly parallel flow, and the Ekman boundary layer, which is a nonparallel flow. The modified Orr-Sommerfeld equation for the asymptotic suction velocity profile, which is the linearized stability equation for this flow, is analyzed and it is shown to have finitely many eigenvalues. In addition, the Ekman boundary layer is considered and the modulation equation for this nonparallel flow is derived for the first time.     

Asymptotic and numerical solutions of three‐dimensional boundary‐layer flow past a moving wedge

We consider a laminar boundary‐layer flow of a viscous and incompressible fluid past a moving wedge in which the wedge is moving either in the direction of the mainstream flow or opposite to it. The mainstream flows outside the boundary layer are approximated by a power of the distance from the leading boundary layer. The variable pressure gradient is imposed on the boundary layer so that the system admits similarity solutions. The model is described using 3‐dimensional boundary‐layer equations that contains 2 physical parameters: pressure gradient (β) and shear‐to‐strain‐rate ratio parameter (α). Two methods are used: a linear asymptotic analysis in the neighborhood of the edge of the boundary layer and the Keller‐box numerical method for the full nonlinear system. The results show that the flow field is divided into near‐field region (mainly dominated by viscous forces) and far‐field region (mainstream flows); the velocity profiles form through an interaction between 2 regions. Also, all simulations show that the subsequent dynamics involving overshoot and undershoot of the solutions for varying parameter characterizing 3‐dimensional flows. The pressure gradient (favorable) has a tendency of decreasing the boundary‐layer thickness in which the velocity profiles are benign. The wall shear stresses increase unboundedly for increasing α when the wedge is moving in the x‐direction, while the case is different when it is moving in the y‐direction. Further, both analysis show that 3‐dimensional boundary‐layer solutions exist in the range −1<α<∞. These are some interesting results linked to an important class of boundary‐layer flows.     

An Exact Navier-Stokes Solution for Three-Dimensional,Spanwise-Homogeneous Boundary Layers

The plane stagnation flow onto (Hiemenz boundary layer, HBL) and the asymptotic suction boundary layer flow over a flat wall (ASBL) are two boundary layer flows for which the incompressible Navier-Stokes equations are amenable to exact similarity solutions. The Hiemenz solution has been extended to swept Hiemenz flows by superposition of a third, spanwise-homogeneous sweep velocity. This solution becomes singular as the chordwise, tangential base flow component vanishes. In this limit, the homogeneous ASBL solution is valid, which however cannot describe the swept Hiemenz flow, because it does not contain any chordwise velocity. This work presents a generalized three-dimensional similarity solution which describes three-dimensional spanwise homogeneously impinging boundary layers at arbitrary wall-normal suction velocities, using a rescaled similarity coordinate. The HBL and the ASBL are shown to be two limits of this solution. Further extensions consist of oblique impingement or different boundary suction directions, such as slip or stretching walls. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Examples of boundary layers associated with the incompressible Navier-Stokes equations

The author surveys a few examples of boundary layers for which the Prandtl boundary layer theory can be rigorously validated. All of them are associated with the incompressible Navier-Stokes equations for Newtonian fluids equipped with various Dirichlet boundary conditions (specified velocity). These examples include a family of (nonlinear 3D) plane parallel flows, a family of (nonlinear) parallel pipe flows, as well as flows with uniform injection and suction at the boundary. We also identify a key ingredient in establishing the validity of the Prandtl type theory, i.e., a spectral constraint on the approximate solution to the Navier-Stokes system constructed by combining the inviscid solution and the solution to the Prandtl type system. This is an additional difficulty besides the wellknown issue related to the well-posedness of the Prandtl type system. It seems that the main obstruction to the verification of the spectral constraint condition is the possible separation of boundary layers. A common theme of these examples is the inhibition of separation of boundary layers either via suppressing the velocity normal to the boundary or by injection and suction at the boundary so that the spectral constraint can be verified. A meta theorem is then presented which covers all the cases considered here.     

Uniform Flow Past an Axially Uniform Viscous Vortex

It is believed that the flow past a tornado causes the formation of smaller vortices which produce the “suction spots” observed along the path of destruction. Here we develop a greatly simplified mathematical model to investigate this phenomenon. An axially uniform vortex is developed by visualizing a circular tube with uniform surface suction of fluid possessing circulation at infinity. This vortex is then perturbed by a uniform flow past it. An inner asymptotic expansion of an E^1/3 radial boundary layer is matched to an outer expansion to obtain a solution. The results show that a stagnation point developing into a secondary vortex is formed in a free shear layer at critical flow conditions. However, it is difficult to apply our results quantitatively because of the difficulty in comparing the axially uniform vortex with a real tornado vortex.     

Qualitative Properties of Separating Flows in the Prandtl Boundary Layer

We study the behavior of solutions to the system of Prandtl boundary layer equations beyond the separation point of the boundary layer. We obtain conditions on the positive pressure gradient which guarantee the attachment of the boundary layer to the streamlined surface after separation. We prove the possibility of controlling the boundary layer by alternating suction and injection.     

On the Generation of Mean Flows by the Interaction of Görtler Vortices and Tollmien-Schlichting Waves in Curved Channel Flows

There are many fluid flows where the onset of transition can be caused by different instability mechanisms which compete in the nonlinear regime. Here the interaction of a centrifugal instability mechanism with the viscous mechanism which causes Tollmien-Schlichting waves is discussed. The interaction between these modes can be strong enough to drive the mean state; here the interaction is investigated in the context of curved channel flows so as to avoid difficulties associated with boundary layer growth. Essentially it is found that the mean state adjusts itself so that any modes present are neutrally stable even at finite amplitude. In the first instance the mean state driven by a vortex of short wavelength in the absence of a Tollmien-Schlichting wave is considered. It is shown that for a given channel curvature and vortex wavelength there is an upper limit to the mass flow rate which the channel can support as the pressure gradient is increased. When Tollmien-Schlichting waves are present then the nonlinear differential equation to determine the mean state is modified. At sufficiently high Tollmien-Schlichting amplitudes it is found that the vortex flows are destroyed, but there is a range of amplitudes where a fully nonlinear mixed vortex-wave state exists and indeed drives a mean state having little similarity with the flow which occurs without the instability modes. The vortex and Tollmien-Schlichting wave structure in the nonlinear regime has viscous wall layers and internal shear layers; the thickness of the internal layers is found to be a function of the Tollmien-Schlichting wave amplitude.     

On the dynamics in a transitional boundary layer

IntroductionIll 1883 Professor Osborne Reynolds published in Philosopl1ical Transactions of the RoyalSociety the outcomes of his flow visua1ization at Manchester. These had shown that whetherthe flow in a pipe was direct to sinuous (or, as nowadays we would say, laminar to turbulent)depended on its Reynolds number. Transition from Iaminar to turbuIent flow becomes animportant probIem i1l fluid mechanics, which has attracted the interest of investigators fOrmore than l00 years. The partic…     

Transonic viscous inviscid interactions in narrow channels

Unsteady as well as steady transonic flows through channels which are so narrow that the classical boundary layer approach fails are considered. As a consequence the properties of the inviscid core and the viscosity dominated boundary layer region can no longer be determined in subsequent steps but have to be calculated simultaneously. The resulting interaction problem for laminar flows is formulated for both perfect and dense gases under the requirement that the channel is sufficiently narrow so that the flow outside the viscous wall layers becomes one-dimensional in the leading order approximation. The latter allows an interpretation of the flow in the core region by means of the theory of one-dimensional transonic inviscid flow through a Laval nozzle while preserving the essential features of the interaction problem associated with the internal structure of pseudoshocks. The sensitivity of a separation bubble caused by a pseudoshock of sufficient strength to perturbations under the condition of choked flow will be demonstrated. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spatial eigenmodes of a boundary layer with a triple-deck velocity field

The beforehand unclear relation between the viscous-inviscid interaction and the instability of viscous gas flows is illustrated using three-dimensional boundary-layer perturbations in the case of sub- and supersonic outer flows. The assumptions are considered under which asymptotic boundary layer equations with self-induced pressure are derived and the excitation mechanisms of eigenmodes (i.e., Tollmien-Schlichting waves) are described. The resulting dispersion relations are analyzed. The boundary layer in a supersonic flow is found to be stable with respect to two-dimensional perturbations, whereas, in the three-dimensional case, the modes become unstable. The increment of growth is investigated as a function of the Mach number and the orientation of the front of a three-dimensional Tollmien-Schlichting wave.     

Effects of suction/blowing on the lower branch modes of the incompressible boundary layer flow due to a rotating-disk

In this study a theoretical approach is pursued to investigate the effects of suction and blowing on the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the incompressible von Karman’s boundary layer flow induced by a rotating-disk. Particular interest is placed upon the short-wavelength, non-linear and nonstationary crossflow vortex modes developing within the presence of suction/blowing at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic framework introduced in [1], the role of suction on the non-linear disturbances of the lower branch described first in [2] for the stationary modes only, is extended in order to obtain an understanding of the behavior of non-stationary perturbations. The analysis using the rational asymptotic technique based on the triple-deck theory enables us to derive initially an eigenrelation which describes the evolution of linear modes. The asymptotic linear modes calculated at high Reynolds number limit are found to be destabilizing as far as the non-parallelism accounted by the approach is concerned, and they compare fairly well with the numerical results generated directly by solving the linearized system with the usual parallel flow approximation. An amplitude equation is derived next to account for the effects of non-linearity. Even though the form of this equation is the same as that of found in [2] for no suction, it is under the strong influence of suction and blowing. This amplitude equation is shown to be adjusted by a balance between viscous and Coriolis forces, and it describes the evolution of not only the stationary but also the non-stationary modes for both suction and injection applied at the disk surface. A close investigation of the amplitude equation shows that the non-linearity is highly destabilizing for both positive and negative frequency waves, though finite amplitude growth of a disturbance having positive frequency close to the neutral location is more effective at destabilization of the flow under consideration. Finally, a smaller initial amplitude of a disturbance is found to be sufficient for the non-linear amplification of the modes in the case of suction, whereas a larger amplitude is required if injection is active on the surface of the disk.     

High performance computation for DNS/LES

The paper is focused on high-order compact schemes for direct numerical simulation (DNS) and large eddy simulation (LES) for flow separation, transition, tip vortex, and flow control. A discussion is given for several fundamental issues such as high quality grid generation, high-order schemes for curvilinear coordinates, the CFL condition for complex geometry, and high-order weighted compact schemes for shock capturing and shock–vortex interaction. The computation examples include DNS for K-type and H-type transition, DNS for flow separation and transition around an airfoil with attack angle, control of flow separation by using pulsed jets, and LES simulation for a tip vortex behind the juncture of a wing and flat plate. The computation also shows an almost linear growth in efficiency obtained by using multiple processors.     

Investigation of secondary flow on an infinite yawed cylinder with 15 per cent. Thick Joukowski profile as section

This paper contains a theoretical investigation of the secondary flow in the laminar incompressible boundary layer on an infinite yawed cylinder with chordwise section as Joukowski profile of 15 per cent. thickness at zero incidence and with homogeneous suction, the suction mass flow coefficient being equal to 0·2085. The secondary flow profiles are obtained at different points of the wing section and for various angles of sweepback. It is found that in favourable pressure gradients and at pressure minimum, the secondary flow profiles have negative values. In regions of adverse pressure gradients after the pressure minimum the secondary flow changes sign from negative to positive values and have points of inflexion. The change of sign starts from the surface and extends to the edge of the boundary layer downstream. At some points in adverse pressure gradients the secondary flow profiles have double points of inflexion and values of both signs simultaneously. It is also found that an adverse pressure gradient produces more powerful secondary flow than a favourable pressure gradient of the same strength.     

Observations on the Role of Vorticity in the Stability Theory of Wall Bounded Flows

The linearized equations for the evolution of disturbances to four wall bounded flows are treated. The flows are plane Couette flow and plane Poiseuille flow, Hagen-Poiseuille pipe flow, and the asymptotic suction profile. By looking at the vorticity it is proved simply that plane Couette flow and Hagen-Poiseuille flow are linearly stable. Further study is made of the structure of the disturbance equation by the introduction of a special vorticity adjoint.     

On the unsteady three-dimensional boundary layer freely interacting with the external stream

Asymptotic equations that define unsteady processes in a three-dimensional boundary layer with self-induced pressure are derived. The pressure gradient under conditions of free interaction is, as usually, calculated not by the solution of the external problem of flow over a body, but on the assumption that it is due to growth of streamline displacement thickness near the body surface. Besides the principal terms, terms of second order of smallness are retained in asymptotic sequencies. If the characteristic dimensions of the free interaction region are the same in all directions in the plane tangent to the body surface, the system of equations defining the thin layer next to the wall must be integrated together with the system which defines the nonviscous stream.