On the initial phase of the triple deck stage in marginally separated flows

The method of matched asymptotic expansions is used to investigate marginally separated boundary layer flows (laminar or alternatively transitional separation bubbles) at high Reynolds numbers. Typical examples include, among others, the flow past slender airfoils at small to moderate angels of attack and channel flows with suction. As is well-known, classical (hierarchical) boundary layer computations usually break down under the action of an adverse pressure gradient on the flow, a scenario associated with the appearance of the Goldstein separation singularity. If, however, the parameter controlling the strength of the pressure gradient (the angle of attack or the relative suction rate in the examples mentioned above) is adjusted accordingly, the application of a local viscous-inviscid interaction strategy is capable of describing localized boundary layer separation. Moreover, taking into account unsteady effects and flow control devices allows the investigation of the conditions leading to forced or self-sustained vortex generation and the subsequent evolution process culminating in bubble bursting. Within the asymptotic formulation of this stage bubble bursting is associated with the formation of finite time singularities in the solution of the underlying equations and a corresponding break down. The distinct blow-up structure gives rise to a fully non-linear triple deck interaction stage featuring shorter spatio-temporal scales characteristic of the successive vortex evolution process. The paper will focus on the numerical treatment of the initial phase of the latter stage. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

On higher order effects in marginally separated flows

If the angle of attack α of a slender airfoil reaches a critical value α_s flow separation is known to occur at the upper s surface. Further increase of α initially leads to the formation of a short laminar separation bubble which has an extremely weak influence on the external flow field – a phenomenon known as marginal separation – but then rather rapidly causes a severe change of the flow behaviour, leading to leading edge stall. According to the asymptotic theory of marginal separation holding in the limit of large Reynolds numbers Re, the flow in the neighbourhood of the separation bubble is governed by an integro-differential equation. This so-called interaction equation contains a single controlling parameter which relates the angle of attack to the Reynolds number, with a value Γ_s corresponding to α_s. Some recent results concerning higher order s s corrections to this theory and their effect on the stability of steady solutions will be presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Wave propagation in a mixed convection boundary‐layer over a horizontal surface

The propagation of disturbances in a mixed convection boundary‐layer flow over a horizontal plate is described by a triple deck problem in the case of the buoyancy parameter being small. The pressure correction in the lower deck consists of two parts: One due to the buoyancy effects in the main deck and one due to the displacement of the outer flow field. The response of the boundary layer flow to an oscillator of frequency ω will be computed and upstream travelling waves will be identified.     

Receptivity of the boundary layer with separation bubble to external sound waves

The receptivity problem of laminar separated two-dimensional boundary-layer under influence of external acoustic waves is considered. Basic features of this theory were formulated by Ruban (1984) and then Goldstein (1985), but these investigations were limited by small perturbations of steady parallel flow only. The current paper takes into account essentially nonparallel regimes of base flow, including attached flows, flow with separated bubbles and the base flow at marginal value of parameter. The steady flow near a corner point of profile is of considerable interest and a question naturally arises about the sensitivity of separation bubble inside the boundary layer to external disturbances such as sound waves. Another question arises about the sensitivity of near critical base flow, namely if corner is concave there is a marginal angle which limits existence of such base flow. This fact usually is linked with sudden reconstruction of full flow pattern past a profile. Development of perturbations inside the interaction region may give some answers about the appearance of turbulence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unified description of bifurcation processes associated with laminar boundary-layer separation

The presentation will concentrate on flows where no steady state exists if an appropriately defined controlling parameter exceeds a critical value while non-uniqueness is observed for sub-critical values of this parameter. Special attention is placed on flow phenomena which are associated with the passage through criticality. Based on a triple deck analysis it found that they can be described as solutions of differential equations of Fisher type which are better known from evolution studies of gene populations. Special examples which will be discussed include 2D marginally separated flows, weakly 3D transonic flows in slender channels and fully 3D subsonic flow past expansion ramps. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shock boundary layer interactions in dense gases

The concept of triple deck theory is applied to study laminar interacting boundary layers of dense gases in external purely supersonic flow. An impinging shock is generated at distances which are large compared to the upper deck thickness. As predicted by weakly nonlinear theory such a discontinuity may disintegrate into a sonic shock and an associated wave fan depending on its amplitude and the magnitude of the so called fundamental derivative. Incoming and outgoing waves are computed analytically by means of the method of multiple scales taking into account that mutual interaction effects between them are restricted to the linear upper deck region. The lower deck problem is solved numerically. The results show that it is possible to reduce the size of the separation bubble or even to avoid the occurrence of flow separation by choosing an optimal thermodynamic state. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Long Eddies in Sheared Flows

The characteristic feature of the wide variety of hydraulic shear flows analyzed in this study is that they all contain a critical level where some of the fluid is turned relative to the ambient flow. One example is the flow produced in a thin layer of fluid, contained between lateral boundaries, during the passage of a long eddy. The boundaries of the layer may be rigid, or flexible, or free; the fluid may be either compressible or incompressible. A further example is the flow produced when a shear layer separates from a rigid boundary producing a region of recirculating flow. The equations used in this study are those governing inviscid hydraulic shear flows. They are similar in form to the classical boundary layer equations with the viscous term omitted. The main result of the study is to show that when the hydraulic flow is steady and contained between lateral boundaries, the variation of vorticity ω(ψ) cannot be prescribed at any streamline which crosses the critical level. This variation is, in fact, determined by (1) the vorticity distribution at all streamlines which do not cross the critical level, by (2) the auxiliary conditions which must be satisfied at the boundaries of the fluid layer, and by (3) the dimensions of the region containing the turned flow. If at some instant the vorticity distribution is specified arbitrarily at all streamlines, generally the subsequent flow will be unsteady. In order to emphasize this point, a class of exact solutions describing unsteady hydraulic flows are derived. These are used to describe the flow produced by the passage of a long eddy which distorts as it is convected with the ambient flow. They are also used to describe the unsteady flow that is produced when a shear layer separates from a boundary. Examples are given both of flows in which the shear layer reattaches after separation and of flows in which the shear layer does not reattach. When the shear layer vorticity distribution has the form ωαyⁿ, where y is a distance measure across the layer, the steady flows are of Falkner-Skan type inside, and adjacent to, the separation region. The unsteady flows described in this paper are natural generalizations of these Falkner-Skan flows. One important result of the analysis is to show that if the unsteady flow inside the separation region is strongly sheared, then the boundary of the separation region moves upstream towards the point of separation, forming large transverse currents. Generally, the assumption of hydraulic flow becomes invalid in a finite time. On the other hand, if the flow inside the separation region is weakly sheared, this region is swept downstream and the flow becomes self-similar.     

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.     

Structure of weak laminar hydraulic jumps in single layer and two layer fluids

We consider the occurrence of hydraulic jumps in near critical single layer and two layer flows under the assumption that viscous effects are confined to a thin laminar boundary layer adjacent to the solid boundary. In the limit of large Reynolds number this leads to a structure problem formed by the classical triple deck equations supplemented with a novel nonlinear coupling condition which allows for the passage through the critical state. In the case of positive hydraulic jumps this passage is achieved by the local thickening of the boundary layer which acts as a viscous hump. Conversely, the pressure drop at the wall associated with negative hydraulic jumps causes the boundary layer to decrease locally thereby forming a local indentation required for the Froude number to pass through one in this case.     

Spatial Stability for Mixed Convection Boundary Layer over a Heated Horizontal Plate

The spatial stability properties of a mixed convection boundary layer developing over a heated horizontal plate is studied here under linear and quasi-parallel flow assumption. The main aim of the present work is to find out if there is a critical buoyancy parameter that would indicate the importance of heat transfer in destabilizing mixed convection boundary layers, when the buoyancy effect is given by Boussinesq approximation. The undisturbed flow used here is that given by the similarity solution of [ 1 ] that implies the wall temperature to vary as the inverse square root of the distance from the leading edge of the plate. The stability of this flow has been investigated by using the compound matrix method (CMM)—that allows finding all the modes in the chosen range in the complex wave number plane for spatial stability analysis. Presented neutral curves for mixed convection boundary layer show the existence of two types of disturbances present simultaneously, for large buoyancy parameter. One notices very unstable high-frequency mode when the buoyancy parameter exceeds the above-mentioned critical value. This unstable thermal mode is in addition to the hydrodynamic mode of isothermal flow given by corresponding similarity profile. The calculated critical buoyancy parameter is shown to qualitatively match with experimental results.     

Effect of surface elasticity on the transformation of acoustic disturbances into Tollmien-Schlichting waves in a boundary layer at transonic free-stream velocities

Within the framework of the triple deck theory, the effect of an elastic surface on the characteristics of a wave packet generated by acoustic disturbances in a boundary layer at transonic free-stream velocities is investigated.     

DNS of laminar separation nubble flows with free-stream fluctuations

A series of Direct Numerical Simulations (DNS) of Laminar Separation Bubble (LSB) flow in the presence of external disturbances has been performed. In all simulations, the primary mechanism for the transition to turbulence was found to be a two-dimensional Kelvin-Helmholtz (KH) instability of the separated shear layer that was triggered by the external fluctuation(s). The KH instability caused the shear layer to roll up. Inside the rolled up shear layer, entrained disturbances triggered elliptic instabilities which led to a rapid transition to fully three-dimensional (3D) turbulence. In the simulations with inflow oscillations the streamwise variation of the location of transition was found to decrease with the amplitude of the inflow oscillation. In the simulations with free-stream turbulence the size of the separation bubble – measured by the shape factor – was found to drastically decrease with increasing free-stream turbulence level. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Numerical and asymptotic study of non‐axisymmetric magnetohydrodynamic boundary layer stagnation‐point flows

Both numerical and asymptotic analyses are performed to study the similarity solutions of three‐dimensional boundary‐layer viscous stagnation point flow in the presence of a uniform magnetic field. The three‐dimensional boundary‐layer is analyzed in a non‐axisymmetric stagnation point flow, in which the flow is developed because of influence of both applied magnetic field and external mainstream flow. Two approaches for the governing equations are employed: the Keller‐box numerical simulations solving full nonlinear coupled system and a corresponding linearized system that is obtained under a far‐field behavior and in the limit of large shear‐to‐strain‐rate parameter (λ). From these two approaches, the flow phenomena reveals a rich structure of new family of solutions for various values of the magnetic number and λ. The various results for the wall stresses and the displacement thicknesses are presented along with some velocity profiles in both directions. The analysis discovered that the flow separation occurs in the secondary flow direction in the absence of magnetic field, and the flow separation disappears when the applied magnetic field is increased. The flow field is divided into a near‐field (due to viscous forces) and far‐field (due to mainstream flows), and the velocity profiles form because of an interaction between two regions. The magnetic field plays an important role in reducing the thickness of the boundary‐layer. A physical explanation for all observed phenomena is discussed. Copyright © 2017 John Wiley & Sons, Ltd.     

On Transonic Marginal Separation

Laminar boundary‐layer separation near the leading edge of a thin airfoil is one of the principal factors that limits the lift force acting on the airfoil. Marginal separation, in particular, denotes the onset of separation which is accompanied by the formation of a short separation bubble. Using asymptotic analysis this effect is studied in the limit of high Reynolds number and for transonic external flow conditions. It is assumed that the fluid under consideration is a perfect gas and the airfoil surface is taken to be thermally insulated. Results to be presented include the analytical investigation of the emerging three layer structure, the associated transonic far field and the calculation of representative wall shear stress distributions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hydromagnetische Stabilität laminarer Grenzschicht an einer längsangeströmten ebenen Platte

Summary The purpose of the paper is to consider the stability for wavelike disturbances in the steady, twodimensional, laminar boundary layer of a magnetic field, which is applied uniformly normal to the flat plate. The results show that the critical Reynolds number (R _c ^* ) increases remarkably with the characteristic parameter (). The increase of the critical Reynolds number depends not only on the shape parameter of the velocity distribution in the boundary layer but also on the peculiarity of the velocity profile. It is also found that the boundary layer holds itself laminar all over the flat plate, when the magnetic parameterN is greater than 1.25×10^–7, then a reduction of the skin-frictin drag might be expeced.     

Numerical study of forced and free convective boundary layer flow of a magnetic fluid over a flat plate under the action of a localized magnetic field

The two-dimensional, steady, laminar, forced and free convective boundary layer flow of a magnetic fluid over a semi-infinite vertical plate, under the action of a localized magnetic field, is numerically studied. The magnetic fluid is considered to be water-based with temperature dependent viscosity and thermal conductivity. The study of the boundary layer is separated into two cases. In case I the boundary layer is studied near the leading edge, where it is dominated by the large viscous forces, whereas in case II the boundary layer is studied far from the leading edge of the plate where the effects of buoyancy forces increase. The numerical solution, for these two different cases, is obtained by an efficient numerical technique based on the common finite difference method. Numerical calculations are carried out for the value of Prandl number Pr =  49.832 (water-based magnetic fluid) and for different values of the dimensionless parameters entering into the problem and especially for the magnetic parameter Mn, the viscosity/temperature parameter Θ _r and the thermal/conductivity parameter S*. The analysis of the obtained results show that the flow field is influenced by the application of the magnetic field as well as by the variation of the viscosity and the thermal conductivity of the fluid with temperature. It is hoped that they could be interesting for engineering applications.