Stuart, which uncovered physical mechanisms of nonlinearities that are crucial to the present developing boundary layers supporting streamwise vortices and their efficient scalar transporting mechanisms. Gortler vortices in a boundary layer in fluid dynamics, gortler vortices are secondary flows that appear in a boundary layer flow along a concave wall. Saric mechanical and aerospace engineering, arizona state university, tempe, arizona 852876106 key words. Boundary layer flows are of critical importance to the motion and physics of a fluid flow. For example, in the case of a boundary layer flow over a threedimensional obstacle or the flow over a. In this paper, we study secondary instability of hypersonic gortler vortices using twodimensional linear stability analysis and numerical simulations. May 21, 2008 the basic underlying pinions of the nonlinear hydrodynamic stability problem are the analyses of j. Control of gortler vortices in highspeed boundary layers. Secondary instability of crossflow vortices in falkner. Nonlinear stability of nonstationary crossflow vortices in.
Pdf a formal analysis of goertlertype instability is presented. This type of vortices will be called longitudi nal vortices in shear layers. There has been little previous theoretical work directed toward this problem and here we are able to characterize the important features of vortices over the complete wavenumber spectrum. Transition experiments in a boundary layer with embedded. We apply the integral method, used already by aihara for the taylorgortler instability,4 and we assume the constant boundarylayer thickness. The development of gortler vortices in boundary layers over curved walls in the nonlinear regime is investigated. Nonlinear development of viscous gertler vortices in a three.
Primary and secondary instabilities of the asymptotic suction boundary layer on a curved plate by daniel s. Related content a new kind of hairpinlike vortical structure induced by crossinteraction of sinuous streaks in turbulent channel jian li, gang dong, hui zhang et al. The particular model we consider consists of the boundary layer flow over a heated or cooled curved rigid body. The formal analysis of the gortlertype instability is presented. Depending on various geometrical and flow conditions, these vortices can significantly distort the baseline flow and lead to secondary instabilities and ultimately to transition. Numerical works on gortler vortex instability were initially focused to find a. Curved ramp models with a radius of curvature of 25. It is well known that the flow within a boundary layer over a curved plate is susceptible to longitudinal vortices. On the existence and character of gortler vortices in.
The wavelength of vortices was preset by vertical wires of 0. This distorted flow is susceptible to strong, high frequency secondary instability that leads to the onset of transition. Temporal stability of hypersonic boundary layer on porous wall. To investigate the stability of vortices generation in realworld case, i tracked a ringlike vortex and a leglike vortex over time from our direct numerical simulation dns data. Stability of gortler vortices in boundary layers aiaa. Shear layer vortices and longitudinal vortices in the near. It is commonly thought that coherent vorti cal structures passing near the wall can induce eruptions see, for example, robinson. Secondary goertler instability in hypersonic boundary layers. The vortical field was generated by a series of jets at 45 degrees with respect to the wall and in a plane transversal to the mean flow direction.
Turbulent boundary layer to gortler vortices on a concave wall. An important contribution was made by hall 29, when he first proposed a more rigorous approach to this problem. Analogy with the longitudinal momentum problem for unity prandtl and schmidt numbers is explored. Stability of boundary layers along curved surfaces nasaads. The inviscid and viscous stability problems are addressed for a boundary layer which can support both gortler and crossflow vortices. A sar image scaled for presentation acquired from radarsat1 cband, hh on 17 january 1997 at 2242 utc off the east coast of the united states when the mabl stratification was unstable. Virginia polytechnic institute and state university.
These vortices are called gortler vortices and affect the flow instability in boundary layers along the concave surface. On the other hand, if the boundary layer thickness is comparable to the radius. In hall 2 the linear evolution of these modes within the context of boundary layers was solved. The nonlinear development of inviscid gortler vortices in a threedimensional boundary layer is considered. It has also been shown that if the underlying basic flow is threedimensional, these modes will eventually be extinguished perhaps in favour of crossflow vortices. Taylorgortler vortices in fully developed or boundary. Gortler instability of compressible boundary layers. Taylor gortler vortices are streamwise vortices having alternate right and lefthand rotation that may develop in the laminar boundary layer along a concave surface. The stability of a boundary layer along a curved wall was first studied by gortler 27 in 1940.
Gortler had used a simple approximation which allowed him to reduce the problem to a set of ordinary differential equations. We do not follow the classical approach of weakly nonlinear stability problems and consider a mode which has just become unstable. Gortler vortices have been studied experimentally and numerically since gortler. Stability of gortler vortices in boundary layers aiaa journal. Pdf stability of gortler vortices in boundary layers researchgate. Rotation can be either stabilizing or destabilizing corresponding to whether a rotation number is negative or positive, respectively. Vortices and stability in superfluid boundary layers. We confine our attention to vortex type instabilities commonly referred to as gortler vortices which have been identified as one of the dominant mechanisms of instability in both centrifugally and buoyancy driven boundary layers. Hairpin vortices in turbulent boundary layers to cite this article. The change in structure of gortler vortices is found when the parameter representing the degree of threedimensionality of the basic boundarylayer flow under consideration is increased. Liepmann i conducted boundary layer transition ex periments. Nonlinear stability of nonstationary crossflow vortices. The leading edge receptivity problem, linear growth and the nonlinear breakdown stage 26 february 2010 mathematika, vol. Gortler instability of compressible boundary layers aiaa.
Transition is not caused directly by these structures, but by the growth of small. It is shown that for vortices of wavelength comparable to the boundary layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. Boundarylayer stability and transition the proceedings. New stability equations for disturbances experiencing streamwise variation were derived by employing the propagation theory under the linear stability analysis. The onset of convective instability of the laminar boundary layer flow over a slightly curved wall was analyzed under the nonparallelism of a base flow. Growth and walltranspiration control of nonlinear unsteady. Instead we extend the method of blackaby, dando, and hall 1992, which considered the closely related nonlinear development of disturbances in stratified. Liepmann i conducted boundarylayer transition ex periments on concave walls and showed that the crit. Gortler vortices were always present when the wall boundary layers were laminar. Rng modeling techniques for complex turbulent flows 3544. Primary and secondary instabilities of the asymptotic. These vortices evolve over a similar scale to the mixing layer and have spanwise wavelength commensurate with the layers thickness, which renders the governing equations parabolic in nature.
Based on feedback from you, our users, weve made some improvements that make it easier than ever to read thousands of publications on our website. The equations which describe a boundary layer on a curved wall in a rotating system are derived and a linear stability analysis of a basic blasius velocity profile is performed. A formal analysis of goertlertype instability is presented. The greater the degree of deceleration, the larger the growth rate of the disturbance. The present paper is a description of measurements of the mean and turbulent properties of decaying vortices and vortex pairs. May 17, 2012 gortler instability of compressible boundary layers. This study investigates the stability and transition of gortler vortices in a hypersonic boundary layer using linear stability theory and direct numerical simulations. The vortices appear to be completely immersed in the boundary layer, whose thickness for the natural flow is about 43 mm at x 250 mm. Figure 4 b, which again shows growth rates for various values of the buoyancy parameter but this time gloc d.
Vortex instabilities in threedimensional boundary layers. In this paper, receptivity of gortler and other instability waves in hypersonic boundary layers are investigated using direct numerical simulation dns. One of the main differences between the stability of two and threedimensional incompressible boundary layers is the occurrence of crossflow instability in the latter. Stability of boundarylayer flows over curved walls.
The basic underlying pinions of the nonlinear hydrodynamic stability problem are the analyses of j. Gortler vortices have been studied experimentally and numerically since gijrtler found them in 1940 l61. Aps fluid dynamics, november 2018 e ect of wall temperature on gortler vortices november 2018 1 16. Intermittent instability induced by longwavelength klebanoff modes. The boundary layer equations and the disturbance equations are formulated in a general, orthogonal, curvilinear system of coordinates constructed from the inviscid flow over a curved surface. Experimental results provide a somewhat inconsistent picture, but in general suggest the existence of vortices for the curvature parameter 5r. The result shows that, with the generation over time, both ringlike vortex and leglike vortex become more stable in the fact of decreasing positive eigenvalues. Pdf stability of gortler vortices in boundary layers. In superfluid helium ii, the problem of boundary layer flow highlights a significant difference between the two fluid components that make up the current model of helium ii. Images in planes perpendicular and parallel to the wall were. The growth of the boundary layer makes a parallelflow analysis impossible except in the highwavenumber regime so in general the instability equations must be integrated numerically. Forcing disturbances are introduced into the hypersonic boundary layers by receptivity to free stream disturbances and blowing and suction. Development and breakdown of goertler vortices in high speed.
The nonlinear development of gortler vortices in growing. Dimensional boundary layer nonlinear development of viscous gertler vortices in a three. Temporal stability of multiplecell vortices shear flow. Gortler vortices in boundary layer flows over concave surfaces are caused. Pdf linear and nonlinear instabilities of blasius boundary. An important feature of steady streamwise vortices within a shear layer is the convection of streamwise momentum in the normal and spanwise directions by very weak velocity components of the vorticity.
The nonlinear evolution of inviscid goertler vortices in. The generation, nonlinear evolution, and walltranspiration control of unsteady gortler vortices in. Effect of wall transpiration and heat transfer on nonlinear g\o. The stability of a boundary layer on a concave wall to gortler vortices that propagate downstream is also considered. A formal analysis of gortlertype instability is presented. Development and breakdown of gortler vortices in high speed. Stability of gortler vortices in boundary layers aiaa arc. Pdf hypersonic boundary layer transition on a concave. Scalar transport in a longitudinal vorticity system, exemplified by nonlinear spatially developing gortler vortices, is discussed.
The boundary layer and disturbance equations are formulated in a general. Pdf stability of boundarylayer flows over curved walls. Gortler vortices in curved mixing layers and their effect on. Marine atmospheric boundary layer cellular convection and longitudinal roll vortices 323 figure 14. Imbedded longitudinal vortices in turbulent boundary layers. Effect of wall temperature on the growth of gortler.
The growth of the boundary layer makes a parallel flow analysis impossible except in the high wavenumber regime so in general the instability equations must be integrated numerically. Following a brief overview of sweptwing instability mechanisms and the crossflow problem, a summary of the important findings of the 1990s is given. Secondary instability of stationary crossflow vortices in. The boundarylayer and disturbance equations are formulated in a general. The wellknown gortler instability 7 is an important mode within curved boundary layers. Direct numerical simulation of gortler instability in. Choudhari, and pedro paredesgonzalez langley research center, hampton, virginia lian duan missouri university of science and technology, rolla, missouri national aeronautics and space administration langley research center.
The effects of gortler vortices on boundary layer development, heat transfer. Transient growth of gortler vortices in twodimensional. The two counter rotating vortices in the central part of the flat plate, approximately symmetric with respect to the z 0 plane, are visible. Secondary instability of stationary crossflow vortices in mach 6 boundary layer over a circular cone fei li, meelan m. Here the spanwise dependence of the basic flow is described using fourier series expansion whilst. Developing an accurate and efficient method for viscous compressible flow simulations an example of cfd in aeronautics 522. Nonlinear instability of developing streamwise vortices. Nonlinear instability of developing streamwise vortices with. Zang7 t computational methods branch, nasa langley research center, hampton, va 23665, usa 1department of mathematics, exeter university. The onset of gortler vortices in laminar boundary layer flow. In the rapidexpansion slotted nozzle the vortices persisted far downstream of the transition point when the bleed slot control valves were open. Linear instability the pioneering work for boundary layers on constantcurvature concave walls was carried out by g.
Compressible stability of growing boundary layers using parabolized stability equations. The second and more important objective of the study is to determine the secondary linear instability properties of the resulting temporally evolving gortler vortices. Coriolis effects on gortler vortices in the boundarylayer. Longitudinal counter rotating vortices appear in boundary layer flow along the concave surface.
Stability and transition analysis code lastrac version 1. Turbulent boundary layer manipulation by longitudinal. On the secondary instability of gortler vortices in three. Vortices were generated inside the boundary layer by means of roughness elements arranged in a regular array along the spanwise z. Scalar transport in a longitudinal vorticity system in.
For high mach number flows, the boundary layer is also subject to the second mode instability. Pdf on the nonlinear interaction of goertler vortices and. These modes are found to be more stable than the usual timeindependent modes and they propagate downstream with the speed of the basic flow in the critical layer. He considered the stability of a parallel blasius boundary layer developing over a. The study of vortex induced erup tions in laminar flow could lead to insights into the regen eration process of turbulent boundary layers. American institute of aeronautics and astronautics 12700 sunrise valley drive, suite 200 reston, va 201915807 703. Nonlinear stability and transition in 3d boundary layers. With the bleed valves closed, the wall boundary layer was tripped by. This produces large changes in the mean velocity profiles. Nonlinear gortler vortices and their secondary instability in. The boundarylayer and disturbance equations are formulated in a general, orthogonal, curvilinear.
The boundarylayer and disturbance equations are formulated in a general, orthogonal, curvilinear system of coordinates constructed from the inviscid flow over a curved surface. Linear and nonlinear instabilities of blasius boundary layer perturbed by streamwise vortices. Our concern here is with the stability of fully threedimensional boundary layers rather than twodimensional boundary layer flows. Primary and secondary instabilities of the asymptotic suction. Close to the wall, a region of colliding spanwise jets is evident for. Zurigat 1 mechanical engineering department university of jordan, amman, jordan m. If the boundary layer is thin compared to the radius of curvature of the wall, the pressure remains constant across the boundary layer. Boundarylayer flow over a concave wall is subject to the gortler instability.
Gajjar institute for computational mechanics in propulsion lewis research center cleveland, ohio and the university of manchester manchester, england may 1995 nasacr1b91b4 nonlinear stability of nonstationary crossflow vortices in compressible. Effects of curvature on the boundarylayer flow are analyzed. That is, the skewinduced vortices can influence the flow for a very long distance downstream. Nonlinear mechanics of wavy instability of steady longitudinal vortices and its effect on skin friction rise in boundary layer flow physics of fluids, vol. Interaction of spanwise vortices with a boundary layer.
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