Instabilities in compressible attachment–line boundary layers (original) (raw)
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On the stability of attachment-line boundary layers. Part 1. The incompressible swept Hiemenz flow
Journal of Fluid Mechanics, 1996
The stability of the incompressible attachment-line boundary layer is studied by solving a partial-differential eigenvalue problem. The basic flow near the leading edge is taken to be the swept Hiemenz flow which represents an exact solution of the Navier-Stokes (N-S) equations. Previous theoretical investigations considered a special class of two-dimensional disturbances in which the chordwise variation of disturbance velocities mimics the basic flow and renders a system of ordinarydifferential equations of the Orr-Sommerfeld type. The solution of this sixth-order system by showed that the two-dimensional disturbance is stable provided that the Reynolds number R < 583.1. In the present study, the restrictive assumptions on the disturbance field are relaxed to allow for more general solutions. Results of the present analysis indicate that unstable perturbations other than the special symmetric two-dimensional mode referred to above do exist in the attachment-line boundary layer provided R > 646. Both symmetric and antisymmetric two-and three-dimensional eigenmodes can be amplified. These unstable modes with the same spanwise wavenumber travel with almost identical phase speeds, but the eigenfunctions show very distinct features. Nevertheless, the symmetric twodimensional mode always has the highest growth rate and dictates the instability. As far as the special two-dimensional mode is concerned, the present results are in complete agreement with previous investigations. One of the major advantages of the present approach is that it can be extended to study the stability of compressible attachment-line flows where no satisfactory simplified approaches are known to exist.
Transient Growth Analysis of Compressible Boundary Layers with Parabolized Stability Equations
54th AIAA Aerospace Sciences Meeting, 2016
The linear form of parabolized linear stability equations (PSE) is used in a variational approach to extend the previous body of results for the optimal, nonmodal disturbance growth in boundary layer flows. This methodology includes the non-parallel effects associated with the spatial development of boundary layer flows. As noted in literature, the optimal initial disturbances correspond to steady counter-rotating streamwise vortices, which subsequently lead to the formation of streamwise-elongated structures, i.e., streaks, via a lift-up effect. The parameter space for optimal growth is extended to the hypersonic Mach number regime without any high enthalpy effects, and the effect of wall cooling is studied with particular emphasis on the role of the initial disturbance location and the value of the spanwise wavenumber that leads to the maximum energy growth up to a specified location. Unlike previous predictions that used a basic state obtained from a self-similar solution to the boundary layer equations, mean flow solutions based on the full Navier-Stokes (NS) equations are used in select cases to help account for the viscous-inviscid interaction near the leading edge of the plate and also for the weak shock wave emanating from that region. These differences in the base flow lead to an increasing reduction with Mach number in the magnitude of optimal growth relative to the predictions based on self-similar meanflow approximation. Finally, the maximum optimal energy gain for the favorable pressure gradient boundary layer near a planar stagnation point is found to be substantially weaker than that in a zero pressure gradient Blasius boundary layer.
On perturbed boundary layer flows
1962
The study ranges from the fundamental considerations regarding the qualitative and quantitative effects of the boundary layer to the explicit evaluation of these effects in problems of current interest. The theory for the laminar boundary layer with free-stream Mach number ranging ...
An asymptotic approach to compressible boundary-layer flow
International Journal of Heat and Mass Transfer, 1987
A regular perturbation approach is applied to take into account variable property effects. Of special interest is the effect of the pressure-dependent density. In the framework of an asymptotic theory, compressibility effects are considered as variable property effects. By means of this asymptotic theory the deviation of skin friction and heat transfer results from their incompressible, isothermal values are determined for laminar Falkner-Skan boundary layers. As far as laminar flow is concerned there is no need for any empirical information.
2012
Flow instability in the non-orthogonal swept attachment-line boundary layer is addressed in a linear analysis framework via solution of the pertinent global (BiGlobal) partial differential equation (PDE)-based eigenvalue problem. Subsequently, a simple extension of the extended Görtler-Hämmerlin ordinary differential equation (ODE)based polynomial model proposed by for orthogonal flow, which includes previous models as special cases and recovers global instability analysis results, is presented for non-orthogonal flow. Direct numerical simulations have been used to verify the analysis results and unravel the limits of validity of the basic flow model analysed. The effect of the angle of attack, AoA, on the critical conditions of the non-orthogonal problem has been documented; an increase of the angle of attack, from AoA = 0 (orthogonal flow) up to values close to π/2 which make the assumptions under which the basic flow is derived questionable, is found to systematically destabilize the flow. The critical conditions of non-orthogonal flows at 0 AoA π/2 are shown to be recoverable from those of orthogonal flow, via a simple algebraic transformation involving AoA.
Stability of the compressible laminar boundary layer
Journal of Fluid Mechanics, 1962
S 0"-' "In previous theor tical treatments of the stability of the compressible * laminar boundary layer the effect of the temperature fluctuations on the "0."viscous" (rapidly-vary ) disturbancestoseither Ignored (Lees-Lin), or 1.4 ' is accounted for incom 7 tely (Dunn-Lin). A thorough reexamination of A this problem shows tha temperature fluctuations have a profound influence on both the "inviscid" (slowly-varYing) and viscous disturbances above a 04 Mach number of about 2. 0. -Th*-0e. analysis Includes the effect of 1043 _ :temperature fluctuations on the viscosity and thermal conductivity, and also S~thnoa r.•clt reatm ent a. SSome important results of the present study are: (1) tnstead of being nearly constant across the boundary layer the amplitude of the . inviscid pressure fluctutions decrease* markedly with distance outward
On the stability of attachment-line boundary layers. Part 2. The effect of leading-edge curvature
Journal of Fluid Mechanics, 1997
The stability of the incompressible attachment-line boundary layer is studied by solving a partial-differential eigenvalue problem. The basic flow near the leading edge is taken to be the swept Hiemenz flow which represents an exact solution of the Navier-Stokes (N-S) equations. Previous theoretical investigations considered a special class of two-dimensional disturbances in which the chordwise variation of disturbance velocities mimics the basic flow and renders a system of ordinarydifferential equations of the Orr-Sommerfeld type. The solution of this sixth-order system by showed that the two-dimensional disturbance is stable provided that the Reynolds number R < 583.1. In the present study, the restrictive assumptions on the disturbance field are relaxed to allow for more general solutions. Results of the present analysis indicate that unstable perturbations other than the special symmetric two-dimensional mode referred to above do exist in the attachment-line boundary layer provided R > 646. Both symmetric and antisymmetric two-and three-dimensional eigenmodes can be amplified. These unstable modes with the same spanwise wavenumber travel with almost identical phase speeds, but the eigenfunctions show very distinct features. Nevertheless, the symmetric twodimensional mode always has the highest growth rate and dictates the instability. As far as the special two-dimensional mode is concerned, the present results are in complete agreement with previous investigations. One of the major advantages of the present approach is that it can be extended to study the stability of compressible attachment-line flows where no satisfactory simplified approaches are known to exist.
Effect of suction and cooling on the stability of subsonic and supersonic boundary layers
1989
An investigation is conducted into the effect of cooling and suction on the stability of subsonic flows over two-dimensional roughness elements and°s upersonic flows over flat plates. First, the effect of wall cooling on the two-dimensionaI linear stability of subsonic flows over two-dimensiona| surface imperfections is investigated. Results are presented for flows over smooth humps and backward-facing steps with Mach numbers up to 0.8. The results show that, whereas cooling decreases the viscous instability, it increases the shear-layer instability and hence it increases the growth rates in the separation region. The coexistence of more than one instability mechanism makes a certain degree of wall cooling most effective. For the Mach numbers 0.5 and 0.8, the optimum wall temperatures are about 80% and 60% of the adiabatic wall temperature, respectively. Increasing the Mach number decreases the effectiveness of cooling slightly and reduces the optimum wall temperature. Second, the effect of suction on the stability of compressible flows over backward-facing steps is investigated. Mach numbers up to 0.8 are hydrodynamic applications. For example, for a vehicle having a moderate Reynolds number, application of Iaminar flow control provides a lucrative increase in fuel efficiency". As the time passed by, the prospect of making LFC practical have increased because of many factors that include production of advanced high strength materials, modern fabrication and manufacturing techniques, and super-critical airfoilsß. For attached flows, Iaminar flow control can be obtained by one or a combination of the following methods: suction, heating in water, cooling in air, favorable pressure gradients in two-dimensional or axisymmetric flows, and convex curvature. Good reviews of these techniques and their applications can be found in Refs. 7 and 8. These techniques are also used in the area of boundary-layer control. To efficiently apply LFC one needs to understand how transition occurs. Experiments performed on flat plates identified one possible route for transition from Iaminar flow into a fully developed turbulent flow'. First, two-dimensional Tollmien-Schlichting (T-S) waves grow downstream. Second, three-dimensional unstable waves and vortices develop in the flow. Third, secondary instabilities take place, resulting in either spikes or low-frequency modulations. Finally, turbulent spots form. These spots then get closer to each other, forming a fully developed turbulent boundary layer. Although many characteristics of the second stage can be explained by the secondary instability theory°•'°, the first stage is the only one that is fully understood. The linear stability theory can accurately predict the shape and the growth rate of instability waves. More importantly the linear theory, and especially the non-parallel theory" , can accurately predict the critical locations where these Introduction 2