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Stabilization of unstable two-point boundary value problems, noting field method for avoiding computer storage problems during integration of related differential equations
The linear stability problem for the compressible Ekman boundary layer common to rotating fluids is formulated and the stability properties determined numerically. Three classes of unstable waves are identified (called class A, B, and C), their properties are described. The class C waves have only recently been reported in the literature and are present only in compressible Ekman boundary layers. Most of the calculations presented here are for uranium hexafluoride gas; however, critical Reynolds numbers are also computed for air and ammonia gas. Compressibility is generally found to decrease the critical Reynolds number for each class of wave. A comparison of results for the three different gases shows the stability to be largely unaffected by changes in the gas properties. Maximum growth rate calculations for each wave show the class A and B waves to be the dominant instabilities.
The receptivity to freestream acoustic disturbances and the stability properties of hypersonic boundary layers are numerically investigated for boundary-layer flows over a 5 straight cone at a freestream Mach number of 6.0. To compute the shock and the interaction of the shock with the instability waves, the Navier-Stokes equations in axisymmetric coordinates were solved. In the governing equations, inviscid and viscous flux vectors are discretized using a fifth-order accurate weighted-essentially-non-oscillatory scheme. A third-order accurate total-variation-diminishing Runge-Kutta scheme is employed for time integration. After the mean flow field is computed, disturbances are introduced at the upstream end of the computational domain. The appearance of instability waves near the nose region and the receptivity of the boundary layer with respect to slow mode acoustic waves are investigated. Computations confirm the stabilizing effect of nose bluntness and the role of the entropy layer in the delay of boundary-layer transition. The current solutions, compared with experimental observations and other computational results, exhibit good agreement.
The effect of stationary, finite-amplitude, linear and nonlinear optimal perturbations on the modal disturbance growth in a Mach 6 axisymmetric flow over a 7 deg. half-angle cone with 0:126 mm nose radius and 0:305 m length is investigated. The freestream parameters (M = 6, Re(exp 1) = 18 x 10(exp. 6) /m) are selected to match the flow conditions of a previous experiment in the VKI H3 hypersonic tunnel. Plane-marching parabolized stability equations are used in conjunction with a partial-differential equation based planar eigenvalue analysis to characterize the boundary layer instability in the presence of azimuthally periodic streaks. The streaks are observed to stabilize nominally planar Mack mode instabilities, although oblique Mack mode and first-mode disturbances are destabilized. Experimentally measured transition onset in the absence of any streaks correlates with an amplification factor of N = 6 for the planar Mack modes. For high enough streak amplitudes, the transition threshold of N = 6 is not reached by the Mack mode instabilities within the length of the cone; however, subharmonic first-mode instabilities, which are destabilized by the presence of the streaks, do reach N = 6 near the end of the cone. The highest stabilization is observed at streak amplitudes of approximately 20 percent of the freestream velocity. Because the use of initial disturbance profiles based on linear optimal growth theory may yield suboptimal control in the context of nonlinear streaks, the computational predictions are extended to nonlinear optimal growth theory. Results show that by using nonlinearly optimal perturbation leads to slightly enhanced stabilization of plane Mack mode disturbances as well as reduced destabilization of subharmonic first-mode disturbances.
Streamers can extend to many solar radii but the closed field regions, or helmets, probably never reach higher than 2-4 solar radii. The brightness boundary defining streamers therefore is a boundary between different flow regimes rather than between static plasma and expanding solar wind. It is reasonable to assume that this boundary divides fast coronal hole wind from slow wind. Flow inside this boundary can be studied using simple MHD models and is a type of stagnation flow. We will present examples of what this flow can be like. The flow effects the stability of the underlying helmet, which can be subject to leakage out the cusp similar to the small mass releases observed with the SOHO/LASCO coronagraph. It can also cause the helmet to be more or less susceptible to being carried away in a coronal mass ejection.
Streamers can extend to many solar radii but the closed field regions, or helmets, reach no higher than 2-4 solar radii. The brightness boundary defining streamers is therefore a boundary between different flow regimes rather than between static plasma and expanding solar wind. It is reasonable to assume that this boundary divides fast coronal hole wind from slow wind. Flow inside this boundary can be studied using MHD models and is a type of stagnation flow. We describe such a model that is essentially analytic and show examples of flow solutions within the context and assumptions of the model. The flow affects the stability of the underlying helmet, which can be subject to a leakage out of the cusp that is similar to the small mass releases observed with the SOHO/LASCO coronagraph. It can also cause the helmet to be susceptible to being carried away in a coronal mass ejection. The model therefore also offers a way to study streamer stability.
An investigation is conducted into the determination of the credibility of interacting boundary layers in predicting compressible subsonic flows over smooth surface imperfections. The case of smooth backward-facing steps is considered. The predicted mean flows are compared with those obtained using a Navier-Stokes solver. Moreover, the linear 2-D compressible stability characteristics of both mean flows are compared. The results show that the interacting boundary-layer formulation produces accurate mean flows that yield accurate linear stability characteristics, such as growth rates and amplification factors.
An investigation is conducted into the determination of the credibility of interacting boundary layers in predicting compressible subsonic flows over smooth surface imperfections. The case of smooth backward-facing steps is considered. The predicted mean flows are compared with those obtained using a Navier-Stokes solver. Moreover, the linear 2-D compressible stability characteristics of both mean flows are compared. The results show that the interacting boundary-layer formulation produces accurate mean flows that yield accurate linear stability characteristics, such as growth rates and amplification factors.
Technical memorandum discusses stability of numerical solutions involving semidiscrete approximations to hyperbolic partial differential equations in initial-and-boundary-value problems. Topic of practical significance because hyperbolic partial differential equations arise in mathematical modeling of waves and blasts. Solutions often needed over restricted regions of unbounded spaces. Outer boundaries artificial, introduced only to limit domains of numerical solutions. Conditions at such artificial boundaries cause numerical instabilities that degrade computed solutions.
With a view to the elaboration of the design of Mach 0.83 and 0.97 cruise-speed long-range aircraft employing LFC, a study is conducted of the laminar flow characteristics of supercritical airfoils of blunt leading-edge X88 type, for the case of lightly loaded wings that dispense with leading-edge flaps for low-speed operations. The boundary layer crossflow in the front acceleration zone of these airfoils' upper surface is optimally stabilized by suction in the upstream portion of the zone, yielding a crossflow that is neutrally stable.
Linear stability theory for hypersonic boundary layers is presented. The theory is used to study the effects of real gas (under the assumption of local chemical equilibrium) and small nose bluntness on hypersonic boundary layer stability. It is found that chemical reactions have a stabilizing effect on the first mode instability and a destabilizing effect on the second mode instability in hypersonic boundary layers. There is also a tendency for the second-mode instability to shift to lower frequencies. The effect of small nose bluntness is found to be stabilizing.
The stability of a high-speed, axisymmetric boundary layer is investigated using secondary instability theory and direct numerical simulation. Parametric studies based on the temporal secondary instability theory identify subharmonic secondary instability as a likely path to transition on a cylinder at Mach 4.5. The theoretical predictions are validated by direct numerical simulation at temporally-evolving primary and secondary disturbances in an axisymmetric boundary-layer flow. At small amplitudes of the secondary disturbance, predicted growth rates agree to several significant digits with values obtained from the spectrally-accurate solution of the compressible Navier-Stokes equations. Qualitative agreement persists to large amplitudes of the secondary disturbance. Moderate transverse curvature is shown to significantly affect the growth rate of axisymmetric second mode disturbances, the likely candidates of primary instability. The influence of curvature on secondary instability is largely indirect but most probably significant, through modulation of the primary disturbance amplitude. Subharmonic secondary instability is shown to be predominantly inviscid in nature, and to account for spikes in the Reynolds stress components at or near the critical layer.
The stability of a high-speed, axisymmetric boundary layer is investigated using secondary instability theory and direct numerical simulation. Parametric studies based on the temporal secondary instability theory identify subharmonic secondary instability as a likely path to transition on a cylinder at Mach 4.5. The theoretical predictions are validated by direct numerical simulation at temporally-evolving primary and secondary disturbances in an axisymmetric boundary-layer flow. At small amplitudes of the secondary disturbance, predicted growth rates agree to several significant digits with values obtained from the spectrally-accurate solution of the compressible Navier-Stokes equations. Qualitative agreement persists to large amplitudes of the secondary disturbance. Moderate transverse curvature is shown to significantly affect the growth rate of axisymmetric second mode disturbances, the likely candidates of primary instability. The influence of curvature on secondary instability is largely indirect but most probably significant, through modulation of the primary disturbance amplitude. Subharamonic secondary instability is shown to be predominantly inviscid in nature, and to account for spikes in the Reynolds stress components at or near the critical layer.
Boundary-layer stability experiments are conducted in the Arizona State University Unsteady Wind Tunnel on a 45 deg swept airfoil. The pressure distribution and test conditions are designed to suppress Tollmien-Schlichting disturbances and provide crossflow-dominated transition. The surface of the airfoil is finely polished to a near mirror finish. Under these conditions, submicron surface irregularities cause the naturally occurring stationary crossflow waves to grow to nonuniform amplitudes. Spanwise-uniform stationary crossflow disturbances are generated through careful control of the initial conditions with full-span arrays of micron-high roughness elements near the attachment line. Detailed hot-wire measurements are taken to document the stationary crossflow structure and determine growth rates for the total and individual-mode disturbances. Naphthalene flow visualization provides transition location information. Roughness spacing and roughness height are varied to examine the effects on transition location and all amplified wavelengths. The measurements show that roughness spacings that do not contain harmonics equal to the most unstable wavelength as computed by linear stability theory effectively suppress the most unstable mode. Under certain conditions, subcritical roughness spacing delays transition past that of the corresponding smooth surface.
The stability of viscous, incompressible flow along a streamwise corner, often called the corner boundary layer problem is examined. The semi-infinite boundary value problem satisfied by small amplitude disturbances in the "bending boundary layer' region is obtained. The mean secondary flow induced by the corner exhibits a flow reversal in this region. Uniformly valid "first approximations' to solutions of the governing differential equations are derived. Uniformity at infinity is achieved by a suitable choice of the large parameter and use of an approximate Langer variable. Approximations to solutions of balanced type have a phase shift across the critical layer which is associated with instabilities in the case of two dimensional boundary layer profiles.
The stability of viscous, incompressible flow along a streamwise corner, often called the corner boundary layer problem is examined. The semi-infinite boundary value problem satisfied by small amplitude disturbances in the 'bending boundary layer' region is obtained. The mean secondary flow induced by the corner exhibits a flow reversal in this region. Uniformly valid 'first approximations' to solutions of the governing differential equations are derived. Uniformity at infinity is achieved by a suitable choice of the large parameter and use of an approximate Langer variable. Approximations to solutions of balanced type have a phase shift across the critical layer which is associated with instabilities in the case of two dimensional boundary layer profiles. Previously announced in STAR as N84-17532
Implicit, noniterative, finite difference schemes were recently developed by several authors for multidimensional systems of nonlinear hyperbolic partial differential equations. When applied to linear model equations with periodic boundary conditions those schemes are unconditionally stable (A-stable). As applied in practice the algorithms often face a severe time step restriction. A major source of the difficulty is the treatment of the numerical boundary conditions. One conjecture was that unconditional stability requires implicit numerical boundary conditions. An apparent counter example was the space time extrapolation considered by Gustafsson, Kreiss, and Sunstrom. Spatial (implicit) and space time (explicit) extrapolation using normal mode analysis for a finite and infinite number of spatial mesh intervals are examined. The results indicate that for unconditional stability with a finite number of spatial mesh intervals, the numerical boundary conditions must be implicit.
The linear stability equations for compressible, three-dimensional laminar boundary layer flow are derived in an orthogonal curvilinear coordinate system. The system of equations is solved using a finite difference scheme in order to study the effects of streamline and surface curvature and compressibility on the stability of the flow past a swept wing. It is known that convex surface curvature can have a stabilizing effect on the laminar boundary layer. Conversely, concave surface curvature can be destabilizing. The magnitude of these effects for swept wings is determined. Results indicate that amplification rates and hence, N-factors, for the flow over the convex upper surface of a swept wing can be reduced by about 15 to 45 percent when curvature effects are included in the linear stability analysis. The results of the calculations show that concave curvature destabilizes crossflow type disturbances with a significant increase in amplification rate. In addition, comparisons are made with some experimental results on a swept concave-convex surface. Calculated velocity vector plots show good agreement with observed disturbances in the laminar boundary layer over the concave surface.