Stability analysis of numerical boundary conditions and implicit difference approximations for hyperbolic equations
Previously announced in STAR as N81-33870
Beam, R. M.↗
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Stability analysis of intermediate boundary conditions in approximate factorization schemes
The paper discusses the role of the intermediate boundary condition in the AF2 scheme used by Holst for simulation of the transonic full potential equation. It is shown that the treatment suggested by Holst led to a restriction on the time step and ways to overcome this restriction are suggested. The discussion is based on the theory developed by Gustafsson, Kreiss, and Sundstrom and also on the von Neumann method.
South, J. C., Jr.↗
Stability of 3D wing boundary layer on a SST configuration
Tollmien-Schlichting (TS) and cross-flow (CF) instability growth characteristics were studied in three dimensions, for the case of a Mach 2.4 SST with double-delta planform whose inboard leading-edge is subsonic and outboard leading-edge is supersonic. Attention is given to the requirements for supersonic speed laminarization of both highly swept, rounded leading-edge wings and moderately-swept, sharp leading-edge wings. Suction requirements for the control of both TS and CF instabilities are calculated; it is found that while mild suction and surface cooling are effective in TS-instability damping, the CF influence of such techniques is rather weak. CF instability control must be via pressure-distribution tailoring and suction.
Parikh, P. G.↗
Stability of a supersonic boundary layer along a swept leading edge
The instability of an attachment-line boundary layer formed on a swept cylinder in a supersonic freestream is considered in the linear regime. The supersonic attachment-line boundary layer is shown to be susceptible to oblique Tollmien-Schlichting wave instability which may be controlled by wall cooling. The critical Reynolds number based upon momentum thickness is found to be about 230. The onset of transition in the attachment-line boundary layer is also studied using the e (sup N) method and results are compared with the experimental data obtained at M (sub infinity) = 3.5 in the absence of any trips or spanwise contamination.
Malik, Mujeeb R.↗
Linear and nonlinear stability of the Blasius boundary layer
Two new techniques for the study of the linear and nonlinear instability in growing boundary layers are presented. The first technique employs partial differential equations of parabolic type exploiting the slow change of the mean flow, disturbance velocity profiles, wavelengths, and growth rates in the streamwise direction. The second technique solves the Navier-Stokes equation for spatially evolving disturbances using buffer zones adjacent to the inflow and outflow boundaries. Results of both techniques are in excellent agreement. The linear and nonlinear development of Tollmien-Schlichting (TS) waves in the Blasius boundary layer is investigated with both techniques and with a local procedure based on a system of ordinary differential equations. The results are compared with previous work and the effects of non-parallelism and nonlinearity are clarified. The effect of nonparallelism is confirmed to be weak and, consequently, not responsible for the discrepancies between measurements and theoretical results for parallel flow.
Bertolotti, F. P.↗
Investigation of the stability of the laminar boundary layer
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Lees, Lesters↗
The Stability of the Laminar Boundary Layer in a Compressible Fluid
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Lees, Lester↗
A study of the stability of the laminar boundary layer as affected by changes in the boundary-layer thickness in regions of pressure gradient and flow through the surface
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Tetervin, Neal↗
Stability of separating subsonic boundary layers
The primary and subharmonic instabilities of separating compressible subsonic two-dimensional boundary layers in the presence of a two-dimensional roughness element on a flat plate are investigated. The roughness elements considered are humps and forward- and backward-facing steps. The use of cooling and suction to control these instabilities is studied. The similarities and differences between the instability characteristics of separating boundary layers and those of the boundary layer over a flat plate with a zero pressure gradient are pointed out and discussed. The theoretical results agree qualitatively and quantitatively with the experimental data of Dovgal and Kozlov. Cooling and suction decrease the growth rates of primary and subharmonic waves in the attached-flow regions but increase them in the separated-flow regions.
Masad, Jamal A.↗
The linear stability of three-dimensional boundary-layers over axisymmetric bodies at incidence
The linear stability of the fully three-dimensional boundary layer formed over a 6:1 prolate spheroid at 10 deg incidence is investigated using a modified version of the linear stability code COSAL. For this case, both Tollmien-Schlichting and cross flow disturbances are relevant in the transition process. The predicted location of the onset of transition using the e exp N method compares favorably with experimental results of Meier and Kreplin (1980). Using a value of N = 10, the predicted location is located approximately 10 percent upstream of the experimentally determined location. Results also indicate that the direction of disturbance propagation is dependent upon the type of disturbance, and hence the dimensional frequency.
Spall, R. E.↗
The linear stability of a flat plate boundary-layer approaching a cylindrical obstacle
The linear stability of the low-speed three-dimensional flow over a flat plate with an attached cylinder is studied. The region of interest is upstream of the initial separation point and includes the effects of both adverse and favorable pressure gradients, as well as crossflow. The resulting boundary-layer is subject to both the Tollmien-Schlichting (TS) and crossflow instabilities. Linear stability calculations, using N-factor correlations, indicate that the transition process would be dominated by TS instabilities, although for low frequencies crossflow-type disturbances are important.
Spall, R. E.↗
Influence of Electrotactile Tongue Feedback on Controlling Upright Stance during Rotational and/or Translational Sway-referencing with Galvanic Vestibular Stimulation
Integration of multi-sensory inputs to detect tilts relative to gravity is critical for sensorimotor control of upright orientation. Displaying body orientation using electrotactile feedback to the tongue has been developed by Bach-y-Rita and colleagues as a sensory aid to maintain upright stance with impaired vestibular feedback. MacDougall et al. (2006) recently demonstrated that unpredictably varying Galvanic vestibular stimulation (GVS) significantly increased anterior-posterior (AP) sway during rotational sway referencing with eyes closed. The purpose of this study was to assess the influence of electrotactile feedback on postural control performance with pseudorandom binaural bipolar GVS. Postural equilibrium was measured with a computerized hydraulic platform in 10 healthy adults (6M, 4F, 24-65 y). Tactile feedback (TF) of pitch and roll body orientation was derived from a two-axis linear accelerometer mounted on a torso belt and displayed on a 144-point electrotactile array held against the anterior dorsal tongue (BrainPort, Wicab, Inc., Middleton, WI). Subjects were trained to use TF by voluntarily swaying to draw figures on their tongue, both with and without GVS. Subjects were required to keep the intraoral display in their mouths on all trials, including those that did not provide TF. Subjects performed 24 randomized trials (20 s duration with eyes closed) including four support surface conditions (fixed, rotational sway-referenced, translating the support surface proportional to AP sway, and combined rotational-translational sway-referencing), each repeated twice with and without GVS, and with combined GVS and TF. Postural performance was assessed using deviations from upright (peak-to-peak and RMS sway) and convergence toward stability limits (time and distance to base of support boundaries). Postural stability was impaired with GVS in all platform conditions, with larger decrements in performance during trials with rotation sway-referencing. Electrotactile feedback improved performance with GVS toward non-GVS levels, again with the greatest improvement during trials with rotation sway-referencing. These results demonstrate the effectiveness of tongue electrotactile feedback in providing sensory substitution to maintain postural stability with distorted vestibular input.
Wood, Scott J.↗
Stability of a diffuse linear pinch with axial boundaries
A formulation of the stability behavior of a finite-length pinch is presented. A general initial perturbation is expressed as a uniformly convergent sum over a complete discrete k set. A variational calculation is then performed, based on the energy principle, in which the end-boundary conditions appear as constraints. The requisite Lagrange multipliers mutually couple the elemental periodic excitations. The resulting extended form of delta-W still admits a proper second-variation treatment so that the minimization and stability considerations of Newcomb remain applicable. Comparison theorems are discussed as is the relevance of this end-effect model to the stability of solar coronal loops.
Einaudi, G.↗
Numerical-perturbation technique for stability of flat-plate boundary layers with suction
A numerical-perturbation scheme is proposed for determining the stability of flows over plates with suction through a finite number of porous suction strips. The basic flow is calculated as the sum of the Blasius flow and closed-form linearized triple-deck solutions of the flow due to the strips. A perturbation technique is used to determine the increment a(ij) in the complex wavenumber at a given location x(j) due to the presence of a strip centered at x(i). The end result is a set of influence coefficients that can be used to determine the growth rates and amplification factors for any suction levels without repeating the calculations. The numerical-perturbation results are verified by comparison with interacting boundary layers for the case of six strips and the experimental data of Reynolds and Saric for single- and multiple-strip configurations. The influence coefficient form of the solution suggests a scheme for optimizing the strip configuration. The results show that one should concentrate the suction near branch I of the neutral stability curve, a conclusion verified by the experiments.
Reed, H. L.↗
Streak Instabilities on HIFiRE-5 Elliptic Cone
The Hypersonic International Flight Research and Experimentation (HIFiRE) 5 flight experiment by the Air Force Research Laboratories and Australian Defense Science and Technology Organization yielded inflight, hypersonic transition data for a canonical, 3D configuration in the form of an elliptic cone at various combinations of angle-of-attack and yaw. Previous analysis has shown that boundary-layer stability calculations for the as-flown, zero-yaw flight conditions are able to capture the second mode dominated transition along the attachment line as well as the crossflow induced transition in between the attachment line and the minor axis symmetry plane. However, there is still a need to understand the transition in the vicinity of the minor axis plane of symmetry, where the boundary layer thickness exhibits an azimuthally compact bulge due to the collision of the secondary flow approaching from the attachment lines on both sides. This paper reports a computational analysis of the stability of the boundary layer flow along the bulge regions on both the leeward and windward sides of the vehicle surface. At a representative flight condition corresponding to a time instant of t = 514.813 seconds, computations are performed to identify several different unstable modes within the bulge regions, with the eventual goal of developing a linear stability correlation for the transition onset near the minor axis symmetry plane. The most amplified flow instabilities in the pretransitional region correspond to a “stem” mode that is concentrated within a pair of nearly vertical shear layers below the cap of the mushroom shaped bulge in the mean velocity contours. An excellent agreement is found between the predictions of instability amplification based on the plane marching parabolized stability equations (PSE) and the direct numerical solutions (DNS). Given the large number of unstable modes within the bulge region, the observed agreement demonstrates, for first time, that the PSE can provide accurate predictions for the development of flow instabilities in streak-like flow features over three-dimensional bodies. The peak N-factor of approximately 15 at the transition location estimated from the thermocouple measurements along the leeward centerline is close to the transition N-factor for Mack mode transition during the HIFiRE-1 circular cone flight experiment with nearly zero degrees angle of attack.
Meelan Choudhari↗
Stability of the solutions of elliptic partial differential equations with general boundary conditions
Existence, uniqueness, stability, and asymptotic stability conditions for solution of elliptic partial differential equations with general boundary conditions
Reiser, E. J.↗
New stability criteria for difference approximations of hyperbolic initial-boundary value problems
New convenient stability criteria are provided for a large class of finite difference approximations to initial-boundary value problems associated with the hyperbolic system u sub t = Au sub x + Bu + f in the quarter plane x greater than or equal to 0, t greater than or equal to 0. The criteria are used to easily establish stability for numerous combinations of well known basic schemes and boundary conditions, thus generalizing many special cases studied in the recent literature. A number of examples are examined, including the unitary unconditionally stable Crank-Nicholson scheme and an almost-dissipative unconditionally stable backward Euler scheme.
Goldberg, M.↗
On the stability of Galerkin methods for initial-boundary value problems for hyperbolic systems
The stability of approximating the solution of mixed initial-boundary value problems for hyperbolic systems by semidiscrete Galerkin methods is studied. It is shown that a particular straightforward Galerkin method yields an unstable approximation, and that this numerical instability is caused by an improper treatment of the boundary. Stable schemes are then presented, one of which differs from the unstable scheme only insofar as the treatment of the boundary is concerned. These stable schemes make use of a particular matrix which symmetrizes the differential system. It is therefore shown that the use of this matrix is crucial to the stability of the computations as well as for obtaining a priori bounds on the energy of the continuous system. This symmetrizing matrix is also related to the diagonalizing matrix for the system of hyperbolic equations and to the Liapunov matrix for the system of ordinary differential equations resulting from the application of Galerkin's method.
Gunzburger, M. D.↗