The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

https://ntrs.nasa.gov/api/citations/20010018414/downloads/20010018414.pdf

Reduced-order modeling: new approaches for computational physics

Review of state of the art in smart rotor control research for wind turbines

Structures, Structural Dynamics and Materials Conference

A new method for constructing reduced-order models (ROM) of unsteady small-disturbance e ows is presented. The reduced-order models are constructed using basis vectors determined from the proper orthogonal decomposition (POD) of an ensemble of small-disturbance frequency-domain solutions. Each of the individual frequencydomain solutions is computed using an efe cient time-linearized e ow solver. We show that reduced-order models can be constructed using just a handful of POD basis vectors, producing low-order but highly accurate models of the unsteady e ow over a wide range of frequencies. We apply the POD/ROM technique to compute the unsteady aerodynamic and aeroelastic behavior of an isolated transonic airfoil and to a two-dimensional cascade of airfoils.

/pdf/proper-orthogonal-decomposition-technique-for-transonic-1qcwd6gv8a.pdf

Proper Orthogonal Decomposition Technique for Transonic Unsteady Aerodynamic Flows

System identification techniques in common use for extracting stability derivatives from flight test data have been adapted for application to data obtained from aeroelastically-scaled flutter models flown in a wind tunnel on a cable mount system. The concept has been applied with reasonable success to data from rigid models of a Space Shuttle Orbiter and a fighter tested in the NASA Langley transonic dynamics tunnel. Further application of this technique should permit extraction of derivatives that include scaled flexibility effects, thereby obtaining additional information from the testing of expensive flutter models.

A wind tunnel technique for determining stability derivatives from cable mounted aeroelastic models

Steady and unsteady measured pressures for a Rectangular Supercritical Wing (RSW) undergoing pitching oscillations have been presented. From the several hundred compiled data points, 27 static and 36 pitching oscillation cases have been proposed for computational Test Cases to illustrate the trends with Mach number, reduced frequency, and angle of attack. The wing was designed to be a simple configuration for Computational Fluid Dynamics (CFD) comparisons. The wing had an unswept rectangular planform plus a tip of revolution, a panel aspect ratio of 2.0, a twelve per cent thick supercritical airfoil section, and no twist. The model was tested over a wide range of Mach numbers, from 0.27 to 0.90, corresponding to low subsonic flows up to strong transonic flows. The higher Mach numbers are well beyond the design Mach number such as might be required for flutter verification beyond cruise conditions. The pitching oscillations covered a broad range of reduced frequencies. Some early calculations for this wing are given for lifting pressure as calculated from a linear lifting surface program and from a transonic small perturbation program. The unsteady results were given primarily for a mild transonic condition at M = 0.70. For these cases the agreement with the data was only fair, possibly resulting from the omission of viscous effects. Supercritical airfoil sections are known to be sensitive to viscous effects (for example, one case cited). Calculations using a higher level code with the full potential equations have been presented for one of the same cases, and with the Euler equations. The agreement around the leading edge was improved, but overall the agreement was not completely satisfactory. Typically for low-aspect-ratio rectangular wings, transonic shock waves on the wing tend to sweep forward from root to tip such that there are strong three-dimensional effects. It might also be noted that for most of the test, the model was tested with free transition, but a few points were taken with an added transition strip for comparison. Some unpublished results of a rigid wing of the same airfoil and planform that was tested on the pitch and plunge apparatus mount system (PAPA) showed effects of the lower surface transition Strip on flutter at the lower subsonic Mach numbers. Significant effects of a transition strip were also obtained on a wing with a thicker supercritical section on the PAPA mount system. Both of these flutter tests on the PAPA resulted in very low reduced frequencies that may be a factor in this influence of the transition strip. However, these results indicate that correlation studies for RSW may require some attention to the estimation of transition location to accurately treat viscous effects. In this report several Test Cases are selected to illustrate trends for a variety of different conditions with emphasis on transonic flow effects. An overview of the model and tests is given and the standard formulary for these data is listed. Sample data points are presented in both tabular and graphical form. A complete tabulation and plotting of all the Test Cases is given. Only the static pressures and the real and imaginary parts of the first harmonic of the unsteady pressures are available. All the data for the test are available in electronic file form. The Test Cases are also available as separate electronic files.

Test Cases for a Rectangular Supercritical Wing Undergoing Pitching Oscillations

The initial application of the CAP-TSD computer program for wing flutter analysis is presented. Computational Aeroelasticity Program - Transonic Small Disturbance (CAP-TSD) is based on an approximate factorization (AF) algorithm that is stable and efficient on supercomputers with vector arithmetic. CAP-TSD was used to calculate steady and unsteady pressures on wings and configurations at subsonic, transonic, and supersonic Mach numbers. However, the CAP-TSD code has been developed primarily for aeroelastic analysis. The initial efforts for validation of the aeroelastic analysis capability is presented. The initial applications include two series of symmetric, planar wing planforms. Well defined modal properties are available for these wings. In addition, transonic flutter boundaries are available for evaluation of the transonic capabilities of CAP-TSD.

/pdf/initial-application-of-cap-tsd-to-wing-flutter-2xyfjfd47p.pdf

Initial application of CAP-TSD to wing flutter

The primary issues involved in the generation of linear, state-space equations of motion of a flexible wind tunnel model, the Active Flexible Wing (AFW), are discussed. The codes that were used and their inherent assumptions and limitations are also briefly discussed. The application of the CAP-TSD code to the AFW for determination of the model's transonic flutter boundary is included as well.

https://ntrs.nasa.gov/api/citations/19910013020/downloads/19910013020.pdf

Aeroelastic modeling of the active flexible wing wind-tunnel model

Transonic shock-boundary-layer oscillations occur on rigid models over a small range of Mach numbers on thick circular-arc airfoils. Extensive tests and analyses of this phenomena have been made in the past, but essentially all of them were for rigid models. A simple flexible wing model with an 18% circular-arc airfoil was constructed and tested in the Langley transonic dynamics tunnel to investigate the dynamic characteristics that a wing might have under these conditions. In the region of shock-boundar y-layer oscillations, buffeting of the first bending mode was obtained. This mode was well-separated in frequency from the shock-boundary-layer oscillations. A limit-cycle oscillation was also measured in a "third bending-like" mode, involving wing vertical bending and splitter plate motion, which was in the frequency range of the shock-boundary-layer oscillations. Several model configurations were tested, and a few potential fixes were investigated. Nomenclature BM = model wing-root bending moment c = wing chord k = reduced frequency based on semichord, coc/2V q = dynamic pressure S = wing area V = freestream velocity a) = frequency of oscillation, rad/s

Robert M. Bennett

Papers

A wind tunnel technique for determining stability derivatives from cable mounted aeroelastic models

Test Cases for a Rectangular Supercritical Wing Undergoing Pitching Oscillations

Initial application of CAP-TSD to wing flutter

Aeroelastic modeling of the active flexible wing wind-tunnel model

Transonic shock-induced dynamics of a flexible wing with a thick airfoil