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

Transonic aeroelastic solutions based upon the transonic small perturbation potential equation were studied. Time-marching transient solutions of plunging and pitching airfoils were analyzed using a complex exponential modal identification technique, and seven alternative integration techniques for the structural equations were evaluated. The HYTRAN2 code was used to determine transonic flutter boundaries versus Mach number and angle-of-attack for NACA 64A010 and MBB A-3 airfoils. In the code, a monotone differencing method, which eliminates leading edge expansion shocks, is used to solve the potential equation. When the effect of static pitching moment upon the angle-of-attack is included, the MBB A-3 airfoil can have multiple flutter speeds at a given Mach number.

/pdf/time-marching-transonic-flutter-solutions-including-angle-of-2d7gr7ddeb.pdf

Time-marching transonic flutter solutions including angle-of-attack effects

The application and assessment of the recently developed CAP-TSD transonic small-disturbance code for flutter prediction is described. The CAP-TSD code has been developed for aeroelastic analysis of complete aircraft configurations and was previously applied to the calculation of steady and unsteady pressures with favorable results. Generalized aerodynamic forces and flutter characteristics are calculated and compared with linear theory results and with experimental data for a 45 deg sweptback wing. These results are in good agreement with the experimental flutter data which is the first step toward validating CAP-TSD for general transonic aeroelastic applications. The paper presents these results and comparisons along with general remarks regarding modern wing flutter analysis by computational fluid dynamics methods.

/pdf/modern-wing-flutter-analysis-by-computational-fluid-dynamics-3lk99fnji4.pdf

Modern wing flutter analysis by computational fluid dynamics methods

A transonic unsteady aerodynamic and aeroelasticity code has been developed for application to realistic aircraft configurations. The new code is called CAP-TSD which is an acronym for Computational Aeroelasticity Program - Transonic Small Disturbance. The CAP-TSD code uses a time-accurate approximate factorization (AF) algorithm for solution of the unsteady transonic small-disturbance equation. The AF algorithm is very efficient for solution of steady and unsteady transonic flow problems. It can provide accurate solutions in only several hundred time steps yielding a significant computational cost savings when compared to alternative methods. The new code can treat complete aircraft geometries with multiple lifting surfaces and bodies including canard, wing, tail, control surfaces, launchers, pylons, fuselage, stores, and nacelles. Applications are presented for a series of five configurations of increasing complexity to demonstrate the wide range of geometrical applicability of CAP-TSD. These results are in good agreement with available experimental steady and unsteady pressure data. Calculations for the General Dynamics one-ninth scale F-16C aircraft model are presented to demonstrate application to a realistic configuration. Unsteady results for the entire F-16C aircraft undergoing a rigid pitching motion illustrated the capability required to perform transonic unsteady aerodynamic and aeroelastic analyses for such configurations.

/pdf/unsteady-transonic-flow-calculations-for-realistic-aircraft-4hjvoni765.pdf

Unsteady Transonic Flow Calculations for Realistic Aircraft Configurations

The Structural Dynamics Division at NASA Langley Research Center has started a wind tunnel activity referred to as the Benchmark Models Program. The primary objective of this program is to acquire measured dynamic instability and corresponding pressure data that will be useful for developing and evaluating aeroelastic type computational fluid dynamics codes currently in use or under development. The program is a multi-year activity that will involve testing of several different models to investigate various aeroelastic phenomena. This paper describes results obtained from a second wind tunnel test of the first model in the Benchmark Models Program. This first model consisted of a rigid semispan wing having a rectangular planform and a NACA 0012 airfoil shape which was mounted on a flexible two degree of freedom mount system. Experimental flutter boundaries and corresponding unsteady pressure distribution data acquired over two model chords located at the 60 and 95 percent span stations are presented.

/pdf/naca-0012-benchmark-model-experimental-flutter-results-with-1r8z13ip8p.pdf

NACA 0012 benchmark model experimental flutter results with unsteady pressure distributions

The motivation for Computational Aeroelasticity (CA) and the elements of one type of the analysis or simulation process are briefly reviewed The need for streamlining and improving the overall process to reduce elapsed time and improve overall accuracy is discussed Further effort is needed to establish the credibility of the methodology, obtain experience, and to incorporate the experience base to simplify the method for future use Experience with the application of a variety of Computational Aeroelasticity programs is summarized for the transonic flutter of two wings, the AGARD 4456 wing and a typical business jet wing There is a compelling need for a broad range of additional flutter test cases for further comparisons Some existing data sets that may offer CA challenges are presented

https://ntrs.nasa.gov/api/citations/20040090449/downloads/20040090449.pdf

Robert M. Bennett

Papers

Time-marching transonic flutter solutions including angle-of-attack effects

Modern wing flutter analysis by computational fluid dynamics methods

Unsteady Transonic Flow Calculations for Realistic Aircraft Configurations

NACA 0012 benchmark model experimental flutter results with unsteady pressure distributions

An Overview of Recent Developments in Computational Aeroelasticity