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

A transonic unsteady aerodynamic and aeroelasticity code CAP-TSD (Computational Aeroelasticity Program-Transonic Small Disturbance) has been developed for application to realistic aircraft configurations. The code uses a time-accurate approximate factorization (AF) algorithm for solution of the unsteady transonic small-disturbance potential equation. The paper gives an overview of the CAP-TSD code development effort and reports on recent algorithm modifications. The algorithm modifications include: an Engquist-Osher (E-O) type-dependent switch to treat regions of supersonic flow, extension of the E-O switch for second-order spatial accuracy, nonisentropic effects to treat strong-shock cases, nonreflecting far field boundary conditions for unsteady applications, and several modifications to accelerate convergence to steady state. The modifications have significantly improved the stability of the AF algorithm and hence the reliability of the CAP-TSD code in general. Calculations are also presented from a flutter analysis of a 45-deg sweptback wing which agree well with experimental data. The paper present descriptions of the CAP-TSD code and algorithm details along with results.

/pdf/steady-and-unsteady-transonic-small-disturbance-analysis-of-54pp7we96g.pdf

Steady and unsteady transonic small disturbance analysis of realistic aircraft configurations

With the continuous progress in hardware and numerical schemes, Computational Unsteady Aerodynamics (CUA), that is, the application of Computational Fluid Dynamics (CFD) to unsteady flowfields, is slowly finding its way as a useful and reliable tool (turbulence and transition modeling permitting) in the aircraft, helicopter, engine and missile design and development process. Before a specific code may be used with confidence it is essential to validate its capability to describe the physics of the flow correctly, or at least to the level of approximation required, for which purpose a comparison with accurate experimental data is needed. Unsteady wind tunnel testing is difficult and expensive; two factors which dramatically limit the number of organizations with the capability and/or resources to perform it. Thus, unsteady experimental data is scarce, often classified and scattered in diverse documents. Additionally, access to the reports does not necessarily assure access to the data itself. The collaborative effort described in this paper was conceived with the aim of collecting into a single easily accessible document as much quality data as possible. The idea is not new. In the early 80's NATO's AGARD (Advisory Group for Aerospace Research & Development) Structures and Material Panel (SMP) produced AGARD Report No. 702 "Compendium of Unsteady Aerodynamic Measurements", which has found and continues to find extensive use within the CUA Community. In 1995 AGARD's Fluid Dynamics Panel (FDP) decided to update and expand the former database with new geometries and physical phenomena, and launched Working Group WG-22 on "Validation Data for Computational Unsteady Aerodynamic Codes". Shortly afterwards AGARD was reorganized as the RTO (Research and Technology Organization) and the WG was renamed as AVT (Applied Vehicle Technolology) WG-003. Contributions were received from AEDC, BAe, DLR, DERA, Glasgow University, IAR, NAL, NASA, NLR, and ONERA. The final publication with the results of the exercise is expected in the second part of 1999. The aim of the present paper is to announce and present the new database to the Aeroelasticity community. It is also intended to identify, together with one of the groups of end users it targets, deficiencies in the compendium that should be addressed by means of new wind tunnel tests or by obtaining access to additionally existing data.

/pdf/a-new-compendium-of-unsteady-aerodynamic-test-cases-for-cfd-1m096hu1cs.pdf

A New Compendium of Unsteady Aerodynamic Test Cases for CFD: Summary of AVT WG-003 Activities

An experimental effort was implemented in aeroelasticity called the Benchmark Models Program. The primary purpose of this program is to provide the necessary data to evaluate computational fluid dynamic codes for aeroelastic analysis. It also focuses on increasing the understanding of the physics of unsteady flows and providing data for empirical design. An overview is given of this program and some results obtained in the initial tests are highlighted. The tests that were completed include measurement of unsteady pressures during flutter of a rigid wing with an NACA 0012 airfoil section and dynamic response measurements of a flexible rectangular wing with a thick circular arc airfoil undergoing shock boundary layer oscillations.

The benchmark aeroelastic models program: Description and highlights of initial results

As computational fluid dynamics methods mature, code development is rapidly tran-sitioning from prediction of steady flowflelds to unsteady flows. This change in emphasisoffers a number of new challenges to the research community, not the least of whichis obtaining detailed, accurate unsteady experimental data with which to evaluate newmethods. Researchers at NASA Langley Research Center (LaRC) have been activelymeasuring unsteady pressure distributions for nearly 40 years. Over the last 20 years,these measurements have focnsed on developing high-quality datasets for use in code eval-uation. This paper provides a sample of unsteady pressure measurements obtained byLaRC and available for government, university, and industry researchers to evaluate newand existing unsteady aerodynamic analysis methods. A number of cases are highlightedand discussed with attention focused on tile unique character of the individual datasetsand their perceived usefulness for code evaluation. Ongoing LaRC research in this areais also presented.

A Sample of NASA Langley Unsteady Pressure Experiments for

A finite difference technique is used to solve the transonic small disturbance flow equation making use of shock capturing to treat wave discontinuities. Thus the nonlinear effects of thickness and angle of attack are considered. Such an approach is made feasible by the development of a new code called CAP-TSD (Computational Aeroelasticity Program - Transonic Small Disturbance), and is based on a fully implicit approximate factorization (AF) finite difference method to solve the time dependent transonic small disturbance equation. The application of the CAP-TSD code to the calculation of low to moderate supersonic steady and unsteady flows is presented. In particular, comparisons with exact linear theory solutions are made for steady and unsteady cases to evaluate shock capturing and other features of the current method. In addition, steady solutions obtained from an Euler code are used to evaluate the small disturbance aspects of the code. Steady and unsteady pressure comparisons are made with measurements for an F-15 wing model and for the RAE tailplane model.

/pdf/calculation-of-steady-and-unsteady-pressures-at-supersonic-4bm08dywxd.pdf

Robert M. Bennett

Papers

Steady and unsteady transonic small disturbance analysis of realistic aircraft configurations

A New Compendium of Unsteady Aerodynamic Test Cases for CFD: Summary of AVT WG-003 Activities

The benchmark aeroelastic models program: Description and highlights of initial results

A Sample of NASA Langley Unsteady Pressure Experiments for

Calculation of steady and unsteady pressures at supersonic speeds with CAP-TSD