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

▪ Abstract In this review we describe the aerodynamic problems that must be addressed in order to design a successful small aerial vehicle. The effects of Reynolds number and aspect ratio (AR) on the design and performance of fixed-wing vehicles are described. The boundary-layer behavior on airfoils is especially important in the design of vehicles in this flight regime. The results of a number of experimental boundary-layer studies, including the influence of laminar separation bubbles, are discussed. Several examples of small unmanned aerial vehicles (UAVs) in this regime are described. Also, a brief survey of analytical models for oscillating and flapping-wing propulsion is presented. These range from the earliest examples where quasi-steady, attached flow is assumed, to those that account for the unsteady shed vortex wake as well as flow separation and aeroelastic behavior of a flapping wing. Experiments that complemented the analysis and led to the design of a successful ornithopter are also described.

Aerodynamics of small vehicles

A discrete adjoint method is developed and demonstrated for aerodynamic design optimization on unstructured grids. The governing equations are the three-dimensional Reynolds-averaged Navier-Stokes equations coupled with a one-equation turbulence model. A discussion of the numerical implementation of the flow and adjoint equations is presented. Both compressible and incompressible solvers are differentiated and the accuracy of the sensitivity derivatives is verified by comparing with gradients obtained using finite differences. Several simplyfying approximations to the complete linearization of the residual are also presented, and the resulting accuracy of the derivatives is examined. Demonstration optimizations for both compressible and incompressible flows are given.

/pdf/aerodynamic-design-optimization-on-unstructured-meshes-using-1f4pe2dpn1.pdf

Aerodynamic Design Optimization on Unstructured Meshes Using the Navier-Stokes Equations

Low Speed Aerodynamics

This paper describes a series of experiments on the SD7003 airfoil, conducted in three different facilities: a low-turbulence wind tunnel, a water tunnel, and a tow tank. An attempt was made to achieve commonality of model geometry, Reynolds number and test conditions, in order to directly compare the experimental results. The SD7003 airfoil was chosen because of its robust, thin laminar separation bubble, which challenges the capability of current Particle Image Velocimetry (PIV) methods to accurately resolve the near-wall flowfield. Chordwise locations of separation, reattachment, and transition were compared. Since the time-averaged bubble geometry is a strong function of the flowfield ambient turbulence level, comparison of bubble geometry gives some assessment of facility flow quality, and hence of facility suitability for low Reynolds number testing in general.

Comparison of Laminar Separation Bubble Measurements on a Low Reynolds Number Airfoil in Three Facilities

Unsteady aerofoil flows are often characterized by leading-edge vortex (LEV) shedding. While experiments and high-order computations have contributed to our understanding of these flows, fast low-order methods are needed for engineering tasks. Classical unsteady aerofoil theories are limited to small amplitudes and attached leading-edge flows. Discrete-vortex methods that model vortex shedding from leading edges assume continuous shedding, valid only for sharp leading edges, or shedding governed by ad-hoc criteria such as a critical angle of attack, valid only for a restricted set of kinematics. We present a criterion for intermittent vortex shedding from rounded leading edges that is governed by a maximum allowable leading-edge suction. We show that, when using unsteady thin aerofoil theory, this leading-edge suction parameter (LESP) is related to the term in the Fourier series representing the chordwise variation of bound vorticity. Furthermore, for any aerofoil and Reynolds number, there is a critical value of the LESP, which is independent of the motion kinematics. When the instantaneous LESP value exceeds the critical value, vortex shedding occurs at the leading edge. We have augmented a discrete-time, arbitrary-motion, unsteady thin aerofoil theory with discrete-vortex shedding from the leading edge governed by the instantaneous LESP. Thus, the use of a single empirical parameter, the critical-LESP value, allows us to determine the onset, growth, and termination of LEVs. We show, by comparison with experimental and computational results for several aerofoils, motions and Reynolds numbers, that this computationally inexpensive method is successful in predicting the complex flows and forces resulting from intermittent LEV shedding, thus validating the LESP concept.

Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding

An inviscid theoretical method that is applicable to non-periodic motions and that accounts for large amplitudes and non-planar wakes (large-angle unsteady thin airfoil theory) is developed. A pitch-up, hold, pitch-down motion for a flat plate at Reynolds number 10,000 is studied using this theoretical method and also using computational (immersed boundary method) and experimental (water tunnel) methods. Results from theory are compared against those from computation and experiment which are also compared with each other. The variation of circulatory and apparent-mass loads as a function of pivot location for this motion is examined. The flow phenomena leading up to leading-edge vortex shedding and the limit of validity of the inviscid theory in the face of vortex-dominated flows are investigated. Also, the effect of pitch amplitude on leading-edge vortex shedding is examined, and two distinctly different vortex-dominated flows are studied using dye flow visualizations from experiment and vorticity plots from computation.

An unsteady airfoil theory applied to pitching motions validated against experiment and computation

The limits of linear superposition in two-dimensional high-rate low-Reynolds-number aerodynamics are examined by comparing the lift-coefficient history and flowfield evolution for airfoils undergoing harmonic motions in pure pitch, pure plunge, and pitch―plunge combinations. Using quasi-steady airfoil theory and Theodorsen's formula as predictive tools, pitching motions are sought that produce lift histories identical to those of prescribed plunging motions. It follows that a suitable phasing of pitch and plunge in a combined motion should identically produce zero lift, canceling either the circulatory contribution (with quasi-steady theory) or the combination of circulatory and noncirculatory contributions (with Theodorsen's formula). Lift history is measured experimentally in a water tunnel using a force balance and is compared with two-dimensional Reynolds-averaged Navier―Stokes computations and Theodorsen's theory; computed vorticity contours are compared with dye injection in the water tunnel. Theodorsen's method evinces considerable, and perhaps surprising, resilience in finding pitch-to-plunge equivalence of lift-coefficient―time history, despite its present application to cases in which its mathematical assumptions are demonstrably violated. A combination of pitch and plunge motions can be found such that net lift coefficient is nearly identically zero for arbitrarily high reduced frequency, provided that amplitude is small. Conversely, cancellation is possible at large motion amplitude, provided that reduced frequency is moderate. The product of Strouhal number and nondimensional amplitude is therefore suggested as the upper bound for when superposition and linear predictions remain valid in massively unsteady two-dimensional problems.

Investigations of Lift-Based Pitch-Plunge Equivalence for Airfoils at Low Reynolds Numbers

A design philosophy for low Reynolds number airfoils that judiciously combines the tailoring of the airfoil pressure distribution using a transition ramp with the use of boundary-layer trips is presented Three airfoils with systematic changes to the shape of the transition ramp have been designed to study the effect of trips on the airfoil performance The airfoils were wind-tunnel tested with various trip locations and at Reynolds numbers of 100,000 and 300,000 to assess the effectiveness of the design philosophy The results show that the design philosophy was successfullyusedin integratinga boundary-layertrip from theoutsetin theairfoildesignprocessFortheReynolds numbers and the range of airfoil shapes considered, however, airfoils designed with trips do not hold any clear advantage over airfoils designed for good performance in the clean condition

/pdf/design-of-low-reynolds-number-airfoils-with-trips-58ucne8kem.pdf

Design of Low Reynolds Number Airfoils with Trips

A novel scheme is presented for an iterative decambering approach to predict the post-stall characteristics of wings using known section data as inputs. The new scheme differs from earlier ones in the details of how the residual is computed. With this scheme, multiple solutions at high angles of attack are brought to light right during the computation of the residual for the Newton iteration. As with earlier schemes, multiple solutions are obtained for wings at high angles of attack and the resulting converged solution depends on the initial conditions used for the iteration. In general, the new scheme is found to be more robust at achieving convergence. Results are presented for a rectangular wing with two different airfoil lift curves and for a wing-tail configuration.

/pdf/poststall-prediction-of-multiple-lifting-surface-50ra2k6ks6.pdf

Ashok Gopalarathnam

Papers

Discrete-vortex method with novel shedding criterion for unsteady aerofoil flows with intermittent leading-edge vortex shedding

An unsteady airfoil theory applied to pitching motions validated against experiment and computation

Investigations of Lift-Based Pitch-Plunge Equivalence for Airfoils at Low Reynolds Numbers

Design of Low Reynolds Number Airfoils with Trips

Poststall prediction of multiple-lifting-surface configurations using a decambering approach