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

Recent progress in flapping wing aerodynamics and aeroelasticity

Interest in novel forms of marine propulsion and maneuvering has sparked a number of studies on unsteadily operating propulsors. We review recent experimental and theoretical work identifying the principal mechanism for producing propulsive and transient forces in oscillating flexible bodies and fins in water, the formation and control of large-scale vortices. Connection with studies on live fish is made, explaining the observed outstanding fish agility.

Hydrodynamics of Fishlike Swimming

Vortex breakdown: a review

1. Introduction 2. Fixed, rigid wing aerodynamics 3. Flexible wing aerodynamics 4. Flapping wing aerodynamics.

/pdf/aerodynamics-of-low-reynolds-number-flyers-4v0j5o71pp.pdf

Aerodynamics of Low Reynolds Number Flyers

Effect of Spanwise Flexibility on Flapping Wing Propulsion

Water tunnel experiments on a flexible airfoil plunging with constant amplitude have been carried out for Reynolds numbers of 0 to 27000. Peaks in thrust coefficient at intermediate values of airfoil stiffness were observed at both zero and non-zero Reynolds numbers, indicating that a degree of flexibility is beneficial at low Reynolds numbers. Time-averaged velocity fields and momentum flux data revealed a broader, higher-velocity jet in cases of optimum airfoil stiffness. Stronger vortices, separated by a larger lateral distance, characterised the corresponding instantaneous velocity fields. The flexibility causes the airfoil to pitch passively; the phase angle of the pitch was found to lead the plunge. Pitch amplitude and trailing-edge amplitude were found to be single-valued functions of pitch phase angle. The shape characteristics of the airfoil could therefore be described by the pitch phase angle only. Thrust coefficient was found to be a function of only two parameters: Strouhal number and pitch phase angle. For each Strouhal number, a peak in thrust coefficient was observed at a particular value of the pitch phase angle. The optimum pitch phase angle was found to tend to a limit of 105±5 degrees at very large Strouhal numbers. A significant thrust benefit was observed over very stiff airfoils when the optimum flexibility is utilized.

Flexible flapping airfoil propulsion at low Reynolds numbers

https://purehost.bath.ac.uk/ws/files/508962/Progress%2520in%2520Aerospace%2520Sciences%2520review%2520article%2520final%2520version.pdf

Unsteady aerodynamics of nonslender delta wings

Thrust generation for an airfoil plunging at zero freestream velocity, the case relevant to hovering birds and insects, has been studied. The objective was to investigate the effect of airfoil stiffness. Particle image velocimetry and force measurements were taken for three airfoils of relative bending stiffnesses 1:8:512 in a water tank. The deformation of the flexible airfoils produces an angle of attack that varies periodically with a phase angle with respect to the plunging motion. Amplitude and phase of this combined plunging/pitching motion play a major role in the flowfield and thrust generation. Vortex pairs or alternating vortex streets were observed depending on the amplitude and phase lag of the trailing edge. The strength of the vortices, their lateral spacing, and the time-averaged velocity of the induced jet were found to depend on the airfoil flexibility, plunge frequency, and amplitude. Direct force measurements confirmed that at high plunge frequencies the thrust coefficient of the airfoil with intermediate stiffness was greatest, although the least stiff airfoil can generate larger thrust at low frequencies. It is suggested that there is an optimum airfoil stiffness for a given plunge frequency and amplitude. The thrust/input-power ratio was found to be greater for the flexible airfoils than for the rigid airfoil.

Flexible Flapping Airfoil Propulsion at Zero Freestream Velocity

Nomenclature AR = aspect ratio; amplitude ratio B = probability density function of velocity c = root chord length f = frequency k = reduced frequency; axial wave number n =w ave number in angular direction P = probability p = pressure fluctuation Re =R eynolds number based on chord length S = spectral density s = local semispan T = period t = time U∞ = freestream velocity u = axial velocity v = swirl velocity x = streamwise distance xbd = breakdown location y = spanwise distance z =v ertical distance above wing surface α = angle of attack � = circulation δ = flap angle � = sweep angle ν = kinematic viscosity τ = time constant � =fi nangle ω =v orticity; radial frequency

Ismet Gursul

Papers

Effect of Spanwise Flexibility on Flapping Wing Propulsion

Flexible flapping airfoil propulsion at low Reynolds numbers

Unsteady aerodynamics of nonslender delta wings

Flexible Flapping Airfoil Propulsion at Zero Freestream Velocity

Review of Unsteady Vortex Flows over Slender Delta Wings