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

Part I. The Surface Vorticity Boundary Integral Method of Fluid Flow Analysis: 1. The basis of surface singularity modelling 2. Lifting bodies, two-dimensional aerofoils and cascades 3. Mixed-flow and radial cascades 4. Bodies of revolution, ducts and annuli 5. Ducted propellers and fans 6. Three-dimensional and meridional flows in turbo-machines 7. Free vorticity shear layers and inverse methods Part II. Vortex Dynamics and Vortex Cloud Analysis: 8. Vortex dynamics in inviscid flows 9. Simulation of viscous diffusion in discrete vortex modelling 10. Vortex cloud modelling by the boundary integral method 11. Further development and applications of vortex cloud modelling to lifting bodies and cascades 12. Use of grid systems in vortex dynamics and meridional flows Appendix.

Vortex Element Methods for Fluid Dynamic Analysis of Engineering Systems: Vortex cloud modelling by the boundary integral method

Aerodynamic performance improvement of wind turbine blade by cavity shape optimization

Trapping of vortices in a cavity has been explored in recent years as a drag reduction measure for thick airfoils. If, however, trapping fails, then oscillation of the cavity flow may couple with elastic vibration modes of the airfoil. To examine this scenario, the effect of small amplitude vertical motion on the oscillation of the shear layer above the cavity is studied by acoustic forcing simulating a vertical translation of a modified NACA0018 profile. At low Reynolds numbers based on the chord (O(104)), natural instability modes of this shear layer are observed for Strouhal numbers based on the cavity width of order unity. Acoustic forcing sufficiently close to the natural instability frequency induces a strong non-linear response due to lock-in of the shear layer. At higher Reynolds numbers (above 105) for Strouhal number 0.6 or lower, no natural instabilities of the shear layer and only a linear response to forcing were observed. The dynamical pressure difference across the airfoil is then dominated by added mass effects, as was confirmed by numerical simulations.

/pdf/flow-around-a-naca0018-airfoil-with-a-cavity-and-its-1vjcl505i4.pdf

Flow around a NACA0018 airfoil with a cavity and its dynamical response to acoustic forcing

This paper reports a detailed numerical study of the aeroacoustics of a NACA 0018 airfoil with a cavity at zero angle of attack, with chord-based Reynolds and freestream Mach numbers at 2×104 and 0...

Aeroacoustics of NACA 0018 Airfoil with a Cavity

A new wing design has been the subject of study in the European project VortexCell2050. For several reasons (structural and fuel load) it is desirable to use relatively thick wings. However, thick wings promote flow separation and/or massive vortex shedding, reducing flight performance. The new design airfoil is equipped with a cavity ("vortex cell") in the wing in order to prevent massive flow separation. This thesis aims to obtain insight into the dynamical behaviour of such a wing with a cavity and to explore which numerical methods are suitable for estimating the unsteady forces. In this thesis experiments and computations are presented, using a geometry that was inspired by cavity shapes which are considered in the VortexCell2050 project. For an airfoil with a cavity, oscillations of the shear layer are expected at Strouhal numbers of order unity, based on the width of the cavity opening. For the size of the cavities considered this implies high values, O(10), of the reduced frequency, based on the chord length of the airfoil. In order to conduct experiments in this high reduced frequency range, a new experimental method has been developed. In this experimental setup the airfoil is fixed to the wind tunnel wall and the first acoustic transversal eigenmode of the wind tunnel test section is used to drive an oscillating flow. In the conventional method the airfoil is oscillating. The main fundamental difference between the two methods is the presence of a time dependent uniform pressure gradient, which drives the oscillating flow, in the new method. The results obtained with both methods are equivalent after correcting for an effective buoyancy force induced by this driving pressure gradient. The new method avoids the use of a complex mechanical system to drive the oscillation of the airfoil. The acoustical forcing amplitude is very easy to vary within two orders of magnitude. The method appears to be most suitable for the conduction of experiments at high values of the reduced frequency. The new measurement method is validated by means of experiments on a standard NACA0018 airfoil complemented with two-dimensional Euler simulations. Thereafter two airfoils with slightly different cavity geometries are investigated in the wind tunnel. These experiments consist of measurements of local surface pressures. For the case of an airfoil with a cavity the Euler equations are not suitable. Two-dimensional simulations using the frictionlass flow approximation approach a so-called Batchelor flow, with a uniform rotation in the cavity. This flow is not observed in experiments. For this reason two-dimensional incompressible Navier–Stokes simulations at a Reynolds number, based on the chord length of the airfoil, of 2 · 104 are performed and indicate shear layer oscillations. In order to validate these low Reynolds number numerical results and to gain more insight in the flow physics, flow visualisations are performed in a waterchannel at the same Reynolds numbers as the numerical simulations. The visualisations also show oscillations of the shear layer at the first and second hydrodynamic mode, this is confirmed by hot-wire measurements in the wind tunnel at low Reynolds numbers. The hot-wire measurements also demonstrate that the expected lock-in of the shear layer does occur in a limited range of Reynolds numbers, based on the chord, sufficiently low such that no turbulence is generated, but higher than a critical value. Experiments and two-dimensional Navier–Stokes simulations indicate that for values of the reduced frequency, in the range of 2–10, no significant deviations in the unsteady lift force occur between an airfoil with cavity and the same airfoil without cavity. The cavity does display shear layer oscillations around the expected Strouhal numbers, however, the associated fluctuations in the lift coefficient appear to be neglegible. For the geometries considered the pressure differences over the airfoil are dominated by the added mass of the airfoil.

/pdf/influence-of-a-cavity-on-the-dynamical-behaviour-of-an-36ipplrod9.pdf

Wfj Jurriën Olsman

Papers

Influence of a cavity on the dynamical behaviour of an airfoil