Dynamics of leading edge separation

If the angle of attack a of a slender airfoil reaches a critical value a s , flow separation is known to occur at the upper surface. Further increase of a initially leads to the formation of a short laminar separation bubble which has an extremely weak influence on the external flow field but then rather rapidly causes a severe change of the flow behavior. The latter phenomenon - known as leading edge stall - is accompanied by an abrupt increase of drag and loss of lift and, therefore, has to be strictly avoided under real flight conditions. An asymptotic theory holding in the limit of large Reynolds number which describes the formation of the short bubble, now commonly referred to as the theory of marginal separation, was developed independently by Ruban and Stewartson, Smith and Kaups. According to this theory the flow in the neighborhood of the separation bubble is governed by an integro-differential equation. It contains a single controlling parameter G which is related to the angle of attack. Numerical investigations have shown that solutions to this equation exist up to a critical value G c only and this has been taken as an indication that a substantial change of the flow field must take place as G exceeds G c, thereby invalidating the theoretical basis of the theory and heralding the onset of stall. This point of view has recently be contested by Braun and Kluwick, who considered unsteady three-dimensional perturbations of a two-dimensional steady marginally separated boundary layer. Specifically, they showed that the (generalized form of the) integro-differential equation reduces to a nonlinear diffusion equation of Fisher type in the limit |G - G c| -> 0. Preliminary analysis strongly suggests that solutions of the Fisher equation which lead to finite-time blow-up may be extended beyond the blow-up time, thus generating moving singularities which can be interpreted as vortical structures qualitatively similar to those emerging in direct numerical simulations of near critical (i.e. transitional) laminar separation bubbles. While this phenomenon requires a certain finite perturbation level if GG c. This then indicates that the passage though G c is associated with a gradual rather than an abrupt change of the flow field. Its main effect is to transform the boundary layer downstream of the separation bubble from a laminar to a turbulent state which can sustain a further increase of the angle of attack without immediate onset of catastrophic stall. Only the first steps have been taken so far in exploiting the capability of Fisher`s equation as a means to describe the complicated dynamics of leading edge separation. Significant further efforts will be necessary to obtain a more complete picture of the associated phenomena which are both of fundamental interest and practical importance. This is the aim of the proposed research which will focus on three topics: (i) a detailed investigation of its properties with special emphasis on solutions which are of relevance for flow control and in the turbulence context, (ii) the mechanism through which the hydromechanic motion inside the boundary layer governed by Fishers`s equation is transformed into aerodynamic sound and (iii) the question if and how finite-time singularities can be resolved by considering smaller scales in time and space where the theory of Braun and Kluwick loses its validity locally.

If the angle of attack a of a slender airfoil reaches a critical value a s , flow separation is known to occur at the upper surface. Further increase of a initially leads to the formation of a short laminar separation bubble which has an extremely weak influence on the external flow field but then rather rapidly causes a severe change of the flow behavior. The latter phenomenon - known as leading edge stall - is accompanied by an abrupt increase of drag and loss of lift and, therefore, has to be strictly avoided under real flight conditions. An asymptotic theory holding in the limit of large Reynolds number which describes the formation of the short bubble, now commonly referred to as the theory of marginal separation, was developed independently by Ruban and Stewartson, Smith and Kaups. According to this theory the flow in the neighborhood of the separation bubble is governed by an integro-differential equation. It contains a single controlling parameter G which is related to the angle of attack. Numerical investigations have shown that solutions to this equation exist up to a critical value G c only and this has been taken as an indication that a substantial change of the flow field must take place as G exceeds G c, thereby invalidating the theoretical basis of the theory and heralding the onset of stall. This point of view has recently be contested by Braun and Kluwick, who considered unsteady three-dimensional perturbations of a two-dimensional steady marginally separated boundary layer. Specifically, they showed that the (generalized form of the) integro-differential equation reduces to a nonlinear diffusion equation of Fisher type in the limit |G - G c| 0. Preliminary analysis strongly suggests that solutions of the Fisher equation which lead to finite-time blow-up may be extended beyond the blow-up time, thus generating moving singularities which can be interpreted as vortical structures qualitatively similar to those emerging in direct numerical simulations of near critical (i.e. transitional) laminar separation bubbles. While this phenomenon requires a certain finite perturbation level if GG c. This then indicates that the passage though G c is associated with a gradual rather than an abrupt change of the flow field. Its main effect is to transform the boundary layer downstream of the separation bubble from a laminar to a turbulent state which can sustain a further increase of the angle of attack without immediate onset of catastrophic stall. Only the first steps have been taken so far in exploiting the capability of Fisher`s equation as a means to describe the complicated dynamics of leading edge separation. Significant further efforts will be necessary to obtain a more complete picture of the associated phenomena which are both of fundamental interest and practical importance. This is the aim of the proposed research which will focus on three topics: (i) a detailed investigation of its properties with special emphasis on solutions which are of relevance for flow control and in the turbulence context, (ii) the mechanism through which the hydromechanic motion inside the boundary layer governed by Fishers`s equation is transformed into aerodynamic sound and (iii) the question if and how finite-time singularities can be resolved by considering smaller scales in time and space where the theory of Braun and Kluwick loses its validity locally.