This work deals with the estimation of surface stresses in immersed-boundary methods. Interpolation methods based on inverse distances labeled. For viscous stress reconstruction, two approaches have been formulated; while one method uses the velocity interpolated at a point on the line normal to the surface, the second method uses the interpolated gradients of velocity at the surface point to construct the viscous stress at the surface. These reconstruction strategies for pressure and shear stress are incorporated into an in-house parallel finite-volume solver (REACTMB) that is coupled to a direct-forcing immersed-boundary method.
The pressure reconstruction methods are verified using an exact solution of inviscid flow through a CD nozzle, for which the analytical solution consists of a normal shock. The viscous stress reconstruction methods are also verified for shear stress reconstruction at the wall for laminar flow past a flat plate using an approximate analytical solution of the velocity profile. Validations for the interpolation methods are presented for steady flow cases: Mach 0.5 laminar flow past a NACA 0012 airfoil, Mach 2 supersonic laminar flow past a circular cylinder, and Mach 0.8 transonic laminar flow past a NACA 0012 airfoil. In these cases, it is seen that the choice of the pressure interpolation method does not affect the results. However, the shear stress estimation using interpolated surface velocity gradients produces smoother Cf plots and a more accurate prediction of loads. Validations for moving boundary flow simulations are also presented and include non- periodic plunging NACA 0012 airfoil, and a flapping elliptic airfoil.
The validated surface stress reconstruction and integration methods are then applied to a problem of flapping flight to study the sources of lift enhancement. The study undertaken here aims to understand the physics of lift enhancement for a specific flapping motion, such that at a given point in the flapping cycle, the airfoil only either translates or rotates and does not perform both the motions simultaneously. This helps in the demarcation of the lift response of the airfoil into a translational and rotational phase and hence facilitates comparison of lift forces attributed to translation and rotation, which are two of the three commonly believed lift enhancing mechanisms. For the cases considered for this study, the lift during translation is significantly higher than the lift during rotation. The third lift enhancing mechanism, which is the wing-wake interaction, is evaluated by comparing the lift response of the airfoil performing pure translation and pure rotation with that of the flapping motion. It is seen that the wing-wake interactions affect the translational response more than the rotational response, although the rotation rate determines the extent of wing-wake interactions on the translational phase. Investigations with an overlap in translation and rotation phases of the flapping cycle show that overlap brings down the magnitudes of the positive and negative lift peaks. A trend that suggests the existence of an optimum overlap for obtaining maximum mean lift is also observed.