show spilling wave near the toe which surface-tracking method does not capture. The overall ... surface-piercing NACA 0024 foil, which also has ...... 124, 2002, pp. 91-101. [28] Lin, P., and Li, C. W.: Wave-Current Interaction ... Report No. 432 ...
VORTICAL AND TURBULENT STRUCTURES AND INSTABILITIES IN UNSTEADY FREE-SURFACE WAVE INDUCED SEPARATION Manivannan Kandasamy, Tao Xing, Robert Wilson, Frederick Stern IIHR-Hydroscience & Engineering C. Maxwell Stanley Hydraulics Laboratory The University of Iowa ABSTRACT Free-surface wave-induced separation caused by interactions of free-surface waves and wall boundary layers has relevance to ship and platform hydrodynamics with regard to resistance and propulsion, stability, and signatures. This paper uses URANS and DES with complimentary EFD to investigate the separation patterns and Strouhal numbers for the different instability mechanisms on a surface piercing NACA0024 foil for Fr=0.37 and 0.55. Instability studies include laminar cases too. Frequency analysis and vortex core detection reveal the nature of freesurface wave induced separation. Normalized Strouhal numbers for shear layer instability (Stθ scaled by momentum thickness at separation θ), Karman instability (Sth scaled by wake thickness h), and flapping instability (StR scaled by mean reattachment length XR) show the effects of free surface and Fr on the instability mechanisms. Stθ (~0.013) for the laminar cases compare well with existing values for airfoils. Stθ for the turbulent cases have lower values that decrease with Fr. Stθ =0.0077 and 0.0052 for Fr=0.37 (Re=1.5e6) and 0.55 (Re=2e6) respectively. Sth for free-surface Karman shedding lies in a reduced range 0.065-0.069 compared to flows without free surface (0.07-0.09). StR = 0.2 lies in similar range as that for backward facing steps and blunt cylinders. A theoretical model constructed based on simple harmonic motion gives a good prediction of the flapping frequency. Unlike RANS, DES predicts nonreattaching separation for Fr=0.37. DES captures the same frequency for shear layer instability along with a much broader range of frequencies. DES provides a broader range of vortical structures in the separated regions and resolves more turbulent structures than URANS. Preliminary results using DES with single-phase level-set method show spilling wave near the toe which surface-tracking method does not capture. The overall results provide credible description of the flow physics.
INTRODUCTION Three-dimensional flow separation, in general by itself, is a vast area of study and there are still a lot to understand. The free surface adds to the complications due to free-surface deformations, freesurface-boundary layer interactions, turbulence, air/water interface, bubble entrainment, surface tension effects, and free-surface vortex interactions. In recent years, researchers have started paying increasing attention to small-scale processes that arise on or near the surface and the associated interactions between the surface motions and the underlying flows. Gaining insight into the fluid mechanics of the above-mentioned areas would be of both fundamental and practical interest, especially regarding applications to ship hydrodynamics. Relatively less work has been done in the area of free surface wave induced separation compared to other three dimensional separations. Chow [1] first
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discovered this phenomenon using a surface-piercing foil designed for insignificant separation for the deep condition. Stern et al. [2] made similar observations in their study of the effects of waves on the boundary layer of a surface-piercing flat plate with an upstream horizontal foil generating the Stokes waves in a towing-tank. For medium Fr separation initiated just beyond the wave trough and extended to the foil trailing edge. On the free surface, the separation region was wedge shaped, broken, and turbulent with vortical flow upstream towards the plate. Zhang and Stern [3] performed steady RANS simulations using free surface tracking and Baldwin-Lomax turbulence modeling along with wave-profile experiments for a surface-piercing NACA 0024 foil, which also has insignificant separation at large depths. The separation region on the free surface and plate is qualitatively similar in shape and depth as described by [1, 2]. The separation was naturally unsteady with two major vortices in the separation region, one going up the free surface and the other into the wake.
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There are some general approaches to study threedimensional flow separation. One is the study of the flow topology by examining the skin friction lines [9]. The other is by a) visualization and categorization of the gross separation patterns, b) identifying and analyzing the different frequencies associated with the separated shear layer and vortex structures and c) analyzing their interactions with the body and/or interactions with each other. According to Kiya [10], separated flows fall into two categories for this type of analysis: 1. Separation with reattachment (the separating shear layer reattaches to the body in backward facing step, leading edge separation bubbles of airfoils, blunt plates, or blunt cylinders) characterized by interactions between vortices and the solid surface and 2. Separation without reattachment (cylindrical bluff bodies like circular cylinders, spheres, and normal plates) characterized by interactions between opposite signed vortices shed from the separation points. Researchers have identified three main instabilities in separated flows: the initial shear layer instability, which causes the roll up of the separated shear layer, the Karman type instability due to the interaction between opposite signed vortices, and a low frequency instability ranging from 1/3 to 1/10 of the vortex shedding frequency. The frequency of the initial instability (f SL) which causes roll up of the separating shear layer to form vortices scales inversely with the momentum thickness at separation (θ) and directly with the convection velocity (UC) of the separated shear layer. Stθ = f SL θ / UC.
(1)
UC is approximately US/2 at separation, US given by Bernoulli’s equation using base pressure at separation Michalke [11] used inviscid linearised spatial stability theory to find the growth rates in a free boundary layer of axisymmetric and plane jets and found good match with the experimental values with Stθ = 0.017. Bloor [12] demonstrated that the shear layer instability of a cylinder normalized by the cylinder diameter as the length scale (fSLD/U) scaled approximately with Re-1/2 and suggested that θ varies with Re1/2 thus ensuring Re independence of fSLθ/U. Early work by Freymuth [13] on a separating laminar boundary layer also show that by using θ as the length scale the instability properties of the disturbed shear layer are independent of Re. Later, Gaster [14] and
5th Osaka Colloquium, Mar 14-15, 2005, OSAKA, JAPAN
Pauley et al. [15] found similar Re independency for the shear layer roll up of leading edge laminar separation bubbles in airfoils. Pauley et al. [15] simulated three cases for Re=59629, 120544 and 238515. For all cases, the Stθ was 0.0137. Ripley and Pauley [16] geometry gave a Stθ of 0.011 for Re = 364747, 209038 and 113928. Conclusion was that Stθ is Re independent, but depends on the nondimensional pressure distribution. Chart 1 provides a list of Stθ for laminar and turbulent cases for geometries investigated until now. The other major instability in separated flows, the most widely recognized and studied, is the Karman instability. Von Karman in 1912 investigated the linear stability of point vortex configurations and showed that two rows of oppositely signed vortices were unstable in both symmetric and asymmetric configurations. Almost all flows past bluff bodies exhibit this type of instability. The symmetric configuration occurs in reattaching flows where the interaction of the vortices is with the mirror image on the wall. In all symmetric type vortex shedding investigated the Karman shedding process is as follows; the shear layer vortices merge to form a large vortex that impinges on the body, interacts with its mirror image, and sheds. Here the successive merging of the shear layer vortices are the precursor to the Karman type vortex shedding. Backward facing step and blunt cylinders demonstrate this type of shedding. Similarly, for asymmetric vortex shedding which occurs in non-reattaching flows like in circular cylinders, the shear layer vortices amalgamate into the primary Karman vortex, before the vortex sheds downstream. But unlike flows with reattaching shear layers this amalgamation is not the precursor to the Karman shedding, evident from the fact that the shear layer vortices were not detectable below Re=1200 [17], but Karman shedding is still observed at Re as low as 150 [18]. Rosko [19] explained this lack of shear layer vortices as follows. There are two criteria for the shear-layer instability vortices to appear in the near wake. First, at least one wavelength has to fit into the formation length, wavelength given by λ/D ~ 37 / Re1/2. Second, the instability should have reached a significant level of amplification within the formation length. The transition region XT for the required amplification is XT/D ~ 75/Re1/2 [20]. The first criteria is not met for Re
