Effects of Grid Interfaces/Quality on Wide-Stencil High Order Accurate Simulations of Vortex-Wake

This paper focuses upon the use of fifth/seventh order spatially accurate, Essentially Non-Oscillatory (ENO), and weighted ENO (WENO) schemes for capturing vortices while minimizing numerical dissipation. ENO/WENO schemes are not compact, and hence a wide stencil of information is required to construct a high fidelity solution. However, the accuracy of the captured physics of vortex dynamics is affected by the how the solution is constructed near inter-grid interfaces, and the quality of the underlying grids. Recent research at Georgia Tech indicates that grid quality -especially grid stretching plays a crucial role in correctly convecting vortex wake associated with a UH60A rotor-blade in hover, when wide-stencil high order schemes are used. A controlled study of selfconvecting vortex system using high order ENO/WENO schemes in relation to grid interfaces/stretching will provide insights that can be useful for larger hover computations. The objective of this study is to characterize the effects of grid interfaces, and grid quality on the accurate computation of convection of vortices in external and self-induced flow fields, using high order schemes. Results indicate that the computed strength, and convection speed of a system of Lamb’s vortices is strongly affected by grid interfaces, and stretching. Nomenclature |A| = Roe’s dissipation matrix F = flux vector for inviscid fluxes Fv = flux vector for viscous fluxes q = vector of primitive flow variables qL = left hand side flow vector at a given face qR = right hand side flow vector at a given face ∆S = face surface area VF = fluid velocity VG = grid velocity ΩJ = cell volume A(.),E(.), R(.) = Operators τ = time vi = induced velocity Ω = rotor angular velocity R = rotor disk radius Mtip = tip Mach number (for rotor blade) CT = thrust coefficient λ = advance ratio M∞ = freestream Mach number α = angle of attack ξ,η,ζ = structured grid generalized coordinate directions i,j,k = Cartesian directional unit vector 1.0 Introduction The flow field around a rotor, whether in forward flight or hover, is difficult to model due to the presence of strong vorticity. The flow phenomena for a rotor differ from that for a wing-inforward-flight, because of the differing influence of their respective wakes. For a wing in forward flight, the generated tip vortex and the vortex sheet are quickly convected away from the wing, and the influence of the shed wake on the flow field in the vicinity of the wing is small. For an adequate numerical simulation of a wing in forward flight, it is sufficient to capture the generated tip vortex in the vicinity of the wing. In contrast, in the flow field around a rotor, a strong vortex wake system lingers in the vicinity of the rotor. In hover, the wake vortex coils beneath the rotor, and significantly alters the effective angle of attack of the rotor. 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 5 8 January 2009, Orlando, Florida AIAA 2009-48 Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 2 Accurate numerical prediction of rotor-blade aerodynamic parameters such as thrust coefficient and induced torque coefficient requires an accurate modeling of the wake vortex. In forward flight the entire vortex system is swept back leading to strong interactions between the blade-tip vortices and the successive blades, a phenomenon known as the bladevortex interaction (BVI). These blade vortex interactions result in rapid changes in local flow conditions and are a major source of aerodynamic noise and structural vibration. The underlying issue in modeling rotorcraft aerodynamics is the necessity to fully account for the complex vortex system generated by the rotor blades. Researchers in the past two decades have adopted a broad class of methodologies with various levels of complexity to model the vortex system. Until recently this representation was externally input from empirical/analytical models because full Euler/NavierStokes (NS) computations were infeasible. With the enormous advances in computational methodologies and computational power researchers have been adapting Euler/NS techniques for the study of the rotorflowfield. These solvers are particularly useful in analyzing new or complex rotor blades where no experimental data is available. Studies by Srinivasan and Ahmad [1], Strawn and Barth [2], and Duque [3, 4] have used a variety of strategies such as unstructured methodologies, and overset methodologies to tackle this problem. An excellent survey article by McCroskey [5] gives a comprehensive review of modern computational strategies for rotor applications. Such methodologies that solve for the flow field from the basic conservation laws without using additional information (information from analyses such as other numerical formulations, analytical formulations, or experimental observations) are generally referred to as "first-principles" based methods. Traditional low-order spatially accurate Euler/Navier-Stokes computational methodologies tend to dissipate the vortex wake system due to the high numerical dissipation inbuilt in such numerical schemes [5]. Some amount of dissipation is essential for numerical schemes to damp high frequency oscillations. However, such a procedure should not diffuse legitimate flow features –such as vorticitythat exhibit sharp gradients in flow properties. Essentially Non-Oscillatory (ENO) schemes developed by Shu and Osher [6, 7], Harten [8] offer an elegant approach to constructing high-order accurate, low dissipation numerical schemes. One of the earliest, relatively successful, attempts by Hariharan and Sankar [9, 10] to capture 3-D rotary-wing vortex wakes utilized the ENO methodology. In recent years various other approaches to building high order schemes for rotorwake capturing have also been proposed. Schemes such as DRP schemes [11] and projected MUSCL [12] have been tried out for studying rotorcraft wakes with varying degrees of success. Compact high order schemes – schemes that use information only from any given cell and its neighbors – such as Discontinuous Galerkin [13] have been moderately successful for 3D unsteady vortex capturing so far. More research is needed to refine high order methods further to arrive at an optimal scheme applicable for 3D calculations. This paper focuses specifically upon the use of fifth/seventh order spatially accurate, Essentially NonOscillatory schemes for capturing vortices while minimizing numerical dissipation. ENO schemes are not compact, and hence a wide stencil of information is required to construct a high fidelity solution. This becomes a problem when large scale simulations are broken into parts and distributed over a computing network, making it necessary to have sufficient overlap between the grid-parts to construct uniformly high order accurate solution. Recent research by Vasilescu et al. [14 ], and Sankar et al. [15] also indicates that grid quality -especially grid stretching plays a crucial role in correctly convecting vortex wake associated with a UH60A rotor-blade in hover, when wide-stencil high order schemes are used. Figure 1a shows the firstpassage tip vortex computed using highly stretched grid cells (from Reference [15]). The first-passage tip vortex is incorrectly predicted to go over the subsequent blade. Figure 1b shows a similar plot when the computation was augmented using adaptive mesh refinement which alleviated the excessive grid stretching (from Reference [15]). The first-passage tip vortex is correctly predicted to go underneath the subsequent blade. Figure 1c shows a similar comparison with a closer view of the first-passage tip vortex. The objective of this study is the analysis of the quality of the vortex preservation, and stability of computed self-induced vortex wake systems when wide-stencil high order schemes are used, in relation to the underlying grid quality (mismatched grid oversetting, grid-stretching). First, an ENO based algorithmic strategy to ensure smooth transfer of vortices across grid interfaces is described. Next, a study on how such vortex-transfers transfers are affected when overset grids of varying grid densities are involved is presented. Finally, a controlled study of self-convecting vortex system using high order ENO/WENO schemes in relation to grid stretching/quality is presented.. The next section briefly describes the formulation of typical ENO schemes. Section 3 presents results.