G. A. Osswald

Simulation of dynamic stall phenomenon using unsteady Navier-Stokes equations☆

A non-inertial body-fixed generalized coordinate frame is employed to develop an unsteady Navier-Stokes (NS) analysis for arbitrary maneuvering bodies. A fully implicit direct numerical simulation (DNS) methodology is implemented for the solution of the governing equations. A velocity-vorticity (disturbance stream function - vorticity in 2-D) formulation is shown to simplify both the theoretical and numerical analysis of maneuvering flight when compared with the primitive-variable formulation. The resulting method is highly vectorizable and currently achieves a computational index of 6 micro-seconds per time step per mesh point using a single processor on the CRAY Y-MP 8864 at the Ohio Supercomputer Center. Vectorization has been achieved satisfactorily, with the code currently performing in the range of 130–180 Mflops. The rapid pitch-up maneuver of a NACA 0015 airfoil is wxamined at Re=1000 and Re=10 000, with particular emphasis being placed on the development and evolution of the dynamic stall vortex.

Analysis of two-dimensional incompressible flow past airfoils using unsteady Navier-Stokes equations

The flow over streamlined lifting airfoils has been a subject of considerable interest to fluid dynamicists, and to date, significant progress has been made towards the design of airfoils, wings, etc., by drawing together resources from experimental, numerical, analytical, and empirical studies. The detailed flow structure of airfoils and wings near maximum lift in low-to-high Reynolds-number (Re) flows still remains unresolved. The increasing interest in these flows stems from the desire for better control in civilian aircraft, and for high maneuvering capability in high-performance military aircraft. The improved performance can be realized from the potential of increasing maximum lift and simultaneously reducing drag under this condition. For some combination of flow parameters, the flow field around an airfoil experiences significant separation, which degrades its performance and leads to stall. The nature of the stall may be characterized by various phenomena such as separation, unsteadiness, transition, and turbulence. The present study is directed towards accurately simulating this flow field and providing further insight into this class of flows. Other important fluid-dynamics applications involving unsteady flows include blade rows in turbomachinery, marine propellers, helicopter rotor blades, and bluff bodies such as buildings, towers, underwater cables, etc., in cross flows. For this class of bluffy-body flows, understanding the vortex-shedding characteristics is very significant. The simulation technique presented here can also provide guidelines for analyzing some of these flow fields.

Velocity-Vorticity Simulation of Unsteady 3-D Viscous Flow within a Driven Cavity

A velocity-vorticity formulation of the unsteady three dimensional Navier-Stokes equations has been used to solve for the incompressible viscous flow within a driven cavity of spanwise aspect ratio 3:1 at a Reynolds number Re = 3200 on a non-uniform (65 × 65 × 49) grid covering one half of the span. An efficient Alternating-DirectionImplicit method which treats all cross derivative terms implicitly and which requires only six scalar tridiagonal sweeps has been developed for the vorticity transport equation. The divergence-curl formulation for the elliptic velocity problem is solved at each time step by a Multi-Grid Distributive Gauss-Seidel iterative scheme. The pressure is solved, only when desired, from the three-dimensional Pressure Poisson problem using a Multi-Grid Gauss-Seidel iterative scheme. Velocity, vorticity and pressure results are given for a characteristic time t = 50 after the upper surface of the cavity is impulsively started from rest.

Physics of forced unsteady flow for a NACA 0015 airfoil undergoing constant-rate pitch-up motion

The unsteady Navier-Stokes (NS) analysis of Osswald, Ghia and Ghia in velocity-vorticity variables is modified to study the dynamic stall phenomenon for a NACA 0015 airfoil undergoing constant Ω0 pitch-up maneuvers at Reynolds number Re = 10 000 and 45000. The use of third-order accurate biased upwind differencing for the nonlinear convective terms in the vorticity transport equation removes the spurious oscillations observed in the earlier studies by the authors for these values of Re. The fully implicit and vectorized ADI-BGE method of the authors is used to solve the unsteady NS equations. Instantaneous inertial surface vorticity, which is an invariant of the choice of reference frame selected, is employed to determine the location of separation of the boundary-layer flow on the suction surface; also a separation bubble embedded within the boundary layer is observed for both cases somewhere between the leading edge and the quarter-chord point. Primary, secondary, tertiary and quarternary vortices have been observed before the dynamic-stall vortex evolves and gathers its maximum strength.

Simulation of Self-Induced Unsteady Motion in the Near Wake of a Joukowski Airfoil

Recent impetus for research in unsteady separated flows stems from a wide range of applications from low- to high- Reynolds number, Re. The physics of high-Re flows, in general, is quite complex and often involves multiple nonuniqueness and chaos, beyond simple unsteady separation. For the low-Re case, e.g. in the manuevering of fighter aircraft at high angle-of-attack in near- and post-stall regime, the vortex interaction dominates the flow field. The passage of vortices over the suction surface and their subsequent shedding leads to self-excited persistently unsteady flows. This flow field is extremely complicated due to the global effect of unsteady separated flow, coupled with the presence of hydrodynamic instabilities which may trigger transition and eventually lead to chaos. Besides supermaneuverability, interest also lies in this low-Re case because of the need for design of efficient airfoil sections for Re in the range of 105–106, for improving the performance of mini-RPV’s (remotely piloted vehicles) operating at low altitudes, jet engine compressor and turbine blades, helicopter rotor blades, etc.

Study of Low-Reynolds Number Separated Flow Past the Wortmann FX 63-137 Airfoil

Flow past a Wortmann FX 63-137 airfoil is analyzed using the unsteady NavierStokes (NS) analysis developed earlier by the authors. The analysis is formulated using conservation form of the governing NS equations in terms of stream function, ψ, and vorticity, ω, in generalized coordinates. A Schwarz-Christoffel mapping technique is developed to provide surface-oriented coordinates for the Wortmann FX 63-137 airfoil. Suitable 1-D clustering transformations are used, not only to obtain a bounded computational domain, but, also to resolve the dominant length scales of the problem. Direct numerical simulation (DNS) methodology is used to obtain flow results at Re = 1,000 with αf varying from −5° to 10°; simulations are also performed at αf = 0° with Re = 10,000 and 100,000 without any turbulence model.

Direct Solution Methodologies for the Unsteady Dynamics of an Incompressible Fluid

A unified mathematical formulation of the unsteady two- and three- dimensional incompressible Navier-Stokes (NS) equations is presented. It contains the theoretical development from which efficient direct inversion methodologies are then developed. The numerical analysis leads to a single-pass direct inversion of the nonlinear partial differential operator (PDO) governing the unsteady motion of an incompressible viscous fluid within arbitrarily shaped boundaries. The method is uniformly second-order accurate both spatially and temporally. It is possible to extend this formulation to include the effect of compressibility.

Simulation of Separated Flow Past a Bluff Body Using Navier-Stokes Equations

An analysis is developed for simulating two-dimensional flow past a bluff body, using the incompressible unsteady Navier-Stokes equations in terms of vorticity and stream function. The conservation-law form of the equations is employed, in generalized orthogonal curvilinear coordinates, using the contravariant component of the vorticity. The fully implicit time-marching alternating-direction implicit-block Gaussian elimination (ADI-BGE) method employed is a direct method, with second-order spatial accuracy and, hence, avoids introduction of any artificial viscosity. This unsteady analysis has been carefully applied to simulate flow past a circular cylinder with and without symmetry, requiring the use of either the half or the full cylinder, respectively. For the latter configuration, at early time levels, the evolution of some flow characteristics in the near wake is compared extensively with existing experimental data, in order to partially validate the present analysis. For the half-cylinder configuration, the asymptotic flow structure predicted by the present analysis conforms, to some extent, with the existing numerical and analytical solutions. This analysis has the potential advantage of computing symmetric unstable modes as well as solution bifurcations which may result for flow up to Reynolds number Re = 104 or even Re = 105.

Efficient computation of unsteady vortical flow using flow-adaptive time-dependent grids

The objective of this study is to efficiently simulate vortex-dominated highly unsteady flows. In such flows, the locations as well as the extent of the regions requiring fine-mesh resolution vary with time. A technique has been developed to simulate these flows on a temporally adapting grid in which the adaption is based on the evolving flow solution. The flow in an axisymmetric constriction has been selected as an illustrative problem. The multiple and disparate length scales inherent in this complex flow make this problem ideally suited for evaluating the adaptive-grid technique. Adaption is based on the equidistribution of a weight function, through the use of forcing functions. The significance of this is that the method can be implemented into existing flow-analysis systems with minimal changes. The grid-generation equations developed are viewed as grid-transport equations. The time-dependent control functions perform the role of the convective speed in this transport mechanism. The equations provide the efficiency and flow tracking capability of parabolic equations, while maintaining the smoothness of computationally expensive elliptic equations. The efficiency and flow tracking capability of the approach is demonstrated for both steady and unsteady flows.

Analysis and control of dynamic stall phenomenon using Navier-Stokes formulation involving vorticity, stream function and circulation

AbstractAn unsteady Navier-Stokes (ns) analysis is developed for studying flow past a maneuvering body. The present inclusion ofcirculation in the earlierns analysis of the authors makes it feasible,for the first time, to accurately simulate the asymptotic far-field boundary condition. In the overall analysis, a clustered conformal mesh withC-grid topology is used, with the governing differential equations being solved using the implicitadi-bge technique. The effects of grid size and clustering on the flow solution, and the effect of grid stretching on the far-field solution, are studied using the flow configuration with Re=45,000 and constantrate pitch-up motion, with