Analytical Prediction of Trajectories of Ice Pieces after Release in an Airstream

lf ice accretes on an airframe ahead of the turbine engines, it may cause engine damage if it enters the inlet after being shed. An analytical method was developed to predict the trajectories of ice pieces that are shed from an airframe into the surrounding flow field to determine if there is a probability that the ice will enter the engine. The approach taken in this study was to model the aerodynamics and kinematics of a piece of ice. After the ice enters the flow field, the aerodynamic forces and moments will cause it to tumble, accelerate across the streamlines, and accelerate to the freestream velocity. Gravity will also influence the trajectory. By systematically varying the size, thickness, initial angle of attack, and rotational damping, it is possible to predict a maximum locus of trajectories for a given geometry and flight condition. NOMENCLATURE C Pitch damping coefficient CD Drag coefficient CL Lift coefficient D Drag L Lift m Mass of ice piece S Area of ice piece V Relative speed VT Airplane true airspeed Vx Speed in the longitudinal direction VY Speed in the lateral direction Copyright © 2000 The American Institute of Aeronautics and Astronautics, Inc. AH rights reserved. Vz Speed in the vertical direction X Location along the longitudinal axis y Incidence angle of the ice piece p Air density 9 Angular position of the ice piece 9 Angular velocity of the ice piece 9 Angular acceleration of the ice piece INTRODUCTION When ice accretes on an airplane surface upstream of an engine inlet, it is possible for pieces of ice to damage the engine if the ice is shed, usually by a deicing system or an increase in ambient temperature above the freezing temperature. No damage will occur if the pieces of ice are small or if the trajectory is such that the ice pieces miss the engine inlet. When designing airplanes, it often necessary to predict the trajectories of released pieces of ice. This paper presents a method of predicting the trajectories of individual pieces of ice after they leave the surface upon which they accrete. It is well known that when ice breaks off a surface, an individual piece of ice will traverse an apparently random path downstream. Slow motion views of ice leaving a surface in flight show that the pieces often begin rotating as soon as they leave the surface and continue rotating in a nearly stable fashion until they disappear from view. Sometimes the pieces appear to randomly tumble upon release. Even though the movement of an individual piece of ice appears random, the locus of the trajectories is confined to a relatively small region of space (References 1 and 2). This locus can be determined by computing the 1 American Institute of Aeronautics & Astronautics c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. trajectories of individual ice pieces that represent the sizes of concern and the range of probable aerodynamic characteristics. The approach taken in this study was to create a simple model of the aerodynamic and kinematic characteristics of a generic square piece of ice with area and thickness as the independent variables. This shape was chosen because it is capable of generating significant lift forces. A random shape that approximates a spherical lump will be influenced primarily by drag and will therefore be accelerated down stream and fall due to gravity. There will be little lateral dispersion. The square piece of ice is released into a uniform flow field with an initial angle of attack such that the lift is in the lateral direction and the axis of rotation remains fixed in the vertical direction. Rotation of the ice shape is calculated based on the aerodynamic pitching moment, aerodynamic damping, and the moment of inertia. The initial forces acting on the piece will be lift, drag, pitching moment, and gravity. The square pieces will be unstable in pitch because the location of the center of pressure will be ahead of the center of mass. Therefore, the piece of ice will begin to rotate. As it rotates, the angle of incidence between the ice piece and the relative wind changes as it accelerates and moves downstream. This entire dynamic process was modeled to calculate individual trajectories. The assumption that rotational motion is confined to the vertical axis is conservative because if the piece of ice begins to roll or yaw, as it is likely to do, it will decrease the forces perpendicular to the assumed axis of rotation and reduce the dispersion distance in that direction. AERODYNAMIC FORCES AND MOMENTS The aerodynamic forces that act on the ice shape are lift and drag. The distribution of these forces along the surface of the ice shape determines the pitching moment applied to the piece of ice. Lift and drag are calculated using the standard equations Lairplane. As the piece of ice moves downstream, its relative velocity decreases and changes direction. The lift coefficient was based on the lift produced by a low aspect ratio flat plate as described by Hoerner in Reference 1. At zero angle of attack, the lift produced is zero. Low aspect ratio flat plates typically have a low lift curve slope and high stalling angle of attack because the flow around the edges of the plate dominates the flow field. As the angle of attack increases, separation on the suction side occurs, but the plate still produces some lift. At an angle of attack of 90 degrees, no lift is produced. For this study, the following was chosen to describe the lift curve: CL = 1.31 tan^, for y 35° Drag is clearly related to angle of attack. At a low angle of attack, the drag due to lift dominates; at a high angle of attack, pressure drag dominates. To model these characteristics with a continuous relationship, the following relationship was used: CD = 1.25(0.01 + 0.531CL), for y 35° These equations are based on fundamental drag relationships from Hoerner in Reference 2. There also exists experimental data on tumbling objects. Raglan, et al, give overall drag values for objects injected into a furnace in Reference 4. To match these data, a multiplier (1.25 in this case) is applied to these equations so that continuous rotation of a simulated object will give the same drag as the comparable shape as reported by Raglan, et al. The pitching moment calculation depends on the location of the center of pressure on the piece of ice. At low angles of attack, the center of pressure is located at the quarter chord point, but as the angle of attack increases, the center of pressure moves aft. The following equations were used to describe the pitching moment relation: When the piece of ice is initially released into the airstream, its relative velocity is the speed of the American Institute of Aeronautics & Astronautics c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.