Simulation of Supercooled Large Droplet Impingement via Reduced Order Technology

A IRCRAFT flying through clouds of supercooled liquid droplets (SLD) can be subjected to in-flight ice accretion. Surface tension prevents the expansion of the droplets that would occur with phase change, forcing them to remain in liquid form even though their temperature is below the freezing point. When the droplets hit an aircraft’s surfaces, the surface tension decreases at the contact point, and theymay freeze completely on impact if the temperature is very low or freeze partially at higher temperatures, whereas the remaining liquid portion runs back on the surface, transported by the pressure gradient and the shear stress of the airflow. If no ice protection is provided, the aerodynamic characteristics of the aircraft and its handling can be severely degraded when ice accretes. The increased drag generated by the roughness of the ice can lead to flow separation, reduction of the stall margins, control reversals, and engine blockages [1]. Airworthiness of transport airplanes in icing conditions is demonstrated by compliance with certification standards (Appendix C of the FAA Federal Aviation Regulations, part 25) set by agencies, such as the Federal Aviation Administration, European Air Safety Association, Transport Canada, etc. These standards, frozen for 50 years, will soon undergo significant revisions with the adoption of Appendix O to address the icing threat posed by SLD conditions. Unlike smaller droplets, SLD can distort, break into smaller droplets, splash, bounce off surfaces, get carried downstream by the flow, and reimpinge, increasing the potential for ice contamination on unprotected surfaces [2–4]. Nowadays, wind tunnel tests, icing tunnel tests, and computer simulations play major complementary roles in the process of certifying a new aircraft [5,6]. Advances in modeling capabilities have created the conditions to accurately simulate the ice accretion process in a realistic three-dimensional (3-D) context [7,8]. Unfortunately, the computational cost associated with performing a multitude of 3-D simulations of various aeroicing conditions somewhat limits the widespread use of computational fluid dynamics (CFD), even if advanced computational resources are available [9,10]. To overcome this difficulty, mostly low-cost and, consequently, low-fidelity tools are usually employed. These may be based on empirical correlations, two-dimensional (2-D) approximations, inviscid or incompressible flow assumptions, and/or other simplifications that result in limited accuracy and realism. A viable alternative is the reduced-order modeling (ROM) approach [11,12], which dramatically reduces the cost of highfidelity simulations while providing solutions of superior accuracy to low-fidelity methods because it preserves the detailed physical modeling of the problem under consideration [13–16]. Although the use of this approach in the aeroicing environment is in its pioneering phase, recent results support the effectiveness of this methodology as a valuable tool in the context of a multicondition, multiparameter certification process [17–20]. In a framework of ice accretion simulation as the succession of airflow, water concentration, and heat transfer calculations, the most time-consuming part is obtaining the water impact patterns. The common practice is to assume a distribution of discrete droplet diameters, compute the impingement distribution of each diameter class, andweight-average thesemonodispersed solutions. In the case of the droplet sizes of Appendix C, seven distinct monodispersed sizes (Langmuir-D distribution) are used to compute the overall impingement distribution. In the SLD regime, however, due to the complex phenomena of droplet breakup, splashing, and bouncing, distributions containing up to 27 diameters of monodispersed droplets are needed to obtain realistic impingement predictions [5–7]. The computational cost of an SLD simulation is hence four times that of a Langmuir-D distribution. In the presentwork, it will be shown that the ROM approach can dramatically reduce cost with only a very modest degradation of the overall accuracy of the simulation. The essentials of the ROM approach used to extract the solution eigenfunctions (or modes) and compute solutions at unknown states are introduced in sections II and III of this paper, whereas section IV illustrates the interpolation technique used to compute the surrogate solutions. Finally, 2and 3-D results and comparison with experiments and other methods are presented to validate the present methodology. Received 29 July 2011; revision received 14 September 2011; accepted for publication 15 September 2011. Copyright © 2011 by Wagdi Habashi. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0021-8669/12 and $10.00 in correspondence with the CCC. Postdoctoral Research Associate, Mechanical Engineering, Computational Fluid Dynamics Laboratory, 688 Sherbrooke Street West; mfossati@cfdlab.mcgill.ca. Member AIAA (Corresponding Author). Professor and Director, NSERC-J. Armand Bombardier-Bell HelicopterCAE Industrial Research Chair of Multidisciplinary Computational Fluid Dynamics Laboratory, Department of Mechanical Engineering, 688 Sherbrooke Street West, Fellow AIAA. Research &Development Director, 688 Sherbrooke StreetWest, Member AIAA. JOURNAL OF AIRCRAFT Vol. 49, No. 2, March–April 2012