Modelling and Simulation in Engineering

Volume 2018, Article ID 9638430, 14 pages

https://doi.org/10.1155/2018/9638430

## Parametric Studies of Flat Plate Trajectories Using VIC and Penalization

^{1}TFT Laboratory, Mechanical Engineering Department, École de Technologie Supérieure, Montréal, QC, Canada^{2}Bordeaux INP, IMB, UMR 5251, 33400 Talence, France^{3}Inria Bordeaux Sud-Ouest, Team CARDAMOM, 200 Avenue de la Vieille Tour, 33405 Talence, France

Correspondence should be addressed to François Morency; ac.ltmste@ycnerom.siocnarf

Received 21 September 2017; Accepted 11 December 2017; Published 1 February 2018

Academic Editor: Zhiping Qiu

Copyright © 2018 François Morency and Héloïse Beaugendre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Flying debris is generated in several situations: when a roof is exposed to a storm, when ice accretes on rotating wind turbines, or during inflight aircraft deicing. Four dimensionless parameters play a role in the motion of flying debris. The goal of the present paper is to investigate the relative importance of four dimensionless parameters: the Reynolds number, the Froude number, the Tachikawa number, and the mass moment of inertia parameters. Flying debris trajectories are computed with a fluid-solid interaction model formulated for an incompressible 2D laminar flow. The rigid moving solid effects are modelled in the Navier-Stokes equations using penalization. A VIC scheme is used to solve the flow equations. The aerodynamic forces and moments are used to compute the acceleration and the velocity of the solid. A database of 64 trajectories is built using a two-level full factorial design for the four factors. The dispersion of the plate position at a given horizontal position decreases with the Froude number. Moreover, the Tachikawa number has a significant effect on the median plate position.

#### 1. Introduction

Flying debris, such as ice fragment, can cause serious damage to structures. One way to mitigate the risks associated with their impacts is to use numerical tools to predict their trajectories. However, ice fragment trajectories are stochastic, as many parameters affect the flying path and the flow [1]. In particular, trajectories are sensitive to the size, shape, location, and initial velocity of ice fragments [2]. Consequently, probability distribution maps are built to show the likely path of such fragments, which can lead to the development of risk mitigation solutions. A large number of trajectories are needed to obtain statistically relevant maps in 3D. Mathematical models enable the simulation of multiple trajectories for various initial conditions, which then allows the computation of probability maps around a geometry [3, 4]. The motion of the ice piece is driven mostly by aerodynamic forces. Research works covering wind engineering have established that three dimensionless parameters play a role in the motion of flying debris, apart from the Reynolds number that may play a role in aerodynamic forces [5–7].

For low fidelity models, motion equations are solved with a Runge-Kutta method in a Lagrangian frame of reference attached to the debris. The aerodynamic forces are computed using the static lift, drag, and moment coefficient correlations for the studied debris shape, neglecting the shape effects on the flow field. The two most studied shapes are the sphere [8] and the rectangular flat plate [6]. For the flat plate, the coefficient correlations depend only on the angle of attack, and additional aerodynamic terms must be added to take into account debris rotation [9]. For the sphere, the coefficient correlations depend only on the Reynolds number.

High fidelity models involve a tight coupling of the flow field to the flying debris motion. These models use a computational fluid dynamic (CFD) to solve the Navier-Stokes equations around the moving debris in an Eulerian frame of reference [7, 10], and the forces and moments are computed from the flow solution. The computational costs of such models are higher than those for models uncoupled from the flow field, especially as the flow Reynolds number increases. However, high fidelity models allow the computation of the trajectory of any kind of shape and do not require experimental databases or correlations for aerodynamic forces and moments. Also, high fidelity models take into account Reynolds number effects and lift and drag fluctuations (10% to 20%) caused by vortex shedding [11, 12] in the case of flow separation around debris. Both models are complementary and need research developments to better assess the risk of impacts.

The goal of the present paper is to investigate the relative importance of four dimensionless parameters: the Reynolds number, the Froude number, the Tachikawa number, and the mass moment of inertia parameter. A secondary objective of the paper is to study the effect of launching time on the trajectories. The focus is on the start of the motion, when aerodynamic forces on the ice fragment are high. A trajectory database is built for a 2D rectangular plate at Reynold numbers around 7000, but with other parameter values typical of inflight aircraft deicing. The plate makes an initial angle with the flow field such that flow separation occurs.

The CFD solver uses an immersed boundary method (IBM) to model the flow around the plate geometry [17]. A Cartesian grid is used for the computational domain. The plate boundaries are modelled by adding continuous forcing terms directly to the flow equations using the penalization method [18–20].

The paper is organized as follows. First, the penalized Navier-Stokes equations are presented in Section 2, together with the numerical method based on a Vortex-In-Cell (VIC) scheme. The model used to compute the forces and moments on the plate, along with the appropriate governing equations for fluid-solid interactions, is also described. In Section 3, test cases are presented to validate computations of forces and moments against literature results. In Section 4, a database is created to study the influences of the four parameters and initial angle of attack on the trajectories of a 2D rectangular plate.

#### 2. Mathematical Model and Numerical Method

The fluid-solid interaction model consists of a set of equations for the fluid flow and another set for the solid motion. The effects of the flow on the solid motion are imposed through aerodynamic forces and moments, while those of the solid on the flow are imposed by penalization.

##### 2.1. Fluid Motion

The incompressible laminar Navier-Stokes equations model the flow into the computational domain . A rigid solid , moving at velocity in the domain , is defined with a level set function . In this work, is the signed distance function to , negative inside the solid and positive outside. The boundaries of are located where the sign of the level set function changes from positive to negative. Figure 1 illustrates the signed distance function around the plate. A Cartesian grid is used to discretize the domain .