Modelling and Simulation in Engineering

Volume 2015, Article ID 565417, 13 pages

http://dx.doi.org/10.1155/2015/565417

## Numerical Investigation on Vortex Shedding from a Hydrofoil with a Beveled Trailing Edge

^{1}Research Institute of Marine Systems Engineering, Seoul National University, Seoul 08826, Republic of Korea^{2}Department of Naval Architecture and Ocean Engineering, Seoul National University, Seoul 08826, Republic of Korea

Received 16 May 2015; Revised 10 August 2015; Accepted 17 August 2015

Academic Editor: Jean-Michel Bergheau

Copyright © 2015 Seung-Jae Lee et al. 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

To better understand the vortex shedding mechanism and to assess the capability of our numerical methodology, we conducted numerical investigations of vortex shedding from truncated and oblique trailing edges of a modified NACA 0009 hydrofoil. The hybrid particle-mesh method and the vorticity-based subgrid scale model were employed to simulate these turbulent wake flows. The hybrid particle-mesh method combines the vortex-in-cell and the penalization methods. We have implemented numerical schemes to more efficiently use available computational resources. In this
study, we numerically investigated vortex shedding from various beveled trailing edges at a Reynolds number of 10^{6}. We then compared the numerical results with the experimental data, which show good agreement. We also conducted numerical simulations of wakes behind the hydrofoil at rest in periodically varying flows. Results reveal that vortex shedding is affected by the periodicity of a free-stream flow, as well as the trailing-edge shape.

#### 1. Introduction

The vortex shedding phenomenon is encountered in many practical engineering applications and physical sciences, and it is an important characteristic of flows past a bluff body. A vortex sheet shed from a solid body consists of alternating vortices of strength that produce their own velocity field, superimposed on the free-stream velocity. These shedding vortices cause an oscillating force component perpendicular to the direction of flow. This force can induce a vibration in the body. Vortex shedding behind circular and square cylinders is a benchmark problem that has been well investigated and is addressed in a vast amount of literature. Several scholarly reviews [1–4] are entirely devoted to the state of the art of this problem. In contrast, vortex shedding from a streamlined body such as a hydrofoil has been studied to a much lesser extent despite its being of direct relevance to practical engineering problems in hydraulic turbines, pumps, and marine propellers. Propeller-singing is well known as a critical vibration phenomenon generated by the interaction between a Kármán vortex-shedding mechanism from the trailing-edge of a blade and its natural frequencies [5].

Some laboratory experiments have been conducted on vortex shedding from the trailing edge of a hydrofoil. Bourgoyne et al. [6, 7] experimentally investigated the dominant features of flows over a two-dimensional (2D) hydrofoil with an antisinging trailing edge at Reynolds numbers ranging from to . They concluded that turbulent fluctuations in the near wake are Reynolds-number dependent because of the varying strengths of the structured near-wake vortex shedding. Mulvany et al. [8] numerically simulated turbulent flows around a modified NACA 16 hydrofoil using four turbulent models available in the commercial computational fluid dynamics (CFD) code FLUENT. They compared their numerical results with those from the experimental data of Bourgoyne et al. [6]. The authors mainly discussed the capabilities of the four different models and did not report any vortex shedding phenomenon. Ausoni et al. [9] performed laboratory-scale experiments to investigate the von Kármán vortex street generated in the wake of a 2D hydrofoil with a truncated trailing edge. Their experiments were performed at a zero angle of attack for a Reynolds number ranging from to , based on the hydrofoil chord length [9, 10]. Their experimental results showed that a lock-in phenomenon occurs, where the vortex shedding frequency is locked onto the natural frequency of a hydrofoil. Interestingly, it has been reported that cavitation has a minor effect on the wake dynamic. Recently, Zobeiri et al. [11] investigated two NACA 0009 hydrofoils with blunt and oblique trailing edges, respectively, for Reynolds numbers ranging from to and conducted high-speed visualization and flow-induced vibration measurements. They confirmed experimentally that flow-induced vibration was significantly reduced with an oblique trailing edge compared with a truncated edge. They also concluded that the collision between upper and lower vortices and the resulting vorticity redistribution were the main reasons for the obtained vibration reduction with the oblique trailing edge.

This study numerically investigated vortex shedding from truncated and oblique trailing edges of a hydrofoil to better understand the vortex shedding mechanism and to assess the capability of our numerical methodology. We employed the hybrid particle-mesh method and the vorticity-based subgrid scale model to undertake a numerical flow simulation. The hybrid particle-mesh method is a combination of the vortex-in-cell (VIC) method (see [12–15] and references therein) and the penalization technique [16], which has been developed in recent years (see [17–24] for a review). The hybrid particle-mesh method enables the use of fast and efficient techniques for computing differential operators, thereby enabling large scale simulations. However, techniques to save computational memory and CPU time consumption are still required. Thus, we have introduced the implementations of numerical schemes to more efficiently use available computational resources [25, 26]. With these numerical schemes, we numerically investigate vortex shedding from various beveled trailing edges of a NACA 0009 hydrofoil at a Reynolds number of . We then compare our numerical results with experimental data [10, 11] to assess the capability of our numerical methodology. In addition, we investigate the influence of periodically varying free-stream flows on vortex shedding to better understand the vortex shedding mechanism.

The organization of the remainder of this paper is as follows: we present a brief description of the numerical methods used for the vortex shedding simulation in Section 2. We describe our implementation of numerical schemes to efficiently use available computational resources in Section 3 and the computational procedure followed in Section 4. We present out vortex shedding results from a variety of beveled trailing edges and discuss the influence of periodic free-stream flows in Section 5. In Section 6, we provide a summary of this study and some perspectives for future work.

#### 2. Governing Equations and Numerical Methods

The vorticity-velocity formulation of the Navier-Stokes equations allows a purely kinematical problem to be decoupled from the pressure term, which is eliminated by applying the curl operator. Pressure which can be evaluated in an explicit manner using the identified vorticity and velocity fields [27] is not part of the solution algorithm. For a 2D flow parallel to the -plane, the vorticity transport equation in terms of the vorticity can be expressed aswhere is the scalar plane component of the vorticity vector; that is, in two dimensions. The evolution of a flow is considered in discrete time steps. The viscous splitting algorithm can be expressed in a Lagrangian framework aswhere is the material derivative. discrete Lagrangian fluid particles with a finite core size are linearly superposed to approximate the vorticity field aswhere is a mollification of the Dirac-delta function [12]. Finite core size vortices are used instead of point vortices. Each particle is characterized by its position and its strength . For a given vortex particle, the circulation is identical to the product of the vorticity and the area of the vortex particle which also represents the contribution of the vortex particle to the vorticity field, . The vorticity-carrying fluid particle is advanced with the velocity and is gradually diffused because of viscous effects.

##### 2.1. Vortex-in-Cell (VIC) Method for Convection

Since the continuity equation and the definition of vorticity lead to and , the velocity of (2a) can be expressed as the gradients of the stream-function . The velocity vector is expressed aswhere is the free-stream velocity and is the rotational velocity. The Poisson equation for the stream-function is . In two dimensions, the stream function is scalar, , and only the vorticity component perpendicular to the plane is nonzero . The vector Poisson equation reduces to a scalar Poisson equation:Rotational velocity is defined by and . In the VIC method, stream functions to evaluate velocities are computed using a uniform Cartesian mesh. The fast Fourier transform- (FFT-) based Poisson solver reduces the computational cost of obtaining the velocity field to where is the number of grid points [12]. To compute the stream functions using the FFT-based Poisson solver, the vorticity is interpolated onto equally spaced Cartesian grid usingwhere is the mesh spacing, using the following third order interpolation kernel [28, 29]: in each coordinate direction. This kernel conserves the 0th, 1st, and 2nd order moments. The subscripts and denote the grid and particle quantities, respectively. A vortex particle contributes to the nearest 16 nodes through the scheme, and the total vorticity at each node is obtained by summing the vorticity contributions of all the vortex particles.

Boundary conditions for the stream function are required to solve the Poisson equation. If the computational domain boundaries are far enough from the particles, homogeneous Dirichlet boundary conditions () may be used. However, using a larger domain is inefficient since it requires a regular grid with too many points. To compactly place a square FFT-domain , we used nonhomogeneous Dirichlet boundary conditions in this study. Unknown stream functions at the domain boundaries () can be directly obtained from Green’s function solution [27], wherebywhere , , and . denotes the number of domain boundary points and the circulation strength of each vortex particle . We note here that all particle vorticity values should be considered. Fast evaluation of the stream functions at the domain boundaries is further discussed in Section 3.3.

##### 2.2. Brinkman Penalization Method for Diffusion

The penalization method was initially designed to take into account solid obstacles in fluid flows [16]. The main point of the penalty term is to replace the usual vorticity creation algorithm for enforcing the no-slip condition for a solid body. By adding the penalization term to (1), the vorticity transport equation becomeswhere is the velocity of the solid body and denotes a mask function that yields in a fluid and in a solid [16, 18]. indicates the penalization parameter with the dimension . To avoid too small time step, the penalization term is evaluated by the implicit expression [16, 18, 30]:whereThis scheme is unconditionally stable [19]. Equation (9) is rewritten by replacing its terms with those in (10) and (11) as follows:The penalization and diffusion terms of (12) are consecutively evaluated on a temporary grid. To reduce any error due to the penalization term, we used a mask function with second-order accuracy, for which the boundary is located at the midpoint of the grid, where the mask function jumps from 0 to 1 [18, 31].

##### 2.3. Smagorinsky Turbulence Model

The filtered vorticity transport equation for a Lagrangian, vorticity-based large-eddy simulation (LES) in two-dimensions can be expressed as [32]where the bar denotes spatial filtering at length scale . The subgrid-scale vorticity stress term , which accounts for the effect of unresolved velocity and vorticity fluctuations, can be defined as . Smagorinsky eddy-viscosity model is one of most commonly used subgrid-scale (SGS) turbulence models. In spite of its simplicity, the Smagorinsky model is still extensively used in practice and has served as a basic tool for the developments of LES modeling [33]. In this model, eddy viscosity is defined bywhere is a nondimensional Smagorinsky constant and is a subgrid characteristic length scale. is the resolved strain-rate tensor and is its magnitude. Here in incompressible flows (), using the Caswell formula [34]. Finally, the vorticity transport equation for the LES in two dimensions is rewritten as In 2D LES models, the filtering process is applied in only two directions, so is defined by the grid spacing. The Smagorinsky constant may vary with the type of flow and the Reynolds number. Values ranging from 0.1 to 0.24 have been suggested in the literatures [33, 35, 36]. The constant is prescribed and its standard value is typically 0.15 [37, 38].

#### 3. Efficient Implementation of Numerical Methods

##### 3.1. Particle-Based Domain Decomposition

Parallel machines and algorithms enable significant reduction in computing time and greater amount of available memory. For high-performance computing, we used the Message Passing Interface (MPI) for our distributed systems, in which each processor has its own local memory. The appropriate decomposition of particles and/or grids should be employed. In domain or spatial decomposition, a given domain is geometrically decomposed by splitting it into several subdomains. Typically, the subdomains are uniform and particles are not evenly distributed among the subdomains. Differences in the number of particles assigned to each processor cause computational load imbalances. As some processors will sit idly while waiting for others, poor performance can result.

We introduce a simple idea to achieve a better load balance during the domain decomposition. In vortex particle methods, fluid particles must periodically be redistributed since their accumulation leads to inaccuracies in numerical solutions. This process is called particle remeshing or redistribution. The remeshing step creates a new set of particles on a uniform Cartesian grid, and then the randomly spaced old particles are removed. All the new particles are reindexed by from upstream to downstream; that is, higher-indexed particles are located further downstream. Due to the position-dependent index of the particles, the entire spatial domain can be easily split to ensure an equal amount of particles in each subdomain (each processor). As a result, the horizontal extent of the domain is unequal, but the particles are evenly distributed in subdomains of different sizes. In every remeshing step, the entire domain is dynamically partitioned. In this way, each subdomain contains an equal number of particles and better load balance is maintained during simulation. This approach is very simple and effective. No additional algorithms are required to evenly distribute the particles or to identify the position of the particles. As shown in Figure 1, we obtained an approximately 30% reduction in computation time simply by load balancing during parallel computation. This technique is also valuable for the efficient use of available computational memory.