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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.

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

Some laboratory experiments have been conducted on vortex shedding from the trailing edge of a hydrofoil. Bourgoyne et al. [

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 [

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

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 [

Since the continuity equation and the definition of vorticity lead to

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 (

The penalization method was initially designed to take into account solid obstacles in fluid flows [

The filtered vorticity transport equation for a Lagrangian, vorticity-based large-eddy simulation (LES) in two-dimensions can be expressed as [

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

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

Computation times for equal- and unequal-sized subdomains.

To avoid an excessive domain size, we considered multiple domains. In this study, the entire domain

A substantial part of the overall computation time is spent on the calculation of the boundary stream functions to solve Poisson’s equation for the stream function. This requires the order of

In this study, we attempted to reduce the number of boundary nodes

First, the entire simulation domain

Particle-based domain decomposition: as discussed in Section

Particle-to-grid interpolation: each processor interpolates the vorticity

Calculation of the boundary condition: each processor computes the boundary stream functions by direct calculation and interpolation using cubic splines, as discussed in Section

Evaluation of the velocity field: the stream function on the grid is computed with the FFT-based Poisson solver, from an open-source library called FFTW (fastest Fourier transform in the west) [

Evaluation of the vorticity field: each processor evaluates the evolution of the vorticity field in time as follows:

The penalization term, that is, the second term on the right hand side of (

The diffusion term, that is, the first term on the right hand side of (

The turbulent viscosity

Grid-to-particle interpolation: each processor interpolates the velocity and vorticity on the grid back to its own particle positions through the

Vortex particles are advanced in time using a midpoint predictor-corrector method as follows: particle positions are predicted by their velocities,

We selected a NACA 0009 hydrofoil with a truncated trailing edge for the numerical simulations. This hydrofoil has the same cross-section as the experimental model used by Ausoni et al. [

Trailing-edge shape. Note that the points A, B, and C indicate the knuckles of the chamfer and trailing-edge wedge, and the bevel angle is defined as

Numerical parameters can be determined using the stability condition

For an impulsively started flow past a hydrofoil at a chord-based Reynolds number of

Wake patterns of the NACA 0009 hydrofoil with a truncated trailing-edge (

Symmetric wake at

Symmetry-breaking transition at

Nonperiodic vortex shedding at

Fully developed periodic shedding at

Time evolution of the lift and drag coefficients for the impulsively started NACA 0009 hydrofoil with a truncated trailing edge (

From our simulation, the Strouhal number based on the trailing-edge thickness,

We conducted numerical simulations to observe vortex shedding from trailing edges with six different bevel angles of

Instantaneous vorticity distributions at

Bevel angle

Bevel angle

Bevel angle

Bevel angle

Bevel angle

Bevel angle

Power spectral density (PSD) of the lift and drag coefficients.

Truncated trailing edge,

Beveled trailing edge,

At a bevel angle of

Figure

Shedding frequency of vortices from the different beveled trailing edges for

Mean and standard deviation of velocities at

Normalized mean of streamwise velocity

Normalized mean of transverse velocity

Normalized standard deviation of streamwise velocity

Normalized standard deviation of transverse velocity

The propeller is generally located behind the ship’s hull and is subjected to a nonaxisymmetric wake field. Considering a 2D flow, the propeller section effectively experiences periodic variation in flow incidence. We conducted numerical simulations to investigate vortex shedding with respect to the sinusoidal motions of the free-stream flow. The hydrofoil was at rest and the flow became the oscillator under these conditions. In the current study, the angle of attack varied sinusoidally with time

Figure

Effect of the incoming flow frequency on vortex shedding from a beveled trailing edge of

Steady flow,

Incoming flow frequency

Incoming flow frequency

Incoming flow frequency

Incoming flow frequency

We simulated impulsively started flows past a modified NACA 0009 hydrofoil at a Reynolds number of

In this study, the numerical simulation results nearly match the experimental results. This demonstrates that vortex shedding from the trailing edge of a 2D hydrofoil can be reasonably simulated with a 2D LES. It is due to the fact that the flow is characterized by quasi-two dimensionality with vortex shedding from the trailing edge of the hydrofoil. We have considered that 2D vortex shedding simulations should not be dismissed so easily. With two-dimensionality, it is possible to solve the problem much less expensively, since 3D problems require significant computational resources. However, the LES should be 3D since turbulence is inherently 3D. While we accept that more realistic 3D simulation is required physically, a 2D LES can be used as a basis for comparison prior to 3D implementation. Once the capability of a numerical algorithm is evaluated in two dimensions, it can be easily extended into three dimensions. This paper is intended as a guide for researchers and designers who are working on a similar problem: vortex shedding from a hydrofoil at high Reynolds numbers.

We found in this study that the vortex formation characteristic in periodically varying inflows remains unchanged. A certain frequency of free-stream flow oscillation tends to make vortex shedding more regular again. This means that the periodicity of flow incidence can stimulate an inherent frequency of vortex shedding from a hydrofoil. However, the effect of inflow oscillation on vortex shedding is not yet fully understood. Future work will include experimental investigations, as well as a 3D flow simulation, to better understand the relationship between inflow oscillation and vortex shedding.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was supported by the Korean Institute of Ocean Science & Technology sponsored by the Ministry of Trade, Industry & Energy (no. 10033668) and the Industrial Convergence Strategic Technology Development Program (no. 10044499) funded by the Ministry of Trade, Industry & Energy (MI, Korea).