Vukčević, Vuko and Cvijetić, Gregor and Gatin, Inno and Jasak, Hrvoje
(2016)
*The harmonic balance method for temporally periodic free surface flows in marine hydrodynamics.*
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*The harmonic balance method for temporally periodic free surface flows in marine hydrodynamics.*
In: 11th OpenFOAM Workshop, 26-30.06.2016., Guimaraes, Portugal.

## Abstract

The Harmonic Balance Method (HBM) for temporally periodic, non–linear, turbulent, free surface flows is presented in this work. The methodology is primarily targeted to efficient simulations related to wave–structure interaction in naval and offshore hydrodynamics. Transient flows in marine hydrodynamics are often periodic, e.g. due to ocean waves (wave propagation and diffraction, seakeeping of a ship) and rotating propellers. Such flows often have well–defined base frequency: wave frequency or rotational frequency of the propeller. In fully non–linear, two–phase state–of–the–art CFD algorithms, such flows are almost exclusively resolved in time domain. Transient simulations usually require a large number of periods in order to achieve harmonically steady (purely oscillatory) solution. Due to its spectral decomposition, HBM allows us to efficiently model flow effects up to a specified order, without performing a fully transient simulation. Hence, a substantial performance improvement is expected, with almost negligible decrease in accuracy for flows with a well–defined base frequency. Due to the steady–state mathematical formulation of the HBM, authors believe that the method is highly suitable for adjoint optimisation regarding seakeeping of marine objects in the ship–building industry. This suitability has been recently confirmed by Huang and Ekici [1], who developed an adjoint shape optimisation tool based on the HBM for turbomachinery applications. The HBM [2] has been originally developed to tackle periodic single–phase turbomachinery flows in an efficient way. This paper presents an extension of the single-phase HBM [3, 4] to two–phase free surface flows, comparing the results and computational efficiency with a transient solution. The implementation is carried out in a second– order accurate, polyhedral Finite Volume framework developed within foam-extend, a community driven fork of the OpenFOAM software. In the HBM, a transient governing equation set is replaced with a specified number of coupled steady state problems, each represented by an equation for a unique time instant. The method simulates a periodic flow by evaluating the temporal derivative via spectral decomposition, yielding flow solution at discrete instants in time simultaneously. Multi–mode transformation from a transient to a set of coupled steady state problems is achieved by a Fourier transform, assuming temporally periodic flow. The accuracy of the model is controlled by a specified number of harmonics to allow efficient capturing of higher order flow effects. Generally, specified number of harmonics n, yields solutions at 2n + 1 discrete time instants (n for real and imaginary parts and 1 for the mean component). As an example, HBM simulation with n = 1 yields solution at t = 1=3T ; 2=3T and T, where T is the specified period. As a post processing step, the three solutions for n = 1 can be used to transform the results into frequency domain, up to first order. In the present study, HBM is applied to Navier-Stokes equations and Level Set interface capturing equation, yielding a coupled set of two-phase flow equations for discrete instants of time within one period. In addition, SWENSE decomposition is used to facilitate incident wave propagation. Preliminary results are presented for a 2D free–surface flow over a ramp, Figure 1, where the inlet free surface height, h1 is set to 1 m. ~U is the periodic inlet velocity with mean value ~U0 = (6 ; 0 ; 0), first order amplitude ~U1 = (1 ; 0 ; 0) and frequency f = 0:5 Hz. Density ratio of water and air is 1000, and the CFD domain extends 1 m above the initial free surface position. Gravity is set to ~g = (0 ;

Item Type: | Conference or Workshop Item (Lecture) |
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Keywords (Croatian): | harmonic balance method; free surface flows; marine hydrodynamics; wave modelling; foam-extend |

Subjects: | TECHNICAL SCIENCE > Shipbuilding |

Divisions: | 500 Department of Energy, Power Engineering and Environment > 530 Chair of Turbomachinery |

Date Deposited: | 02 Mar 2018 16:02 |

Last Modified: | 02 Mar 2018 16:02 |

URI: | http://repozitorij.fsb.hr/id/eprint/8502 |

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