The effect of non-linearity at the free surface on flow past a submerged cylinder

1965 ◽  
Vol 22 (2) ◽  
pp. 401-414 ◽  
Author(s):  
E. O. Tuck
Keyword(s):  
Free Surface ◽  
Surface Condition ◽  
Second Order ◽  
Order Effects ◽  
Surface Boundary ◽  
First Order ◽  
Flow Past ◽  
Free Surface Boundary Condition ◽  
Non Linear ◽  
Wave Induced

Plane potential flow past a circular cylinder beneath a free surface under gravity is investigated in order to determine the importance or otherwise of non-linear effects from the free-surface boundary condition. It is shown that non-linear second-order corrections to the first-order linearized expressions for the wave-induced forces on the cylinder are considerably larger than second-order effects which are present even with a linear free-surface condition. Further evidence for the importance of non-linearity is presented in the form of streamline plots of the first-order solution showing strange behaviour at wave crests.

Waves and wave resistance of thin bodies moving at low speed: the free-surface nonlinear effect

1975 ◽  
Vol 69 (2) ◽  
pp. 405-416 ◽  
Author(s):  
G. Dagan
Keyword(s):  
Free Surface ◽  
Froude Number ◽  
Surface Condition ◽  
Perturbation Expansion ◽  
Wave Resistance ◽  
The Body ◽  
Thin Body ◽  
Two Dimensional ◽  
First Order ◽  
Flow Past

The linearized theory of free-surface gravity flow past submerged or floating bodies is based on a perturbation expansion of the velocity potential in the slenderness parameter e with the Froude number F kept fixed. It is shown that, although the free-wave amplitude and the associated wave resistance tend to zero as F → 0, the linearized solution is not uniform in this limit: the ratio between the second- and first-order terms becomes unbounded as F → 0 with ε fixed. This non-uniformity (called ‘the second Froude number paradox’ in previous work) is related to the nonlinearity of the free-surface condition. Criteria for uniformity of the thin-body expansion, combining ε and F, are derived for two-dimensional flows. These criteria depend on the shape of the leading (and trailing) edge: as the shape becomes finer the linearized solution becomes valid for smaller F.Uniform first-order approximations for two-dimensional flow past submerged bodies are derived with the aid of the method of co-ordinate straining. The straining leads to an apparent displacement of the most singular points of the body contour (the leading and trailing edges for a smooth shape) and, therefore, to an apparent change in the effective Froude number.

A Second-Order Theory for the Potential Flow About Thin Hydrofoils

1984 ◽  
Vol 28 (01) ◽  
pp. 55-64
Author(s):  
Colen Kennell ◽  
Allen Plotkin
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Potential Flow ◽  
Order Theory ◽  
Wave Drag ◽  
Second Order ◽  
The Body ◽  
Surface Shape ◽  
Surface Boundary ◽  
Free Surface Boundary Condition

This research addresses the potential flow about a thin two-dimensional hydrofoil moving with constant velocity at a fixed depth beneath a free surface. The thickness-to-chord ratio of the hydrofoil and disturbances to the free stream are assumed to be small. These small perturbation assumptions are used to produce first-and second-order subproblems structured to provide consistent approximations to boundary conditions on the body and the free surface. Nonlinear corrections to the free-surface boundary condition are included at second order. Each subproblem is solved by a distribution of sources and vortices on the chord line and doublets on the free surface. After analytic determination of source and doublet strengths, a singular integral equation for the vortex strength is derived. This integral equation is reduced to a Fredholm integral equation which is solved numerically. Lift, wave drag, and free-surface shape are calculated for a flat plate and a Joukowski hydrofoil. The importance of free-surface effects relative to body effects is examined by a parametric variation of Froude number and depth of submergence.

Accuracy Effects of Low Frequency Wave Loads on Ships Offshore to LNG Marine Terminal Design

2004 ◽  
Author(s):  
Monica J. Holboke ◽  
Robert G. Grant
Keyword(s):  
Free Surface ◽  
Shallow Water ◽  
Low Frequency ◽  
Second Order ◽  
Surface Boundary ◽  
Wave Loads ◽  
Wave Load ◽  
Frequency Wave ◽  
First Order ◽  
Terminal Design

This paper presents the results of a two-body analysis for a moored ship sheltered by a breakwater in shallow water with and without free surface forcing in the low frequency wave load calculation. The low frequency wave loads are determined by second order interactions from the first order. The free surface forcing term arises from the free surface boundary condition, which is trivial to first order but is not at second order. We demonstrate in the frequency domain the importance of this term in a two-body analysis. Additionally, we show how inaccurate calculations of the off-diagonal terms of the Quadratic Transfer Function can translate to over or under prediction of low frequency wave loads on moored ships sheltered by breakwaters in shallow water. Low frequency wave load accuracy has direct consequence for LNG marine terminal design. Generally, LNG marine terminals are sited in sheltered harbors, however increasingly they are being proposed in offshore locations where they will require protection from persistent waves and swells. Since breakwaters typically cost twice as much as the rest of the marine facilities, it is important to optimize their size, orientation and location. In a previous paper we described this optimization process [1], which identified a key step to be the transforming of waves just offshore the breakwater into wave loads on the moored ships. The ability to do this step accurately is of critical importance because if the loads are too large, the breakwater will be larger and more expensive than necessary and if the loads are too small, the terminal will experience excessive downtime and loss of revenue.

Review of Approximations to Evaluate Second-Order Low-Frequency Load

2012 ◽  
Author(s):  
Guillaume de Hauteclocque ◽  
Flávia Rezende ◽  
Olaf Waals ◽  
Xiao-Bo Chen
Keyword(s):  
Free Surface ◽  
Shallow Water ◽  
Low Frequency ◽  
Second Order ◽  
The Body ◽  
Difference Frequency ◽  
Surface Boundary ◽  
First Order ◽  
Potential Part ◽  
The Difference

The second order low-frequency loads are one of main sources of excitation for moored systems. These loads are usually decomposed into the quadratic part, contributed only by first order quantities and potential part contributed by the second order potentials. In shallow water the second order incoming and diffracted potentials give a significant contribution to the low frequency forces. Therefore, the accuracy on the determination of this parcel of the low-frequency loads is a key issue for the assessment of mooring lines and operability of systems moored in shallow water area, as for example LNG terminals. Due to the complexity in computing the second order diffraction potential, which would involve a non-homogeneous free surface boundary condition, the so-called Pinkster approximation has been proposed. This approximation is based on the assumption that the major contribution to the potential part of low-frequency loads is given by the second order potential of the undisturbed incoming waves. The methods to compute the wave forces related to the second order potentials are based on scaling of the first order wave induced forces. Another approximation recently formulated in Chen and Rezende consists of developing the second-order bi-frequency load into a series of different orders of the difference frequency. The potential contribution to the term proportional to the difference-frequency can be evaluated efficiently by involving an integral over a small zone on the free surface around the body. In the present paper, the existing approximations are revisited and compared to analytical solution of exact second-order load on a vertical cylinder and for the case of floating body (LNG) in shallow water. Some guidelines in the practical use of different approximations will be derived.

A numerical free‐surface condition for elastic/viscoelastic finite‐difference modeling in the presence of topography

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1921-1934 ◽  
Author(s):  
Johan O. A. Robertsson
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Surface Condition ◽  
Finite Difference Schemes ◽  
Surface Boundary ◽  
Computationally Efficient ◽  
Free Surfaces ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition ◽  
Irregular Topography

An accurate free‐surface boundary condition is important for solving a wide variety of seismic modeling problems. In particular, for earthquake site studies or shallow environmental investigations the surface of the earth may have a significant impact on the outcome of simulations. Computations based on several elastic/viscoelastic flat horizontal free‐surface conditions are compared and benchmarked against an analytical solution. An accurate and simple condition is found and then generalized to allow for irregular free surfaces. This new method is simple to implement in conventional staggered finite‐difference schemes, is computationally efficient and enables modeling of highly irregular topography. The accuracy of the method is investigated and criteria for sampling of the wavefield are derived.

Analyses of Waves and the Wave Resistance due to Transom-Stern Ships

1969 ◽  
Vol 13 (02) ◽  
pp. 155-167
Author(s):  
Bohyun Yim
Keyword(s):  
Boundary Condition ◽  
Free Surface ◽  
Wave Resistance ◽  
Second Order ◽  
Surface Boundary ◽  
Exact Formulation ◽  
Wave Problem ◽  
Ship Wave ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition

Waves and the wave resistance due to a ship with a transom stern are analyzed starting with an exact formulation of the general ship wave problem. Green's theorem is utilized together with the well-known Green's function which satisfies the linear free-surface boundary condition on the free surface. Here it is required to take at least the second-order terms in the formal development. It is found that the immersed transom stern acts to cancel stern waves. Using a limited form of Michell's wave-resistance formula, the wave resistance of the transom-stern ship is analyzed

Comments on Second-Order Wave Effects on Large-Diameter Vertical Cylinder

1978 ◽  
Vol 22 (04) ◽  
pp. 266-268
Author(s):  
Subrata K. Chakrabarti
Keyword(s):  
Free Surface ◽  
Large Diameter ◽  
Vertical Cylinder ◽  
Wave Interaction ◽  
Scattered Wave ◽  
Second Order ◽  
Surface Boundary ◽  
Order Solution ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Conditions

In a recent paper, Raman et al (RJV) [1] 2 presented a second- order solution for the wave interaction with a large-diameter, bottom-mounted, surface-piercing, fixed vertical cylinder. Earlier, Chakrabarti [2,3] obtained an approximate solution for the vertical cylinder due to Stokes's fifth-order wave. Later, Yamaguchi and Tsuchiya (YT) [4] proposed a second-order solution in a closed form for the wave interaction problem with the vertical cylinder which was examined by Chakrabarti [5]. All these solutions are different due to the assumptions regarding the frequency of the scattered wave and the kinematic free-surface boundary condition. RJV [1] claim that their solution is superior to the earlier solutions because, according to them, it satisfies the nonlinear free-surface boundary conditions completely.

Linear and non-linear potential-flow calculations of free-surface waves with lift and induced drag

2000 ◽  
Vol 214 (6) ◽  
pp. 801-812
Author(s):  
C-E Janson
Keyword(s):  
Free Surface ◽  
Potential Flow ◽  
Lift Force ◽  
Radiation Condition ◽  
Linear Potential ◽  
Surface Boundary ◽  
Model Approximation ◽  
Non Linear ◽  
Lifting Surfaces ◽  
The Waves

A potential-flow panel method is used to compute the waves and the lift force from surface-piercing and submerged bodies. In particular the interaction between the waves and the lift produced close to the free surface is studied. Both linear and non-linear free-surface boundary conditions are considered. The potential-flow method is of Rankine-source type using raised source panels on the free surface and a four-point upwind operator to compute the velocity derivatives and to enforce the radiation condition. The lift force is introduced as a dipole distribution on the lifting surfaces and on the trailing wake, together with a flow tangency condition at the trailing edge of the lifting surface. Different approximations for the spanwise circulation distribution at the free surface were tested for a surface-piercing wing and it was concluded that a double-model approximation should be used for low speeds while a single-model, which allows for a vortex at the free surface, was preferred at higher speeds. The lift force and waves from three surface-piercing wings, a hydrofoil and a sailing yacht were computed and compared with measurements and good agreement was obtained.

On the choice of CFD codes in the design process of planing sailing yachts

2009 ◽  
Author(s):  
Jérémie Raymond ◽  
Jean-Marie Finot ◽  
Jean-Michel Kobus ◽  
Gérard Delhommeau ◽  
Patrick Queutey ◽  
...  
Keyword(s):  
Free Surface ◽  
Design Process ◽  
Finite Differences ◽  
Potential Flow ◽  
Surface Condition ◽  
Cpu Time ◽  
Free Surface Condition ◽  
Finite Differences Method ◽  
Non Linear ◽  
Ranse Solver

The discussion is based on results gathered during the first two years of a 3 years research program for the benefits of Groupe Finot-Conq, Naval Architects. The introduction presents the objectives of the program: Setting up a practical method using numerical and experimental available tools to design fast planing sailing yachts. The aim of this paper is to compare advantages and disadvantages of four different kinds of CFD codes which are linear and non-linear potential flow approach, RANSE solver using finite differences method and RANSE solver using volume of fluid method. The Fluid Mechanics Laboratory of the Ecole Centrale de Nantes (France) has developed those three approaches so those homemade codes will be used for this study. The first one is REVA, a potential flow code with a linearised free surface condition. ICARE is a RANSE solver using finite differences method with a non linear free surface condition. It is extensively used for industrial projects as for sailing yachts projects (ACC for example). ISIS-CFD is a RANSE solver using finite volume method to build the spatial discretization of the transport equations with unstructured mesh. The latter is able to compute sprays for fast planing ships but is also the slower in terms of CPU time. In addition, we had the opportunity to test FS-FLOW which is a potential flow code with a non linear free surface condition distributed by FRIENDSHIP CONSULTING. Numerical results for the four codes are compared with the other codes' results as with tank tests data. Those tank tests were made using captive model test technique on two Open60' models. Reasons of the choice of the captive model technique are explained and experimental procedures are briefly described. Comparisons between codes are mainly based on the easiness of use, the cost in CPU time and the confidence we can have in the results as a function of the boat speed. Flow visualizations, pressure maps, free surface deformation are shown and compared. Analysis of local quantities integrated or by zone is also presented. Results are analyzed focusing on the ability of each code to represent flow dynamics for every speed with a special attention to high speeds. The practical question raised is to know which kind of answers each code can bring in terms of tendencies evaluation or sensitivity to hull geometry modifications. The main goal is to be able to judge if those codes are able to make reliable and consistent comparisons of different designs. Conclusion is that none of the codes is perfect and gather all the advantages. It is still difficult to propose a definitive methodology to estimate hydrodynamic performances at every speed and at every stage of the design process. Knowing each code limitations, it appears more coherent to use each of them at different stages of the design process: the quickest and less reliable to understand the main tendencies and the longest and more precise to validate the final options.

scholarly journals Statistical analysis of second order effects variation with the stories height of reinforced concrete buildings

2017 ◽  
Vol 10 (2) ◽  
pp. 333-357
Author(s):  
D.M. OLIVEIRA ◽  
N.A. SILVA ◽  
C.C. RIBEIRO ◽  
S.E.C. RIBEIRO
Keyword(s):  
Reinforced Concrete ◽  
Statistical Analysis ◽  
Second Order ◽  
Order Effects ◽  
Reinforced Concrete Buildings ◽  
First Order ◽  
Concrete Buildings ◽  
Ideal Situation ◽  
Second Order Effects ◽  
The Ideal

Abstract In this paper the simplified method to evaluate final efforts using γ z coefficient is studied considering the variation of the second order effects with the height of the buildings. With this purpose, several reinforced concrete buildings of medium height are analyzed in first and second order using ANSYS software. Initially, it was checked that the (z coefficient should be used as magnifier of first order moments to evaluate final second order moments. Therefore, the study is developed considering the relation (final second order moments/ first order moments), calculated for each story of the structures. This moments relation is called magnifier of first order moments, "γ", and, in the ideal situation, it must coincide with the γ z value. However, it is observed that the reason γ /γ z varies with the height of the buildings. Furthermore, using an statistical analysis, it was checked that γ /γ z relation is generally lower than 1.05 and varies significantly in accordance with the considered building and with the presence or not of symmetry in the structure.

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