Complementarity Problem Formulation of Three-Dimensional Frictional Contact

1991 ◽  
Vol 58 (1) ◽  
pp. 134-140 ◽  
Author(s):  
B. M. Kwak
Keyword(s):  
Complementarity Problem ◽  
Frictional Contact ◽  
Nonlinear Complementarity Problem ◽  
Three Dimensional ◽  
Linear Complementarity Problems ◽  
Problem Formulation ◽  
Linear Complementarity ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Special Case

A general three-dimensional frictional contact problem has been formulated in the form of a complementarity problem in an incremental analysis setting. The derivation is straight forward and very natural. It is shown that the complementarity problem for a three-dimensional case is inherently nonlinear, not like the two-dimensional problem where a linear complementarity problem formulation is possible. The two-dimensional case is a special case of the three-dimensional formulation. Approximate linear complementarity problems of the nonlinear complementarity problem by a Newton approach, or by introducing polyhedral law of friction, have been proposed for efficient numerical implementations.

Three-dimensional disturbances to a mixing layer in the nonlinear critical-layer regime

1995 ◽  
Vol 291 ◽  
pp. 57-81 ◽  
Author(s):  
S. M. Churilov ◽  
I. G. Shukhman
Keyword(s):  
Travelling Waves ◽  
Mixing Layer ◽  
Three Dimensional ◽  
Critical Layer ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
New Type ◽  
Two Stages

We consider the nonlinear spatial evolution in the streamwise direction of slightly three-dimensional disturbances in the form of oblique travelling waves (with spanwise wavenumber kz much less than the streamwise one kx) in a mixing layer vx = u(y) at large Reynolds numbers. A study is made of the transition (with the growth of amplitude) to the regime of a nonlinear critical layer (CL) from regimes of a viscous CL and an unsteady CL, which we have investigated earlier (Churilov & Shukhman 1994). We have found a new type of transition to the nonlinear CL regime that has no analogy in the two-dimensional case, namely the transition from a stage of ‘explosive’ development. A nonlinear evolution equation is obtained which describes the development of disturbances in a regime of a quasi-steady nonlinear CL. We show that unlike the two-dimensional case there are two stages of disturbance growth after transition. In the first stage (immediately after transition) the amplitude A increases as x. Later, at the second stage, the ‘classical’ law A ∼ x2/3 is reached, which is usual for two-dimensional disturbances. It is demonstrated that with the growth of kz the region of three-dimensional behaviour is expanded, in particular the amplitude threshold of transition to the nonlinear CL regime from a stage of ‘explosive’ development rises and therefore in the ‘strongly three-dimensional’ limit kz = O(kx) such a transition cannot be realized in the framework of weakly nonlinear theory.

On the Hydrodynamic Interaction Between Ship and Free-Surface Motions on Vessels With Moonpools

2019 ◽  
Author(s):  
Senthuran Ravinthrakumar ◽  
Trygve Kristiansen ◽  
Babak Ommani
Keyword(s):  
Free Surface ◽  
Flow Separation ◽  
Coupling Effect ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Navier Stokes ◽  
Linear Potential ◽  
Two Dimensional ◽  
Sloshing Mode ◽  
Vessel Motion

Abstract Coupling between moonpool resonance and vessel motion is investigated in two-dimensional and quasi three-dimensional settings, where the models are studied in forced heave and in freely floating conditions. The two-dimensional setups are with a recess, while the quasi three-dimensional setups are without recess. One configuration with recess is presented for the two-dimensional case, while three different moonpool sizes (without recess) are tested for the quasi three-dimensional setup. A large number of forcing periods, and three wave steepnesses are tested. Boundary Element Method (BEM) and Viscous BEM (VBEM) time-domain codes based on linear potential flow theory, and a Navier–Stokes solver with linear free-surface and body-boundary conditions, are implemented to investigate resonant motion of the free-surface and the model. Damping due to flow separation from the sharp corners of the moonpool inlets is shown to matter for both vessel motions and moonpool response around the piston mode. In general, the CFD simulations compare well with the experimental results. BEM over-predicts the response significantly at resonance. VBEM provides improved results compared to the BEM, but still over-predicts the response. In the two-dimensional study there are significant coupling effects between heave, pitch and moonpool responses. In the quasi three-dimensional tests, the coupling effect is reduced significantly as the moonpool dimensions relative to the displaced volume of the ship is reduced. The first sloshing mode is investigated in the two-dimensional case. The studies show that damping due to flow separation is dominant. The vessel motions are unaffected by the moonpool response around the first sloshing mode.

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