Twenty-First Symposium on Naval Hydrodynamics (1997) / Chapter Skim
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A Multigrid Velocity-Pressure-Free Surface Elevation Fully Coupled Solver for Calculation of Turbulent Incompressible Flow around a Hull
A Multigrid Velocity-Pressure-Free Surface Elevation Fully Coupled Solver for Calculation of Turbulent Incompressible Flow around a Hull
Pages 328-345
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From page 328... ...
This evolution of hydrodynamic softwares seems reasonable but free surface boundary conditions reveal some unpredictable difficulties. 328 The use of kinematic equation for free surface height calculation when velocity field is known necessarily involves adding of non physical boundary conditions which do not allow neither resolution of tangential dynamic conditions nor introduction of viscosity or surface tension effects.
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From page 329... ...
Transport equations are written on the nodes of the mesh' the pressure equation is solved at the centre of elementary volumes and normal dynamic free surface condition at the cerise of the free surface interfaces. The two tangential dynamic conditions and the kinematic condition form the set of velocity boundary conditions on the free surface.
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From page 330... ...
Boundary conditions on Es are one kinematic condition, two dynamic tangential conditions and one normal dynamic condition. The kinematic condition, coming from the continuity hypothesis, expresses that the fluid particles of free surface stay on it.
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From page 331... ...
The two tangential dynamic conditions associated with kinematic condition give the 3 boundary conditions on the velocity at the free surface and normal dynamic condition allows to compute the free surface elevation. With this formulation, we linearise the kinematic condition as a implicit relation between the 3 velocity components and the free surface elevation: hl + A1U1 + A2U2 _ u3 = Alu1 A2U2 (16)
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From page 332... ...
and (18) are free surface velocity field boundary conditions.
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From page 333... ...
, boundary conditions on velocities and the implicit relations for secondary velocity unknowns on ~Q, the normal dynamic condition on 352si and the pressure equation on Qv (24~. It is important to note that no condition on pressure or free surface elevation is needed.
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From page 334... ...
Unfortunately the fully coupled linear system for the velocities, pressure and free surface elevation unknowns is very ill-conditionned, particulary the Amp block concerning the incompressible pressure equation, that means the mass conservation. So classical iterative methods are not available to solve this very large system.
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From page 335... ...
The solution on the fine grid is corrected using prolongation of coarse grid residuals. Generally algebraic interpolations work well but in the case of boundary unknowns located at the centre of each cells (the pressure in this case)
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From page 336... ...
A shock is created on the free surface and this requires an important subrelaxation of the velocity field and free surface elevation to converge. In order to avoid this problem the calculation simulates an uniform acceleration to reach the final speed.
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From page 337... ...
Figure 8, turbulent viscosity on Be x/1 = 0.4 section 8.2 Free surface field The originality of the method is its ability to take into account the exact free surface conditions in the whole space and particularly near the hull. Additional non physical boundary conditions, like Neumann conditions, are not required and the kinematic condition is solved, without smoothing, up to the first free surface elevation control point next to the body.
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From page 338... ...
I ~ I ~ I of . ~0.4 -0.2 0.0 Xle1 0.2 0.4 Figure 10, free surface elevation along the hull Figure 11 presen t a compari son between measured and computed free surface field on the fine grid.
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From page 339... ...
It has been shown that the influence of grid size on free surface elevation is light despite a small damping of oscillations for the coarse grid. On the contrary the grid size is very important to compute well the velocity profiles connected to a good calculation of turbulent values in the boundary layer.
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From page 340... ...
and n~casurcd (1~) velocity field at x/l = -O.S section U 1 0~161 1 on 1 Q~29 ~S1~3 0~97 ~1 o.soems 0.~8648 omo~z Q~e DJC Q141 ~11n42 ~WM~ o.`loasc4s .O - 61S 1 ~,8 1~8 1 ~Z74194 1 ~OB1411211 1~ ~1 a0~4 aO241S194 ~1 40145161 4~71 ~e2ss ~.0728BW 411129 1 ~Q1W~ 1 415 Figure 17 Computed (right)
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From page 341... ...
velocity field at xil = -0.3 section U 1 ~ 0.9741" 1 o,~e 1 0~4 0~916129 0.~774 tk87741 9 0~ 0~81 SASS - 0.8 341 _~ V ~ 0.0449s~s 1 - 0CQ741534 ~3 0~010~ 4006451 m — ~ 4~71 ~ -0.0572581 · ~.0741 - S ~ Q - 11" · 41 _ ~Q125 _ _ ~ _ W 1 · Q0~42 1 ~q 1 1 onto - 1 0.~419 0. Q048SB71 887097 0.01~ Q~7742 - o Figure 19 Computed (right)
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From page 342... ...
and measured (1~11) velocity field at x/l = +0.4 section .: .;.,]
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From page 343... ...
However the finest grid size seems too coarse to compute well free surface elevation, pressure velocity field and especially turbulent quantities. The continuation of this work will consist in perform calculation with finer mesh to obtain grid independency results (perhaps around 500 000 or 600 000 nodes)
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From page 344... ...
methods, which can use block-structured and unstructured grids with cell-wise local refinement. Especially the local refinement capability is essential if a high accuracy is to be achieved in calculating flows around ship hulls, since very fine grid is needed close to the hull and in the wake while much coarser grid can be used elsewhere.
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From page 345... ...
The physics does not follow the continuity hypothesis, and particularly the kinematic condition seems to be unverified. The problem is not to predict the meniscus and the dynamic contact angle at a very small scale, but to ensure the contact line progression with classical boundary conditions.
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