Eighteenth Symposium on Naval Hydrodynamics (1991) / Chapter Skim
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A Numerical study of Three-Dimensional Viscous Interactions of Vortices with a Free Surface
A Numerical study of Three-Dimensional Viscous Interactions of Vortices with a Free Surface
Pages 727-788
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From page 727... ...
At intermediate Froude numbers, the normal incidence of a vortex ring with a clean free surface results in the formation of secondary vortex rings. Numerical studies of vortex tubes interacting with free surfaces show two possible mechanisms for the reconnection of vorticity with a free surface including primary and secondary vorticity reconnections.
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From page 728... ...
The three-dimensional interactions of these flow fields with the free surface are very complicated, but it is our hope that simple models will provide much needed insight. Some simple models of the largescale vertical flows that occur in ship wakes include two-dimensional vortex pairs, ring vortices, and three-dimensional vortex tubes.
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From page 729... ...
Our numerical simulations for the wall case show the formation of secondary and tertiary vortex rings. In fact, the primary vortex ring is wrapped in a sheath of counter-sign vorticity.
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From page 730... ...
. The bbserstlons in these expert meets seems to locate that vortex reconnection takes place as a rest It of canceH,tlon of vortlc1ty as two eposltely signed vortex tubes lutersect.
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From page 731... ...
At the low Froude number, the height of the upwelling is significantly reduced and the vortex pair splits apart upon impact with the free surface. Secondary vortices form and partially wrap themselves around the primary vortex, but the Reynolds number is 50 low that the vortices lose too much energy for significant inviscid interactions to take place.
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From page 732... ...
Our numerical studies of vortex rings interacting with a free surface show that secondary vortex rings may be torn away from the free-surface boundary layer at intermediate Froude numbers, Ad at lower Froude numbers the free surface acts like a free-slip wall. Our numerical simulations at low Froude numbers agree qualitatively with the vortex ring experiments of Bernal, et al, (1989)
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From page 733... ...
Note that Kim & Main, just like Ghia, et al, (1977) , use Neumann boundary conditions in the Poisson equation for the pressure, and as a result, they must satisfy a numerical solvability condition.
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From page 734... ...
the translation and diffusion of a two-dimensional vortex. The numerical schemes are used to study several physical problems including vortex rings and tubes impacting walls and free surfaces.
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From page 735... ...
Under these assumptions, the exact (nonlinear) field equations are still valid outside the boundary layer, but the nonlinear kinematic and dynamic free-surface boundary conditions (Wehausen & Laitone, 1960)
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From page 736... ...
Unlike a free-slip wall, the vorticity vector must be parallel to a no-slip wall, w = (~v, O) on z = 0, and as a result a vortex tube cannot terminate normal to a no-slip wall.
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From page 737... ...
The validation studies include numerical simulations of axisymmetric stagnation flows, attenuation of axisymmetric standing waves, and translation and diffusion of two-dimensional vortices. 3.1 A Semi-Implicit Time-stepping Scheme We derive here a first-order semi-implicit scheme for solving the Navier-Stokes equations.
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From page 738... ...
Note that the equations for the error terms (20) and the residual errors (21)
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From page 739... ...
For larger time steps, howe~rer, the diffusion terms dominate, the system of equations are elliptic, and multigrid iteration is much more effective. Finally, if the diffusion terms dominate and if the grid spacing along one or two axes is much smaller than the spacings along the other axes, the system of equations are anisotropic and a line Gauss-Seidel sol~rer should be used in conjunction with a multigrid solver.
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From page 740... ...
Once the pressure has been updated, the NewtonRaphson iteration may continue to reduce the residual errors or the solution scheme may progress to the next time step. The stability and accuracy of this numerical scheme are discussed later.
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From page 741... ...
3.4 Numerical Implementation of the Boundary Conditions Our numerical simulations use periodic, free-slip, no-slip, arid free-surface boundary conditions. To illustrate how these boundary conditions are implemented, we assmne that one of these types of boundary conditions is imposed on the plane z=0.
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From page 742... ...
As a result, a sheet of vorticity is instantaneously formed on the no-slip wall. In our simulations of vortex rings impacting the wall, the rings are initially far enough away from the wall such that the strength of the vorticity on wall is less than one percent of the core strength.
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From page 743... ...
The numerical simulations of two-dimensional vortices orbiting in a box test the temporal and spatial accuracy of our vortex formulations. Finally, numerical simulations of the attenuation of a~cisym~netric standing waves validates the formulation of the free-surface boundary conditions for low Reynolds numbers and small wave amplitudes.
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From page 744... ...
Par both computer codes the parameters Each control the namer of sweeps over the West grid level me as Buoys: number of Newton-Raphsou lteratlons, al; namer of ~tlgrld lteratlons, ^~'=6; namer ofOauss Seldel lteratlons, =6; ad namer of Jacot1 iterations ~=2. par ~ on n~rlc~ studies, the namer of Jacobi iterations is Amps Wed at Waco = 2.)
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From page 745... ...
The first problem is the decay of a two-dimensional rectilinear Gaussian vortex tube. Since the free decay of this vortex in an unbounded domain is relatively trivial and does not involve the nonlinear convective terms, we consider instead the vortex tube placed asymmetrically inside a rectangular domain.
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From page 746... ...
For the no-slip wall cases the radial expansions of the primary vortex rings are stopped by the growth of the boundary layer on the wall. As in the expert meets of Walker, et al, (1987)
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From page 747... ...
Here, we refer to the original vortex ring as the primary vortex ring, and any vortex rings that are ejected from the boundary layer of the primary vortex as secondary and tertiary vortex rings depending on when they are formed. The secondary and tertiary vortex rings have a sense of rotation that is opposite to the primary vortex ring, but as we observe in Figure (11)
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From page 748... ...
reconnection of the secondary vorticity that is generated in the freesurface boundary layer and swept up by the primary vortex. We hypothesize that reconnection of the primary vortex tubes may occur for deeply submerged trailing vortices that have enough time to develop large undulations prior to their impact with the free surface.
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From page 749... ...
provides a schematic of the helical vorticity spiraling off of the primary vortex tube. As the helical vortices are swept around the primary vortex, they attach themselves to the free-surface boundary layer.
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From page 750... ...
, the free surface acts like a free-slip wall, and the image of the vortex tube above the free surface induces high horizontal velocities in the region where the primary vortex tube is connecting with the free surface. The connecter region moves faster than the deeply submerged portions of the primary vortex tube so that as the vortex tube brealts apart, a cusp pattern is formed.
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From page 751... ...
At earlier time steps, which correspond to the onset of a helical instability, the sheets of helical vorticity are very tightly wound around the primary Vortex tube. Dritschel's (1989)
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From page 752... ...
, the tips of the U-shaped vortex tubes are wrapped around the primary vortex to the point where contact may occur with the free surface. In addition, as the secondary vorticity is wrapped around the primary vortex core, the velocities induced by the secondary vorticity on the primary vortex can cause the primary vortex to collide with the wall again.
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From page 753... ...
One type of primary vortex reconnection forms a cusp pattern on the free surface, and secondary vorticity reconnection should appear as paired dimples on the free surface. The essential aspects of the reconnection of secondary vorticity with the free surface include the generation of U-shaped vortices by helical vorticity shed from the primary vortex tube, the wrapping of the U-shaped vortices around the primary vortex tube, and the reconnection of the bases and tips of the U-shaped vortices with the free surface.
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From page 754... ...
. On the interaction of vortex rings and padre with a free surface for varying amounts of surface active agent.
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From page 755... ...
(1971~. Motion of distorted vortex rings.
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From page 756... ...
(1988~. Cut-and-connect of two antiparallel vortex tubes.
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From page 757... ...
Submitted for publication. Appendix A: Axisymmetric Flow Equations Our studies of vortex rings interacting with walls and free-surfaces are based on an axisymmetric formulation.
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From page 758... ...
If a won problem is being solved, subroutine FEASIBLE is called to make the source terms realizable at each of the coarse grid levels. The fine-to-coarse grid iterations continue until the coarsest grid level is reached, and then the coarse-to-fine grid iterations begin.
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From page 759... ...
. The pointer array IPOINT stores grid points sequentially as a function of the boundary type and grid level.
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From page 760... ...
Note that PSOU may also be used to store inhomogeneous Neumann boundary conditions. The inner DO-loop vectorizes because ISTEP=2 ensures that no diagonal element is a function of another diagonal element until the DOloop is exited.
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From page 761... ...
The number of Newton-Raphson iterations, multigrid iterations, and Gaus~Seidel iterations are respectively NATO ~ NmU`', and N'aU, . The number of time steps and sweeps over the finest grid level are respectively Name and Noper .
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From page 762... ...
The initial core radius and peals ~rorticity are denoted by rc and we. The vortex core is centered at (To' Zo)
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From page 763... ...
Runs 1 thru 6 use free-slip boundary conditions on the vertical walls, and Run 7 uses periodic boundary conditions along the :z-axis. The axes of the vortex tubes are aligned with the ~axis.
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From page 764... ...
3.0 3.5 4.0 4.5 5.0 Figure 1: The spatial accuracy as a function of grid Reynolds number for a jet impinging a wall. The logic maximum absolute errors in the (a)
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From page 765... ...
3._ 4.0 Figure 3: The values of stability parameters at the maximum allowable time step as a function of grid Reynolds number for a FTCS scheme in two dimensions. The parameters are labeled as follows: (-1 )
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From page 766... ...
l 0 0 2 4 6 8 RFigure 4b: T = 10. Figure 4: The relative error in the maximum vor ticity as a function of grid Reynolds number for a Gaussian vortex core orbiting a boo.
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From page 767... ...
as a function of grid Reynolds number for a Gaussian vortex core orbiting a boa. The results are plotted for At = 0.01 ~ 1-~ ; At = 0.02 ~2- ~;andAt=0.05( 3 ~.
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From page 768... ...
~ ~ r LFREE-SLIP WALL ~ ~ ~(FRONT FACE) ~TO VORTEX TUBE\ ~ Figure 8: The numerical simulation of a vortex tube impinging a boundary.
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From page 769... ...
Figure 10: A ring vortex impinging a no-slip Tall. The w=-1 contours of the primary vortex core are plotted at different instants of time: t=O, 2, 4, 6, 8, 12, 16, & 20.
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From page 770... ...
4 . 6 Figure 12: The orbit of a secondary vortex ring.
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From page 771... ...
The numerical parameters for these runs are provided in Runs 2 & 6 of Table (2~. I , 1 ~ ~ I I , 1 ,~ 1 , ili = ~~ High resolution | CONTOUR FROM -1 TO .25 BY 1 .25 _ '1 ~1 .2 1.4 1 .6 1.8 2.0 2.2 2.4 2.6 2.8 3 0 r Figure 13: The formation of a tertiary vortex ring.
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From page 772... ...
The ring vortex starts at the bottom of Part (b) and expands radially outward.
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From page 773... ...
The ring vortex starts at the bottom of Part (b) and expands radially outward up to r ~ 2.25.
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From page 774... ...
.., .,, ., ~ ~ .. I ~5 1.0 1 5 2.0 2.5 3.0 3.5 4.0 4.5 r Figure 19: The shedding of a tertiary vortex ring by a free surface at an intermediate Froude number.
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From page 775... ...
Although the figures on this page have been adapted from a numerical simulation of a vortex tube impinging on a no-slip wall, we expect similar behavior for a contaminated freesurface. vortex sheet Cross-section of attached U-shaped vortex tube Sheath of secondary vorticity wrapping around primary vortex Unattached U-shaped vortex tube Base of U-shaped vortex tube 775 Figure 20: The unwinding of helical vorticity around a primary vortex: tube.
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From page 776... ...
The submerged portion of the primary vortex tube is moving from left to right across the page. To emphasize the cusp pattern this image has been reflected across the symmetry boundary at y = 4.
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From page 777... ...
Figure 23b: two = 1.25 isosurfaces at t=5. ., Figure 23c: ALAS-1 25 - .
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From page 778... ...
Figure 23e: 1~1 = 1.25 isosurfaces at t=15.
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From page 779... ...
t=20. The view is from below a no-slip wall, and the primary vortex is moving into and to the left of the page due to its images across the centerplane and above the wall.
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From page 780... ...
Figure 24b: `/~ = 0.5 isosurfaces at t= 7.5. _~ Figure 24b: ~-0.5 isosurfaces at t=7.5.
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From page 781... ...
Figure 24c: `~ = 0.5 isosurfaces at t=10. Figure 24c: ~ = 0.5 isosurfaces at t=10.
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From page 782... ...
Figure 24d: ) ~2 + W2 = 0.5 isos~faces at t=12.5.
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From page 783... ...
1 Figure 24e: I+ = 0.5 isosurfaces at t-15 Figure 24e: iw~ = 0.5 isosurfaces at t=15. Figure 24e: 1~',1 = 0.5 isosurfaces at t=15.
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From page 784... ...
Figure 24f: ,~ = 0.5 isosurfaces at t_17.5. Figure 24f: 1wl = 0.5 isosurfaces at t=17.5.
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From page 785... ...
Figure 24g: x/~ = 0.5 isosurfaces at t=20. Figure 24g: ~ = 0.5 isosurfaces at t=20.
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From page 786... ...
To emphasize the U-shaped feature this image has been reflected about its midspan relative to the images appearing in Figures (24~. The numerical parameters for this run are provided in Run 3 of Table (4~.
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From page 787... ...
t=20. Observe that the features are oriented normal to the axis of the primary vortex tube which is parallel to the Taxis.
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From page 788... ...
of Song, et al's (1990) paper shows a U-shaped feature wrapped around the primary vortex tube.
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