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Fluid Dynamics Research 3 (1988) Ill-114North-Holland

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Vortex shedding from edges includingviscous effectsJ.M.R. GRAHAM and P.D. COZENS

Abstract. This paper describes results obtained by using the inviscid Cloud-in-Cell vortex method to model thevortex sheet which is shed and rolls up from a single sharp edge. There is good agreement between these resultsand previous (Pullin 1978) computations of the development of the sheet in impulsively started incompressibleinviscid flow. The Cloud-in-Cell method has been modified to include viscous diffusion calculated by finitedifferences on the mesh to give a mixed Eulerian-Lagrangian Navier-Stokes solver. This method has beenshown to model the diffusing free vortex and the Stokes boundary layer quite accurately. It is used to computeimpulsively started flow past sharp right-angled edges and edges with small rounding. The effect of viscousdiffusion on the development of the shed vortex is discussed.

The method is also used to study the effect of rounding on the vortex shedding from a right-angled edge inoscillatory flow. This problem is particularly important in determining the roll damping and hence response ofcertain types of ship hull in waves. It is shown that the strength and effect of the shed vortices rapidly decreaseas the ratio of the edge radius to the oscillation amplitude increases, and that at larger values of this ratio themode of shedding changes from two vortices per cycle from one edge to a more complicated mode. Thecomputed results are compared with flow visualisation using dye and neutrally buoyant particles in water flowaround an oscillating edge.

The Cloud-in-cell method (see Christiansen 1973) is a mesh method in which a discretemoving point vortex representation of the vorticity field is transferred to a fixed mesh. Anumerical approximation to the velocity field is calculated on this mesh and transferred back tothe moving points as a convection velocity. This process limits the instability in representationsof vortex sheets to wavelengths greater than twice the mesh spacing. In the present paper theCloud-in-cell method has been adapted by including viscous diffusion of the vorticity on themesh so that a numerical approximation to the full Navier-Stokes equations is solved. Thisgives a mixed Eulerian-Lagrangian method somewhat simiiar to that described by Farmer(1986).

This method has been used here to study steady and oscillatory flow past 90* edges bytransforming the flow to a rectangular domain. The development of the vorticity (w) field isdescribed by the equations:

4,V, + qV,, = -a, (1)(4 + 4,,% - +Xwy= 4%. + @.,,>* (2)

where 4 is the streamfunction.The convection part of eq. (2) is modelled by the moving point vortices carrying circulationthrough the mesh. The diffusion is carried out directly by finite differences on the fixed mesh.The circulation distribution is projected from the moving point vortices onto the fixed meshand back again by using a projection function which conserves the moments of the distribu-tion: CT,, z3xjr, and Cy,f,, and also minimises numerical diffusion.0169-5983/88/$2.00 0 1988, The Japan Society of Fluid Mechanics

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112 J. M. R. Graham, P. D. Cozens / Voriex shedding /mm edges

2. Results2. I. Impulsively started flow past a sharp edge

Pullin (1978) has computed the strength and shape of the vortex sheet resulting from flowround sharp edges with a range of internal edge angles 6. Fig. 1 shows the streamlinescomputed by Pullin for the case 6 = 90 compared with those computed by the Cloud-in-Cellmethod with effective zero viscous diffusion. The value of the total shed circulation r(t) whichgrows as t I2 for inviscid flow is shown for both methods in fig. 2. The effect of viscosity on thesame impulsive edge flow is to reduce the initial rate of growth of circulation as also shown infig. 2. By making the inviscid similarity substitutions:

w = Q/t, X = VW/WtV4X, Y = f,73./4[V4t3/4y, $ = V3/2/~/2t~Plpeqs. (1) and (2) become

and ?F;i,+ \E/,,= -D (3)-52+(~II,-~X)9x-(\II,+aY)s2,=R~(t)(a,,+~;2,.)-1Ra,, (4)

where V and 1 are velocity and length scales of the flow and R is the instantaneous Reynoldsnumber V32f1/2t1/2 /v. Therefore the non-dimensionalised shed circulation F for example,evolves as a function of R(t) only. Fig. 2 shows a plot of this. The fact that the curve for theviscous case does not become asymptotic to the inviscid value as R (or t) + a may be dueeither to the influence of secondary separation or to uncertainty in distinguishing between shedand boundary-layer vorticity.2.2. Oscillatory flow past sharp and rounded edges

This flow has been studied previously (Graham 1980) using the discrete vortex method foroscillatory flow and a range of edge angles. Fig. 3 shows an example of flow visualisation of thevortex pairing which occurs at an edge with a bilge keel. Computation of oscillatory inviscid

Fig. 1. Impulsively started flow past a 90 edge. (a) Pullin, (b) Present computation. - streamlines, - - - -vortex sheet.

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J. M . R. Graham, P. D. Cozens / Vor t ex shedding f r om edges 113

i-1vv+2 __- _~_- r- - - - - - 7=- _~- _- - - _- _- - -________- _____p- -FYT - - - - +- -

Fig. 2. Impulsively started flow past a 90 o edge. Circulation growth for inviscid and viscous flow. Inviscid - - - - -(Pullin), .-.-. (Graham), - - - present method; viscous - present method.

Fig. 3. Oscillatory flow past an edge with bilge keel. Flow visualisation in water.

Fig. 4. Oscillatory flow past edges of increasing radius of curvature. Computation.

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Fig. 5. Oscillatory flow past a rounded edge. Flow vtsualization in water.

flow past a sharp 90 edge by the present method generates similar but weaker vortex pairs. Avortex formed during the first half of a flow cycle passes back over the edge to pair up with thevortex of opposite sign forming in the second half. The vortices convect rapidly away from theedge when their strengths are approximately equal. This may leave residual vorticity whichforms a pair on the next flow reversal. Thus one or two vortex pairs may be formed per cycle.Viscosity does not appear to have a strong effect on this flow pattern provided the length scaleof the vortices is considerably larger than the thickness of the Stokes layers on the sides of theedge. Diffusion does reduce the circulation in the shed vortices with time by mixing but theseparation point remains at the edge. However in the case of a rounded edge the ratio of rc, theradius of curvature of the edge, to the amplitude I of oscillation of the flow has a significanteffect. Fig. 4 shows examples of this for increasing values of r,/l. As this parameter increasesthe shedding pattern becomes more confused and irregular from cycle to cycle. Vortices start toform at positions away from the edge due to the effect of shed vortices passing close to the wallcausing large scale separation of the boundary layer. The vortex shedding and resultant inducedforce decrease in strength.

These computations were all carried out under the same applied flow conditions with aReynolds number based on typical vortex scales of order 104. Fig. 5 shows for comparison aflow visualization in water using dye tracer of a corresponding flow induced by rolling of amodel barge of rectangular cross-section with a rounded edge equivalent to the largest roundingshown in fig. 4.

AcknowledgementsThis work was partly sponsored by BMT Ltd. and partly by the M.T.Ds Fluid Loading

Programme, a programme of research jointly funded by SERC, the Department of Energy andthe Offshore industry.

ReferencesChristiansen, J.P. (1973) Numerical simulation of hydrodynamics by a method of point vortices. J. Camp Ph.w 13. 363.Farmer, C.L. and Norman, R.A. (1986) The implementation of moving point methods for convection diffusion

equations. Numer. Methods for Fluid Dynamics II, I.M.A. Conf Serres.Graham, J.M.R. (1980) The forces on sharp-edged cylinders in oscillatory flow at low Keulegan-Carpenter number. J.

Fluid Mech. 97, 331.Pullin, D.I. (1978) The large scale structure of unsteady self-similar rolled up vortex sheets, J. Fluid Mrch. 88. 401.