SStart-up vortex flow past an accelerated flat plate
Ling Xu a) and Monika Nitsche b) Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303,USA Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131,USA (Dated: 18 April 2014)
Viscous flow past a finite flat plate moving in direction normal to itself is studied numerically. The platemoves with velocity at p , where p = 0 , . , ,
2. We present the evolution of vorticity profiles, streaklines andstreamlines, and study the dependence on the acceleration parameter p . Four stages in the vortex evolution,as proposed by Luchini & Tognaccini (2002), are clearly identified. The initial stage, in which the vorticityconsists solely of a Rayleigh boundary layer, is shown to last for a time-interval whose length shrinks to zerolike p , as p →
0. In the second stage, a center of rotation develops near the tip of the plate, well beforea vorticity maximum within the vortex core develops. Once the vorticity maximum develops, its positionoscillates and differs from the center of rotation. The difference between the two increases with increasing p , and decreases in time. In the third stage, the center of rotation and the shed circulation closely satisfyself-similar scaling laws for inviscid flow. Finally, in the fourth stage, the finite plate length becomes relevantand the flow begins to depart from the self-similar behaviour. While the core trajectory and circulationclosely satisfy inviscid scaling laws, the vorticity maximum and the boundary layer thickness follow viscousscaling laws. The results are compared with experimental results of Pullin & Perry (1980), and Taneda &Honji (1971), where available.Keywords: Starting vortex; power law; viscous flow; separation; Reynolds; streaklines; vortex center I. INTRODUCTION
This paper presents numerical simulations of the start-ing vortex flow at the edge of an accelerating finite flatplate. The plate is assumed to have zero thickness, andmoves with speed U ( t ) = at p in direction normal to it-self. The flow is nondimensionalized based on the platelength and the parameter a , yielding a characteristicflow Reynolds number Re . Here we study the effect ofthe acceleration parameter p for fixed Reynolds number Re = 500. The effect of varying Re for fixed p is pre-sented elsewhere .Being of intrinsic interest in fluid dynamics, the start-ing vortex flow has been the focus of many researchworks, beginning with the work of Prandtl in 1904 .Most relevant to the accelerated case considered here arethe following. Taneda & Honji presented experimen-tal results for both uniform and accelerated flow pasta plate, and observed an apparent scaling of the vortexsize. Pullin and Perry performed experiments of flowpast finite wedges, with wedges as small as 5 ◦ , whichserve as a basis of comparison for our present results.They reported detailed measurements of the vortex coretrajectory at early times, and compared their observa-tions to inviscid similarity theory results obtained byPullin . The theory holds for inviscid vortex sheet sep-aration at the edge of a semi-infinite plate. In view ofthe absence of a plate or viscous length scale, in that a) Electronic mail: [email protected] b) Electronic mail: [email protected] case the vortex center trajectory and the shed circulationsatisfy inviscid scaling laws in time that were already re-ported by Kaden in 1931. Pullin computed the time-independent self-similar shape using an iterative scheme.Scaling laws also exist for viscous flow past a semi-infinite plate, and follow from dimensional analysis.However, in this case the solution depends on the vis-cous length scale. The viscous scaling was exploited byLuchini and Tognaccini , who computed flow past a semi-infinite plate in a self-similar reference frame. The finiteplate case was studied numerically by Koumoutsakos andShiels . They computed flow past a plate moving with ei-ther impulsively started velocity or constant acceleration.For their accelerated flow, they observe a shear layer in-stability similar to that observed by Pierce and Lianand Huang . These numerical results are given mostlyfor relatively large times.Here, we use highly resolved simulations to present asystematic study of the dependence on p over a largerange of times. We present the evolution of vorticityprofiles, streaklines and streamlines, track vortex coretrajectories and vorticities, and compute the shed circu-lation following the approach taken in Xu and Nitsche .Luchini and Tognaccini propose four different timeregimes for viscous flow past finite plates, and it is in-teresting to identify these regimes using the present sim-ulations. We observe and present the timescales of aninitial Rayleigh flow regime. We study the emergenceof a vortex core in a second regime, and compare twopossible definitions, namely the center of rotation andthe position of the vorticity maximum. We observe theself-similar scaling laws in a third regime, and estimatethe time at which the finite plate length dominates, in a a r X i v : . [ phy s i c s . f l u - dyn ] A p r FIG. 1. Sample vorticity contours for p = 0 at a relativelylarge time, with positive vorticity in black, negative vorticityin grey. fourth regime. All results are computed with Re = 500,with the exception of a comparison with experimentalresults by Pullin and Perry, for which Re = 6000, and p = 0 .
45. The results are also compared with the scalingbehaviour proposed by Taneda and Honji .The paper is organized as follows. Sections II and IIIpresent the problem and the numerical method used, sec-tion IV presents the numerical results, section V summa-rizes the observations. II. PROBLEM FORMULATION
A finite plate of length L and zero thickness, immersedin a homogeneous viscous fluid, is accelerated in directionnormal to itself with speed (cid:98) U ( (cid:98) t ) = a (cid:98) t p , (1)where a is a dimensional constant. Here and through-out the paper, the hat symbol denotes that the variablesare dimensional. We consider p = 0, 0.5, 1, 2. Theseinclude impulsively started flow ( p = 0), uniform accel-eration ( p = 1), and linear acceleration ( p = 2). Theflow is assumed to be two-dimensional, and to remainsymmetric about the centerline at all times. We choosea reference frame fixed on the plate, in which the plate ishorizontal and the driving velocity moves upwards, ap-proaching parallel flow in the far field. Figure 1 illus-trates the fluid vorticity some time after the beginningof the motion. As fluid moves from upstream (below theplate) to downstream (above the plate), boundary layersof vorticity form along the plate walls which eventuallyseparate and form a pair of counterrotating vortices. Wenote that changing to an accelerated time-dependent ref-erence frame does not affect the vorticity dynamics, butonly the pressure and thus the forces acting on the plate.The results below are therefore the same, up to a trans-lation, as those for a moving plate in a reference framefixed at infinity. The flow is nondimensionalized with respect to theplate length and the parameter a , yielding a characteris-tic timescale T = (cid:18) La (cid:19) / ( p +1) , (2)and flow Reynolds number Re = L νT = a / ( p +1) L (2 p +1) / ( p +1) ν , (3)where ν is the kinematic fluid viscosity. The problem isdescribed in nondimensional time t and Cartesian coor-dinates ( x, y ), chosen so that the plate lies on the x-axis,centered at the origin, at { ( x, | x ∈ [ − / , / } . Thefluid velocity and scalar vorticity are ( u ( x, y, t ) , v ( x, y, t ))and ω ( x, y, t ). The relation between dimensional andnondimensional variables is, for example t = (cid:98) tT , x = (cid:98) xL , U = (cid:98) U TL , ω = (cid:98) ωT . (4)The nondimensional far field velocity is (0 , t p ).The flow is governed by the incompressible Navier-Stokes equations, DωDt = 1 Re ∇ ω , ∇ ψ = − ω , (5)( u, v ) = ∇ ⊥ ψ = (cid:18) ∂ψ∂y , − ∂ψ∂x (cid:19) . It is initially irrotational, ω ( x, y,
0) = 0, with boundaryconditions ψ = 0 and u = 0 on the plate, and ψ ( x, y, t ) → ψ ∞ as | ( x, y ) | → ∞ . Here, ψ ∞ is the potential flowthat induces the far field velocity, given by the complexpotential W ∞ ( x, y, t ) = t p (cid:114) − z = φ ∞ + iψ ∞ , (6)where z = x + iy . III. NUMERICAL METHOD
The governing equations (5) are solved in a finiterectangular domain in the right half plane, [0 , x max ] × [ y min , y max ], using a time-splitting mixed finite-differenceand semi-Lagrangian scheme. The domain in space andtime is discretized using a uniform mesh, ∆ x = ∆ y = h ,with constant timestep ∆ t , over a given time interval.The solution is advanced from time t n to t n +1 by con-vecting the current vorticity according to DωDt = 0 (7)using a semi-Lagrangian scheme; using the updated vor-ticity to obtain updated interior and boundary stream-function, velocity and vorticity values; and then solving ∂ω∂t = 1 Re ∇ ω (8) TABLE I. List of parameters in computations: the mesh size h , time step ∆ t , starting/ending time t start / t end , computationaldomain [0 , x max ] × [ y min , y max ], the number of grid points N x × N y , the value(s) of p and the Reynolds number Re . h ∆ t [ t start , t end ] [0 , x max ] × [ y min , y max ] N x × N y p Re × − [0, 0.0004] [0, 0.55] × [-0.05, 0.1] 2816 ×
768 0 5001/2560 4 × − [0, 0.005] [0, 0.55] × [-0.05, 0.1] 1408 ×
384 0 5001/1280 5 × − [0, 0.1] [0, 0.75] × [-0.125, 0.25] 960 ×
480 0, 0.5, 1, 2 5001/640 1 × − [0.1, 0.7] [0, 0.1] × [-0.25, 0.75] 480 ×
480 0, 0.5, 1, 2 5001/320 2 × − [0.7, 3] [0, 1.5] × [-0.25, 2.75] 480 ×
960 0, 0.5, 1, 2 5001/160 4 × − [3, 4] [0, 1.5] × [-0.5, 5.5] 240 ×
960 0 5001/1280 5 × − [0, 0.75] [-0.1, 0.4] × [0, 0.5] 960 ×
640 0.45 6000 using an implicit Crank-Nicolson method. The methodis described in detail in Xu & Nitsche and is based onthe work in Xu .Table I lists all parameters used in the present compu-tations for various values of p . In several cases, in order tocompute the flow to large times, the computations wereperformed using a fine mesh until some early time, sub-sampling that result and continuiung on a coarser mesh,and repeating this process. The time intervals used areindicated in the table. For example, the result for p = 0at t = 4 is obtained using h = 1 / t ∈ [0 , . t = 0 . h = 1 / t = 0 .
7, etc. The case of p = 0 is more difficult tocompute in view of the initial singularity at the plate tipdiscussed in . Therefore, finer resolutions are used inthis case than for p > IV. NUMERICAL RESULTSA. Vorticity, streamlines, and streaklines for p = 1 Figure 2 presents the computed flow evolution for p = 1 (constant acceleration) and Re = 500. The fig-ure shows vorticity contours (left), streaklines (middle),and streamlines (right) at t =0.04, 0.2, 0.5, 1, 1.5, 2,as indicated. Positive vorticity contours are shown inblack, negative ones in a lighter shade of grey. Resultsare plotted in the right half plane, x ≥
0. By symmetry,the vorticity and streamfunction in x ≤ y = 0, as can be seen infigure 2 at t = 0 .
04. The corresponding streamlines, inthe right column, are also practically symmetric across the plate. It is noteworthy to remark that this almostsymmetric flow regime is not present for the impulsivelystarted case presented in . This will be discussed inmore detail below.As time evolves, vorticity upstream from the plate isconvected downstream, concentrating near the tip as avortex that grows in time and breaks the approximatesymmetry of the initial boundary layers. At the sametime that the vorticity concentrates, a region of recircu-lating flow forms near the tip. The recirculating region isbounded by the zero level streamline and is seen in figure2 as early as t = 0 .
2, in the right column. Within the re-circulation region and close to the wall, fluid moves in theopposite direction of the background flow, thereby gener-ating a region of opposite signed boundary layer vorticity.This negative wall vorticity appears in the fluid flow at t ≈ t ≥ . x = 0,together with the recirculation region. At t =1, the down-stream wall vorticity is all negative, and separates theplate from the original positive boundary vorticity. Theremaining positive boundary vorticity diffuses and van-ishes, as seen here at times t = 1 . ,
2, leaving the startingvortex clearly separated from the original boundary layervorticity. The negative vorticity near the tip is entrainedby the vortex, which grows and convects in the down-stream direction.The streaklines shown in the middle column in figure 2are computed by releasing fluid particles from the platetip at each time step, and computing their evolution withthe fluid velocity. At any given time, the figure 2 showsthe current position of all particles released previously,mimicking what can be visualized in laboratory exper-iments. The particles rotate around the vortex center,with particles that have been released earlier travelingcloser to the center than those released later. This givesthe resulting streaklines their spiral shape. We note that y − y − y − y − y − y x0 0.8 − x0 0.8 t=0.04 0.2 2 1.5 1 0.5 1.3 -‐0.1 0 0.8 0 0 0.8 0.8 FIG. 2. Vorticity (left), streaklines (middle) and streamlines (right) at t = 0.04, 0.2, 0.5, 1, 1.5, 2, for p = 1, Re = 500.Vorticity contours level are ± j , j = − , . . . , the tip, which is the point at which the particles are re-leased, is also the point at which the flow vorticity ismaximal. The particles thus approximate the convectionof the maximum vorticity, and thus approximate the cen-terline of maximum vorticity in the separated shear layer. However, the size of the spiral streakline is not represen-tative of the size of the vortex recirculation region. y p=0 − x y − − y p=1 p=1/2 x0 0.5 1 x0 0.5 1 p=2 x0 0.5 1 FIG. 3. Vorticity( top ), streaklines( middle ) and streamlines( bottom ) for fixed displacement d = 1, and p = 0 , / , ,
2, asindicated.
B. Dependence on p at fixed displacement d = 1 The parameter p describes the driving velocity (0 , t p )in the far field. In a reference frame fixed at infinity,the plate moves downward with velocity (0 , − t p ). Thesolution for varying p at a fixed time t varies greatlysince the plate has travelled significantly different dis-tances d = p +1 t p +1 at equal time t , resulting in vorticesof significantly different size. It is more meaningful tocompare solutions with varying p at times at which theplate displacement d is equal. All results shown hereinthat compare the solution for various p are therefore plot-ted in reference to the displacement d , instead of time t .Figure 3 compares the vorticity profiles (top), streak-lines (middle) and flow streamlines (bottom) at fixed dis-placement d = 1 for all p =0, 1/2, 1, 2 computed, as in-dicated. As p increases, the vorticity contours show thatthe vortex decreases slightly in size, and the core vorticitybecomes more uniform. Furthermore, the wall boundarylayer thickness decreases slightly. The vorticity profileas a function of p is more clearly shown in figure 4(a), which plots the vorticity at d = 1 along the vertical line x = x m through the vorticity maximum ( x m , y m ) in thevortex center, as a function of y , for all p computed. Itshows that the profile is flatter near the core for largervalues of p . The outermost shear layer turn is strongerfor larger p , with larger maximal vorticities, and peaksat a smaller value of y , reflecting the smaller shape seenin the vorticity contours.The spiral streaklines plotted in the middle row of fig-ure 3 show that as p increases, the spiral size decreases,and the roll-up away from the center is less tight. Thatis, for larger p , more particles released at early times endup near the spiral center. This is caused by the fact thatfor larger p , the vortex has travelled less far from theplate at early times. As p increases, the spiral shape ismore elliptical and less round. It also leans further to theleft, with the line from the spiral center to the plate tipsubtending a smaller angle with the plate.The streamlines plotted in the bottom row in figure3 show that as p increases, the size of the recirculationregion decreases. From the streamline density we deducethat the velocity gradients in the core first decreases as y ω (a) p=0p=1/2p=1p=2 (b) u y FIG. 4. (a) Vorticity and (b) velocity at d = 1 along a verticalline x = x m through the vorticity maximum, vs. y , for p =0 , / , ,
2, as indicated. p increases from 0 to p = 1 /
2, but then increase as p increases past 1/2. Figure 4(b) plots the velocity along x = x m , and shows that this is indeed the case. Theprofiles also show the decreasing value of y of the pointwith u = 0, near the rotation center. C. Comparison with laboratory experiments
The most detailed experimental results available forcomparison are those of Pullin & Perry of flow past awedge of angle β . Their streakline visualization for theirsmallest wedge angle used, β = 5 o , is shown in figure5, left column. The photographs show snapshots of anexperiment performed in a rectangular tank, in whichwater flows from left to right past a planar wedge ofheight h = 12 . U = at p , with p = 0 . a = 0 .
86 cm/sec p +1 . By sym-metry, the flow is comparable to flow past a plate with L = 2 h . At water temperature of 24 o C, the correspond-ing Reynolds number as defined in equation (3) above is Re = 6621.The experimental results are compared with the FIG. 5. Left column: Streaklines for flow past a wedge ofangle β = 5 o , at t =1s, 1.6s, 2.8s and 4s, for p = 0 .
45 and Re = 6621, obtained by Pullin and Perry from laboratoryexperiments (reproduced with permission from the Journal ofFluid Mechanics). Right column: Numerical simulations forflow past a plate ( β = 0 o ) at the same times, for p = 0 .
45 and Re = 6000. present numerical simulations of flow past a plate ( β =0 o ), with p = 0 .
45 and Re = 6000. The right columnin figure 5 shows the computed streaklines at the nondi-mensional times corresponding to those shown in the leftcolumn, using the timescale in equation (2). The com-puted results are shown in a rotated frame, for bettercomparison with the experiments. The figure also plotsthe position of particles initially placed along a verti-cal line below the plate (in the rotated frame) to betterreproduce the experiment. These particles form the out-ermost turn of the spiral streakline.Good agreement between experimental and numericalresults is observed for the spiral streakline size, the over-all spiral shape, and the spiral center position. The spi-ral centers will be compared in more detail in section E − − − − (a) (b) (c) (d) S ψ x y (x c , y c ) ω=ω c (x m , y m ) ω=ω m y FIG. 6. Four stages in the vorticity evolution, illustrated here for p = 1 /
2, at times (a) d = 0 . t = 0 . d = 0 . t = 0 . d = 0 .
310 ( t = 0 .
6) and (d) d = 0 .
88 ( t = 1 . x c , y c ), the position of the vorticity maximum ( x m , y m ), the corresponding vorticities ω c and ω m , and the vortex size s ψ are indicated in plot (d). −4 −3 −2 −1 m = 3p T R h=1/640h=1/1280 FIG. 7. Duration T R of the Rayleigh stage, as a function of p , computed with two resolutions h = 1 / h = 1 / m . below. One difference is observed in the spiral turn em-anating from the plate tip, which displays small vorticesin the experiment, reflecting an instability that is notseen in the computed results at these times. Schneider et al give evidence that the shape of a finite thicknessplate tip can contribute to these oscillations. Anotherdifference is observed in the outer spiral turn, which hasmoved further to the right in the experiments than in thecomputations. This difference may be due to differencesin the initial particle positions below the plate, for whichthe experimental data is not available. It may also bedue to differences in the wedge angle between the exper-iment and the computation, whose effects remains to bestudied. D. Four stages in the vortex evolution
Luchini and Tognaccini propose four stages in the evo-lution of the starting vortex. It is interesting to identifythem here, as illustrated in figure 6. The figure plots vor-ticity contours for p = 1 / y P=0 P=1/2 P=1 P=2 FIG. 8. Trajectories of the rotation center ( x c , y c ) (solid line)and the position of the vorticity maximum ( x m , y m ) (dashedline) for various p , as indicated, for d ∈ [0 , ψ = 0 that bounds the region of recirculating flow, aswell as a streamline close to the center of rotation withinthis region.In the first stage, referred to as the Rayleigh stageand illustrated in figure 6(a), the vorticity consists of analmost symmetric boundary layer of uniform thicknessaround the whole plate, without any apparent separatedflow. In the second stage a region of recirculating flowhas formed near the tip of the plate, containing a well-defined center of rotation, as seen in figure 6(b) and theassociated boundary layer of negative vorticity. In thisstage the vortex center grows, but does not satisfy scalinglaws. Luchini and Tognaccini refer to it as the viscousstage. In the third stage, the self-similar stage, looselyrepresented by figure 6(c), the vortex center grows closelysatisfying the self-similar scaling for inviscid separation inthe absence of a plate length scale. The observed scalingbehaviour is the subject of the next section. In the laststage, the ejection stage, illustrated in figure 6(d), thevortex departs from the self-similar growth, and the finiteplate length noticeably affects the flow.It is interesting to note that the Rayleigh stage is not m = −3/4−1/61/8 5/12 (a) −2 −1 d ω c p=0 p=1/2p=1 p=2 m = 2/3 (b) −2 −1 d −3 −2 −1 y c (c) d x c − . −3 −3 m = 2/3 m = −3/4(d) −2 −1 d ω m m = 2/3 (e) −2 −1 d −3 −2 −1 y m (f) d x m − . FIG. 9. (a-c) Vorticity and coordinates of the rotation center, vs. displacement d . (d-f) Vorticity and coordinates of the corevorticity maximum center, vs. displacement d . Results are shown for p =0, 1/2, 1, 2, as indicated in the legend in (a). Thedashed lines have the indicated slopes. observed for the impulsively started flow, p = 0. In thatcase (see also ), the recirculation region near the platetip and the associated negative vorticity appear withinthe first 10-15 timesteps, that is, in time O (∆ t ). Figure7 plots the duration T R of the Rayleigh stage, definedby the time at which the recirculation region and nega-tive vorticity first appear, as a function of p . The resultswhere computed using two resolutions, h = 1 / / p = 1 / , / , / , ,
2. While theconvergence is slower for smaller values of p , the interpo-lating curves appear to converge to T R ≈ . p . (9)Thus, T R grows as p increases, and vanishes for p = 0,consistent with our observations. E. Vortex core trajectory and vorticity At t = T R , a region of recirculating flow forms withan associated center of rotation, which can be used todefine the center of the starting vortex. Alternatively,the vortex center can be defined as the position of thevorticity maximum near its center. However, as can beseen in figure 6(b), at early times when the recircula-tion region is already well established, no local vorticitymaximum has yet formed within the starting vortex. Atthese early times, the vorticity grows along a curved ridgestarting at its maximum value at the plate tip, but theridge does not develop a local maximum along it untilmuch later. Once the local vorticity maximum appears, as in figure 6(c,d), it is not at the same location as thecenter of rotation. We denote the positions of the centerof rotation and the vorticity maximum by ( x c , y c ) and( x m , y m ), respectively, and the corresponding vorticityat those points by ω c and ω m (see figure 6(d).Figure 8 compares the trajectory of the rotation cen-ter (solid) and the core vorticity maximum (dashed) for p = 0 , / , ,
2, computed for displacements d ∈ [0 , p increases, the vorticity maximum appears later and at afurther distance from the plate tip. Furthermore, as p in-creases, the difference between the two points increases.Finally, while the rotation center travels on a monotonicpath inward from the plate tip for most of the inter-val shown, the vorticity maximum oscillates as it trav-els downstream. It is possible that such oscillations inthe core vorticity is partially responsible for oscillationsalong the separated shear layer often observed in labora-tory experiments and in computations . However,this issue remains to be investigated.Figure 9 plots the vortex core vorticity and coordinatesas a function of the displacement d , on a logarithmicscale, in order to reveal their scaling behaviour. The toprow, figures 9(a,b,c), shows the results ω c , y c , x c for therotation center, which is the first to form. The bottomrow, figures 9(d,e,f), shows the results ω m , y m , x m for thevorticity maximum.We first discuss the results for core vorticity shownin figures (a,d). As noted in , for p = 0 the vorticityclosely follows the viscous scaling found in flow past asemi-infinite plate, over several decades in time. Con-sider flow past a semi-infinite plate driven by (cid:98) ψ ∞ = A (cid:98) t p (cid:98) r / cos( θ/ (cid:98) r, θ are the polar coordinates of apoint with origin at the plate tip, and A is a dimensionalconstant. Due to the absence of a plate length scale, itfollows from dimensional analysis that the dimensionalflow streamfunction has the form (cid:98) ψ ( (cid:98) x, (cid:98) y ) = A (cid:98) t p ( ν (cid:98) t ) / f ( (cid:98) x √ ν (cid:98) t , (cid:98) y √ ν (cid:98) t ) . (10)In our case, this scaling can be expected to be a goodapproximation at early times, as long as the vortex sizeis small relative to the finite plate length. By takingsecond derivatives, one finds that the corresponding vor-ticity scales as (cid:98) ω ∼ A (cid:98) t p ( ν (cid:98) t ) − / . In our nondimensionalvariables, using A = L / a , this implies that ω ∼ t p (cid:16) tRe (cid:17) − / ∼ d p − / p +1 = d α , (11)where α = − / − /
6, 1 /
8, 5 /
12 for p = 0 , / , , p = 0 thisscaling is observed over a large range of times, both in ω c and ω m . For p >
0, the scaling is observed in ω c , butover a much smaller time interval, starting after an ini-tial transition region and ending aproximately at d = 0 . p >
0, the vorticity maximum ω m has not even yet formed during those times. It ap-pears only around d = 0 . y c , y m ofthe rotation center and the vorticity maximum, respec-tively, as a function of the plate displacement d , respec-tively. For p = 0, both variables closely satisfy y c , y m ∼ d / (12)over several decades in time, until about d = 1. For p >
0, the data shows that y c approximates the samescaling quite well for roughly d > .
2, after an initialtransition period. As already noted, the values of y m donot exist during this transition period. They are in fairagreement with y c afterwards. as shown in figure 9(e).Figures 9(c,f) plot the horizontal displacement of thevortex center from the plate tip, x c − . x m − . d . Figure(c) shows that the center of rotation first moves mono-tonically towards the axis, as could also be seen in figure8. For larger values of p , it moves further to the left. Ap-proximately around d = 1 .
5, for all p , the vortex turnsaround and moves outwards. The inset in figure (c) plots | x c − . | on a logarithmic scale, and shows that duringthe inward motion, the scaling | x c − . | ∼ d / , (13) x , y ( c m ) y (P&P exp) y (P&P self − similar) y (simulation) x (P&P exp) x (P&P self − similar) x (simulation) y ⌃ y ⌃ y ⌃ x ⌃ x ⌃ x ⌃ x , ⌃ y ⌃ t ⌃ FIG. 10. Comparison of dimensional vortex core coordinates( (cid:98) x, (cid:98) y ) relative to the plate tip, for p = 0 .
45. Experimental re-sults of Pullin & Perry ( Re = 6621), inviscid similarity the-ory results , and numerical results for the rotation center( Re = 6000) are shown. is approximately satisfied. For p >
0, it is satisfied wellafter an initial transition period. The data for p = 0appears to have a slightly smaller slope, between 2 / /
2. The reason for this apparent jump between p = 0and p = 0 . x m , shown infigure 9(f), shows that the vorticity maximum oscillatesas it moves inward towards the axis, before it moves backout, as was already observed in figure 8. The oscillationamplitude increases as p increases.Note that the observed scaling in equations (12,13) isnot the one that follows for viscous flow, from the ar-gument leading to (10). Instead, it is the scaling foundfor inviscid flow in the absence of a length scale, alsobased on dimensional analysis. This case was consid-ered by Pullin , who derives the self-similar scaling forthe vortex sheet separation and spiral roll-up at theedge of a semi-infinite plate driven by accelerating back-ground flow, and computes the time-independent self-similar shape using an iterative scheme. His results showthat the coordinates of the spiral center satisfy (cid:98) x c + i (cid:98) y c = Ω (cid:16) A (cid:98) t p p (cid:17) / (14)where the complex number Ω depends little on p . In thedimensionless variables used here, and again making thecorrespondence A = L / a , equation (14) is equivalentto equations (12,13).Next we compare not only the scaling behaviour, butthe actual values of the core coordinates with experi-mental data of Pullin & Perry (P&P) and with simi-larity theory results given therein. Figure 10 plots thedimensional coordinates ( (cid:98) x, (cid:98) y ) of the vortex center rel-ative to the plate tip as a function of dimensional time (cid:98) t . Consistent with the rest of this paper, (cid:98) x refers to the0 − y Ω (a) d Γ (b) −2 −1 −4 −2 p=0 p=1/2p=1 p=2 d Γ α ___ (1+p) γ (c) −3 −1 −4 −2 m=1 FIG. 11. (a) Sketch showing domain Ω used to define shed circulation Γ, following . (b) Γ vs displacement d . (c) Γ /α / (1+ p ) γ vs d . where α = p p ) , γ = p p . Results are shown for p = 0, 1 /
2, 1 and 2, as indicated in (b). The dashed line in (c) hasthe indicated slope. vortex displacement parallel to the plate, while (cid:98) y refersto the displacement normal to the plate. The experi-mental measurements obtained by P&P (closed and opencircles) represent the spiral center of the experimentallyobserved streakline, for flow past a wedge with β = 5 ◦ .The similarity theory results obtained by Pullin (thickand thin dashed curves) also correspond to flow past awedge with β = 5 o . The computed results (thick andthin solid curves) denote the dimensional position of therotation center relative to the plate tip, for a plate with β = 0, Re = 6000, p = 0 .
45. They are dimensionalizedappropriately, (cid:98) x = L (0 . − x c ) , (cid:98) y = Ly c (15)and shown at dimensional times (cid:98) t = T t . The times (cid:98) t ∈ [0 ,
6] shown in the figure correspond to nondimensionaltimes t ∈ [0 , . d ∈ [0 , . x c (see figure 9c), that is, increasingvalues of (cid:98) x .Figure 10 shows that the computed values for the com-ponent (cid:98) y normal to the plate is in excellent agreementwith both the experimental data and similarity theory,for relatively long times (cid:98) t ≤ d ≤ . (cid:98) x tangent to the plate is in less agree-ment with the experimental data. The experiments showlarger deviation from the tip. However, the computedvalues of (cid:98) x , corresponding to β = 0 o , are in surprisinglygood agreement with similarity theory results for β = 5 o . F. Vortex circulation
This section presents the shed circulation Γ as a func-tion of p . No vorticity has separated for t < t R , duringwhich there is a Rayleigh boundary around the wholeplate, as shown in figure 6(a). After this time, a regionof recirculating flow has formed near the tip which onecan associated with a starting vortex. However, the cor-responding vorticity is embedded in the boundary layervorticity, and it is not clear a priori how to define shedvorticity. In order to distinguish separated from attachedvorticity, we use the fact that for t > t R , the vorticitycontours above the plate develop a point of maximal cur-vature. These high curvature points closely follow a slantline to the left of the vortex, as shown in figure 11(a).We follow our earlier work and define the separatedvorticity to be that enclosed in the region Ω shown infigure 11(a). It includes all vorticity to the right of thetip, and all positive vorticity to the left of the slant line.The negative vorticity attached to the wall is excluded,although it is convected into Ω at later times, when itenters x ≥ .
5, such as in figures 6(c,d). This definitionhas a continuous and natural extension to later times,see . The shed circulation is defined to beΓ( t ) = (cid:90) Ω( t ) ω ( · , t ) dA . (16)Figure 11(b) plots the circulation Γ, computed withthis definition, as a function of the displacement d , for p = 0 , / , ,
2, as indicated. Each curve begins at thevalue of d corresponding to t = t R . For each p , the curvesclosely follow a straight line in the logarithmic scaleshown, indicating a power law behaviour Γ ∼ d β . Theslopes β increase with increasing p . They are in fact inclose agreement with inviscid similarity theory. Pullin shows that for the self-similar inviscid vortex sheet sep-aration at the edge of a semi-infinite plate, driven by apower law background flow, the separated vortex sheet1circulation Γ vsh satisfiesΓ vsh ≈ J t p (1 + p ) / (17a)where J is only weakly dependent on p . This equationcan be rewritten as Γ /αvsh (1 + p ) γ ≈ J /α d (17b)where α = p p ) and γ = 1 − α = p p . To comparethe present results for viscous vortex separation with theinviscid similarity theory, figure 11(c) plots Γ /α / (1+ p ) γ ,vs the displacement d . The curves for all p are almostparallel and have slope approximately equal to 1, showingthat the power law scaling (17b) is closely satisfied, withΓ ∼ p ) / t p . (18)However the constant of proportionality depends signif-icantly on p and increases by a factor of almost 10 as p decreases from p = 2 to p = 0. The dependence on p increases as p decreases, with a large difference between p = 0 and p = 1 /
2, while the results for p ≥ / G. Characteristic sizes
To conclude, this section presents two characteristicsizes of the flow: the boundary layer thickness δ and thesize of the recirculation region s Ψ . Figure 12 plots δ p +1 vs displacement d , where δ is the vertical thickness of theregion below the plate with ω ≥ − , at x =0.25. Thethickness decreases noticeably as p increases. The figureshows that δ p +1 ∼ d or δ ∼ t (19)for approximately d ∈ [0 , . − . p . The thickness at othervalues of x is qualitatively similar. Thus, the boundarylayer grows as expected, satisfying the viscous scaling ofequation 10.The size of the vortex pair separating at the edge ofa finite plate has previously been reported in terms ofthe quantity s ψ , defined as shown in figure 6(d) to be theheight of the recirculation region on the axis of symmetryalong the middle of the plate. Note that this quantityis defined only after the recirculation region has formedand has reached the axis of symmetry, as for example infigure 6(b). Figure 13 plots s ψ as a function of d , for p = 0 , / , ,
2, as indicated. After an initial transitionperiod, the curves for all p approach the same commoncurve. For approximately d > .
5, this curve closelysatisfies s ψ ∼ d / (20) m = 1/2 p=0 p=1/2p=1 p=2 d δ p + −5 −3 −1 −5 −3 −1 FIG. 12. Boundary layer thickness δ of the region with ω ≥ − at x = 0 .
25, below the plate. The quantity δ p +1 isplotted vs displacement d , for p =0, 1/2, 1 and 2, as indicated.The dashed line has the indicated slope. and is in good agreement with the observations byTaneda & Honji based on laboratory experiments. Theinitial transition period is not a viscous or finite plate ef-fect and thus the scaling does not hold asymptotically as d → y c , which alsoscales as d / . −1 −3 −2 −1 d s ψ m = 2/3 p=0p=1/2p=1p=2 FIG. 13. Vortex size s ψ vs displacement d , for p =0, 1/2, 1and 2, as indicated. The dashed line has the indicated slope. V. SUMMARY
This paper presents a numerical study of viscous flowpast a finite flat plate moving with accelerated velocity U = t p ( (cid:98) U = a (cid:98) t p ) in direction normal to itself. The focusis on the effect of the parameter p , for p = 0 , / , , Re = 500. Allresults are computed and presented in a reference framefixed on the plate. Most results are reported as func-tions of the plate displacement d , which was found to2more concisely reflect the dependence on p . We reporton the vorticity contours, velocity profiles and streaklinesat fixed displacement d , and on the evolution of the vor-tex core trajectory, maximum vorticity, and circulationas functions of d .At fixed displacement d , the acceleration parameter p affects the vorticity distribution within the core. Forexample, at d = 1, for larger values of p , the outer turnsof the shear layer rollup are stronger, and the vorticityprofiles near the center are flatter. The spiral roll-up ofparticle streaklines is more concentrated near the center,with fewer outer turns. The streaklines and the vortexcenter are in good agreement with available experimentaldata for p = 0 . , can be identified. In the initialRayleigh stage the vorticity consists of an almost uniformlayer around the whole plate, including the tip, with noapparent separation. This stage lasts for a time T R thatscales surprisingly well with p , as p , and vanishes as p →
0. In the second stage a recirculation region has formednear the plate tip, with an enclosed rotation center whosetrajectory transitions towards self-similar growth. Thethird stage is defined by self-similar growth. In the fourthstage the finite plate length significantly affects the flowand the trajectory departs from self-similar. We notethat for p > L ∼ √ νt , ω ∼ t p / ( νt ) / , (21)respectively. For inviscid flow, dimensional analysis im-plies that length scales and shed circulation grow as L ∼ t p ) / , Γ ∼ t (1+4 p ) / (22)respectively. For the viscous vortex separation computedhere some quantities closely satisfy the viscous scaling,while others are largely independent of viscosity andclosely satisfy the inviscid scaling.For example, the boundary layer thickness closely fol-lows the viscous scaling laws until at least d = 0 .
1. Themaximum core vorticity for p = 0 is in excellent agree-ment with the viscous laws as well, for practically the whole range computed d ∈ [10 − ,
1] (see also ). For p >
0, the viscous scaling is visible for a short time in-terval only, early on. The vortex center trajectory andcirculation on the other hand are largely independent ofviscosity and closely follow the inviscid laws after theinitial transition time, until relatively large times with d ≈
1. As a result, the vortex size, defined as the heightof the recirculation region on the axis, also follows theinviscid scaling after it has reached a value comparableto the vertical vortex displacement.The changes in the computed solutions between p = 0and p = 0 . p = 0 . p = 2. The flow behaviour for p ∈ (0 , .
5) remains to bestudied more closely, as does the dependence on Re for p > J. D. Anderson Jr , Ludwig Prandtls Boundary Layer,
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