Numerical study of viscous starting flow past a flat plate
UUnder consideration for publication in J. Fluid Mech. Numerical study of viscous starting flow pasta flat plate
L I N G X U A N D
M O N I K A N I T S C H E Department of Mathematics and Statistics, Georgia State University, Atlanta, GA30303,USA Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM87131, USA(Received 16 January 2014)
Viscous flow past a finite flat plate which is impulsively started in direction normal to it-self is studied numerically using a high order mixed finite difference and semi-Lagrangianscheme. The goal is to resolve details of the vorticity generation at early times, and to de-termine the effect of viscosity on flow quantities such as the core trajectory and vorticity,and the shed circulation. Vorticity contours, streaklines and streamlines are presentedfor a range of Reynolds numbers Re ∈ [250 , t ∈ [0 . ,
1. Introduction
Vorticity separation in flow past sharp edges is a fundamental process of intrinsic inter-est in fluid dynamics. The boundary layer vorticity is convected around the edge, whereit concentrates and forms a vortex. The vortex grows in strength and size, eventuallycausing the boundary layers to separate as a shear layer that rolls up in a spiral shapearound the vortex core. The starting vortex flow has been the focus of many experimen-tal, analytical and numerical studies, beginning with the work of Prandtl (see Lugt 1995).This paper concerns flow past a finite flat plate of zero thickness which is impulsivelystarted in direction normal to itself. Closely related laboratory experiments include theworks of Pierce (1961), Taneda and Honji (1971), Pullin and Perry (1980), Lian andHuang (1989), and Lepage et al (2005). They visualize the rolled-up layer and yield dataon the vortex size, core trajectory, core vorticity distribution, and the onset of an insta-bility along the outer spiral turns. Related numerical results include the simulations ofWang (2000) and Eldredge (2007) for viscous flow past thin rounded plates, and those ofHudson and Dennis (1985) and Dennis et al (1993), followed by Koumoutsakos and Shiels(1996), and Luchini and Tognaccini (2002), for flow past plates of zero thickness. Thefirst three of these consider finite plates, while Luchini and Tognaccini (2002) computeflow past a semi-infinite plate. These works report vortex fields, vortex core trajectoriesand induced forces at intermediate to relatively large times. a r X i v : . [ phy s i c s . f l u - dyn ] J a n L. Xu and M. Nitsche
The main goal of this paper is to complement these earlier works with numerical resultsthat yield new information about the flow, in particular on quantities that may be moredifficult to measure experimentally. Our focus is to resolve the flow over several decadesin time for a range of Reynolds numbers, show details of the vorticity generation andstudy the effect of viscosity on various flow quantities. Specific quantities of interestinclude the vortex trajectory, the forces induced by the wall vorticity, and the shedcirculation. Computing shed circulation requires defining the region of entrainment ofthe starting vortex, which is not clearly apparent in the early formation stages. Oncethis region is determined, we investigate convective and diffusive contributions to thecirculation shedding rate, and thereby obtain detailed insight into how viscosity affectscirculation shedding. The computations also yield several scaling laws that show thedependence on time and Reynolds number for the corresponding quantities. While theresults pertain to an idealized flow past a plate of zero thickness, they can be used as abasis of comparison to evaluate widely used low order models for separated flows, suchas point vortex models (eg., Cortelezzi and Leonard 1993, Michelin and Llewellyn Smith2009, Eldredge and Wang 2010, Ysasi et al 2011), or vortex sheet models (eg., Krasny1991, Nitsche & Krasny 1994, Jones 2003, Jones and Shelley 2005, Alben and Shelley2008, Shukla and Eldredge 2007), which are all based on simple approximations for thecirculation shedding rate. The results also provide a basis of comparison to determine,for example, the effect of finite plate thickness, the shape of the plate tip (Schneider etal Re = 500, including detailsof the vorticity near the boundary. Following results using Re ∈ [250 , Re . Based on the computed profiles, we define the separated vorticity atearly times, when it is not clearly differentiated from boundary vorticity, and use thisdefinition to compute shed circulation, as well as convective and diffusive componentsof the vorticity flux into the starting vortex. The results show that the chosen vortexboundary indeed bounds separated vorticity from vorticity that remains attached. Theyalso show that viscous diffusion contributes significantly to the circulation shedding rate,but its contribution depends surprisingly little on the value of the Reynolds number.The shed circulation and the vortex core trajectory are found to follow inviscid scalinglaws (Kaden 1931, Pullin 1978) over several decades in time. The computed trajectoryis also in good agreement with experimental results of Pullin and Perry (1980). The corevorticity and dissipation, and the induced drag and lift forces, are found to follow scalinglaws that define their dependence on time and on Re .The paper is organized as follows. Section 2 describes the problem of interest andthe governing equations. Section 3 presents the numerical method, its accuracy, and theresolution obtained. Section 4 presents the numerical results, including the evolution intime for fixed Re , the dependence at a fixed time on Re , the core trajectory and vorticity, (a) (b) Figure 1: Sample solution at a relatively large time. (a) Streamlines. (b) Vorticity field.Positive vorticity contours are shown in black, negative ones in a lighter shade of grey.the circulation and circulation shedding rates, and the induced drag and lift forces, inthat order. The results are summarized in section 5.
2. Problem Formulation
Problem description
A finite plate of length L and zero thickness is immersed in viscous fluid and impulsivelystarted to move from zero velocity to a constant velocity U > L as the characteristiclength scale, and U as the characteristic velocity. An alternative nondimensionalization,appropriate in the absence of a length scale, or at very early times, is given by using ν/U instead of L as the characteristic length scale, where ν is the kinematic fluid viscosity.This alternative is discussed in the Appendix.The flow is assumed to be two dimensional. It is described in nondimensional Cartesiancoordinates x = ( x, y ), and time t , with fluid velocity u ( x , t ) = (cid:10) u ( x, y, t ) , v ( x, y, t ) (cid:11) .We choose a reference frame moving with the plate, in which the plate is positionedhorizontally on the x-axis, centered at the origin, at S = (cid:110) ( x, y ) : x ∈ (cid:2) − , (cid:3) , y = 0 (cid:111) , (2.1)the plate velocity is zero, and the far field velocity points upwards, u ∞ ( t ) = (cid:10) , (cid:11) . (2.2)To illustrate, figure 1 plots the streamlines and vorticity field at some relatively large timepast the start of the motion. Here and throughout the paper, positive vorticity contoursare shown in black, negative contours are shown in a lighter grey scale. The flow isassumed to remain symmetric about x = 0, since at the times considered here symmetrybreaking instabilities are not expected to significantly affect the flow. Hereafter, resultsare only shown for the right half plane, x ≥ ψ ∞ ( x, y ), is given by the complex potential W ∞ ( x, y ) = (cid:114) − ( x + iy ) = φ ∞ + iψ ∞ . (2.3) L. Xu and M. Nitsche
This flow is induced by a vortex sheet in place of the plate whose strength is such that noflow passes through the plate. For higher generality, instead of using equation (2.3), weapproximate ψ ∞ using a sufficiently fine discretization of the vortex sheet, following theapproach taken in Nitsche & Krasny (1994), which can be applied to other geometriesas well, even if the analytic expression for the potential is not known.2.2. Governing equations
The fluid flow is modeled by the incompressible Navier Stokes equations with constantdensity. The governing equations, given in terms of the fluid vorticity ω ( x , t ) = v x − u y and stream function ψ ( x , t ), are(a) ∂ω∂t + ( u · ∇ ) ω = 1 Re ∇ ω, (2.4 a )(b) ∇ ψ = − ω, with ψ = 0 on S and ψ → ψ ∞ as x → ∞ , (2.4 b )(c) u = ∇ ⊥ ψ, with u = 0 on S , (2.4 c )where ∇ ⊥ ψ = (cid:10) ∂ψ∂y , − ∂ψ∂x (cid:11) , Re = LU/ν , and ν is the kinematic fluid viscosity.
3. Numerical Approach
Numerical Method
The numerical method is based on fourth order finite difference approximations of thegoverning equations on a regular grid. The computational domain is the rectangularregion D = [0 , x max ] × [ y min , y max ] , (3.1)with symmetry imposed across x = 0. The interior of the domain is given by the interiorof D \ S , where S is the plate position given in (2.1). The computational boundary consistsof ∂D ∪ S . Here x max > / y min < y max > ω effectively vanishes on ∂D for all the times computed. The domain isdiscretized by ( N x + 1) × ( N y + 1) equally spaced gridpoints ( x i , y j ), where x i = ih , i = 0 , . . . , N x , h = x max /N x , (3.2 a ) y j = y min + jk , j = 0 , . . . , N y , k = ( y max − y min ) /N y , (3.2 b )and N x , N y are chosen so that h = k . Similarly, time is discretized as t n = n ∆ t , n = 0 , . . . , N , ∆ t = T fin /N , (3.2 c )where T fin = t N is the final time. Streamfunction, velocity and vorticity are carried onthe gridpoints, with ψ ni,j , (cid:104) u, v (cid:105) ni,j , ω ni,j approximating ψ ( x i , y j , t n ), (cid:104) u, v (cid:105) ( x i , y j , t n ), and ω ( x i , y j , t n ).The boundary stream function at time t n is ψ nbd . The values of ψ bd on the plate S are zero, which ensures that on the plate, v = 0. The values of ψ bd on the remainingboundaries of ∂D are obtained numerically, as explained below. The boundary vorticityat time t n is ω nbd . The values of ω bd on ∂D are zero. The values of ω bd on the upstreamand downstream sides of the plate S , denoted by ω n + and ω n − , respectively, are obtainedby enforcing that u = 0 on the plate, see below. The initial conditions are given by zerovorticity in the interior of the domain.The vorticity at time t n is updated to time t n +1 by solving equation (2.4) in two steps: Step 1:
The interior vorticity is convected by solving the equation
DQDt = 0 subject to Q ( t n ) = ω n , (3.3 a )for one timestep and setting ω ∗ = Q ( t n +1 ). The values of ω ∗ are then used to obtainupdated interior and boundary values of the stream function, velocity and vorticity, ψ n +1 , (cid:104) u, v (cid:105) n +1 and ω n +1 bd . Step 2:
The interior vorticity is diffused by solving the equation ∂Q∂t = 1 Re (cid:53) Q subject to Q ( t n ) = ω ∗ , Q bd ( t n +1 ) = ω n +1 bd , (3.3 b )for one timestep and setting ω n +1 = Q ( t n +1 ).Several details in each of the two steps above remain to be explained. Step 1a:
Equation (3.3 a ) is solved using a semi-Lagrangian scheme which is secondorder in time and fourth order in space, as follows. For each interior grid point ( x i , y j ),first find the location of a particle at t n that travels with the fluid velocity, and ends upat ( x i , y j ) at t n +1 . This is equivalent to solving d x dt = u ( x , t ) , x ( t n +1 ) = ( x i , y j ) (3.4)for x ( t n ), where x = ( x, y ). Equation (3.4) is solved to second order in time usingvelocity values at the current and previous timestep, (cid:104) u, v (cid:105) n − and (cid:104) u, v (cid:105) n . Then, obtainthe vorticity of the particle at t n , ω ( x ( t n ) , t n ), from vorticity values at nearby grid pointsusing a fourth order bi-cubic interpolant. This step uses interior and boundary values ofvorticity at t n . Finally, set ω ∗ i,j = ω ( x ( t n ) , t n ). Details can be found in the paper by Xu(2012). Step 1b:
Updated interior values of the stream function at t n +1 are obtained bysolving ∆ ψ n +1 = ω ∗ in interior , ψ = ψ n +1 bd on ∂D ∪ S . (3.5)Here, the Laplace operator is approximated by a compact fourth order finite differencescheme (Strikwerda 1989, equation 12.5.6). The boundary values ψ bd are given by ψ bd = 0 on S ∪ { x = 0 } . On the remaining three sides of ∂D , ψ bd is computed using an integral formulation. Weuse the domain specific Green’s function G S G S ( x , x o ) = log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:113) x + iy − / x + iy +1 / − (cid:113) x o + iy o − / x o + iy o +1 / (cid:113) x + iy − / x + iy +1 / − (cid:113) x o + iy o − / x o + iy o +1 / ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (3.6 a )where x = ( x, y ), x o = ( x o , y o ), and ∗ denotes the complex conjugate, and compute ψ bd ( x bd , t ) = ψ ∞ ( x bd , t ) + (cid:90) D/S ω ( x o , t ) G S ( x bd , x ) d x , (3.6 b )where x bd ∈ ∂D . Alternatively, one can use the free space Green’s function G ∞ , andsimulate the effect of the plate by a vortex sheet in its place. In either of these approaches,one needs to compute an area integral (cid:82) ω ( x , t ) G ( x bd , x ) d x . In practice, we only integrateover the region in which | ω | ≥ − , and use the fourth order Simpson’s method. Thelinear system for ψ n +1 ij obtained by discretizing (3.5) is solved using the conjugate gradientmethod. L. Xu and M. Nitsche
Step 1c:
Updated interior values of velocity at t n +1 are obtained by solving (cid:104) u, v (cid:105) n +1 = ∇ ⊥ ψ n +1 (3.7)using fourth order centered difference approximations. Boundary values of velocity areonly needed on the plate, where they vanish, and on the axis x = 0, where they areobtained by centered differences from ψ n +1 and use of symmetry. The updated velocityis used at the next timestep, in Step 1a. Step 1d:
Updated boundary vorticity values at t n +1 are obtained from the updatedstream function by enforcing the no-slip boundary condition on the plate. The boundarycondition φ = 0 ensures that v = 0 on the walls. The boundary vorticity is ω n +1 bd = − ∂ ψ n +1 ∂y . This equation is discretized so that ψ = 0 and ∂ψ/∂y = 0 on the wall, and thus u = 0.Here, we use a fourth order version of the Thomas formula, known as Briley’s formula,following E and Liu (1996) (their equation 2.11). The updated vorticity values are usedin Step 2, below, as well as at the next timestep, in Step 1a. Step 2:
Equation (3.3 b ) is solved by discretizing the Laplace operator with the 4thorder compact finite difference scheme also used for equation (3.5) and then applying animplicit Crank-Nicolson method which is second order in time and fourth order in space(Fletcher 1991, page 255ff). The resulting linear system for the vorticity ω n +1 is solvedusing the conjugate gradient method.This completes the description of the numerical method. In order to visualize streak-lines as may be observed in laboratory experiments, particles are also initially placednear the plate and passively transported by the fluid flow. Their position x ( t ) is given by d x dt = u ( x , t ) , x (0) = x o , (3.8)where u is the fluid velocity. The velocity at the current particle position is obtained byinterpolation, and the equation is solved using the second order explicit Adam-Bashforthscheme, for a range of initial positions.3.2. Resolution and Convergence
To test this numerical scheme, Xu (2012) applied it to the driven cavity problem of E andLiu (1996), and reproduced their results. For smooth cavity lid motion (see also Johnston1999), the method was confirmed to converge to 4th order in space, and to first order intime. The slow convergence in time is a property of standard splitting schemes. Here, wediscuss the performance of the method applied to the more singular case of impulsivelystarted flow past a sharp edge.To illustrate the effect of resolution in space and time, figure 2 plots vorticity contourscomputed for Re = 500 at t = 0 .
05, with various values of the meshsize h and timestep∆ t . The resolution is coarsest in figure 2(a), finest in figure 2(d), as given in the caption.The figure shows contours ω = ± − in a region close to the tip of the plate, withpositive vorticity in black, negative vorticity in a lighter shade of grey, and the plate asa black line. The zero vorticity contour level appears as a thick dark curve which in factconsists of many positive and negative vorticity contour levels of small magnitude. Wefirst describe the well-resolved result in figure 2(d). Recall that the background drivingvelocity flows from bottom to top. This causes the formation of a boundary layer ofpositive vorticity around the plate. The maximum vorticity and velocity magnitude, ω max and U max , occur at the tip of the plate, one gridpoint away from it, at all times. At (a) (b) (c) (d) x (x c ,y c ) ω = ω c u |=U max | ω = ω max Figure 2: Vorticity contours ω = ± − at t =0.05 for Re =500 for the problem ofimpulsively started flow past a finite plate. The mesh size h and time step ∆ t for eachfigure are (a) h =1/160, ∆ t =4 × − , (b) h =1/320, ∆ t =2 × − , (c) h =1/640, ∆ t =1 × − , (d) h =1/1280, ∆ t = 5 × − .the time shown here, positive vorticity has moved upward to form a concentrated vortexon the downstream side of the plate, with a second local maximum in the core vorticity,at ( x c , y c ) with magnitude ω c . This vortex induces positive flow on the dowstream platewall, which causes the formation of a thin region of opposite signed, negative vorticityalong the wall. The negative vorticity is entrained into the leading vortex.If h, ∆ t are too large, as in figure 2(a), the lack of resolution is evidenced by alternatelayers of positive and negative vorticity that form outside the leading vortex. If the flowis only slightly underresolved, as in figure 2(b), ripples in the vorticity are first visiblebelow and to the right of the tip. As the resolution increases, as in figure 2(c), the ripplesdisappear and the vorticity is smooth. Finer resolution, as in figure 2(d), leaves the resultspractically unchanged.We found this to be the case at all times computed: at all times, an instability isapparent for large enough values of h, ∆ t . If the resolution is sufficiently fine, the resultsare smooth and remain unchanged to the eye under further refinement. The values of h, ∆ t required for smooth results are smaller at earlier times. We thus take the followingapproach: results at a given time are computed with a value of h, ∆ t sufficiently small sothat the vorticity contours with | ω | ≥ − appear resolved. Table 1 lists the meshsizesand timesteps used in time intervals [0 , T fin ], for different values of T fin . The range given L. Xu and M. Nitsche h ∆ t T fin [ x min , x max ] × [ y min , y max ]1/160 (4-5) × − × [-0.50,5.50]1/320 2 × − × [-0.25,0.75]1/640 (0.5-2) × − × [-0.25,0.50]1/1280 (2-5) × − × [-0.25,0.50]1/2560 (4-5) × − × [-0.05,0.10]1/5120 2 × − × [-0.05,0.10] Table 1: The mesh size h , time step ∆ t , final time T fin , computational domain used forthe range of Re = ∈ [250 , Re , a givenvalue of h is used with the smaller values of ∆ t and to smaller final times T fin . h e hc e hω e hψ,x e hψ,y along x = 0 . y = 0 . . × − . × − . × − . × − . × − . × − Table 2: Errors e hω , e hc , e hψ,x , and e hψ,y , at t = 0 .
05, computed with Re =500.for ∆ t and T fin reflects values used for different Reynolds numbers Re . For larger Re,a given time requires a smaller value of h and ∆ t . For example, h = 1 /
320 is used forthe runs with T fin = 0 . Re = 250, but is required for much larger T fin = 3 for Re = 2000.An estimate of the order of convergence of the method for impulsively started flow isobtained from Table 2. The table lists the errors in the position ( x hc , y hc ) and vorticitymagnitude ω hc of the vortex core at t = 0 .
05 (see figure 2(d)), as well as errors in thestream function along horizontal and vertical lines near the vortex core. The errors arecomputed relative to the results with h = 1 / e hω = | ω hc − ω / c || ω / c | , e hc = || x hc − x / c || || x / c || (3.9 a ) e hψ,x = || ψ h − ψ / || ∞ along x = 0 . b ) e hψ,y = || ψ h − ψ / || ∞ along y = 0 .
048 (3.9 c )where x c = (cid:104) x c , y c (cid:105) . The data in table 2 is summarized in figure 3, together with a linewith slope m = 2. Even though the amount of data points is rather limited, the data isconsistent with second order or better rate of convergence, with faster convergence awayfrom the tip. 3.3. Singular initial flow
The difficulty in resolving the flow is largely due to the singular nature of the initial flow.To illustrate, figure 4 plots the maximum velocity U max , and the maximum absolutevorticity ω max , vs. time t . In each case, results for all values of h used are plotted, asindicated. −2.8 −2.6 −2.4 −2.2 −4 −3 −2 h e rr o r s m=2 e c e ω e ψ ,x e ψ ,y Figure 3: Errors e hc , e hω , e hψ,x , e hψ,y , vs. h , as indicated. A line of slope m = 2 is also shown. −6 −5 −4 −3 −2 −1 U m a x t t −1/4 (a) h=1/5120h=1/2560h=1/1280h=1/640h=1/320h=1/160 −6 −5 −4 −3 −2 −1 t ω m a x t −1 (b) Figure 4: (a) Maximum velocity U max and (b) maximum absolute vorticity ω max , vs. t ,for Re =500, computed with the indicated values of h .The maximal absolute velocity U max , plotted in figure 4(a), becomes unbounded as t approaches 0. Recall that the initial potential flow has unbounded velocity at the tipof the plate. The maximum velocity U max is bounded at all positive times, and decaysfast initially. However, because of the initial singularity, it is not possible to resolve theflow until after some small initial times. Figure 4(a) shows that with smaller values of h , U max can be computed smoothly during earlier times, with U max ≈
200 at t ≈ − .The results with varying h appear to converge to a line with slope − /
4, indicating that U max decays as U max ∼ t − / . (3.10)Figure 4(b) plots the maximum absolute vorticity ω max . Its values are of order 10 atthe earliest time shown, and are fairly well resolved with the smallest values of h shown.The maximum vorticity decreases in time approximately as ω max ∼ t − . (3.11)0 L. Xu and M. Nitsche
We note that the maximum velocity and vorticity occur one gridpoint away from thetip of the plate. Thus, unlike figure 3, the results in figure 4 do not represent a study ofpointwise convergence at a fixed point. They do illustrate the singular nature of the flowand the extent to which it is recovered by the finite numerical resolution.
4. Numerical Results
Vorticity, streaklines, streamlines, Re = 500This section describes the evolution of the flow near the plate, for the case of fixed Re = 500, unless noted. Figure 5 shows vorticity contours (left column), streaklines(middle column) and streamlines (right column), at the indicated times. At each time,the results shown are computed with the finest resolution listed in table 2 for that time.The vorticity contours are ω = ± [ − , with positive contours in black, negative onesin grey.The driving far field flow u ∞ is the potential flow moving upwards past and around theplate. Initially, the flow generates a boundary layer of positive vorticity along both theupstream and downstream sides of the right half-plate. Upstream vorticity is convecteddownstream, concentrating near the tip as a vortex that grows in time. The vortexentrains nearby vorticity, while vorticity further away is swept from the vortex towardsthe axis, thus depleting the region in between. As a result, the leading vorticity, whichinitially is connected to the downstream boundary layer, begins to separate from it.At some time between t=0.2 and 0.5, the positive vorticity in the leading vortex hascompletely separated from the positive boundary layer vorticity, resulting in a moreclearly defined starting vortex.The leading vortex induces a region of recirculating flow that can be seen in thecorresponding streamlines. The region of recirculating flow forms immediately after themotion begins. The fluid within this region, below the center of rotation, flows in directionopposite to the starting flow, and generates negative vorticity attached to the wall. In thecomputations, the negative vorticity is observed already after a few timesteps. It is barelyvisible in figure 5, at t=0.005, but grows in time and is clearly discernible by the greycontours at later times. The negative vorticity region grows horizontally along the plate,away from the starting vortex, until it reaches the axis at t ≈ . t = 1 . t = 3, the positive vorticity in thestarting vortex has reached the axis of symmetry, x = 0. It diffuses out of the recirculationregion, so that at t = 5, much vorticity is outside the enclosing streamline and movesupwards away from the plate. The results at the larger times presented here are in goodagreement with results shown by Koumoutsakos and Shiels (1996).The streaklines, shown in the middle column of figure 5, are obtained by releasing aparticle at a point near the tip at each timestep, and computing its evolution with thefluid velocity. At a given time, the figure shows the position of all particles at that timethat were released previously. The streakline plots thus mimic what one would observein a laboratory experiment if dye were continously released at a point near the tip. Eachreleased particle circulates around the vortex center. Particles that have been releasedearlier travel closer to the center and thus, the resulting streakline has a spiral shape.The spiral tightens near the center and the number of spiral turns increase in time. Themaximum vorticity near the tip of the plate is convected with the particles along the1 −0.05 0.11 y t=0.005 −0.06 0.12 y −0.08 0.13 y −0.10 0.22 y −0.13 0.35 y −0.15 0.60 y −0.15 0.95 y x x x (for caption see next page)streakline, and diffuses. Thus the streakline is a good indicator of the centerline of theseparated shear layer, but not of the overall vortical region, or of the recirculation region,both which extend beyond the region occupied by the spiral. At the times shown, thespiral center is a good indicator of the vorticity maximum in the vortex core, and of thecenter of fluid rotation.2 L. Xu and M. Nitsche −0.2 2.0 y t=3.0 −0.2 2.9 y x x x Figure 5: Vorticity, streaklines, and streamlines, for Re = 500 at a sequence of times, t = 0 . , . , . , . , . , . , . , . , .
0, as indicated. The vorticity contour levels are ω = ± [ − . The stream function levels are ψ = [ − .
05 : 1] for t ≤ ψ = [ − . t = 3 , ψ = 0 streamline, which leaves the tip of the plate and reattacheson the downstream side, at a short distance behind the vortex. As the recirculationregion grows, the enclosing streamline ψ = 0 first reaches the axis, between time t = 0 . .
2, and then continues to move up along the centerline, x = 0. It then forms thefamiliar rounded symmetric recirculation bubble downstream of the plate, as observedexperimentally and computationally, before the flow looses its symmetry at later times(see, eg, van Dyke 1982, figure 64, and Koumoutsakos & Shiels 1996, figure 18).Figure 6 shows a closeup of the vorticity contours near the tip of the plate, plottedas the black and grey solid contour lines. It also shows two of the flow streamlines as3Figure 6: Closeup of vorticity contours near the plate tip, for Re = 500, p = 0, at (a) t = 0 . t = 0 . t = 0 .
008 and (d) t = 0 .
1. Vorticity contour levels are ± [ − .dashed curves. One is the ψ = 0 streamline enclosing the region of recirculating flow,the other is a closed streamline near the center of rotation. Figure 6(a) shows that veryearly, at t = 0 . t = 0 . ω + on the upstream side of the plate at a sequence of times. Thevorticity is initially unbounded at the plate tip, zero at the axis, and positive, increasing,in between. As time increases, it remains positive but decreases in magnitude. Figure 7(b)plots the wall vorticity ω − on the downstream side of the plate. At t = 0+, the vorticityis also positive everywhere, but negative unbounded at the tip. The interval along theplate in which the vorticity is negative grows, and the maximal magnitude decreases.The wall vorticity values are responsible for the drag forces normal to the wall parallel tothe background flow, presented later in § L. Xu and M. Nitsche x ω + (a) x ω − (b) Figure 7: Wall vorticity (a) ω + ( x ), and (b) ω − ( x ), on the upstream and downstreamsides of the plate, respectively, vs. x , at a sequence of times t =0.0002, 0.0004, 0.001,0.002, 0.005, 0.01, 0.02, 0.04, 0.08, 0.2, 0.5, 1, 3, 5, for Re = 500. The vorticity decreasesin magnitude as time increases. −0.02 −0.01 0 0.01 0.02−500−400−300−200−1000100200300400 y ω (a) Figure 8: Vorticity t = 0 .
04 along the lines x =0.4, 0.425, 0.45 and 0.475, as functions of y . The arrows indicate increasing values of x .t=0.04, along four vertical lines crossing the region of negative vorticity, x = 0 .
4, 0.425,0.45, 0.475, as a function of y . The arrows indicate the direction of increasing x . Thevorticity is positive on the upstream side of the plate ( y < y > x increases towards the tip, the wall vorticity and their gradients ∂ω/∂y increase in magnitude. The wall vorticity gradients are responsible for the lift forcesparallel to the wall, normal to the background flow, presented in § L neg and a characteristic thickness H neg , both of which are illustrated in figure 9(a), andthe integral negative vorticity Γ neg . For later reference, results are plotted for variousReynolds numbers, as indicated in the legend in figure 9(b). The length L neg , plottedin figure 9(b), is also the length of the recirculation region. It increases until it reaches L neg = 0 .
5, which is when the recirculation region reaches the axis. For Re = 500, thisoccurs at t ≈ . Re , it occurs sooner. After this time, negative vorticitycovers all of the downstream plate wall. The thickness H neg , plotted in figure 9(c), ischosen to be the thickness of the negative vorticity region at x = 0 .
4. It increases initially,reaches a maximum, and then decreases again as the negative vorticity is entrained by theleading vortex. Larger Reynolds number flows have thicker regions of negative vorticity.5 y x L neg (a) H neg L neg t (b) H neg t (c) Γ neg t (d) −0.1−0.2−0.3 Figure 9: Quantifiers of negative vorticity region. (a) Negative vorticity contours andstreamlines at t = 0 .
04, showing definition of length L neg and thickness H neg . (b) Length L neg vs. t. (c) Thickness H neg vs. t . (d) Integral negative vorticity Γ neg vs. t . Figures(b,c,d) show results for Re = 250 , , , x = 0 . neg = (cid:90) ω< x ≥ ω dA , (4.1)is plotted in figure 9(d). Remarkably, even though the layer thickness H neg dependsheavily on Re , its integral circulation is practically independent of Re , at least untilabout t = 3. After that, diffusion causes the integral negative vorticity to decrease inmagnitude, with larger decrease for lower Re .4.2. Dependence on Re Figure 10 shows the dependence of vorticity contours, streamlines and streaklines at afixed time t = 1 on Reynolds number, for Re ∈ [250 , Re increases, the vorticitycontours show well-known features: the wall boundary layer thickness decreases; theseparated shear layer thickness decreases, and its spiral roll-up becomes more evident;the thickness of the negative vorticity region decreases, as already seen in figure 9(c). Forlarger Re , the separated vorticity is supported in a smaller, more compact region.Some dependence on Re is also observed in the spiral streaklines. Most noticeably, thespiral roll-up near the center is tighter for larger Re , and there are more spiral turns. Thespiral size does not depend much on Re , and is, in particular, not a good indicator of thesize of the vortex structure. The size of the recirculation region does not depend much on Re either. The streamline density within the vortex increases with Re , indicating largergradients, that is, larger fluid velocities.A measure of the boundary layer thickness is given by the thickness δ of the positive6 L. Xu and M. Nitsche −0.15 1.0 y −0.15 1.0 y −0.15 1.0 y −0.15 1.0 y x x x Figure 10: Vorticity, streak lines and streamlines at t =1 for Re =250, 500, 1000 and 2000.Vorticity contour levels are ± [ − , and streamlines contour levels are [ − . x = 0 .
2, plotted in figure 11. Here, δ is the thickness of the region with ω ≥ − . The figure shows that when plotted against7 −7 −6 −5 −4 −3 −2 −3 −2 −1 t/Re δ m = 1/2 Re=250Re=500Re=1000Re=2000
Figure 11: Boundary layer thickness δ vs. t/Re . Results are shown for Re =250, 500, 1000and 2000, as indicated. (a) −0.005 0.016 y (b) (c) −0.005 0.016 y x (d) x Figure 12: Closeup of vorticity contours near the plate tip, at t = 0 . Re =250, (b) Re = 500, (c) Re = 1000, (d) Re = 2000. Vorticity contour levels are ± [ − . t/Re , the results for all Re collapse onto a line of slope 1 /
2, and thus δ ∼ ( t/Re ) / . (4.2)asymptotically, as t →
0. This is in agreement with results for self-similar flow pastinfinite plates. The thickness at other values of x is qualitatively similar.Figure 12 shows a closeup of the vorticity at a fixed early time, for different Reynoldsnumbers. With low Re , as in figure 12(a), the vorticity has not yet formed a local maxi-8 L. Xu and M. Nitsche −4 −3 −2 −1 t ω c (a) Re=250Re=500Re=1000Re=2000 −7 −6 −5 −4 −3 −2 t/Re ω c m = −3/4 (b) t y c (c) −4 −3 −2 −1 −3 −2 −1 m = 2/3t y c (d) t x c − . (e) −4 −3 −2 −1 −4 −3 −2 −1 t | x c − . | m = 2/3 (f) Figure 13: Core vorticity and trajectory, for Re = 250 , , , ω c vs. t . (b) Core vorticity ω c vs. scaled time t/Re . (c,d) Vertical coreposition y c vs. t . (e,f) Horizontal core displacement x c − . Re , the core vorticity increases, the negative vorticity region lengthens and rollsup around the spiral. These features at a fixed time, as Re increases, are similar to thefeatures observed with fixed Re , as time increases (see figure 6), up to scale, which is asexpected at early times in which the presence of a length scale is not yet noticeable.94.3. Core vorticity and trajectory
The core position and vorticity are defined as the coordinates ( x c , y c ) and vorticity ω c atthe local vorticity maximum in the leading vortex (see figure 2d). They are defined onlyafter this local maximum away from the tip of the plate has formed. As seen in figure12, this occurs earlier for higher Re . This section investigates their scaling behaviour anddependence on Re . We note that alternatively, the core position can be defined as thecenter of rotation, which exists at all times, but this option is not explored here.Figure 13 plots the core coordinates and vorticity computed for all Re , as indicated inthe legend in figure 13(a), and all values of h used, as given in table 2. Thus, each subplotin figure 13 shows results for about 20 different time series, computed with different Re and resolutions. Figure 13(a) plots the values of ω c vs. t. The values show remarkablylittle dependence on the resolution: only for the largest Reynolds numbers do we see smalljumps in the values of ω c as h is doubled. The values ω c appear linear in the logarithmicscale, with a vertical shift between different Re . By plotting the results versus t/Re , infigure 13(b), all the data collapse onto one curve, and agree for over more than 5 decadesin time with the approximation, ω c ≈ . (cid:18) Ret (cid:19) / . (4.3)Thus, at any fixed time, the core vorticity increases as Re / . For fixed Re , it decreasesin time as t − / . We do not know of an analytical result that explains this observation.Figures 13(c,d) plot the vertical displacement y c of the vortex core from the plate, ona linear and a logarithmic scale respectively, versus t. Here again, the values for all Re and all resolutions computed collapse onto one curve, with no apparent dependence on h , and only a small dependence on Re visible at later times. The data for all Re scalesfor about 4 decades as y c ≈ . t / (4.4)with deviations from this line visible after approximately t = 1. This scaling agrees withthe self-similar inviscid spiral roll-up of semi-infinite free vortex sheets, or of separatedvortex sheets at the edge of a semi-infinite plate (Kaden 1931, Pullin 1978).Figure 13(e) plots the horizontal displacement x c − . y c , and less well resolved,with the dependence on h more visible. To within the available resolution, the resultsdepend little on Re . The linear scale shows that x c − . t = 2, the symmetric vortices at each end of the plate more slightly inward, as they wouldin the self-similar inviscid case. After that time they begin moving outwards again, with x c > . t = 5. The logarithmic scale shows that until about t = 1, x c also satisfies the self-similar inviscid scaling, with x c − . ≈ − . t / . (4.5)The closest data on viscous flow at early times available for comparison in the litera-ture are the experimental measurements of Pullin & Perry (1980), who measured vortexcore positions in flow past wedges and compared them to similarity theory. Figure 14 re-produces their results for the smallest wedge considered, of wedge angle β = 5 o , togetherwith our computed results. The experimental data span an early time interval t ∈ [0 , . y c is in quite good quantitative agreement with the computedvalues. The horizontal displacement x c overlaps with the present better resolved valuesat the lower Reynolds numbers, except for the last two data points. The experimental0 L. Xu and M. Nitsche t y c (a) Re=250Re=500Re=1000Re=2000P&P Exp t x c − . (b) Figure 14: Comparison of vortex core coordinate with experimental data of Pullin & Perry(1980). (a) Vertical displacement y c . (b) Horizontal displacement x c . The computed datafor all Reynolds number and the experimental data is shown, as indicated in the legend.The vertical bars through the experimental data are the error bars indicated in P&P.data was obtained at larger Reynolds number of Re ≈ Re , as is clearly the case for the values of y c . We cannotexplain the deviation of the computation from the last two experimental data points infigure 14(b).With knowledge of the core vorticity scaling, we now plot scaled vorticity, as well asdissipation. Figure 15 shows color coded contours of the scaled fluid vorticity, ω/Re / , at t = 1, for the indicated values of Re (left column), and of the corresponding dissipation∆ ω/Re (right column). For clarity, we note that the plots consist of equally spacedcontour curves of A tanh( q/A ), for a chosen value of A below the true maximum, where q is either the scaled vorticity or dissipation. As a result, contours of arbitrarily largelevels can be shown. Also as a result, the values of the quantity q shown in the attachedcolorbars are not equally spaced in the color level.At this time, the vorticity attached to the upper plate wall is all negative, and allremaining vorticity is positive. The vorticity contours show the decrease of boundaryand shear layer thicknesses, and the increasingly visible spiral shear layer roll-up as Re increases. In a crossection at any point through the shear layer, the vorticity is largest inthe middle of the layer, and decreases to local minima in between different spiral turns.We note that at this rather late time, the vorticity maximum only approximately satisfiesthe scaling ω c ∼ Re / , and the scaled values plotted in the figure decrease from 0.359to 0.250.The dissipation plot indicates the magnitude of viscous diffusion in the flow. Thedissipation at the vortex core, where the vorticity has a maximum, is negative, whichcauses the maximum vorticity to decrease. The largest absolute values at the core increaseas Re increases, from 16.16 to 54.96, in a manner consistent with the scaling∆ ωRe ∼ Re / . (4.6)Across each of the spiral shear layer turns the dissipation changes sign twice. It is positiveat the local vorticity minima in between spiral turns, causing these minimum values toincrease, and it is negative at the local maxima in the middle of the layer, causing thesemaxima to decrease. As a result, the spiral turns are more clearly visible in the dissipation1Figure 15: Scaled vorticity ω/Re . (left column) and dissipation ∆ ω/Re (right column)at t = 1, for Re =250, 500, 1000 and 2000, from top to bottom. The colorbar is the samefor all values of Re , and shown for highest Re only.2 L. Xu and M. Nitsche Ω C + C s C o (a) Ω C + C − C s C o (b) Ω C + C − C ax C o (c) Figure 16: Sketch defining the domain Ω and portions C + , C − , C s , C o , C ax of its bound-ary with nonzero circulation flux.plots than in the vorticity contours. The dissipation is largest in magnitude near the tipof the plate, where it reaches values well above 200.4.4. Shed vortex circulation
Definitions
One of our main interest in performing the present simulations was to obtain circula-tion shedding rates for the viscous flow, for which little data is available in the literature.We are interested in resolving the shed circulation over large time scales, including earlytimes. However, at early times the vorticity in the leading vortex is not clearly distin-guished from the boundary layer vorticity. Here, we define the shed circulation to beΓ = (cid:90) Ω( t ) ω dA , (4.7)where the region Ω( t ) is defined in figure 16. The circulation is normalized by U L .The definition of Ω is defined slightly differently in different time-regimes of the flow,with continuous transitions between them. At early times, when the region of negativevorticity has not yet been entrained past the vertical line x = 0 .
5, we follow the sketchin figure 16(a). On the upstream side ( y < C + . On the downstream side ( y > C o that separates negative from positive vorticity, and by a slant line C s through points of high curvature visible in the vorticity contours. That is, the regionΩ is defined to include all vorticity to the right of the tip, to exclude the negativeboundary layer vorticity, and is limited on the left by the slant line. Figure 16(b) showsthe vorticity at intermediate times, when the negative vorticity on the downstream sidehas been entrained past the vertical line x = 0 .
5. Here we include all vorticity, positiveor negative, to the right of the vertical line through the tip, which introduces a verticalpiece of boundary C − above the plate. Figure 16(c) shows the vorticity at later times,when all the positive boundary layer vorticity on the downstream wall has diffused andeffectively vanished. At this time the vortex is bounded on the left not by the slant line,but by the axis C ax . Each subplot in figure 16 shows a typical length scale, indicatingthat the three regimes in time span three decades of length scales in space.In order to determine the effect of viscous diffusion relative to inviscid convectionof vorticity into Ω, and also to determine the suitability of the definition above, weconsider shedding rates of vorticity through the various components of the boundary of3Ω. By applying the Transport Theorem, the Navier Stokes Equations, and the DivergenceTheorem, one finds that d Γ dt = ddt (cid:90) Ω( t ) ω ( x , t ) dA = (cid:90) Ω( t ) ∂ω∂t dA + (cid:90) ∂ Ω ω ( u bd · n ) ds = (cid:90) Ω( t ) (cid:20) − ( u · ∇ ) ω + 1 Re ∆ ω (cid:21) dA + (cid:90) ∂ Ω ω ( u bd · n ) ds = (cid:90) ∂ Ω( t ) (cid:20) − ω u · n + 1 Re ∇ ω · n + ω ( u bd · n ) (cid:21) ds = d Γ c dt + d Γ d dt + d Γ m dt (4.8)where n is the outward normal, and u bd is the velocity of the boundary. That is, acrosseach piece of the boundary there is a contribution to the vorticity due to convection,diffusion, and the moving boundary. We denote these components by subscripts c , d ,and m respectively. Notice that the only moving boundary portions are C o and C s , andthe latter has no nonzero vorticity moving with it or convecting through it. Similarly,there is no convection of vorticity through C ax . Thus the nonzero contributions to thecirculation shedding rate are the 9 components d Γ + c dt , d Γ − c dt , d Γ sc dt , d Γ + d dt , d Γ − d dt , d Γ od dt , d Γ sd dt , d Γ axd dt , d Γ sm dt where the superscript refers to the portion of the boundary, and the subscript refersto the component of the shedding rate. For conciseness, we combine two of the viscouscomponents into one: d Γ − d dt + d Γ od dt → d Γ − d dt . Below, we first investigate the 8 circulation shedding rates for Re = 500, and thendetermine dependence on Re .4.4.2. Re = 500Figures 17(a,b) plot the shed circulation computed using the above definition forRe=500. The logarithmic scale in figure 17(b) shows that the circulation satisfies thescaling behaviour predicted by inviscid similarity theory (Pullin 1978) surprisingly well,Γ( t ) ∼ t / . Moreover, it shows that the circulation is resolved over more than 4 decadesin time. Finally, we note that just as the data for ω c and y c , the circulation data showsremarkable independence of the meshsize used in the computation. Even though the fig-ure plots the results for all meshsizes and time intervals given in table 2, the data is analmost continuous function of the meshsize.Figures 17(c,d) show some of the circulation flux components. The largest flux into theregion is the convective component through the vertical C + on the upstream side of theplate, d Γ + c /dt , shown as the thickest curve in figure 17(c). It is larger than d Γ /dt , shownas the dashed curve. The viscous flux components are negative and reduce the totalcirculation. Of these, largest in magnitude is the viscous flux through C − ∪ C o , d Γ − d /dt ,shown as the curve of medium thickness. We conclude that d Γ + c /dt is most significant,but the contribution to the total flux due to viscous diffusion is nonnegligible.Figure 17(d) shows the three flux components through the slant line. To note is, first,that these are much smaller than the the largest components shown in figure 17(c) anddo not contribute significantly to the circulation, and second, that they vanish quickly4 L. Xu and M. Nitsche t Γ (a) −4 −3 −2 −1 −1 t Γ m=1/3(b) d -d / dt d -d / dt d / dt c d / dt m d / dt S c d / dt S d d / dt S Figure 17: Circulation and shedding rates for Re = 500. (a) Circulation Γ( t ), linear scale.(b) Circulation Γ( t ), logarithmic scale. (c) Components d Γ + c ( t ) /dt , d Γ + d ( t ) /dt , d Γ − d ( t ) /dt , d Γ /dt , on a logarithmic scale (d) Components d Γ sc ( t ) /dt , d Γ sd ( t ) /dt , d Γ sm ( t ) /dt .and are negligibly small after about t = 0 . Dependence on Re In order to determine the dependence of the shed circulation on Re , figure 18 plotsseveral circulation flux components for Re = 250 , , , Re , but is compar-atively very small. Similarly, the convective component through the vertical C − on thedownstream side of the plate, shown in figure 18(b), depends on Re but is small. Thelargest component, the convective component through the vertical C + on the upstreamside of the plate, shown in figure 18(c), is completely independent of Re . The next largestcomponent, the viscous component of vorticity diffusion through C − ∪ C o , shown in fig-ure 18(d), appears to be quite independent of Re at early times, based on the resultsfor Re = 250 , , Re = 2000, the results are not fully resolved, due to thedifficulty in resolving the large vorticity gradients present near the tip of the plate.We conclude from figure 18 that the flux is essentially independent of Re . The convec-tive flux is clearly so, and the dominant diffusive flux, even though it is large, is largelyindependent on Re . We attribute the latter to the fact that as Re increases, vorticitygradients increase, but are offset by the 1 /Re factor in the diffusive term of equation5 axd d / dt t d Γ − c / d t (b) −3 −2 −1 −1 t d Γ + c / d t (c) Figure 18: Dependence of circulation shedding rate components on Re, for Re =250 , , , d Γ axd /dt (b) d Γ − c /dt , (c) d Γ + c /dt , (d) d Γ − d /dt , vs. t .(4.8), 1 Re (cid:90) ∇ ω · n ds . (4.9)Figure 19 plots the circulation Γ( t ) for Re = 250 , , , Re at earlytimes. As time increases, differences between Re increase slightly. The largest differenceover the range of Re considered here occurs at the last time computed, t = 5, and is lessthan 5% of the circulation at that time. The logarithmic plot in figure 19(b) shows thatall 20 time series computed with different values of Re and meshsizes collapse onto onecurve, which is well approximated byΓ( t ) ≈ t / as t → . (4.10)4.5. Vortex forces
The vorticity profiles on the plate wall induce drag and lift forces. Here we compute thedrag and lift, F D and F L , defined to be total force parallel and normal to the backgroundflow, in the right half of the plate only. They are given by (Eldredge 2007) F D ( t ) = 2 Re (cid:90) / [ ω + ( x, , t ) + ω − ( x, , t )] dx , (4.11 a )6 L. Xu and M. Nitsche t Γ (a) Re=250Re=500Re=1000Re=2000 −4 −3 −2 −1 −1 t Γ m=1/3(b) Figure 19: Dependence of total shed circulation Γ( t ) on Re , for Re = 250 , , , t F L (a) h=1/1280h=1/640h=1/320h=1/160 t F D (b) KSh=1/320h=1/160
Figure 20: Forces (a) F L and (b) F D for Re = 1000, computed with the indicated valuesof h . For comparison, (b) includes the data by Koumoutsakos & Shiels (1996, denotedby KS). F L ( t ) = 2 Re (cid:90) / x (cid:20) ∂ω + ∂y ( x, , t ) + ∂ω − ∂y ( x, , t ) (cid:21) dx , (4.11 b )where all forces are normalized by ρU L . By symmetry, the overall lift force acting onthe whole plate is zero, and the total drag is 2 F D .Computing the forces using formulation (4.11) is sensitive to discretization errors, sincethe formulation depends on values of the vorticity and its derivatives on the plate wall.These values are large and difficult to compute accurately, specially near the tip. Toillustrate, figure 20 plots the computed lift and drag for Re = 1000 for various values of h used and shows that the convergence in h is slow. An alternative formulation for thedrag force is used by Koumoutsakos and Shiels (1996) (KS), who compute drag as thetime derivative of an area integral, see their equations (40-41). The value of F D givenby (4.11b) corresponds to the variable c D plotted in KS in their figure 13. Figure 20(b)in this paper compares the drag computed here with the values computed by KS, for Re = 1000. The figure shows that the values are in fairly good agreement, althoughdifferences exist, mainly in the decay rate at early times.Figures 21(a,b) plot the lift and drag force on a logarithmic scale, for all Reynolds7 −2 −3 −2 −1 Re*t F L (a)m=−0.55 Re=250Re=500Re=1000Re=2000 −4 −2 t F D (b)m=−0.58 Figure 21: (a) Lift force vs. Re · t . (b) Drag force vs. t . Results are shown for Re =250 , , , h used. They show that the results at earlytimes collapse quite well onto a common curve. Figure 21(a) plots the lift force versus ascaled time Re · t . In these variables, the data at early times collapses onto a curve thatdecreases in time and in Re approximately as F L ( t ) ≈ ( Re · t ) − / . (4.12)The drag, on the other hand, plotted in figure 21(b) versus time t , appears to be almostindependent of Re , and is approximately given by F D ( t ) ≈ t − / . (4.13)These results suggest that at a fixed time, the lift decays significantly faster, as 1 /Re / ,than the drag, which remains almost constant in Re for early times. In view of equation(4.11), this in turn indicates that the wall vorticity grows as Re / , while the wall vorticitygradients grow faster, almost linearly in Re . At later times the drag force decreases as Re increases, consistent with the results shown by Dennis et al (1993) and by KS.
5. Summary
Viscous flow past a flat plate of zero thickness which is impulsively started in directionnormal to itself is studied using highly resolved numerical simulations for a range ofReynolds numbers Re ∈ [250 , Re were found forseveral quantities in the flow, over several decades in time. Some quantities clearly dependstrongly on Re , such as the core vorticity, the boundary layer thickness, and the lift force8 L. Xu and M. Nitsche over the half-plate, with ω c ( t ) ≈ . (cid:16) Ret (cid:17) / , δ ( t ) ∼ (cid:16) tRe (cid:17) / , F L ( t ) ≈ ( Re · t ) − / , (5.1 a )all as t →
0. Other quantities of the early starting flow are largely independent on Re ,such as the core trajectory, the shed circulation, the integral negative vorticity, and thedrag force over the half-plate, with y c ( t ) ≈ . t / , x c ( t ) − . ≈ − . t / , (5.1 b )Γ( t ) ≈ t / , F D ( t ) ∼ t − / . (5.1 c )The scaling for x c , y c and Γ are in excellent agreement with the inviscid scaling laws forself-similar roll-up.One of the main contributions is to define and compute the viscous shed circulation,specially at early times, when the shed vorticity is not clearly separated from the bound-ary layer vorticity. Our definition is validated by plots of the circulation shedding ratesacross various portions of the boundary defining the shed vorticity, which show that prac-tically no vorticity enters or leaves the boundary by convection except near the tip, whereit is convected into the vortex by the separating boundary layers. With this definition,the shed circulation satisfies the self-similar scaling laws for more than three decades intime.The presentation of several components of the circulation shedding rate gives insightinto the effect of viscosity on the total circulation. The largest component is the gainof circulation due to convection of vorticity from the upstream boundary layer into theleading vortex. This component is highly independent of the Reynolds number. It is offsetby loss of circulation due to viscous diffusion of vorticity out of the boundary, which isof opposite sign but also significant in magnitude. Interestingly, this diffusive componentalso depends little on Re at early times, consistent with the fact that the overall shedcirculation is basically independent of Re at these times. This observation suggests thatthat as Re increases, vorticity gradients responsible for viscous diffusion grow in sucha way that the quotient ∇ ω/Re changes little. We conclude that the effect of viscousdiffusion on the overall shed circulation is significant, but its contribution depends littleon the value of Re . Appendix
In this appendix we compare the scaling laws observed in this paper using two alternatenondimensionalizations. For clarity, in this appendix only, let all variables without a hator double hat, such as x, t, Γ , ω, F D , denote the original dimensional variables. Let allvariables with one hat denote variables nondimensionalized by length and time scales L and L/U . For example, (cid:98) x = xL , (cid:98) t = U tL , (cid:98)
Γ = Γ
LU , (cid:98) ω = LωU , (cid:99) F D = F D LU . ( A ν/U and ν/U . For example, (cid:98)(cid:98) x = U xν , (cid:98)(cid:98) t = U tν , (cid:98)(cid:98) Γ = Γ ν , (cid:98)(cid:98) ω = νωU , (cid:99)(cid:99) F D = F D νU . ( A (cid:98)(cid:98) x = Re · (cid:98) x , (cid:98)(cid:98) t = Re · (cid:98) t , (cid:98)(cid:98) Γ = Re · (cid:98) Γ , (cid:98)(cid:98) ω = (cid:98) ωRe , (cid:99)(cid:99) F D = Re · (cid:99) F D . ( A (cid:98)(cid:98) ω c ≈ . Re / (cid:98)(cid:98) t − / , ( A. a ) (cid:98)(cid:98) δ ∼ (cid:98)(cid:98) t / , ( A. b ) (cid:99)(cid:99) F L ≈ Re · (cid:98)(cid:98) t − / , ( A. c ) (cid:98)(cid:98) y c ≈ . Re / (cid:98)(cid:98) t / , ( A. d ) (cid:98)(cid:98) x c ≈ − . Re / (cid:98)(cid:98) t / , ( A. e ) (cid:98)(cid:98) Γ ≈ Re / (cid:98)(cid:98) t / , ( A. f ) (cid:99)(cid:99) F D ∼ ( Re/ (cid:98)(cid:98) t ) / , ( A. g )all as (cid:98)(cid:98) t →
0. Notice that of all these, one variable, namely the boundary layer thickness (cid:98)(cid:98) δ , scales as a function of (cid:98)(cid:98) t independent of the value of Re . One may therefore suggestthat the double-hat nondimensionalization is more natural for this variable. Using thesame argument one may say that the single-hat nondimensionalization is more naturalto describe the vortex coordinates and circulation, as in equations (5.1bc). The essenceis that the variables depend on L, U, ν as described by either equations (A.3) or (5.1).
REFERENCESS. Alben and M. J. Shelley,
Physical Review Letters , , 074301. L. Cortelezzi and A. Leonard,
Fluid Dyn. Res. , , 264-295. S. C. R. Dennis, W. Qiang, M. Coutanceau and J.-L. Launay,
J. Fluid Mech. , , 605-635. W. E and J. G. Liu,
J. Comput. Phys. , , 122-138. J. D. Eldredge,
J. Comput. Phys. , , 626-648. J. D. Eldredge and C. Wang,
AIAA , , 1–19. C. A. J. Fletcher,
Springer Verlag ,New York.
J. D. Hudson and S. C. R. Dennis,
J. Fluid Mech. , , 369–383. H. E. Johnston,
PhD. thesis , Uni-versity of Maryland.
M. Jones,
J. FluidMech. , , 405–441. M. A. Jones and M. J. Shelley,
J. Fluid Mech. , , 393-425. H. Kaden,
Ing. Arch , 140 (Englishtrans. R.A.Lib.Trans. no.403). L. Xu and M. Nitsche
P. Koumoutsakos and D. Shiels,
J. Fluid Mech , , 177–277. R. Krasny,
Lectures in AppliedMathematics , 385–402. C. Lepage, T. Leweke and A. Verga
Phys. Fluids , 031705. Q. X. Lian and Z. Huang,
Exp Fluids , , 95–103. P. Luchini and R. Tognaccini,
J.Fluid Mech , , 175–193. H. J. Lugt,
S. Michelin and S. G. Llewellyn Smith,
Theor. Comput. Fluid Dyn. , , 127–153. M. Nitsche, R. Krasny,
J. Fluid Mech , M. Nitsche, M. A. Taylor and R. Krasny,
Computational Fluid and Solid Mechanics , Oxford:Elsevier Science.
D. Pierce,
J. Fluid Mech. , , 460–464. D. I. Pullin,
J. Fluid Mech , , 401–430. D. I. Pullin and A. E. Perry,
J. Fluid Mech , , 239–255. D. I. Pullin and Z. J. Wang,
J. Fluid Mech. , , 1–21. K. Schneider, M. Paget-Goy, A. Vega and M. Farge,
M. Seaid,
CMAM , , 392–409. R. K. Shukla and J. D. Eldredge,
Theor. Comput. Fluid Dyn. , , 343–368. A. Staniforth, J. Cote,
Monthly Weather Review , , 2206–2223. J. C. Strikwerda,
Wadsworthand Brooks/Coles . S. Taneda and H. Honji,
Journal of the Physical Society of Japan , , 262–272. A. Ysasi, E. Kanso, and P. K. Newton,
Fluid Dynamics: From Theory to Experiment , , 1574–1582. M. Van Dyke,
Parabolic Press , Stanford, CA.
Z. J. Wang,
J. Fluid Mech. , , 323-341. L. Xu,