NUMERICAL SIMULATION OF VORTEX SHEDDING FROM AN INCLINED FLAT PLATE

Engineering Applications of Computational Fluid Mechanics Vol. 4, No. 4, pp. 569 579 (2010) NUMERICAL SIMULATION OF VORTEX SHEDDING FROM AN INCLINED FLAT PLATE K. M. Lam* and C. T. Wei Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China * E-Mail: kmlam@hku.hk (Corresponding Author) ABSTRACT: Vortex shedding flow from a flat plate inclined to a uniform flow at an angle of attack between 20 o and 45 o is simulated with a finite volume CFD code with RNG k- turbulence model. The unsteady flow simulation at Re=2 10 4 with RANS shows two trains of vortices shed from the two different edges of the plate forming a vortex street in the wake of the plate. The computed results provide support to previous experimental observations that in this asymmetric flow geometry, the two trains of vortices in the vortex street possess different vortex strengths. There is further evidence that the vortex from the plate leading edge is actually shed from a location near the trailing end of the plate. The computed flow at successive phases of a vortex shedding cycle show different development and shedding mechanisms for the two trains of vortices. The study also explores the generation mechanism of the fluctuating lift and drag on the plate and its relationship with the vortex shedding processes. Keywords: CFD, vortices, inclined flat plate, circular cylinder 1. INTRODUCTION Vortex shedding flow from a bluff body has attracted many experimental and numerical investigations due to its rich fluid dynamical phenomena and important engineering applications such as unsteady fluid loading of structures, flow-induced vibration and flow noises. Most of the studies investigate flow over a twodimensional bluff body such as a circular cylinder, a square cylinder and a flat plate normal to the flow. From these bodies with a symmetric geometry, periodic vortices are shed alternatively in the form of two trains of opposite-sign but equal-strength vortices. The mean flow is symmetric about the wake centerline with zero mean lift and the fluctuating lift oscillates at the vortex shedding frequency. Flow over a bluff body with an asymmetrical geometry generates lift and examples include a flat plate or an aerofoil inclined to the flow, an asymmetric aerofoil at zero angle of attack, or a rotating circular cylinder. The first author has been investigating how the periodic vortex shedding and the vortex street are affected by the degree of wake asymmetry (e.g., Lam, 1996 and 2009; Lam and Leung, 2005). For the present problem of flow past an inclined flat plate, the mean flow is asymmetric about the wake centerline and there exists a non-zero mean lift on the plate. A number of past experimental investigations studied the mean wake and vortex shedding from flat plates at normal incidence, that is =90 o, to plates inclined to the flow at a relatively small angle of attack, down to =30 o (e.g., Fage and Johansen, 1927; Perry and Steiner, 1987; Knisely, 1990). In the wake of these inclined plates, a vortex street was observed with evidence of vortex shedding alternatively from the two edges of the plates. It was also found that the vortex shedding frequency, f, scales with the projected width B of the plate normal to the freestream. The Strouhal number is approximately constant at St =fb /U 0.15 for =30 o to 90 o, U being the free-stream velocity. The issue of any asymmetry of the vortex street was not targeted in these previous studies, although Knisely (1990) reported that as becomes smaller than 30 o, St increases sharply and the wake becomes dominated by vortices from the trailing edge of the plate. It is the aim of the first author to study in details any asymmetry in the dynamics of the two trains of vortices shed from the different edges of the inclined plate. Lam (1996) started to measure some details of the vortex street behind a flat plate inclined at =30 o. Lam and Leung (2005) further studied the vortex dynamics in the wake of an inclined flat plate at between 20 o and 30 o. Phase-locked vortex patterns were obtained with velocity measurement by particle-image velocimetry (PIV). The vorticity contours showed that at same axial distances, the train of vortices from the trailing edge of the plate possess higher magnitudes of peak vorticity levels than the train of leading edge vortices. This asymmetry in the vortex street behind a plate at high incidence is Received: 21 Jan 2010; Revised: 27 Apr. 2010; Accepted: 29 Jun. 2010 569

not very significant and has been shown so far in experiments by the first author. To seek additional evidence to the experimental observation of a street of alternating vortices of unequal strengths, the present authors attempt a computational fluid dynamics (CFD) study of flow over an inclined flat plate at a number of between 20 o and 45 o. The focus is on the dynamics of vortices shed from the two edges of the plate, any asymmetry in the vortex street and the relationship with lift production. This paper is to report the CFD results, a part of which have been included in a conference paper (Lam and Wei, 2006) which is not widely available. There have been numerical computation studies on vortex shedding from bluff bodies and this problem is often used as a benchmark test for different CFD approaches. Traditionally, the Reynolds-averaged Navier-Stokes (RANS) equations are the least computational demanding approach to the simulation of engineering turbulent flow. Vortex shedding from circular and square cylinders has been modelled with some success with unsteady RANS solutions employing the standard or modified k- turbulence model (e.g., Bosch and Rodi, 1998; Iaccarino, et al., 2003; Shao and Zhang, 2006). With the advance of computational resources, the more powerful approaches of large-eddy simulation (LES) and direct numerical simulation (DNS) are increasingly used in the predictions of engineering flows (e.g., Breuer, 1998; Rodi, 2006). Compared to the square and circular cylinders, there have been few CFD studies on flow over a flat plate. Lasher (2001) computed the flow over a normal flat plate with RANS but the main emphasis was on the drag on the plate. The most notably CFD study of flow past an inclined flat plate is probably that of Breuer et al. (2003). That study chose the inclined flat plate at =18 o as a representative of high-lift aerodynamic flows with a massive separation region. Thus, the focus was on the large-scale separation mainly from the leading edge of the plate and the prediction performance by RANS, LES and detached eddy simulation. There was little information on the dynamics of the shed vortices in the downstream wake. As stated earlier, the main objective of the present RANS-based CFD study is to confirm whether there exist differences in the dynamics of the two trains of vortices in the wake of the plate. It is not intended to discuss the accuracy of the CFD approaches as in Breuer el al. (2003). As a closing note to this introduction, the asymmetry in the vortex street behind an inclined plate is illustrated by the picture of oil spill in the recent incident of a stranded ship in the Great Barrier Reef shown in Fig. A of the Appendix. 2. COMPUTATIONAL SCHEME Flow computation was carried out for a number of flow cases including flow over an inclined flat plate at four different angles of attack, =20 o, 25 o, 30 o and 45 o. The main flow parameters are summarized in Table 1. Prior to the inclined plate flow, the validity of the CFD approach is studied by modeling vortex shedding from a circular cylinder at different Reynolds numbers (Re) and the computational cases are shown in Table 2. All the flow cases in Tables 1 and 2 are two- Table 1 Computed global wake characteristics of flow past an inclined flat plate at Re=2 10 4 for two-dimensional turbulent flow cases with 62 10 3 computational cells. fb/u St F L C L C L C D C D 1 2 2 U B 20 o 0.49 0.169 1.11 3.24 0.08 1.20 0.03 25 o 0.39 0.165 1.31 3.11 0.20 1.47 0.09 30 o 0.35 0.174 1.51 3.02 0.26 1.76 0.15 45 o 0.23 0.166 1.75 2.47 0.32 2.48 0.30 Table 2 Computation of flow past a circular cylinder on effect of grid size and Re for two-dimensional flow cases. Re Computation Grid fineness St C D C L (total no. of cells) 10 3 2D, laminar Coarser: 11 10 3 0.23 1.35 0.81 10 3 2D, laminar Finer: 97 10 3 0.23 1.65 1.06 10 3 2D, laminar Standard: 24 10 3 0.23 1.64 1.05 10 4 2D, turbulent Standard: 24 10 3 0.29 0.43 0.062 10 5 2D, turbulent Standard: 24 10 3 0.28 0.29 0.072 570

dimensional. This study used the finite volume code FLUENT (Fluent, Inc., 2003) to solve the incompressible continuity and momentum equations in two dimensions (Patankar, 1980). For the turbulent flow cases, the equations were Reynolds-averaged and the k- model was used for turbulence closure. This study adopted the Re-Normalization Group (RNG) extension of k- model (Yakhot and Orszag, 1986). The RNG k- model has been shown by some workers to yield improvement over the standard k- model for recirculating flows and flows with strongly curving streamlines (e.g., O Shea and Fletcher, 1994; Pagageorgak and Assanis, 1999, Ferreira et al., 2002) and is often used in CFD for wind engineering problems including the authors team (Stathopoulos, 2006; Lam and To, 2006). Standard values were used for model constants: C =0.0845, C 1 =1.42 and C 2 =1.68. The flow equations and closure equations were solved to obtain solutions of the six flow variables, namely, pressure, three velocity components, k and. The solution scheme made use of the SIMPLEC algorithm (Van Doormaal and Raithby, 1984) for pressure-velocity coupling and the QUICK scheme (Leonard, 1979) for convective transport modelling. To study vortex shedding, unsteady solutions of the equations were sought. It can always be criticised that compared with the more advanced simulation by LES and DNS, the unsteady RANS method cannot accurately reproduce separated flow and vortex shedding. However, the main focus of the present study is a relative comparison between the features and strengths of the two trains of vortices. With quantitative data of the flow of secondary concern, the more economical RANS computation was acceptable to meet the objective of the study. The two-dimensional flow was modelled with a rectangular computational domain inside which a circular cylinder of diameter D=10 cm or an inclined flat plate of breadth B=15 cm was placed. The width of the domain was 8D for the cylinder flow and 5B for the inclined plate flow (Fig. 1). The length of the domain was from 8D to 16D or 6B to 14B, respectively. Uniform smooth flow of air at velocity U entered the inlet of the computational domain and all flow was made to exit through the downstream end of domain. The sides of the domains were set to permit no flow across them. The surface of the cylinder or plate was set as a solid wall with the no-slip condition and the standard wall function. Time-dependent solutions were sought at different times. We used a time step of 1/20 of the expected vortex shedding period. Truncation-induced flow asymmetry in the computed flow is sufficient to trigger initial instability and subsequent vortex shedding in the solution and the solutions became periodic after a few vortex shedding periods. Convergence of the unsteady solutions was normally achieved after about 3,000 iterations. Grid dependence was studied through computation of flow over a circular cylinder at Re=1,000 using three sets of grids (Table 2). The Reynolds number is Re=U D/, where is the kinematic viscosity of air. In each of the CFD results, vortex shedding was modeled. The key information of the flow, including the mean drag coefficient, C D and the root-mean-square (rms) lift coefficient, C L on the cylinder and the vortex shedding frequency, in the form of Strouhal number, St=fD/U, were calculated and used to test grid independence. As shown in Table 2, the coarser grid with 11,000 meshes was not sufficiently fine for grid independence. The finer grid with 97,000 meshes produced essentially the same solutions of St, C D and C L as the standard grid of 24,000 meshes. Thus, the standard grid setting was judged to produce grid-independent solutions. In the standard grid setting for the circular cylinder, the cylinder circumference was divided into 120 grid points, as compared to 90 or 240, respectively, in the coarser or finer grid. The inlet or outlet of the computational domain was divided into 120 meshes and the length into 160 meshes. For flow over the inclined plate, an even denser computational mesh was used. There were 60 grid points on one face of the plate. The twodimensional computation meshes were shown in Fig. 1. The numbers of grids on the end and side faces of the computational domain were 160 and 360, respectively and the total number of meshes was 62,000. Fig. 1 Computational meshes for flow past inclined flat plate. Total number of computational cells =62 10 3. 571

3. VALIDATION CASES OF FLOW OVER CIRCULAR CYLINDER Validation of the present CFD approach is made by carrying out computation at three values of Re, Re=10 3, 10 4 and 10 5. This is achieved by changing U. At the lowest Re, the flow is computed assuming laminar flow. Fig. 2 shows the computed vorticity contours for the vortex shedding flow behind the cylinder at the two lower Re, at the instants of peak lift in either direction. For the turbulent flows, very similar patterns are found at Re=10 5 as at Re=10 4, while the laminar flow at Re=10 3 shows a wider wake. Thus, results at the highest Re are not shown for brevity. In all Re, the computed lift force on the cylinder exhibits sinusoidal fluctuations at the vortex shedding frequency. There is a sinusoidally fluctuating component in the computed drag force at double the vortex shedding frequency. The values of St determined from the lift curve have been listed in Table 2. For the laminar flow at Re=10 3, the computed value of St=0.23 is in reasonable agreement with the experimental observations (Fey et al., 1998). At the higher Re, much higher values of St are computed and these depart more from the experimental results. Table 2 also shows that higher drag coefficients are found for the laminar and lower Re turbulent flows. We can also observe in Fig. 2 the effect of Re on the vortex formation length. At Re=10 3, isolated regions of vorticity concentration are detached from the attached separation shear layer as shed vortices at x/d 1.5. The vortex formation length at Re=10 4 or 10 5 is much longer at x/d 2 to 2.5. Vorticity contours in Fig. 2 are at the instants of peak upward and downward lift. At these instants, the two regions of vorticity concentration at two sides of the cylinder are found to swing to their extreme lateral positions. Nishimura and Yaniike (2001) investigated experimentally the relationship between vortex shedding and generation of fluctuating lift and drag on a circular cylinder. They suggested that shedding of a vortex leads to a rotation of both separation points on the cylinder surface away from the cylinder side where the vortex is shed. Essentially, the two shear layers are pushed to the other side of the cylinder. This rotates the total force on the cylinder from the alongwind direction towards one side. The lateral component of the total force leads to an increasing lift while the drag, being the alongwind component, drops. Alternative vortex shedding results in sideway oscillations of the total fluid force about the mean flow direction (Drescher, 1956; Norberg, 2003). The lift thus oscillates at the vortex shedding frequency while the drag fluctuates at twice that frequency. The same sideways flipping of the shear layers is Fig. 2 Computed vortex patterns at instants of peak upward and downward lift on circular cylinder. Re: (a) 10 3 ; (b) 10 4. Left panels: peak upward lift; right panels: peak downward lift. Velocity vectors and contours of non-dimensional vorticity shown at = D/U ={±0.25, ±0.5, ±1, ±2, ±3, }. 572

observed in our CFD results. In this case of a cylinder wake, the fluid force is contributed by suction pressure caused by fluid entrainment from both shear layers. This is different from the mechanism of lift production on an inclined plate where the flow separation points are fixed to the plate edges and the suction on the plate surface is contributed almost entirely by the shear layer from the leading edge of the plate. The details will be discussed in the next section. 4. FLOW PAST INCLINED FLAT PLATE Computation of flow past an inclined flat plate is carried out at four angles of attack, =20 o, 25 o, 30 o and 45 o. The Reynolds number is Re=U B/ =2 10 4, with the free-stream velocity at 2 m/s and B=15 cm. The Reynolds numbers of our previous experimental studies are Re=3 10 4 in Lam (1996) and Re=5.3 10 3 in Lam and Leung (2005). At all four values of, periodic vortex shedding is computed. Fig. 3 shows the computed vorticity fields at these four angles of attack at the instants of maximum and minimum lift. Alternating vortices of opposite senses are shed from the plate in the form of a vortex street. The sizes of the wake, vortices and the vortex street Fig. 3 Computed vortex patterns of flow past inclined flat plate. Angle of attack, : (a) 20 o ; (b) 25 o ; (c) 30 o ; (d) 45 o. Left panels: maximum upward lift; right panels: minimum lift. Velocity vectors and contours of non-dimensional vorticity shown at = B 1 /U ={±0.2, ±0.5, ±1, ±2, ±3, }. 573

C L C D 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 Fig. 4 0 5 10 15 20 25 tu /B 1 Fluctuating lift and drag coefficients on inclined flat plate. : solid line: 20 o ; broken line: 25 o ; +: 30 o ; : 45 o. C L =C D at =45 o. clearly scale with the projected plate width. In the following sections, solutions of flow velocities, vorticity and pressure at successive phases of a vortex shedding cycle will be analysed to investigate the shedding process of the vortices and the relation to the oscillating lift. 4.1 Strouhal number, lift and drag coefficients The solution of the pressure field in the flow can be used to compute the drag and lift forces on the plate. For all four plates, there is a non-zero mean lift plus a periodically fluctuating lift at the vortex shedding frequency. As expected, the mean lift force on the plate, or the lift coefficient based on the constant plate width B, is found to increase mildly with (Table 1). For an inclined flat plate, the projected plate width B 1 =B sin has been shown to be a better characteristic length for the wake (Fage and Johansen, 1927; Knisely, 1990; also see Fig. 3). Thus, unless specified otherwise, B 1 is used to calculate the Strouhal number for vortex shedding, St=fB 1 /U, and the lift (and drag) 2 coefficients, such as C 1 L F L ( 2 U B 1 ) (where F L is the lift force on unit length of the plate). Table 1 shows that the lift coefficient, based on the projected plate width, exhibits lesser variations with than the lift force itself. It is worth noting that C L increases with decrease in but the rms lift coefficient C L of the lift fluctuations (based on B 1 ) decreases as becomes smaller. When decreases from 25 o to 20 o, C L drops by more than one half. This suggests that the strength of the vortex street, say at =45 o, is much larger than that at, say =20 o. As the angle of attack becomes smaller, the wake width becomes smaller and the vortices have C D smaller length scales (Fig. 3). Thus, the vortex shedding frequency, f, as determined from the time history of the lift fluctuations, becomes higher. This is shown by the value of fb/u in Table 1. When scaled with the projected plate width, the Strouhal number for vortex shedding shows similar values at St 0.17±0.005 for different. Previous experiments found that St has similar values near 0.15 for 30 o and there is a sharp increase in St at smaller (Knisely, 1990). Lam and Leung (2005) found that St=0.15 at =25 o and St=0.18 at =20 o. For the drag on the plate, the mean drag coefficient is found to increase with even when the projected plate width is used to define the coefficient. The rms drag coefficient is also higher at larger. One notable observation about the drag force is shown in Fig. 4. It is evident that the drag oscillates at the vortex shedding frequency just as the lift. This is in contrast to the flow past a circular cylinder where the drag force oscillates at twice the vortex shedding frequency (which is also observed in our CFD results). Furthermore, the fluctuations in drag and lift are in phase, that is, the maximum drag occurs at the same instant as the maximum lift. Fig. 4 also shows that the fluctuating lift or drag starts to develop a sub-harmonic when increases to 30 o and this sub-harmonic is obviously observed at =45 o. An explanation for the above observations will be suggested in the following section. 4.2 Vortex shedding process and fluctuating lift production Fig. 5 shows in details the flow past the inclined plate at =30 o at successive phases in one vortex shedding cycle, starting and ending with the instant of maximum lift on the plate. The development of the clockwise-rotating vortex from the trailing edge of the plate is relatively simple as compared with the counterclockwiserotating vortex from the plate leading edge. Flow separates from the trailing edge and the separation shear layer rolls up into a vortex which grows on being attached to the plate edge. The vortex grows to maturity just before the instant of maximum lift (Fig. 5(f)) and is then shed from the plate trailing edge (Figs. 5(a) and (h)). An important observation is that associated with this shedding is the movement of the separation shear layer from the plate leading edge towards the plate. The shear layer originates from flow separation at the leading edge and extends over and beyond the plate length. As a new trailing edge vortex grows, say, from Fig. 5(b) onwards, 574

this separation shear layer from the plate leading edge is observed to be pushed more away from the upper surface of the plate. As the shear layer entrains fluid and thus produces suction pressure on the upper plate surface (Luo et al., 1994), its outward movement leads to lower suction pressure and consequently lower lift and drag force on the plate. The minimum lift (and drag) on the plate occurs at the instant between Figs. 5(c) and 5(d). After the instant of minimum lift, the trailing edge vortex starts to move away from the plate edge but remains attached to it. At the same time, the middle part of the upper shear layer starts to move towards the plate upper surface and makes its way into the lower shear layer. This increases the suction pressure on the plate upper surface and raises the lift and drag. Eventually, the upper shear layer intrudes into the space between the plate trailing edge and the trailing edge vortex. This breaks off the vortex from its attachment to the plate trailing edge and the lower shear layer, thus causing the vortex to be shed in Fig. 5(h). This stage of vortex shedding mechanism has been well documented for vortex shedding from a bluff body (e.g., Cantwell and Coles, 1983; Lam, 2009). The formation and shedding of a vortex from the leading edge can be observed from Fig. 5(a) onwards. The vortex originates from the separation shear layer at the plate leading edge and it grows mainly in length. During the growth, it moves downwards to trigger the shedding of the trailing edge vortex (Fig. 5(a)) but it continues to grow in length afterwards. In Fig. 5(b), a new trailing edge vortex rolls up but the upper shear layer continues to elongate. In Fig. 5(c), it extends Fig. 5 Computed vorticity at successive phases of a vortex shedding cycle. Inclined flat plate at =30 o. Phases shown are in parts of a period of one shedding cycle from the instant of maximum lift. Velocity vectors and vorticity contours shown at = B 1 /U ={±0.25, ±0.5, ±0.75, ±1, ±1.25, }. 575

to a length of almost 2B without showing any eminent breaking. It is until some growth of the trailing edge vortex which is about to intrude into the elongated upper shear layer that the shear layer eventually breaks it into two parts (Figs. 5(e)-(g)), with the detached part becoming a shed leading edge vortex. The actual shedding of the leading edge vortex takes place not near the plate edge but is at a location similar to that of the trailing edge vortex. This is in agreement with the experimental findings of Lam and Leung (2005) and the LES simulation of Breuer et al. (2003). Lam and Leung (2005) discussed that for the development and shedding of the leading edge vortex, the upper surface of the inclined plate acts like an after body part of a bluff body and thus the shedding location is not from the fixed separation point of the plate leading edge. On the contrary, the plate surface has a much weaker effect on the development of the trailing edge vortex and the singular plate edge condition leads to a simpler shedding location. This discussion is supported by the present CFD result in Fig. 5. It should, however, be noted that the actual formation of the leading edge vortex involves large-scale flow separation which cannot be well modelled by the present RANS computation. The LES simulation of Breuer et al. (2003) showed a more complex process but the focus of this study is on the vortex street downwards and the production of fluid force. Fig. 6 shows the computed flow and distributions of pressure coefficient above the inclined plate at the instants of maximum and minimum lift. It is evident that the maximum lift occurs when the trailing edge is shed from the plate trailing edge (Figs. 6(a) and (b)). Although the vortex is associated with negative pressure (suction), its location is completely downstream of the plate such that the negative pressure does not contribute to fluid force on the plate. Instead, the upper separation shear layer from the plate leading edge moves to the closest distance from the upper plate surface. The region above the plate is entirely under the effect of this shear layer which entrains air from the region. There is recirculating flow in this region with uniformly negative pressure. This produces the largest normal force on the plate within the vortex shedding cycle. Since both the plate lift and drag are components of this force, they reach their maximum values together (Fig. 4). About half a cycle later, the next trailing edge vortex grows to maturity. Being remained attached to the plate, it occupies the largest area above the trailing edge of the plate. However, the associated negative pressure region still cannot produce a noticeable force on the plate due to its location at the plate trailing edge and thus downstream of the plate (Fig. 6(d)). Instead, its fluid circulation induces some reverse flow to hit back onto the plate surface (Fig. 6(c)) and causes some pressure recovery. The presence of the vortex also pushes the upper shear layer from the plate leading edge farther away from the plate. The upper plate surface is thus under low negative pressure, leading to the smallest normal force on the plate. Fig. 6 suggests that the flow and pressure on the lower side of the plate do not experience large changes during a cycle of vortex shedding. Fig. 6 Computer flow vectors and pressure distribution around inclined flat plate at =30 o. (a) instant of maximum lift; (b) instant of minimum lift. Contours of pressure coefficient shown at C p ={±0.2, ±0.4, ±0.6, }. Solid contours for negative pressure; broken contours for positive pressure. 576

' 6 4 2 0 Fig. 7 : Trailing edge vortex. : leading edge vortex. Solid symbols: present CFD data. Open symbols: PIV data of Lam and Leung (2005) with adjustment. 0 4 8 12 x/b' Magnitudes of peak vorticity levels of leading and trailing edge vortices behind inclined flat plate at =30 o. Fig. 4 shows that at the largest 30 o, the lift (and drag) experiences a short reduction after it reaches the maximum. It have been discussed in Fig. 5 that the production of fluctuating normal force on the upper plate surface is affected by the sideways swaying movement of the upper separation shear layer in relation to the growth and shedding of the trailing edge vortex. At =30 o (Fig. 5) and smaller values of (Fig. 3), the movement of the upper shear layer does not appear to reach the plate surface even in the extreme instant of its intrusion into the lower shear layer to shed the trailing edge vortex. However, the CFD results for =45 o (e.g., Fig. 3) suggest that upon shedding of the trailing edge vortex, the intrusion of the upper shear layer does reach the downstream part of the upper plate surface. It is believed that there is some partial pressure recovery associated with this flow hitting the plate surface and this is responsible for the subharmonic variation in the lift curve at =45 o (Fig. 4). 4.3 Strengths of vortices in vortex street One objective of the present CFD study is to provide evidence to the experimental observations of Lam and Leung (2005) that the trailing edge vortices possesses higher vorticity levels than the leading edge vortices at the same axial locations. Fig. 7 shows the magnitudes of peak vorticity inside the leading and trailing edge vortices in Fig. 5 plotted against the axial locations of the vortex centres after shedding. The computed nondimensional vorticity, = B 1 /U, is shown in the figure. It is evident that the CFD results also show higher peak vorticity levels for the trailing edge vortex. The experimental data of (x) are included in Fig. 7 but they have been multiplied by a constant adjustment factor of value 2. The unsteady RANS solutions exhibit almost perfectly periodic flow patterns but jitter among cycles was present due to random turbulent fluctuations in the experiments. The jitter resulted in reduced values of the phase-averaged peak vorticity levels. This is why an adjustment is necessary to bring the CFD data and experimental data to similar levels. The lower peak vorticity levels in the experimental data are also believed to be caused by the much larger spatial resolution of the velocity vector data from PIV as compared to those in the fine computational grid. Notwithstanding this adjustment, the two sets of data show very similar trends of vorticity change with axial distance. The CFD results for the plate inclined at other values of show the same observation of different strengths between the two trains of vortices. The details are not shown for brevity. 5. CONCLUSIONS The vortex shedding flow past an inclined flat plate at Re=2 10 4 is simulated with unsteady RANS computation. The primary objective is to provide evidence to the previous experimental finding of two trains of unequal-strength vortices behind a flat plate at high incidence (Lam, 1996; Lam and Leung, 2005). Computation is performed at four values of =20 o, 25 o, 30 o and 45 o and vortex shedding in the form of a vortex street is observed from each edge of the inclined plate. The global features of the wake from the computational results are in agreement with previous experimental investigation. These include Strouhal number and the scaling of vortices and wake on the projected plate width. The computed vorticity field shows different formation and shedding stages between the vortices from the leading edge and the trailing edge of the plate. While these main findings are in agreement with the experiments, the additional information of flow-induced pressure and force in the CFD results shows the relationship between vortex shedding and the generation of fluctuating lift and drag forces on the plate. The trailing edge vortex develops from the rollup of the separation shear layer at the plate trailing edge. Although it remains attached to the plate during its growth, it does not contribute much to the fluid force on the plate due to its location downstream of the plate. The fluctuating force on the plate is largely caused by the pressure on the upper plate surface which is governed by the location, length and fluid entrainment of the upper shear layer from 577

the plate leading edge. These activities of the upper shear layer are found to exhibit interrelationship with the growth of the trailing edge vortex. The maximum lift occurs during the shedding of the trailing edge vortex when the upper shear layer moves closest to the plate to trigger the shedding. On the other hand, when the trailing edge vortex grows to maturity but remains attached to the plate, the upper shear layer is pushed farthest away from the plate and the plate is under the minimum lift. The CFD results provide support to the experimental observation that the leading edge vortex is formed from a break off of the upper shear layer. The actual shedding location is near the trailing end of the plate. A comparison between the computed peak vorticity levels at the vortex centres shows again that at the same downstream locations, the trailing edge vortex has higher vorticity levels that the leading edge vortex. REFERENCES 1. Bosch G, Rodi W (1998). Simulation of vortex shedding past a square cylinder with different turbulence models. Int. J. Numer. Meth. Fluids 28:601 616. 2. Breuer M (1998). Large eddy simulation of the subcritical flow past a circular cylinder: numerical and modeling aspects. Int. J. Numer. Meth. Fluids 28:1281 1302. 3. Breuer M, Jovicic N, Mazaev K (2003). Comparison of DES, RANS and LES for the separated flow around a flat plate at high incidence. Int. J. Numer. Meth. Fluids 41:357 388. 4. Cantwell B, Coles D (1983). An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder. J. Fluid Mech. 136:321 374. 5. Drescher H (1956). Messung der auf querangestr. omte Zylinder ausgeubten zeitlich veranderten Drucke. Zeitschrift fur Flugwissenschaften und Weltraumforschung 4:17 21. 6. Fage A, Johansen FC (1927). On the flow of air behind an inclined flat plate of infinite span. Proc. R. Soc., London, Ser. A 116:170 197. 7. Ferreira AD, Sousa ACM, Viegas DX (2002). Prediction of building interference effects on pedestrian level comfort. J. Wind Eng. Ind. Aerodyn 90:305 319. 8. Fey U, Konig M, Eckelmann H (1998). A new Strouhal-Reynolds-number relationship for the circular cylinder in the range 5 47 Re 2 10. Phys. Fluids 10(7):1547 1549. 9. Fluent Inc. (2003). Fluent 6.1 User s Manual, www.fluent.com. 10. Iaccarino G, Ooi A, Durbin PA, Behnia M (2003). Reynolds averaged simulation of unsteady separated flow. Int. J. Heat and Fluid Flow 24:147 156. 11. Knisely CW (1990). Strouhal numbers of rectangular cylinders at incidence: a review and new data. J. Fluids Struct. 4:371 393. 12. Lam KM (1996). Phase-locked eduction of vortex shedding in flow past an inclined flat plate. Phys. Fluids 8(5):1159 1168. 13. Lam KM (2009). Vortex shedding flow behind a slowly rotating circular cylinder. J. Fluids Struct. 25(2):245 262. 14. Lam KM, Leung MYH (2005). Asymmetric vortex shedding flow past an inclined flat plate at high incidence. Euro. J. of Mech.- B/Fluids 24:33 48. 15. Lam KM, To AP (2006). Reliability of numerical computation of pedestrian level wind environment around a row of tall buildings. Wind Struct. 9:473 492. 16. Lam KM, Wei CT (2006). Characteristics of vortices shed from a circular cylinder and an inclined flat plate. Proc. 4th Int. Sym. Comput. Wind Eng., Yokohama, July 2006:649 652. 17. Lasher WC (2001). Computation of twodimensional blocked flow normal to a flat plate. J. Wind Eng. Ind. Aerodyn. 89:493 513. 18. Leonard BP (1979). A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comp. Meth. in Appl. Mech. and Engrng. 19:59 98. 19. Luo SC, Yazdani MG, Chew YT, Lee TS (1994). Effects of incidence and afterbody shape on flow past bluff cylinders. J. Wind Eng. Ind. Aerodyn. 53:375 399. 20. Nishimura H, Taniike Y (2001). Aerodynamic characteristics of fluctuating forces on a circular cylinder. J. Wind Eng. Ind. Aerodyn. 89:713 723. 21. Norberg C (2003). Fluctuating lift on a circular cylinder: review and new measurements. J. Fluids Struct. 17:57 96. 22. O Shea N, Fletcher CAJ (1995). Prediction of turbulent delta wing vortex flows using an RNG k model. Lecture Notes in Physics 453:496 501. 23. Papageorgakis GC, Assanis DN (1999). Comparison of linear and nonlinear RNSbased k models for incompressible turbulent flows. Num. Heat Transfer 35:1 22. 578

24. Patankar SV (1980). Numerical heat transfer and fluid flow. Hemisphere Publ. Corp., Washington D.C. 25. Perry AE, Steiner TR (1987). Large-scale vortex structures in turbulent wake behind bluff bodies. Part 1. Vortex formation processes. J. Fluid Mech. 174:233 270. 26. Rodi W (2006). DNS and LES of some engineering flows. Fluid Dyn. Res. 38:145 173. 27. Shao J, Zhang C (2006). Numerical analysis of the flow around a circular cylinder using RANS and LES. Int. J. Comput. Fluid Dyn. 20:301 307. 28. Stathopoulos T (2006). Pedestrian level winds and outdoor human comfort. J. Wind Eng. Ind. Aerodyn. 94:769 780. 29. Van Doormaal JP, Raithby GD (1984). Enhancement of the SIMPLE method for predicting incompressible fluid flows. Num. Heat Transfer 7:147 163. 30. Yakhot V, Orszag SA (1986). Renormalization-Group analysis of turbulence. Phys. Rev. Lett. 57:1722 1724. APPENDIX In April 2010, a Chinese bulk carrier ran aground near the Great Barrier Reef and caused oil spill (http://en.wikipedia.org/wiki/2010_great_barrier _Reef_oil_spill). It is interested to note that the flow of ocean current past the stranded ship on the sea surface bears close resemblance to the flow over an inclined plate. The flow was clearly visualized by the 3.7 km length of leaked oil slick and detergents. There are video clips on the patterns from the above quoted webpage and a photograph is reproduced here in Fig. A. One can clearly observe the difference between the masses contained within the leading edge vortices and the trailing edge vortices. Fig. A Oil slick pattern from a stranded ship. 579