Numerical Heat Transfer, Part A, 41: 469±490, 2002 Copyright # 2002 Taylor & Francis 1040-7782 /02 $12.00 + .00
APPLICATION OF A FRACTIONAL-STEP SCHEM E AND FINITE-VOLUM E M ETHOD FOR SIM ULATING FLOW PAST A SURFACE-M OUNTED M IXING TAB Suchuan Dong and Hui Meng Department of Mechanical & Aerospace Engineering, State University of New York at Buffalo, Buffalo, New York, USA
Rodney O. Fox Department of Chemical Engineering, Iowa State University, Ames, Iowa, USA The fractional-step scheme and nite-volume method are applied on a structured body- tted grid to simulate the ow passing over a trapezoidal tab mounted on a at plate. The implementation of boundary conditions on tab surfaces is greatly simpli ed with this grid system. Due to grid nonorthogonality, however, discretization of Navier–Stokes equations leads to linear systems with complicated coecient matrices. For the problem size in this work, performance data indicate that parallel operations occupy about 98.38% of the simulation, giving rise to a maximum parallel speedup of S p; max º 61:73. The ow assing over the trapezoidal tab is simulated at a Reynolds number Re = 600 based on the inlet free-stream velocity and the tab height, and the results are compared with a particle image velocimetry (PIV) measurement with the same parameters. The simulation successfully captures the vortex structures in the tab wake as observed in the experiments. Comparisons of the instantaneous ow patterns, the mean velocity, and second-order moments also show good agreement. The simulation and PIV experiment also produce a similar shear-stress distribution along the streamwise direction at the at plate.
INTRODUCTION The surface-mounted trapezoidal mixing tab has found increasing application in heat transfer and low-viscosity liquid and gas mixing due to its ability to generate large-scale vortical structures that promote cross-stream mixing. It has also attracted fundamental ¯uid dynamics research because of the topological resemblance of these vortical structures to those observed in natural boundary layers.
Received 24 October 2000; accepted 15 August 2001. This project was partially supported by the NSF grant CTS-9625307. It was also supported by the National Center for Supercomputing Applications (NCSA) and the Center for Scienti®c Supercomputing (CSS) at Kansas State University. Address correspondence to Hui Meng, Department of Mechanical & Aerospace Engineering, State University of New York at Bu alo, 342 Jarvis Hall, Bu alo, NY 14260-4400 , USA. E-mail:
[email protected] alo.edu 469
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NOMENCLATURE A; B; C; D H L N P Q R S Sp T U; V; W Uave Xt a b h ~ n p s t u¤ ui u; v; w
coe cient matrices nonlinear term length of simulation domain integral form of nonlinear term number of processors height of simulation domain right-hand side of linear equation width of simulation domain parallel speedup time velocity components outlet average velocity coordinate in streamwise direction relative to tab tip width of tab bottom edge width of tab top edge height of tab top edge above the wall unit vector pressure grid step size time friction velocity velocity component in i direction velocity components
xi x; y; z y¤ F; f D a b g
v r t j x
Superscripts max in 0 w ‡
^ n i t
coordinate in i direction coordinates in three directions inner layer length scale pseudopressure step size tab inclination angle trapezoidal taper angle projected angle (90¯ 7 ) on the ¯at plate kinematic viscosity density of ¯uid shear stress generic variable percentage of parallel operations and Subscripts maximum inlet free stream wall in wall units intermediate variable nth time step ith component relative to tab tip
Gretta [1] systematically measured the in¯uence of the geometry parameters of the trapezoidal tab and in-¯ow conditions on the wake mixing characteristics and heat transfer. He found that the in-¯ow condition (laminar or turbulent) did not a ect the cross-stream penetration notably, but the tab inclination angle in¯uenced the mixing e ciency signi®cantly. Gretta and Smith [2] observed a sequence of largescale hairpin vortices shed from the tab with scales comparable with the tab height (distance between the tab top edge and the wall, denoted by h) along with a counterrotating vortex pair (CVP) induced by the pressure di erence across the tab. To examine the generation and evolution of hairpin structures, Meng and Yang [3] and Yang et al. [4] analyzed a large quantity of instantaneous PIV velocity ®eld realizations along the center plane and two horizontal planes in the wake of a trapezoidal tab. They observed that the hairpin vortices increased in strength along the streamwise direction until a distance of about 3h from the tab tip, and the hairpin vortices persisted until as far as 20h. Secondary hairpin vortices and reverse vortices (with sign of vorticity opposite that of hairpin vortices) were also observed in the tab wake in their experiments. They found that the passage of the hairpin vortices coincided with the in¯ection points of the mean velocity pro®les and with locations of high Reynolds stress, indicating that the hairpin vortices were responsible for crossstream transport in the region where they dominated. To distinguish the roles of CVP and hairpin structures in di erent downstream locations, Elavarasan and Meng [5] carried out detailed ¯ow visualization of the tab wake ¯ow using the planar laser-
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471
induced ¯orescence (PLIF) technique. Their results indicated that the CVP and hairpin vortices dominated the ¯ow in di erent regions. The CVP only existed in the near-tab region, where their counter-rotating motion pumped the low-speed ¯uid from the boundary layer to the high-speed free-stream ¯ow and vice versa. Beyond the near-tab region the CVP diminished and the ¯ow was taken over by hairpin structures, which were responsible for the mixing in the rest of the wake. Based on the ¯ow visualization and PIV results, Meng and Yang [3] proposed a conceptual model for the coherent structures in the tab wake, in which the CVP evolved topologically into the legs of the hairpin vortices. All the past experimental studies were based upon single-point or 2D measurements or visualizations. Questions remain unanswered as to the exact 3D topology of the vortex structures as well as the relationship between and the dynamics of the di erent elements. To complement experimental studies and provide details about the 3D topological and dynamical characteristics of the vortical structures in the tab wake and the mechanisms responsible for the mixing enhancement, numerical simulations of the tab ¯ow based on the second-order fractional step scheme [6] and ®nite-volume method have been carried out. The objective of this article is to evaluate the performance of the numerical scheme and to establish its reliability by comparing the simulation results with PIV measurement under the same conditions. Originally introduced independently by Chorin [7] and Temam [8] and later extended by Kim and Moin [6], Zang and Hussaini [9], and Yamenko [10], the fractional-step scheme splits the numerical operators and achieves pressure-velocity coupling through solving a Poisson-like elliptic equation for pressure. However, no consensus exists concerning the boundary conditions for intermediate variables and implementation details. Di erent boundary conditions for the velocity and pressure have been proposed to improve the accuracy [6, 11, 12]. Later Perot [13, 14] showed that no boundary conditions were necessary for the intermediate velocity or pressure variables if the fractional-ste p method was viewed as an approximate block LU factorization of the fully discretized equations on a staggered grid. The fractionalstep scheme has been applied in conjunction with di erent spatial discretization methods, including ®nite-di erence [6, 15±17], ®nite-element [18±21], ®nite-volume [22±25], and spectral methods [26]. In this article we apply the fractional-step scheme in conjunction with the ®nite-volume method to simulate the ¯ow passing over a surface-mounted trapezoidal tab. The boundary-®tted grid used in the simulation greatly simpli®es the implementation of the boundary conditions on tab surfaces. However, grid nonorthogonality in the region close to the tab leads to complicated linear equation systems. Iterative solvers BiCGSTAB [27] and GMRES(m) [28] are employed to solve the linear equation systems. Employing this scheme, we simulated the ¯ow passing over the trapezoidal mixing tab mounted on a ¯at plate at Re = 600 based on the free-stream velocity and the tab height, and compared the results with PIV measurements recently conducted in our laboratory. The numerical scheme is presented in the next section, where we discuss in detail the discretization, boundary conditions, grid nonorthogonality , and the parallel performance. The simulation results for the tab ¯ow and the comparison with PIV experiments are presented in the subsequent section.
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NUM ERICAL ALGORITHM Flow Configuration and Grid The geometric con®guration and coordinate system used in the computation are shown in Figure 1, in which a trapezoidal mixing tab is mounted on the ¯at plate with an inclination angle a . The ¯at plate coincides with the plane y = 0. Table 1 lists the geometric parameters including the width of the bottom and top edges (a; b), two tapering angles ( b 1 ; b 2 ), tab inclination angle a , tab height h, and the inlet boundary layer thickness d in . A zero-thickness tab is assumed in the simulation. The structured boundary-®tted grid used for the simulation (Figure 2a) is nonorthogona l around the tab, which greatly simpli®es the implementation of no-slip boundary conditions on tab surfaces. Both the experiments and preliminary simulations of the ¯ow passing over the tab indicate that there exists a strong shear layer behind the top edge of the tab with a small upward tilting angle. To resolve these characteristics the grid is re®ned in all three directions around the tab, and ®ner grids are placed near the ¯at plate and the tab.
Figure 1. Geometric con®guration. Dashed lines mark the domain of simulation.
Table 1. Flow parameters in PIV experiment and the simulation
PIV=Simulation
a (cm)
b (cm)
b 1 (¯ )
b 2 (¯ )
a (¯ )
h (cm)
Re
din =h
2.9
2.1
7.6
7.6
24.5
1.24
600
0.7
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Figure 2. Grid system. (a) Structured boundary-®tted mesh for the tab ¯ow; (b) staggered arrangement of variables.
Computation M ethodology The fractional step scheme of Kim and Moin [6] and the ®nite-volume method are employed to discretize the incompressible Navier±Stokes equations in the temporal and spatial directions, respectively. For incompressible viscous ¯ow we have the Navier±Stokes equations (repeated indices imply summation): 1 q q qui q qp ‡ ui uj = – ‡ ui qt q xj q xi Re q xj qxj
(1)
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q ui = 0 q xi
(2)
where ui (i = 1; 2; 3) are velocity components and p is the pressure. The velocity components are normalized by the inlet free-stream velocity U0 , and the spatial coordinates are normalized by the tab height h. So the Reynolds number for this problem is de®ned by Re = (U0 h=v), where v is the kinematic viscosity of the ¯uid and is assumed to be a constant. The two-step time-advancemen t scheme for (1) and (2) can be written as u^i – uni 1 1 2 = (3H ni – Hin– 1 ) ‡ H ( u^i ‡ uni ) 2 2Re Dt
(3)
uni ‡ 1 – u^i q n‡ 1 = – f Dt q xi
(4)
q uni ‡ 1 = 0 q xi
(5)
and
where u^i are the intermediate velocity components; f is the pesudopressure and p= f‡
Dt 2 H f 2Re
(6)
Hi = –
q ui uj q xj
(7)
Expressed in integral form for the ®nite-volume method, Eqs. (3), (4), and (5) become R R ³I ´ I u^i dv – uni dv 1 1 n n– 1 n = (3Ni – Ni ) ‡ ~ n ¢ H u^i ds ‡ ~ n ¢ H ui ds (8) 2 2Re Dt R n‡ 1 R I ui dv – u^i dv ‡ = – (9) f n 1 ni ds Dt I ui ni ds = 0 (10)
H where ~ n is the outward unit vector of the grid cell and Ni = – ui V ¢ ~ n ds. Under a fully staggered arrangement of ¯ow variables (Figure 2b), the volume and surface integrals in Eqs. (8) and (9) are evaluated on the grid cell corresponding to ui , while the surface integral in (10) is evaluated on the grid cell for pseudopressure f. The velocity components and the pseudopressure are arranged into vector forms, denoted as Ui and F , respectively. Then Eqs. (8)±(10) are reduced to the following linear systems: ^i = Ri Ai U
i = 1; 2; 3
(11)
FLOW PAST A SURFACE-MOUNTED MIXING TAB
^ i – D tC– 1 Bi F n‡ 1 Uni ‡ 1 = U i 3 X
i = 1; 2; 3
Di Uni ‡ 1 = 0
475
(12) (13)
i= 1
^i ; Ci are diagonal where Ai are the coe cient matrices for velocity component U matrices consisting of the grid cell volumes, and Ri =
Z
uni dv ‡
Dt
2
1 (3Nni – Nn– )‡ i
I
Bi F = 3 X
Di Ui =
i= 1
Dt
2Re
I
n ¢ Huni ~
(14)
f ni ds
(15)
I
(16)
ui ni ds
Using (12) and (13) we derive the Poisson equation for the pseudopressure F : Á
3 X
Di Ci– 1 Bi
i= 1
!
F n‡ 1 =
3 1 X ^i Di U D t i= 1
(17)
Equations (6), (11), (12), and (17) are solved for the velocity and pressure as follows. ^ i . The First, the linear system, Eq. (11), is solved with an iterative solver to ®nd U ^ right-han d side of Eq. (17) is then evaluated with the newly computed Ui , and Eq. (17) is solved for the pesudopressure F n‡ 1 . Finally, the velocity and pressure at the new time step are computed from Eqs. (12) and (6), respectively. Grid Nonorthogonality The no-slip boundary condition on the tab surface is implemented in a straightforward way because the tab surface coincides with one face of the surrounding grid cells. However, the three grid lines (i = const, j = const, k = const) are not orthogonal to one another in this region of the grid. As a result, the coe cient matrices exhibit more complex patterns compared with the implementation of the same schemes on regular grids. Grid nonorthogonality also causes one of the grid lines to be only piecewise continuous between di erent regions of the grid, which complicates the discretization process and requires special care. With the ®nite-volume method we need to approximate the ®rst derivatives of velocity components arising from the viscous term. It turns out to be rather involved to approximate the derivatives directly with velocity values on neighboring grid points because the grid lines are not perpendicular to one another. We derive the derivatives from the directional derivatives along the grid lines. Consider a unit vector ~ n and the ®eld j (x; y; z). The derivative of j along direction ~ n can be expressed as
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qj qj qj qj = ~ n ¢ Hj = n1 ‡ n2 ‡ n3 qn qx qy qz
(18)
With derivatives of j along three distinct directions, the partial derivatives of j can be determined from Eq. (18). The three grid lines are the natural choice of the distinct directions. For example, consider a portion of the grid for velocity component u. Projections of the grid onto x±y and x±y planes are shown in Figure 3a, where black solid lines mark the boundary of the grid cell. In a staggered variable arrangement, the velocity u(x; y; z) is de®ned at point P, and the neighboring points are denoted by E; W; T; B; S; N; TE; TW; BE; BW; ES; WS; EN; WN. Consider the derivatives qu=q x, qu=q y, and qu=q z at point e, the center of the right face of the grid cell. The grid lines passing through point e, PeE, (be)e(te), and (en)e(es), are chosen for determining the partial derivatives. Let ~ n1 , ~ n2 , and ~ n3 denote the direction vectors of these three grid lines, respectively. They can be expressed as ~ n1 = sin g k‡ 1=2~ i – cos g k‡ 1=2 k ~
(19)
1 ~) (sin g k‡ 1=2~ n2 = q i – cos g k‡ 1=2 k ~ 1 ‡ tan2 a i‡ 1=2 sin2 gk‡ 1=2 sin g k‡ 1=2 tan a i‡ 1=2
~ ‡ q j 1 ‡ tan2 a i‡ 1=2 sin2 g k‡ 1=2 n3 = k~ ~
(20)
(21)
where g is the angle between the z axis and the grid plane k = const, and a is the angle between the x axis and the grid plane i = const. Then the derivatives can be expressed as qu qu = qz q n3
(22)
1 1 qu qu qu = ‡ q x sin g k‡ 1=2 qn1 tan gk‡ 1=2 q n3
(23)
1 qu qu = – ‡ sin gk‡ 1=2 tan a i‡ 1=2 q n1 qy
q 1 ‡ tan2 ai‡ 1=2 sin2 g k‡ 1=2 qu sin g k‡ 1=2 tan a i‡ 1=2
q n2
(24)
The discretization results in coe cient matrices containing 15 nonzero diagonals for the velocity (u; v; w) and 27 nonzero diagonals for the pressure. In contrast, for second-order schemes on a regular grid in three dimensions the coe cient matrix contains only seven nonzero diagonals. As a result it is extremely di cult to devise e cient preconditioners for this system, especially when parallelization is taken into
477
Figure 3. Grid nonorthogonality: (a) nonorthogonal grid around the tab. The top ®gure is the projection onto x±z plane of the grid plane n±n in the bottom ®gure; the bottom ®gure is the projection onto x±y plane of the grid plane m±m in the top ®gure. (b) Boundary grid between di erent grid regions. The top ®gure is the projection onto x±z plane of the grid plane n±n in the bottom ®gure; the bottom ®gure is the projection onto x±y plane of the grid plane m±m in the top ®gure.
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account. In our implementation the Jacobi and incomplete LU factorization preconditioners are used to accelerate the convergence. The boundary grids between di erent grid regions require special care during discretization because of the piecewise continuous nature of one of the grid lines. Consider a portion of the boundary grid between the inlet region (regular) and the region close to the tab (nonregular) for velocity component v. Projections of the grid onto the x – y and x – z planes are shown in Figure 3b, in which the direction of the grid line corresponding to index i is not continuous. We need to evaluate the derivative qv=q x at point e. The directional derivative method described above cannot be used directly because of the discontinuity of the grid line at point e. We approximate qv=q x as follows: vE0 – vP qv = q x (hij ‡ hi‡ 1j )=2
(25)
where hij is the grid step size in the x direction at the point (i; j; k). vE0 is determined by interpolating the velocity on surrounding points. For boundary cells between other grid regions the combination of the approximation described here and the directional derivative method may be necessary for approximating the derivatives. The grid quality, mainly the grid skewness, a ects the accuracy and stability of the solution. The grid skewness depends on the tab inclination angle, a , and the two bottom angles of the tab, b 1 and b 2 . The most severe skewness occurs when a ! 0 and b 1 ; b 2 ! 90¯ . For these extreme cases (a = 0 or b 1 ; b 2 = 90¯ ), the velocity component stored at the cell face in the staggered arrangement of variables is parallel to the face and hence makes no contribution to the mass ¯ux through that face. This defeats the strong coupling between the pressure and the velocity found on regular staggered grids. In addition, numerical experiments indicate that due to the skewness the grid resolution in all three directions must satisfy certain constraints on the boundaries between di erent grid regions to ensure the stability of the scheme. The ¯ow dynamics critically depend on the grid re®nement at the tab tip due to the grid skewness. Numerical experiments indicate that the re®nement of the grid in the streamwise and spanwise directions does not a ect the wake vortex dynamics (rolling up of shear layer, formation of hairpin vortices) so much as that in the direction normal to the ¯at plate. The simulation fails to capture the hairpin vortices if the grid is not su ciently re®ned to capture the shear layer behind the tab tip. Boundary Conditions At the ¯at plate and tab surfaces no-slip boundary conditions are applied. A convective boundary condition [29] is imposed at the outlet of the ¯ow domain as follows: q ui qui ‡ U ave = 0 qt qx
(26)
where Uave is chosen such that the total mass conservation in the ¯ow domain is preserved. Periodic boundary conditions are applied at z = 0 and z = S. On the upper boundary a no-stress wall condition is imposed:
FLOW PAST A SURFACE-MOUNTED MIXING TAB
v= 0
qu qw = = 0 qy qy
479
(27)
A laminar boundary layer pro®le with thickness din is prescribed at the inlet. Performance The linear system arising from the momentum equation (1) is solved with the BiCGSTAB [27] solver and has extremely fast convergence. The Poisson equation (17) for the pressure is solved with the GMRES(m) [28] solver because the BiCGSTAB solver is found to lead to larger numerical errors due to large irregularities of the residual. More than two thirds of the CPU cycles required for the solution of the equations are consumed to solve the Poisson equation. Parallelism is implemented with the POSIX thread model [30]. Figure 4 shows the parallel speedup with the timing data collected on the SGI Cray Origin 2000 at the National Center for Supercomputing Applications (NCSA) and HP-Convex Exemplar SPP 2000 at Kansas State University. According to Amdahl’s law [31] Sp =
1 1 – x ‡ x =P
Figure 4. Relationship between speedup Sp and number of processors P.
(28)
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where Sp is the speedup, we have that P is the number of processors used, and x is the percentage of parallel part in the implementation. Parallel operations account for x º 98:38% of the simulation on average, which gives rise to a maximum speedup Sp;max = 1=1 – x º 61:73 for the problem size considered in this article. This implies that in order to maintain good parallel e ciency (parallel e ciency Eff = Sp =P) no more than 62 processors should be used in the simulation for the current grid size. (The e ciency decreases to about 50% for processors.) The maximum parallel speedup Sp;max increases with a larger problem size [31]. SIM ULATION AND RESULTS Simulation Configuration The ¯ow passing over the trapezoidal mixing tab with an inclination angle of 24:5¯ is simulated for comparison with PIV experiments. Flow parameters are chosen in accordance with PIV experiments, as listed in Table 1. We use a computation domain of size 10h £ 4h £ 6h with a grid of 161 £ 65 £ 81 (in x, y, and z directions) for a Reynolds number Re = (U 0 h=v) = 600 based on the inlet freestream velocity U0 and the tab height h, resulting in ap wake region with a length of about 5:8h. Based on the inlet friction velocity, u¤ = tw =r ) (where tw is the wall shear stress at the inlet), and the inner layer length scale, y¤ = v=u¤ , the grid spacings in the streamwise and spanwise directions are D x‡ = D x=y¤ = 2:68 and D z‡ = Dz=y¤ = 1:73. The ®rst mesh point above the ¯at plate is a y‡ = Dyw =y¤ = 0:229, and the resolution at the top edge of the tab is D y‡ = D y=y¤ = 0:564. Maximum grid spacing occurs at the upper boundary (free surface) of the domain with D y‡max = Dymax =y¤ = 9:87. Because of grid nonorthogonalit y and the re®ned mesh near the tab top edge, the maximum time step requirement is quite stringent and is ®xed at Dt = 0:0025h=U0 . The total simulation time is about ttotal = 35(h=U 0 ). Approximately two ``¯ow-through’’ times (= 20(h=U0 )) of the simulation are discarded to allow for the passage of initial transients. The statistical data are then accumulated over the remaining time period D T = 15(h=U0 ). Due to the highly 3D nature of the tab ¯ow and the complex geometry a formal grid test is extremely expensive and impractical. To evaluate if the current grid was su cient to capture the physics, we increased the total number of grid points by three times. The new simulation results demonstrated the same ¯ow characteristics. The mean ¯ow on these two grids shows little di erence in magnitude, although the Reynolds stress áu0 v0 ñ (u0 ; v0 denoting ¯uctuating streamwise and vertical velocities) shows a slightly larger di erence in magnitude than the mean velocity. This indicates that the current simulation has captured the characteristics of tab wake.
Flow Characteristics from Simulation Results To demonstrate the ¯ow characteristics we show in Figure 5 a set of instantaneous streamlines passing through a horizontal rake at the inlet at a height of 0:23h above the ¯at plate. Figures 5a and 5b give the perspective view and the front view of the streamlines, respectively. As the ¯ow passes over the tab, a strong shear
FLOW PAST A SURFACE-MOUNTED MIXING TAB
Figure 5. Instantaneous streamlines from the simulation: (a) perspective view; (b) front view.
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layer is generated behind the tab, and a complicated wake region develops that grows spatially along the streamwise direction. On both sides the ¯ow wraps around the edges of the tab and sweeps inward toward the center plane due to the pressure di erence between the front and back tab surfaces. As a result, a pair of streamwise counter-rotatin g vortices is formed behind the tab with length scales comparable to the tab height. These are evident from the streamlines in Figure 5b, and the velocity vector patterns in the cross-stream plane at x = 5:3 h in Figure 6a. The counterrotating vortex pair generates an energetic upward motion in the plane of symmetry. The combination of the upward motion in the center plane and the sweeping motion on both sides induces the ¯ow that goes in a spiral pattern in the wake (Figures 5a, 5b). The shear layer that wraps around the tab edges becomes unstable at about h ¹ 2h behind the tab, rolls up, and forms hairpin-like vortices that have been described as being periodically shed from the tab [2, 4]. Figures 6a and 6b show the velocity vector patterns in two cross-stream planes: x = 5:3h (1:1h behind the tab tip) and x = 7:5h (3:3h behind the tab tip). While Figure 6a clearly demonstrates the counter-rotatin g vortex pair in the near-tab wake, the cross-stream plane at x = 7:5h (Figure 6b) cuts through two consecutive hairpin vortices that wrap around each other. Figure 6b also shows a secondary streamwise vortex pair below the hairpin vortex legs that rotate in opposite directions and cause a downward motion in the center plane near the ¯at plate. These simulation results agree with Elavarasan and Meng’s [5] observation that at this Reynolds number the CVP dominates in the neartab wake region and hairpin vortices exert more in¯uence farther downstream. The vortical structures in the ¯ow can also be visualized with the pressure distribution because vortex cores usually possess low values of pressure. The hairpin vortex structures are periodically shed from the tab and generate a sequence of alternating low- and high-pressure core regions in the tab wake. Figure 7 shows the instantaneous isosurface of the low-pressure core regions in the wake, which demonstrate a sequence of hairpin-like structures with the legs entangling with one another. Comparison w ith PIV Experiments The simulation has been compared with PIV measurements of the same ¯ow that were conducted in our lab. Comparisons are made in terms of instantaneous ¯ow patterns, mean velocity, second-order moments, and the shear-stress distribution at the ¯at plate. The tab wake is characterized by its unsteady nature with quasiperiodic vortex shedding. A series of instantaneous ¯ow patterns in a cross-stream plane at x = 8h within roughly one vortex-shedding period (about 0:6s) is obtained from both simulation and PIV. The patterns are plotted together in Figure 8 for comparison. Since the train of hairpin vortices continuously passes through the cross-stream viewing plane, Figure 8 approximately indicates the ¯ow patterns at several streamwise locations in the sequence of vortices. At t = 0, the legs of two consecutive hairpin vortices entangling with each other pass through the viewing plane. The cross section of the hairpin vortex legs can be clearly identi®ed in Figure 8. The strong pumping e ect caused by the hairpin vortex legs generates the mushroom-shape d velocity vector pattern shown in the ®gure. The velocity ®elds at
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483
Figure 6. Instantaneous velocity ®eld in cross-stream planes from the simulation: (a) x = 5:3h; (b) x = 7:5h.
the next two instants (t = 0:2s, t = 0:4s) in Figure 8 show the ¯ow patterns when the hairpin head, which cannot be clearly seen from the cross-stream view, passes through the viewing plane. At t = 0:6s, the ¯ow patterns are similar to those at the ®rst instant, indicating that the legs of subsequent hairpin vortices are crossing the
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Figure 7. Isosurface of the instantaneous pressure ®eld from the simulation.
viewing plane. Both the simulation and PIV measurements capture these characteristics and produce similar velocity patterns and the same sequence of dynamic events in the ¯ow. Figure 9a shows a comparison of the pro®les of mean streamwise velocity U ¹ y in the center plane. The simulation and PIV measurements have produced the same shape for the mean pro®les with slight di erences in magnitude. Both show the existence of two in¯ection points in the mean pro®le, which are imprints of the hairpin heads and reverse vortices passing the location [3, 4]. In agreement with the observations by Yang et al. [4], the velocity de®cit in the pro®le rises along the streamwise direction, indicating that these vortices move farther away from the ¯at plate downstream. The pro®les of the vertical mean velocity V ¹ y in the center plane are compared in Figure 9b. Both the PIV experiment and the simulation indicate that there exists a strong upward motion in the center plane (positive V) that produces a large peak in the pro®les. This upward motion is caused by the pumping of the hairpin vortex legs, as is evident from the instantaneous ¯ow patterns in Figure 6. In addition, both the experiment and the simulation capture the second, weaker peak above the major one. Nevertheless, larger discrepancies between the simulation and the PIV experiment exist in the V ¹ y pro®le near the ¯at plate. The di erence is caused by the secondary streamwise vortex pair mentioned in the previous section. Both the simulation and the PIV experiment show the secondary streamwise vortex pair rotating in the opposite direction (Figure 6b) below the legs of the hairpin vortices, which induces the downward ¯uid motion or negative values in the V ¹ y pro®le near the ¯at plate. In the experiment this vortex pair is observed at locations farther downstream than in the simulation. This may be due to the limited spatial
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Figure 8. Comparison of instantaneous velocity patterns in the cross-stream plane at x = 8h over one vortex shedding period. Left column: simulation; Right column: PIV.
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Figure 9. Mean pro®les in center plane at x = 7h: (a) mean streamwise velocity U ¹ y; (b) mean vertical velocity V ¹ y.
resolution of PIV near the wall. The PIV measurements are inherently less accurate at solid walls due to light re¯ection at the walls and the averaging e ect over the correlation window. Figure 10 shows a comparison of the pro®les of the Reynolds shear stress áu0 v0 ñ in the center plane. Both the simulation and PIV experiment show that along the wall normal direction across the wake there exist three distinct layers of Reynolds stress values: a thin layer of negative values, a region of positive Reynolds stress, and a region of negative Reynolds stress. The thin layer of negative Reynolds stress next to the wall is caused by the aforementione d secondary streamwise vortex pair, which induces a downward ¯ow in the vicinity of the ¯at plate in the center plane. The layer of positive Reynolds stress in the middle corresponds to the region occupied by the legs of hairpin vortices. The region of strong negative Reynolds stress above the positive layer roughly corresponds to the passage of the heads of the train of hairpin vortices, demonstrating the strong ejection and sweeping e ects caused by the rotation of hairpin vortex heads [4]. Figure 11 shows a comparison of the shear stress distribution at the centerline along the streamwise direction for the ¯at plate from both the simulation and the experiment. The wall shear stress is de®ned as t = m (dU=dy)jy= 0 , where U is the mean streamwise velocity and m is the dynamic viscosity of the ¯uid. Because of the limited PIV resolution at the wall, the experimental results do not provide accurate velocity pro®les near y = 0, and hence it would not be meaningful to directly compare the simulation results with the experiment for the wall shear stress. Instead, the U(y) from the simulation is ®rst interpolated onto the PIV grid points and then is used to compute the shear stress at the ®rst PIV grid point from the wall. This shear stress is then compared with that calculated from PIV results. As shown in Figure 11, both simulation and experiments indicate that the shear stress ®rst decreases in the
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Figure 10. Reynolds stress pro®le áu0 0ñ ¹ y in the center plane: (a) x = 7:6h; (b) x = 9:6h.
streamwise direction until it reaches a minimum at about 1:2h behind the tab, beyond which it increases farther downstream. In summary, the comparisons for the instantaneou s ¯ow dynamics, mean velocity, Reynolds stress, and wall shear stress show good overall agreement between the simulation and PIV measurement. However, we ®nd that some discrepancies between the simulation and the experiment are unavoidabl e due to the following factors: a. Tab thickness. In the simulation the thickness of the tab was assumed to be zero, while in the PIV experiment a ®nite-thickness tab was used. The ®nite thickness and the sharp corners at the edges of the tab could cause additional ¯ow separation and noticeably a ect the statistics of the tab wake. b. In¯ow condition. A Blasius boundary-laye r pro®le without any ¯uctuations was prescribed for the inlet boundary condition in the simulation. The inlet velocity ¯uctuations existing in the PIV experiment may have a ected the statistical values of the tab wake and contributed to the discrepancies. c. Resolution of PIV. In PIV the velocity vector was obtained by computing the correlations between two interrogation windows. Hence, the resulting velocity vectors were inherently spatial averages. Near the wall, due to the image inhomogeneities caused by the re¯ections of the solid boundary the PIV measured velocity was less accurate.
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Figure 11. Nondimensionalized ¯at plate shear stress distribution (Xt ) = Re( where Xt = x – xtip and xtip is the streamwise coordinate of the tab tip.
2 w =U0 )
in the center plane,
d. Boundary condition on sides of the domain. In the simulation, periodic boundary conditions were imposed on both sides of the domain, which assumes the presence of an in®nite number of tabs in the spanwise direction. In contrast, the PIV experiment was conducted with a single tab mounted on the ¯at plate. e. Boundary condition at the outlet. The most commonly used boundary condition at the outlet of the ¯ow (adopted in the current simulation) is the convective boundary condition, which assumes that the viscous di usion is negligible at the outlet and all the ¯ow variables are convected out of the domain at an average velocity. This boundary condition may not be true in real ¯ow situations, which may have caused a deviation of the simulation results from the experimental results, especially near the outlet of the domain. CONCLUSION The fractional-step scheme and ®nite-volume method are applied for simulating the ¯ow passing over a surface-mounte d mixing tab. The boundary-®tted
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nonorthogona l grid greatly simpli®es the implementation of the boundary conditions on the tab surfaces. However, grid nonorthogonalit y leads to much more complicated coe cient matrices, imposing di culty for devising e cient preconditioners. Performance data indicate that parallel operations occupy 98.38% of the simulation, giving rise to a maximum speedup of about 62 for the problem size in the current simulation. The ¯ow with an incoming laminar boundary layer is simulated at a Reynolds number of Re = 600 and compared with the PIV experiment conducted with the same parameters. The comparison shows a good overall agreement in terms of instantaneous ¯ow patterns, mean velocity, Reynolds stress, and streamwise shear stress distribution at the ¯at plate. It indicates that the simulation successfully captures the physics of the ¯ow.
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