Proceedings of the ASME 2012 31st International Conference on Ocean, Offshore and ArcticEngineeringOMAE 2012

June 10–15, 2012, Rio de Janeiro, Brazil

OMAE2012-83730

DRAFT: THREE-DIMENSIONAL EFFECTS OF THE FLOW NORMAL TO A FLATPLATE AT A HIGH REYNOLDS NUMBER

Xinliang Tian∗

State Key Laboratory ofOcean Engineering, Shanghai

Jiao Tong University,Shanghai, China, 200240tianxinliang@sjtu.edu.cn

Muk Chen OngDepartment of Marine Technology,

Norwegian University ofScience and Technology,

NO-7491 Trondheim, Norwaymuk.c.ong@ntnu.no

Jianmin YangState Key Laboratory of

Ocean Engineering, ShanghaiJiao Tong University,

Shanghai, China, 200240jmyang@sjtu.edu.cn

Dag MyrhaugDepartment of Marine Technology,

Norwegian University ofScience and Technology,

NO-7491 Trondheim, Norwaydag.myrhaug@ntnu.no

Gang ChenState Key Laboratory of

Ocean Engineering, ShanghaiJiao Tong University,

Shanghai, China, 200240gangchen@sjtu.edu.cn

ABSTRACTPlate components are often found in offshore and marine

structures, such as heave damping plates and ship stabilizers.Two-dimensional (2D) and three-dimensional (3D) numerical si-mulations are performed to investigate the 3D effects of the flownormal to a flat plate at a high Reynolds number (Re = 1.5×105,based on the height of the plate and the free stream velocity). Theratio of the plate thickness to the plate height is 0.02. The 2Dsimulations are carried out by solving the Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations with the k-ω SST(Shear Stress Transport) turbulence model, while the 3D simu-lations are carried out with the large-eddy simulation (LES)method. The hydrodynamic results (such as time-averaged dragcoefficient, Strouhal number and mean recirculation length) arecompared with the published experimental data. The near-wakeflow structures are also discussed. The 3D simulation results arein good agreement with the published experimental data; howev-er, the 2D simulations show a poor comparison with the experi-

∗Author of correspondence.

mental data. This shows that the 3D effects are important for thehigh Reynolds number flow normal to a flat plate.

Keywords: Three-dimensional effect; Flat plate; Turbulence;URANS; LES

1 INTRODUCTIONTwo-dimensional (2D) numerical simulations are commonly

adopted to calculate the hydrodynamic quantities for the nomi-nally 2D problems, such as flow past or oscillating around an in-finitely long cylinder with circular, spherical, rectangular or thinplate cross-sections, due to the extensive requirement of the com-putational resources in three-dimensional (3D) simulations. Thedrag coefficient and the vortex shedding frequency calculated by2D simulations were reported to agree well with the experimentalresults for the flow around a circular cylinder (see e.g. Rahmanet al. (2007), Re=100, 1000, 3900; Ong et al. (2009), Re=1, 2,3.6×106) or a square cylinder (see e.g. Bosch and Rodi (1998),Re = 2.2× 104; Shimada and Ishihara (2002), Re = 2.2× 104;

1 Copyright c⃝ 2012 by ASME

Tian et al. (2011), Re = 2.14× 104). Here Re = U∞H/ν is theReynolds number, where U∞ is the free stream velocity; H isthe characteristic length of the obstacle, i.e. diameter of circu-lar cylinder, height of the cross-section of an infinitely long plateor rectangular cylinder; ν is the kinematic viscosity of the fluid.However, it has been known that the wake flow shows a 3D fea-ture even at a low Re. The critical Reynolds number (Recrit ) forthe occurrence of the 3D flow depends on the geometry of thebody, but it is close to 200 (Najjar and Vanka, 1995a). For theflow around a circular cylinder, Recrit ≈ 180 (Williamson, 1996).Because the velocity component in the third direction, (e.g. span-wise direction for an infinitely long cylinder) is not consideredin the 2D simulations, the calculated results by 2D simulationsmay not always agree with the results of 3D numerical simu-lations or experiments, even for the drag coefficient or vortexshedding frequency. Najjar and Vanka (1995a) carried out 2Dand 3D direct numerical simulations (DNS) for the flow past anormal flat plate at Re = 1000. They reported large discrepan-cies between the 2D and 3D results for the drag coefficients andthe vortex formation. The drag coefficient calculated by their 2Dsimulation was overpredicted by a factor of 1.6 compared withthe experimental results of Fage and Johansen (1927), while thevalues in the 3D simulations agreed well with the experimentalresults. Lisoski (1993) reported a similar overprediction of thedrag coefficient in his 2D numerical simulation compared withhis experiments. Tian et al. (2011) calculated the flow normalto a series of rectangular cross-sectional cylinders with differentaspect ratios (R = B/H = 1−0.05, where B is the breadth of therectangular cross-section in the streamwise direction) through2D simulations with the k-ω Shear Stress Transport (SST) tur-bulence model. The calculated drag coefficients for the low as-pect ratios (R= 0.4,0.2,0.1,0.05) were significantly overpredict-ed compared with the experimental results (Norberg, 1993; Fageand Johansen, 1927), while the results for the high aspect ratios(R = 1,0.8,0.6) agreed very well with the experimental results(Norberg, 1993; Lyn et al., 1995). Tian et al. (2011) pointed outthat it might be due to that the 3D effects became more signif-icant for the low aspect ratios and for which the 3D numericalsimulations should be carried out.

The discussions on the reliability of 2D simulations havelast for more than two decades (see e.g. Tamura et al. (1990);Sakamoto et al. (1993); Najjar and Vanka (1995a); Mittal andBalachandar (1995); Saha et al. (2003); So et al. (2005)). Itmight be too crude to conclude that the inaccurate predictionof drag in 2D simulations is due to 3D effects. It is notewor-thy that Mittal and Balachandar (1995) investigated the 3D ef-fects in numerical simulations by comparing the results of 2Dand 3D simulations for the flow past a circular cylinder and anelliptic cylinder, respectively, using DNS. They pointed out thatthe differences in the spanwise-plane Reynolds stresses are theprimary cause for the drop in the mean drag forces for 3D simu-lations. In the present study, the discrepancies between 2D and

3D simulations for the flow normal to a flat plate will be inves-tigated. The simulations will be carried out at a high Reynoldsnumber (Re = 1.5×105) which is more relevant for engineeringapplications than the previous low Reynolds number studies (seee.g. Najjar and Balachandar (1998), Re= 250; Narasimhamurthyand Andersson (2009), Re = 750; Najjar and Vanka (1995a),Re = 1000). The results of 2D and 3D simulations will be com-pared with each other, and also with the published experimentaland numerical results. Thus, this study provides a documentationon the reliability of 2D simulation with the k-ω SST model.

2 MATHEMATICAL FORMULATION AND NUMERICALMETHODS

2.1 Mathematical formulationThe flow is assumed to be unsteady, incompressible, viscous

and turbulent. The turbulence flow in 2D and 3D simulationswill be modeled using the k-ω SST model and the large-eddysimulation (LES) method, respectively. The variables ξ and ξare used to represent the time-averaged value of ξ in k-ω SSTmodel and the spatial-filtered value of ξ in LES, respectively.For the convenience of postprocessing, ⟨ξ ⟩t represents the time-average of ξ and ⟨ξ ⟩ represents the time- and spanwise-average(if exists) of ξ .

2.1.1 k-ω SST model (2D simulation) The govern-ing equations for the 2D simulations are the Reynolds-averagedNavier-Stokes (NS) equations, given by

∂ui

∂xi= 0 (1)

∂ui

∂ t+u j

∂ui

∂x j=− 1

ρ∂ p∂xi

+ν∂ 2ui

∂x2j−

∂u′iu′j

∂x j(2)

where i, j=1, 2. Here x1 and x2 denote the stream and cross-stream directions, respectively; u1 and u2 are the correspondingmean velocity components; u′iu

′j is the Reynolds stress compo-

nent, where u′i denotes the fluctuating part of the velocity; p isthe pressure; t is the time; and ρ is the density of the fluid.

The k-ω SST turbulence model (Menter, 1994) combines thek-ω and the k-ε models, with the original k-ω model of Wilcox(1998) in the near-wall region and the standard k-ε model (Jonesand Launder, 1973) in the outer wake region and in free shearlayers. Following Menter et al. (2003), the equations for the SST

2 Copyright c⃝ 2012 by ASME

model is taken as

D(ρk)Dt

= Pk −β ∗ρωk+∂

∂x j

[(µ +σkµt)

∂k∂x j

](3)

D(ρω)

Dt= αρS2 −βρω2 +

∂∂x j

[(µ +σω µt)

∂ω∂x j

]+

+2(1−F1)ρσω21ω

∂k∂x j

∂ω∂x j

(4)

where Pk is given by

Pk = min[

µt∂ui

∂x j

(∂ui

∂x j+

∂u j

∂xi

),10β ∗ρkω

](5)

If ϕ1 represents any constant in the original k-ω model(σk1, . . .) , and ϕ2 represents any constant in the original k-ε mod-el (σk2, . . .) , then ϕ , the corresponding constant of the new modelgiven by Eqs. (3) and (4), is

ϕ = F1ϕ1 +(1−F1)ϕ2 (6)

F1 = tanh(arg41) (7)

arg1 = min

[max

( √k

β ∗ωy,

500νy2ω

),

4ρσω2kCDkω y2

](8)

CDkω = max(

2ρσω21ω

∂k∂x j

∂ω∂x j

,10−10)

(9)

here y is the distance to the nearest wall, and CDkω is the positiveportion of the cross-diffusion term of Eq. (4).

The turbulent eddy viscosity is defined as

νt =a1k

max(a1ω,SF2)(10)

where S is the invariant measure of the strain rate and F2 is givenby

F2 = tanh(arg22),arg2 = max

(2

√k

0.09ωy,

500νy2ω

)(11)

The constants of SST are: β ∗ = 0.09, a1 = 0.31, α1 =0.5532, α2 = 0.4403,β1 = 0.075,β2 = 0.0828, σk1 = 0.85034,σk2 = 1.0, σω1 = 0.5, σω2 = 0.85616.

2.1.2 LES with Smagorinsky model (3D simula-tion) The idea of LES is that the large-scale motions are com-puted explicitly, and the small-scale eddies are filtered by a filteroperation and modeled with a subgrid scale (SGS) model. Thefiltered NS equations can be expressed as follows

∂ ui

∂xi= 0 (12)

∂ ui

∂ t+

∂∂x j

(uiu j) =− 1ρ

∂ p∂xi

+ν∂ 2ui

∂x jx j−

∂τi j

∂x j(13)

where i, j = 1–3; x1, x2 and x3 denote the streamwise, cross-stream and spanwise directions, respectively; u1, u2 and u3 arethe corresponding filtered velocity components; p is the filteredpressure; and the subgrid stress τi j is given by

τi j = uiu j − uiu j (14)

The SGS model proposed by Smagorinsky (1963) is used inthe present study, and τi j is written as

τi j −13

δi jτkk =−2νt Si j (15)

where δi j is the Kronecker delta; νt is the subgrid eddy vis-cosity; Si j = 0.5(∂ ui/∂x j + ∂ u j/∂xi) is the strain rate in the re-solved velocity field.

Using dimensional analysis, the eddy viscosity νt can be cal-culated by the following formula

νt = (Cs∆)2|S| (16)

where |S| = (2Si jSi j); the constant Cs = 0.2; ∆ is thefilter-width. In this study, the Van Driest damping function(Van Driest, 1956) is applied; ∆ = min∆mesh,(κ/C∆)y[1 −exp(−y+/A+)], where ∆mesh is the cubic root of the mesh cellvolume; κ = 0.41 is the von Karman constant; C∆ = 0.158;A+ = 26; y+ is the dimensionless distance to the wall, taken asy+ = u∗y/ν , where u∗ is the wall friction velocity.

3 Copyright c⃝ 2012 by ASME

X

Y

Z

Symmetry

Symmetry

)D3(

0~0~

~

D20

3

2

1

2

1

u

u

Uu

u

Uu

)D3(0~

)D2(0

n

n

i

i

u

u

H

H8

H8

H5.7 H20

)D3(0~)D2(0

i

i

u

u

FIGURE 1. Spanwise view of the computational domain and bound-ary conditions, where n is the normal unit vector.

2.2 Numerical methodsThe open source Computational Fluid Dynamics (CFD)

code OpenFOAM is used here. OpenFOAM is mainly appliedfor solving problems in continuum mechanics. It is built basedon the tensorial approach and object oriented techniques (Welleret al., 1998). The PISO (Pressure Implicit with Splitting of Op-erators) scheme (pisoFoam) is used in the present study. Thespatial schemes for interpolation, gradient, Laplacian and diver-gence are linear, Gauss linear, Gauss linear corrected and Gausslinear schemes, respectively. All these schemes are in second or-der. The second order Crank-Nicholson scheme is used for thetime integration. Further details of these schemes are given inOpenFOAM (2009).

3 COMPUTATIONAL OVERVIEW AND CONVER-GENCE STUDIES

3.1 Computational overviewAs shown in Figure 1, the computational domain in the span-

wise plane is 27.5H by 16H. The origin of the coordinates islocated at the center of the plate. The distances from the platecenter to the inlet and outlet boundaries are 7.5H and 20H, re-spectively. The top and bottom boundaries are located at a dis-tance of 8H to the center of the plate. These distances are largeenough to eliminate the far field effects from the boundaries. In3D simulations, the spanwise length of the computational do-main is 4H. The thickness of the plate is equal to 0.02H. Dueto the difficulties in numerical modeling of the sharp edges, theplate corners are rounded with the circular arcs with a radius of0.01H.

At the inlet boundary, the flow is set as a uniform flow:u1 = U∞, u2 = 0 (2D) and u1 = U∞, u2 = u3 = 0 (3D). Alongthe outlet boundary, the velocity is set as zero normal gradientand the pressure is set as zero. The top and bottom boundaries

TABLE 1. Results of grid and time step convergence studies.

Case Elements ∆tU∞/H ⟨CD⟩t St

2D(I) 53.4K 0.0002 4.58 0.120

2D(II)∗ 75.3K 0.0002 4.61(0.7%) 0.120(0.0%)

2D(III) 75.3K 0.0003 4.63(0.4%) 0.119(0.8%)

3D(I) 1.71M 0.0006 2.20 0.155

3D(II)∗ 9.86M 0.0006 2.20(0.0%) 0.155(0.0%)

3D(III) 9.86M 0.0010 2.23(1.3%) 0.155(0.0%)∗The final case for analyses

are set as symmetry planes. The spanwise planes in the 3D si-mulations are set as periodic boundary conditions. On the platesurface, no-slip boundary condition is prescribed, i.e. ui = 0 (2D)and ui = 0 (3D).

In the spanwise plane, the computational domain is dividedinto 10 parts so as to control the grid distribution. The grids nearthe plate surface and in the wake region are refined. The heightof the first node from the plate surface is fixed at 4×10−5H, en-suring that the average y+ is less than 1. For the 3D simulations,the grids spread uniformly in the spanwise direction.

Each 2D simulation runs at least 10 vortex shedding cyclesafter the flow is fully developed, while each 3D simulation runsfor 200H/U∞ from which the statistic results are obtained.

3.2 Convergence studiesThe convergence studies have been carried out in order to

evaluate the spatial and time resolutions. The time-averaged dragcoefficient ⟨CD⟩t and the Strouhal number St are considered inthe convergence studies. The definition of CD and St are taken as

CD =Fx1

12 ρU2

∞H∆Z(17)

St = f H/U∞ (18)

where Fx1 is the streamwise force acting on the plate (per unitspanwise length in the 2D simulations). It is directly computedfrom the pressure and shear stress distributions over the plate sur-face. ∆Z is the extension of the computational domain in span-wise direction, ∆Z = 1 in 2D simulations and ∆Z = 4H in 3Dsimulations. f is the vortex shedding frequency.

Table 1 shows the results of the grid and time step conver-gence studies for both 2D and 3D simulations. For the 2D simu-lations, the results of the case 2D(II) with 75.3K elements and

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FIGURE 2. Instantaneous vortical structures in the wake: iso-surfaceof λ2 = 1.0.

a non-dimensional time step of ∆tU∞/H = 0.0002, where ∆t isthe dimensional time step, are compared with the results of thecase 2D(I) with a different spatial resolution and the case 2D(III)with a different time step. Only very small differences are ob-served in the values of ⟨CD⟩t and St, i.e. less than 1%. Similarconvergence studies are carried out for 3D simulations, see Ta-ble 1. The 2D(II) and 3D(II) cases give sufficient spatial andtime resolutions. The 2D and 3D results presented in the follow-ing sections are all calculated from the 2D(II) and 3D(II) caseswhich have been marked by ∗ in Table 1.

4 RESULTS AND DISCUSSIONFirst, the instantaneous vortical structures in the wake i-

dentified by the iso-surface of λ2 = 1.0 are shown in Figure 2.Here the λ2 proposed by Jeong and Hussain (1995) is definedas the second largest eigenvalue of the symmetric tensor Si jSi j +Ωi jΩi j, where Si j and Ωi j are the symmetric and antisymmet-ric components of the velocity gradient tensor, respectively. Ahighly 3D and chaotic wake flow is observed even though thegeometry of the plate is nominally 2D.

The hydrodynamic quantities, ⟨CD⟩t , St and ⟨Lw⟩/H, calcu-lated in this study are compared with the published experimentalresults (Fage and Johansen (1927); Leder (1991); Kiya and Mat-sumura (1988)), see Table 2. Here the mean recirculation length⟨Lw⟩ is defined as the streamwise distance from the center of theplate to the position where the mean streamwise velocity changesits sign from negative to positive. Large discrepancies betweenthe present 2D and 3D simulations are observed in the valuesof ⟨CD⟩t , St and ⟨Lw⟩/H. As shown in Table 2, the values of⟨CD⟩t , St and ⟨Lw⟩/H calculated by the present 3D simulationare in good agreement with the experimental results. However,the present 2D simulation gives about two times higher ⟨CD⟩t and⟨Lw⟩/H values than experimental results by Fage and Johansen(1927) and Leder (1991). The St value obtained in the present2D simulation is slightly lower than the experimental results.

TABLE 2. Results of grid and time step convergence studies: (a)Present 2D(II), (b) Present 3D(II), (c) Fage and Johansen (1927), (d)Leder (1991) and (e)Kiya and Matsumura (1988).

Case Method ⟨CD⟩t St ⟨Lw⟩/H Re

(a) 2D(URANS) 4.61 0.120 4.74 1.5×105

(b) 3D(LES) 2.20 0.155 2.28 1.5×105

(c) Exp. 2.13 0.146 - 1.5×105

(d) Exp. - 0.14 2.5 2.8×104

(e) Exp. - 0.146 - 2.3×104

0 50 100 150 2000

1

2

3

4

5 2D 3D

CD

Non-dimensional time, tU /H

FIGURE 3. Time-dependent variations of the drag coefficient.

Figure 3 shows the CD versus tU∞/H for the 2D and 3Dsimulations. It appears that the 2D simulation CD exhibits a pe-riodically repeating feature while the 3D simulation CD appearsto be disordered. Further, a low frequency variation at one tenthof the primary vortex shedding frequency is observed in the 3Dsimulation CD. This low frequency behavior has also been ob-served in the DNS study for the flow normal to a flat plate atRe = 250 by Najjar and Balachandar (1998).

The distributions of the time and spanwise (if exists) av-eraged pressure coefficient ⟨Cp⟩ on the front and back sidesof the plate are shown in Figure 4. The mean pressure co-efficient is defined as ⟨Cp⟩ = 2(⟨p⟩ − ⟨p∞⟩)/(ρU2

∞) (2D) and⟨Cp⟩= 2(⟨p⟩−⟨p∞⟩)/(ρU2

∞) (3D), where the reference pressurep∞ and p∞ are taken as the pressure at the center of the inletboundary. On the front side the flow is not disturbed, and thus thepressure distributions there are quite stable and the results of the

5 Copyright c⃝ 2012 by ASME

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5-5

-4

-3

-2

-1

0

1

Cp

Y/H

Present 2D, Re=1.5 105

Present 3D, Re=1.5 105

Fage & Johansen (1927), Exp. Re=1.5 105

FIGURE 4. Distribution of the mean pressure coefficient, ⟨Cp⟩, on thefront and back sides of the plate.

present 2D and 3D simulations agree well with the experimen-tal results of Fage and Johansen (1927). On the back side, thepresent 3D simulation predicts the constant pressure distributionsuccessfully and provides good agreement with the experimentalresults. However, the pressure distributions on the back side inthe 2D simulation are much lower than that in the 3D simulationand the experiments. The present 2D simulation fails to predictthe constant pressure distribution on the back side of the plate.

The streamlines of the mean velocity field obtained in thepresent 2D and 3D simulations are shown in Figures 5 (a) and(b), respectively. The extension of the bubble in the streamwisedirection corresponds to the mean recirculation length (⟨Lw⟩)given in Table 2. It appears that the recirculation length in the2D simulation is approximately two times larger than that inthe 3D simulation. Furthermore, the cores of the bubble in the3D simulation are located at X ≈ 1.1H which is approximate-ly one half of the mean recirculation length ⟨Lw⟩ = 2.28H, seeFigure 5 (b). The cores of the bubble in the 2D simulation arelocated at X ≈ 0.4H which is approximately one twelfth of themean recirculation length ⟨Lw⟩ = 4.74H, see Figure 5 (a). Theupstream cores in the 2D simulation result in a very high veloc-ity level near the back side of the plate, and thus a low pressuredistribution and a high drag force compared with the 3D case.

5 CONCLUSIONSThe flow normal to a flat plate with an infinite spanwise

length at Re = 1.5× 105 has been studied using 2D and 3D nu-merical simulations. The 2D and 3D simulations are carried outwith the k-ω SST turbulence model and the LES method, respec-

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

-1.0

-0.5

0.0

0.5

1.0

Y/H

X/H

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

-1.0

-0.5

0.0

0.5

1.0(b)

Y/H

X/H

FIGURE 5. Streamlines of the mean velocity field in (a) 2D and (b)3D simulations.

tively. The 3D simulation shows 3D features even though thegeometry of the plate is nominally 2D. The hydrodynamic re-sults of the 3D simulation agree very well with the experimentalresults. Large discrepancies between the 2D and 3D simulationsare observed. The mean drag force and mean recirculation lengthare overpredicted by a factor of approximately 2 in the 2D simu-lation. One might suspect that this poor performance of 2D sim-ulation is due to the limitation of the k-ω SST model and not the2D assumption. However, the k-ω SST model gives very goodprediction of the drag force for the square cylinder (Tian et al.,2011). Also, similar overpredictions of the drag force for a plateat Re = 1000 were reported in the 2D DNS studies by Najjar andVanka (1995b). In conclusion, the reliability of 2D simulationsfor the nominally 2D problem is questionable and more attentionshould be made to explore the 3D effects, especially for the bluntgeometries.

ACKNOWLEDGMENTThis work has been supported by National Natural Science

Foundation of China (Grant No. 50979057). This support isgratefully acknowledged.

6 Copyright c⃝ 2012 by ASME

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