19th Australasian Fluid Mechanics ConferenceMelbourne, Australia8-11 December 2014

High Resolution Hybrid RANS/LES Simulations of Flow Over Surface-mounted Cubes Using aSixth Order Finite Difference Solver

A. Curley1, M. Uddin1 and B. Peters1

1North Carolina Motorsports and Automotive Research CenterThe University of North Carolina at Charlotte, NC 28223, USA

Abstract

The veracity of two popular turbulence models in predictinghighly separated turbulent flows was evaluated in a study of asingle surface-mounted cube and two surface-mounted cubes intandem for various spacings at a Reynolds number of 22,000.Transient hybrid RANS/LES delayed multi-scale simulationsusing the k−ω SST model and the Langtry-Menter SST tran-sition model were performed using a sixth-order finite differ-ence compressible code OVERFLOW on overset chimera gridsgenerated using Chimera Grid Tools. While both turbulencemodels predicted, within an acceptable engineering accuracy,the mean drag coefficient of an isolated cube and that of theupstream cube in case of tandem configurations, neither modelcaptured the dynamics of the wake accurately. The Langtry-Menter model outperformed the k−ω SST model in stagnationregions and provided a closer approximation of the wake evo-lution despite both models’ overall failure in separated regions.However, this improved prediction comes at cost of 20% in-crease in computation time.

Introduction

Turbulent flows around single and tandem bluff bodies exhibitcomplex flow structures of particular interest in engineering ap-plications. Massive separation, multiscale vortical structures,highly turbulent recirculation zones, and periodic vortex shed-ding in the shear layer result in oscillatory pressure distributionson the surrounding geometry and unsteady flow patterns down-stream of the obstacles. Our interest in this area stems froman observation during NASCAR’s 2011 Daytona 500 racingevent when a new style of racing emerged as two cars locked to-gether bumper-to-bumper exhibited a 20 MPH increase in speedcompared to cars spaced out in the standard drafting formation.For those not living in North America, NASCAR is America’sprime motorsports sanctioning body, and the Daytona 500 is itsmost prestigious event. Preliminary computational fluid dynam-ics (CFD) simulations based on two-equation RANS models atUNC Charlotte accurately predicted this speed increase by anaccurate prediction of the reduction in mean drag coefficients;however, these simulations failed to produce a reasonable valuefor the downforce. Thus, this paper investigates the abilities ofthe two-equation models to predict flows around a single cubicbluff body and two cubic bluff bodies in tandem arrangementwith inter-cube spacing, S, ranging from 0.25H to 8H.

Surprisingly, literature presents few credible in-depth experi-mental studies of flow around single bluff bodies. Castro andRobins [1] studied the nature of the wake for a single cubein uniform and turbulent streams with a thick boundary layerand found that the wake decayed within approximately six cubeheights downstream. Further investigations by Lyn et al. [3]in the base region concluded that the periodic component wasdriven by the streamwise size of the vortex formation regionwhile the turbulent component was driven by the wake width.This was based on the observation that the periodic stresses de-cayed much more rapidly than the turbulent stresses in this re-gion. Observations by Martinuzzi and Tropea [6] of a prismatic

obstacle in a fully developed channel flow indicated that for ob-jects with a small aspect ratio, such as a cube, the flow is domi-nated by the interaction of the horseshoe vortex with the cornervortex behind the cube and with the mixing layer in the wake.

More recent studies have looked at the interaction of two closelyspaced bluff bodies. Martinuzzi and Havel [4] found that forsmall separations (S/H ≤ 1.4), the shear layer reattaches on thetop side of the downstream obstacle. This reattachment resultsin a separation zone on the front face of the downstream obsta-cle and a strong stream of fluid directed downward along thefront face. As the spacing between the obstacles increases, theflow begins to reattach to the base wall and a second horseshoevortex in front of the downstream cube emerges. Martinuzziand Havel [5] later discovered that for spacings between 1.5Hand 2.5H, a geometrically locked regime exists in which theStrouhal number based on the cube spacing remains constant.In this locked regime, the periodic vortex shedding results fromthe interaction between the side shear layers and the strong ver-tical stream along the front face of the downstream obstacle.

Computational simulations of S > 2H have begun to assess thecapabilities of turbulence models to predict the characteristicstructures in the flow field. Paik et al. [9] reported the fail-ure of unsteady Reynolds-averaged Navier-Stokes (URANS) tocapture the horseshoe vortex along the base of the upstreamcube and the premature switching to unresolved DNS in refinedregions of both detached eddy simulation (DES) and delayed-detached eddy simulation (DDES). For inter-cube spacings S <2H, a lack of detailed assessment exists for turbulence mod-els and few experimental investigations have reported extensivediscussions of the flow field within the cavity region for smallspacings.

Numerical Model

The compressible unsteady Reynolds-averaged Navier-Stokes(RANS) equations were solved using a sixth-order finite dif-ference code OVERFLOW (version 2.2g) on overset chimeragrids generated using Chimera Grid Tools. The turbulence mod-els implented in the hybrid RANS/LES multi-scale simulationswere the Menter [7] k−ω SST model (SST, hereinafter for sim-plicity) and the Langtry-Menter [2] local correlation-based tran-sition model (LMT). The dimensions of the computational do-main replicated the experimental setup of [5] and a detail of themesh in the immediate region surrounding the cubes is shownin figure 1. The same domain was used for both the single andtandem arrangements.

For the single cube simulations, the mesh consisted of 55 mil-lion vertices with an average y+ value of 0.3178; for the tan-dem cube simulations, the mesh consisted of 71 million verticeswith an average y+ value of 0.3162. For both simulation setups,the immediate area surrounding the cubes (−H ≤ x ≤ 5H) wasresolved to 0.0125H between neighboring vertices and a firstnode height of 0.000277H for the entire computational domain.The upstream boundary was defined as a uniform velocity in-let and was moved further upstream to x = −8H to avoid nu-

X

Z

-2 0 2 4 6 80

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 1.

Figure 1: Computational domain showing the mesh refinementregion. Inlet is at x = −8H; outlet is at x = 25H; and inviscidboundaries are at y =±6H and z = 6H

merical interference. The wall boundaries of this inlet region(−8H ≤ x≤−2H) were modeled as inviscid to avoid boundarylayer growth before reaching the test section of [5]. The inletconditions replicated the experimental setup of [5] which noteda uniform inlet velocity of U0 = 8.8 m/s and a maximum turbu-lent intensity of 1.5% in their tunnel. The downstream boundarywas defined as a simple outflow and the far-field boundaries ofthe domain were defined as inviscid walls. All simulations werecomputed over 3000 timesteps with a nondimensional timestepof 0.176 and a physical timestep of ∆t = 0.0008s. To improveconvergence, low-mach preconditioning was implemented andthe first 500 timesteps were computed using multigrid acceler-ation. The mean flow conditions were averaged over the final1000 timesteps.

The LMT model interacts with the two-equation k−ω SSTmodel by modifying the production term based on the rela-tion between the transition momentum-thickness and strain-rateReynolds number. The production term formulations of the twomodels are briefly described here.

In Menter’s k−ω SST model, to prevent the excessive build-upof turbulence in stagnation regions, the production term in theturbulent kinetic energy equation is described as

P̃k = min(τi j

∂Ui

∂x j,10β∗kω

)(1)

where β∗ ≡ 9100 . Within the wake region, the β∗kω term dom-

inates the evolution of turbulent kinetic energy downstream ofthe cube. The LMT model bases the production term in theturbulent kinetic energy equation on the intermittency for pre-dicting separation induced transition and has the form

P̃k = γe f f Pk (2)

γsep = min(2max

[0,

( Rev

3.235Reθc

)−1]Freattach,2

)Fθt (3)

Freattach = e−( RT20 )4

(4)

The introduction of the intermittency factor γe f f = max(γ,γsep)initiates the production of turbulent kinetic energy (P̃k) down-stream of the transition onset location.

Results and Discussion

To evaluate the performance of SST and LMT models in highlyseparated aerodynamic flows, the flow around a single cubic ob-stacle was investigated in addition to two cubic obstacles in tan-dem arrangement. Various flow properties are compared withthe experimental studies of [1] and [6]; however, the detailedanalysis is limited to the single cube flow in this paper.

CFD predicted drag coefficients, for 0.25 ≤ x/H ≤ 8, obtainedusing SST and LMT models are compared against the experi-mental results of [5] in Figures 2(a) and (b) respectively. Thedifferences between the predictions from the SST and the LMTmodels are not very distinguishable. Clearly, the CFD resultsindicate an inability of both of these models to adequately cap-ture the effects of the upstream cube wake on the downstreamcube for inter-cube spacings less than 2H and greater than 3H.For the smaller inter-cube spacings, both models greatly over-predict CD for the downstream cube; however, as the inter-cubespacing increases, both models begin to underpredict CD for thedownstream cube. For nearly all spacings, both models slightlyoverpredict the CD of the upstream cube, but the mean valuesare within acceptable engineering tolerance; the error is within5% which is again within the experimental uncertainty.

Figure 2: Mean drag coefficient, CD, versus S/H for (a) up-stream and (b) downstream cubes in tandem arrangement.

To further understand the failure of these models to capture thewake of the upstream cube, single cube simulations were runand pressure and velocity distribution results were comparedagainst the experimental studies of [1] and [6]. Predicted dragcoefficient values for a single cube are compared against theexperimental values [8] in table 1. Arguably, both models pre-dicted results with an acceptable level of accuracy.

Figures 3(a) and (b) show the mean pressure coefficient,

Cp = (p− pre f )/(0.5ρU2re f ) (5)

distributions along y = 0 up the front and rear faces of the cube,respectively. Note that these Cp values are based on the refer-ence velocity, Ure f , and pressure, pre f , as used in [1]. It canbe seen from figure 3(a) that along the front face of the cube,the LMT model very accurately captures the mean pressure dis-tribution while the SST model overpredicts this. Now turningto the rear face, in figure 3(b), the LMT model closely predictsCp at the base of the rear face. Further up the rear face, asthe z-position increases and approaches the large separation re-gion, the LMT model begins to diverge from the experimentalresults. This figure also suggests that the prediction of the SSTmodel has a much larger deviation from the experiment. TheSST model overpredicts Cp everywhere on the rear face, and it

appears that this profile shifted vertically from the one obtainedfrom the LMT model. Despite the success of both models toemulate CD, the SST model predicts significantly greater pres-sure magnitudes on both faces. At this stage it is not clear tous despite this large discrepancies in rear face pressure distribu-tion, why the mean drag predictions from both of these modelsare very close to the experiment and, as well as, to each other.However, referencing back to our previous studies on NASCARbump drafting, this inability of the turbulence models to cap-ture the wake correctly contributes to the inaccurate predictionof downforce, but, like this case, predicted the drag values ac-curately.

Experiment [8] Transition Model k−ω SSTCD 1.05 1.026 1.009

Error — 2.3% 2.9%

Table 1: Average Single Cube Drag Coefficients

Figure 3: Cp distribution for a single cube along y/H = 0 (a) upthe front face and (b) up the rear face compared to Castro andRobins [1].

The Cp distribution along the floor is shown in figure 4 whichalso contains the experimental pressure distributions from [6]For this plot, the Cp values are calculated following the con-ventions of [6] where the bulk velocity and the freestream staticpressure are used as the reference velocity and pressure respec-tively. Compared to this experiment, the LMT model predictsa reattachment location further downstream. As discussed ear-lier, the LMT model captures the base pressure at the rear faceof the cube, but in the near wake region it begins to underpre-dict Cp before asymptoting to the experimental results furtherdownstream. Conversely, the SST model overpredicts the basepressure but then captures the near wake region before diverg-ing from experimental results further downstream.

The mean x-velocity profiles at three locations along the y = 0centerline are shown in figures 5(a)–(c). Both SST and LMTpredictions are in close agreement with experimental results;however, very close to the wall as z approaches zero, the com-putational results and the experimental results begin to deviate.The good correlation between experimental and computationalresults of the velocity profiles affirm the effectiveness of a sixth-order solver.

Figure 4: Cp distribution for a single cube along the floor at y/H= 0 compared to Martinuzzi and Tropea [6].

Mean turbulent intensity profiles at the same locations as themean velocity profiles are presented in figures 6(a)–(c). Atz/H = 0.5, both the SST and the LMT models show a muchgreater turbulent intensity compared to experimental results.The location, however, of the peak turbulent intensity at thisposition corresponds to that of the experimental results. Addi-tionally, the peak intensity of the LMT model is significantlygreater than that of the SST model which illustrates the greaterstrength of the wake predicted by the LMT model as discussedearlier.

As the streamwise position increases, both turbulence modelsbegin to underpredict the turbulent intensity with decreasingdiscrepancy between the two models which leads to two obser-vations. First, neither model adequately captures the dissipationof turbulent kinetic energy in the wake region and secondly, thedissipation rate of turbulent kinetic energy for the LMT modeldecreases more rapidly than that of the SST model. The ex-perimental results of [1] show that the peak turbulent intensitydecreases only slightly between x/H = 0.5H and x/H = 2.5H.Conversely, the peak turbulent intensity decreases by a factor ofapproximately 1.8 for the SST model and approximately 2.2 forthe LMT model over this interval.

The excessive decay of turbulent kinetic energy in the wake isa direct result of the production terms shown in equations (1)and (2) for the SST and LMT models respectively. For the SSTmodel, a careful modification to the β∗ model coefficient mayreduce the turbulent kinetic energy dissipation in the wake re-gion. Similarly, the intermittency factor γe f f controls the pro-duction of the LMT model and the model coefficients in equa-tions (3) and (4) may be adjusted to develop a more accuratecorrelation for highly separated flows. The value of 3.235 inequation (3) was adopted by Langtry and Menter [2] to changethe relationship between the vorticity or strain-rate Reynoldsnumber, Rev, and the critical Reynolds number at which inter-mittency begins to increase, Reθc, based on the shape factorat a separation point rather than at that of a Blasius boundarylayer. This value was increased from 2.193 for a Blasius bound-ary layer to the current value of 3.235, effectively reducing thetransition onset Reynolds number. Along this thought and thedata presented herein, this model coefficient may need to beincreased to accurately predict highly separated aerodynamicflows.

Conclusion

The performance of two turbulence models was evaluated inpredicting highly separated aerodynamic flows. It was foundthat the Langtry-Menter transition model reproduced Cp distri-bution along the front face of single cube and overpredicted Cp

Figure 5: Mean x-velocity profiles along y/H = 0 for (a) x/H =0.5H, (b) x/H = 1.5H and (c) x/H = 2.5H compared to theresults of Castro and Robins (1977).

in the large separation region along the rear face while the k−ωSST model significantly overpredicted Cp for both faces. Thiseffect was attributed to an underprediction of turbulent intensitycaused by an overprediction of turbulent kinetic energy dissi-pation in the wake region behind the upstream cube. The datapresented herein offers an explanation of the dilemma in previ-ous CFD simulations that reproduced the mean drag coefficientof two vehicles in tandem drafting but failed to reproduce rea-sonable downforce predictions. The Langtry-Menter transitionmodel presents a baseline for a new physical transition onsetmodeling; however, adjustments to the model correlations arerequired to suit the flow structure development in highly sepa-rated flows.

Acknowledgment

Alex Curley acknowledges the support from UNC CharlotteLevine Scholars Program. Support from UNC Charlotte Uni-versity Research Computing and Mosaic Computing are alsogratefully acknowledged.

References

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Figure 6: Mean turbulent intensity (I) profiles along y/H = 0 for(a) x/H = 0.5H, (b) x/H = 1.5H and (c) x/H = 2.5H comparedto the results of Castro and Robins (1977).

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