Aerodynamic characteristics of circular cylinder and squared

cylinders with and without rounded corners

Hugo Rafael Lopes Abreuhugo.abreu@ist.utl.pt

Instituto Superior Tecnico, Lisboa, Portugal

December 2015

Abstract

Viscous flows around cylinders is a classical research topic in computational fluid dynamics (CFD)with a vast amount of practical applications in the field of aerodynamic and hydrodynamic. Manyoffshore applications use cylinders that range from circular cross-sections to square cylinders withrounded corners. Typical Reynolds numbers of practical applications are in the range of 105 to 106

where the so-called drag crisis occurs. Flow simulations in such conditions are extremely challengingbecause the flow exhibits laminar, transitional and turbulent regions. Furthermore, due to the existenceof vortex shedding, the flow is not statistically steady. Although the Reynolds-Averaged Navier-Stokes(RANS) equations supplemented by eddy-viscosity models have evident shortcomings in such complexflows, there are several attempts published in the open literature to simulate this type of flows withsuch mathematical model. Due to the periodic nature of vortex shedding, ensemble averaging mustbe used for the definition of the mean flow and for the averaging of the mass and momentum balance.Therefore, the RANS equations are not statistically steady, which is usually designated by URANS.We present a study of the calculation of the flow around a square cylinder with rounded corners. Wehave selected the widely used shear stress transport (SST) k − ω eddy-viscosity model to solve theRANS equations with two main assumptions: two-dimensional and incompressible flow. However,before we address Validation (comparison with experimental data), we investigate the influence of thephysical settings of the problem, i.e. size of the computational domain and specification of boundaryconditions. Furthermore, grid and time refinement studies are performed to assess the numericaluncertainty of meaningful flow quantities for practical applications. All calculations are performedwith the ReFRESCO solver.Keywords: 2D square cylinder, rounded corners, URANS, vortex shedding, Reynolds number effect

1. Introduction

Many practical constructions use cylinders thatrange from circular cross-sections to square cylin-ders with rounded corners. The flow around squarecylinders with corner modifications has attracted agreat deal of attention because of its practical sig-nificance in industrial installations. Practical exam-ples are tall buildings, monuments, towers and ca-bles in suspension bridges, and otherwise the poleshave common usage for supporting lights, hoard-ings and signs (civil engineering), landing gearsand fuselage cross section (aeronautical engineer-ing) are permanently exposed to wind. Similarly,heat exchanger pipe bundles (process engineeringand energy conversion) and piers, bridge pillars andlegs of offshore platform (maritime engineering) arecontinuously subjected to loads produced by waterstreams. At the typical Reynolds numbers of prac-tical applications, these bodies create a large regionof separated flow that leads to an unsteady wake

region where vortex shedding occurs. This typeof flow leads to unsteady (periodic) lift and dragforces acting on the cylinders and to fluctuationsin the flow field that are related to the sheddingprocess. Therefore, a flow around a square cylindermay originate damaging oscillations when the natu-ral frequency of the obstacle is close to the sheddingfrequency of the vortices. If the resulting excitationfrequency synchronizes with the natural frequencyof a cylinder, the phenomenon of resonance is theobvious outcome.

Typical Reynolds numbers of practical applica-tions are in the range of 105 to 106 where the so-called ”drag crisis” occurs, [19]. Flow simulations insuch conditions are extremely challenging becausethe flow exhibits laminar, transitional and turbu-lent regions. Furthermore, due to the existence ofvortex shedding, the flow is not statistically steady.Although many of the publications available in theopen literature focus on the circular cylinder, there

1

is a growing interest in square shapes with roundedcorners [5],[6], [4], [7], [17], [16] and [14].

Recently [13], a squared cylinder with roundedcorners of 9% of the diameter has been tested ina towing tank for Reynolds numbers Rn based onthe undisturbed incoming flow U∞ and the cylin-der width D ranging from 7.2 × 104 to 3.1 × 105.The results obtained for the lift and drag force sug-gest the existence of different flow characteristics inthis range of Reynolds numbers, which is in agree-ment with of the studies reported in [5], [6] and[4]. Therefore, it is important to address this prob-lem with mathematical models (instead of physicalmodels) to obtain some insight about the origin ofthese flow changes.

The main objective of this thesis is to study theflow around a squared cylinder with rounded cor-ners using mathematical modelling, i.e. flow simula-tions. The selected geometry was tested in a towingtank and average drag coefficients are reported in[13] suggesting the existing of different flow regimesfor the range of Reynolds numbers tested. Themain goal of this work is to investigate the abil-ity to capture the dependency of the flow regimeson the Reynolds number. However, in order to as-sess the quality of the mathematical modelling, itis also important to address the influence of the dif-ferent components of the model on the outcome ofthe simulations, i.e. the domain size, mass and mo-mentum balance modelled equations and boundaryconditions.

To this end we have chosen the simplest modelto simulate high Reynolds numbers wall-boundedturbulent flows: the Reynolds-Averaged (ensem-ble average) continuity and Navier-Stokes (RANS)equations supplemented by the eddy-viscosity two-equation SST k − ω model [15]. Furthermore, wehave assumed two-dimensional flow. On the otherhand, several sensitivity studies are conducted todetermine the influence of the remaining compo-nents of the mathematical model, namely:

• The size of the domain.

• The specification of the inlet turbulence quan-tities.

• The specification of the pressure boundary con-ditions.

For all these sensitivity test, grid/time refinementstudies are performed to estimate the numerical un-certainty of the solutions [10], which is a fundamen-tal component of any proper Validation procedure[2], i.e. of the estimation of the modelling error.

Although we are aware of the shortcomings of theselected model for such complex flows, the stud-ies described above are extremely time-consuming,

even for this model. Furthermore, the problems ad-dressed in this study are common to more sophisti-cated models that would make this type of investi-gation unaffordable.

To fulfill the main goal of this work, the flowaround the selected geometry is calculated for theReynolds numbers used in the experiments reportedin [13]. In order to assess the influence of the prob-lem settings, two different domains were selected:

• A domain that corresponds to the dimensionsof the towing tank.

• A sufficiently large domain to have a negligibleeffect of the selected pressure boundary condi-tions.

Finally, the ASME V&V 20 Validation procedure[2] is applied to the results obtained in the do-main corresponding to the towing tank to evaluatethe modelling error of the two-dimensional RANSequations supplemented by the SST k − ω model[15]. This exercise also illustrates the requirements(problem settings, experimental and numerical un-certainties,...) for a proper assessment of the mod-elling error.

The paper is organized in the following way: theflow solver is briefly described in section 2 and theproblem settings including domain size, boundaryconditions, grid sets, quantities of interest and cal-culation details are presented in section 3; resultsof the grid/time refinement studies are subsequentlypresented and discussed in section 4, while the mainconclusions are summarized in section 5.

2. Mathematical Model2.1. URANS modelThe present model is intended to deal with turbu-lent flows. Turbulent flows are highly unsteady andseveral time and length scales are usually presentedin realistic turbulent flows.

In a statistically steady, every variable φ can bewritten as the sum of a time-average, −, plus a fluc-tuation, ′, about that value, see for instance [12],

φ (xi, t) = φ (xi) + φ′ (xi, t) , (1)

where

φ (xi) = limT→∞

1

T

∫ t0+T

t0

φ (xi, t) dt. (2)

Here t is the time and T is the averaging interval.The averaged continuity and momentum equationscan, for incompressible flows, be written in tensornotation and Cartesian coordinates as:

∂Uixi

= 0,

∂Ui∂t

+∂

∂xj

(Ui Uj + u′iu

′j

)= −1

ρ

(∂p

∂xi− ∂τ ij∂xj

),

(3)

2

where the τ ij = τ ji are the mean viscous stresstensor components, being i = j = 1, 2, 3 runningindices:

τ ij = 2µSij = µ

(∂U i∂xj

+∂U j∂xi

). (4)

2.2. Turbulence model

In laminar flows, energy dissipation, transport ofmass, momentum and energy normal to the stream-lines are mediated by the viscosity, so it is natu-ral to assume that the effect of turbulence can berepresented as an increased viscosity. This leadsto the eddy-viscosity model (also called Boussinesqapproximation) for the Reynolds stress tensor,

−u′iu′j = νt

(∂U i∂xj

+∂U j∂xi

)− 2

3kδij , (5)

where the k term is the turbulence kinetic energygiven by k = 1

2u′iu′j , νt is the turbulence kinematic

viscosity, and νt = µt

ρ is the turbulent eddy vis-cosity. This model assumes that the νt is approx-imately isotropic, an important simplification wichmay not be realistic for several flows.

In the present work, the turbulence model is usedthe original version of de k − ω SST turbulencemodel by Menter (in 1994), see for instance [15].

2.3. Domain size

The domain for the calculation of the two-dimensional flow around the rounded corner (rc =0.09D) cylinder at a Reynolds number of 1.74×105

is a rectangle of variable size that is illustratedin figure 1. The inlet and outlet boundaries arex = const. planes located Lin and Lout upstreamand downstream of the cylinder centre, respectively.The external boundaries are y = const. planeslocatedLext away from the cylinder centre. Threedifferent domains were tested in the sensitivity stud-ies. The corresponding values of Lin, Lout and Lextare given in table 1.

Figure 1: Domain for the calculation of the two-dimensional flow around a squared cylinder ofrounded corners at Rn = 1.74× 105.

Domain Lin Lout LextDS 5 20 10DM 10 40 20DL 20 80 40

Table 1: Distances of the boundaries to the cylindercentre for the three domain sizes tested in the calcu-lation of the two-dimensional flow around a squaredcylinder with rounded corners at Rn = 1.74× 105.

Domain Lin Lout LextDL 20 80 40DT 5 20 8.5

Table 2: Distances of the boundaries to the cylindercentre for the two domain sizes tested in the calcu-lation of the two-dimensional flow around a squaredcylinder with rounded corners at various Reynoldsnumbers.

The unsteady turbulent flow over square cylin-der with rounded corners was computed at vari-ous Reynolds numbers, ranging from 7.24 × 104 to3.13 × 105. The calculations were computed withtwo different sizes of the computational domain inthe present exercise, the corresponding values ofLin, Lout and Lext are given in table 2. The two do-mains DL and DT are used to investigate the block-age effect in the localization at where ”drag crisis”occurs, in the last the selection of this domain hasbeen guided by the towing tank size observed in theexperiments of the [13].

2.4. Boundary conditionsSeveral alternatives were tested for the specificationof the boundary conditions. Nonetheless, there areboundary conditions that remained fixed for all cal-culations:

• No-slip and impermeability conditions are ap-plied at the cylinder surface (no wall-functions)and the normal pressure derivative is set equalto zero. The turbulence kinetic energy k is setequal to zero and ω is specified at the near-wallcell centre [10].

• Zero stream wise derivatives are assumed forall flow variables at the outlet boundary.

• At the external boundary, zero normal deriva-tives have been applied to the x velocity com-ponent ux and to the turbulence quantities kand ω, whereas the y velocity component uy isset equal to zero.

• At the inlet boundary, ux = U∞, uy = 0 andthe pressure is extrapolated from the interiorassuming zero stream wise derivative.

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Inlet k/U2∞ ωD/U∞ νt/ν I(%)

IL 1.5× 10−4 2609.6 0.01 1IM 3.75× 10−3 6525 0.1 5IH 1.5× 10−2 2609.6 1 10

Table 3: Turbulence quantities specified at the in-let of the domain in the calculation of the two-dimensional flow around a squared cylinder withrounded corners at Rn = 1.74× 105.

Two different alternatives were tested for thepressure at the external boundary:

1. Pressure equal to undisturbed flow pressurep∞, i.e. Cp = 0.

2. Zero normal pressure derivative and Cp = 0 atthe top left corner of the domain.

Three different values were tested for the turbu-lence quantities at the inlet boundary. The selectedvalues of k, ω and the corresponding values of eddy-viscosity νt and turbulence intensity I are given intable 3.

The boundary conditions used for all the simula-tions for various Reynolds numbers are the same asdescribed above. In the case of calculations elabo-rated in the largest domain, the pressure boundarycondition was tested the pressure equal to undis-turbed flow pressure. As stated below, the influenceon the flow quantities to be reduced and providesbenefits to the level of the iterative convergence. Onthe other domain size, the pressure boundary condi-tion was tested the zero normal pressure derivativeand Cp = 0 at the top left corner of the domain.The evaluation of the influence of the blockage ef-fect for the high Reynolds numbers of turbulent flowwas performed for smaller turbulence quantities, IL.

2.5. Quantities of interestThe influence of the domain size and boundaryconditions specification is evaluated for the follow-ing flow quantities: mean drag coefficient (CD)avg,maximum (CL)max, root mean squared (CL)rms ofthe lift coefficient, root mean squared (CD)rms ofthe drag coefficient, Strouhal number St and lengthof the mean recirculation Lrec. Besides these func-tional quantities, we will also analyze the surfacepressure Cp = (p−p∞)/(1/2ρU2

∞) and skin frictionCf = τw/(1/2ρU

2∞) coefficients, where p∞ is the

undisturbed pressure and τw is the shear-stress atthe wall for the sensitivity studies.

3. Numerical Solution3.1. ReFRESCO flow solverReFRESCO is a viscous-flow CFD code that solvesmultiphase (unsteady) incompressible flows us-ing the Navier-Stokes equations,complemented with

turbulence models, cavitation models and volume-fraction transport equations for different phases.The equations are discretized using a finite-volumeapproach with cell-centered collocated variables, instrong-conservation form,and a pressure-correctionequation based on the SIMPLE algorithm is used toensure mass conservation. Time integration is per-formed implicitly with first or second-order back-ward schemes. At each implicit time step, thenon-linear system for velocity and pressure is lin-earized with Picard’s method and either a seg-regated or coupled approach is used. A segre-gated approach is always adopted for the solutionof all other transport equations. The implemen-tation is face-based, which permits grids with el-ements consisting of an arbitrary number of faces(hexahedrals, tetrahedrals, prisms, pyramids, etc.),and if needed h-refinement(hanging nodes). State-of-the-art CFD features such as moving, slidingand deforming grids, as well automatic grid refine-ment are also available. For turbulence modelling,RANS/URANS, SAS and DES approaches can beused (PANS and LES are being currently studied).The code is parallelized using MPI and sub-domaindecomposition, and runs on Linux workstations andHPC clusters. ReFRESCO is currently being devel-oped and verified at MARIN (in the Netherlands)in collaboration with IST (in Portugal), USP-TPN (University of Sao Paulo, Brazil), TUDelft(Technical University of Delft, the Netherlands),UoS (University of Southampton, UK)and recentlyUTwente (University of Twente, the Netherlands)and Chalmers (Chalmers University, Sweden) [1].

3.2. Test case3.2.1 Geometry

The smooth fixed square cylinder simulatedpresents diameter, D, and the corner radius of thecylinder was chosen to be rc = 0.09D. This choicewas made to be able to compare the results of thecomputations with experimental data available atMARIN. The rounded-off square cylinder was de-signed to represent a typical column of a semi-submersible.

3.2.2 Grid generation

The numerical calculation of the flow around com-plex geometries requires the use of ”good quality”grids. The concept of a good quality grid is clearlydependent on the numerical technique applied andon the type of equations solved. Creating a meshfor high Reynolds number flows around rounded-offsquare cylinder is a complex process, because thehigh pressure gradients are often observed close towalls and especially close to the corners of the ge-ometry, which it has necessitated a large discretiza-tion of the geometry, i.e. a high number of cellsalong the cylinder surface. In the areas where the

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Figure 2: Section of the rounded-off square cylinder

pressure gradients are smaller, coarser cells can beused to keep the total number of the cells as low aspossible without affecting the solution.

The grids for the present study are multi-blockstructured, containing five blocks. The block-topology used for building the mesh around a squarecylinder is shown in figure 3. An O-type grid isused around cylinder, covering the zones close tothe body (boundary layer) and close to the bodywake (shear-layer zones). The girds are defined byfunction the number of the cells in the circumfer-ential direction per quadrant of the cylinder. Thecentral block is constituted by the two sub-blocks,wherein on inner block, the mesh is orthogonal andthe cells have constant size along the wall, and onouter block, the mesh is generate with a ellipticalgrid generator with control functions based on theGrape approach, see for instance [8], being that thecells along the outer boundary have uniform size, asillustrated in figure 3.2. And the others blocks, theinterior grid also is generated with a elliptical gridgenerator. The grid node distribution in the nor-mal direction the wall follows a one-sided stretch-ing functions, see for instance [18] are applied toensure the required y+2 for the correct applicationof the no-slip condition, in case the grid node distri-bution along the cylinder wall is equally spaced. Inthe y-direction the mesh is symmetric with respectto y = 0.

No wall-functions are used (the cylinder issmooth and therefore no roughness, nor rough-wall-functions are considered), follows the suggestion of[11], and very fine grid is used close to the bodysurface to maintain (for all grids) a y+ ≤ 1 for walladjacent grid cells, assuring that first nodes werewell inside the viscous sub-layer, thus dispensingwall functions. Away from the cylinder the grid iscoarse, again to save computational resources.

The grid presented in figure 3 has 60 cells perquadrant, i.e. 240 cells along the cylinder wall.However, for each domain size, the coarsest gridwere used for calculations has 720 cells along the

Figure 3: Overview of grid layout (top) and detailof the mesh close to the cylinder (bottom)

cylinder surface, whereas the finest grid has 1440cells. The grid parameter is h = 1/

√Ncells, since

the problem is two-dimensional, because the z com-ponents of the Navier-Stokes equations are notsolved. Therefore, the grid is identified by grid re-finement, rx, is defined by

rxi =hih1

=

((Ncells)1(Ncells)i

) 12

(6)

in which (Ncells)i is the number of the cells of gridused in the calculation to be analysed and (Ncells)1is the finest number of the cells of grid used in thecalculations. Once again, the finest grid is identifiedby rxi = hi

h1= 1, and the others, larger than one.

The grids are utilised for sensitivity studies aboutthe boundary conditions were different of it is usedfor the simulation of the flow around a squaredcylinder of rounded corners at Reynolds numberrange between 104 and 106. The grids were gener-ated for the simulation of the flow around a squaredcylinder with rounded corners at various Reynoldsnumbers, the external boundary of the central block

5

is different of defined previously, because for the fu-ture work, the calculations are extended to anglesof incidence between 0◦ and 45◦, which facilitatesthe process of the grid generation.

3.3. Grid sets and time step

The set of five geometrically similar grids weregenerated for the simulation of the flow around asquared cylinder of rounded corners. The finest gridof the DS domain contains 254016 cells with 1440cells on the cylinder surface. The maximum dimen-sionless distance to the wall of the near-wall cells inthe coarsest grid is (y+2 )max < 0.3.

The time step of each calculation is tuned to ob-tain an average Courant number close to 3 that cor-responds to a maximum Courant number close to23. The time step ∆ti is refined with the same ratioof the grid refinement, i.e. ri = ∆ti/∆t1 = hi/h1.The dimensionless time step of the finest grids cal-culation is ∆tU∞/D = 0.02083 (roughly 335 timesteps per period).

3.4. Calculation Details

The selected discretization schemes are all second-order accurate including the convective terms of thek and ω transport equations. However, flux limitersare applied to all second-order upwind schemes thatare used to discretize the convective terms of allequations solved. The segregated solver is used forthe solution of the momentum and continuity equa-tions at each time step.

All the calculations were performed in doubleprecision to guarantee that the round-off error isnegligible when compared to the discretization er-ror. The iterative convergence criteria applied ateach time step of the last 200 dimensionless timeunits requires a maximum normalized residual ofall equations solved (momentum balance, mass con-servation and k and ω transport equations) belowResmax < 10−6. The normalized residuals areequivalent to the variables change in a simple Ja-cobi iteration. However, to facilitate the start ofthe vortex shedding, all the simulations are startedwith 20 dimensionless time units calculated with aniterative convergence criteria of 10−2.

4. Sensitivity studies

The results demonstrate that the computational do-main size, pressure boundary conditions and inletturbulence quantities may have a significant influ-ence on the predicted flow field around a cylin-der with squared corners. Table 4 presents themaximum differences obtained for the selected flowquantities in percentage of the mean value, ∆φ =200(φmax − φmin)/(φmax + φmin). The resultsare presented for changes in domain size, pressureboundary conditions and inlet turbulence quanti-ties. The final column corresponds to the compari-

ri Ncylinder Ncells ∆tU∞/D1.714 840 86456 0.05711.500 960 112896 0.05001.333 1080 142884 0.04441.200 1200 176400 0.04001.000 1440 254016 0.0333

Table 5: Parameters of the refinement level used tocalculate flow around squared cylinder of roundedcorners for all Reynolds numbers.

son of all calculations performed.

5. Results for different Reynolds number5.1. Numerical resultsThe numerical simulations of the flow aroundsquare cylinder with rounded corners at sixReynolds number have been performed for two do-main sizes. However, the results were presented indetail are calculated with the corresponding size tothe towing tank. For later they were compared withthe published experimental results.

For each Reynolds number, the flow is computedby employing the grid/time refinement specified intable 5. Sensitivity tests have shown the vortexshedding results to be very sensitive to the wall dis-cretization resolution. For this reason, it was rede-fined for the present study.

The time histories of the force coefficients areshown in figure 4 for six values of Rn with finestrefinement level, ri = 1.

Table 6 summarizes the results for the flow quan-tities calculated for the flow around square cylinderwith rounded rounded corners for the finest refine-ment. All these flow quantities were obtained fromdata of the last eight cycles performed in each cal-culation. It can be observed that a ”drag crisis”appears for Rn between 7.24× 104 and 3.13× 105.Across the critical regime the average drag coeffi-cient drops from about 1.591 to 1.486, the root-mean squared of the lift and drag coefficients de-creases significantly up to Rn = 2.34 × 105 andrapidly increases for higher Reynolds numbers, andthe Strouhal number practically doubles jumpingfrom 0.146 to 0.261.

5.2. Experimental dataRecently at MARIN, several experimental cam-paigns have been performed to access the VIV(vortex induced vibrations) phenomenon for fixedand freely vibration, smooth square cylinder withrounded corners, the first measurements beingmade [13]. These experiments were done with theHigh Reynolds VIV test apparatus in the HighSpeed Basin on a 200mm rounded-off square cylin-der with 18mm corner radius and provided timetraces of the forces on the square cylinder, (CD)avg

6

∆φ(%)DomainSize

Pressure BoundaryCondition

Inlet TurbulenceQuantities

Global

(CD)avg 9.33 0.10 5.98 15.2(CL)max 11.6 3.47 31.9 38.3(CL)rms 11.8 3.61 32.4 38.7St 4.96 1.50 1.11 5.63

Table 4: Largest differences of average drag coefficient (CD)avg, maximum (CL)max and root meansquared (CL)rms lift coefficient and Strouhal number St. Calculation of the two-dimensional flow arounda squared cylinder with rounded corners at Rn = 1.74×105 with different domain sizes, pressure boundaryconditions and inlet turbulence quantities.

Rn (CD)avg (CL)rms (CL)max (CL)rms St (Cp)base7.24× 104 1.591 0.938 1.310 0.069 0.146 -1.2249.71× 104 1.581 0.881 1.233 0.054 0.147 -1.2071.74× 105 1.544 0.720 1.013 0.023 0.149 -1.1582.34× 105 1.492 0.560 0.792 0.008 0.150 -1.1032.70× 105 1.486 0.673 0.984 0.116 0.261 -1.7253.13× 105 1.488 0.754 1.127 0.142 0.256 -1.834

Table 6: Flow quantities of the refinement level used to calculate flow around squared cylinder of roundedcorners for all Reynolds numbers.

and St. With respect to the lift coefficient CL, max-imum value is difficult to be defined the irregularbehavior of the measured signals. This is caused bythe three-dimensional effects observed in the basin.Table 7 summarizes the experimental results for theflow quantities, (CD)avg and St.

5.3. Experimental uncertainty

In order to make a consistent validation procedure,we must also determine the uncertainties in the ex-perimental data. However, it is quite rare to seeexperimental uncertainties in publications, thus oneattempt to obtain some estimation of these uncer-tainties from the experimental data. The estima-tion more reliable check of the statistical conver-gence of the mean value of the selected flow quan-tities may be performed with the auto-covariancemethod proposed [3] in that estimates the uncer-tainty of the mean value of a time series as a func-tion of experience time.

5.4. Validation exercise

A validation exercise was also done for the calcu-lations showed herein in order to evaluate our ap-proach. According to [9], the aim of Validation isto estimate the modelling error of the mathemati-cal model to set of experimental data, which repre-sents the physical model. Once validation is done,one can say that the model/code is valid for thatparticular problem and conditions. The procedureproposed by ASME [2]for validation is based on the

comparison of the quantities:

Uval =√U2num + U2

input + U2D, (7)

and

E = S −D. (8)

In Eq. (7) and Eq. (8), Unum is the numerical un-certainty estimated for a certain quantity, Uinput isthe uncertainty of the parameter inputs (which areconsidered negligible for the present exercise)andUD is the experimental uncertainty, S is the nu-merical prediction of the parameter value and D isthe experimental value. The comparison betweenUval and E may lead to two possibilities:

• |E| � Uval means that the comparison is poormost likely because the modelling errors are ofmost importance.

• |E| < Uval means that the solution is withinthe noise imposed by the different sources ofuncertainty. In this case, if E is small enough,then the solution is validated with the experi-ment at Uval precision. Otherwise, the qualityof the numerical solution and/or the experi-ment should be improved for a better compar-ison.

Table 8 shows the results for our validation ex-ercise concerning drag coefficient, considering theforward towing tests for estimates the experimentaluncertainties.

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Forward towing Backward towingRn (CD)avg St (CD)avg St

7.24× 104 1.579681 0.135179 1.634260 0.1338359.71× 104 1.583371 0.133005 1.645611 0.1317771.74× 105 1.689359 0.135462 1.694741 0.1366302.34× 105 1.347625 0.118064 1.361771 0.1238672.70× 105 1.363977 0.116691 - -3.13× 105 1.587285 0.128321 1.608560 0.131954

Table 7: Experimental data for the flow quantities, (CD)avg and St.

Rn Unum (%) UD (%) Uval E7.24× 104 0.113 (7.08) 0.011 (0.72) 0.113 0.0119.71× 104 0.073 (4.63) 0.007 (0.43) 0.074 0.0021.74× 105 0.287 (18.57) 0.007 (0.43) 0.287 0.1462.34× 105 0.049 (3.31) 0.007 (0.50) 0.050 0.1452.70× 105 0.007 (0.45) 0.006 (0.40) 0.009 0.1223.13× 105 0.103 (6.90) 0.017 (1.07) 0.104 0.099

Table 8: Validation results for drag coefficient, (CD)avg.

6. ConclusionsThis thesis presents an investigation of the flowaround a squared cylinder with rounded cornersbased on numerical simulations. The flow ispredicted with one of the simplest mathematicalmodels available: two-dimensional, incompressible,Reynolds-averaged Navier-Stokes equations supple-mented with the SST k − ω eddy-viscosity model.Naturally, ensemble average has been applied dueto the periodic nature of the flow.

Two different topics are addressed in this work:

• The effect of the problem settings on the pre-dicted flow quantities, i.e. the domain size andthe specification of the pressure and inlet tur-bulence quantities boundary conditions.

• The effect of the Reynolds number for therange 7.24 × 104 to 3.13 × 105 on the flowregimes.

The numerical uncertainty of the prediction hasbeen carefully addressed including the assessmentof the iterative and discretization errors contribu-tions to the numerical uncertainty. In order to ob-tain a negligible contribution of the iterative error,the convergence criteria adopted for each time stepmust be significantly more demanding than the tra-ditional three orders of magnitude of residual drop.For the present level of grid refinement, the iter-ative convergence criteria requires that the maxi-mum normalized residual of all transport equationssolved (including turbulence quantities) must besmaller than 10−6. The normalization makes theresiduals of the equations equivalent to the change

in the solution field for a simple Jacobi iteration.Furthermore, the time simulated for each flow con-dition guarantees that the influence of the initialcondition on the last six cycles is negligible.

The results obtained for the investigation of theproblem settings suggest the following conclusions:

• The specification of the pressure boundary con-ditions exhibits the smallest influence on theselected flow conditions. However, for thesmallest domain sizes tested, there is a cleardifference between imposing pressure at the ex-ternal boundaries or at a single point.

• The domain size and the specification of theinlet turbulence quantities have a significantinfluence on the outcome of the simulations.Interestingly, the effect of these choices is notidentical for all the selected flow quantities:

– The strongest influence of the domain sizeis obtained for the average drag coefficientand Strouhal number.

– The specification of the inlet turbulencequantities has the largest effect on themaximum and root mean squared lift co-efficient.

The second exercise presented in this thesisdemonstrates that the present mathematical modelis able to identify two different regimes for the flowaround this squared cylinder with rounded corners:

1. Flow separation occuring at the left corners ofthe cylinder without reattachment on the topand bottom surfaces.

8

2. Separation bubble on the top and bottom sur-faces of the cylinder followed by flow separationat the right corners of the cylinder to generatevortex shedding.

These two regimes are dependent on the Reynoldsnumber and its determination is only possible for acomputational domain that matches the size of thetowing tank where the experiments were performed.The results obtained for the second regime suggestthat this type of flows may become extremely com-plex making the determination of the mean flowsolution a real challenge. Nonetheless, it is clearthat Validation of this type of flows depends on verychallenging experiments that should obtain a com-plete definition of boundary conditions and a carefulassessment of experimental uncertainties.

References[1] Refresco, 2015. http://www.refresco.org.

[2] ASME. Standard for verification and vali-dation in computational fluid dynamics andheat transfer. Technical report, ASME, 2009.ASME Standard.

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Rn = 7.24× 104

Rn = 9.71× 104

Rn = 1.74× 105

Rn = 2.34× 105

Rn = 2.70× 105

Rn = 3.13× 105

Figure 4: Time histories of the drag coefficient(red line) and the lift coefficient (blue line) for six valuesof Rn with finest refinement level, ri = 1.

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