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International Journal of Heat and Fluid Flow

journal homepage: www.elsevier.com/locate/ijhff

Simulation of the flow around a circular cylinder at Re=3900 with Partially-Averaged Navier-Stokes equations

F.S. Pereira⁎,a,b,c, G. Vazb, L. Eçac, S.S. Girimajia

a Texas A&M University, Ocean Engineering Department, College Station, United StatesbMaritime Research Institute Netherlands Academy, Wageningen, The Netherlandsc Instituto Superior Técnico, Mechanical Engineering Department, Lisbon, Portugal

A R T I C L E I N F O

Keywords:Discretization errorModelling errorPartially-Averaged Navier-Stokes equationsCircular cylinder

=Re 3900

A B S T R A C T

This study employs Partially-Averaged Navier-Stokes (PANS) equations to simulate the flow around a smoothcircular cylinder at Reynolds number 3900. It intends to evaluate the importance of discretization and modellingerrors on the accuracy of this mathematical model. Furthermore, the study addresses the effect of the physicalresolution, or fraction of turbulence kinetic energy being modelled fk, on the predictions accuracy. To this end,Validation exercises are carried out using five different values of fk which range from typical values for well-resolved Scale-Resolving Simulations (fk≤ 0.25) to Reynolds-Averaged Navier-Stokes equations ( =f 1.00k ).Naturally, these exercises require the evaluation of numerical errors, i.e. Verification studies. Consequently, andtaking advantage of the ability of PANS to enable the distinction between discretization and modelling errors,spatial and temporal grid refinement studies are carried out to assess the magnitude of the discretization error, aswell as its dependence on fk. The outcome confirms the ability of PANS, in combination with fk < 0.50, tosubstantially decrease the modelling error when compared to =f 1.00k . However, the reduction of fk tends toincrease the model dependence on the spatial and temporal resolution. It is demonstrated that similarly to theeffect of the spatial and temporal grid resolution on the magnitude of the numerical error, the modelling errordiminishes with the physical resolution (fk→ 0). The convergence of the predictions with fk is also illustrated.

1. Introduction

Modelling the turbulence field of statistically non-steady flows is acomplex subject of Computational Fluid Dynamics (CFD) owing to thedependence of the largest and energetic turbulent scales on the boundaryconditions. In such circumstances, mathematical formulations modellingthe entire turbulence field, such as the Reynolds-Averaged Navier-Stokes(RANS) equations, usually lead to modest performances, i.e. to significantmodelling errors. On the other hand, the direct resolution of the entireturbulence field, Direct Numerical Simulation (DNS), is restricted to simplegeometries and low Reynolds number flows due to the numerical andphysical requirements to resolve all turbulent scales. Between the formerdistinct paradigms lie the hybrid and bridging models. These two classes ofmathematical models are aimed to fill in the gap between RANS and DNS,focusing on engineering applications. Hybrid models combine RANS with aScale-Resolving Simulation (SRS) model. Whereas RANS is commonly ap-plied in boundary-layers, a SRS model is used in detached and outer-flowregions. Hence, this technique may improve the simulations accuracywith a substantial reduction of the numerical demands when compared toDNS or Large-Eddy Simulation (LES)1. However, commutation errors

(Ghosal and Moin, 1995), transition between RANS and SRS zones (Spalartet al., 1997; Squires, 2004; Mockett et al., 2015), and the near-wall mod-elling accuracy (Pereira et al., 2016) play an important role on thesemodels. On the other hand, bridging formulations employ the same for-mulation in the entire domain, being able to operate at any degree ofphysical resolution, i.e. from a fully modelled to a fully resolved turbulencefield. Although the computational demands and grid generation complexityare substantially larger, this type of modelling philosophy with constantresolution is not affected by commutation errors nor the transition betweenRANS and SRS zones. Furthermore, it enables the separate assessment ofmodelling and numerical errors which is an important feature to carry outVerification and Validation exercises. Detached-Eddy Simulation (DES)(Spalart et al., 1997) and the constant resolution Partially-Averaged Navier-Stokes (PANS) equations (Girimaji, 2005) are examples of hybrid andbridging formulations, respectively.

Naturally, it is expected that by increasing the physical resolution,these mathematical formulations enhance the numerical simulations fi-delity and so reduce modelling errors. Nonetheless, the further develop-ment of such formulations requires comprehensive exercises to assesstheir modelling accuracy, main features (transition between models,

https://doi.org/10.1016/j.ijheatfluidflow.2017.11.001Received 23 December 2016; Received in revised form 1 November 2017; Accepted 1 November 2017

⁎ Corresponding author at: Instituto Superior Técnico, Mechanical Engineering Department, Lisbon, Portugal.E-mail addresses: filipemsoares@ist.utl.pt (F.S. Pereira), g.vaz@marin.nl (G. Vaz), luis.eca@ist.utl.pt (L. Eça), girimaji@tamu.edu (S.S. Girimaji).

1 LES requires the resolution of, at least, 80% of the turbulence kinetic energy (Pope, 2000).

International Journal of Heat and Fluid Flow 69 (2018) 234–246

0142-727X/ © 2017 Elsevier Inc. All rights reserved.

T

dependence on the underlying turbulence model, etc.), and numericaldemands. The flow around the smooth circular cylinder at Reynoldsnumber 3900 is an example of a relevant benchmark case used to addressthe previous points due to the complexity of its physics: laminarboundary-layer, separation, and free shear-layer; laminar-turbulent tran-sition in the free shear-layer; and turbulent wake. To illustrate the com-plexity of its numerical simulation, Fig. 1 presents a thorough review ofnumerical results available in the literature for the time-averaged dragand base pressure coefficients, CD and Cpb. The data is depicted as afunction of the number of cells, Nc, dimensionless time-step, ΔtV∞/D, andmathematical model: RANS, PANS (fk < 1.0), Hybrid, Implicit LES (ILES,LES with no turbulence model), LES, and DNS. The collected results ex-hibit a wide range of values, confirming the difficulty of simulating suchtype of flows. Evidently, this dispersion of data has four possible sources:numerical errors, modelling errors, statistical errors, and boundary con-ditions (domain dimensions, inflow turbulence content, etc.). The in-formation contained in Fig. 1 addresses the first two sources. The col-lected data evidence the relevance of discretization errors due to the clearreduction of the results dispersion for studies using Nc > 1×106. Thistrend, however, is not observed with the decrease of the dimensionlesstime-step. Nonetheless, it is interesting to observe that for such broadrange of Nc values, the selected values for the dimensionless time-step areconfined to a few values. In addition to discretization errors, the reductionof the data dispersion with the increase of Nc might also be related withthe fraction of scales being resolved (physical resolution). Considering thetraditional direct dependence of SRS models on the spatial resolution,finer spatial resolutions will enable these formulations to resolve a widerrange of scales which tends to improve the predictions accuracy. Overall,the data plotted in Fig. 1 suggest that discretization errors are not neg-ligible, making the evaluation of modelling errors troublesome.

The present work employs PANS to simulate the previous flow problemand evaluate the relevance of discretization and modelling errors on thefidelity of this mathematical model. Moreover, it assesses the effect of thephysical resolution on the simulations accuracy, i.e. the relevance of thefraction of turbulence kinetic energy being modelled, fk. Therefore,Verification and Validation exercises are carried out through spatial andtemporal refinement studies using distinct values of fk. Evidently, theevaluation of numerical and modelling errors should be performed sepa-rately. This is an important property of PANS that results from its ability toselect a constant value for fk (no direct dependence on the grid resolution).The selected fk’s range from typical well-resolved SRS (fk≤ 0.25) to RANS( =f 1.0k ) values: 0.15, 0.25, 0.50, 0.75 and 1.00. On the grids side, fourspatial and temporal resolutions are used, covering a refinement ratio, r, ofapproximately 1.6. All calculations are carried out with the CFD solverReFRESCO (ReFRESCO, 2017).

The remainder of this article is structured as follows: Section 2 de-scribes the mathematical model PANS, while Section 3 introduces thenumerical setup and methods: test-case, computational domain andboundary conditions; numerical settings; ReFRESCO CFD solver; and themethods used to estimate discretization and modelling errors. Thereafter,Section 4 presents the numerical results and their discussion. The articleends with the main conclusions and future work in Section 5.

2. Partially-Averaged Navier-Stokes Equations

The application of an arbitrary (implicit or explicit) constant-pre-serving2 filtering operator that commutes with spatial and temporal

CD

CD

Cpb

Cpb

Fig. 1. Numerical results available in literature (Breuer, 1998; Fröhlich et al., 1998; Kravchenko and Moin, 2000; Lübcke et al., 2001; Tremblay, 2001; Franke and Frank, 2002; Hansenand Long, 2002; Lakshmipathy, 2004; Mahesh et al., 2004; Marongiu et al., 2004; Park et al., 2004; de With and Holdø, 2005; Alkishriwi et al., 2006; Kim, 2006; Park et al., 2006; Dröge,2007; Xu et al., 2007; Young and Ooi, 2007; Ayyappan and Vengadesan, 2008; Parnaudeau et al., 2008; Xu and Ma, 2009; Meyer et al., 2010; Ouvrard et al., 2010; Patel, 2010; Wong andPng, 2010; Afgan et al., 2011; Lakshmipathy et al., 2011; Wornom et al., 2011; Jee and Shariff, 2012; Lysenko et al., 2012; Sidebottom et al., 2012; Han and Krajnović, 2013; Lehmkuhlet al., 2013; Bandringa et al., 2014; Luo et al., 2014; Palkin et al., 2015; D’Alessandro et al., 2016) for the time-averaged drag and base pressure coefficients, CD and Cpb. The data are

depicted as a function of the number of cells, Nc, dimensionless time-step, ΔtV∞/D, and mathematical model: RANS, PANS (fk < 1.00), Hybrid, LES, ILES, and DNS. Experimental valuestaken from Norberg (2002) (see Table 2 for the reported experimental uncertainties).

2 The sum of all filter coefficients for a given cell is equal to one.

F.S. Pereira et al. International Journal of Heat and Fluid Flow 69 (2018) 234–246

235

differentiation (Germano, 1992; Girimaji, 2005) to the incompressiblecontinuity and momentum equations leads to the Partially-AveragedNavier-Stokes (PANS) equations,

∂∂

=Vx

0,i

i (1)

⎜ ⎟= ∂∂

⎡

⎣⎢

⎛⎝

∂∂

+∂

∂⎞⎠

⎤

⎦⎥ +

∂∂

−∂∂

D VDt x

νVx

Vx ρ

τ V Vx ρ

Px

1 ( , ) 1 ,i

j

i

j

j

i

i j

j i (2)

where any instantaneous quantity, Φ, is decomposed into a resolved (orfiltered), ⟨Φ⟩, and a modelled (or residual), ϕ, component:

= +Φ Φ ϕ. In Eqs. (1) and (2), xi are the coordinates of a Cartesiancoordinate system, ⟨Vi⟩ are the resolved velocity components, ⟨P⟩ is theresolved pressure, ν is the molecular kinematic viscosity, ρ is the fluiddensity, and τ(Vi, Vj) is the sub-filter stress tensor. In order to close thissystem of equations, τ(Vi, Vj) is here modelled through the Boussinesq’shypothesis,

= −τ V V

ρν S kδ

( , )2 2

3,ij i j

t ij ij(3)

where νt is the eddy kinematic viscosity, ⟨Sij⟩ is the resolved strain-ratetensor, k is the modelled turbulence kinetic energy, and δij is the Kro-necker symbol. However, the former relation creates two additionalunknowns: k and νt. These are determined by a turbulence model.

PANS relies on an underlying RANS turbulence model to close theprevious equations. In the present study the PANS formulation is basedon the −k ω Shear-Stress Transport (SST) RANS model (Menter et al.,2003). Therefore, this turbulence model is reformulated in order toinclude the parameters fk and fω which define the portion of turbulencekinetic energy, K, and specific dissipation, Ω, being modelled,

=f kK

,k (4)

= =f ω ffΩ

,ωk

ϵ

(5)

where fϵ is the fraction of modelled dissipation ϵ (ϵ∝kω). As a result, thetransport equations for the modelled quantities k and ω are

⎜ ⎟= − + ∂∂

⎡

⎣⎢

⎛⎝

+ ⎞⎠

∂∂

⎤

⎦⎥

DkDt

P β kωx

ν ν σff

kx

* ,kj

t kω

k j (6)

⎜ ⎟ ⎜ ⎟= − ⎛⎝

′ −′

+ ⎞⎠

+ ∂∂

⎡

⎣⎢

⎛⎝

+ ⎞⎠

∂∂

⎤

⎦⎥

+ − ∂∂

∂∂

DωDt

αν

P P Pf

βωf

ωx

ν ν σff

ωx

σω

ff

F kx

ωx

2 (1 ) ,

tk

ω ω jt ω

ω

k j

ω ω

k j j1

2

(7)

where ′ =P αβ k ν* / ,t and νt is given by

=ν a ka ω S Fmax{ ; }

.t1

1 2 (8)

Note that, the present derivation of the PANS −k ω SST model assumesa negligible contribution of the resolved turbulence field, ⟨Φ⟩, to theturbulence transport of the modelled field, ϕ. This assumption is namedzero transport model (Girimaji, 2005). The two auxiliary functionspresent in Eqs. (7) and (8), F1 and F2, are defined as

=⎛

⎝

⎜⎜⎜

⎧

⎨⎪

⎩⎪

⎧⎨⎩

⎫⎬⎭

⎫

⎬⎪

⎭⎪

⎞

⎠

⎟⎟⎟

∂∂

∂∂

−{ }F k

ωdν

d ωρσ k

dtanh min max

0.09; 500 ;

4

max ; 10,ω

ρσω

kx

ωx

1 22 2 10

4

ω

j j

2

2

(9)

= ⎛

⎝⎜

⎧⎨⎩

⎫⎬⎭

⎞

⎠⎟F k

ωdν

d ωtanh max 2

0.09; 500 .2 2

2

(10)

The constants of this model are indicated in Menter et al. (2003). Asmentioned in Pereira et al. (2015a), the present derivation follows therecommendation of Menter (1994) to use νt defined by Eq. (8) in theproduction term of the ω transport equation instead of simply =ν k ω/t .Consequently, the dissipation term of ω is not simplified as done inLakshmipathy and Togeti (2011): ′ =P αβ k ν* / t instead of ′ =P αβ ω* .Furthermore, the Prandtl number σω2 is not recalibrated by fω/fk. In thiswork the parameters fk and fω are set constant in time and space(constant resolution) in order to prevent commutation errors and en-able the study of discretization and modelling errors separately. Thevalues prescribed for fk are 0.15, 0.25, 0.50, 0.75 and 1.00 (RANS). Onthe other hand, it is assumed that the cut-off is not placed in the dis-sipation range (typically for < −f 0.10 0.20k ) so that turbulence dis-sipation occurs entirely in the unresolved scales. In this manner, fϵ is setequal to one (Girimaji, 2005) and the assessment of the selected fk’soccurs under the same conditions. Nonetheless, the effect of fϵ on thesimulations has been investigated in Pereira et al. (2015a). The studydemonstrates that selecting =f 1.0ϵ is the most appropriate strategy tosimulate the present flow with fk≥ 0.25 ( =f 0.15k was not tested). Forinstance, this study shows that simulations employing fk≥ 0.50 andassuming =f fkϵ lead to the suppression of the vortex-shedding due toexcessive diffusion (see Eqs. (6) and (7)).

3. Numerical setup and methods

3.1. Test-case, computational domain, and boundary conditions

The selected test-case is the flow around a smooth circular cylinderat Reynolds number, Re, based on the incoming velocity, V∞, and cy-linder diameter, D, equal to 3900. Experimental investigations of thisflow are reported in Ong and Wallace (1996) andParnaudeau et al. (2008). At this Reynolds number, the flow is in theregime (Zdravkovich, 1997) where turbulent transition occurs in thefree shear-layer (also named the sub-critical regime),350≤ Re≤ 2.0×105, which is featured by a laminar boundary-layer,flow separation, and free shear-layer; transition in the free shear-layer;and turbulent wake. Naturally, the transition location in this regimemoves towards the cylinder with the increase of Re.

The computational domain is a rectangular prism defined in aCartesian coordinate system centred at the cylinder axis and span-wisebottom plane, Fig. 2. The domain has the following dimensions: theinflow and outflow boundaries are located at 10D upstream and 40Ddownstream of the cylinder centre, respectively; the top and bottomboundaries are 12D distant from the cylinder centre, while the span-wise length of the computational domain is 3D. The velocity and tur-bulence quantities are set constant at the inflow boundary, = −x D/ 10,1whereas the pressure is extrapolated from the interior of the domain.The turbulence quantities result from setting the turbulence intensity as

=I 0.2% to match the experimental conditions ofParnaudeau et al. (2008), and a ratio between eddy and molecular ki-nematic viscosities of = −ν ν/ 10t

3. At the outflow boundary, =x D/ 40,1

the stream-wise derivatives of all dependent variables are equal to zero,whereas the pressure is imposed at the top and bottom boundaries,

= ±x D/ 122 . Furthermore, the transverse derivatives of the remainingdependent variables are set equal to zero at = ±x D/ 122 . Symmetryboundary conditions are applied at the span-wise boundaries, =x D/ 03and =x D/ 33 . This option results from fact that neither cyclic norsymmetric conditions are optimal to handle the span-wise instabilitiesof the non-turbulent field (Pereira et al., 2015b). Therefore, consideringthe inherent additional cost (numerical overheads) of cyclic conditions,we impose symmetry conditions. Naturally, no-slip and impermeabilityconditions are prescribed on the cylinder surface. In addition, themodelled turbulence kinetic energy and normal pressure derivative areset equal to zero, while the modelled specific dissipation is given at thecentre of the nearest-wall cell by = −ω νd f80 ω

2 (adapted from the −k ωRANS models condition (Wilcox, 2010)).

F.S. Pereira et al. International Journal of Heat and Fluid Flow 69 (2018) 234–246

236

3.2. Numerical settings

The numerical simulations are carried out on four spatial resolu-tions, ranging from 1.04×106 to 4.55×106 cells, and covering a re-finement ratio, r, of, approximately, 1.64. A similar refinement ratio isapplied to the time-step. Therefore, the dimensionless time-step, ΔtV∞/D, varies between × −8.53 10 3 and × −5.21 10 3. It is acknowledged thatthese grid sizes and refinement ratio are necessarily small due to limitedcomputational resources3. However, they are believed to be sufficientfor drawing general conclusions regarding the dependence of PANS onfk. Furthermore, PANS in a practical context is intended for muchcoarser physical and spatio-temporal grid resolutions compared to LES.The numerical resolution resultant from the combination of the spatialresolution hi with the temporal resolution ti is referred throughout thiswork as grid gi ( = …i 1, ,4). ri is defined as,

= = ≈r tt

hh

NN

ΔΔ

,ii i c

c i1 1

,1

,

3

3 (11)

where h is the typical cell-size, Nc the number of cells, and the index i thegrid number. Table 1 summarizes the main properties of the grids used inthis work: refinement ratio, ri, number of tangential cells on the cylindersurface, Nr, number of cells in each 2D plane, N2D, number of planes on thespan-wise direction, N3, total number of cells, Nc, dimensionless time-step,ΔtV∞/D, and minimum time to converge the flow statistics, ΔTV∞/D. Thetime-averaged maximum Courant number, C , and dimensionless (usingwall coordinates) cell length on the tangential, +x ,t normal, +x ,n and span-wise, +x ,s directions are lower than 2.7, 1.8, 0.38, and 38.9 (based on halfcell length in each direction and g4), respectively. In order to minimizeround-off and iterative errors, all simulations are carried out on doubleprecision and the iterative convergence criterion, cit, requires a maximum,

L∞, normalized residual of −10 5 for all dependent variables at each time-step (equivalent to a L2 norm of approximately −10 7). To this end, anddepending on fk, the solver requires between 30 to 40 iterations at eachtime-step to attain the intended cit. The spatial and temporal discretizationsare second-order accurate, being all convective terms discretized withQUICK (Leonard, 1979). The initialization of the velocity, pressure, andturbulence fields relies on an initial RANS simulation of 200 time-units. Allthe simulations run for more than 500 time-units, being the flow statisticscalculated with a number of time-units ranging between 350 and 450.

3.3. ReFRESCO CFD solver

ReFRESCO (ReFRESCO, 2017) is a community based open-usage CFDcode for the maritime world. It solves multiphase incompressible viscous-flows using the continuity and Navier-Stokes equations (filtered), com-plemented with turbulence models, cavitation models and volume-fractiontransport equations for different phases. The equations are discretized usinga finite-volume approach with cell-centred collocated variables, in strong-conservation form, and a pressure-correction equation based on theSIMPLE algorithm (Patankar and Spalding, 1972) is used to ensure massconservation. Time integration is performed implicitly with first or second-order backward schemes. At each implicit time step, the non-linear systemfor velocity and pressure is linearised with Picard’s method (see for ex-ample Ferziger and Perić, 1997) and either a segregated or coupled ap-proach can be used. A segregated approach is adopted for the solution of allother transport equations.

The implementation is face-based, which permits grids with ele-ments consisting of an arbitrary number of faces (hexahedrals, tetra-hedrals, prisms, pyramids, etc.), and if needed h-refinement (hangingnodes). For turbulent flows, RANS, SAS, DES, XLES, PANS, and LESapproaches can be used (Pereira et al., 2015a; 2015c). State-of-the-artCFD features such as moving, sliding and deforming grids, as well au-tomatic grid adaptation (refinement and/or coarsening) are alsoavailable. Coupling with structural equations-of-motion is also possible.The code is parallelized using MPI and subdomain decomposition, andruns on Linux workstations and HPC clusters. The code is currentlybeing developed, verified and tested at MARIN (the Netherlands) incollaboration with Instituto Superior Técnico (Portugal), Texas A&MUniversity (United States of America), and other universities around theworld (see ReFRESCO, 2017).

3.4. Estimation of discretization and modelling errors

The procedures available in the literature to estimate the numericaluncertainty of any local or functional quantity ϕ require at least threegrids (see for instance Roache, 1998; Stern et al., 2001; Celik et al.,2008; Xing and Stern, 2010; Eça and Hoekstra, 2014). Unfortunately, asit will be demonstrated in Section 4.2, the present two coarsest grids donot enable a proper quantification of the numerical uncertainty withsuch techniques due to insufficient resolution for the lowest values of fk.Therefore, this study estimates directly the discretization error, Ed(ϕ),through a power series expansion (Roache, 1998) and imposing thespatial and temporal order of convergence, = =p p p,h t

Fig. 2. Computational domain and coarsest spatial resolution, h4.

Table 1Grids properties, gi: refinement ratio, ri, number of tangential cells on the cylinder surface,Nr, number of cells in each 2D plane, N2D, number of planes on the span-wise direction,N3, total number of cells, Nc, dimensionless time-step, ΔtV∞/D, and minimum time toconverge the flow statistics, ΔTV∞/D.

gi r Nr N2D N3 Nc ΔtV∞/D ΔTV∞/D

g1 1.00 1,140 94,725 48 4,546,800 × −5.209 10 3 >350g2 1.15 988 71,708 42 3,011,736 × −5.976 10 3

g3 1.35 836 51,051 36 1,837,836 × −7.046 10 3

g4 1.64 640 34,578 30 1,037,340 × −8.526 10 3

3 The calculation time is significantly affected by the required strict iterative con-vergence criterion at each time-step.

F.S. Pereira et al. International Journal of Heat and Fluid Flow 69 (2018) 234–246

237

= − =−−

+

+E ϕ ϕ ϕ

ϕ ϕr

( )1

.d i oi

ip1

1 1

1 (12)

Note that, ϕo is the estimated solution for a zero discretization error.Since the numerical data does not allow the proper estimation of p, thequantification of Ed(ϕ) uses =p 1.0 and =p 2.0. Therefore, whereas

=p 1.0 leads to a conservative estimation of Ed(ϕ), the use of =p 2.0may under-predict its value (the discretization techniques used in thiswork are second-order).

On the other hand, modelling errors are here quantified using theprocedure proposed by The American Society of MechanicalEngineers (ASME) (2009). Therefore, the interval containing the mod-elling error, Em(ϕi), for 95 out of 100 cases is given by,

− ≤ ≤ +E ϕ U ϕ E ϕ E ϕ U ϕ( ) ( ) ( ) ( ) ( ) ,c i v i m i c i v i (13)

where Ec(ϕi) is the comparison error defined as the difference betweenthe numerical, ϕ, and the experimental measurement, ϕe,

= −E ϕ ϕ ϕ( ) .c i i e (14)

Uv(ϕi) is the validation uncertainty, which is a consequence of the factthat neither ϕi nor ϕe are exact, and it is defined as

= + +U ϕ U ϕ U ϕ U ϕ( ) ( ) ( ) ( ) ,v i i i e i n i2 2 2 (15)

where Ui(ϕi), Ue(ϕi), and Un(ϕi) stand for input, experimental and numer-ical uncertainties. Ui(ϕi) results from the existence of several parameters ofthe problem that are not exact, such as the turbulence content at the inflowboundary or the fluid properties. Ue(ϕ) and Un(ϕi), in turn, derive from theexperimental and numerical techniques used to determine the quantities ofinterest, respectively. In the present exercise, we have assumed that Ui(ϕi)is zero. Furthermore, we have assumed that the numerical uncertainty isequal to the estimate of the discretization error to avoid over-conservativeestimates of Un(ϕi) (see Section 4.2).

4. Results

This section presents the results of this study. It starts by reportingrelevant general considerations about the experimental data used toquantify modelling errors, Section 4.1. Afterwards, Section 4.2 ad-dresses the relevance of the numerical resolution on the present simu-lations (Verification), whereas Section 4.3 investigates the modellingaccuracy of PANS to predict the analysed flow through the quantifica-tion of modelling errors (Validation).

4.1. General considerations

The assessment of modelling errors relies on the experimental mea-surements of Norberg (2002; 2003) and Parnaudeau et al. (2008) which aresummarized in Table 2. Note that, some of the experimental uncertaintiesreported in Table 2 only consider the experimental statistical uncertainty.

The selected quantities are the drag and lift coefficients, CD and CL,Strouhal number, St, base pressure coefficient, Cpb, length of the re-circulation bubble at =x D/ 0,2 Lr, minimum stream-wise velocity magni-tude at =x D/ 0,2 ⟨V1⟩min , flow separation angle, θs, pressure distributionon the cylinder surface, Cp(θ), and velocity magnitude, ⟨Vi⟩, variance, vivi,and covariance, vivj, fields. All the selected flow quantities are dimension-less with reference values equal to V∞, D and ρ. For the force coefficients,we selected the time-averaged drag coefficient, C ,D and the root-mean-square of the lift coefficient ′CL

4. All the remaining flow quantities are time-averaged and calculated at the span-wise mid-plane.

The comparison of the numerical (see Section 3) and experimentalapparatus indicates the existence of some mismatches between them:

– The wind tunnels span-wise length, L3/D, is substantially largerthan that used in the numerical simulations. Experimental studiessuch as Norberg (1994, 2003) addressed the importance of thisratio, pointing out that the flow field depends on L3/D. On the otherhand, the DNS study reported in Ma et al. (2000) evidences a rela-tion between L3/D and two distinct flow modes: =L D/ 3.03 leads toa U-shape velocity profile while =L D/ 6.03 to a V-shape profile atthe very near-wake. However, the DNS studies of Wissink and Rodi;(2008) and Lehmkuhl et al. (2013) present small variations in thetime-averaged first and second-order statistics with L3/D. Despitethese negligible differences, Wissink and Rodi (2008) also indicatethat the autocorrelation of ⟨V1⟩ does not converge to zero even with

=L D/ 8.03 . The authors investigated the relevance of L3/D, ob-taining similar trends to those of Wissink and Rodi (2008) andLehmkuhl et al. (2013). Therefore, considering the cost of increasingL3/D, we decided to use =L D/ 3.03 ;– The height of the wind tunnels, L2/D, of Norberg (2002, 2003) ismuch larger than the L2/D of the computational domain or the ex-perimental facility used in Parnaudeau et al. (2008). This resultsinto an increase of the blockage effect which might lead to the nu-merical over-prediction of the quantities measured inNorberg (2002, 2003). In this study we prescribed =L D/ 242 tomatch the experimental setup of Parnaudeau et al. (2008);– The time used to converge the flow statistics in the experiments issubstantially larger than that used in the current work or in themajority of the studies included in Fig. 1. This will certainly addstatistical uncertainty to the numerical results as demonstrated inParnaudeau et al. (2008) or Pereira et al. (2015c);– Especially for mathematical formulations that resolve part of theturbulence field, the turbulence intensity at the inflow plays an im-portant role on the location of the laminar-turbulent transition(Zdravkovich, 1997). Multiple experimental studies (Fage and Warsap,

Table 2Experimental measurements details: fluid, Reynolds number, Re, inflow turbulence intensity, I, wind tunnel transversal and span-wise lengths, L2/D and L3/D, experimental method,dimensionless time used to converge the flow statistics, ΔTV∞/D, measured quantity, ϕ, and respective uncertainty, Ue(ϕ). PIV stands for Particle Image Velocimetry, HWA for Hot-WireAnemometer, and PT for Pressure Tap.

Source Fluid Re I(%) L2/D L3/D Method ΔTV∞/D ϕ Ue(ϕ)

Parnaudeau et al. (2008) Air 3,900 < 0.20 23.3 23.3 PIV 2.08× 105 Vi 1.0%v vi i 4.0%

HWA 2.83× 103 St 0.002

Norberg (2002) Air 4,000 < 0.06 215.5 80.0 PT 2.10× 104 θs 0.5°

CD 1.0%

Cpb 0.01

C θ( )p 0.01

Norberg (2003) Air 4,400 < 0.06 314.1 105.0 PT 3.20× 104 ′CL 4.0%

4 A precise comparison of ′CL would require the use of filtered experimental data tomatch the intended fk in the calculation of ′CL.

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1929; Sadeh and Saharon, 1982; Norberg, 1987; Norberg and Sundén,1987) have demonstrated the relevance of this parameter to the flowstatistics. For example, Norberg (1987) observed at =Re 3000 that areduction of the free-stream turbulence intensity from 1.4% to 0.1%leads to an increase of 11% in the length of the recirculation region.Therefore, the common approach for this test-case of setting an uniformvelocity field will certainty delay transition in the free shear-layer.However, the use of velocity perturbations would require an increase ofthe spatial resolution between the inlet and the body, as well as theexact information about the free-stream turbulence.

The former mismatches will certainly contribute to Ui(ϕi) and Un(ϕi).

However, their magnitudes are expected to be small compared to dis-cretization and modelling errors and so they should not affect theconclusions of the study.

4.2. Numerical errors

The Verification exercises are carried out for functional and localquantities. Fig. 3 presents the predictions of C ,D ′C ,L C ,pb L ,r V ,1 min andθs. The selected flow quantities are plotted as a function of r and fk. Theindependent variables axis are related to numerical (ri) and modelling(fk) errors, i.e. to the numerical and physical resolutions. The estimateddiscretization errors for these quantities and St are presented in Table 3,

Fig. 3. Time-averaged drag coefficient, C ,D root-mean-square lift coefficient, ′C ,L time-averaged base pressure coefficient, C ,pb recirculation length, L ,r minimum stream-wise velocity

magnitude, V ,1 min and flow separation angle, θ ,s as a function of the refinement ratio (grid resolution), ri, and physical resolution, fk.

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which includes the maximum and minimum estimates of the dis-cretization error obtained with spatial and temporal orders of con-vergence of 1 and 2. The present Verification exercises evidence thefollowing trends:

– Most of the quantities measured on the cylinder surface (C ,D ′C ,LC ,pb and θs) present a monotonic behaviour for the five values of fktested. The exception is the Strouhal number (not shown in Fig. 3).Nonetheless, its non-monotonic behaviour for =f 1.00,k 0.25 and0.15 is likely to be the combination of statistical errors, estimationprocedure (Welch’s power spectral density estimate), and small in-fluence of the spatial and temporal resolution;– Although the remaining variables, Lr and V ,1 min also show amonotonic convergence, the data is more noisy;– As discussed in Section 3.4, for the majority of the selected flowvariables and values of fk tested, it is not possible to obtain a reliableestimate of the observed order of spatial and temporal convergence;– For most of the selected flow quantities, there is a significantdifference between the results obtained in the finest spatial andtemporal resolution, g1, and the remaining ones when =f 0.50k .These results suggest that the mathematical model for fk≤ 0.50 isable to capture different aspects of the flow physics if the numerical/discretization error is sufficiently small (this might lead to a cou-pling between numerical and modelling errors which affects esti-mates of Ed(ϕ)). The discussion of such result is out of the scope ofthis exercise. However, such behaviour of the data increases sig-nificantly the error estimates performed for =f 0.50k which aremost likely too conservative;– A similar trend is observed for fk≤ 0.25 between the two finestand the two coarsest spatial and temporal resolutions. Moreover,these differences increase with the reduction of fk. As a result, thedata of Fig. 3 demonstrate that each degree of physical resolutionhas a correspondent numerical resolution which should be fulfilledin order to take advantage of the mathematical model;– The ratio between the two error estimates is slightly larger than 2for most of the selected flow variables and fk’s tested. A conservativeoption to estimate the numerical uncertainty would beUn(ϕ)≃ Ed(ϕ1)max . However, due to the behaviour of the data ob-tained for =f 0.50,k we have assumed Un(ϕ)≃ Ed(ϕ1)min .

Naturally, the time-averaged pressure coefficient distribution on thecylinder surface, C θ( ),p and the stream-wise velocity magnitude pro-files, V ,1 at the very near-wake shown in Fig. 4 agree with the con-clusions previously drawn: for =f 0.50,k there is a significant differencebetween the g1 solution and those obtained in the remaining spatial andtemporal resolutions; the same trend is observed for =f 0.25k and 0.15but with the major change in the solution occurring from g3 to g2.Several studies in the literature (see for instance Kravchenko and Moin,2000; Ma et al., 2000; Parnaudeau et al., 2008; Wissink and Rodi, 2008;Lehmkuhl et al., 2013;) address the shape of the stream-wise velocitymagnitude profile at =x D/ 1.061 . The present results, Fig. 4, indicatethat the shape of this quantity converges from a V-shape into a U-shapewith the reduction of fk (in this comparison we are referring to grid g1).Naturally, the magnitude of the discretization error is also important(see Fig. 4(f) and (h)). Consequently, the shape of the stream-wise ve-locity field at the vicinity of the cylinder is very sensitive to the physicalresolution (fk), which is evidently related to the modelling error.

Overall, the present Verification studies have indicated that for agiven level of spatial and temporal resolution the discretization errortends to increase with the reduction of fk (increase of the physical re-solution). Furthermore, it is also evident that obtaining reliable esti-mates of the discretization error for fk≤ 0.50 would require largerspatial and temporal resolutions than for fk > 0.50. As a consequence,validation uncertainties are still meaningful which makes the quanti-tative assessment of the modelling errors less precise than desirable.Nonetheless, the results obtained in grid g1 are sufficient to at leastperform a qualitative comparison of the modelling errors obtained forthe different fk’s tested.

4.3. Modelling errors

This section discusses the ability of PANS to accurately simulate theselected test-case. The discussion is focused on the results obtained inthe finest grid, g1. The numerical predictions and estimated comparisonerror of C ,D ′C ,L C ,pb St, L ,r V ,1 min and θs are shown in Fig. 3 and Table 4.The data suggest the following conclusions:

– The numerical simulations employing fk≥ 0.75 lead to largecomparison errors for all the analysed quantities. For example, the

Table 3Time-averaged drag coefficient, C ,D root-mean-square lift coefficient, ′C ,L time-averaged base pressure coefficient, C ,pb Strouhal number, St, recirculation length, L ,r minimum stream-wisevelocity magnitude, V ,1 min flow separation angle, θ ,s and respective estimated discretization errors (maximum, =p 1.0, and minimum, =p 2.0), Ed(ϕ), as a function of the physicalresolution fk. Results for g1, and experimental data taken from Norberg (2002, 2003) and Parnaudeau et al. (2008).

fk ϕ CD ′CL − Cpb St Lr V1 min θs

1.00 ϕ1 1.245 0.664 1.405 0.211 0.546 −0.227 84.7|Ed(ϕ1)max | 0.024 0.118 0.306 0.042 1.610 1.239 0.56|Ed(ϕ1)min | 0.011 0.055 0.143 0.019 0.750 0.577 0.26

0.75 ϕ1 1.250 0.668 1.425 0.211 0.530 −0.128 84.8|Ed(ϕ1)max | 0.038 0.018 0.036 0.007 0.056 0.009 0.50|Ed(ϕ1)min | 0.018 0.009 0.017 0.003 0.026 0.004 0.23

0.50 ϕ1 1.036 0.284 1.050 0.214 1.124 −0.273 81.8|Ed(ϕ1)max | 0.678 1.132 1.165 0.030 2.009 0.188 8.29|Ed(ϕ1)min | 0.316 0.528 0.543 0.014 0.935 0.087 3.86

0.25 ϕ1 0.927 0.095 0.864 0.208 1.728 −0.272 80.3|Ed(ϕ1)max | 0.079 0.115 0.118 0.027 0.655 0.057 0.98|Ed(ϕ1)min | 0.037 0.054 0.055 0.013 0.305 0.026 0.46

0.15 ϕ1 0.919 0.077 0.852 0.205 1.772 −0.269 80.3|Ed(ϕ1)max | 0.035 0.064 0.051 0.029 0.259 0.231 0.26|Ed(ϕ1)min | 0.016 0.030 0.024 0.014 0.121 0.107 0.12

Exp. ϕe 0.98 0.096 0.88 0.208 1.51 −0.34 84.0Ue(ϕe) ± 0.01 ±0.004 ±0.01 ± 0.002 ±0.02 ± 0.03 ± 0.5

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′E C( )c L obtained with such values of fk is close to 600% of the ex-perimental value. In addition, the data demonstrate that these largevalues of fk lead to results similar to RANS ( =f 1.00k ), which for

certain quantities ( ′C ,L Cpb or V1 min ) may even exhibit comparisonerrors larger than those obtained with RANS;– Although an apparent deterioration of the modelling accuracy may

x 2/D

x 2/D

x 2/D

x 2/DCp

Cp

Cp

Cp

Fig. 4. Time-averaged pressure distribution on the cylinder surface, C θ( ),p and stream-wise velocity magnitude, V ,1 profiles at the very near-wake ( =x D/ 1.06,1 1.54 and 2.02) as a

function of the grid and physical resolution, gi and fk.

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result from statistical and discretization errors, it should not bediscarded that a reduction of fk will certainly result in a calibrationdeficit of the turbulence model constants. Therefore, the gain inaccuracy resultant from resolving a larger fraction of the turbulencefield should compensate a possible under-performance of the un-derlying turbulence model. Consequently, large values of fk mightresult into a model less accurate than RANS;– The results show the existence of two groups of predictions: thoseusing fk < 0.50 and fk > 0.50. Whereas predictions usingfk < 0.50 show small values of Ec(ϕ), those using fk > 0.50 lead tolarge values of Ec(ϕ);– The data evidence the existence of a threshold value close to

=f 0.50k that leads to the reduction of Ec(ϕ). Naturally, numericalerrors play an important role on the identification of the thresholdvalue for fk. This is suggested by the large differences observed be-tween the predictions in grids g1 and g2 using =f 0.50k . As a result,the definition of a precise threshold value for fk would require si-mulations with finer grids using several values of fk close to 0.50;– In general, the decrease of fk leads to a reduction of the compar-ison error for the selected flow quantities. For example, E C( )c D dropsfrom 27.1% for =f 1.00k to values close to − 6.0% for fk≤ 0.25. Thisdifference is significantly larger than the validation uncertainty.Moreover, for ′CL the reduction of Ec is even larger, from 592% for

=f 1.00k (RANS) to −19.9% for =f 0.15k ; The exception is theseparation angle. Nonetheless, it is important to stress that the se-paration angle is estimated using the same procedure used in theexperiments, i.e. with the inflection point of the pressure coefficientdistribution on the cylinder surface. This technique makes its esti-mation very sensitive to small mismatches between the numericaland experimental measurements (see the small values of E C θ( ( ))c pfor fk < 0.50 in Fig. 5). However, using the traditional =C 0f cri-terion, the values of θs increase to 85.9°, 85.9°, 88.8°, 91.5° and 92.0°for fk’s ranging from 0.15 to 1.00, respectively. The values obtainedwith fk < 0.50 are very close to the measurements of Son andHanratty (1969) ( =Re 5000), which report a separation angle ofapproximately 86°.

Overall, the predictions of the former quantities obtained with va-lues of fk < 0.50 demonstrate a good agreement with experiments.However, it is important to stress that PANS with fk≤ 0.25 is close towell-resolved SRS.

The previous trends are confirmed by the pressure coefficient on thecylinder surface, Fig. 5, and by the velocity magnitude, variance andcovariance profiles at the near-wake, Figs. 6 and 7. These pictures de-monstrate that simulations using fk < 0.50 are in very good agreement

with the experiments, whereas the results for fk > 0.50 lead to largecomparison errors. In particular, Fig. 5 shows a substantial reduction ofE C θ( ( ))c p for fk < 0.50 when compared to fk > 0.50, Fig. 5(b). Fig. 6,in turn, demonstrates that values of fk larger than 0.50 lead to a sig-nificant over-prediction of v vi i and v vi j which evidently results intoinaccurate predictions of Vi . On the other hand, the simulations usingfk < 0.50 show a very good agreement with the experimental mea-surements. These trends are also visible in Fig. 7 where it is shown thetime-averaged stream-wise velocity field. In comparison with the ex-periments of Parnaudeau et al. (2008), =f 1.00k predicts a very smallrecirculation bubble which length tends to grow with the reduction of

Table 4Estimated comparison error, Ec(ϕ), and validation uncertainty, Uv(ϕ), of the time-averaged drag coefficient, C ,D root-mean-square lift coefficient, ′C ,L time-averaged base pressurecoefficient, C ,pb Strouhal number, St, recirculation length, L ,r minimum stream-wise velocity magnitude, V ,1 min and flow separation angle, θ ,s as a function of the physical resolution, fk.Results shown for grid g1 as percentage of the experimental data of Norberg (2002, 2003) and Parnaudeau et al. (2008).

fk ϕ Cd ′CL Cpb St Lr V1 min θs

1.00 Ec(ϕ1) +27.1 +591.8 +59.7 +1.4 −63.8 +33.1 +0.8Uv(ϕ1) ± 1.5 ±57.6 ± 16.2 ± 9.4 ±49.7 ± 169.7 ± 1.0

0.75 Ec(ϕ1) +27.6 +595.6 +62.0 +1.4 −64.9 +62.3 +1.0Uv(ϕ1) ± 2.1 ± 9.7 ±2.2 ± 1.8 ± 2.0 ±1.5 ± 0.7

0.50 Ec(ϕ1) +5.7 +195.7 +19.3 +2.8 −25.6 +19.8 −2.6Uv(ϕ1) ± 32.2 ± 549.7 ± 61.7 ± 6.8 ±62.0 ± 25.7 ± 4.6

0.25 Ec(ϕ1) −5.4 −1.1 −1.8 +0.0 +14.4 +20.0 −4.4Uv(ϕ1) ± 3.9 ±55.9 ±6.4 ± 6.1 ±20.2 ±7.8 ± 0.8

0.15 Ec(ϕ1) −6.2 −19.9 −3.2 −1.4 +17.4 +20.9 −4.3Uv(ϕ1) ± 2.0 ±31.3 ±2.9 ± 6.7 ± 8.1 ± 31.6 ± 0.6

Exp. ϕe 0.98 0.096 0.88 0.208 1.51 0.34 84.0

Fig. 5. Time-averaged pressure coefficient distribution on the cylinder surface, C θ( ),p and

respective comparison error, E C θ( ( )),c p as a function of the physical resolution, fk. Results

for the finest grid, g1, and experimental data taken from Norberg (2002).

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fk. Although the agreement between simulations using fk < 0.50 andexperiments is very good for the V1 field, it is shown that the re-circulation bubble length is slightly larger than that reported in theexperiments. Nonetheless, it is important to stress again that the factthat no turbulence content is used in the inflow condition is certainlydelaying transition and, consequently, it is likely to be the reason forthe larger velocity-deficit and consequent recirculation length(Zdravkovich, 1997). As discussed in Section 4.1, several experimentalinvestigations have demonstrated the dependence of the recirculationregion on the free-stream turbulence intensity. For example,Norberg (1987) observed a variation of 11% in the recirculation lengthwhen the free-stream turbulence intensity was reduced from 1.4% to0.1%. The asymmetries observed in Fig. 7(d) (small) and (e) may alsocontribute to this issue (Parnaudeau et al., 2008), and demonstrate thatthe reduction of fk increases the required simulation time to convergethe flow statistics (simulations using =f 0.15k run for 700 time-units,being only the last 350 time-units used to calculate the flow statistics).

5. Conclusions

This study investigates the simulation of the flow around a circularcylinder at Reynolds number 3900 using PANS. It intends to assess therelevance of discretization and modelling errors on the accuracy of thenumerical predictions. Moreover, it evaluates the effect of the physicalresolution (portion of turbulence kinetic field being modelled), fk, onthe model accuracy. Towards this end, Validation exercises are carriedout using distinct values of fk: 0.15, 0.25, 0.50, 0.75 and 1.00.Naturally, this type of exercise requires the estimation of numericalerrors, i.e. Verification exercises. Therefore, grid refinement studies areexecuted for all the prescribed values of fk using four spatial and tem-poral resolutions. Although it would be desirable to further reduce thenumerical uncertainty, the outcome of this study suggests the following:

– The results demonstrate that PANS is able to substantially improvethe accuracy of the predictions of functional and local quantities

Fig. 6. Time-averaged stream-wise and transversal velocity magnitude, V ,i variance, v v ,i i and covariance, v v ,i j profiles at different x1/D and x2/D locations as a function of the physical

resolution, fk. Results for the finest grid, g1, and experimental data taken from Parnaudeau et al. (2008).

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when compared to RANS ( =f 1.00k );– Nonetheless, the model accuracy depends on the portion of theturbulence kinetic energy being modelled, fk: whereas PANS em-ploying fk > 0.50 leads to performances similar to RANS, simula-tions using fk < 0.50 attain a very good agreement with the ex-periments, i.e. low comparison errors;– Naturally, the enhancement of the modelling accuracy with the re-duction of fk is coupled with an increase of the numerical requisites;– The predictions indicate that similarly to the effect of the spatialand temporal grid resolution on the magnitude of the numericalerror, the increase of the physical resolution (fk→ 0) leads to thereduction of the modelling error. It is shown the convergence of theresults with fk;– The results also suggest the existence of a critical value of fk which

enables the mathematical model to substantially reduce the mod-elling error. Such value is close to =f 0.50,k but its accurate de-termination requires the use of finer spatial and temporal resolu-tions that reduce discretization errors to negligible levels, andseveral values of fk close to 0.50.

This latter topic is currently being addressed by studying the effectof the physical resolution on the flow dynamics, focusing on the re-quired value of fk to resolve the dominant flow instabilities and, con-sequently, to improve the modelling accuracy.

Acknowledgements

The authors would like to thank the Maritime Research Institute

Fig. 7. Time-averaged stream-wise velocity magnitude, V ,1 at the near wake-field as a function of the physical resolution, fk. Results for the finest grid, g1, and experimental data takenfrom Parnaudeau et al. (2008). Red line delimits recirculation region - =V 01 .

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Netherlands and the Laboratory for Advanced Computing of Universityof Coimbra for providing the computational resources to execute thisproject. Moreover, the authors are also grateful to C. Norberg, L. Carlierand E. Lamballais for providing their experimental measurements andfor the interesting discussions about experimental uncertainty.

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