Challenges in Modelling and Simulation of Turbulent Flows with Spatially-Developing Coherent Structures
11 Challenges in Modelling and Simulation ofTurbulent Flows with Spatially-DevelopingCoherent Structures
F.S. Pereira † , L. E¸ca , G. Vaz , and S.S. Girimaji Department of Ocean Engineering, Texas A&M University, College Station, United States ofAmerica Department of Mechanical Engineering, Instituto Superior T´ecnico, Lisbon, Portugal Maritime Institute Research Netherlands, Wageningen, The Netherlands
The objective of this work is to investigate the fundamental challenges encountered inmodelling and simulation of turbulent flows driven by spatially-developing coherent struc-tures. Scale-Resolving Simulations (SRS’s) of practical interest are expressly intendedfor efficiently computing such flows by resolving only the most important features ofthe coherent structures and modelling the remainder as stochastic field. With referenceto a typical Large-Eddy Simulation (LES), practical SRS methods seek to resolve aconsiderably narrower range of scales (physical resolution) to achieve an adequate degreeof accuracy at reasonable computational effort. The success of SRS methods dependsupon three important factors: i ) ability to identify key flow mechanisms responsible forthe generation of coherent structures; ii ) determine the optimum range of resolutionrequired to adequately capture key elements of coherent structures; and iii ) ensurethat the modelled part is comprised nearly exclusively of fully-developed stochasticturbulence. This study considers the canonical case of the flow around a circular cylinderto address the aforementioned three key issues. It is first demonstrated using experimentalevidence that the vortex-shedding instability and flow-structure development involvesfour important stages. The inherent limitations of the Reynolds-Averaged Navier-Stokes(RANS) approach in addressing the flow physics in these four stages of development areexplained. A series of SRS computations of progressively increasing resolution (decreasingcut-off length) are performed. An a priori basis for locating the origin of the coherentstructures development is proposed and examined. The criterion is based on the fact thatthe coherent structures are generated by the Kelvin-Helmholtz (KH) instability. The mostimportant finding is that the key aspects of coherent structures can be resolved only if theeffective computational Reynolds number (based on total viscosity) exceeds the criticalvalue of the KH instability in laminar flows. Finally, a quantitative criterion assessing thenature of the unresolved field based on the strain-rate ratio of mean and unresolved fieldsis examined. Based on the findings, quantitative guidelines for determining the optimaldegree of physical resolution in flows of practical interest are proposed. Key words:
Scale-Resolving Simulation; Coherent Structures; Physical Resolution; FreeShear-Layer; Circular Cylinder. † Email address for correspondence: fi[email protected] a r X i v : . [ phy s i c s . f l u - dyn ] D ec F.S. Pereira, L. Eca, G. Vaz, and S.S. Girimaji
1. Introduction
Turbulent flows with spatially-developing coherent structures constitute an importantclassification of interest in many natural phenomena and engineering applications. Co-herent structures drive the flow dynamics in wakes, jets, mixing layers and internal flows.For example, the vortex-shedding in many wake flows is preceded by Kelvin-Helmholtzrollers that develop in the free shear-layers emerging from the surface of the body. Atypical bluff-body wake in such regime consists of four important flow regions: upstreamlow intensity turbulence; initiation and development of coherent structures in the freeshear-layer; subsequent vortex-shedding; and high-intensity stochastic turbulence in thefar-wake.The velocity or pressure field, Φ , in flows with coherent structures is most convenientlydescribed with a triple decomposition (Hussain & Reynolds 1970; Schiestel 1987): Φ = Φ + ˜ φ + φ (cid:48) . (1.1)Here Φ is the appropriately averaged flow field, ˜ φ is the velocity field associated withthe coherent structures, and φ (cid:48) represents the stochastic background turbulence. Therationale of the triple decomposition can be explained as follows. The coherent velocityfield encapsulates the space and time correlations of the large-scale structures that arespecific to the flow conditions and geometry. The background velocity field representsmore universal stochastic features of a fully-developed turbulent field which is character-ized by shorter spatio-temporal correlation distances. In a typical wake flow, the upstreamand far-wake velocity field may be reasonably represented as fully-developed stochasticturbulence. In the near-wake, the coherent structures clearly dominate the flow dynamics.Predictive computations of flows with spatially-developing coherent structures is rifewith challenges. While Direct Numerical Simulations (DNS) and Large-Eddy Simulations(LES) are suitable for capturing the coherent field physics, they are computationallyprohibitive for many applications. On the other hand, the Reynolds-Averaged Navier-Stokes (RANS) approach is inherently deficient as it does not distinguish between thecoherent and stochastic parts of the velocity field. These closure models cannot accountfor two or multi-point coherence in the velocity field and hence are best suited only forthe stochastic flow fieldOver the last two decades Scale-Resolving Simulations (SRS’s) of practical interest haveemerged expressly to address flows with large-scale coherent structures. The objective isto resolve a small but sufficient range of scales to capture the main aspects of the coherentstructures and model the remainder of the flow field with an appropriate stochasticclosure. These SRS strategies are intended to render a reasonable representation ofthe coherent flow field while resolving a substantially smaller range of scales than in atypical LES. Whereas the SRS rationale is meritorious, its utilization is difficult as thereare no clear guidelines on the precise range of coherent scales that must be resolved.Undoubtedly, the success of the SRS approach depends upon i) the ability to identifythe flow physics underlying the generation of coherent structures; ii) designating theoptimum range of scales to be resolved and iii) ensuring that only the stochastic field ismodelled.The SRS approaches of practical interest available in literature can be broadly classifiedinto two types: hybrid (Spalart et al. et al. et al. et al. et al. odelling and Simulation of Turbulent Flows with Coherent Structures et al. i) identify the key flow physics underlyingthe coherent structures development in a prototypical flow; and ii) develop criteria foroptimum flow resolution. An optimal SRS entails two important factors: i ) the resolutionshould be adequate, but not excessive, to directly compute the key features of the coherentstructures; and ii ) ensure that the unresolved part of the velocity field is exclusively fully-developed stochastic turbulence. The flow past a circular cylinder, an archetypal flow withwake vortices, is employed in the study to address these questions.The remainder of the paper is arranged as follows. In Section 2, the key elements of theVon-K´arm´an vortex-street development in the flow past a circular cylinder are identified.Experimentally measured ranges of important flow features are established. The PANSmethod is described in Section 3. The computational setup is explained in Section 4.PANS circular cylinder simulations are performed with different ranges of resolved scales(degrees of physical resolution). In Section 5, PANS flow fields are analysed using thenon-Newtonian turbulence paradigm and compared against experimental data. Criteriafor optimal degree of coherent structures resolution are developed and demonstrated.The paper concludes in Section 6 with a summary of the major findings.
2. Flow Around Circular Cylinder
The flow around circular cylinders can be classified into distinct regimes based on theReynolds number, Re ≡ V ∞ Dν , (2.1)where V ∞ is the incoming time-averaged stream-wise velocity, D is the cylinder diameter,and ν is the fluid kinematic viscosity. According to Zdravkovich (1997), it is possibleto distinguish five different regimes depending on Re and on the location of turbulenttransition: fully laminar (L, Re < − Re < − Re < . × − . × ); transition inthe boundary-layer (TBL); and fully turbulent boundary-layer (T). All these categoriescontain sub-regimes with well-defined features.In addition to Re , there are other parameters that may influence the limits of suchregimes. The most relevant and common influencing parameters (Zdravkovich 1997) arethe turbulence intensity (Fage & Warsap 1929; Sadeh & Saharon 1982; Norberg & Sund´en1987); aspect ratio (West & Apelt 1982; Szepessy & Bearman 1992; Norberg 1994); wall-blockage (West & Apelt 1982); surface roughness (Fage & Warsap 1929; Achenbach1971; G¨uven et al. b ); and cylinder oscillations (Koopmann 1967; Tokumaru & Dimotakis 1991). Theimportance of the turbulence intensity, I , and aspect ratio, Λ = L/D , is demonstrated intable 1. For example, the data show that the time-averaged vortex-shedding formationlength, L F , may vary up to 11% with I . F.S. Pereira, L. Eca, G. Vaz, and S.S. Girimaji
Re I (%)
Λ St C D C pb L F /D ∗ .
06 5-50 0.194-0.209 - 0.68-0.87 -
Table 1: Experimental measurements of the Strouhal number, St , time-averaged dragcoefficient, C D , base pressure coefficient, C pb , and formation length, L F , in terms offree-stream turbulence intensity, I , and aspect ratio, Λ . Data from Norberg (1987) andNorberg (1994). ∗ denotes interpolated values using data at Re = 3800 and 4400.Each of the flow regimes poses a different set of challenges to modelling and simulation.The focus of this study is the TSL regime which is crucially dependent upon thedevelopment of coherent structures in the free shear-layer coming off the cylinder. Thedetails of this regime are summarized in Section 2.1. A thorough overview of all flowregimes is given in Williamson (1996) and Zdravkovich (1997).2.1. Transition in the Free Shear-Layer Regime
The flow in the TSL regime is characterized by three shear-layers - boundary-layer, freeshear-layer, and wake. In this range of Re , two boundary-layers detach from the cylinder’ssurface creating a recirculation region and two free shear-layers. Kelvin-Helmholtz (KH)rollers are then generated in these shear-layers. Their features govern the flow dynamicsin the free shear-layer. The importance of this coherent structure has motivated multipleinvestigations in the past decades (Roshko 1954; Bloor 1964; Prasad & Williamson 1997 a ;Rajagopalan & Antonia 2005). Figure 1a shows a flow visualization in this regime fromPrasad & Williamson (1997 a ).The Kelvin-Helmholtz instability manifests as intermittent high frequency bursts inthe velocity field with a characteristic frequency f KH - figures 1b and 1c. The irregularbehaviour of this instability increases with Re . Although Reynolds number values rangingfrom 350 to 2600 are also reported in the literature (Rajagopalan & Antonia 2005; Prasad& Williamson 1997 a ), there is a general consensus that this instability can only bereliably observed beyond Re > a ). This phenomenonis also critically dependent on influencing parameters such as ending conditions (Unal& Rockwell 1988; Prasad & Williamson 1997 a ). These small coherent structures, calledtransition waves in Bloor (1964), are responsible for the turbulence onset/transition.After the onset of turbulence, the two free shear-layers roll-up and generate the primarywake instability characterized by a periodic transverse motion with frequency f vs . Suchinstability is variously called vortex-shedding, K´arm´an-B´ernard or simply Von-K´arm´anvortex-street. Throughout this manuscript this instability will be referred as primary orvortex-shedding instability. Figure 1c depicts a representative spectra of a stream-wisevelocity signal in the free shear-layer. It shows that while f vs has a sharp peak, f KH presents a relatively broad band resulting from the intermittency of this instability.Literature reports the existence of more instabilities in this regime. Mansy et al. (1994)and Lin et al. (1995) observed the existence of mode B instabilities up to Re ≈ , f vs and f KH . Lehmkuhl et al. (2013) odelling and Simulation of Turbulent Flows with Coherent Structures (a) Cross-section view.(b) V ( t ).(c) E ( V ). Figure 1: Cross-section view of the flow around a circular cylinder in the near-wake( Re = 10 , V , time trace ( Re = 4000) and spectrum( Re = 3700) in the free shear-layer. Figures taken from Prasad & Williamson (1997 a )and Rajagopalan & Antonia (2005). F.S. Pereira, L. Eca, G. Vaz, and S.S. Girimaji observed the existence of a frequency significantly lower than f vs . Apart from someharmonics, none of these instabilities is expected to have higher frequencies than f KH .In summary, the development of a flow in the TSL regime entails four important steps:(i) onset of the Kelvin-Helmholtz instability in the free shear-layer;(ii) spatial development of the Kelvin-Helmholtz rollers;(iii) breakdown to high intensity turbulence;(iv) turbulent shear-layer roll-up leading to vortex-shedding.It is evident that these flow features must be accurately simulated in order to reasonablypredict the flow statistics for practical applications. We will now develop the criteria foroptimal simulation of these structures and flow features.
3. Closure Modelling and Simulation
As mentioned in the introduction, fluid motions containing coherent structures can bebest described with a triple decomposition (Hussain & Reynolds 1970; Schiestel 1987).Hence, any dependent flow variable Φ is seen as a combination of an appropriate averaged( Φ ), coherent ( ˜ φ ), and stochastic background turbulence ( φ (cid:48) ) parts, Φ = Φ + ˜ φ + φ (cid:48) . (3.1)The coherent field is typically dominated by strong flow-dependent spatio-temporal cor-relations. For this reason, the simulation and modelling of this component is particularlycomplex. The stochastic turbulent field, on the other hand, exhibits more universalfeatures with shorter spatio-temporal correlation distances. Although DNS and LES arecapable of accurately computing such flows, their numerical demands are prohibitivelyexpensive in many cases. A computationally viable approach is therefore often necessary.3.1. Reynolds-Averaged Navier-Stokes Closures
By definition, current one-point turbulence closures account for the effect of theturbulent field on the mean flow equations by employing either an algebraic constitutiverelation or evolution equations for the Reynolds stresses. One-point closures employingalgebraic constitutive relations obtain length and time scales by solving zero, one, and twoextra transport equations. The algebraic constitutive relationships are typically based onone of the following three tenets: molecular analogy (Boussinessq approximation), weak-equilibrium assumption (algebraic Reynolds stress models - Pope (1975); Rodi (1976);Gatski & Speziale (1993) and Girimaji (1996)) or extended thermodynamic concepts(Huang & Rajagopal 1996). The more advanced algebraic models can also account foranisotropic turbulent viscosity and non-linear constitutive relation effects. However, allthese closures assume turbulence in a fully-developed near-equilibrium state (whereinthis is a balance between linear, non-linear and viscous mechanisms). Girimaji (2001)has demonstrated how some transient effects can be included in an algebraic closure. Allthese closures preclude turbulence far from equilibrium and exclude the presence of anyorganized motion or coherent structures in the modelled flow field.Reynolds stress transport models (Launder et al. et al. odelling and Simulation of Turbulent Flows with Coherent Structures i ) theinability to accurately model the physical features of flow instabilities in the unresolvedfield; and ii ) the inadequacy in accounting for transient turbulence effects. These effectsare critical in flows with spatially-developing coherent structures.Multi-point closures can, in principle, account for some degree of spatial coherence inthe flow field. However, at the current time, neither suitable closure models nor viablenumerical approaches have been developed for computing inhomogeneous turbulent flowswith coherent structures.3.2. Scale-Resolving Simulation Approaches
Over the last two decades, hybrid and bridging SRS formulations have emerged for theexpress purpose of simulating practical flows containing coherent structures. The aim ofany SRS formulation of practical interest is to resolve only the coherent component of theflow field and model the stochastic part with a suitable closure - figure 2. The resolution ofthe coherent field obviates the need for accurate closure modelling. Therefore, the extentof resolution is dictated by the complexity of the coherent flow field and the requireddegree of accuracy. The concept of accuracy-on-demand is embedded in these methods.The SRS governing equations are based upon the scale invariance property of thefiltered Navier-Stokes equations (Germano 1992). Let us assume a general filteringoperator, constant preserving, and commuting with spatial and temporal differentiation.This filter decomposes any dependent quantity Φ into a filtered/resolved, (cid:104) Φ (cid:105) , and aunresolved/modelled, φ , component, Φ ≡ (cid:104) Φ (cid:105) + φ . (3.2)Ideally, the resolved flow field (cid:104) Φ (cid:105) comprises both mean and coherent parts. The unre-solved component φ is then modelled as stochastic field. The challenge of SRS lies inguaranteeing that an optimal portion the coherent field is resolved.The application of the aforementioned operator to the incompressible Navier-Stokes F.S. Pereira, L. Eca, G. Vaz, and S.S. Girimaji equations leads to its filtered form (Germano 1992), ∂ (cid:104) V i (cid:105) ∂x i = 0 , (3.3) ∂ (cid:104) V i (cid:105) ∂t + (cid:104) V j (cid:105) ∂ (cid:104) V i (cid:105) ∂x j = − ρ ∂ (cid:104) P (cid:105) ∂x i + ν ∂ (cid:104) V i (cid:105) ∂x j ∂x j + 1 ρ ∂τ ( V i , V j ) ∂x j , (3.4)where V i are the Cartesian velocity components, P is the pressure, ρ is the fluid density,and τ ( V i , V j ) is the generalized central second moment or Sub-Grid Stresses (SGS)tensor which accounts for the effects of the modelled velocity field on the resolved field.Clearly, a closure model for the sub-grid stresses tensor is needed. In LES, Smagorinsky-type closures are often used. Other SRS formulations employ n -equations closures. TheDetached Eddy Simulations of Spalart et al. (1997), and the PANS of Girimaji (2005)and Basara et al. (2011) are cases of such formulations. The use of second momentclosure models is also possible as demonstrated in Chaouat & Schiestel (2005) wheresuch type of closure is adapted to the Partially-Integrated Transport Model (Chaouat &Schiestel 2005; Schiestel & Dejoan 2005). PANS is the SRS approach employed in thisinvestigation. 3.3. Partially-Averaged Navier-Stokes Equations
The quality of a SRS is clearly dependent on the range of resolved scales (physicalresolution or cut-off length-scale), and on the fidelity of the closure model used torepresent the stochastic field. It is expected that the accuracy of a SRS computation willimprove with an increase in the range of resolved scales as the closure model contributionprogressively diminishes.Although one-point closures are inadequate to represent the physics of a coherentvelocity field, they are reasonably accurate for modelling fully-developed stochasticturbulence. Since the intent of PANS is to resolve the coherent structures, it can beargued that the residual unresolved field can be adequately represented by a simpleone-point closure such as the Boussinesq relation: τ ij ( V i , V j ) ρ = 2 ν u (cid:104) S ij (cid:105) − k u δ ij , (3.5)where (cid:104) S ij (cid:105) is the resolved strain-rate tensor, (cid:104) S ij (cid:105) = 12 (cid:18) ∂ (cid:104) V i (cid:105) ∂x j + ∂ (cid:104) V j (cid:105) ∂x i (cid:19) , (3.6) ν u is the turbulent or eddy viscosity of the unresolved field, k u is the modelled turbulencekinetic energy, and δ ij is the Kronecker delta. For the sake of clarity we use the subscript u to denote the unresolved component of equation 3.2 (instead of just lower-case andcapital symbols). The turbulent viscosity and kinetic energy are then calculated througha PANS closure-equations.The range of resolved scales in PANS closures is controlled by two parameters, f k ≡ k u k , f (cid:15) ≡ (cid:15) u (cid:15) , (3.7)which characterize the fraction of turbulence kinetic energy and dissipation rate ( (cid:15) ) beingmodelled. Alternatively, the decomposition can also be effected in terms of f k and f ω : f ω ≡ ω u ω ≡ f (cid:15) f k , (3.8) odelling and Simulation of Turbulent Flows with Coherent Structures ω is the specific dissipation.The k − ω PANS (Lakshmipathy & Girimaji 2006) and the k − ω Shear-Stress Transport(SST) PANS (Pereira et al. Dk u Dt = τ ij ∂ (cid:104) V i (cid:105) ∂x j − β ∗ k u ω u + ∂∂x j (cid:20)(cid:18) ν + ν u σ k f ω f k (cid:19) ∂k u ∂x j (cid:21) , (3.9) Dω u Dt = αν u τ ij ∂ (cid:104) V i (cid:105) ∂x j − (cid:18) αβ ∗ − αβ ∗ f ω + βf ω (cid:19) ω u + ∂∂x j (cid:20)(cid:18) ν + ν u σ ω f ω f k (cid:19) ∂ω u ∂x j (cid:21) + D c . (3.10)In equation 3.10, D c is the cross-diffusion term, D c = , ( k − ω )2 σ ω ω f ω f k (1 − F ) ∂k u ∂x j ∂ω u ∂x j , (SST) , (3.11)and the turbulent viscosity is defined as ν u = k u ω u , ( k − ω ) a k u max { a ω u ; (cid:104) S (cid:105) F } , (SST) . (3.12)The remaining coefficients and auxiliary functions are as given in Wilcox (1988) andMenter et al. (2003).The turbulent viscosity of the unresolved field in a PANS simulation can be approxi-mated as ν u ≈ f k f ω kω ≈ f k f (cid:15) C µ k (cid:15) , (3.13)for one-point closures based on k − ω and k − (cid:15) formulations. In high Reynolds numberflows, all of the dissipation is expected to occur at the unresolved scales, leading to f (cid:15) = 1 .
0. Then, we can write, ν u ≈ f k kω ≈ f k C µ k (cid:15) = f k ν t , (3.14)where ν t is the corresponding turbulent viscosity of modelling the entire turbulent field(RANS). Clearly, the turbulent viscosity vanishes as f k tends to zero. At the otherextreme, the turbulent viscosity of the unresolved fields goes to the RANS values ( ν t )when f k approaches unity. Girimaji & Abdol-Hamid (2005) have proposed the followingrelation to estimate the maximum physical resolution f k that a given spatial gridresolution can support, f k (cid:62) (cid:112) C µ (cid:18) ∆L (cid:19) / . (3.15)In equation 3.15, ∆ is an appropriate measure of the grid spatial resolution, and L is thecharacteristic length of the largest turbulent scales - L = k / /(cid:15) . From this relation, it ispossible to infer that as f k decreases, the computational requirements increase.The flow field of PANS and other SRS methods can be analysed most conveniently asthe direct numerical simulation of a non-Newtonian fluid at a lower effective computa-tional Reynolds number Re e (Reyes et al. Re e ≡ V ∞ Dν + ν u . (3.16)The consequence of this decrease of Reynolds number is the reduction of the numerical0 F.S. Pereira, L. Eca, G. Vaz, and S.S. Girimaji effort. In order to illustrate this, assume a high Reynolds number flow so that alldissipation occurs entirely in the unresolved scales - f (cid:15) = 1 .
00. Then, it is possibleto estimate the range of scales being resolved for a given f k (Reyes et al. Lη u ∼ Re / e = (cid:18) V ∞ Dν + ν t f k (cid:19) / . (3.17)Here, η u is the computational Kolmogorov length scale (smallest scales being resolved)given by η u = (cid:32) ( ν + ν u ) (cid:15) (cid:33) / . (3.18)Note that the length of η u should be of the size of the spatial grid resolution. Fromequations 3.16 and 3.17, it is possible to infer that as f k tends to unity, the simulationapproaches a RANS computation. At the other extreme, as f k vanishes the effectivecomputational Reynolds number approaches the flow Reynolds number, leading to aDNS. Between these extremes of f k , the effective Reynolds number takes intermediatevalues.
4. Problem Setup
The selected test-case is the flow around a circular cylinder at Re = 3900. This flowis representative of the regime described in Section 2.1 where the vortex-shedding andKelvin-Helmholtz instabilities are the dominant spatially-developing coherent structures.It is therefore expected that the findings from this study can be used to establish generalguidelines for the minimum resolution necessary for flows with coherent structures.4.1. Numerical Details
The numerical simulations are carried out with the finite volume solver ReFRESCO(2017) wherein all terms of the closed set of equations are discretized with second-order accurate schemes. The values prescribed in PANS simulations for the fractionof the turbulence kinetic energy being modelled, f k , are 0 .
25, 0 .
50, 0 .
75 and 1 .
00 (0.15is also used for the SST closure). It is assumed that dissipation, (cid:15) , occurs entirely inthe unresolved scales so that f (cid:15) = 1 .
00 ( f ω = f (cid:15) /f k ). The computational domain is arectangular prism centred at the cylinder’s axis with stream-wise, x , transverse, x ,and span-wise, x , lengths of 50 D , 22 D , and 3 D . The boundary conditions are similarto those employed in Pereira et al. (2017): velocity and turbulent quantities are setconstant at the inlet boundary, x /D = −
10, whereas the pressure is extrapolated fromthe interior of the domain. The turbulent quantities k u and ω u result from setting theturbulent intensity as I = 0 .
2% (Parnaudeau et al. ν t /ν = 10 − ; at the outlet, x /D = 40, the stream-wise derivatives of all dependent variables are set equal to zero; atthe top and bottom boundaries, x /D = ±
12, the transverse derivatives are zero and thepressure is imposed; and symmetry boundary conditions are employed in the transversedirection (Pereira et al. , ,
800 volumes and a dimensionless time-step, ∆tV ∞ /D , of 5 . × − .The selection of this spatial-temporal resolution is consequence of the verification studiesexecuted in Pereira et al. (2017). In order to minimize round-off and iterative errors, thecalculations run on double precision and the maximum norm of the normalized residualof all dependent quantities is equal to 10 − at each time-step. The simulated time is 500 odelling and Simulation of Turbulent Flows with Coherent Structures V ∞ , D , and ρ as referencequantities. 4.2. Experimental Measurements
The experimental measurements of Parnaudeau et al. (2008) are used as referencein this study. These experiments were carried out in a wind tunnel at Re = 3900. Theexperimental facility has a square section with length 23 . D , and the incoming flow has aturbulence intensity, I , lower than 0 . (cid:104) V i (cid:105) , varianceand covariance, v i v j , fields. These fields were measured using Particle Image Velocity(PIV), and the dimensionless time, ∆T V ∞ /D , used to converge the flow statistics isapproximately 2 . × time-units. Although at a slightly different Reynolds number, Re = 4000, the experiments of Norberg (2002) are also used as reference for the pressuredistribution on the cylinder’s surface, C p ( θ ). The data were obtained on a wind tunnelwith a 215 . D ( L ) by 80 . D ( L ) section and I < . . × time-units. The root-mean-square liftcoefficient C (cid:48) L measurements of Norberg (2003) are also considered. The main differenceto the experimental setup of Norberg (2002) are the Re , L , L and ∆T V ∞ /D . Theseare, respectively, 4400, 314 . D , 105 . D and 3 . × . The results shown in table 1 arealso used in this study to demonstrate the relevance of influencing parameters.
5. Results and Discussion
This section starts by analysing the influence of the physical resolution (range ofresolved scales) on the fidelity of the simulations. The results obtained at differentresolutions are compared with experimental measurements in Section 5.1. We seek toestablish general convergence of SRS results towards a range of observations over acollection of similar experiments. Thereafter, the physical rationale behind the behaviourof the simulated results is investigated in Section 5.2. Finally, in Section 5.3 we developa set of guidelines for optimal SRS computations of flows with coherent structures.5.1.
Effect of Physical Resolution
The results from different f k simulations of various integral and local flow quantitiesare shown in figures 3 to 6 and table 2. As expected, the data show that as the physicalresolution increases ( f k → f k values capture this quantity quite accurately. On the otherhand, the data demonstrate a clear overprediction of the resistance coefficients ( C D , C (cid:48) L , and C pb ) in low resolution simulations (here denoted as f k > . C (cid:48) L , it is observed that the prediction for this quantity resulting from simulationsat f k = 1 .
00 is nearly five times higher than the experimental observation. Yet, as f k decreases, the results approach the experiments. Similar trends are obtained for thetime-averaged lengths of recirculation and formation of the vortex-shedding instability, L r and L F . The refinement of resolution leads to slightly larger values of L r and L F than those measured experimentally. This result is addressed later. Overall, the data oftable 2 indicates that when the resolution is improved beyond f k = 0 .
50, two importantoutcomes are observed: i) both k − ω and SST results begin to converge - figure 4;2 F.S. Pereira, L. Eca, G. Vaz, and S.S. Girimaji (a) f k = 1 .
00. (b) f k = 0 . f k = 0 .
50. (d) f k = 0 . Figure 3: Instantaneous Q factor iso-surfaces ( Q = 0 .
1, 0 . .
5) coloured with thestream-wise velocity magnitude, (cid:104) V (cid:105) , for different physical resolutions. Results for SSTPANS closure.and ii) the converged values are clearly within the experimental range of the respectivequantities.Next we consider the profiles of the time-averaged pressure coefficient on the cylinder’ssurface, C p ( θ ), stream-wise velocity, (cid:104) V (cid:105) , and stresses, v i v j , at different locations infigure 5. Whereas high resolution (here denoted as f k (cid:54) .
50) simulations closely matchthe experimental measurements of the pressure coefficient distribution, low resolutionsexhibit large discrepancies. Similar conclusions can be inferred for the velocity andstresses profiles. For instance, considering the profiles of (cid:104) V (cid:105) at x /D = 1 .
06, thesemove from a ”V” to an ”U” shape matching the experimental observations. The stressesare clearly overpredicted in low resolution simulations.The time-averaged stream-wise velocity fields on the x − x plane are shown in figure 6for the two closure models at different resolutions. These fields are compared against theexperimental data of Parnaudeau et al. (2008) presented in figure 6e. The recirculationzone and reattachment length can be inferred from the figures. It is once again evidentthat the results generally improve by decreasing f k . The computed reattachment lengthmoves a little further downstream from the experimental location at the finest resolution.Many experimental works (Fage & Warsap 1929; Sadeh & Saharon 1982; Norberg &Sund´en 1987; Norberg 1987) have noted that the reattachment length is a strong functionof the upstream (inflow boundary condition) turbulence intensity, I . Norberg (1987) hasdemonstrated that reducing I from 1 .
4% to 0 .
1% leads to an increase of 11% in therecirculation length - see table 1. In the experiment considered here (Parnaudeau et al. I , but indicate that it is below 0 . odelling and Simulation of Turbulent Flows with Coherent Structures Model f k St C (cid:48) L C D − C pb L r L F k − ω Table 2: Strouhal number, St , root-mean-square lift coefficient, C (cid:48) L , time-averaged dragcoefficient, C D , pressure base coefficient, C pb , recirculation length, L r , and formationlength (second peak in v v at x = 0), L F , for different physical resolutions andPANS closures. Experiments 1 taken from Parnaudeau et al. (2008), Norberg (2002),and Norberg (2003); while experiments 2 denote a data range collected from table 1 at Re = 3000 and 3900.table 1) for the observed differences between PANS simulations at f k (cid:54) .
25 and theexperiments of Parnaudeau et al. (2008).In summary, it is evident that the simulation fidelity varies substantially with im-proving resolution in the range f k > .
50. For f k (cid:54) .
50, however, the results are onlyweakly dependent on resolution, and converge towards the experiments with improvingresolution. We now examine the underlying physics to explain the convergence beyond f k equal to 0 .
50. 5.2.
Physics of Scale-Resolving Simulation Flow
Experiments (discussed in Section 2) clearly indicate that the flow comprises fourcritical stages: i) onset of the Kelvin-Helmholtz (free shear-layer) instability; ii) spatialdevelopment of the Kelvin-Helmholtz rollers; iii) breakdown to high-intensity turbulence;and iv) vortex-shedding. All these stages must be reasonably replicated in a simulationin order to achieve an adequate agreement with experiments. Furthermore, experimentalobservations indicate that this process is strongly related with the magnitude of Re .Most notably, the Kelvin-Helmholtz rollers that dictate the flow dynamics in the freeshear-layer are only observed beyond a Reynolds number of 1200. We will now examinehow the closure model and the range of resolved scales influence these flow features,and ultimately lead the simulation data to be in agreement with the experiments. Inthe remainder of this sub-section we will examine i ) how the instability is modelled atdifferent resolutions, and ii ) the manner in which spatially-developing turbulence whichis far from equilibrium is represented.It has been well known for a long time (Rayleigh 1883) that the Kelvin-Helmholtz in-stability originates and develops along the locus of the background velocity field inflectionpoint - the so-called inflection line. It is therefore of primary importance in a numerical4 F.S. Pereira, L. Eca, G. Vaz, and S.S. Girimaji (a) C (cid:48) L . (b) C D .(c) C pb . (d) L r . Figure 4: Convergence of the root-mean-square lift coefficient, C (cid:48) L , time-averaged dragcoefficient, C D , pressure base coefficient, C pb , and recirculation length, L r , with thephysical resolution for different PANS closures (data connected through splines).simulation to accurately replicate the location and development of the inflection line.In figure 7, the inflection lines computed from simulations at different resolutions arecompared against that from the experimental measurements of Parnaudeau et al. (2008).All simulations accurately capture the initial location of the inflection line. However, thesubsequent development changes as a function of the resolution f k . For the f k = 1 . .
75 cases, the inflection lines curve inward in comparison to the experimental data.On the other hand, the f k = 0 .
50 and 0 .
25 cases follow the experimental trajectory quiteaccurately. This observation holds for both k − ω and SST closures. Yet, the deviation ismore pronounced in the SST case. As a result of the distinct change in the behaviour ofsimulations at f k > .
50 and f k (cid:54) .
50, we now analyse them individually.5.2.1.
Low Resolution Simulations
The evolution of the inflection line for high f k (including RANS) cases clearly indicatesthat the closure model is inadequate in this region. As mentioned in Section 3, theRANS models, by design, are best suited for regions of fully-developed turbulence. Toexamine the state of turbulence in the inflection-line region, we present the contours ofthe time-averaged strain-rate ratio of mean and unresolved fields, (cid:104) S (cid:105) k u /(cid:15) u , in figure8. High values of this quantity (typically (cid:104) S (cid:105) k u /(cid:15) u >
8) indicate that turbulence isdominated by linear processes and the instabilities are still growing. Therefore, turbulenceconstitutive relations are elastic rather than viscous in the linear regime, leading to theoverprediction of turbulent viscosity in these regions. On the other hand, low magnitudesof (cid:104) S (cid:105) k u /(cid:15) u (between 3 and 6) denote fully-developed turbulence. Boussinesq relationstypically provide reasonable closure only in these regions. odelling and Simulation of Turbulent Flows with Coherent Structures (a) C p ( θ ). (b) (cid:104) V (cid:105) .(c) v v . (d) v v . Figure 5: Time-averaged pressure distribution on the cylinder’s surface, C p ( θ ), andstream-wise velocity magnitude, (cid:104) V (cid:105) , variance, v v , and covariance, v v , in the near-wake for different physical resolutions. Experiments taken from Norberg (2002) ( C p ( θ ))and Parnaudeau et al. (2008) ( (cid:104) V (cid:105) , v v , and v v ). Results for k − ω PANS closure.Both f k = 1 .
00 and 0 .
75 simulations exhibit large areas where (cid:104) S (cid:105) k u /(cid:15) u >
8. Thismeans that these regions where turbulence is developing and far from equilibrium arebeing represented by closures that are only appropriate for fully-developed turbulence.Consequently, figure 8 demonstrates that RANS and coarse resolution SRS tend tosignificantly overestimate the turbulent viscosity in the vicinity of the inflection line.This evidently stems from the unphysically large levels of production and turbulencekinetic energy predicted by the closure model in these regions.The first and most significant consequence of the high turbulent viscosity level isthat the breakdown to high-intensity turbulence occurs prematurely, well ahead ofthe experimental location. To illustrate this point, figure 9 presents the time-averagedvariance (or normal stress) of the stream-wise velocity field in the near-wake at x /D =0 .
0. The location of the second and highest peak marks the beginning of vortex-shedding,whereas the second the onset of instabilities (in Norberg (1998), it is compared to thelocation of the breakdown to high intensity turbulence). The data shown in these figuresexhibit a single peak for the low resolution simulations which moves upstream with theincrease of f k . This is a direct consequence of the high level of turbulent viscosity. Asa result, RANS and low resolution SRS do not distinguish between the onset of theKelvin-Helmholtz instability and the breakdown to turbulence.The overproduction of turbulence kinetic energy and turbulent viscosity causes thereduction of the effective computational Reynolds number shown in figure 10. For the twolowest resolutions tested, it is evident that the effective computational Reynolds number Re e is lower than 1200 (the lower limit to experimentally observe the Kelvin-Helmholtz6 F.S. Pereira, L. Eca, G. Vaz, and S.S. Girimaji (a) f k = 1 .
00. (b) f k = 0 . f k = 0 .
50. (d) f k = 0 . et al. (2008). Figure 6: Time-averaged stream-wise velocity magnitude, (cid:104) V (cid:105) , for different physicalresolutions and PANS closures. White line delimits recirculation region, (cid:104) V (cid:105) = 0.Experiments taken from Parnaudeau et al. (2008).rollers). In fact, figures 10a and 10b show that the start of the inward curving of theinflection lines is related with the point where the inflection lines experience Re (cid:54) (cid:104) ω (cid:105) , field in the near-wake, the temporalevolution of the resolved stream-wise velocity at P = ( x /D = 0 . x /D = 0 . f k , and that the developmentof the Kelvin-Helmholtz rollers is suppressed since neither the vorticity contours or thespectra evidence the presence of such instability. odelling and Simulation of Turbulent Flows with Coherent Structures (a) k − ω .(b) SST. Figure 7: Time-averaged inflection line of the free shear-layer, ∂ (cid:104) V (cid:105) /∂x = 0, fordifferent physical resolutions and PANS closures. Experiments taken from Parnaudeau et al. (2008). P ( x /D = 0 . x /D = 0 .
66) denotes the location of the probe used infigures 11 and 12.5.2.2.
High Resolution Simulations
Although the complex nature of turbulence along the inflection line region limits theapplication of the PANS closures, the increase of resolution reduces the model influenceon the simulations. This permits capturing the appropriate physics in the resolved scales.For this reason, simulations employing f k (cid:54) .
50 yield (cid:104) S (cid:105) k u /(cid:15) u (cid:54) (cid:104) S (cid:105) k u /(cid:15) u occurs coincides with the inflection line.The increase in the extent of the linear-physics region results in a more accurate rep-resentation of the turbulence kinetic energy production. The formation length increaseswith the reduction of f k and gets closer to the experimental measurements - figure 9. Thestream-wise normal stresses in the centreline now exhibit two peaks. This indicates thatthe aforementioned four stages process is better represented, especially for the f k = 0 . Re e contours of figure 10, the results confirm that the extension of thefree shear-layer is closely related to the magnitude Re e . Whereas for the f k = 0 .
25 casethe inflection line never experiences Re e (cid:54) f k = 0 .
50 that occurs much furtherdownstream than for the low resolution simulations. The high values of Re e guaranteethe adequate replication of the four stages of the vortex-shedding structure development.8 F.S. Pereira, L. Eca, G. Vaz, and S.S. Girimaji (a) f k = 1 .
00. (b) f k = 0 . f k = 0 .
50. (d) f k = 0 . Figure 8: Time-averaged mean-to-unresolved field strain-rate ratio, (cid:104) S (cid:105) k u /(cid:15) u for differentphysical resolutions and PANS closures. White/black lines represent the stream-wisevelocity inflection point, ∂ (cid:104) V (cid:105) /∂x = 0.Figure 9: Time-averaged stream-wise velocity variance, v v , at x /D = 0 for differentphysical resolutions. Results for k − ω PANS closure. odelling and Simulation of Turbulent Flows with Coherent Structures (a) f k = 1 .
00. (b) f k = 0 . f k = 0 .
50. (d) f k = 0 . Figure 10: Time-averaged computational Reynolds number, Re e , in the near-wake fordifferent physical resolutions and PANS closures. White/black lines represent the stream-wise velocity inflection point, ∂ (cid:104) V (cid:105) /∂x = 0, while white line delimits Re e = 1200.The consequences of all these results are evident in figure 12. At the f k = 0 .
25 case itis now possible to visualize the Kelvin-Helmholtz rollers, and evidently detect them inthe velocity field and correspondent spectrum. Note that the similarities between figures1, 12a and 12e are remarkable. For the f k = 0 .
50 case, however, this instability is lesspronounced, but still visible in the spectrum. Overall, although f k = 0 .
50 and f k = 0 . f k = 0 .
50 may besuffice for most engineering applications.5.3.
Towards Criteria for Optimal SRS Resolution
Our study demonstrates that successful SRS computations of engineering flows withspatially-developing coherent structures depend upon three key factors: i ) identificationof the physical mechanisms responsible for generating coherent structures; ii ) optimalresolution of the key features of the coherent flow field that are not amenable to modelling;and iii ) modelling only the fully-developed (stochastic) portion of the turbulence flowfield which lends itself to tractable closures. Based on the findings of the current study,we propose the following criteria for achieving optimal SRS computations to provide areasonable balance between accuracy and computational effort.The first task is to identify the region in the flow field wherein the coherent structuresare generated. In general, the origin of the coherent structures can be traced back toan underlying instability in the flow field. In the present case, the Kelvin-Helmholtzinstability is the mechanism responsible for generating coherent structures in the flow. It0 F.S. Pereira, L. Eca, G. Vaz, and S.S. Girimaji (a) (cid:104) ω (cid:105) at f k = 1 .
00 (SST). (b) (cid:104) ω (cid:105) at f k = 0 .
75 (SST).(c) (cid:104) V (cid:105) ( t ) at f k = 1 .
00 (SST). (d) (cid:104) V (cid:105) ( t ) at f k = 0 .
75 (SST).(e) E ( (cid:104) V (cid:105) ) at f k = 1 .
00. (f) E ( (cid:104) V (cid:105) ) at f k = 0 . Figure 11: Schematic of the instantaneous span-wise vorticity, (cid:104) ω (cid:105) , field, time-trace ofthe stream-wise velocity, (cid:104) V (cid:105) ( t ), field and respective frequency spectrum, E ( (cid:104) V (cid:105) ), at P ( x /D = 0 . x /D = 0 .
66) for f k = 1 .
00 and 0 .
75 and different PANS closures.is well-known that this instability occurs along the inflection-point locus of the velocityfield. Thus, we search the resolved field to identify the region wherein the secondderivative of the time-averaged stream-wise velocity vanishes - figure 7. For other typesof instabilities, the corresponding instability criteria must be used to locate the regionof coherent-structure generation.The second step is to ensure that the instability is fully manifested, leading tothe natural spatial development of coherent structures. It is evident from the resultspresented in this paper that the instability is fully established if, and only if, the effectivecomputational Reynolds number exceeds the critical Reynolds number of that instability(
Re > f k > . odelling and Simulation of Turbulent Flows with Coherent Structures (a) (cid:104) ω (cid:105) at f k = 0 .
50 (SST). (b) (cid:104) ω (cid:105) at f k = 0 .
25 (SST).(c) (cid:104) V (cid:105) ( t ) at f k = 0 .
50 (SST). (d) (cid:104) V (cid:105) ( t ) at f k = 0 .
25 (SST).(e) E ( (cid:104) V (cid:105) ) at f k = 0 .
50. (f) E ( (cid:104) V (cid:105) ) at f k = 0 . Figure 12: Schematic of the instantaneous span-wise vorticity, (cid:104) ω (cid:105) , field, time-trace ofthe stream-wise velocity, (cid:104) V (cid:105) ( t ), field and respective frequency spectrum, E ( (cid:104) V ), at P ( x /D = 0 . x /D = 0 .
66) for f k = 0 .
50 and 0 .
25 and different PANS closures.to 6 and table 2. For finer resolutions ( f k (cid:54) . Re > f k (cid:54) .
50) is not significant enoughto justify the added computational burden - figures 3 to 6 and table 2. Thus, the optimalresolution in the region of coherent structure development is such that the computationalReynolds number is marginally higher than the critical Reynolds number. It must alsobe noted that the resolution outside the coherent flow region does not appear to play asignificant role. These findings also support the paradigm that the coherent structurescan be considered as instabilities in a non-Newtonian fluid whose viscosity is given bythe effective computational viscosity.The third task is to confirm that the modelled flow field component is comprised only2
F.S. Pereira, L. Eca, G. Vaz, and S.S. Girimaji of fully-developed (stochastic) turbulence. This can be adjudged by examining the strain-rate ratio of the mean and unresolved fields ( (cid:104) S (cid:105) k u /(cid:15) u or (cid:104) S (cid:105) /ω u ) throughout the flowfield. The SRS can be considered physically reasonable if, and only if, the strain-rate ratiois below 6 which is an indication that the modelled field is fully-turbulent or stochastic.If the computed ratio exceeds a value of about 6, that indicates that the turbulence inthat region is experiencing linear effects and the closure model is likely to be inadequate.In low-resolution simulations ( f k > . (cid:104) S (cid:105) k u /(cid:15) u >
8) over significant regions of coherent structure development - figure8. In these cases, the SRS computation attempts to model the coherent flow leadingto a poor agreement with the experimental observations. In fine resolution simulations( f k (cid:54) .
6. Conclusions
We investigate the effectiveness of the Scale-Resolving Simulation approach in com-puting flows with spatially-developing coherent structures. SRS methods of practicalinterest seek to resolve only the key features of coherent structures and represent theresidual stochastic turbulent field with closure models. The objective of this study is toidentify the closure modelling/simulation challenges and develop criteria for a reasonableresolution that provides an optimal balance between accuracy and computational effort.The selected canonical case is the flow around a circular cylinder at Re = 3900 whichcomprises several complex flow features. The SRS approach employed is the PANSmethod of Girimaji (2005).SRS computations are performed at four levels of physical resolution (cut-off lengthscale) designated according to the fraction of unresolved turbulence kinetic energy f k = 1 .
00, 0 .
75, 0 .
50 and 0 .
25. It is important to note that f k = 1 .
00 correspondsto a RANS (Reynolds-Averaged Navier-Stokes) computation and decreasing f k impliesincreasing resolution. It is exhibited that the quality of the results improves with increas-ing resolution (decreasing f k ) but begins to approach the experimental observation onlybeyond f k (cid:54) .
50. It is shown that a good agreement with the experimental measurementscan be achieved only if all the stages of the coherent structure development are adequatelyresolved. Four key stages are identified: Kelvin-Helmholtz instability onset in the shear-layer behind the cylinder; gradual spatial development of the Kelvin-Helmholtz rollers;breakdown to high intensity turbulence; and vortex-shedding.The behaviour of the SRS flow fields at different resolutions is examined to proposegeneral guidelines for optimal SRS of flows with coherent structures: i ) The critical instability region of coherent structure generation and developmentmust be first identified. Here, the resolved field is investigated to locate the region whereinthe second derivative of the time-averaged stream-wise velocity vanishes. This region isdesignated as the “critical” zone in which the resolution must be sufficient in order toreasonably replicate the flow structures; ii ) The effective computational Reynolds number in the coherence region must exceedthe critical Reynolds number needed for the onset of the responsible instability. In thepresent case, the responsible mechanism is the Kelvin-Helmholtz instability and thecritical Reynolds number is 1200. It is shown that coarser resolutions cause a truncateddevelopment of the instability leading to a poor agreement of the flow statistics withthe experiments. On the other hand, finer resolutions with larger effective computationalReynolds number ( f k (cid:54) . Re > odelling and Simulation of Turbulent Flows with Coherent Structures iii ) Finally, it must be confirmed using the strain-rate ratio of mean and unresolvedfields analysis that the modelled field is comprised mostly of fully-developed (stochastic)turbulence for acceptable accuracy. Here, it is demonstrated that for coarse resolutioncases ( f k > . Re <
Acknowledgements
The authors would like to express their sincere gratitude to J. Carlier, R. Govardhan,E. Lamballais, C. Norberg, S. Rajagopalan, and C.H.K. Williamson for providing theexperimental measurements used in this study. The authors also gratefully acknowledgeJ. de Wilde for the interesting discussions, as well as the Maritime Research Institute ofNetherlands (MARIN) for providing the necessary HPC resources.
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