Guidelines for implementing Detached-Eddy ... - Science Direct

Aerospace Science and Technology 11 (2007) 376–385 www.elsevier.com/locate/aescte

Guidelines for implementing Detached-Eddy Simulation using different models ✩ Anwendung unterschiedlicher Modelle bei der Detached-Eddy Simulation Ulf Bunge a,∗ , Charles Mockett b , Frank Thiele b a IVM Automotive Wolfsburg GmbH, Wolfsburger Landstraße 22, D-38442 Wolfsburg, Germany b Technische Universität Berlin, Sekr. HF 1, Straße des 17 Juni 135, D-10623 Berlin, Germany

Received 7 November 2005; received in revised form 1 February 2007; accepted 1 February 2007 Available online 13 February 2007

Abstract This paper presents a range of guidelines that the authors have come to formulate during research work related to Detached-Eddy Simulation (DES). In the course of this work, DES has been implemented, calibrated and tested for three different linear and non-linear eddy viscosity turbulence models (EVM) of varying degrees of complexity, thorough literature researches were conducted and the authors participated in two European research projects involving DES. The key steps along the path to implementation of DES in existing Reynolds-Averaged Navier–Stokes (RANS) solvers are outlined, and open questions regarding the DES technique are identified. © 2007 Elsevier Masson SAS. All rights reserved. Zusammenfassung Im Rahmen von Forschungsarbeiten zum Thema Detached-Eddy Simulation (DES) haben die Autoren eine Reihe von Richtlinien erarbeitet, die in dieser Veröffentlichung dargestellt werden. In diesem Zusammenhang wurde die DES für drei unterschiedliche lineare und nicht-lineare Wirbelviskositäts-Turbulenzmodelle von variierender Komplexität implementiert, kalibriert und getestet. Darüberhinaus sind umfangreiche Literaturrecherchen durchgeführt worden, und die Autoren haben im Rahmen zweier europäischer Forschungsvorhaben zum Thema DES mitgearbeitet. Die entscheidenden Schritte zur Implementierung der DES-Methode in einen vorhandenen Löser auf der Basis der Reynolds-gemittelten Navier– Stokes (RANS) Gleichungen werden aufgezeigt und offene Fragen im Zusammenhang der DES identifiziert. © 2007 Elsevier Masson SAS. All rights reserved. Keywords: Turbulence modelling; Hybrid RANS-LES; Detached-Eddy Simulation; FLOMANIA; DESider Schlüsselwörter: Turbulenzmodellierung; Hybride RANS-LES; Detached-Eddy Simulation; FLOMANIA; DESider

1. Introduction Hybrid approaches with the goal of combining Large-Eddy Simulation (LES) with models for the Reynolds-Averaged Navier–Stokes (RANS) equations have become increasingly popular in recent years, as they offer a reduced computational effort in comparison to LES while retaining much of the physi✩

This article was presented at the German Aerospace Congress 2005.

* Corresponding author. Tel.: +49 5362/17 301; fax: +49 5362/17 370.

E-mail addresses: [email protected] (U. Bunge), [email protected] (C. Mockett). 1270-9638/$ – see front matter © 2007 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ast.2007.02.001

cal accuracy of the method. Detached-Eddy Simulation (DES), is a prominent representative, if not the most popular of these methods, chiefly because of the simplicity of its implementation to a wide range of existing RANS models in industrial and commercial CFD codes. The emphasis of this paper is accordingly weighted towards users considering the addition of DES capability in RANS solvers, rather than the alternative of attempting to use DES for RANS wall modelling in LES codes. The basic concept of DES was published in 1997 [21] based on a formulation employing the popular Spalart–Allmaras (SA) one-equation turbulence model. This was followed by a more

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general discussion and definition in [24]. According to this, DES is defined as a three-dimensional, unsteady numerical solution using a single turbulence model, which functions as a subgrid-scale model in regions where the grid density is fine enough for LES, and as a RANS-model in regions where it is not. In order to achieve this, the length scale in the underlying turbulence model is replaced by the DES length scale: LDES = min(LRANS , CDES Δ),

(1)

where CDES is a model constant similar to that of the Smagorinsky constant in LES, the calibration of which is described later. LDES is the turbulence length scale of the RANS model to be applied in the DES and Δ is an appropriate grid scale, e.g. the maximum distance between vertex nodes defining a control volume. This takes the role of the filter width in LES mode, and as this is directly based on the local grid dimension, the filtering can be described as implicit. The aim is to achieve a RANS simulation in the vicinity of walls and LES in areas of massive flow separation outside of the boundary layer. RANS models are usually calibrated and well understood for wall-bounded flows, which underscores the basic motivation for this approach. It is also worth emphasising that the grid design for a correctly-executed DES should differ from that of LES only in the boundary layer [20], where the DES operates in RANS mode. This is the only reason why the overall requirements of spatial resolution are much lower for DES than LES. This does not apply to the grid regions where LES-mode operation is intended, i.e. those characterised by highly complex, unsteady and three-dimensional flow phenomena. Here, the grid resolution must conform to the requirements of LES (however unclear such requirements are, as discussed in [13]). The division of the computational domain into target regions for different modelling strategies is concisely summarised in [19]. The principle step in DES implementation, i.e. the replacement of the length scale (1) in the model equation, is simple; however it gives rise to potential problems in practical application. A comprehensive discussion of this is to be found in [20], whereby two possible failure mechanisms are identified: The first is modelled stress depletion (MSD), which occurs when excessive tangential grid refinement causes the LES length scale to become active inside an attached boundary layer, extreme cases of which lead to what has become known as grid-induced separation (GIS). The second is an unphysical reduction of eddy viscosity due to the activation of low Reynolds number wall-damping terms in fine grid areas far from the wall. As protection against such failures, a GIS shield and a low-Re term correction should be incorporated. The term ‘shield’ here reflects the concept that the boundary layer is shielded against the DES modification in a model. A GIS shield was first proposed by Menter et al. in [12], based on DES using the SST RANS model. In [20], generalised modifications are proposed to remedy both of the above problems, resulting in a variation on the original DES formulation known as “Delayed DES” (DDES). Analogous model-specific modifications are presented here for the CEASM-based DES, the performance of which is demonstrated to underline the importance of taking such steps for practical application.

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In addition to these robustness enhancing modifications, several further steps are essential for the validation and calibration of a reliable DES implementation, the details of which will be presented. The principle concern is to validate the LES capability of the code, and to calibrate the CDES constant for each RANS model to be used in DES. Some exemplary results will furthermore be shown to illustrate the superiority of correctly applied DES over conventional RANS simulations. 2. Method In this section, the numerical method used to compute the presented results is only briefly described. All results were obtained with the in-house code ELAN presented below. 2.1. Numerical procedure The flow is computed numerically using an in-house finitevolume based code which has been in use for around 10 years. The code solves either the unsteady Reynolds-averaged or spatially-filtered Navier–Stokes equations in case of a RANS or LES, respectively. The procedure is implicit and of second order accuracy in space and time [27]. All scalar quantities as well as the Cartesian components of tensor quantities are stored in the cell centres of arbitrarily curvilinear, semi-structured grids that can capture complex geometries and allow for local refinement. Diffusive terms are approximated with central schemes, whereas convective terms can be treated with central or upwindbiased limited schemes of higher order. A hybrid blending of both approaches for Detached-Eddy Simulation as suggested by [25] is used. The formulation of the blending function is taken directly from [25], and is designed to deliver upwind differencing (σ = 1) in areas of unidirectional flow or relatively coarse grids, whereas central differencing (σ = 0) is used for finer grids, higher vorticity and lower strain. As such, in areas where LES is possible and desirable, low-dissipative central differencing is achieved, and more stable upwind differencing is used elsewhere. The investigations of decaying, isotropic turbulence in Section 4.1 emphasise the importance of a lowdissipative convection scheme for the LES-mode of DES. Although other possible hybrid formulations are conceivable, the authors have found that of [25] to perform satisfactorily for the flows so far investigated. The linearised equations are solved sequentially and the pressure is iterated to convergence using a pressure-correction scheme of the SIMPLE type which assures mass conservation as the pressure equation is derived from the continuity equation [9]. A generalised Rhie and Chow interpolation is used to avoid an odd-even decoupling of pressure, velocity and Reynolds-stress components [14]. 2.2. Turbulence models Turbulence is handled using three RANS turbulence models of different degrees of complexity. The simplest is a modification of the SA model, the Strain-Adaptive Linear SpalartAllmaras model (SALSA) [16], which is used without the

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trip function present in the original model. The main modifications compared to the original SA model are a production term based on the shear rate together with a sensitisation of production and dissipation towards non-equilibrium flows. The second model, the Linear Local Realisable (LLR) k–ω model [17] is a local linear two-parameter model derived from realisability and non-equilibrium turbulence constraints. The coefficients of the stress-strain relation and the turbulence transport equations are all functions of the non-dimensional invariants of the mean strain and vorticity rates. The approach tries to accomplish consistent stress-strain distributions not only in plane shear flow, but also in more general flow situations. The third model, the compact explicit algebraic stress model (CEASM) [11], is based on the LL k–ε model [10] as a background model. The Reynolds stresses are computed by an expression derived by projection of the algebraic stress model into a 5-generator integrity basis. Coefficients and generators are carefully chosen to yield physically correct results for 3D flows, such as wall-bounded jets whilst preserving the simplicity of the quadratic formulation through restriction to the leading order terms. Whereas the implementation of DES for the SALSA and LLR k–ω models are straightforward, strongly resembling the already published implementations for the SA and the SST model, the CEASM-based DES is formulated in a way that deviates somewhat from the standard methods, as shall be discussed later. 2.3. DES implementations The aforementioned three models are modified to create DES models through substitution of the model length scale by the DES length scale (1) as described in the introduction. The constant CDES is calibrated by computation of the decay of isotropic turbulence (DIT) as described by [1] and [2], and in the following results chapter. For the SALSA model, the substitution of the length scale (the wall normal distance) follows the same procedure as for the standard DES based on the SA model. For the two-equation model by contrast, the turbulence length scale is a function of the model turbulence variables (the length scale for the LL k–ε model is given in Eq. (6)), and appears in more than one term of the model equations. Although the standard approach is to substitute this in the dissipation term of the k-equation as is done in the precursor SA-DES [1], a degree of freedom exists as to the term chosen for substitution, as has been demonstrated in [28]. In this study, three alternative implementations of DES based on the Wilcox k–ω model were examined, and despite varying levels of theoretical viability and strongly varying calibrated CDES values produced indiscernible results for the NACA0012 flow case. Both a wall normal distance and a locally-determined length scale are present in the background model of the third model, the CEASM. This feature offers certain practical advantages concerning the avoidance of the problems outlined in the introduction. As the DES implementation differs from the standard method, a description is given here. Due to the fact that the non-linear part of the model is not modified by the DES implementation, attention can be confined to the background model

of the CEASM, the LL k–ε model [10]. For a complete summary of the equations, tensor representations and constants, the reader is referred to either [11] or [1]. The transport equations for the LL k–ε model are:    μt ∂k ∂ ∂ρk ∂(ρkui ) μ+ , (2) + = ρP − ρε + ∂t ∂xi ∂xi Prk ∂xi ∂ρε ∂(ρεui ) + ∂t ∂xi    μt ∂ε ε  ε2 ε) + ∂ μ+ . (3) = ρ (Cε1 P − C k ∂xi Prε ∂xi In the above equations k denotes the turbulent kinetic energy, ε the turbulent dissipation, ui a component of the velocity vector in the i-direction. ρ represents density, P production and μ the molecular viscosity. The eddy viscosity is denoted by μt , and the Prandtl number by Pr. Unsteady and convective terms on the left hand side of the equations are balanced by production, dissipation, molecular and turbulent diffusion on the right hand side. Coefficients necessary for the discussion are:   ε2 = Cε2 (1 − 0.3e−Re2t ), ε1 = Cε1 1 + P , C C P √ 3/2  kLn Cε2 k k2 2 P = e−Aμ Rek , Ret = , Rek = , Cε1 Lε νε ν 3/4 Lε = κcμ Ln (1 − e−0.263Rek ). (4) Aμ = 0.00222, In the above Eqs. (4), the constants Cε1 = 1.44, Cε2 = 1.92 and cμ = 0.09 are those of the ‘standard’ k–ε model [8]. The LL k–ε model [10] is based on the idea of two-layer models [15] so as to conform with turbulent length-scale constraints inherent to Wolfshtein’s one-equation model [26]. This basic concept presents certain advantages for the application in a DES formulation, as will become apparent. The DES modification is performed by the following lengthscale replacement in the dissipation term of (2), which includes the GIS-shield function:    1 1  3/2 2 1 − tanh(AL−ν ) ; (5) ρε → ρk max LRANS CDES Δ with the RANS-model length-scale k 3/2 ε and GIS-shield function   3/2 k νk ; 2 . AL–ν = max 2 εLn εLn LRANS =

(6)

(7)

In the original RANS model, the two constants Cε1 and Cε2 are manipulated depending on the wall-normal distance Ln , this non-local quantity being introduced as an additional lengthscale similar to that of the SA model. These manipulations ensure that far away from a wall, the standard k–ε model is used: ε1 = Cε1 , lim C

Ln →∞

ε2 = Cε2 . lim C

Ln →∞

(8)

This is of course the area in which the DES modification is intended to become active, and the behaviour of (7) should not be

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disrupted by this. To ensure that this is the case, the wall-normal distance length scale is not replaced by the DES length scale (1) as is the case for the SA-DES [21]. An important advantage of this simple decision is that an undesired activation of the low-Re near-wall terms in areas of fine grid and low velocity far from the wall cannot occur, thereby addressing the second problem outlined in the introduction. This problem can occur with SAbased DES, and a complex function designed to prevent this was presented in [20] as a necessary robustness modification. Such considerations are therefore rendered unnecessary for the presented CEASM-DES implementation. In addition to this, the wall-normal distance, or any other quantity related to it, can than be used to formulate a GIS shield to protect the RANS-mode boundary layer from an encroachment of the LES-mode. The method chosen for the CEASMDES uses exclusively model variables already present in the two-layer background turbulence model, as shown in Eqs. (5) and (7). In this modification the ratio of the turbulence length scale to the wall-normal distance is used together with the ratio of molecular to eddy viscosity to ensure the presence of RANSmode inside the entire attached boundary layer. This method is based on that of Menter for the SST-DES [12], and constitutes only one of a multitude of possible methods to achieve a GIS shield. The functionality and necessity of the GIS shield is demonstrated for the simplified case of a two-dimensional developing flat plate boundary layer. Grids with varying stream-wise resolution were computed using CEASM-RANS as well as CEASM-DES with and without the shield function active. For grids with very coarse stream-wise resolution, the results of all methods coincide, as the LES length scale does not become active inside the boundary layer. For the fine stream-wise resolution however, the standard DES results show a strong degradation due to the occurrence of MSD, as shown in Fig. 1. The DES with GIS shield on the other hand agree very well with the RANS results, proving that the modification has effectively shielded the boundary layer from the encroachment of LES mode. The shield function is seen to be active (equal to 1) from the buffer layer, throughout the logarithmic layer to the edge of the boundary layer at around y + = 1000.

Fig. 1. Velocity profile and GIS shield for flat plate boundary layer for RANS and DES-like simulations.

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Other previous formulations based on the terms shown in Eq. (4) have proven to be more complex and less reliable. Therefore, former suggestions as presented by Bunge et al. [3] are seen as unfeasible. 3. Results The first part of this chapter is dedicated to the calibration computation procedure for DES models, whereby the results for the CEASM-DES variant are shown and discussed. Constant values calibrated as such form the basis for all further results obtained by the authors. Moreover, this calibration process is considered to be an essential prerequisite for successful DES computations, as it offers a validation of the capability of the modified RANS model and underlying numerics to partially resolve turbulence. 3.1. Decay of isotropic turbulence (DIT) Three different grid levels are used for these computations consisting of 163 , 323 and 643 rectangular and equidistant control volumes. The physical domain represents a rectangular box of edge length 2π for which energy spectra versus wavenumber for the velocity field can be evaluated. All results are presented for a non-dimensional time of t = 2.0 and are compared to the benchmark experiment of Commte-Bellot and Corrsin [4]. A number of alternative methods for calibration using DIT have recently come to the authors’ attention, including the use of DNS simulations of DIT, such as those collated in the AGARD LES validation database [7]. Although offering some potential advantages for a more detailed analysis (velocity field can be directly imported, a higher Reynolds number), no difference in the calibrated constant value was found to emerge in a direct comparison. It should also be clearly pointed out that these computations differ significantly from the way this case would be computed for calibration of a standard RANS model. This is usually conducted in order to obtain compatible constants [8], such that the exponential decay of turbulent quantities can be computed correctly. However, the flow field for such computations is Reynolds-averaged, and therefore constant in space and time. The computations for DES calibration partially resolve the three-dimensional and unsteady turbulent fluctuations in the velocity field dependent on the spatial resolution and therefore constitute Large-Eddy Simulations. DIT is therefore also a validation of the pure LES-mode of DES. As such, all RANS-mode operation of the DES must be disabled by setting the DES length scale (1) directly equal to CDES Δ. To demonstrate the importance of the DES modification, Fig. 2 shows the resulting turbulence kinetic energy spectra for a resolved DIT simulation using the unmodified RANS equations. It is clear that an excessive dissipation of the small scales takes place. By contrast, Fig. 3 shows the behaviour of the Smagorinsky LES model, which clearly successfully captures the correct spectral character. These computations were conducted on each grid level for different values of the model constant, CS , which

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Fig. 2. RANS results of a DIT computation using the unmodified model (CEASM).

Fig. 3. LES results for DIT on different grids using varying values for Smagorinsky constant CS .

determines the proportionality between the subgrid eddy viscosity νt and the strain Sij in the resolved velocity field according to: (9) νt = (CS Δ)2 2Sij Sij . It is evident that the Smagorinsky constant value exhibits a strong influence on the modelled turbulent diffusion resulting in different levels of damping of the smaller scales apparent in the spectra. Furthermore, the optimal value of CS is highly sensitive to the grid resolution, decreasing with increasing resolution. This can be explained by the increasing proportion of resolved turbulence, since in the limit of direct numerical simulation (DNS) the modelled part should vanish even with a finite Δ. Therefore, the assumption that this factor of proportionality is a constant is wrong. Despite this, the simplicity of the Smagorinsky model allows a simple test of the code’s basic LES capability, as an optional precursor to DES tests and calibration. Before the calibration of DES using DIT can be executed, one additional problem must be addressed concerning the initialisation of the subgrid eddy viscosity field. For the Smagorinsky model, this is no problem as the eddy viscosity can be explicitly determined from the velocity field (9). However, this becomes more complicated for DES models, as they generally

Fig. 4. LES results for DIT on medium grid with varying flux-blending parameters.

have a time-dependent term and are not explicit. One approach to resolve this problem is to make use of the Smagorinsky model to obtain an initial eddy viscosity field which is then transferred to the DES model in the first time step. However, it has been shown that the result even at advanced time steps shows a strong dependence on the initial choice of CS , which therefore becomes an unwanted additional parameter of influence. This can be eliminated using an alternative technique to obtain the initial eddy viscosity from the DES model itself. The technique involves the steady calculation of the “frozen” initial velocity field using the chosen DES model, solving only the turbulence model equations (i.e. with the momentum and pressure calculations disabled). The ease of convergence on the frozen field, and the role played by any numerical parameters also delivers important information as to the numerical robustness of the DES SGS model. Direct comparison of both techniques has shown that the resulting calibrated value is not affected, as long as a “sensible” value of the Smagorinsky constant has been chosen. This can be obtained from a precursor calculation with the Smagorinsky model as shown in Fig. 3. The relatively high value of CS = 0.20 was chosen to reflect the fact that with the curvilinear grids used for practical applications, additional numerical dissipation is present. A related factor that must be taken into account is the numerical diffusion inherent in any simple numerical approximation. The effect of this is shown in Fig. 4 by employing different flux blending factors and a constant modelling constant. The numerical diffusion is low for 100% central differencing (CDS) and increases when higher (although small) percentages of upwinding (UDS) are used. The hybrid nature of the eddy viscosity modelling places conflicting demands on the numerical convection scheme, whereby diffusive upwind schemes are preferable for stability considerations in the RANS-mode and coarse grid zones, and low-diffusive central schemes are desired in the LES-mode regions. This is therefore addressed by a hybrid numerical blending scheme, as has been outlined in the method, Section 3.1. As the hybrid scheme produces a very high proportion of central differencing, a high flux blending parameter is used for the calibration. Additionally the hybrid scheme it-

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Fig. 5. DES based on CEASM with varying CDES parameter on all grids.

Fig. 6. DIT based on calibrated CEASM-DES with hybrid numerical scheme for convective terms.

self has been tested in comparative DIT calculations (presented later). The important point is that it is essential that the model be calibrated against the same numerical framework to be used in practical applications. DES calibration computations based on the CEASM model are given in Fig. 5, in which a similar influence of the CDES value to that seen with CS for the Smagorinsky subgrid-scale model is apparent. One key difference, with important implications for practical applications, is that the level of grid dependency is much lower for the DES, the high wavenumber ends of the curves for the different grids almost overlapping. This is seen with all DES models, and would seem to suggest that contrary to the simple Smagorinsky model, some kind of dynamic character is inherent in DES SGS models. This point is the subject of ongoing research. Owing to the fairly high number of influence parameters, as well as the high level of abstraction between isotropic turbulence and practical turbulent shear flows, it would be overzealous to state the calibrated values to a high degree of accuracy. Consequently, a range of optimal values for CDES are preferably given which is found to lie within 0.6 < CDES < 0.7 for the CEASM-DES implemented in the ELAN CFD code. To demonstrate that the calibrated CDES values remain valid in the practical numerical framework, Fig. 6 shows DIT spectra obtained using the hybrid convection scheme mentioned ear-

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Fig. 7. Experimental data for NACA0012 and NACA0021 profile, summarised from [18,22,23].

Fig. 8. Effect of spanwise domain extent on DES results, NACA0012, 45◦ angle of attack [5].

lier. These compare with a flux blending parameter of between 95% and 100% CDS, showing that the hybrid blending function selects the correct low-diffusive convection scheme. As a further test, the computation is conducted with and without enforcement of LES-mode in the DES length scale. The resulting spectra coincide precisely, showing that the model’s natural state is indeed LES mode for the DIT case. 3.2. Massively separated wake flow A popular and suitable test case for DES is the flow around a symmetric NACA0012 profile at high angles of attack and Re = 100,000 as presented in [22]. Here, an angle of attack of α = 60◦ is chosen, which is far beyond the maximum lift angle, cf. Fig. 7. The experimental data for this case (as used in [22]) are limited to time-averaged force coefficients. More recently, experimental data for the thicker NACA0021 airfoil at Re = 270,000 have become available, which also include timeresolved force data [23]. To make use of the deeper level of comparison this offers, additional calculations have also been conducted for this case. The wake of the airfoil at such a high angle of attack cannot be considered two-dimensional, therefore both the DES and (U)RANS computations are all performed on three-dimensional grids with one chord length in the spanwise direction. In a re-

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Table 1 Grids used for NACA0012 test case, O and H-topologies Grid level

NCircum .

NNormal

NZ

NXY Z

coarse (c) medium (m) fine (f) vfine-o (vf-o) vfine-h (vf-h)

141 141 141 282 n/a

61 81 81 162 n/a

25 25 30 60 48

215 025 285 552 342 630 2 741 040 1 463 000

cent related study by Guenot [5], DES of a NACA0012 airfoil at 45◦ angle of attack was computed for a wide variation of different spanwise domain extents. The results of this are shown in Fig. 8, which show that with the considered one chord length, a quantitative comparison with the experimental data is not valid. The most that can be said in relation to the experimental averaged force coefficients, is that they should be over-predicted by the simulation setup using a confined spanwise extent. This trend has been reproduced by computations for the NACA0021 airfoil, presented later. Therefore, the averaged force coefficients are only compared between computations. For the NACA0012 profile, five different grids are used in order to determine the level of grid dependence of the solution. The coarsest three grids were provided in the framework of the FLOMANIA research project [6] as used in [22], and are of O-topology. Because the difference in refinement between these is not very great, two further highly-refined grids were constructed based on different topologies. The first corresponds to a factor two refinement of the fine (f) grid in every direction using an identical topology and point distribution, and is referred to as “vfine-o” (vf-o). In an attempt to focus even more points in the “focus region” [19] directly behind the rear surface of the airfoil, a second highly-refined grid was constructed using a more complex combination of different C and H grid topological blocks. This grid is referred to as “vfine-h” (vf-h). Due to the different topology, the circumferential and normal point numbers are not relevant in comparison with the O-grids. The newly constructed grids are compared in Fig. 9. The results for all NACA0012 calculations are summarised in the chart of Fig. 10, where the grids are arranged in order of increasing focus region refinement. Demonstration of the correct implementation is given by comparison of the force values for DES and URANS with those of previously published calculations on the same grids taken from references [6,13,22]. A clear difference between URANS and DES values can be seen, as well as the lower level of model dependency of the DES. This is to be expected, as the RANS modelling only plays a dominant role in the boundary layer of the airfoil, which is of minor impact compared to the turbulent wake at this angle of attack. The DES data for the first three grids also shows much lower levels of grid dependency, the weak variations apparent between the coarsest three grids falling within the level of statistical error expected. This is because the difference in the refinement between the first three grids is very small. The nature of the focus region refinement effect on the forces is therefore first clearly demonstrated in the steps from f to vf-h, for which a significant increase in the force coefficients is evident. The

Fig. 9. Highly-refined grid topologies for the NACA0012 airfoil.

Fig. 10. Collated time-averaged force coefficient data for all grids and models.

computational expense associated with the highly-refined grids has unfortunately excluded calculation with all models, however the good agreement between the results for LLR-DES and SALSA-DES for vf-h suggest that this trend is model independent, as would be expected. It has therefore been clearly observed that a strong level of dependence on the LES region refinement exists in DES. This remains despite the finding that all grids except the coarsest have a focus region resolution satisfying existing LES grid criteria, as noted in [13]. Unfortunately as discussed, no concrete comparison with the experiment is possible, so no statement can be made as to the accuracy of these grid predictions. This level of dependence of the result on the grid refinement is clearly a disadvantage of the method, and forms the subject of ongoing research. On the positive side, the much weaker model dependence of DES for a given grid represents a clear advantage over URANS. Further advantages of the DES approach over URANS can be demonstrated when we turn to the NACA0021 case, for which more detailed time-resolved experimental data exists. Computations were conducted on an O-topology grid similar in form and fineness to the “fine” NACA0012 grid, with a total of 476,000 points, as well as a variant with an extended spanwise domain of 3.24 chord lengths (1.54 million points), in order

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Fig. 13. Visualisation of LES-RANS interface for SA-DES (left) and CEASM-DES (right).

Fig. 11. Lift coefficient vs. time for NACA0021 profile at Re = 270,000, angle of attack α = 60◦ : experiment [23] (upper), CEASM-DES (middle), CEASM-URANS (lower).

Fig. 12. Spectral content of lift and drag signals from experiment [23] (dots), DES with 1c spanwise domain (dashed lines) and 3.24c spanwise domain (solid lines).

to account for the effect seen by Guenot [5]. Fig. 11 shows a comparison of the lift coefficient signal of the experimental data with DES and URANS calculations based on the CEASM model for the NACA0021 profile (1c in spanwise direction). The URANS produces a highly periodic signal, which contrasts strongly with the more stochastic DES signal. The high qualitative agreement in the nature of the DES and experimental signals serves as strong evidence that a more accurate representation of the wake flow physics is delivered by the DES. This is confirmed by analysis of the frequency content of the signals; Fig. 12 shows a comparison of the lift and drag spectra from the experiment and DES. With the spanwise domain of 1c, a clear shift in the Strouhal shedding frequency is seen. The agreement is much better with the results of an equivalent cal-

culation using a spanwise domain of 3.24c. When the Strouhal number, as defined in [23] using the frontal height of the airfoil, obtained from the URANS and DES are compared, the experimental value of 0.17 is heavily under-predicted by all URANS calculations (SALSA, LLR and CEASM), with values varying between 0.10–0.13, whereas the DES calculations show much better agreement with values between 0.16–0.17. An outcome of the higher level of stochastic content in the signal is that much longer time series must be calculated in order to obtain an adequate level of statistical convergence. However, one would hesitate to state this as a disadvantage of DES, as the highly stochastic signal with strong long period modulations represents the true nature of the flow in question. It does however measure as an issue when considering the practical aspects of DES. The promising level of physical accuracy of DES for this flow case has motivated the use of DES to examine physical aspects of the vortex shedding process. An outline of this ongoing work is introduced in [13], whereby some light is shed on the processes giving rise to the phenomenon of weak shedding cycles (the non-periodic temporary collapse of shedding visible in the experimental and DES time traces of Fig. 11). A further interesting aspect that has been revealed by these computations employing two-equation models in the context of DES is the different concept of the interface between the RANS and LES-mode regions. For the ‘standard’ DES based on the SA one-equation model, this interface is directly set and controlled by the grid fineness, as the turbulence length-scale in the model is the wall normal distance. For the other two models applied, the length scale is defined by the local turbulence model variables (e.g. Eq. (6)). Thus, the length scale, and consequently, the interface location, is part of the solution and a function of space and time, as shown in Fig. 13. It is important at this stage to point out that the idea of an interface involving any discontinuities is a misconception for DES in general, as only one model or eddy viscosity is used in all regions. The equations remain the same, the sole modification is a limitation of the dissipation or destruction mechanism in the applied transport equations and the interface is simply found where this limitation is naturally

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triggered on or off; there are no additional conditions or constraints applied on or at the interface. 4. Conclusions The results and practical observations derived from the investigations and considerations summarised here clearly demonstrate that a first and vital step towards the implementation of DES in an existing RANS solver is to test the ability of the numerical framework to perform a LES. The minimum requirement is a computation of the decay of isotropic turbulence with a Smagorinsky subgrid-scale model; however, this case also represents the most reliable method for calibrating the constant CDES which must afterwards remain untouched. Another important ingredient is a hybrid treatment of the convection scheme, which in the view of the authors is an essential modification to the background numerical framework of DES. The scheme proposed by [25] has so far proven successful in this respect. The correct functionality of the GIS shield modification introduced in the CEASM DES implementation has been demonstrated. Although this case represents an “artificially enforced” occurrence of the MSD problem, it must be emphasised that practical occurrence of MSD cannot be purely attributed to poor grid design. In many cases, geometric details may require such high tangential grid resolutions, and particularly in internal flows, boundary layers can grow to considerable thicknesses. The introduction of such a shield function is therefore also regarded as very important to the practical application of DES. The DES method has been shown to yield much superior results over conventional (U)RANS approaches for massively separated flows. Furthermore, for the case presented where the boundary layer prediction plays a subordinate role in the flow physics, this improvement is found regardless of the turbulence model used. The superiority of advanced RANS modelling in a suitable DES case where complex boundary layer physics plays a key role has yet to be demonstrated. At this stage then, the selection of RANS model is dominated rather by practical implementation issues. For example, it has been demonstrated that the CEASM model requires less “intervention” to achieve a robust and reliable procedure. That this model also involves the most advanced RANS modelling would suggest an advantage should such a complex case described above be tested. A general guideline is therefore that two-equation models are preferable, especially when they are formulated on a twolayer concept. This is true for the SST Model [12] as well as the LL k–ε background model of the CEASM [11]. As discussed, this recommendation is not yet based upon improved results, rather upon the enhanced ease by which a GIS shield to tackle the problem of “modelled stress depletion” (MSD) [20] can be constructed, as well as the lack of necessity of modification of the low Reynolds number terms. Another important outcome of the NACA0012 grid refinement study, is that a relatively strong dependency upon the fineness of the LES-mode region exists in DES. This somewhat contradicts the finding from the DIT study, which show

a much lower level of grid sensitivity than the LES Smagorinsky subgrid-scale model. However, it must be appreciated that DIT represents a significant abstraction from practical flow cases, both in terms of the simplified turbulence physics as well as in terms of the orthogonal uniform grid. Future studies in the field of DES should address the grid sensitivity issue, which has important practical implications. An approach to this may include more precise examination of the LES-mode of the model, incorporating more complex pure-LES cases than DIT. Another approach may be to look at alternatives to implicit filtering, which perhaps contribute to the lack of grid convergence so far observed. Regarding complex industrial applications, a case requiring complex RANS-mode modelling should be investigated, in order to determine the level of advantages offered by more complex turbulence models to be used in a DES framework. Acknowledgements Some of the more time consuming computations for the NACA0012 time series were conducted on the IBM pSeries 690 at the Zuse-Institut Berlin (ZIB) and the Norddeutschen Verbund für Hoch- und Höchstleistungsrechnen (HLRN). The authors acknowledge the partial funding of the work presented here by the European Community during the FLOMANIA and DESider projects. The FLOMANIA project (Flow Physics Modelling – An Integrated Approach) is a collaboration between Alenia, AEA, Bombardier, Dassault, EADS-CASA, EADS-M, EDF, NUMECA, DLR, FOI, IMFT, ONERA, Chalmers University, Imperial College, TU Berlin, UMIST and St. Petersburg State University. The project is funded by the European Union and administrated by the CEC, Research Directorate-General, Growth Program, under Contract No. G4RD-CT2001-00613. The DESider project (Detached Eddy Simulation for Industrial Aerodynamics) is a collaboration between Alenia, ANSYS-CFX, Chalmers University, CNRS-Lille, Dassault, DLR, EADS-M, EUROCOPTER Germany, EDF, FOI-FFA, IMFT, Imperial College London, NLR, NTS, NUMECA, ONERA, TU Berlin, and UMIST. The project is funded by the European Community represented by the CEC, Research Directorate-General, in the 6th Framework Program, under Contract No. AST3-CT-2003-502842. Particular thanks also go to K. Swalwell of Monash University for the very informative discussions and the very helpful provision of additional experimental data not included in the original publication. References [1] U. Bunge, Numerische Simulation turbulenter Strömungen im Kontext der Wechselwirkung zwischen Fluid und Struktur, Dissertation, Technische Universität Berlin, 2004. [2] U. Bunge, C. Mockett, F. Thiele, Calibration of different models in the context of Detached-Eddy Simulation, AG STAB Mitteilungen, DGLR, Göttingen, 2003. [3] U. Bunge, C. Mockett, F. Thiele, New background models for detachedEddy simulation, in: Proceedings of the Deutscher Luft- und Raumfahrtkongress, DGLR-JT2005-212, Friedrichs-hafen, 2005.

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