Computers & Fluids 48 (2011) 1–15
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Computers & Fluids j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p fl u i d
Zonal Detached Eddy Simulation of a spatially developing ﬂat plate turbulent boundary layer Sébastien Deck ⇑, Pierre-Élie Weiss, Mathieu Pamiès, Eric Garnier ONERA, Applied Aerodynamics Department, 8, rue des Vertugadins, F-92190 Meudon, France
a r t i c l e
i n f o
Article history: Received 22 September 2010 Received in revised form 3 February 2011 Accepted 8 March 2011 Available online 23 March 2011 Keywords: DES Turbulent boundary layer RANS/LES
a b s t r a c t The present work presents a Zonal Detached Eddy Simulation (ZDES) to simulate a spatially developing turbulent boundary layer over a smooth ﬂat plate at Reh = 2900. Results are compared with the experimental data of De Graaff and Eaton . A synthetic reconstruction method for the pseudo-viscosity ﬁeld m~ is proposed in the frame of a synthetic eddy method. First, it is shown that ZDES amounts to LES with a plausible one-equation subgrid scale model and wall modeling. More precisely, both the mean and second-order ﬁeld are well predicted compared with the experiment and a reference LES with the mixedscale-model. The separate effect of the streamwise (respectively spanwise) resolution on skin friction and turbulence is then evaluated. A measure of the global error which is based on the error on the friction and on the turbulent shear stresses has been deﬁned. It is observed that without ﬁxing the height of the RANS-LES interface, the error does not vary monotonicaly with the resolution. Conversely, ﬁxing the interface height to 50 or 100 wall unit brings both an intuitive reduction of the error with the resolution and a global reduction of the error level with respect to the aforementioned case. Furthermore, it is outlined in this study of spatially developing boundary layer that the potential computational effort reduction brought by RANS-LES approaches depends not only on the grid resolution but also on the establishment distance of the solution. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Computational power has dramatically increased over the last decades with unsteady 100 million points grid simulations being now conducted with increasing regularity. In a recent discussion on the status of advanced modeling for aerodynamics, Sagaut and Deck  emphasized that the next foreseen challenges in applied numerical aerodynamics will be ﬁrstly the capture of the boundary-layer dynamics including transition and pressure-gradientdriven separation issues and secondly the capability to handle accurately geometrically complex conﬁgurations with validated unsteady tools. The ability of the Large Eddy Simulation (LES) technique to correctly simulate the spatio-temporal dynamics of turbulent ﬂows is now well acknowledged. This technique relies on a decomposition of the aerodynamic ﬁeld between the large scales (responsible for turbulence production) and the small scales of the ﬂow, the former being directly resolved while the effect of the latter is taken into account through the use of a model. The primary obstacle to practical use of LES on industrial ﬂows which involve wall boundary-layers at high Reynolds number remains computational resources. Indeed, LES aims at capturing the scales of motion accountable for turbulence production which imposes severe ⇑ Corresponding author. E-mail address: [email protected]
(S. Deck). 0045-7930/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compﬂuid.2011.03.009
demands on the grid resolution near solid-walls. As an example, the grid extension sizes Dx+ 50, Dy+ 1 and Dz+ 15 are those classically retained in LES to capture the near wall structures. Hybrid RANS/LES was invented to alleviate this resolution constraint in the near-wall regions. For instance, Detached Eddy Simulation (DES97), proposed by Spalart et al. , is well understood in thin boundary layers where the model acts in RANS mode and in massive separation where the model degenerates toward a subgrid scale model. Nevertheless, and from the beginning, a special concern was devoted to the region, named ‘‘gray-area’’, where the model switches from RANS to LES, and where the velocity ﬂuctuations, the ‘‘LES-content’’, are expected to be not sufﬁciently developed to compensate for the loss of modeled turbulent stresses. This can lead to unphysical outcomes, like an underestimation of the skin friction [4,5] which, at worst, can lead to artiﬁcial separation denoted as ‘‘Grid-Induced-Separation’’ (GIS). In order to get rid of this drawback, Spalart et al.  proposed a modiﬁcation of the model length scale presented as a Delayed Detached Eddy Simulation (DDES) to delay the switch into the LES mode to prevent ‘‘model-stress depletion’’ (MSD). In a different spirit, Deck [7,8] proposed a Zonal Detached Eddy Simulation (ZDES) approach, in which RANS and DES domains are selected individually. These methods can be considered as weak RANS-LES coupling methods since there is no mechanism to transfer the modeled turbulence energy into resolved turbulence energy. In practice, the eddy vis-
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cosity remains continuous across the RANS/LES interface but the rapid decrease of the level of RANS eddy viscosity enables the development of strong instabilities. This family of techniques is well adapted to simulate massively separated ﬂows characterized by a large scale unsteadiness dominating the time-averaged solution (see [9,10] for further discussion). However, these methods may not be adequate in situations where the ﬂow is sensitive to the history of the upstream turbulence like a shallow separation bubble on a smooth surface induced by a moderate adverse pressure gradient. As a consequence, the simulation has to be fed with turbulent ﬂuctuations to match the low-order statistics given for example by a RANS calculation. In this framework, the hybrid RANS/LES model can be considered as a LES where the RANS model plays the role of a wall-layer model. Piomelli et al.  studied more deeply this intermediate blending region within the DES approach. They performed LES of a plane channel ﬂow at high Reynolds number by varying the location and extent of this blending layer. Their study shows that the ‘‘DES buffer layer’’ is characterized by very long eddies, with unphysical long time-scales. Improvements were obtained by reducing the value of CDES to bring the outer-ﬂow eddies closer to the wall. They also used a stochastic forcing by adding Gaussian noise in the transition region but an ad hoc tuning of the magnitude was required as the grid was changed. More recently, Keating and Piomelli  used a dynamic method to calculate the magnitude of this stochastic forcing to introduce small ﬂuctuations in the transition region with the object to generate resolved ﬂuctuations. Another approach based on additional ﬁltering near the RANS/LES interface has been proposed by Hamba  to eliminate the loglayer mismatch. Templeton et al.  also suggest that RANS and LES modes need to be coupled via boundary conditions. Both latter methods were again assessed on plane channel ﬂow conﬁgurations. In addition, Shur et al.  improved the DDES model to permit wall modeling in LES (WMLES). These authors evaluated their model in a plane channel ﬂow and in a plane hydrofoil with trailing edge separation. In this latter calculation, inﬂow turbulent content was generated from another precursor ﬂat plate calculation which itself uses a recycling procedure. Similar inﬂow techniques have been used by Xiao et al.  and Choi et al.  in the frame of hybrid RANS/LES calculations based on two-equation models. It is worth noting that the generation of unsteady turbulent inﬂow quantities (like eddy viscosity) is rarely addressed in the literature. Technically however, two major difﬁculties relative to recycling methods remain its initialization and its adaptation to relevant geometries. As noted by Spalart , the generation of resolved turbulence in attached boundary layers needs to become routine and efﬁcient. Besides, the majority of previous studies concerned the fully developed channel ﬂows instead of a spatially developing boundary layer over a smooth ﬂat plate due to the important computational resources required to simulate the latter case. Conversely to channel ﬂow conﬁguration, the unsteady character of the ﬂow ﬁeld is not imposed by a forcing term in the movement equations associated to periodic boundary conditions. In addition, Marusic et al.  reviewed the signiﬁcant differences between channel and boundary layer ﬂows even in the inner region. It thus appears to our view more challenging. Besides, the wider scope of practical applications in various domains of aeronautics has also motivated our choice. Indeed, we consider that the capability to simulate a spatially developing boundary layer is a mandatory milestone in the assessment of an hybrid RANS/LES method intended to simulate complex ﬂows of practical interest. In contrast to the high quality DNS datasets ([20–22]), there exist very few publications where the hybrid RANS/LES results (even in pure LES studies) are compared with experimental data (see ) in the case of a spatially developing boundary layer. Note that in their recent paper, Wu
and Moin  emphasized that the smooth ﬂat plate boundary layer is still an important problem in ﬂuid mechanics. Of interest, these authors presented one of the largest DNS study to date of a spatially developing boundary layer that avoids the use of spectral methods. The inﬂow was a laminar Blasius boundary layer that undergoes transition and the highest momentum thickness Reynolds number simulated was Reh = 940. This value of Reh is far from the Reynolds numbers reached in technical applications and shows that an emerging theme is the generation of turbulence inside an attached boundary layer without computing explicitly the laminar/turbulent transition. These are the reasons why modeling is still necessary (and for a long time) prior to solving the Navier–Stokes equations for technical ﬂows. 2. Organization of the paper In this paper, we assess the ZDES technique in a framework where the realistic turbulent structures are imposed in the inﬂow. The main difference with classical DNS/LES approaches relies on the size of the computational cells which can reach values as high as 200 wall units in the streamwise direction and 100 wall units in the spanwise direction. Such a coarse resolution is mandatory to resolve high Reynolds number wall-bounded ﬂows turbulence with current supercomputer capacities. Our efforts to address this main goal are detailed in the present paper which is organized as follows. Section 2 is devoted to the description of the test case of the spatially developed boundary layer chosen for this study. The code and the meshes used to assess the grid sensitivity of the results are presented. Section 3 is devoted to the turbulence modeling. After a presentation of the ZDES approach, we summarize the principle of the newly developed synthetic eddy method which aims at generating realistic inﬂow turbulence without suffering from the limitations attributed to recycling methods. An original contribution of the present study appears at this stage since we propose here a procedure for the reconstruction of the turbulent viscosity, a prescribed velocity ﬁeld being given. Section 4 is dedicated to the results presentation. This section aims at answering three important questions relative to the use of ZDES for simulating of wall-bounded ﬂows. A ﬁrst natural interrogation is related to the capacity of ZDES to perform as an accurate subgrid scale model. This question is addressed comparing to experiment ZDES and classical LES on a standard LES grid. The main part of the paper addresses the important issue of the coarsest resolution usable with ZDES preserving an acceptable level of error. Friction coefﬁcient and cross Reynolds stress have here been selected as optimality criteria to be optimized. In ﬁne, we try in the last part to improve the results prescribing the height of the RANS/LES interface. The idea is here to protect a RANS zone in which the friction is supposed to be accurately estimated. Conclusions of the present study are drawn in Section 5. 3. Numerical procedure 3.1. Test case The test case is a spatially developing zero-pressure gradient turbulent boundary layer over a smooth ﬂat plate. The free stream velocity is Ue = 70 m s1, the static pressure is set to Pe = 99,120 Pa, the temperature equals 287 K leading to a Reynolds number per meter Re = 4.72 106 m1. The initial boundary layer thickness is d0 = 3.8 mm so that Red0 ¼ 18; 000. The Reynolds number based on friction velocity and initial boundary layer thickness (respectively initial momentum thickness) is Res = 750 (respectively Reh = 1750).
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The computational domain sizes in the streamwise, wallnormal and spanwise directions are respectively Lx = 62d0, Lz = 4d0 and Ly = 10d0 so that the range of Reynolds number covered by the simulation is 1750 6 Reh 6 4000 (750 6 Res 6 1300). Note that for x/d0 > 48 (where Reh 3500), mesh cells are stretched in the streamwise direction so that the turbulent ﬂuctuations are progressively damped. This procedure is common in such simulations and ensures that the outﬂow condition does not pollute the ﬂowﬁeld in the domain of interest. The choice of these parameters is motivated ﬁrstly by the fact that we want to assess if a turbulent boundary layer would be sustained (i.e. without any streamwise artiﬁcial relaminarisation) and secondly to evaluate the accuracy of the proposed method by comparing the results with the previously published experimental data by De Graaff and Eaton  at Reh = 2900. We also investigate the effect of the grid extension. Indeed, in the framework of hybrid RANS/LES devoted to practical applications, the ambition for wall modeling is to remove limits in terms of grid spacing: Dx+ and mainly Dz+. The major parameters of grid resolution are gathered in Table 1.
3.2. Simulations The FLU3M code developed by ONERA solves the compressible Navier–Stokes equations on multiblock structured grids. The time integration is carried out by means of the second-order-accurate backward scheme of Gear. The spatial scheme is a modiﬁed AUSM+(P) scheme (). Further details concerning the numerical method and implementation of turbulence models can be found in references ([24,25]). The accuracy of the solver for DNS, LES and hybrid RANS/LES purposes has been assessed in various applications including transitional ﬂows (), wall-bounded turbulent ﬂows ([26,27]), massively separated ﬂows ([28–30]) and ﬂow control applications ([31,32]). In these last references, the numerical results are thoroughly compared with the available experimental data including spectral and second-order analysis. The CPU cost per cell and per inner-iteration is less than 0.3 106 s. The simulations are performed on a single processor of a NEC-SX8 supercomputer and the code is running approximately at 8 109 ﬂoating-point operations per second. The time-step is ﬁxed to DtCFD = 0.32 ls which corresponds to a non-dimensionalized time-step D~t ¼ Dt CFD U e =d0 ¼ 5:9 103 . Temporal accuracy of the calculation was checked during the inner-iteration process (four Newton-like inner-iterations are used to reach second-order time accuracy). A decrease of the inner-residuals of at least one order is obtained. The Courant-Friedrich-Levy (CFL) number based on the maximum acoustic velocity (U + a) is lower than 13. After the transient phase, the real unsteady calculation begins allowing to collect statistics. The average procedure is performed in time during the calculation over a total of 900d0/Ue inertial times. The computation and post-processing of the results are based on dimensionless ﬂow quantities.
Table 1 Parameters of the spatially developing ﬂat plate boundary layer. Nx, Ny and Nz are the grid sizes along the axes and the D+’s are the corresponding resolutions. Grid name +
50 12 50+50+ 100+50+ 50+100+ 200+50+ 200+100+ 200+100+L
Nx Ny Nz
791 81 241 791 81 61 396 81 61 791 81 31 199 81 61 199 81 31 434 81 31
50 50 100 50 200 200 200
1 1 1 1 1 1 1
12 50 50 100 50 100 100
4. Turbulence modeling The generation of inlet conditions for spatially developing turbulent ﬂows remains one of the challenges that must be addressed prior to the application of LES and hybrid RANS/LES to technical ﬂows. In the frame of hybrid RANS/LES, the reconstruction of turbulent ﬂuctuations in the outer layer of the boundary layer is of primary importance. A wide range of methods has been developed in the framework of LES and DNS (see the review in [33,9]) but, as reminded in the introduction, few of them have been designed to ~ needs operate in a DES framework where the pseudo-viscosity m to be reconstructed. In this paper, the SEM method is extended in the framework of a Zonal Detached Eddy Simulation (ZDES). In the following, a brief description of the ZDES model is ﬁrst given before explaining the extension of the SEM method to the treat~. ment of m 4.1. Zonal Detached Eddy Simulation (ZDES) The original Detached Eddy Simulation due to Spalart et al.  and denoted as DES97 in this article is based on the Spalart–Allmaras RANS-model. Here we brieﬂy mention its salient features necessary for the present study and refer to the original paper () for a full description. The model is based on the transport equation ~ and obeys the following equation: of a pseudo-viscosity m
2 i ~ Dm 1h m~ ~þ r:ððm þ m~ÞÞ þ cb2 ðrm~Þ2 cw1 fw ¼ cb1 e Sm Dt r dw
dw denotes the distance to the wall and the eddy viscosity is ﬁnally ~. Functions fv1, fv2 and fw are near-wall correcdeﬁned as mt ¼ fv 1 m tions. Let us just remind that the damping function fv1 is designed ~ equals j.usy in the log-layer, in the buffer layer to ensure that m and in the viscous layer. The vorticity magnitude S is modiﬁed such that e S maintains its log-layer behavior e S ¼ us =j:y which is accomplished with the help of the fv2 function. The function fw was designed to obtain a faster decaying behavior in the outer region of the boundary layer. The basic idea of DES97 was to modify the destruction term so that the RANS model is reduced to a LES subgrid scale one in the detached ﬂows. These authors proposed to replace the distance ~ deﬁned by: dw with d
~ ¼ minðdw ; C DES DÞ d
where D = max(Dx, Dy, Dz) is the characteristic mesh length and CDES = 0.65. When production and destruction terms are balanced, the eddy viscosity scales with the mesh length D and the local vorticity modulus: mt SD2 then adopts the form of Smagorinsky’s subgrid scale viscosity. The deviatoric part of the SGS stresses are given by hu0i u0j id ¼ mt ð@ui [email protected]
þ @uj [email protected]
Þ where ui is the resolved ﬁeld. Zonal Detached Eddy Simulation (ZDES) ([7,8]) differs from DES97 by the fact that within ZDES, the user has to select individual RANS and LES domains while standard DES97 is a non-zonal approach. Within ZDES, the subgrid length-scale is given by the 1 cube root of the cell D ¼ ðDx; Dy; DzÞ3 and the near-wall functions of the RANS model are disabled in LES mode (i.e. for dw > CDESD): fv1 = 1, fv2 = 0 and fw = 1. In practice, ZDES switches very quickly in LES mode, thus limiting the extent of the gray area responsible for the delay in the formation of instabilities in free shear layers. The motivation of ZDES was to be fully safe from MSD and GIS when treating technical ﬂows featuring massive separation and to clarify the role of both RANS and LES region. ZDES is free from MSD only when it is forced to operate in RANS mode. It can be achieved by specifying the domain list operating in RANS mode or/and by specifying a boundary layer height below which a RANS behavior is forced. The ZDES approach has been successfully used
S. Deck et al. / Computers & Fluids 48 (2011) 1–15
to predict the transonic wing buffet phenomenon ([7,35]), the ﬂow around a high-lift conﬁguration () as well as to investigate both subsonic ([29,36]) and supersonic base ﬂow aerodynamics under highly compressible conditions ([28,37]) and the mixing enhancement process in a supersonic mixing layer (). In the previous applications, the attached boundary layers are systematically treated in RANS mode (natural use of DES-type methods). One of the objective of the present study is to broaden the application area of such methods by permitting the activation of LES inside the boundary layer. The zonal aspect of the method enables one to prescribe the RANS interface at a given altitude dw/d0.
ui ðx; y; z; tÞ ¼ U i ðyÞ þ
where the aij(y) correspond to the Cholesky decomposition of a prescribed Reynolds tensor Rij(y). As wall-bounded ﬂows are populated with eddies which sizes depend on their distance to the wall ([41– 43]), an adaptation of the Synthetic Eddy method to such ﬂows has been proposed by Pamiès et al. . It consists in the speciﬁcation of P modes (typically P 6 4), each of which are scaled depending on their wall distance:
~j ¼ u
~ jp is deﬁned as the superimposition of turbulent strucEach mode v tures with prescribed time and length scale and geometrical shape (i.e. vorticity content) which are randomly positioned in the inlet plane. The idea is to reproduce realistically the distribution of scales in the wall-normal direction (see  for a thorough presentation of the procedure). This synthetic turbulence method (denoted as SEM in the following) has then been adapted to ZDES simulation, where the eddy viscosity ﬁeld is reconstructed from the synthetized velocity ﬁeld as follows:
mt ðtÞ ¼
l :D2 : < S > if dw < C DES D ðC S DÞ2 SðtÞ if dw P C DES D
0.8 0.6 0.4 0.2
Recycling methods use a modiﬁcation of periodic boundary conditions that takes into account the inhomogeneity of the ﬂow in the streamwise direction. This kind of methods has nevertheless two main weak points for their use in applied aerodynamics. Firstly the recycling procedures introduce a spurious periodicity in the ﬂow direction which is problematic when this spurious periodicity becomes comparable with the frequencies of interest in the application. Secondly, the ﬂow needs to be initialized to turn the inﬂow generation on which is far from being trivial in the framework of technical ﬂows. As regards these two aspects, synthetic methods are feedforward and their random nature is expected to suppress any periodicity. Indeed, inﬂow methods relying on synthetic turbulence generation are based on the assumption that turbulence can be represented by a superimposition of coherent structures and thus can be speciﬁed by using only low-order statistics. Jarrin et al. [39,40] proposed the Synthetic Eddy Method (SEM) where the coherent eddies are randomly generated both in time and space in the inﬂow plane. Following this principle, a stochastic signal is built for each velocity component and superimposed to a mean. A renormalization condition ensures that the signal has a unit variance. The signal is then linearly transformed to match prescribed Reynolds stresses using a Cholesky decomposition:
~ j ðx; y; z; tÞ i ¼ 1; 2; 3 aij u
4.2. Synthetic eddy viscosity reconstruction
y/δ Fig. 1. Analytical inlet eddy viscosity proﬁle in RANS mode.
where CS = 0.1 is the Smagorinsky constant and S(t) (respectively hSi) the magnitude of the instantaneous vorticity built from the synthetic velocity ﬁeld given by Eq. (3) (respectively the vorticity calculated from the mean inﬂow velocity proﬁle). l is the mixing length valid over the entire boundary layer which was proposed by Michel et al.  (see also ):
l ¼ 0:085d tanh
dw 0:085 d
j ¼ 0:41
Note that l jdw in the internal zone and l 0.085d in the external zone where l is proportional to the boundary layer thickness d (which is calculated from the imposed inlet mean velocity proﬁle). The damping function D adopted here is given by D ¼ þ 1 expðdw =26Þ. It is an explicit version of the one originally deﬁned by Michel et al. , which includes dependence on the eddy pﬃﬃ n and molecular viscosities lt and l: DMQD ðlt Þ ¼ 1 expð 26jÞ with 2 þ n ¼ ql llþ2lt S. DMQD tends to zero for dw > 500 leading to a faster de-
cay of the eddy viscosity in the outer-part of the boundary layer than with the Van Driest function. In practice, and as shown by the eddy viscosity proﬁle in Fig. 1, our slight adaptation of the original model does not alter signiﬁcantly the eddy viscosity proﬁle especially near the wall. As a result, the present turbulence model is based on a turbulent viscosity which is deﬁned algebraically according to a mixing length scale valid over the whole boundary layer thickness. It has the advantage of being more robust and more computationaly efﬁcient than others (one might consider for instance the model proposed by Nolin et al. , which requires the inversion of a linear system). One other asset of our model is that the pseudo-eddy viscosity can be computed explicitly from the turbulent one. This comes from the use of the Spalart–Allmaras turbulence model, which al~: lows to get the following fourth-order polynomial equation in qm
~Þ4 lt ðqm ~Þ3 l3 lt c3v 1 ¼ 0 cv 1 ¼ 7:1 ðqm
The solving of this fourth-order polynomial equation is described in Appendix A. 5. Results and discussion 5.1. Assessment of ZDES as a subgrid scale model Hybrid methods are expected to accurately simulate wallbounded ﬂows at high Reynolds number. In this section, the objective is to evaluate the capability of ZDES to operate accurately as a
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WMLES model. Hence, ZDES is compared with a reference LES with the Mixed Scale Model (MSM) () as well as to another reference simulation which uses the recycling procedure of Lund et al. . The grid resolution Dx+ = 50, Dy+ = 1 and Dz+ = 12 (see Table 1) is retained in this section. In addition, the mean and rms proﬁles needed for the SEM method are extracted from the reference LES simulation so that all calculations have the same statistical inﬂow information. Let us just be reminded that the recycling method consists in taking a plane (noted rec) from a location x = Lr several boundary layer thicknesses d0 downstream from the inﬂow. These data are then rescaled with proper scaling laws for mean ﬂow and ﬂuctuations and reintroduced at the inﬂow (noted in). In this study, the extraction plane is positioned at Lr = 8d0 in accordance with the recommendation of Lund et al. . The recycling approach (denoted as RECY in the following) has been successfully used in previous Direct and Large Eddy Simulation but not so many publications are devoted to its use in hybrid RANS/LES. Analogous to the treatment of the velocity ﬁeld, the scaling of the pseudo y þ þ ~ is given by: m ~in ~rec ~in eddy viscosity m inner ðy Þ ¼ bminner ðy Þ and mouter d ¼ y ~rec bm outer ðdÞ where b is the rescaling factor and the subscripts inner and outer refer to the inner and outer layer of the boundary layer which are rescaled separately. Complete eddy viscosity proﬁles valid over the entire inﬂow boundary layer are obtained by the tanh weighting function introduced by Lund et al.  which joins continuously inner and outer regions together. In addition, the rescaling procedure concerns the whole pseudo-eddy viscosity (no distinction between mean and ﬂuctuating parts of the turbulent quantity as in Xiao et al. ). Two scaling factors have been as rec 18 in
us sessed namely b1 ¼ ddin and b2 ¼ urec . These scalings are motis
~ ¼ jus y vated by the linear behavior of the pseudo-eddy viscosity m from the log-layer to the wall thanks to the fv1 damping function of the Spalart–Allmaras (SA) turbulence model. As discussed in , minor differences are observed between the scaling factors b1 and b2. Therefore, b1 is retained analogous to the scaling factor for the velocity ﬁeld. Fig. 2 presents the contours of the streamwise component of the velocity and pseudo-viscosity of the ZDES-SEM computation compared with the ZDES-RECY case. One can notice that ZDES-SEM method already provides a high degree of realism of the inﬂow data which helps to speed its achievement up. Indeed, Fig. 3 pre-
Fig. 3. Reynolds stresses at different stations downstream from the inﬂow (ZDESSEM calculation).
~=m (lower part) in the inlet plane. Left part: Fig. 2. Comparison of the instantaneous streamwise velocity ﬁeld (upper part) and instantaneous pseudo-eddy viscosity ﬁeld m ZDES recycling method. Right part: ZDES synthetic eddy method.
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sents the Reynolds stresses at different streamwise locations for the ZDES-SEM case and shows that proﬁles at x/d0 = 6 are already close to those at x/d0 = 12. The power spectral density function of the streamwise velocity component at y+ = 400 is displayed in Fig. 4 for both ZDES calculations at x/d0 = 6. Both spectra exhibit the classical broadband aspect according to which the large scales (low frequencies) mainly contribute to the energy before cascading to smaller scales (high frequencies) until the cut-off frequency implicitly induced by the grid is reached. It is worth noting that the RECY case introduces non-physical frequencies fk = k.0.9Ue/Lr due to the artiﬁcial streamwise periodicity introduced by the method. These frequencies are the result of data convection at velocity 0.9Ue, i.e. the advection velocity of the largest scale in the outer-part of the boundary layer in the present case. These unphysical energy-carrying frequencies are potentially problematic if close to those of the physical phenomena of interest located further downstream. Conversely, the ZDES-SEM calculation presents a much smoother spectrum in the low frequency range. The performance of the ZDES-SEM method is then assessed by measuring its ability to reproduce a correct mean velocity proﬁle and skin friction coefﬁcient. Fig. 5 shows an overall comparison of both experimental and RANS data with all calculations in terms of mean velocity proﬁle and skin friction coefﬁcient. One can notice that for a same subgrid scale model (LES-MSM or ZDES), minor differences are observed between the synthetic turbulence method and the recognized recycling procedure. As far as the streamwise evolution of the skin friction coefﬁcient is concerned, a converged state is reached near 10d0 downstream from the inlet. The establishment distance is different between SEM and RECY methods, the latter technique recovering quickly the expected value. In , a grid sensitivity has been conducted with recycling methods and a relaminarisation has been observed for coarse grids while the SEM method does not suffer from this important limitation. Besides, the error on skin friction coefﬁcient is of the order of 4% for ZDES and 8% for LES-MSM compared with the RANS value. The preponderant terms of the Reynolds stress tensor are then plotted inpFig. 6.ﬃ As suggested by De Graaff and Eaton , the scalﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ing hu0i u0j i Cf =2 in mixed-outer coordinates is used. The justiﬁcation for this mixed velocity scaling comes from the energy balance of the boundary layer since the total power dissipated by the boundary layer scales on Uesw. For a constant density ﬂow, the total rate of energy dissipation by turbulence depends on both Ue and us. From Fig. 6, one can notice that all the present calculations (i.e. with the recycling procedure or synthetic turbulence) are in an overall good agreement with the experiments of De Graaff and Eaton . However, the behaviors of LES-MSM and ZDES are
Fig. 5. Comparison between the ZDES results, LES and experimental data at Reh = 2900. Upper part: mean velocity proﬁle. Lower part: Streamwise evolution of the skin friction coefﬁcient. The dashed line indicates the location of the beginning of mesh stretching.
Fig. 6. Comparison of the Reynolds stresses between the ZDES results, LES and experimental data at Reh = 2900. The vertical solid line separates the inner and outer scalings, respectively y+ and y/d. The dashed line indicates the location deﬁned by the value of CDESD = dw.
Fig. 4. Power spectral density for the streamwise velocity at 6d0 downstream from the inﬂow for the ZDES-SEM case.
slightly different depending on the region of the boundary layer. In its inner part, notably, the 15% overestimation of the maximum of streamwise velocity ﬂuctuations by LES-MSM is not present in ZDES calculations. In the outer-part of the boundary layer (i.e. for y/d > 0.2), both LES-MSM ans ZDES are hardly underestimating streamwise and wall-normal velocity ﬂuctuations, but the ZDES
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Fig. 7. Time averaged spanwise spectra of the streamwise velocity component at Reh = 2900 (same legend as in Fig. 6).
method succeeds in reproducing the same behavior as LES, which is our target in the frame of a hybrid RANS/LES simulation. Spanwise spectra evidence how the energy of ﬂuctuations are distributed in wave number space for several altitudes in the boundary layer. Especially, Fig. 7 identiﬁes the highest length scales (kzd > 4) responsible for the extra energy in the plane y+ = 18. At higher altitude y+ = 100, both ZDES spectra (i.e. for the RECY and SEM cases) corroborate with those of the standard LES results with the MSM subgrid scale model. Otherwise, both approaches are similar in terms of energy levels and scales distribution. In the outer-part of the boundary layer (y/d = 0.8), energy is conﬁned near kzd 1 which simply reminds us that energy is carried by the largest scales whose sizes are of the order of d. Minor discrepancies are observed at highest wave numbers but their contribution to the total energy is negligible. As a ﬁrst conclusion, the above analysis shows that ZDES amounts to LES with a plausible one-equation SGS model and wall modeling since both the mean and second-order ﬁeld are well pre-
Fig. 8. Iso-surface of QU 2e =d20 ¼ 0:01: (a) 50+12+, (b) 50+50+, (c) 50+100+, (d) 100+50+, (e) 200+50+ and (f) 200+100+.
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dicted and the energy of ﬂuctuations is properly distributed in spectral space. This result broadens Kikitin et al. ’s pioneering work. From now on, only the ZDES-SEM model is retained for the following calculations. 5.2. Effect of grid density A tangential objective of the present work is to stimulate discussion on the objectives of the simulation (skin friction, velocity ﬂuctuations) and limitations of the method, in terms of ﬁdelity of the resolved ﬁeld and correlatively in terms of grid spacing. As one of the objective of hybrid RANS/LES simulations aims at describing the outer part of the boundary layer, the results are normalized by the characteristic scales in this region, namely Ue and d. An overview of the turbulent content generated by the ZDES simulations is evidenced in Fig. 8 by showing the iso-surfaces of the Q criterion  Q = 1/2(kSkkXk) with S and X denoting respectively the strain and the rotation tensor. It is worth noting that turbulence is sustained in the outer layer even for the coarsest resolutions 200+100+ over the whole computational domain. A similar observation has been made by Nikitin et al.  with even coarser resolutions in the case of channel ﬂows. However, in the frame of spatially developing ﬂat plate boundary layer, the unsteady character of the ﬂow ﬁeld is not imposed by periodic boundary conditions like in channel ﬂows, which makes the present results even more remarkable. Fig. 9 displays the mean velocity proﬁles and indicates how the grid resolution affects the velocity gradient at the wall and the logarithmic region. To further investigate this effect on the velocity proﬁle, Fig. 10a compares the viscous, turbulent (resolved + modeled) and total (viscous + turbulent) shear stresses for grid 100+50+ with the RANS proﬁles. Conversely to the outer-part of the boundary layer, the turbulent stresses differ signiﬁcantly from the Reynolds shear stress provided by the RANS calculation in the inner part of the boundary layer. Indeed, in a resolved LES the ﬁne turbulent structures near the wall generate high strain rates and thus an important part of the skin friction. On coarse grids, these structures cannot be properly represented numerically (see Fig. 11). The decay in vertical velocity hv0 v0 i ﬂuctuations (see Fig. 10b) affects directly the production term in the Reynolds shear stress budget which in planar case is given by:
Phu0 v 0 i ¼ hv 0 v 0 i
Fig. 10. Effect of the resolution: (a) Shear stresses for grid 100+50+: RANS (solid)/ ZDES (symbol). (b) Vertical velocity ﬂuctuations proﬁles.
For grid 100+50+, the RANS interface is located deeply into the boundary layer near y/d = 0.01. In practice, the eddy viscosity levels
Fig. 11. Spanwise two-point autocorrelations of streamwise velocity: 50+12+ (symbol) 100+50+ (solid line).
Fig. 9. Effect of the grid resolution on the mean velocity proﬁle at Reh = 2900. The dashed lines represent the altitude of the interface deﬁned by the value of CDESD/d for each grid.
are too low so that the modeled part does not compensate for the loss of resolved stresses induced by the mesh resolution. This is referred to as ‘‘Modelled Stress Depletion’’ by Spalart et al. . Since the total stress is imposed, the viscous stress m < @u/@y> must compensate. As a result, the mean velocity is itself incorrect in that region. In addition, it is evidenced in Fig. 11 that the spanwise
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correlation of the longitudinal velocity within the near-wall region is higher for the 100+50+ grid than for the 50+12+ one. More precisely, the autocorrelation length in this region scales more like the grid spacing than like the viscous unit m/us as one would expect. This suggests that the coarser grid does not allow for the correct discretization of the near-wall structures. On the other hand, the largest scales populating the outer-part of the boundary layer are well represented in the 100+50+ simulation. According to this result, it is quite clear that the turbulence in the outer region of the boundary layer will survive even if the interface between RANS and LES modes is prescribed explicitly. The interest and effect of prescribing the location of the interface will be evidenced in the next section. From Fig. 12, one can clearly see that the spanwise resolution affects dramatically the value of skin friction (see for instance results on grids 50+12+, 50+50+ and 50+100+). Moreover, it is observed that the adaptation distance increases as the grid spacing in the streamwise direction Dx+ raises. As an example, a stabilized behavior of Cf is reached after 15 initial boundary layer thicknesses downstream from the inlet plane on the grid 50+50+ while 25d0 and 45d0 are necessary for the grid 100+50+ and 200+50+ respectively. Correlatively, the CPU cost saving for the grid 200+100+ is potentially 4 (respectively 2) compared with the grid 100+50+ (respectively 200+50+) but one has to dedicate an adaptation distance three times (respectively 1.7) longer. In other words compared to grid 50+50+ the effective CPU saving, in terms of number of points in the streamwise direction, is 1.3 (instead of 4) for the grid 200+50+ and 1.2 (instead of 2) for the grid 100+50+ in our case. In addition, one can notice that no converged state is reached for the grid 200+100+ within the present computational domain (i.e. for x/d < 48). A simulation on a longer grid in the streamwise direction (as will be discussed in the last section) indicates that 60d0 are necessary to get a stabilized state. An other quantitative insight is provided by the error analysis between the ZDES cases and the RANS case at station Reh = 2900 where experimental data are available. Such error is measured according to Eq. (9) which are adapted from Keating et al. :
Ehu0 v 0 i ðDxþ ; Dzþ Þ ¼
jhu v i hu v iðDx ; Dz Þjdy Rd jhu0 v 0 i jdy 0:2d 0
As the skin friction constitutes a primary quantity of interest, it is worthwhile to assess the error on the skin friction as follows:
ECf ðDxþ ; Dzþ Þ ¼
where hu0 v0 i⁄ is the target RANS shear stress and hu0 v0 i is the total shear stress (i.e. modeled + resolved) calculated in the spatially developing case. Note that the bounds of the integral are limited to 0.2d d since, as reminded earlier, the objective of hybrid RANS/LES simulations is precisely to describe the outer-part of the boundary layer.
Fig. 12. Effect of the grid resolution on the streamwise evolution of the skin friction coefﬁcient. The dashed line indicates the location of the beginning of mesh stretching.
jCf ðDxþ ; Dzþ Þ Cf j Cf
where Cf⁄ is the skin friction of the RANS result. Fig. 13 shows the effect of grid resolution on both total shear stress and skin friction errors. First, ZDES yields the smallest error (less than 4%) on both indicators at Dx+ = 50 and Dz+ = 12 showing again ZDES amounts to LES with a plausible subgrid scale model compared with the established Mixed Scale Model. Of interest, this ﬁgure highlights the separate effect of the streamwise (respectively spanwise) resolution on skin friction and turbulence. Indeed, let us ﬁrst consider the results on grid 50+50+. One can notice that for the same spanwise resolution doubling the streamwise resolution (i.e. grid 100+50+) results in a decrease (by a factor near 2) on the error of the total shear stress without modifying the error on the skin friction. Similarly for grid 50+100+, a twice higher spanwise resolution Dz+ increases (by a factor near 2) the error on the skin friction reaching 30% while maintaining the same error on the total shear stress. Otherwise, results can be very different for a same area Dx+Dz+. Besides, the coarsest grid investigated in this study (i.e. grid 200+100+) provides an error on the total shear stress lower than 10% despite an error on the skin friction near 25%. However, the worst result (ECf 28% and Ehu0 v 0 i 22%) are obtained with a four times more important grid, namely grid 50+100+. Even if the dominant shear stress is the one in the direction of the mean skin friction, getting the right shear stress does not necessarily mean that the skin friction is well calculated. This result is reminiscent to what Meyers and Sagaut  observed in case of channel ﬂows. Indeed, these authors investigated several combinations Nx Ny and observed that the skin friction does not converge monotonously and that ECf can be close to zero for certain coarse resolutions. In order to account for both error indicators, we deﬁne:
dðDxþ ; Dzþ Þ ¼
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E2Cf þ E2hu0 v 0 i
As d = 0 corresponds to the exact solution, d is interpreted as the distance between the hybrid calculation and RANS results which are fully consistent with the experiment while giving information about hu0 v0 i. Let us be reminded that Fukagata et al.  have shown that the skin friction coefﬁcient depends on a weighted
Fig. 13. Effect of the grid resolution on ECf and E<u0 v 0 > at Reh = 2900 without prescribed interface. : LES (MSM) 50þ 12þ ; : 50þ 12þ ; . : 50þ 50þ ; : 50þ 100þ ; } : 200þ 50þ ; : 200þ 100þ ; D : 100þ 50þ .
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integral of the Reynolds stress distribution. Thus, the skin friction coefﬁcient and the Reynolds stresses are not completely independent variables. This quantity d, computed for Reh = 2900, is displayed in Fig. 14 as a function of the grid resolution. In this plot, the error is represented by a circle whose size is set proportional
to d. One can notice that the global error is non monotonous with the grid resolution since it is higher with grid 50+100+ compared with grid 200+100+. However, let us be reminded that in all previous grids the switching to LES mode occurs inside the boundary ~ ¼ C DES D branch of Eq. (2) intrudes the boundary layer (i.e. the d layer). As an example, the DES interface is located near 20+ for the grid 200+100+ when using the cubic root as ﬁlter width (instead of max(Dx+, Dy+, Dz+) as in DES97). Otherwise, the eddy viscosity is not high enough for modeled stresses to compensate for the loss of resolved stresses. As shown in , the deﬁnition of the subgrid length-scale D appearing in the DES length scale is far from being trivial since the cube-root appears suited when ﬁne grids (in a well-resolved LES sense) are used but can lead to ‘‘modeledstress-depletion’’ for larger grid resolutions. The authors of DES97  who initially advocated the use of the maximum of the three cell dimensions have also modiﬁed the deﬁnition of the subgrid length-scale in their IDDES approach  which is now capable of running in LES mode when the resolution is sufﬁcient. A deep discussion concerning the deﬁnition of the subgrid length-scale is beyond the scope of this paper but the effect of the RANS-LES interface is investigated in the next section. 5.3. Effect of a prescribed interface
Fig. 14. Effect of the grid resolution on the global error d at Reh = 2900. The center of the circle corresponds to the resolution of the mesh (see Table 1) and its diameter is deﬁned by Eq. (11).
In the previous section, it has been shown on the one hand that for coarse resolutions (in a resolved LES sense), the depletion of
Fig. 15. Effect of the RANS/LES interface on the streamwise evolution of the skin friction coefﬁcient (left part) and on the mean velocity proﬁle at Reh = 2900 (right part): (a) Dx+ = 50, Dz+ = 50; (b) Dx+ = 100, Dz+ = 50. : Exp.  (Reh = 2900), –: RANS-SA, j: no ﬁxed interface, N: ﬁxed interface at 50+, I: ﬁxed interface at 100+, : ﬁxed interface at 200+. The dashed line indicates the location of the beginning of mesh stretching.
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modeled stresses corrupts the velocity proﬁle. Especially, the error on the skin friction increases dramatically when increasing the spanwise resolution Dz+. On the other hand, the spanwise twopoint autocorrelation function (see Fig. 11) indicates that the eddies populating the outer-part of the boundary layer can be well prescribed though a poor near-wall layer description. In this analysis, it was not possible to distinguish between the effect of the resolution in the wall parallel direction and the associated change of RANS/LES interface height. In order to assess the consequence of such change, we investigate here the effect of a prescribed RANS/ LES interface by forcing the treatment of the inner layer of the boundary layer in URANS mode. Three interface locations namely y+ = 50, 100, 200 which correspond respectively to the beginning, the inner and outer parts of the logarithmic region are investigated ~ ¼ dw if dw 6 dprescribed ). (i.e. Eq. (2) reads as d w Fig. 15 gives again a clear impression of the non-homogenous convergence behavior of the skin friction with the height increase of the RANS-LES interface. For ﬁxed interface at 100+ and to a lesser extent at 50+, convergence towards the RANS reference is observed compared to the case without a prescribed interface. The amplitude in the peak of Cf downstream from the inﬂow is reduced and the adaptation distance to get a stabilized behavior of Cf is increased. Conversely, when the interface is prescribed at 200 wall unit, a different behavior is observed. As an example grid 200+100+ does not allow to get a stabilized behavior of the skin friction within the computational domain (i.e. for x/d0 < 48). Hence, additional simulations are performed on a longer grid denoted as 200+100+L (see Table 1). As already mentioned in the previous section, near 60 initial boundary layer thicknesses are necessary to get a stabilized behavior of Cf (see Fig. 16a) while 75d0 (respectively 110d0) are necessary when prescribing the interface at y+ = 100 (respectively y+ = 200). Otherwise, one has to dedicate the same number of points Nx Nz between grid 100+50+ and grid 200+100+L (with a prescribed interface at y+ = 200) to get a stabilized skin friction coefﬁcient. In addition, Fig. 16 indicates that one has to be particularly cautious when assessing the performance of an hybrid model on coarse resolutions since getting the right velocity proﬁle is not enough. Indeed from Fig. 16b, one can notice that the velocity proﬁle compares fairly well with the reference RANS one for a ﬁxed interface at y+ = 100. Nevertheless, the skin friction is not stabilized at the location corresponding to Reh = 2900 (see Fig. 16b at x/d0 = 46). More precisely, the ﬁrst location where the skin friction is established corresponds to Reh = 3500. When the interface is ﬁxed at y+ = 200, the skin friction coefﬁcient is not stabilized before Reh = 4400. In order to further analyze the effect of a prescribed interface on the ZDES simulation, both modeled and resolved Reynolds stresses are plotted in Fig. 17 for the 100+50+ case. The prescribed interface at 50, 100 and 200 wall unit corresponds respectively for the chosen Reynolds number to 3, 6 and 12 percents of the local boundary layer thickness. It is also worthwhile to notice that without prescribed interface, the natural transition between RANS and LES occurs at y = 0.011d. An important result from these simulations is that the resolved stresses are still signiﬁcant under the height of the prescribed interface. Quantitatively, when the interface is imposed at y+ = 200, the resolved stress represents about 20% of the total stress at altitude y+ = 100 and 40% at altitude y+ = 200. The interface can then be considered as very permeable. As expected, a high altitude interface is associated to a low level of resolved stress. The strong discontinuity observed in the modeled stress when the interface is not prescribed is not distinguishable for every speciﬁed interface position. This can be easily understood looking at the variation of the fv1 function which passes abruptly from a value of about 0.4 at the height of the natural transition between RANS and LES to 1 above this limit. In every other cases the
Fig. 16. Effect of the RANS/LES interface for grid 200+100+: (a) skin friction; (b) mean velocity proﬁle.
behavior of the fv1 function is such that its value is already equal to one at the interface height. The sum of the modeled and resolved shear stress as a function of grid resolution and location of the RANS/LES interface is compared with the reference Reynolds shear stress in Fig. 18. As expected, the discontinuity in the modeled resolved stress for the non forced situation is also apparent in the total stress. On the ﬁner grid, the agreement with the RANS result is better with a prescribed interface located at 100+ or 200+ than with the interface positioned at y+ = 50. This result illustrates the beneﬁt of specifying the height of the zone where RANS modeling is enforced. The latter procedure permits to compensate for the limitation of LES approaches when the severe constraints on the grid construction needed for these techniques are not satisﬁed. The results of the 200+100+ case are atypical and evidence a marked overestimation of the shear stresses in the outer part of the boundary layer. This overestimation is directly related to an error on the resolved stress the modeled one being negligible at this altitude. This is likely to be associated to the occurrence of superstreaks in the outer-part of the boundary layer. The inﬂuence of the interface height on the error both on the shear stress and on the skin friction is presented in Fig. 19 for grids 50+50+, 100+50+ and 200+100+. Note that for the latter grid the case with an interface at 200+ must be interpreted cautiously since as already mentioned the skin friction evolution is not stabilized at Reh = 2900. To study the trend of the error d, one can either con-
S. Deck et al. / Computers & Fluids 48 (2011) 1–15
Fig. 17. Effect of the RANS/LES interface on the resolved turbulent stresses (top) and on the modeled stresses at Reh = 2900 (bottom) for the resolution Dx+ = 100, Dz+ = 50.
sider the ﬁrst station where the solution is in equilibrium (which would mean considering different Reynolds numbers) or the station where Reh = 2900 (with the possible drawback of getting a non equilibrium stage). We made the choice of computing the error d at Reh = 2900 for two reasons. First, because we think that it is questionable to compute the error based on different Reh. Secondly, experimental data are available for Reh = 2900. However, the counterpart is that a monotonic trend cannot be straightforwardly highlighted. One can observe that if one gives the same weight to the two parameters (using Eq. (11)), the computations with speciﬁed interface surpass the ones where the natural interface given by Eq. (2) is adopted. In the 50+50+ grid, the error of the case with an interface ﬁxed at 100+ is only half the one of the reference case. This procedure thus permits to recover an intuitive grid dependence of the error. This represents a signiﬁcant improvement with respect to the observations of Fig. 14, a ﬁner resolution being now associated to a lower error level. On the 100+50+ grid, the maximum error level (identiﬁed by the dashed circle) is smaller than 17% independently of the interface height. However, for an equivalent error level the establishment distance of the solution may be a parameter to consider in the decision process. For this grid, the case with an interface located at 50+ is the ﬁrst for which the skin friction adopts a physical behavior (at x/d0 = 30). It thus minimizes the total cost of the simulation. Globally, the computational effort is potentially divided by a 4 factor with respect to the LES, the factor
Fig. 18. Effect of the RANS/LES interface on the total stresses: (a) Dx+ = 50, Dz+ = 50; (b) Dx+ = 100, Dz+ = 50; (c) Dx+ = 200, Dz+ = 100. : Exp.  (Reh = 2900), – –: RANSSA, j: no ﬁxed interface, N: ﬁxed interface at 50+, I: ﬁxed interface at 100+, : ﬁxed interface at 200+.
8 on the number of points being partially compensated by a factor 2 for the establishment distance of the solution. On the coarsest grid, it is striking to observe that low heights (50+ or less) of the interface are associated to a low error on ﬂuctuations and a large error on the friction whereas the opposite is evidenced with a higher interface altitude (100+). Even if this shows
S. Deck et al. / Computers & Fluids 48 (2011) 1–15
Fig. 19. Effect of the RANS/LES interface on the global error d at Reh = 2900 (right part): (a) Dx+ = 50, Dz+ = 50 ; (b) Dx+ = 100, Dz+ = 50; (c) Dx+ = 200, Dz+ = 100. : ZDES 50+12+, no prescribed interface, h: no prescribed interface, D: ﬁxed interface at 50+, .: ﬁxed interface at 100+, }: ﬁxed interface at 200+.
that it is conceivable to perform some optimization according to a given objective, the global error level reaches at least 21% and this grid resolution is certainly the coarsest possible one.
6. Conclusions Zonal Detached Eddy Simulation has been used to simulate a spatially developing turbulent boundary layer over a smooth ﬂat plate at Reh = 2900. Results have been compared with the experimental data of De Graaff and Eaton . The ZDES model has been retained because its ability to simulate three-dimensional technical ﬂows (attached boundary layers treated in RANS mode) has already been demonstrated thanks to thorough comparison with the available experimental data including spectral and second-order analysis. One of the objectives of this study was to broaden the application area of such hybrid RANS/LES methods by permitting the activation of LES inside the boundary layer. So, our study has followed an evolutionary path. First a synthetic reconstruction ~ has been proposed in the method of the pseudo-viscosity ﬁeld m
frame of a synthetic eddy method. The RANS part of the model depends also on an explicit mixing length, proposed by Michel et al. , valid over the entire boundary layer thickness. A thorough comparison with experiment and reference LES (with the mixedscale-model) have been conducted. The new approach has also been compared with the time-honored recycling method including the rescaling of the eddy viscosity. It has been shown that ZDES amounts to LES with a plausible one-equation SGS model and wall modeling. More precisely, both the mean and second-order ﬁelds have been well predicted compared with the experiment and to a reference LES with the mixed-scale-model. Compared to recycling techniques, no initialization of the ﬂow is needed and the simulation is operational after a small transient corresponding to the time needed by the smallest scales to reach the end of the computational domain. The method provides a high degree of realism of the inﬂow data which allows to get converged rms velocities after about 10d0 downstream from the inﬂow for the ﬁnest grid. The effect of both grid resolution and prescribed interface on the numerical solution have been then discussed. Some ﬁgures of merit used to assess the numerical results have been proposed. It
S. Deck et al. / Computers & Fluids 48 (2011) 1–15
has been shown that the error on the skin friction coefﬁcient increases dramatically when increasing the spanwise resolution Dz+ while increasing the streamwise resolution Dx+ results in a longer adaptation distance. In addition, getting the right velocity proﬁle is not enough since one has also to check that the skin friction is stabilized. In the same way, the maps of global error d have evidenced that the user can build its own norm according to the weight he puts either on the turbulent shear stresses or on the skin friction coefﬁcient. One of the most remarkable result of this study came from the fact that, once given a ﬁgure of merit (Eq. (11)), error reduction can be obtained by specifying the height of the RANS/ LES interface to values which can be as high as 100+ from the wall. This interface is permeable to turbulent ﬂuctuations, and if it is imposed sufﬁciently far from the wall, discontinuities are absent in the total shear layer proﬁles. With this strategy, resolution of 100+50+ should allow for treating complex conﬁgurations at reasonable Reynolds number with the currently available computation power (see for instance  for an application to threedimensional shock/boundary layer interaction). It is however worthwhile to notice that the limitation of the method has been identiﬁed since the combination of a very coarse resolution (200+100+) and a prescribed interface at 200+ from the wall leads to very long transient which makes this technique non-applicable in practice. More generally, it has been observed in this study of spatially developing boundary layer that the potential computational effort reduction brought by RANS-LES approaches depends not only on the grid resolution but also on the establishment distance of the solution. We plan to explore possible improvements of the method in order to minimize the adaptation distance for the largest grid resolutions. Its use in applied aerodynamics needs also the extension of the method to compressible ﬂows in the frame of three-dimensional applications. Appendix A
X 4 þ aX 3 þ b ¼ 0
qm~ l ; a¼ t l l
and b ¼ c3v 1
The method proposed by Ferrari allows to factorize the above equation thanks to the introduction of a parameter Y. At ﬁrst, Ferrari applies an afﬁne variable change to get rid of the third order term:
Z 4 þ pZ 2 þ qZ þ r ¼ 0
3 a3 a4 Z ¼ X þ a=4; p ¼ a2 ; q ¼ and r ¼ b 3 8 8 16 The parameter Y is then introduced according to Z2 (A.2) can be factorized if:
Z2 + Y. Eq.
p 4rp q2 Y 3 Y 2 rY þ ¼0 2 8
It is possible to ﬁnd at least one real root of this third order equation, thanks to the method proposed by Cardan. In detail, Eq. (A.3) reduces to Eq. (A.4) with the afﬁne variable change Y = K + p/6:
K 3 þ lK þ m ¼ 0 2
4 3 l 27
Viewed as a function of lt/l, and assuming lt/l > 0, DC is always strictly positive (it is zero when lt/l = 0, and strictly growing for lt/l > 0). It implies that Eq. (A.4) has one real solution K0, which reads:
K0 ¼ u þ v
pﬃﬃﬃﬃﬃﬃ 1=3 u ¼ m=2 þ DC =2 and
pﬃﬃﬃﬃﬃﬃ 1=3 DC =2
Thus, Eq. (A.2) can be factorized if Y = Y0 = K0 + p/6, and reads:
ðZ 2 þ Y 0 a0 Z b0 ÞðZ 2 þ Y 0 þ a0 Z þ b0 Þ ¼ 0
a0 ¼ p þ 2Y 0 and b0 ¼ q=ð2a0 Þ ~ can be Eq. (A.7) admits only one positive real root, from which qm deduced as:
qm~ ¼ l
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a0 =2 þ a20 þ 4ðb0 a0 Þ=2 a=4
It must be noted that the intermediate variables used to solve Eq. (A.1) can be very sensitive to numerical errors. Powers to rational numbers or substractions may require cut-offs to avoid numerical overﬂows. Namely, the authors have restricted lt/l to the range [0.02; 1], which accounts for usual values of lt with a signiﬁcant safety factor. References
The fourth order Eq. (7) to solve reduces to:
DC ¼ m2 þ
where l = p /12 r and m = p/54(9r + p /2) + rp/2 q /8. The roots of the above equation depend on the sign of its determinant DC:
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