Key factors in the use of DDES for the flow around a simplified car

International Journal of Heat and Fluid Flow 54 (2015) 236–249

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Key factors in the use of DDES for the flow around a simplified car N. Ashton ⇑, A. Revell Modelling & Simulation Centre, School of Mechanical, Aerospace & Civil Engineering, University of Manchester, UK

a r t i c l e

i n f o

Article history: Received 21 July 2014 Received in revised form 2 March 2015 Accepted 2 June 2015 Available online 23 June 2015 Keywords: Hybrid RANS–LES DDES Elliptic relaxation Ahmed body Zonal methods Synthetic turbulence

a b s t r a c t The Ahmed car body represents a generic vehicle exhibiting key aspects of the 3D flow arising due to standard automobile designs. It is recognised to be a challenging test case for the turbulence modelling community; combining strong separation with a pair of counter-rotating vortices, which interact to produce a downstream recirculation region. In recent years this case has been extensively studied using a range of methods, with varying success. In general, conclusions have been made on the basis of the standard form of each model, while in the present work we focus on variants of the common Delayed Detached-Eddy Simulation (DDES) approach, in order to demonstrate its sensitivity to commonly varied aspects of its usage. We demonstrate that variations in the usage of a single approach can easily be of the order of those observed when using different approaches. Previous studies, reconfirmed here, indicate that the majority of standard single point closure turbulence models are unable to provide a satisfactory prediction of the recirculating flow region aft of the body. This holds regardless of mesh resolution, model selection or numerical scheme. These models under-predict levels of turbulence over the slanted back, leading to over-prediction of the size of the separation region. DDES can offer an improved prediction although, while better than URANS, the use of DDES in its standard form still falls short of equivalent results obtained from either wall modelled or wall resolved Large Eddy Simulation. In the present work we investigate four aspects of DDES in an attempt to identify mechanisms for improving DDES for this representative case: (1) the underlying RANS model, (2) mesh resolution, (3) numerical scheme and (4) the addition of turbulent fluctuations. We observe that with insufficient mesh resolution the DDES models produce worse results than the URANS models. While first order methods are inappropriate, the more commonly selected second order upwind scheme is also demonstrated to have substantial adverse impact. As mesh resolution is increased the influence of the underlying RANS model diminishes. In a zonal RANS–DDES approach, the domain was split in two just before the rear slant, and the upstream RANS solution is used to inform a DDES calculation via the superposition of synthetic fluctuations at the interface. This technique is demonstrated to substantially improve the prediction, whilst also reducing the overall simulation cost by virtue of a smaller domain size. The injection of synthetic fluctuations provides a more accurate level of turbulence at the onset of separation and thereby overcomes the lack of resolved turbulence in the initial separated shear layer. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction Computational Fluid Dynamics has increasingly provided an important design tool for the automotive industry, used as a supplement to experimental studies. With a desire to reduce noise levels and improve fuel efficiency, reliable CFD simulations of the complex separated turbulent flow around vehicles is becoming an ever more crucial goal. Obtaining accurate estimates for

⇑ Corresponding author. E-mail addresses: [email protected] (N. Ashton), alistair.revell@ manchester.ac.uk (A. Revell). http://dx.doi.org/10.1016/j.ijheatfluidflow.2015.06.002 0142-727X/Ó 2015 Elsevier Inc. All rights reserved.

aerodynamic forces is increasingly relevant with more and more focus on electric powered vehicles where drag reductions offer increased vehicle economy. The validation of turbulence simulation methods for such detailed geometries is difficult, since adequate mesh refinement generally remains beyond the reach of most users. As such, clear conclusions are rarely found in these cases. The choice between RANS and LES methods is dependant on many factors, such as accuracy requirements, level of available computer resource, of the level required physics. For flows that are largely steady and for which only mean quantities are of interest, RANS modelling is often a suitable choice. Indeed in many

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cases where only moderate accuracy is required (i.e to within 5–10% of an integral force) or when the objective is simply to capture general trends, then RANS models offer a fast and much simplified process of mesh creation and computation. Many large aerospace and automotive companies continue to use steady RANS models (of the one or two equation variety) as the backbone of their CFD process precisely for these reasons. The Ahmed body (Ahmed et al., 1984) is a generic car geometry, as illustrated in Fig. 1; comprising a flat front with rounded corners and a sharp slanted rear upper surface. More recently the same geometry has been tested by Lienhart and Becker (2003), who performed a detailed experimental study at a lower Reynolds number, which included LDA measurements of the mean and fluctuating velocities as well as on-surface oil flows. This was undertaken for slant back angles of both 25 and 35 , and the results confirmed many of the observations made by Ahmed. This test case has proven popular amongst the turbulence modelling community, largely on account of it’s industrial relevance, but also because of the significant challenge it poses to standard RANS based closures. For this reason it has been the focus of many validation studies in the literature, both for RANS (Jakirlic´ et al., 2002; Manceau et al., 2002; Haase et al., 2006) and Hybrid RANS–LES (Mathey and Cokljat, 2005; Serre et al., 2013; Caridi et al., 2012) methods alike. While representing a much simplified version of a car, The Ahmed body provides many of the flow features found due to everyday automobiles, such as the large 3D separation region behind the car body and the roll up of vortices at the rear corners (see Fig. 1). The wake behind the car body is a complex interplay between the counter-rotating vortices and the highly turbulent recirculating flow. The angle of the rear slant was found to be influential in the structure of the wake and the reattachment point. At 35 the counter-rotating vortices are weaker, which results in the flow being completely separated over the entire slant back of the vehicle. As the angle is reduced, the strength of the counter-rotating vortices is increased relative to the inertia of the recirculating turbulent flow. By 25 the vortices are strong enough to divert sufficient momentum back into the separation region, so that the flow is able to reattach half way down the slant. 1.1. Previous studies on this case The Ahmed body case has been the focus of several CFD workshops and Project Consortia; most notably the 9th and 10th ERCOFTAC workshops on refined turbulence modelling (Jakirlic´ et al., 2002; Manceau et al., 2002), where a comprehensive range of RANS models were applied to this flow. It was also selected as one of the test cases examined within the EU funded projects ‘FLOMANIA’ (Haase et al., 2006) and ‘DESider’ (Haase et al., 2007), in which both RANS and hybrid RANS–LES approaches were investigated. Success of the RANS-based studies (Jakirlic´ et al., 2002; Manceau et al., 2002; Haase et al., 2006) was found to be strongly dependent on the slant angle. At 35 , where separation occurs over the entire slant back, most of the RANS approaches, simple and more complex alike, captured the correct level of the turbulent stresses and indicated good agreement with experimental results. At 25 , where there is partial reattachment, the majority of RANS models systematically failed to predict the flow correctly. In general, they either failed to predict any separation at all, or when they did not predict the correct location of separation point and were thus unable to capture the correct size of the recirculation region. The source of this problem was identified in many instances to be an under-prediction of the turbulent stresses, resulting from the inability to correctly account for the turbulence resulting from the interaction of non-local, inertial range, turbulent structures.

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In addition to testing a broad range of RANS models, the DESider project (Haase et al., 2007) reported results from DES or DDES approaches (Spalart et al., 1997, 2006) based upon either the SST (Menter, 1994) or SA (Spalart and Allmaras, 1994). The focus was placed on the more challenging 25 case, initially employing relatively coarse grids in the range of 3–5 million cells. None of the methods were able to obtain good agreement with the experimental data, and substantial dependence on the underlying RANS model was demonstrated. The flow was also reported to be sensitive to small changes in the computational set-up, such as the grid or inlet conditions; though absolute conclusions were complicated by the range of codes used by the project partners (Haase et al., 2007). Interestingly, the importance of the flow immediately upstream of the rear slant was emphasised with respect to it’s role in the subsequent development of the separated shear layer, and the strength of the counter-rotating vortices. It was highlighted that the underlying RANS model can play a crucial role in this regard, since it is largely responsible for the flow up to this point; though as indicated later, this is role likely to be reduced with more adequate mesh refinement. Further studies for this case using DES variants have been reported by Kapadia et al. (2003), Serre et al. (2013), and Guilmineau et al. (2011), though none of were able correctly to convincingly match the experimental data, particularly in the immediate vicinity of the rear slant. In Serre et al. (2013) the separation region was over predicted for the 25 case with the SST–DES model even when an unstructured mesh of 21 million cells was employed. The source of the disappointing performance of DES approaches observed in some of these cases is difficult to pin down, though likely contributing factors include mesh resolution, model formulation (i.e DES or DDES) and numerics. An interesting study by Fares (2006), employed a Lattice Boltzmann (LB) code to the same case (at both slant angles) and demonstrated that this increasingly popular method is capable of similar results to those with more common methods; overcoming the standard uniform mesh limitation of LB by combining regions of embedded refinement and a novel interpolation scheme at a wall. The turbulence modelling scheme is reported as a Very Large Eddy Simulation VLES, and solves the turbulent transport equations discretised using a finite difference scheme. A number of LES studies have also been performed, again predominantly choosing to focus effort on the 25 case (Krajnovic and Davidson, 2004; Hinterberger et al., 2004; Minguez et al., 2008; Serre et al., 2013; Lehmkuhl et al., 2012). These studies were performed with a range of sub-grid scale models and wall-treatments and, while some were more successful than others, many failed to obtain the highest level of accuracy one might expect from this level of closure. The high-Reynolds number (Re ¼ 7:68  105 ) motivated meshes of up to 48 million cells, although even this falls short of the ideal resolution for a wall-resolved LES (Serre et al., 2013). Indeed, in summarising their comprehensive study, Serre et al. indicate that hybrid RANS–LES methods represent an attractive alternative, although mesh generation for these approaches requires particular care with regards to the RANS–LES interface. In summary, it is clear that the series of collaborative workshops and projects on this case have been invaluable in providing focus and a high degree of interrelated consistency, and have helped to forge many conclusions about the relative merits and downfalls of various approaches. Nevertheless, there remain some open questions, and several sources of error that are often impossible to negate; such as those arising from different codes and different user procedures. The issue of underlying numerics is particularly troublesome for approaches such as DDES, which must first be calibrated to each code before reliable results can be obtained (Haase et al., 2007). Such ambiguity raises the possibility

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x

orte

top v

corn

er v

base

orte

x

x

vorte

Fig. 1. (top) Dimensions of geometry (Serre et al., 2013); (btm) flow schematic for 25 case.

that conclusions about different model developments or refinements might previously have been formed on the basis of an incomplete set of observations.

LDDES ¼ LRANS  f d max ð0; LRANS  LLES Þ; 0" #3 1 m þ m t A; f d ¼ 1  tanh @ 8 pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 U i;j U i;j j y2

2. Variants of DDES used in the present work

where LLES ¼ WC DDES D and C DDES is an empirical parameter which needs to be calibrated. W is a correction term to ensure the model returns to the classical Smagorinsky form when in LES mode. Finally D is the filter width for LES, commonly taken to be the cube root of the cell volume for unstructured grids (although the max definition could also be used where, D ¼ maxðDx; Dy; DzÞ). In the expression for f d ; j is the von Kármán constant, and y is the distance to the wall. The function f d takes the value 1 in the LES region and 0 elsewhere. While DES and DDES are then Non-Zonal methods, the location of the switch from RANS to LES can be closely controlled by the user, e.g. via targeted regions of local mesh refinement at focal points in the domain. The original DES method was based on the Spalart–Allmaras (SA) model, as this was seen as ‘the most convenient length scale to inject D and turn a RANS model into a SGS model’ (Travin et al., 2000). Since then, many RANS models have been applied to DES & DDES, the most popular of which are the SA and k—x Shear-Stress Transport (SST) DDES models. In the present study an alternative version of DDES is also used; the u—f DDES model (Ashton et al., 2013) which uses the u—f underlying RANS model (Laurence et al., 2004). This has previously shown promising performance on several canonical flows and is used to assess the influence of the underlying RANS model on the predictive accuracy. The Improved Delayed Detached-Eddy Simulation (IDDES) method proposed by Shur et al. (2008) was introduced as a ‘Wall Modelled LES’ approach to prevent erroneous double counting of turbulence (from modelled and resolved sources) in the vicinity of the switch from RANS to LES. While the approach has been demonstrated to perform very well in a number of confined flows, it’s benefit for external flows remains to be clearly established. The recent EU ATAAC project (Advanced Turbulence Simulation for

Whilst wall-resolved LES is a far more accurate alternative to RANS modelling for unsteady flows, the method incurs a much higher cost; indeed prohibitively so for the practical study of high-Reynolds numbers flow. Hybrid RANS–LES methods solve RANS equations in regions where the flow is attached and thus more simple to model; switching to an LES formulation in regions where resolved content is required. The many approaches developed in this vein may be broadly split into two groups, Zonal and Non-Zonal methods, and the review paper of Fröhlich and von Terzi (2008) provides an excellent overview of recently developments in the context of this dichotomy. In Zonal methods one explicitly defines a certain portion of the flow as RANS and other as LES; usually via the non-dimensional wall distance, yþ , or an additional blending function. In contrast Non-Zonal methods employ an intrinsic function, based on a flow quantity or the mesh itself, to automatically switch between a RANS or LES approach. One of the more common hybrid RANS–LES methods in usage is Delayed Detached-Eddy Simulation (DDES) (Spalart et al., 2006), which is an improved version of the original Detached-Eddy Simulation (DES) (Spalart et al., 1997). DES is effectively super-ceded by DDES in the sense that spurious ‘Grid Induced Separation’ problems identified is the former are substantially reduced in the latter. DDES can be seen as a three-dimensional, unsteady model based on an underlying ‘off-the-shelf’ RANS model. It seamlessly joins a sub-grid scale model in regions where the grid is fine and outside of the attached boundary layer to a RANS model in all other regions. The principle of DDES is to modify the RANS length scale, LRANS , in such a way that it is instead based on the grid size and a blending function, f d :

ð1Þ

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Aerodynamic Application Challenges) (Schwamborn and Strelets, 2012) showed that in some cases IDDES performed worse than DDES and concluded that the superiority of the former over the latter remains case-specific. It was also noted that the interaction of solver-specific numerics with the quite detailed empirical functions within IDDES might lead to different behaviour in different codes. In the present study we test 6 models on the Ahmed body case on two different meshes, as indicated in Table 1. There are a vast array of similar methods that are available to the industrial practitioner, and while there is merit in benchmarking one against another, we have instead set out to provide a systematic study of a single approach in a single code. We selected the DDES model for its prominence amongst codes used in industry, and for its relative ease of implementation, with the intention that our findings might provide a benchmark for practitioners in the evaluation of the accuracy of their own results.

2.1. Embedded DDES In the present work we also test a zonal RANS–DDES methodology which we refer to as Embedded-DDES (E-DDES), whereby we use synthetic turbulence at the interface to pass from RANS to DDES, in a decoupled fashion. This is a development of previous work in this area by the authors (Poletto et al., 2012), and is motivated by the observation that there is insufficient turbulence at the point of flow separation at the rear slant. The idea also springs from the impressive results reported with the Zonal Detached Eddy Simulation (ZDES) approach of Deck (2005, 2011), and Laraufie et al. (2012), who have demonstrated LES accuracy with substantial cost savings for a range of compressible flow aerospace applications. Of particular note is the so-called ‘mode 3’ of ZDES, which is applied when the flow does not exhibit a large enough separation to generate sufficient resolved content, and thus requires the injection of synthetic turbulence upstream of the region of interest. In the present work we make no adjustment to the standard formulation of DDES, in contrast to mode 3 of ZDES, which sets an alternative definition of the filter width to reduce modelled content. We do not modify the length scale with respect to the standard form of DDES although we do use the cube root of the cell volume for the filter width as mentioned previously. For the inlet fluctuations, we use the recently developed Divergence-Free Synthetic Eddy Method (DFSEM) (Poletto et al., 2013); based on the methodology described in Jarrin (2008). A set of synthetic eddies are convected through a box that entirely surrounds the inlet plane; at which the mean quantities are prescribed by the precursor RANS model. This method, or similar, is already available in the most widely used CFD codes and as such is assumed to accessible to most industrial practitioners. An overview of the algorithm is provided below: 1. The user selects the location of the interface between RANS and DDES, defined by the surface, X. 2. Mean quantities are extracted from the precursor RANS; i.e. mean velocity, UðxÞ turbulence kinetic energy, kðxÞ, and an associated turbulent lengthscale rðxÞ, for x  X. 3. The ‘eddy box’ is defined as: maxfx þ rg; minfx  rg for x  X. 4. A suitable number N eddies are each assigned a position xk and intensity ak at random. 5. Eddies are convected through the eddy box, by xk ¼ xk þ Ub  Dt, where Ub is the bulk velocity calculated from the user imposed average velocity. Eddies that leave the Bounding Box are re-generated at the opposite surface. 6. u0 ðxÞ calculated on X and superimposed to u to generate the inlet condition.

Table 1 Summary of simulations. Mesh: (C) Coarse, (F) Fine. Numerics: (CD) Central Differencing (SOLU) Second Order Linear Upwind. Run

Method

RANScomponent

Mesh

Numerical scheme

Domain size

1 2 3 4 5 6 7 8 9 10 11 12

URANS URANS URANS URANS DDES DDES DDES DDES DDES DDES E-DDES E-DDES

SST SST u—f u—f SST SST SST u—f u—f u—f SST u—f

C F C F C F F C F F F F

CD CD CD CD CD CD SOLU CD CD SOLU CD CD

Full Full Full Full Full Full Full Full Full Full Zonal Zonal

The reader is referred to Jarrin (2008) and Poletto et al. (2013) for a more comprehensive description.

3. Computational grid and boundary conditions The geometry, as defined by Ahmed is illustrated in Fig. 1; the body has a length of L=H ¼ 3:625 with a height, H ¼ 0:288 m. In the experimental study, the body was mounted on four stilts, at a height of y=H ¼ 0:174, to permit flow under the car which is important for ‘ground effect’. To avoid additional mesh complexity, the stilts were here neglected, in line with previous numerical studies in the literature. Thus the body is fixed at the same height, y=H ¼ 0:174, above the floor. In view of the partial flow reattachment, we here focus on the 25 rear slant as it presents the greater modelling challenge. Two structured meshes were used to enable us to assess the influence of spatial resolution on model performance (shown in Figs. 2 and 3). We did not set out to undertake our own mesh independence study, since this has been performed extensively at previous workshops (e.g. Jakirlic´ et al., 2002; Manceau et al., 2002; Haase et al., 2006). Instead we have selected two of the more common mesh sizes, one representing a ‘Coarse’ mesh and a second representing a ‘Fine’ mesh. Both meshes have previously been employed in earlier studies.1 The Coarse mesh has 2.7 million cells while the Fine mesh has a total of 16 million cells. In addition, the Fine mesh has a greater level of refinement directed in the wake. The flow is at a Reynolds number of Re ¼ 7:68  105 based on the body height H and the free-stream velocity U 1 ¼ 40 m s1 . An inlet condition is imposed x=H ¼ 7:3 upstream of the body and an outlet condition is imposed x=H ¼ 20:3 downstream. A no-slip wall condition is imposed on the ground floor and car body, with slip walls applied to the wind tunnel walls. In the case of the simulations with the synthetic eddy method then the whole geometry was cut at x=H ¼ 1:73 and fluctuating velocity components were imposed based upon the time averaged velocity and turbulent quantities from a pre-courser RANS simulation. The time step is set to DtU 1 =L ¼ 2  104 and 1  104 for the URANS and DDES simulations respectively, which ensures a maximum CFL number of less than one. Each simulation was run for a total of 30 convective transit times (=TU 1 =L); time-averaging began after the initial 10 transit times. Time resolution and averaging period have been selected to as far as possible eliminate these factors from the ensuing sensitivity study. 1 The Coarse mesh is that used in FLOMANIA (Haase et al., 2006) and the Fine mesh was kindly shared by Prof. Krajnovic of Chalmers University (Krajnovic and Davidson, 2004).

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Fig. 2. (top) Coarse and (btm) fine mesh. Dashed line indicates zonal interface.

Fig. 3. (left) Coarse and (right) fine meshes at the symmetry plane (top) and x=H ¼ 0:5 (bottom).

All calculations were performed using the open-source software Code Saturne, developed by EDF R&D (Archambeau et al., 2004) and highly optimised for HPC (Fournier et al., 2011). The temporal discretization is second order, while a hybrid numerical scheme based on a blend of central differencing (for the LES zones) and upwinding (for the RANS zones) is used to discretize spatially the convective terms (Ashton et al., 2011). The use of the hybrid numerical scheme is essential as applying a pure central differencing scheme with some percentage of upwinding would add unwanted numerical dissipation in the LES regions. With the hybrid numerical scheme we ensure low numerical dissipation in the LES regions, but allow additional numerical dissipation in the form of a 2nd order upwind scheme in the boundary layer and all other regions in RANS mode.

3.1. Mesh refinement 3.1.1. Near wall mesh refinement We first assess the influence of the mesh resolution for the methods applied to the present test case. The primary metric for measuring the quality of the numerical grid is generally the size of the wall-adjacent cells in dimensionless wall units. For a wall-resolved LES (WRLES) of a turbulent channel flow, this constraint is reported by Piomelli and Chasnov (1996) to be 50 < xþ < 150; yþ < 2; 15 < zþ < 40, where xþ ; yþ ; zþ refer to the dimensionless stream-wise, wall-normal and lateral components respectively. Although these values might be expected to be somewhat increased for usage with DDES, since only the RANS model should be active adjacent to a wall, there is some difficulty in

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N. Ashton, A. Revell / International Journal of Heat and Fluid Flow 54 (2015) 236–249 Table 2 Mesh resolution assessment ranges for the Ahmed car body. Mesh

xþ

yþ

zþ

D=g

mt =m

WRLES (Piomelli and Chasnov, 1996; Fröhlich et al., 2005) Coarse (present) Fine (present)

50–150 30–500 10–150

<2 0.1–0.9 0.1–0.5

15–40 5–80 5–70

12 120–250 40–110

1–10 140 40–50

prescribing an equivalent set of ‘Best Practice’ values. Instead, one can consider the values required for WRLES to be something of an upper limit on the resolution that one would expect to need for DDES. In the present work, these values have been extracted from the results reported by the SST–DDES model for the rear slant, and are listed in Table 2 for comparison. The near-wall resolution of the Fine mesh appears to be almost sufficient for WRLES, while those from the Coarse mesh also satisfy the constraint in the wall-normal and span-wise directions. The Coarse mesh is however noticeably less resolved in the stream-wise direction; by a factor of greater than 3 in the worst case. Overall both might a priori be deemed satisfactory for a hybrid RANS–LES approach, and one might even expect the Fine mesh suitable to do reasonably well for an LES. However, while the size of the wall-adjacent cells is an important metric, it is not sufficient alone. 3.1.2. Interior mesh refinement Another measure of the mesh refinement must be provided for the interior regions of the domain, particularly those where significant turbulence is expected such as a separation zone. Following Šaric´ et al. (2006), a suitable metric for this is the quantity D=g, which represents a ratio of the filter width D to an estimate of 1=4

the Kolmogorov length-scale, g ¼ ðm3 =eÞ . In computing the latter quantity, the dissipation rate can be taken using the length-scale determining equation, e or e ¼ 0:09kx, though clearly this remains a significant approximation. As reported in Fröhlich et al. (2005) the target value for this ratio with LES is D=g < 12, so as to ensure that a suitable amount of the dissipation range is resolved. For the fine mesh, over the slant back of the car body (0:52 < x=H < 0), this ratio reaches as high as D=g ¼ 40 in the initial separated shear layer and D=g ¼ 110 behind the car body. For the coarse mesh this is D=g ¼ 120 in the initial separated shear layer, rising to D=g ¼ 250 behind the car body. For both meshes, the number of cells covering the vorticity thickness in the initial separated shear layer is approximately 15, although the coarse mesh has fewer cells further downstream of this region. A further measure of the suitability of the interior grid refinement is the ratio of the modelled turbulent viscosity to the molecular viscosity mt =m, which provides an indication of the ratio of the modelled and resolved contributions to the dissipation (Fröhlich et al., 2005). For the fine grid this parameter reaches mt =m ¼ 50 in the initial separation region and reduces to approximately mt =m ¼ 40 further downstream of the body. Ideally this value should be as close to 1 as possible (Fröhlich et al., 2005). However for the coarse mesh, it rises to mt =m ¼ 140 in the initial separation region. As expected, the relative low resolution of the coarse grid at the rear of the body implies that the underlying RANS model retains a significant contribution to the overall turbulence level. Thus the coarse nature of this grid means that the refinement observed at the wall is not sustained as the mesh expands away from the wall. Such a high value is not ideal and suggests that the LES mode will not be fully capable of resolving the flow and additional dissipation from the sub-grid scale model (the underlying RANS model) is expected. While the wall resolution is satisfactory, the interior mesh resolution is then demonstrated to fall significantly short of that which would ensure an accurate WRLES. One would expect a

suitable WRLES computation of a flow at this Reynolds number to require a grid size of the order of 50–100 M cells. This demonstrates the need to extend any claims about mesh resolution beyond solely the measures obtained from the wall-adjacent cells. With these factors in mind it becomes more apparent as to why a wall-resolved LES is expensive for complex high-Reynolds number flows of this nature and, as such, demonstrates the practical need for Hybrid RANS–LES methods. 4. Results In the following we summarise our findings with respect to different aspects of the use of DDES for the Ahmed Body Flow: sensitivity to mesh, numerics and underlying RANS model. We also consider results using a zonal RANS–DDES approach, Embedded-DDES. 4.1. Sensitivity study Centreline profiles of mean stream-wise velocity and turbulence kinetic energy (TKE) (modelled + resolved components) are plotted for both sets of RANS and DDES models in Figs. 4 and 5. In the case of the RANS results, only a small variation is observed between the coarse and fine mesh, suggesting that a reasonable level of mesh convergence has been reached. This is particularly true in the vicinity of the rear slope where mesh density is likely to be higher. Indeed the differences are amplified in the region downstream of the body, Fig. 4(b) and (d), where the mesh is beginning to coarsen more rapidly. The difference between the two RANS models is relatively small, considering the spread of RANS results from the ERCOFTAC workshops on refined turbulence modelling (Jakirlic´ et al., 2002; Manceau et al., 2002). However we are using the same code, mesh and numerics, and both are low-Reynolds number models with limited sensitivity to streamline curvature. Despite quite different formulations, they are implemented with similar (standard) model limiters and numerical options. They both fail to correctly predict the level of turbulence at the point of separation. Significantly, it can be seen from Fig. 4c) that there is a large under-prediction of TKE at the centreline versus the experiment. For the DDES results there is a dramatic shift from the coarse to fine meshes as shown in Fig. 5. The mean velocity profiles for the coarse mesh are particularly far from the experimental values. The level of turbulence over the rear slant is initially under-predicted such that there is very little modelled or resolved turbulence in this region. The consequence of this under-prediction of the turbulence level is to have almost no turbulent mixing, and as a result the flow stays completely separated over the rear slant. This under-prediction of both the modelled and resolved turbulence is what has become known as the ‘grey-area’ problem; i.e. when the model operates in LES mode as a function of insufficient resolution, rather than because of a realistic representation of the turbulent length scale (see Eq. (1)). In the vicinity of the separated flow over the rear slant, the modelled turbulence would be expected to be low and the resolved turbulence level high. If the grid is too coarse, the numerical scheme too dissipative

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500

380 360

Ux

(b)

400

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Z (mm)

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(a) U x

300

Exp. SST RANS COARSE SST RANS FINE ϕ − f RANS FINE ϕ − f RANS COARSE

280 260 240 -240

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0 0

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400

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(c) k

300

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Fig. 4. RANS results: Mean stream-wise velocity (top row) and mean turbulent kinetic energy (btm row) on both coarse and fine meshes.

380

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Ux

400

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Exp. SST DDES COARSE SST DDES FINE ϕ − f DDES COARSE ϕ − f DDES FINE

280 260 240 -240

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Z (mm)

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(c) k

300

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X (mm)

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X (mm)

Fig. 5. DDES results: Mean stream-wise velocity (top row) and mean turbulent kinetic energy (btm row) on both coarse and fine meshes.

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With regards to the sensitivity to the underlying RANS model, little differences have been observed between the SST DDES and /—f DDES formulations tested here. This is perhaps not surprising given that these models on their own return fairly similar predictions (Fig. 4). Certainly, we expect the influence of the underlying RANS model to diminish with increasing mesh resolution, since its region of activation will be reduced. Within the parameters studies in this work, the results presented above (e.g. Figs. 5 and 6) indicate that the mesh is the dominant factor, followed by the numerical scheme. The underlying RANS model is here the least influential factor in the simulations. Fig. 7 plots the mean streamlines at the centreline for all six models tested; 3 approaches (URANS, DDES and E-DDES) for both RANS variants; SST and /—f . All are from the fine mesh using the hybrid numerical scheme (UDS for RANS and CDS for LES regions). There are clear differences between URANS, DDES and E-DDES, while underlying RANS model again appears to have a far lower impact. The main feature of note is the size of the first recirculation bubble along the rear slant; only with E-DDES is this captured correctly. In the next section we compare the three approaches in more detail; although since differences between the RANS variants is low, we restrict comparison to SST-based models, since their performance is marginally superior to the /—f based models. 4.2. Comparison of different methods With the objective of evaluating differences between SST versions of URANS, DDES and E-DDES, we compare profiles of mean TKE, stream-wise velocity, wall-normal velocity and wall-normal Reynolds Stress in Fig. 8, and mean TKE, stream-wise velocity and turbulent viscosity ratio in Fig. 9. Additionally we show contour plots for mean TKE and stream wise velocity in Figs. 10 and 12 respectively and Iso-contours of the Q-criterion in Fig. 13. Fig. 8 offers a comparison of the centreline values of mean velocity and TKE in the context of the figures reported in the last section. The benefit of incorporating fluctuations upstream of the separation point is clear; E-DDES is the only method able to provide the correct level of turbulent kinetic energy, with a marked improvement in both plots over DDES. This in turn leads to a very good prediction of the size of the recirculation region as indicated in Fig. 7. Fig. 9 illustrates this clearly, where only E-DDES captures the correct velocity profile and turbulence levels. It is noticeable that DDES has a higher level of turbulent viscosity ratio, which may be damping the transition from modelled to resolved turbulence. Fig. 10 provides indication of the fraction of turbulence which is resolved by the model, kr =k, as well as how the total turbulence k ¼ km þ kr from each method compares with the

380

380

360

360

340

340

320

Z (mm)

Z (mm)

or the DES blending function too strong, the separated shear layer does not generate resolved content, but as it is in LES mode the turbulence viscosity ratio is computed to be very low. For the coarse mesh this is very clearly observed from plots of TKE in Fig. 5. Fig. 5(a) and (b) presents a stark difference in the prediction of the flow between the 2.7 M and the 16 M cell meshes, whereby the coarse mesh returns a worse solution than either of the URANS models on the same mesh. This reiterates an important message to potential industrial users. The authors believe that for this flow the coarse grid is too coarse to be considered a proper DDES. In regions where LES mode is active, the mesh resolution should follow as much as possible the same guidelines as a standard LES. DDES exhibits here a strong mesh dependency, and there is an ‘accuracy drop-off’ at low resolutions. The method can only deliver improvements over standard URANS models when the grid resolution is sufficient. Table 2 indicates that for the coarse mesh the turbulent viscosity ratio reaches a value of 150 near wake region; while a well resolved LES would be expected to be in the region 1–10. The authors observations of industrial application of (D)DES would suggest that a value is 150 is commonly observed, and thus the use of URANS would be preferable at this level of refinement. An example of this finding in an applied context is reported clearly in the work of Gant (2010). The numerical scheme for the discretisation of the convective terms also has a noticeable impact on the accuracy of the simulation. While it is commonly accepted that 1st order schemes in space and time are inadequate for turbulence simulation approaches, there is more ambiguity about the effect of second order upwind schemes, which the authors have observed in industrial application of DDES. Fig. 6 provides a comparison of the mean stream wise velocity and TKE, on the fine mesh for both RANS versions of DDES. Results clearly demonstrate the adverse effect of using second order linear upwind (SOLU) over the hybrid numerical scheme (which uses Central differencing (CD) in the LES region), this difference is of a similar order of magnitude for both RANS-variants of DDES. The use of a 2nd order upwind scheme adds additional numerical dissipation, as can be observed from the diffused profiles of TKE in Fig. 6. Since the resolved turbulence is damped, the total level of turbulence is lower, which in turn increases the size of the recirculation region. It is expected that for coarser meshes the effect of the numerical scheme would be even more noticeable. As previously mentioned, the inclusion of a small degree of upwinding is not uncommon amongst industrial application of (D)DES, since complex meshes often converge more readily in this case; although this is at the cost of solution accuracy. Clearly, in such cases of complexity and limited resources, one has to consider that URANS may be the better option.

Ux

300 280 260 240 -240

Exp. SST DDES SST DDES SOLU ϕ − f DDES ϕ − f DDES SOLU

-200

-160

-120

320

k

300 280 260

-80

-40

240 -240

0

X (mm) Fig. 6. Sensitivity of DDES to 2

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-120

X (mm) nd

order numerical schemes (on fine mesh).

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(a) Experiment [2]

(b) SST URANS

(c) SST DDES

(d) SST E-DDES

(e)

(f)

(g)

− f URANS

− f DDES

− f E-DDES

380

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Exp. SST URANS SST DDES SST E-DDES

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k

300 280

Exp. SST URANS SST DDES SST E-DDES

280

Z (mm)

Z (mm)

380

Z (mm)

Z (mm)

Fig. 7. Mean streamlines over the Ahmed car body, on fine mesh with central differencing.

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-40

0

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-80

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0

320 300 280 260

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-40

0

X (mm)

240 -240

X (mm)

Fig. 8. Comparison of different methods based on SST (URANS, DDES, E-DDES).

experiment. In the case of the former, a zero value indicates that all turbulence is modelled, while unity indicates that all is resolved. The results from the URANS model indicate that almost all

turbulence is modelled, suggesting that a steady state model is sufficient for this flow, as can be expected. In contrast, Fig. 10 indicates that for E-DDES, virtually all turbulence is resolved, i.e.

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0.0

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(j) x/ H = −0.688

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N. Ashton, A. Revell / International Journal of Heat and Fluid Flow 54 (2015) 236–249

340

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(k) x/ H = −0.618

0.0

50

100

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(l) x/ H = −0.553

Fig. 9. Mean profiles of stream wise velocity, turbulent kinetic energy and turbulent viscosity ratio for different methods based on SST (RANS, DDES, E-DDES).

kr =k > 0:9 for most of the flow. In the case of DDES, this ratio is on a par with that from the E-DDES across most of the wake. Crucially though, kr =k falls below 0:6 at the beginning of the rear slant, indicating that in this region there is less resolved content. This is a clear example of the ‘grey area problem’ of standard DDES, in that there is a noticeable transition period where resolved turbulence is insufficiently high (Mockett, 2009). It is then unsurprising that the addition of turbulence fluctuations as in E-DDES helps to circumvent this issue by providing the correct levels of resolved turbulence. The absolute levels of TKE can be seen from the second row of Fig. 10 and the different responses to a flow separation are clearly indicated. The URANS produces insufficient turbulence, while the DDES model is excessive. The E-DDES results are again in very good agreement with the experiment, indicating a narrow

peak of turbulence close to the start of the rear slant. Close examination, in conjunction with Fig. 8 reveals the extent of this narrow band to be 200 < X=H < 100, and the lack near wall turbulence predicted by the DDES model in this region is clear. In order to investigate the upstream influence of the RANS model further, Fig. 11 provides a plot of f d (see Eq. (1)) which blends between RANS and LES operation of the model. Surprisingly there is little difference between DDES and E-DDES along the rear-slant and in the region immediately upstream. In this area f d ¼ 1 away from the wall and f d ¼ 0 for a thin boundary surrounding the wall, where the model is in RANS mode. For E-DDES the introduction of synthetic turbulence reduces f d in the development region but it recovers to similar values as DDES prior to the separation point. This reduction in the f d function for

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URANS

DDES

EDDES

EXPT

Fig. 10. Iso-contours of (top): ratio of resolved to total TKE (kr =k). (btm) Total TKE (k ¼ km þ kr ). SST model variants compared to Expt (Lienhart and Becker, 2003).

(a) DDES

(b) E-DDES

Fig. 11. (top) Mode of operation Black:RANS, White:LES and (btm) DDES blending function f d for both SST DDES model and SST E-DDES model.

E-DDES may also contribute to the lower turbulent viscosity observed in Fig. 9. Fig. 12 displays contours of stream-wise velocity compared to experiment, for three YZ planes perpendicular to the flow direction, commencing just after the end of the Ahmed body. The plot of iso-contours of Q-criterion in Fig. 13 is also helpful in understanding the predicted development of these vortices. It can be seen that the URANS model under-predicts the strength of the counter-rotating vortices, a key feature of the flow at 25 , since it is responsible for the partial reattachment further down the slant. This is in part down to a well documented failing of the eddy viscosity assumption for streamline curvature effects. Insufficient physics renders the model unable to correctly respond to the solid body rotation, which should act to re-laminarise the turbulence and help prolong the duration of the vortex. Instead the turbulence is allowed to grow exponentially and the vortex is diffused. Another example of this effect is found in recent work on the NACA0012 wingtip (Craft et al., 2006; Revell et al., 2006) which demonstrated the advantages of using Reynolds Stress Transport models in this context.

From Fig. 13 it appears that while the URANS results indicate the formation of these counter-rotating vortices, they appear somewhat reduced in magnitude (thinner) compared to the equivalent structures from DDES and E-DDES, which is consistent with the previous remarks. There are also very few resolved structures visible from the URANS, in contrast to the other models. The equivalent plot for DDES indicates the formation of structures just downstream of the rear slant, while for E-DDES, the small synthetic turbulent structures introduced at the zonal interface are clearly visible. Synthetic structures introduced further away from the body, where the mesh is coarse, will dissipate quickly and do not appear to adversely impact the prediction accuracy, though further work is currently in progress to limit the inclusion of these synthetic structures to relevant regions of the boundary layer. 4.3. Computational expense vs. accuracy Table 3 shows the computational expense for each method on the fine mesh. The u—f model incorporates an additional transport equation for /, the normalised wall-normal velocity fluctuations,

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URANS

DDES

E-DDES

EXPT

Fig. 12. Iso-contours of mean velocity at (top to bottom); x=H ¼ 0; x=H ¼ 0:27 and x=H ¼ 0:69 for the SST model variants.

(a) URANS

(b) DDES

(c) E-DDES

Fig. 13. Iso-surfaces of the Q-criterion coloured by mean stream-wise velocity over the Ahmed car body for the SST model variants. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 3 Computational expense of each RANS and DDES formulation (based on fine mesh and 384 CPU cores). Method

Wall time per iteration (s)

Total compute time

Relative cost

SST URANS

2  10

4

8.1

337 h (14 days)

1

SST DDES

1  104

7.2

600 h (25 days)

1.78

SST E-DDES

1  104

6.3

525 h (21 days)

1.557

4

9.2

383 h (16 days)

1.14

1  104

8.1

675 h (28 days)

2.00

1  104

7.1

591 h (24 days)

1.75

u—f URANS u—f DDES u—f E-DDES

Time step

2  10

as well as an equation for the f, the elliptic operator. As such, it is on average 13% more costly than the SST based models. Both were used with the same time-step. Clearly the URANS cost can be reduced considerably if a steady solver is used instead; here we continued the averaging simply to ensure that each case was run for the same simulation time. Since the DDES models naturally resolve turbulent structures of higher frequency than the URANS, the time step was reduced to ensure the CFL was below unity. This reduction in time step implies additional iterations to achieve the same total simulation time, and the simulation time itself must also be increased if time-averaged quantities are sought. In the case of the E-DDES computations, the DFSEM incurred additional

CPU time compared to the non-embedded approach but remained faster than the DDES approach on account of the 25% reduction in grid size in this case, compared to the full domain as used with DDES alone. 5. Conclusions The Ahmed car body is a classic example of a complex 3D turbulent flow in action. The link between the strength of the counter-rotating vortices and the initial turbulence in the separated shear layer give rise to the differences observed between the DDES formulations.

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The results presented in this work, through an analysis of the modelled and resolved turbulence levels point to three main conclusions. Firstly the two RANS models, tested with sufficient near-wall resolution and interior refinement, are unable to capture the correct level of turbulence in the initial separated shear layer. The consequence of this under-prediction of the turbulence is reduced turbulent mixing and an over-prediction of the separated region over the slant of the car body. Secondly a DDES simulation (regardless of the underlying URANS model) with insufficient grid resolution produces worse results than either URANS model. The grey area problem is clearly observed, with a lack of both modelled and resolved turbulence in the initial separated shear layer, i.e. the transition to resolved turbulence is delayed such that an over-prediction of the separation region is observed. The DDES simulation on a finer mesh improves the prediction of the overall turbulence in the initial separated shear layer (due to a faster production of resolved turbulence than the coarse mesh) compared to both the URANS and coarse DDES results, though there is still a delay in the simulation of resolved content at the onset of flow separation. In addition the adverse effect of using a second-order upwind is demonstrated. Finally it is shown that an embedded one-way coupling method based upon the framework of DDES, supplies the missing resolved turbulence and enables an accurate prediction of the level of turbulence kinetic energy in the initial separated shear layer. Compared to non-embedded DDES and URANS, the injection of synthetic turbulence upstream of the separation point allows for the correct level of flow separation and thus the correct balance with the counter-rotating corner vortices in this flow. The resulting separation zone is correctly predicted and the grey-area problem is reduced. In addition to improving predictive performance of DDES, E-DDES provides a means for a considerable reduction in computational resource, since the domain size can be shortened. Furthermore, it is introduced using existing modelling approaches, already available in commercial and open-source CFD codes alike, which should encourage other users to investigate this methodology further. We believe that zonal methods where RANS and hybrid RANS– LES regions are manually specified can be useful for industrial applications where the flow physics is largely known a priori. Non-zonal methods such as DDES can still be used in specified regions, as these models offer automatic selection of RANS or LES in the near-wall region. Ongoing work is planned to refine the E-DDES approach to limit the production of synthetic turbulence to regions where they are needed. We also plan to evaluate its effectiveness at different locations upstream of the onset of flow separation, The modelled turbulence is almost gone but the resolved turbulence does not develop quickly enough to produce the correct level of turbulence at the very beginning of the separated region. A longer term objective is to introduce a full two-way coupling between the RANS and LES zones as this would be required for more complex geometries.

Acknowledgements The authors gratefully acknowledge computational support from Barcelona Supercomputer Centre (BSC) and also to the Hartree and STFC for the use of the Blue Joule Blue Gene Q machine. Part of this work was carried out under the EU project Go4Hybrid funded by the European Community in the 7th Framework Programme under Contract No. APC3-GA-2013-605361Go4Hybrid. The authors would like to acknowledge the assistance given by IT Services and the use of the Computational Shared Facility at The University of Manchester. The work also made use of the facilities of N8 HPC, provided and funded by the N8

consortium and EPSRC (Grant EP/K000225/1), and coordinated by the University of Leeds and the University of Manchester.

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