Improved Delayed Detached-Eddy Simulation of Turbulent Vortex Shedding in Inert Flow over a Triangular Bluff Body

Abstract

The Improved Delayed Detached-Eddy Simulation (IDDES) is a modification of the original Detached-Eddy Simulation (DES) design to incorporate Wall Modeled Large Eddy Simulation (WMLES) capabilities and to extend the class of flows suitable for this methodology. For thin attached boundary layers, typically seen in external aerodynamic flows, the DES branch of the model is active, whereas with thick boundary layers, typically seen in internal flows and also wake flows, the WMLES branch is active, thus providing a numeric method suited to handling most flow cases automatically. The flow over a triangular bluff body is used to validate the suitability of the IDDES model and compare the results with experimental, DDES, and LES data. The IDDES model is found to be relatively accurate when compared with the experimental results, with recirculation length, streamwise velocity, and Reynolds stresses all showing good agreement with the experimental data. However, when compared with the DDES model, there is a ~4% overprediction of the recirculation length using the same mesh and numerical scheme. The code, with its extra complexity, is also ~3% slower to solve. The IDDES model has also been tested against different meshes, and the results show that even for a coarse mesh, there is still good agreement with the experimental data.

1. Introduction

Triangular bluff bodies are typically used as flame holders for industrial combustion systems. When a triangular bluff body is placed within a flow stream, the velocity in the wake behind the body is smaller relative to the whole flow field. A fuel can be ignited in this region, and the resultant flame will propagate within the combustion chamber. The flow field downstream is highly turbulent due to the vortex shedding caused by the bluff body.

Sjunnesson et al. conducted experimental results for this test case on a test rig at Volvo [1,2,3], and it has been added to the European Research Community On Flow, Turbulence, And Combustion (ERCOFTAC) knowledge base as a combustion application challenge. For this study, combustion has not been investigated, and only the inert flow experiment is considered for this analysis. Johansson et al. [4] conducted one of the earlier studies of the inert flame holder using the k-ε turbulence model and found very good agreement with respect to the streamwise velocity profile at the centerline, the profiles for velocity, and turbulent kinetic energy. However, being a RANS simulation, it was only able to recover time-averaged results.

Hasse et al. [5] performed a DES involving an Unsteady Reynolds-Averaged Naiver–Stokes (URANS) SST model and Large Eddy Simulation (LES) combination. Their results generally agree better with the experimental results [1,2,3] than the URANS simulations, particularly for the length of the recirculation zone.

Lysenko et al. [6,7] conducted a comprehensive numerical study for the inert and reactive flow conditions of this experimental test case using Unsteady Reynolds-Averaged Naiver–Stokes (URANS), Scale-Adaptive Simulation (SAS), and Large Eddy Simulation (LES). The standard k-ε [8] model was used in the URANS simulations. For SAS, the model of Menter and Egorov [9] is based on the k-ω SST [10] with updated coefficients [11] and High-Wave-Number damping [12] based on the Wall-Adapting Local Eddy-viscosity (WALE) model [13]. Finally, for the LES calculations, the k-equation eddy viscosity subgrid-scale model [14] and the Smagorinsky model [15] were used. Results were compared with the earlier work of Hasse et al. [5], who compared Detached Eddy Simulation (DES) and LES. The results of these studies show that for the inert flow, the recirculation length (L_rc) behind the triangular bluff body was predicted with a reasonable level of accuracy between all models, as was streamwise velocity and turbulent kinetic energy.

Spalart et al. [16] first coined the term DES in 1997. The central idea is that it is expensive to conduct LES in the attached boundary layer due to the requirement that the grid cells for LES need to be isotropic. As the boundary layer becomes thinner at higher Reynold’s numbers, the grid requirement scales accordingly, making LES prohibitively expensive for most practical applications. It is, however, inexpensive to solve the free flow turbulence away from the walls with LES. RANS does not have the same boundary layer requirement and, as such, is relatively cheap for solving thin boundary layers with a reliable prediction of separation. With this in mind, it can be seen that a model with LES for the areas of strongly separated flows, combined with a RANS model for the reduced grid requirements of the attached boundary layer, is an appealing combination.

This original proposal used the one-equation model of Spalart and Allmaras [17] with a modification to a DES-length scale. This means that in the boundary layer, the model will operate in a RANS mode, and outside the turbulent boundary layer, it will operate in LES.

Spalart foresaw a key weakness in the original DES concept where too fine grid resolution in the boundary layer would cause the length scale to switch from RANS to LES. This would result in LES being solved on a grid not sufficiently fine enough to resolve the turbulent scales and the modeled stress dropping below RANS levels, causing a phenomenon known as Modeled Stress Depletion (MSD). The result of MSD is a strong under-prediction of the skin friction, which can lead to very poor prediction of the flow separation, known as Grid-Induced Separation (GIS) [18]. Menter et al. [19] first proposed shielding the boundary layer regardless of grid size with a DES variant based on the k-ω SST turbulence model [11,20], and the first major modification to the methodology was published by Spalart et al. [21], called “Delayed” DES (DDES). This is now the default over the initial DES model and has been shown to greatly improve its robustness.

While the intention of DDES is to maintain RANS in the boundary layer and LES in regions of massive separation, there was a view that DES could work as a Wall-Modeled Large Eddy Simulation (WMLES) [22]. WMLES aims to resolve the largest structures of the boundary layer and, as such, is considered a higher fidelity method compared with DES. Building from this, Shur et al. [23] worked on a new DES variant, which blended DDES with WMLES, called Improved Delayed Detached-Eddy Simulation (IDDES). This model uses empirical blending function to switch between DDES and WMLES depending on the flow conditions and, like DDES, has been formulated to include the k-ω SST model [24].

In most engineering companies RANS using commercial code is the staple approach applied in CFD. More computer-intensive approaches of LES or even DNS can provide more accurate results; however, they are computationally more expensive and are typically reserved for academic or research rather than industrial applications. Indeed, it has been predicted that LES will not be ready for full-scale engineering applications until 2045 [25]. This means that companies that are limited in the time and money they can invest in CFD are limited to the less accurate RANS approach, and this is likely to remain the case even after 2045 due to the computational costs of LES. Recent advances such as IDDES offer a compromise between the efficiency of RANS and the accuracy of LES and offer an opportunity to replace industrial applications RANS in some areas. To this end, this paper will evaluate the accuracy that can be obtained using IDDES using a relatively modest mesh to ensure that the approach can be replicated within industry.

The objective of this study is to expand the DES methodology that has previously been conducted on this test case by Hasse et al. [5] to include the subsequent developments of the technique incorporated in IDDES, such as the inclusion of WMLES, so the use of IDDES in problems of this type involving vortex shedding at sharp corners was evaluated. This was performed using the bluff body, triangular cylinder geometry of the flame holder [1,2,3]. This study recreates the DDES study of Hasse et al. [5], the LES results of Lysenko et al. [6,7], and conducts a new study using IDDES methodology. Different grid resolution and numerical schemes are tested with the aim of developing a method for best practice for this type of test case. The results are validated against the experimental data from Sjunnesson et al. [1,2,3].

2. Numerical Methodology

For this study, the commercial code Star CCM+ version 12 is used for all simulations. The Navier–Stokes equations are discretized on a structured grid strategy using the finite volume method. All numerical simulations are incompressible implicit unsteady and are second-order accurate in time and space. A time step of 10⁻⁵ s ensures that the CFL number is below 1 in most of the domain. Both the Bounded Central (BC) [26] (a second-order central difference scheme blended with a second-order upwind scheme) and the Hybrid Bounded Central (HBC) numerical scheme are used within this study, where the HBC is defined as

$$F_{\mathrm{H}\mathrm{B}\mathrm{C}}=\left(1-\sigma \right)F_{\mathrm{C}\mathrm{D}\mathrm{S}}+\sigma F_{\mathrm{U}\mathrm{D}\mathrm{S}}$$

(1)

where $F_{\mathrm{C}\mathrm{D}\mathrm{S}}$ and $F_{\mathrm{U}\mathrm{D}\mathrm{S}}$ are the central and upwind estimates, and σ is designed as a limiter based on a tanh function that changes rapidly from around 1 to around 0. When σ is approximately 1, an upwind scheme is applied for the RANS simulation, while when σ is around 0, a central difference scheme is applied for the LES simulation. A complete description of σ is neglected here for the sake of simplicity, but a full description is provided by Travin et al. [27].

All simulations were run on the SCIAMA HPC facility at the University of Portsmouth. Each simulation used 20 cores and at maximum had 135 × 10³ cells per core.

2.1. Turbulence Modeling

Two turbulence models were tested within the present study, both of which are modifications of the original DES model by Spalart et al. [16] based on the one-equation model by Spalart and Allmaras [17]. The central idea behind DES, as explained in the section above, is to use a RANS model as the subgrid-scale model for LES by taking the minimum of the RANS turbulence length scale d and the cell length Δ = max(Δx, Δy, Δz) so that

$$\tilde{d}=\mathrm{m}\mathrm{i}\mathrm{n}(d,C_{\mathrm{D}\mathrm{E}\mathrm{S}}∆)$$

(2)

the constant C_DES is usually equal to 0.65 and was calibrated from decaying isotropic turbulence [28]. This means that in the boundary layer where d < C_DESΔ, the model will operate in a RANS mode, and outside the turbulent boundary layer where d > C_DESΔ, it will operate in LES.

Travin et al. [27] then modified the original DES model for the two-equation k-ω SST model. This model behaves in the same way as the original DES model, but the formulation will solve RANS k-ω in the boundary layer and a blended mix of LES with RANS k-ε elsewhere, where the RANS turbulence length scale reads as

$$l_{k-\omega }=k^{1/2}/\left({\beta }^{\ast }\omega \right)$$

(3)

when the RANS turbulence length scale is larger than the local grid spacing Δ, the model will switch to an LES model. Here, k is the turbulent kinetic energy, $\omega$ is the specific dissipation rate of turbulent kinetic energy, and ${\beta }^{\ast }$ = 0.09 is a constant. The only term that requires modification for DES is the dissipative term in the k-transport equation for RANS:

$$D_{\mathrm{R}\mathrm{A}\mathrm{N}\mathrm{S}}^k=\rho {\beta }^{\ast }k\omega ={\displaystyle \frac{\rho k^{\frac{3}{2}}}{l_{k-\omega }}},$$

(4)

where $\rho$ is the density of the fluid. This leads to a substitution of the length scale to modify the equation for DES:

$$D_{\mathrm{D}\mathrm{E}\mathrm{S}}^k=\rho k^{3/2}/\tilde{l}$$

(5)

where

$$\tilde{l}=\min \left(l_{k-\omega },C_{DES}∆\right),$$

(6)

and the $C_{\mathrm{D}\mathrm{E}\mathrm{S}}$ term is formulated using k-ω and k-ε models that are blended using Menter’s blending function [10]:

$$C_{\mathrm{D}\mathrm{E}\mathrm{S}}=\left(1-F_1\right)C_{\mathrm{D}\mathrm{E}\mathrm{S}}^{k-\epsilon }+F_1{C}_{\mathrm{D}\mathrm{E}\mathrm{S}}^{k-\omega }$$

(7)

and F₁ is a blending function between the k-ω and k-ε models. A complete description of F₁ is neglected here for the sake of simplicity, but a full description is provided by Menter [10].

The ‘delayed’ in DDES comes from an addition limiter that is introduced in order to reduce the risk of MSD and GIS. This modification to the equation is defined as

$$D_{\mathrm{D}\mathrm{E}\mathrm{S}}^k=\rho {\beta }^{\ast }k\omega ·F_{\mathrm{D}\mathrm{E}\mathrm{S}},$$

(8)

where

$$F_{\mathrm{D}\mathrm{E}\mathrm{S}}=\mathrm{m}\mathrm{a}\mathrm{x}\left({\displaystyle \frac{l_{k-\omega }}{C_{\mathrm{D}\mathrm{E}\mathrm{S}}∆}}\left(1-F_{\mathrm{S}\mathrm{S}\mathrm{T}}\right),1\right),$$

(9)

and F_SST can be taken as F₁ or F₂ [10] in the SST model.

The ‘improved’ in the IDDES of Shur et al. [23] builds another layer of complexity into the modeling strategy, as both WMLES and DDES branches are added to the code. The length scale $l_{\mathrm{I}\mathrm{D}\mathrm{D}\mathrm{E}\mathrm{S}}$ is modified to accommodate these two branches, defined as

$$l_{\mathrm{I}\mathrm{D}\mathrm{D}\mathrm{E}\mathrm{S}}={\tilde{f}}_d·\left(1+f_e\right)·l_{\mathrm{R}\mathrm{A}\mathrm{N}\mathrm{S}}+\left(1-{\tilde{f}}_d\right)·l_{\mathrm{L}\mathrm{E}\mathrm{S}}$$

(10)

The two blending functions complete the model. The first blends between DDES and WMLES and is defined as

$${\tilde{f}}_d=\mathrm{m}\mathrm{a}\mathrm{x}\left[\left(1-f_{dt}\right),f_b\right].$$

(11)

The second blending function is aimed at preventing the excessive reduction in the RANS Reynolds stresses, which is a known problem between RANS and LES interfaces. This elevating function is considered instrumental in combating log-layer mismatch and is defined as

$$f_e=f_{e2}·\max \left[\left(f_{e1}-1.0\right),0.0\right].$$

(12)

Here, $f_{dt},$ $f_b,$ $f_{e1}$, and $f_{e2}$ are functions to switch between the WMLES and DDES. For a full derivation of these blending functions, the reader is encouraged to view the work of Shur et al. [23] and Gritskevich et al. [24].

In summary, there are three switches that occur within the IDDES simulation: between RANS and LES, between k-ω and k-ε, and between WMLES and DDES. The switch between RANS and LES is covered by the length scale in Equation (6), which depends on the k and ω but is primarily controlled by Δ, that is, the mesh spacing, with LES occurring where the mesh is sufficiently fine. Coupled with this, an upwind scheme is applied for RANS simulations that is too dissipative for LES. Thus, there is an associated switch from upwind to central difference, following Equation (1) [27]. The switch between k-ω and k-ε is determined by F₁ in Equation (7), with a full discussion and definition provided in [10]. The switch from this DDES model and WMLES is the final component of IDDES. This is performed through Equations (10)–(12), with full details provided in [23,24].

2.2. Domain and Boundary Conditions

The computational domain for this study is shown in Figure 1, following the experimental setup in [1,2,3]. The triangular flame holder, with a length L = 0.04 m at its rear face, spans the computational domain. The inlet is placed 25 L upstream of the X = 0 L position and is set as a velocity inlet with an average velocity of 16.6 m/s. The actual velocity distribution and turbulence profiles of the inlet are obtained from a precursor RANS simulation of a channel flow to match the experimental domain length. The outlet is placed 25 L downstream of the X = 0 L position and is set as an atmospheric pressure outlet, with a reference pressure of 101 kPa. The lateral boundaries in the Z direction have a span of 1 L and are set as symmetry planes with a periodic condition. The bottom-to-top boundary is at Y = 0 L and Y = 3 L, and these are set as no-slip walls, as is the flame holder itself. The boundary layer thickness on the top and bottom walls in the region of the flame holder and where the subsequent results are presented was found to be no greater than 0.4 L. The red lines are the positions where subsequent results are presented.

2.3. Computational Mesh

In this study, all meshes are created within Star CCM+ version 12 using a 3D multi-block-structured grid strategy. Three different grids were designed using the best-practice guidance for DES from Spalart and Streett [29], with the coarsest grid having a cell count of 621,000 cells, a medium resolution grid consisting of 1,276,000 cells, and a fine grid with 2,700,000 cells. The turbulence length scale was obtained from the precursory RANS simulations following the methodology of turbulent flows by Pope [30], and the LES mesh size was selected to be 1/6 of this for the fine grid. The coarser meshes were also considered to evaluate the performance of the IDDES approach in cases with a lower computational cost. A zoomed-in cross-section of the coarsest grid is shown in Figure 2.

For all grids, a total of 45 grid cells are placed in the Z direction to ensure that the grid is as isotropic as possible in the shear layer region. These meshes were designed in order to resolve the boundary layer, with the cells close to the walls being refined to ensure that the boundary layer is suitably captured and a first cell wall normal distance of y⁺ < 1.

3. Results and Discussion

The results are presented here in terms of the effect of the different models and the computational mesh refinement.

3.1. Effect of Turbulence Modeling

Table 1. Case descriptions.

Case Name	Numerical Scheme	Nx	Ny max	Nz	Ntotal (×106)	Lrc	% Difference
Experiment	-	-	-	-	-	1.33	-
LES	BC	450	150	45	2.70	1.14	19.00
DDES Fine	HBC	450	150	45	2.70	1.20	13.26
IDDES Coarse	HBC	270	60	45	0.62	1.20	12.94
IDDES Medium	HBC	360	90	45	1.28	1.27	6.00
IDDES Fine	HBC	450	150	45	2.70	1.16	17.23

shows the grid sizes for the whole mesh for various simulations, along with the predicted length of recirculation for the flame holder and the percentage difference between the experimental results. The experimental data from ERCOFTAC are provided without any experimental uncertainty. Subsequent IDDES results presented for the fine mesh LES simulations are generally considered accurate, are expected to be in excellent agreement with the experimental data, and are often used as a measure to compare approaches such as IDDES when experimental data are not available. However, here, the ‘fine’ mesh is not sufficiently fine to perform a standard LES simulation and is included for a direct comparison between IDDES and LES on the same mesh. DDES and IDDES simulations are presented on the same fine mesh to enable a comparison between the methods. One cannot mesh converge an LES study in the same way as a RANS simulation. The fine grid here performs LES in all domains, but it is not refined enough in the Y direction to give an accurate result. For the course grid, the resolution is not sufficient to perform a good LES simulation.

When comparing the experimental results [1,2,3] against the LES, DDES, and IDDES results of this study, it can be seen in Figure 3 that all models manage to capture all of the main flow physics with a very good level of accuracy. Figure 3 also shows the recirculation length at the back of the flame holder, and it can be seen that both DDES and IDDES are nearly identical, which is to be expected as the IDDES model uses the DDES branch of the model. However, there are two main areas of difference between the two, the first being from X/L = 0 to X/L = 1, where the DDES model follows the experimental data slightly better than IDDES, and the opposite can be said for X/L = 4 to X/L = 6, where IDDES is in better agreement with the experimental data. In regards to recirculation length, as shown in Table 1, the DDES and IDDES models show a ~13% and ~17% difference, respectively. This gives the IDDES model a 4% overprediction in the recirculation length compared with the simpler DDES code. The LES model underpredicts the recirculation length by 19% compared with the experimental results; this is most likely related to the inlet conditions of the synthetic turbulence generator used and an under-resolved mesh that would not be considered suitable for a typical LES simulation.

The velocity in the U direction and the resolved stresses for U and V are shown in Figure 4, Figure 5 and Figure 6, respectively. These profiles show that IDDES and DDES have near identical results, which is to be expected, as discussed previously, and both match the experimental data with a very good level of accuracy. The LES results in these figures show that there is not a great difference between the hybrid modeling and the LES at this grid resolution.

The Q criterion for both DDES and IDDES is shown in Figure 7 and Figure 8. There is a noticeable difference in the breakdown of the turbulence, with the wake of the DDES model showing more turbulent structures than that of IDDES, and the vortex cores are being sustained further downstream of the triangular bluff body. Figure 9 and Figure 10 show the RANS-LES and upwind-bounded central blending functions for the DDES and IDDES simulations. When looking at the numerical scheme and how it is used within the domain, it can be seen by comparing the DDES in Figure 9 and IDDES in Figure 10 that with IDDES, there is more upwind within the wake region downstream, and this is causing the vortex cores to be damped out. It is also shown that the regions of LES and RANS have been calculated very differently: with IDDES, there is a large clear area of LES in the wake, which is being solved predominantly by an upwind scheme. The DDES model seems to be able to pair the numerical scheme with the blending function better than IDDES.

3.2. Effect of Computational Mesh

Three different computational meshes were tested to ascertain the performance of the IDDES model. These different mesh cases are shown in Table 1 and in the streamwise velocity results in Figure 11. It can be seen that all three meshes perform reasonably well and manage to capture the general flow physics. The fine mesh is the only one to reach the correct velocity for the minimum behind the bluff body, whereas the medium and coarse meshes overpredict this by ~30%, which would not be considered acceptable. We also note that refining the grid does not necessarily improve the results obtained, for example, the recirculation length in Table 1 is most closely predicted using the medium mesh. This could be due to the fine mesh not being sufficiently fine for a good-quality LES simulation, while the medium mesh restricts the regions where LES is performed or the inlet conditions of the synthetic turbulence generator.

Figure 12, Figure 13 and Figure 14 show that for all meshes, there is very good agreement for the streamwise velocity and resolved stresses, with the only discrepancy being that the fine mesh overpredicts the resolved stresses on most of the profiles. Table 1 shows that the recirculation length for the course mesh is approximately the same as the DDES result, although the DDES result was able to better capture the magnitude of the velocity, which contributes to the overall accuracy. The medium mesh has the best recirculation length with a 6% difference from the experiments, but as with the coarse mesh, it still suffers from underprediction of the magnitude of the maximum velocity. The Power Spectral Density (PSD) of all three meshes is shown in Figure 15.

In terms of computation effort required, there has always been a concern that as these types of models become more complex, they will inherently be more expensive to run.

Table 2. Computational time for different cases.

Case Name	Solver Elapsed Time per Time Step (s)	Total Time (hrs)	% Difference to RANS
Typical RANS	6.1	113	-
DDES	6.4	119.5	5.4
IDDES	6.6	123	8.1
LES	4.6	86	−31.4

shows typical run times for each code against a base RANS simulation. When comparing just the solver elapsed time per time step, it can be seen that there is a ~5% difference in DDES with a typical RANS solution and a ~8% difference with IDDES, which, considering the extra information included within the solution, could be considered a worthwhile expense. In regards to the difference between DDES and IDDES, the ~3% difference does not improve the solution, and in the test case, the DDES model outperforms IDDES in speed and accuracy. The LES results are considerably better than those of all models due to its simplicity, but the overall accuracy on this mesh is not considered to be sufficient, and the mesh would need to be further refined to improve the results to an acceptable level. Additionally, the time step would have to be reduced as a consequence, both of which will have the obvious effect of increasing the computational time.

4. Conclusions

The k-ω SST IDDES model used within this study has been shown to perform well against the experimental results, with very good agreement for all the main flow features. The IDDES model overpredicts the recirculation length by ~4% compared with DDES and is ~3% slower in computing the solution.

The testing of different grids shows that IDDES can still perform relatively well on a coarse mesh, with a generally good agreement found with the velocity and Reynolds stresses. The recirculation length shows an underprediction of the magnitude of the time-averaged velocity in the streamwise direction through the center of the domain as the grid is coarsened. We also note that the recirculation length is most accurately predicted by the medium mesh, rather than the fine mesh. This may be due to the different balance of LES and RANS on the two meshes or the synthetic turbulence in the inlet conditions; however, fully explaining this is a topic for future research.

The IDDES model is considered the default within Star CCM+ version 12, and although it can compute the solution with relatively good accuracy for this test case, it is not an improvement over the DDES model, and careful model selection should be considered before implementing a DES strategy.