A new hybrid modelling method termed improved scale-adaptive simulation (ISAS) is proposed by introducing the von Karman operator into the dissipation term of the turbulence scale equation, proper derivation as well as constant calibration of which is presented, and the typical circular cylinder flow at Re = 3900 is selected for validation. As expected, the proposed ISAS approach with the concept of scale-adaptive appears more efficient than the original SAS method in obtaining a convergent resolution, meanwhile, comparable with DES in visually capturing the fine-scale unsteadiness. Furthermore, the grid sensitivity issue of DES is encouragingly remedied benefiting from the local-adjusted limiter. The ISAS simulation turns out to attractively represent the development of the shear layers and the flow profiles of the recirculation region, and thus, the focused statistical quantities such as the recirculation length and drag coefficient are closer to the available measurements than DES and SAS outputs. In general, the new modelling method, combining the features of DES and SAS concepts, is capable to simulate turbulent structures down to the grid limit in a simple and effective way, which is practically valuable for engineering flows.

Large-scale separations coupled with chaotic, nonlinear phenomena are often strongly pronounced in aerodynamic and industrial turbulent flows; the wide investigations for accurately predicting such flows have proven one of the most challenging CFD tasks, which is mainly related to the complex boundary layer and the troublesome broad turbulence spectra [

In contrast to DES which employs grid size as the deciding scale to produce LES content for unstable flows, the concept of scale-adaptive simulation (SAS) characterized with a second-order von Karman length scale (_{vk}) was firstly conducted by Menter et al. [

At this point, two appealing hybrid methods regarding DES and SAS have been briefly reviewed. It was expected that the LES solutions are still too expensive for industrial flows with high Reynolds number while the feasible RANS/LES approaches will dominate turbulence modelling in the next few decades [_{vk} scale with concept of scale-adaptive enables the solution smoothly varying from RANS region to LES-like region, which allows easier grid generation especially in the complex applications. Nevertheless, the current SAS mechanism by an additional source term is troublesome and may be less efficient to obtain a convergent resolution. Furthermore, the von Karman scale and grid scale are proportional to the shear layer thickness at near-wall region, while at the separated region, both of them are proportional to the resolved scales and responsible for the generation of spectral content with LES-level viscosity [_{vk} scale into the one-equation SA-DES model [

In this paper, a new hybrid model named improved scale-adaptive simulation (ISAS) is proposed by replacing the grid scale of the DES with the von Karman scale from SAS. The derivation and the constant calibration of the ISAS are described in the following sections. As for the model validation, the typical bluff body flow over a circular cylinder at Re = 3900 was selected mainly due to the rich flow physics and various references regarding both experimental reports and numerical studies. The ISAS performance was tested by numerical comparisons with available experimental data on both fine and coarse grids, and the main objective is to verify the capability of the underlying method in predicting the massively separated flows. It is worth noting that all the hybrid formulations for the following simulations are based on the two-equation shear stress transport (SST) model [

Benefiting from

All model parameters _{1} is defined as
_{1} = 0.5556, _{2} = 0.44, _{1} = 0.075, _{2} = 0.0828,

The turbulent eddy viscosity is expressed as
_{ij}_{1} = 0.31, and _{2} is a second blending function defined as

It is important to note that a limit function is used in recent production terms to fairly avoid the build-up issue in stagnation regions. The production terms are expressed as follows:

The transformation of the DES term to the SST model is firstly introduced by Strelets [_{DES}) directly acting on the dissipation term of the _{DES} is the model constant and is equal to 0.61, as the limiter should only be active in the _{kω}

As shown above, in DES, the switch from RANS to LES-like modes depends on the ratio between the turbulent length scale (modelled) and grid scale (resolved). When near walls, _{kω}_{DES}Δ and _{DES} = 1.0, the model function is passive and therefore suitable for the stable flows in a RANS mode, whereas, when typically in the separating regions with refinement grids, _{kω}_{DES}Δ and _{DES} > 1.0, a LES-like mode is gained.

The formulation of the SST-SAS model differs from the pure SST model by the additional source term (_{SAS}) in the _{kω}_{vk}) is achieved by

It is worth mentioning that the second derivative (_{vk}) composed of the first and second velocity derivatives is not sensitive to grid efforts but dynamically adjusts the resolution to the spectra content. Thus, it is an appealing element that has a strong theoretical foundation and enables flow mode to smoothly vary from RANS to LES-like mode. Moreover, when in the stable region, _{kω}_{vk} and _{SAS} = 0, returning a pure RANS mode, while when in the unstable region, _{kω}_{vk} and _{SAS} > 0 and the mode will turn to LES-like mode.

To combine the DES and SAS concepts based on the SST model, a simple way is to replace the grid scale in (

As known, using Reynolds averaging method, the standard two-equation models such as SST always produce a turbulent length scale (_{vk}, respectively, used in DES and SAS allow the solution to adapt to the resolved structures in a classical LES-like behavior and gradually reach to convergent solutions. The current relations between different length scales can be expressed as
_{1} and _{2} are of the order of one. Moreover, considering the equilibrium assumptions (balance between production and destruction of the turbulence energy), the eddy viscosity of each method is of the following relation [

It is shown that both the two formulations have the similar structure as the LES viscosity:
_{s}_{vk} scales are well compatible with each other and it is theoretically acceptable to the replacement.

Through the brief derivations above, the new proposed hybrid function employs von Karman scale instead of grid scale in (_{ISAS}) of which is reformulated as
_{ISAS} is the control parameter with the value of 1.67 calibrated optimal in the next section.

With the modification in (_{vk} scale enables the model to dynamically provide the eddy viscosity with the concept of scale-adaptive; thus, the underlying method is named improved scale-adaptive simulation (ISAS).

In (_{DES} = 0.61. In (

Assuming that SAS and DES models produce the same energy dissipation, we can obtain _{s}_{vk} instead of Δ in (

To obtain the optimal value of _{ISAS}, a recommended way is to calculate the simple decaying homogeneous isotropic turbulence (DHIT), for example, the classical test case performed by Comte-Bellot and Corrsin [_{ISAS} and the line with −5/3 slope are also given for comparison. As displayed, the energy spectra change as using different _{ISAS} values; concretely, the lower the value, the less the dissipation. However, the curve with _{ISAS} = 1.67 best matches the measurement and provides the spectral slope close to the −5/3 near the cutoff wave number, and the declared _{ISAS} value fairly fits the rough estimate above.

Comparison of energy spectra between ISAS simulations and DHIT data.

Through the simple validation in the freely decaying isotropic turbulence, it turns out that, with a proper constant, the ISAS model allows for the formulation of the turbulence spectrum, which is contrary to URANS but similar to DES and SAS. As a result, the _{ISAS} value 1.67 is proven optimal, and more comprehensive validation will be presented in the next section.

Flow over a circular cylinder is a typical case of bluff body flows, which is often involving complex vortex motion and rich physics such as separation and reattachment. Because of the simple shape and representative flow phenomena, the cylinder flow is often favorable for the validation of turbulence models. As mentioned above, the flow over a circular cylinder at Re = 3900 (corresponding to cylinder diameter and freestream velocity) is selected for the thorough validation of the proposed ISAS model, mainly due to the large amount of experimental and numerical data in this particular subcritical Reynolds number.

All three-dimensional simulations were performed on a multiblock grid extruding from the cross-sectional plane, displayed in Figure ^{+} values well below 1.0 at the wall, the minimum wall grid spacing is set to 2 × 10^{−3}

The design of cross-sectional grid (a) overview and (b) near-wall region (top, fine grid; bottom, coarse grid).

Zoom of the near-wall region

Concerning the boundary conditions, to prevent the undesirable effort from inflow and outflow, a fixed velocity field (0.05 Ma) was used at inflow plane and a distance of 20

All the solutions presented in this paper are based on a 3-D incompressible solver. The space discretization of the governing equations was performed using the popular second-order finite volume method (FVM), and a second-order implicit backward method was used for time integration coupled with the dynamically adjustable stepping technique [^{−6} for pressure and 10^{−7} for other dependent variables.

It is well known that periodical vortex shedding exists in the cylinder flow at current Re, and sufficiently long physical time integration is very essential for the accurate predictions of the average surface forces. Therefore, in order to get reliable statistics, a total time of about 60 shedding cycles (_{∞}) was taken for each simulation, and the final time-spatial averaged data were recorded over approximate 50 cycles with the initial 10 cycles ignored. The focused time is more than 40 shedding cycles suggested by Franke and Frank [

Figure _{d}) and lift (_{l}) coefficients for the ISAS simulation performed on the fine grid. As shown, the lift amplitude is unstable but appears periodical wave, while the drag magnitude appears strong fluctuation near an average value. Moreover, the Strouhal number (St) corresponding to shedding frequency is also given using a fast Fourier transform (FFT) to the lift signal with time. In addition, the mean force coefficients as well as Strouhal numbers of all simulations are listed in Table

Time history of force coefficients using ISAS model and fine grid (lift, _{l}; drag, _{d}).

Summary of the statistical flow quantities for the flow over a cylinder flow at Re = 3900.

Case | _{d} |
_{r}/ |
−_{min}/_{∞} |
−_{pb} |
St | _{sep} |
---|---|---|---|---|---|---|

Exp. [ |
— | 1.51 | 0.34 | — | 0.21 | — |

Exp. [ |
0.98 | — | — | 0.94 | — | — |

ISAS1 | 1.00 | 1.496 | 0.296 | 0.915 | 0.214 | 87.24 |

DES1 | 0.95 | 1.857 | 0.308 | 0.878 | 0.207 | 86.95 |

SAS1 | 1.01 | 1.485 | 0.285 | 0.964 | 0.212 | 87.90 |

ISAS2 | 1.03 | 1.348 | 0.264 | 1.002 | 0.208 | 88.10 |

DES2 | 1.06 | 1.271 | 0.287 | 1.037 | 0.203 | 88.47 |

SAS2 | 1.06 | 1.213 | 0.294 | 1.044 | 0.205 | 88.68 |

DNS [ |
— | 1.41 | 0.291 | — | 0.20 | — |

LES [ |
1.04 | 1.35 | 0.37 | 0.94 | 0.21 | 88 |

LES-SMAG [ |
1.18 | 0.9 | 0.26 | 0.8 | 0.19 | 89 |

LES-THE [ |
0.97 | 1.67 | 0.27 | 0.91 | 0.21 | 88 |

SST-DES [ |
1.01 | 1.46 | 0.29 | 0.89 | 0.20 | 86.4 |

SST-PANS [ |
1.06 | 1.20 | 0.28 | 0.96 | 0.20 | 87.1 |

SA-DES, [ |
1.20 | 0.85 | 0.3 | 1.08 | 0.22 | 89.3 |

SA-DDES [ |
0.97 | 1.37 | 0.28 | 0.97 | 0.22 | 88.3 |

SA-IDDES [ |
1.02 | 1.42 | 0.28 | 0.89 | 0.22 | 87 |

^{2}- |
0.99 | 1.68 | 0.4 | 0.83 | 0.21 | 86.4 |

In this section, all numerical results with available experimental data are presented and detailed comparisons are conducted between ISAS, DES, and SAS models. It is worth mentioning that the following terminologies will be used for the subsequent simulations with the goal of briefly and avoiding confusion: the results based on the fine grid for the three models are, respectively, named ISAS1, DES1, and SAS1 while those based on the coarse grid are named ISAS2, DES2, and SAS2.

To compare the efficiency of different models, all simulations were performed on the same workstation with 40 cores. It should be emphasized that all the simulations used the same numerical setup and the iteration in every time step was ensured convergent; thus, the model efficiency just depends on the solving speed of the turbulent equations.

The computational efficiency of ISAS, DES, and SAS methods in both fine and coarse grids is discussed in Figure _{∞}/_{SAS} in SAS is more complex than that only using the ratio of turbulent length scale to the von Karman length scale in ISAS, which leads the SAS equations more expensive to be solved, while with a similar formulation, there is a tiny gap between ISAS and DES efficiency though the _{vk} scale is somewhat more complicated than grid scale.

Computational efficiency of ISAS, DES, and SAS.

Fine grid

Coarse grid

It is well known that the thin shear layers with about one cylinder diameter length are the distinct feature of the cylinder flow at current Re = 3900 [

Separating shear layers displayed by the mean vorticity magnitude.

ISAS1

DES1

SAS1

ISAS2

DES2

SAS2

Figures

PIV (particle image velocimetry) data measured by Parnaudeau et al. [

HWA (hot-wire anemometry) data measured by Ong and Wallace [

Distribution of mean streamwise velocity along the wake centerline.

Fine grid

Coarse grid

Mean velocity distributions at different sections (for details, see the caption for Figure

Streamwise velocity (fine grid)

Transverse velocity (fine grid)

Streamwise velocity (coarse grid)

Transverse velocity (coarse grid)

Figure _{r}/

Combining the analysis of Figures _{r}/_{min}/_{∞}) are listed in Table

From the above, we have simply discussed the relation between the simulation of boundary layer and grid using different turbulent models, while the ability of ISAS to simulate the separated flows is primitively validated. The further explanations for the LES-like resolution are based on the theoretical formulation and the physical mechanism of the three models. It is well known that the hybrid methods adjust the viscosity to different levels regarding the near-wall region and separated region in different ways. Considering the proposed ISAS model in this paper, it has a theoretical mechanism similar to DES but a physical mechanism like SAS.

Noting the different scales on the colour bar, the instantaneous ratio of the modelled turbulent length scale to the dominant length scale for different models is displayed in Figure _{vk}) rather than the grid scale used in DES. Both the _{vk} and Δ scales are passive in the boundary layer and active in the detached region, which is distinguished by the dividing factor _{vk} = 1.0 or _{vk} scale, which is determined by the local flow physics and provides a finer adjusting effort for the flow unsteadiness. It should also be mentioned that the _{ISAS} scale distribution (using fine mesh) fits the mechanism of hybrid approach very well with no very large value occurred, as plotted in Figure

Contours of the dominant turbulent length scales.

ISAS1

DES1

SAS1

ISAS2

DES2

SAS2

Time-averaged contours of the dominant turbulent length scales (ISAS1).

Both treating _{vk} as the defining scale, the adjusting effort of ISAS is visually similar to SAS as shown in Figures _{SAS}) in the second _{vk} = 0. By contrast, the ISAS adjusting formulation directly acting on the dissipative term of

The mean turbulent viscosity (_{t}

Contours of mean turbulent viscosity (top, fine grid; bottom, coarse grid).

ISAS viscosity

DES viscosity

SAS viscosity

As the additional information to the first-order velocity magnitude, the time and space averaged Reynolds stresses are presented in Figure

Mean Reynolds stress in different sections (for details, see the caption for Figure

Streamwise Reynolds stress (fine grid)

Transverse stress (fine grid)

Streamwise Reynolds stress (coarse grid)

Transverse stress (coarse grid)

Figure _{p}) and the vorticity magnitude (mag|Ω|). It should be noted that the angle

Pressure measurements conducted by Norberg [

Vorticity measurements conducted by Ma et al. [

Comparison of the mean wall data.

Mean wall pressure coefficient (fine grid)

Mean wall vorticity magnitude (fine grid)

Mean wall pressure coefficient (coarse grid)

Mean wall vorticity magnitude (coarse grid)

Concerning the mean pressure distribution first, as shown in Figures _{pb}, _{p} at _{sep}) of the attached layer are also listed in Table

Several important integral flow statistics are listed in Table _{d}) and the recirculation length (_{r}/_{d} and excessively longer _{r}/_{vk} scale which will not introduce unpredictable efforts into the detachment mechanism of the shear layers. When using the coarse grid, the _{d} and _{r}/_{min}/_{∞}), the ISAS results are not the best compared with DES and SAS results but still comparable with most of numerical references, while there is minor difference between the DES and SAS results. As for the base pressure coefficient (−_{pb}), the ISAS results are better than the other two simulations, while between the DES and SAS results, it is still shown that each of them is better on the fine or coarse grids. Finally, the discrepancy from the three models is very small concerning the Strouhal number (St) and the separation angle (_{sep}), and all results are comparable with the other numerical results.

The capabilities of current models in reproducing turbulent unsteadiness are displayed in Figure

Instantaneous turbulent structures identified by the

ISAS1

DES1

SAS1

ISAS2

DES2

SAS2

Finally, the one-dimensional energy spectrum is plotted in Figure _{vs} (_{vs} = St _{∞}/

One-dimensional spectra of the transverse velocity at

In this paper, by introducing the von Karman factor into the scale-determining equation, a new hybrid turbulent model combining the features of DES and SAS methods is proposed for massively separated flows, and with the concept of scale-adaptive, the underlying method is termed improved scale-adaptive simulation (ISAS). The derivation and calibration of the transformed formulation are presented, and the cylinder flow at Re = 3900 is selected as the benchmark case for the model validation. Detailed numerical comparisons with the available experimental data are conducted to study the properties of the proposed ISAS model. As expected, the ISAS model is capable to simulate turbulent details down to the grid limit in a simple and efficient way and, furthermore, appears some advantages over the original DES and SAS models.

From the analysis to the energy spectrum of the freely decaying isotropic homogeneous flow and the cylinder flow, it is shown that the ISAS model allows for the turbulent spectra close to the Kolmogorov’s energy spectrum, demonstrating the capability of which in representing the resolvable contents with lower frequency.

By comparison with DES model, with similar formulation, the efficiency of ISAS in providing a convergent simulation is very close to that of DES. The _{vk} length scale, according to the local flow physics, enables ISAS model to dynamically adjust the resolution to resolved spectra and appear comparable with DES in capturing the transient unsteadiness. Meanwhile, benefiting from the grid-insensitive character of _{vk} scale, it enables ISAS model to operate a smooth switch from stable mode to unstable mode and avoid the grid-dependent issues such as gray area and modeled stress depletion appeared in DES simulations.

For the comparison with SAS model, with the reformulation in ISAS, the concept of scale-adaptive is made more intuitive and efficient than SAS. The resolving speed of SAS is noticeably improved through the simplified process. With a more flexible adjusting mechanism, the ISAS model provides a more feasible dissipative level and captures finer turbulent structures than SAS.

Concerning the benchmark cylinder flow, with an applicable viscosity level, the development of the shear layers is reproduced very well by the ISAS model. Thus, the focused quantities of industrial interest such as the drag coefficient and recirculation length are in the best agreement with the experimental data on fine grid. All the numerical accuracy from the coarse grid is reasonably decreased for the comparative ISAS, DES, and SAS models, though not always the best such as the minimum back flow velocity and Reynolds stress, most of the ISAS outputs are still closer to the measurements.

In general, the establishment of ISAS is a positive attempt to the combination and remedy of current hybrid methods considering the still unaffordable LES situation. The proposed method allows for a good compromise regarding the numerical accuracy, efficiency, and robustness, which is beneficial for the development of turbulence models with practical value for the engineering flows.

The authors declare that they have no conflicts of interest.

The authors wish to acknowledge the National Natural Science Foundation of China, Award no. 11202101.

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