International Journal of Aerospace Engineering

Volume 2018, Article ID 5231798, 16 pages

https://doi.org/10.1155/2018/5231798

## An Improved Scale-Adaptive Simulation Model for Massively Separated Flows

School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China

Correspondence should be addressed to Xiaorong Guan; nc.ude.tsujn@rxg

Received 30 November 2017; Accepted 31 January 2018; Published 22 March 2018

Academic Editor: William W. Liou

Copyright © 2018 Yue Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1, 2]. Thus, at least a partial resolution of the turbulence spectrum has been the main topic of turbulence modelling for technical simulations in the last decades. In recent years, the hybrid RANS/LES methods are of great interest to both academic institutions and industry applications with the motivation that it is powerless to solve the problems at hand when using the LES or RANS method alone [3–5]. As the continuing theoretical progress and from a practical standpoint, many scholars recognize that whether a turbulence model can predict turbulent structures relies on the available eddy viscosity level when using the eddy viscous method. Following the LES method, a typical hybrid LES/RANS model termed detached eddy simulation (DES) was firstly proposed by Spalart et al. [6]. It is well known that the DES concept combines the advantages of RANS and LES methods in a simple and effective way by using the grid spacing as an adjusting scale, which has been proven promising for engineering flows, and a variety of hybrid approaches have since appeared [3, 7]. However, the DES models are not without problems [8, 9], and the main stumbling stone for DES is the ambiguous response to the grid refinement. When exceeding a critical level of grid refinement near walls, the standard DES formulation may negatively impact the RANS performance and arise some issues, such as, gray area, modeled stress depletion (MSD), and grid-induced separation (GIS) [10]. Various remedial measures such as DDES [11, 12], IDDES [13–15], and zonal DES [16] have been conducted to protect the boundary layer from the DES limiter; nevertheless, the remedies do not entirely avoid the issues appearing either conservative or inefficient [10, 17].

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 (*L*_{vk}) was firstly conducted by Menter et al. [18]. The higher derivative reflects the local flow physics and gradually adjusts the resolution to the resolved scales with grid independency, which is thereby viewed the main advantage of SAS over the other hybrid methods. Again because of grid independence, the SAS is often treated as an advanced unsteady RANS (URANS) [3]. Since the first one-equation KE1E-SAS was introduced, several transformations based on different RANS background have been conducted by the Menter team [19–21]. In recent years, the SAS models have been extensively validated in numerous test cases and continuously undergone certain evolution, which dramatically shows its potential in the prediction of actual turbulent flows [22–24].

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 [10, 17]; thus, the modification and improvement for the existing hybrid methods are very essential. For this perspective, we naturally intend to combine the formulations of DES and SAS in this paper. On one hand, the simple modelling mechanism using the ratio of modelled scale to grid scale enables DES method to be the most widely used hybrid method. But the strict grid requirement of which is difficult to cover the entire boundary layer [25]. On the other hand, the *L*_{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 [21, 26]. It thereby means the two scales are theoretically compatible with each other. In fact, there have been several reformulations by introducing *L*_{vk} scale into the one-equation SA-DES model [27, 28], while an even more interesting example is the “turbulence-resolving RANS” (TRANS) approach based on the two-equation *k*-*ω* model [29]. The TRANS model uses the ratio between strain and vorticity (*S*/Ω) as the spatial operator rather than the high derivative, which also allows the reproduction of unsteady content and makes SAS more intuitive, so the model is practical but without a clear physical mechanism between the eddy viscosity and resolved scales.

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 [30], which is demonstrated in robustly and accurately predicting the boundary layers with adverse pressure gradients.

#### 2. Turbulence Modelling

##### 2.1. SST Model Formulation

Benefiting from *k*-*ω* model in calculating the viscous sublayer as well as *k*-*ε* model in better dealing with the fully turbulent flows away from walls, Menter’s two-equation SST model combines the best features of standard *k*-*ω* and *k*-*ε* models, which is very popular in the prediction of aeronautics flows and treated as a main platform for the hybrid formulations. In this paper, the target SST model is the version of Menter [31], and the two independent transport equations of which are written as follows:
where *k* is the turbulence kinetic energy, *ω* is the specific dissipation rate, *ρ* is the density, **U** is the velocity vector, and *μ* is the dynamic viscosity.

All model parameters *φ* are obtained by an applicable compromise of *k*-*ω* and *k*-*ε* constants via , in which the blending function *F*_{1} is defined as
with and *d* is the distance from the nearest solid surface. The adopted model constants are , *α*_{1} = 0.5556, *α*_{2} = 0.44, *β*_{1} = 0.075, *β*_{2} = 0.0828, , , , and .

The turbulent eddy viscosity is expressed as
where *S* is the scalar invariant of the strain rate (*S _{ij}*), ,

*α*

_{1}= 0.31, and

*F*

_{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:

##### 2.2. DES Model Formulation

The transformation of the DES term to the SST model is firstly introduced by Strelets [32] and simplified by Menter et al. [17]. Here, the SST-DES model of Menter is viewed as the standard DES formulation, and the distinguishing factor over the background SST model lies in a hybrid function (*F*_{DES}) directly acting on the dissipation term of the *k*-scale equation, which is expressed as
where *C*_{DES} is the model constant and is equal to 0.61, as the limiter should only be active in the *k*-*ε* region, *l _{kω}* is the turbulent length scale defined by , and is the local grid scale reading Δ = max(∆

*x*, ∆

*y*, ∆

*z*).

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, *l _{kω}* <

*C*

_{DES}Δ and

*F*

_{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,

*l*>

_{kω}*C*

_{DES}Δ and

*F*

_{DES}> 1.0, a LES-like mode is gained.

##### 2.3. SAS Model Formulation

The formulation of the SST-SAS model differs from the pure SST model by the additional source term (*Q*_{SAS}) in the *ω*-scale equation [20]:
where similar to DES, *L _{kω}* is the modelled turbulence length scale and reads , while the distinctive von Karman length scale (

*L*

_{vk}) is achieved by . The model constants are , , , , and .

It is worth mentioning that the second derivative (*U*″) is used instead of the third derivative, appearing more sufficient in the 3-D simulation [21, 22, 33]. The von Karman length scale (*L*_{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, *L _{kω}* <

*L*

_{vk}and

*Q*

_{SAS}= 0, returning a pure RANS mode, while when in the unstable region,

*L*>

_{kω}*L*

_{vk}and

*Q*

_{SAS}> 0 and the mode will turn to LES-like mode.

##### 2.4. ISAS Model Formulation

To combine the DES and SAS concepts based on the SST model, a simple way is to replace the grid scale in (6) with the von Karman scale.

As known, using Reynolds averaging method, the standard two-equation models such as SST always produce a turbulent length scale (*L*) proportional to the thickness of the boundary layer (*δ*), which is suitable for the stable shear flows near walls but depresses the resolved scales in the unsteady regions. When using the hybrid methods such as DES and SAS, the relation between *L* and *δ* in the RANS regions is still [20]
whereas in the separated regions, the grid scale Δ and the von Karman scale *L*_{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
where the model constants *C*_{1} and *C*_{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 [21, 26]:

It is shown that both the two formulations have the similar structure as the LES viscosity:
where the *C _{s}* value is also of the order of one and thus the Δ and

*L*

_{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 (6), and the dominant factor (*F*_{ISAS}) of which is reformulated as
where *C*_{ISAS} is the control parameter with the value of 1.67 calibrated optimal in the next section.

With the modification in (13), the new model can effectively adjust the turbulent fluctuations by directly operating on the *k* equation like DES, and more importantly, the *L*_{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).

##### 2.5. ISAS Model Calibration

In (10), the DES constants , *C*_{DES} = 0.61. In (11), the SAS parameter appears in [21] associated with high wave number damping. We just calibrate the whole part in the DHIT case like that done in [34]. Finally, we found 0.42 is feasible for the proportional relation. From the above derivation, we get the scale relation in the following:

Assuming that SAS and DES models produce the same energy dissipation, we can obtain and . If simply replace the grid scale in (6) with von Karman scale and avoid excessive dissipation, an approximate range will be conducted. Meanwhile, one similar relation based on (11) and (12) has been built in [21] with an optimal *C _{s}* = 0.11. We assume the SAS method matches the same energy spectrum of LES and the equation can be gained. Again, roughly use

*L*

_{vk}instead of Δ in (6) and avoid more dissipative action, and we can get . It is very close between the two rough results.

To obtain the optimal value of *C*_{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 [35], just as that did for most LES and hybrid models calibrations [19, 29, 32, 36]. The experimental equipment was simplified to a cubical box with periodic boundary conditions for the ISAS simulations, and a uniform grid with 32 × 32 × 32 points was used. Moreover, the numerical strategies such as the initial conditions and splitting schemes are drawn from the similar DES calibrations [29, 32]. Figure 1 shows the spectra of the resolved kinetic energy for the selected DHIT case at the normalized time *t* = 2; meanwhile, the ISAS outputs with different *C*_{ISAS} and the line with −5/3 slope are also given for comparison. As displayed, the energy spectra change as using different *C*_{ISAS} values; concretely, the lower the value, the less the dissipation. However, the curve with *C*_{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 *C*_{ISAS} value fairly fits the rough estimate above.