Dynamic Nonlinear Algebraic Models With Scale-Similarity Dynamic Procedure For Large-Eddy Simulation of Turbulence

A dynamic nonlinear algebraic model with scale-similarity dynamic procedure (DNAM-SSD) is proposed for subgrid-scale (SGS) stress in large-eddy simulation of turbulence. The model coe ﬃ cients of the DNAM-SSD model are adaptively calculated through the scale-similarity relation, which greatly simpliﬁes the conventional Germano-identity based dynamic procedure (GID). The a priori study shows that the DNAM-SSD model predicts the SGS stress considerably better than the conventional velocity gradient model (VGM), dynamic Smagorinsky model (DSM), dynamic mixed model (DMM) and DNAM-GID model at a variety of ﬁlter widths ranging from inertial to viscous ranges. The correlation coe ﬃ cients of the SGS stress predicted by the DNAM-SSD model can be larger than 95% with the relative errors lower than 30%. In the a posteriori testings of LES, the DNAM-SSD model outperforms the implicit LES (ILES), DSM, DMM and DNAM-GID models without increasing computational costs, which only takes up half the time of the DNAM-GID model. The DNAM-SSD model accurately predicts plenty of turbulent statistics and instantaneous spatial structures in reasonable agreement with the ﬁltered DNS data. These results indicate that the current DNAM-SSD model is attractive for the development of highly accurate SGS models for LES of turbulence.


Introduction
Turbulent flows involve a wide range of length scales across several orders of magnitude, therefore the direct numerical simulation (DNS) of turbulence at high Reynolds number is impractical to solve all flow scales ranging from inertial to viscous ranges [1,2,3]. Large-eddy simulation (LES) is an effective approach which adopts the coarse mesh to merely resolve the large flow scales and model the effect of residual subgrid scales (SGS) on the resolved large scales [4,5,6,7]. Extensive SGS models are proposed to reconstruct the unclosed SGS stress in previous works, including the Smagorinsky model [8,9,10], the velocity-gradient model (VGM) [11], the scale-similarity model [12,13], the implicit LES (ILES) [14,15,16], the Reynolds-stress-constrained LES model [17], the datadriven models [18,19,20,21,22,23,24,25], etc. The Smagorinsky model is one of the commonly-used SGS models whose model coefficient for the original version is statically adjusted by the experimental and DNS data in the early stage.
Germano et al. [26] and Lilly [27] pioneered the development of a dynamical procedure based on the Germano identity through the least-squares algorithm, which makes the parameter of the dynamic Smagorinsky model (DSM) dynamically determined as the flow changes. Subsequently, the dynamical versions of some conventional SGS models with the Germano-identity based dynamic procedure (GID) were proposed [4,5,6,7], including the dynamic mixed model (DMM) [28,29,30,31], the dynamic Clark model [32], the dynamic localization model [33], etc.
The Smagorinsky model [8,9,26,27] constructs the SGS stress with the linear constitutive relation based on the Boussinesq hypothesis, which requires the alignment between the SGS stress and the filtered strain-rate tensor. Pope [34]d e r i v e d the general expression between the Reynolds stress and the averaged strain-rate and rotation-rate tensors with eleven integrity basis tensors based on the theory of invariants. Due to the expensive calculations of the high-order basis tensors in the general expression, the numerical verification of Pope's general viscous hypothesis was only limited to the two-dimensional turbulence [34]. Lund and Novikov [35] showed that the sixth invariant can be expressed as the ratio of the other five invariants, and reduced the original eleven polynomial basis tensors to five, which greatly simplified the computational complexity of the nonlinear algebraic SGS model in LES calculations. Especially, the anisotropic part of the SGS stress can be expressed as the general expression of the resolved strain-rate and rotation-rate tensors with five model coefficients [35]. Speziale et al. [36,37] further simplified the Lund's general expression to a quadratic constitutive relation for the Reynolds stress. The model coefficients of the nonlinear algebraic model were mostly determined by the DNS data in the early research work. Wong [38] proposed a two-parameter dynamic nonlinear algebraic model (DNAM) using the quadratic constitutive relation with the Germano-identity based dynamic procedure. Kosović [39] applied the nonlinear constitutive relation to the shear-driven boundary layers at high Reynolds number. Wang et al. [40,41] proposed a dynamic SGS model based on the quadratic nonlinear constitutive expression with local stability. Marstorp et al. [42] proposed an explicit algebraic SGS stress model with the equilibrium assumptions made on the partial-differential equations of SGS stress, and successfully applied to the rotational channel flow. Recently, a stochastic extension of the explicit algebraic SGS models has been developed by Rasam et al. [43] In our previous research work, a nonlinear algebraic model based on the artificial neural network (ANN-NAM) was proposed [44], whose model coefficients are predicted by the invariant-input ANN with embedded invariance. The ANN-NAM model [44] reconstructs the SGS stress and statistics of velocity with high accuracy both in the ap r i o r iand a posteriori analyses of LES. Wang et al. [45] proposed a ANN-based semi-explicit spatial gradient model with embedded invariance. A dynamic version of the spatial gradient model (DSGM) [46] was proposed for the parameter determination strategy. Yuan et al. developed deconvolutional artificial neural network (DANN) [21] and dynamic iterative approximate deconvolution (DIAD) models [22] to recover the local unfiltered velocity with the neighboring spatial stencils of the filtered velocity. A scale-similarity-based dynamic procedure (SSD) was proposed to adaptively calculate the weights of the spatial stencil [22]. The DIAD model with the SSD procedure is superior to the other conventional dynamic SGS   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 65 models in the reconstruction of the statistics of velocity and transient coherent structures of turbulence [22].
In the current research, a novel dynamic nonlinear algebraic model with scalesimilarity dynamic procedure (DNAM-SSD) is developed for reconstructing the unclosed SGS stress in LES of incompressible turbulence. The performance of the DNAM-SSD model is examined by comparing with those of some classical SGS models both in the ap r i o r iand a posteriori testings of LES at two filter widths ∆ = 16h DNS and 32h DNS with the corresponding grid resolutions of N = 128 3 and 64 3 . The computational accuracy and costs of the newly-proposed scale-similarity based dynamic (SSD) procedure are compared to the conventional Germano-identity based dynamic (GID) procedure. The remainder of the paper is organized as follows. The governing equations of LES will be described in Section 2. The introductions of the conventional SGS models and DNAM models are respectively illustrated in Sections 3 and 4. Sections 5 will conduct the numerical simulation of incompressible isotropic turbulence. The ap r i o r iand a posteriori studies are correspondingly provided in Sections 6 and 7. Conclusions are drawn in Section 8.

Governing equations of large-eddy simulation
The incompressible turbulence is governed by the Navier-Stokes equations, whose dimensionless conservation form is written as [1] where u i denotes the i-th velocity component (i =1, 2, 3re p re s e n t sth eth re ed i re ctions of the Cartesian coordinate system, respectively.), p is the pressure, Re is the Reynolds number, and F i is the i-th large-scale force component. [47,48,21] For brevity and simplicity, we adopt the summation convection for the repeated indices by default in this paper. Besides, the governing dimensionless parameter for the incompressible turbulence, namely, the Taylor microscale Reynolds number Re λ is given by [1] where ⌫ denotes the kinematic viscosity and u rms = p hu i u i i is the root-mean-square (rms) value of the velocity magnitude. Here, "h•i" represents a spatial average over the entire computational domain. In addition, the Taylor microscale is expressed as [1] = u rms p 5⌫/", where " =2⌫ hS ij S ij i denotes the dissipation rate and S ij = 1 2 (@u i /@x j + @u j /@x i ) is the strain-rate tensor.
For the large-eddy simulation, the resolved large scales are separated from the subgrid small scales by the spatial filtering operation, which is introduced as [ where f (x) represents the arbitrary physical variable, and an overbar stands for the low-pass spatial filtering. Here, Ω denotes the entire physical domain, with G and∆ respectively being the spatial filter function and filter width. The governing equations for the LES can be obtained by applying the spatial filtering on the Eqs. (1) and (2), correspondingly, which can be derived as [1,2] @ū i @x i =0, Here, the unclosed SGS stress ⌧ ij in the Eq. (7)i sd e fi n e db y [ 4,5,6] The SGS stress involves the nonlinear interactions between the resolved large scales and under-solved small scales, therefore additional SGS stress modeling is required to close the governing equations of LES. In the following two sections, the conventional SGS models and the proposed dynamic nonlinear algebraic models with scale-similarity dynamic procedure (SSD) are respectively described for the LES computations.

Conventional SGS models
The explicit modeling for the unclosed SGS stress can be divided into the functional modeling and structural modeling. The functional models mimic the forward energy transfer from the resolved large scales to the residual small scales by constructing the explicit eddy-viscosity forms, while the structural modeling is established by the truncated series expansions or the hypothesis of scale similarity to correctly recover the SGS stress with high accuracy. A typical functional model is the dynamic Smagorinsky model (DSM), whose constitutive relation for the deviatoric SGS stress is given by [26,27] where |S| =( 2 S ijSij ) 1/2 is the characteristic filtered strain rate, andS ij = 1 2 (@ū i /@x j + @ū j /@x i ) is the filtered strain-rate tensor. The superscript "A" represents the trace-free part of the arbitrary variables, namely, (•) A ij =(•) ij (•) kk ij /3. Here, the isotropic SGS stress ⌧ kk is absorbed into the pressure term. C 2 S is the Smagorinsky coefficient, which can be determined by the Germano identity dynamic procedure (GID). The test-filter level SGS stress with the double-filtering scale∆=2∆ is expressed as [26,27] where a tilde stands for the test filtering operation at the filter scale∆. The deviatoric part of T ij can be modeled based on the scale-invariance hypothesis, defined by [26,27] These two SGS stresses with different filter scales, namely, ⌧ ij and T ij satisfy the Germano identity, expressed as [26] L where the Leonard stress L ij can be calculated by the resolved filtered field for LES calculations. Therefore, the optimal Smagorinsky coefficient C 2 S can be further determined by the least-squares algorithm, namely [27] where A typical structural model is the velocity gradient model (VGM) based on the truncated Taylor series expansions, given by [11] The dynamic mixed model (DMM) combines the scale-similarity model with the dissipative Smagorinsky term, which can overcome the deficiency of numerical instability in the structural modeling of the SGS stress. The SGS stresses constructed by the DMM model at scales∆ and∆ are expressed, respectively, as [49,12,28] where Here, the hat stands for the test filtering at scale∆=4∆. Similar to   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 the DSM model, the model coefficients C 1 and C 2 are calculated by the Germano identity dynamic procedure, namely [21,22] where

Dynamic nonlinear algebraic models (DNAM)
In the SGS stress modeling, the constitute relation of the unclosed SGS stress can be regarded as the function of the local filtered quantities, i.e., the filtered strain-rate tensorS ij and filtered rotation-rate tensorΩ ij , namely [34,35] ⌧ where the filtered rotation-rate tensorΩ ij = 1 2 (@ū i /@x j @ū j /@x i ). For brevity and simplicity of the tensorial polynomials, the matrix multiplications for the tensor contractions are expressed as [34,35,40] A general expression of the modeled SGS stress [Eq. (19)] can be expanded to the sum of an infinite number of tensorial polynomials with the formS m1Ωn1Sm2Ωn2 ···, where m i and n i are positive integers. The infinite tensorial polynomials can be reduced to a finite number by the Cayley-Hamilton theorem, thus the modeled SGS stress is expressed as the linear combination of the basis tensors formed by the product ofS andΩ, namely [34] where T  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 65 If the model coefficients g n are relaxed as the ratios of polynomials of these integrity invariants, the number of the above basis tensors can be reduced from eleven to five. In accordance with the dimensional consistency, the anisotropic part of the modeled SGS stress can be given by [35] where the characteristic filtered strain rate S =( 2 S ijSij ) 1/2 , and C n are five dimensionless model coefficients. The corresponding basis tensors T (n) ij that satisfy the consistent dimension with the square of the velocity gradient are defined by In the paper, two dynamic procedures are adopted to determine the model coefficients C n of the dynamic nonlinear algebraic models (DNAM). One is the Germano identity dynamic (GID) procedure based on the scale-invariance assumption, and the other is newly proposed scale-similarity dynamic (SSD) procedure in accordance with the scale-similarity relation. The rest of this section will be divided into two subsections to respectively introduce these two different modeling approaches.

DNAM models with scale-similarity dynamic procedure (DNAM-SSD)
In this paper, we propose a novel scale-similarity dynamic procedure for the DNAM model to determine the optimal model coefficients dynamically. The real SGS stress can be regarded as the nonlinear function of the velocity u i and the filter kernel at scale∆, whereas the SGS stress modeled by the the DNAM model has the nonlinear constitutive relation with the local filtered physical quantities (e.g. the filtered strain-rate and rotation-rate tensorsS andΩ), Based on the scale-similarity hypothesis, the modeled SGS stress at the filter scalẽ ∆ shares the consistent model coefficients C n with that at the filter scale∆, namely 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 65 (31) The constitutive equation of the SGS stress is assumed to be invariant to the physical field, therefore we can replace the unfiltered velocity u i with the filtered velocitȳ u i in Eq. (30) and obtain L A ij is the anisotropic part of the resolved Leonard stress. For simplicity, we let ij are both resolved in the filtered field, the model coefficients C n can be determined by the least-squares method, It is worth noting that the DNAM-SSD model only calculates T

Numerical simulation of incompressible isotropic turbulence
In order to validate the performance of the proposed DNAM-SSD model, the numerical simulation of incompressible isotropic turbulence is performed in a cubic box of (2⇡) 3 with periodic boundary conditions at the Taylor Reynolds number Re λ ⇡ 250. The pseudo-spectral approach with the two-thirds dealiasing rule is adopted for the spatial discretization of the governing equation. A second-order explicit Adams-Bashforth scheme [50] is applied to the time advancement. The large-scale forcing is implemented on the two lowest wavenumber shells [48,47,21]t ok e e pt h et u rbulence in equilibrium. In the paper, we use N = 1024 3 uniform grids in the DNS calculation with the grid spacing h DNS =2⇡/1024. The kinematic viscosity is set to ⌫ =1/Re =0.001. The detailed one-point statistics of DNS calculation are summarized in Table 1. Here, k max = 2π 3hDNS represents the largest effective wavenumber after the fully two-thirds dealiasing, and ! rms = p h! i ! i i stands for the root-meansquare value of the vorticity magnitude. The Kolmogorov length scale ⌘ and the 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 integral length scale L I represent the smallest resolved scale and the largest characteristic scale, which are defined, respectively, by where " =2 ⌫ hS ij S ij i is the dissipation rate. The total turbulent kinetic energy , and E(k) stands for the velocity spectrum. The resolution parameter k max ⌘ 2.1 is found to be sufficient enough for the convergence of turbulent kinetic energy at all scales[51, 52].
In the paper, the filtered physical quantities and the real SGS stress ⌧ ij are calculated by a Gaussian filter, which is expressed as [1,2] G r;∆ = We select two filter scales (∆ = 16h DNS and 32h DNS ) for model verification, and the corresponding cutoff wavenumbers are k c = ⇡/∆ = 32 and 16, respectively. Figure 1 shows the velocity spectra of the DNS and filtered DNS at both filter widths (∆ = 16h DNS and 32h DNS ). The filtered velocity spectra almost overlap with the DNS data at the low-wavenumber region satisfying the Kolmogorov k −5/3 scaling, while generally diminish with the increasing of wavenumbers, and drop rapidly at the region larger than the truncated wavenumber k c . More kinetic energy is filtered out at a lager filter scale, therefore the filtered velocity spectrum at∆ = 32h DNS is lower than that at∆ = 16h DNS . Overall 95% and 88% of the turbulent kinetic energy are retained in the filtered velocity field at the filter widths∆ = 16h DNS and 32h DNS ,r e s p e c t i v e l y .

A priori study of the DNAM Models
In the ap r i o r ianalysis, twenty snapshots of DNS data at equal temporal intervals during two large-eddy turnover periods (⌧ = L I /u rms ) are adopted to examine the model accuracy of the DNAM-GID and DNAM-SSD models with several filter scales ranging from∆=4 h DNS to∆ = 64h DNS . Two evaluation metrics are used to quantify the distinction between the real value (Q real ) and the modeled value (Q model ) for targeted variable Q, namely the correlation coefficient C(Q) and the relative error E r (Q), respectively defined by [21,22] 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 65 where h•i represents the ensemble average of total samples. In the ap r i o r istudy, we first investigate the impact of integrity basis tensors T (1),A ij = S S gives the worst predictions with correlation coefficients lower than 30% among these five basis tensors. It is worth noting that the terms T (2),A ij = S 2 A , therefore we keep all five basis tensors without any simplification in the paper. With the increasing of the filter widths, the correlation coefficients between the first basis tensor T (1),A ij and the SGS stress constantly increase, while those of the other four terms gradually drop but still higher than those of the first term. These results indicate that the classical Smagorinsky model (linear relation with only the first basis tensor T (1),A ij ) cannot fully reconstruct the SGS stress. Figures 2 and 3 respectively illustrate the correlation coefficients and relative errors of the normal and shear components of the SGS stress for different SGS models at a number of filter scales ranging from the inertial region to the dissipation range. Here, the VGM model is the velocity gradient model (see Eq. 14) which has high ap r i o r iaccuracy among the classical SGS models. The DNAM-LS model is a DNAM model with ap r i o r iknowledge of DNS data, whose model 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 coefficients are calculated by the least-squares method using the real SGS stress, , (n =1, 2, ··· , 5) . The model coefficients of the DNAM-GID model and DNAM-SSD model are calculated by the dynamic procedure based on conventional Germano identity (cf. Eq. 29) and the newly proposed scale-similarity dynamic procedure (cf. Eq. 33), respectively.
The DNAM-LS model has the highest correlation coefficients and the lowest relative errors with the SGS stress, since the DNS data are used to determine the model coefficients. The correlation coefficients and relative errors predicted by the proposed DNAM-SSD model are very close to the DNAM-LS model at all filter scales, which are much better than the DNAM-GID and VGM models. The DNAM-SSD model predicts the SGS stress accurately with the correlation coefficients overall higher than 92% and the relative errors less than 40% ranging from the viscous region to the inertial region. In contrast, the DNAM-GID model gives the worst prediction among these SGS models with the relative errors approximately over 40%. It is worth noting that the DNAM-SSD model performs better than the conventional VGM model at all filter widths, indicating that the basis tensors of the DNAM model are more complete than the velocity gradient in reconstructing the SGS stress.    3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 65 In order to further quantify the model accuracy of different SGS models in the a priori analysis, we compare the correlation coefficients and relative errors of the SGS stress at filter widths∆ = 16h DNS and 32h DNS listed in Tables 3 and 4,respectively . The DSM and DMM models use the Germano-identity dynamic procedure (see Eqs. 13, 17 and 18) to dynamically determine model coefficients. The DNAM-SSD and DNAM-LS models give the best prediction of the SGS stress with correlation coefficients higher than 95% and 92% as well as relative errors lower than 30% and 38% at corresponding filter widths∆ = 16h DNS and 32h DNS among these SGS models. In contrast, the DSM model performs the worst compared to other SGS models at both filter scales, whose correlation coefficients are lower than 30% and relative errors are nearly 100%. The DNAM-GID model predicts the SGS stress obviously worse than the DNAM-SSD and DNAM-LS models, but still much better than the classical DMM model with the consistent GID dynamic procedure. Besides, the performance of the VGM model in ap r i o r istudy is between the DNAM-GID model and DNAM-SSD model at both filter widths. These results demonstrate that the basis tensors of the DNAM model are more complete in modeling the SGS stress compared to those of the DMM model and VGM model. In the ap r i o r ianalysis, the proposed scale-similarity dynamic procedure (SSD) shows distinct advantages over the conventional Germano-identity dynamic procedure (GID) in determining the model coefficients of SGS models.
Finally, we evaluate the SGS energy transfer for different SGS models by comparing the normalized SGS energy flux Π/✏ DNS ,w h e r eΠ= ⌧ ijSij represents the SGS energy flux and ✏ DNS denotes the dissipation rate calculated by the DNS data. The PDFs of the SGS energy flux reconstructed by the DNAM-SSD and DNAM-LS models coincide with the filtered DNS data at both filter scales∆ = 16h DNS and 32h DNS , which are obviously better than the VGM and DNAM-GID models. In comparison, DNAM-GID model fails to predict the SGS energy transfer, indicating that model-coefficient determination by the proposed SSD procedure is superior to that of the conventional GID procedure, and is well approximated to the DNAM-LS model in the ap r i o r istudy.
The SGS energy flux characterizes the kinetic energy transfer between the resolved large scales and the unclosed subgrid scales. The positive SGS energy flux stands for the forward energy cascade, while the negative SGS energy flux represents the energy backscatters. The PDFs of the normalized SGS energy flux Π/✏ DNS at filter scales∆ = 16h DNS and 32h DNS are plotted in Figure 7. The DSM model only reconstructs the forward SGS energy transfer whose modeled SGS energy flux is always non-negative. The PDFs of the SGS energy flux for the DMM and DNAM-GID models are obviously narrower than the fDNS data. In comparison, the DNAM-SSD model can accurately mimic both the forward SGS energy transfer and the energy backscatter.  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 We further compare the normalized strain-rate tensor reconstructed by different SGS models, whose PDFs of the normal and shear components at both filter widths ∆ = 16h DNS and 32h DNS are illustrated in Figures 8 and 9, respectively. The strainrate tensor represents the local straining of the turbulence flow. ILES overestimates the strain-rate tensor and the predicted PDFs of the strain rate are obviously wider than the fDNS data, while those PDFs predicted by the DSM, DMM and DNAM-GID models are apparently narrower due to the excessive model dissipation. In contrast, the DNAM-SSD model can recover more small scales and performs well on the predictions of the strain-rate tensor.
Figures 10 exhibits the PDFs of the normalized characteristic strain rate for different SGS models at filter scales∆ = 16h DNS and 32h DNS . ILES obviously fails to predict the characteristic strain rate, while the DSM, DMM and DNAM-GID models cannot well capture the peak of the PDFs in reconstructing the characteristic strain rate. In comparison, the PDFs of the characteristic strain rate predicted by the DNAM-SSD model are very close to the fDNS data.
We finally examine the reconstruction ability of the turbulent coherent structure by comparing the transient contours of the normalized vorticity, shown in Fig. 11. The snapshots of LES calculations for different SGS models are selected on an arbitrary XY slice at the consistent time with approximately two large-eddy turnover periods. For the DSM and DNAM-GID models, some small-scale flow structures are excessively dissipated and only large scales are maintained. Compared to the other SGS models, the vortex structures reconstructed by the DNAM-SSD model exhibit more similar spatial distribution with the fDNS data, and more multiple-scale flow structures are accurately recovered by the proposed DNAM-SSD model.

Conclusions
In the current work, we develop a dynamic nonlinear algebraic model with the newly proposed scale-similarity dynamic procedure (DNAM-SSD) for large-eddy simulation of turbulence. In the DNAM-SSD model, the model coefficients are dynamically determined based on the scale-similarity relation, which greatly simplifies the conventional dynamic procedure based on the Germano identity (GID). The ap r i o r i analysis demonstrates that the proposed DNAM-SSD model outperforms the conventional velocity gradient model (VGM) and DNAM-GID model at a number of filter scales ranging from the inertial to dissipation ranges. The DNAM-SSD model gives the best prediction of the SGS stress with correlation coefficients higher than 95% and 92% as well as relative errors lower than 30% and 38% at corresponding filter scales∆ = 16h DNS and 32h DNS in comparison with the dynamic Smagorinsky model (DSM), dynamic mixed model (DMM), VGM model and DNAM-GID model, respectively. The proposed SSD procedure shows significant advantages over the conventional GID approach in determining the model coefficients of SGS models.