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Spectral non-equilibrium property in homogeneous isotropic turbulence and its implication in subgrid-scale modeling
Physics Letters A 379 (2015) 2331–2336
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Physics Letters A www.elsevier.com/locate/pla
Spectral non-equilibrium property in homogeneous isotropic turbulence and its implication in subgrid-scale modeling Le Fang a , Ying Zhu a,b , Yangwei Liu b,∗ , Lipeng Lu b a
Laboratory of Mathematics and Physics, Ecole Centrale de Pékin, Beihang University, Beijing 100191, China National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, School of Energy and Power Engineering, Beihang University, Beijing 100191, China b
a r t i c l e
i n f o
Article history: Received 7 January 2015 Received in revised form 24 April 2015 Accepted 14 May 2015 Available online 18 May 2015 Communicated by F. Porcelli Keywords: Non-equilibrium turbulence Large-eddy simulation Subgrid-scale modeling
a b s t r a c t The non-equilibrium property in turbulence is a non-negligible problem in large-eddy simulation but has not yet been systematically considered. The generalization from equilibrium turbulence to nonequilibrium turbulence requires a clear recognition of the non-equilibrium property. As a preliminary step of this recognition, the present letter defines a typical non-equilibrium process, that is, the spectral non-equilibrium process, in homogeneous isotropic turbulence. It is then theoretically investigated by employing the skewness of grid-scale velocity gradient, which permits the decomposition of resolved velocity field into an equilibrium one and a time-reversed one. Based on this decomposition, an improved Smagorinsky model is proposed to correct the non-equilibrium behavior of the traditional Smagorinsky model. The present study is expected to shed light on the future studies of more generalized nonequilibrium turbulent flows. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The large-eddy simulation (LES) method has achieved great success in engineering applications and therefore is usually expected to be the main Computational Fluid Dynamics (CFD) technique in the future [1–3]. However, LES simulations for capturing largescale coherent structures (e.g., separation, transition and boundary layer) do not usually yield satisfactory results, while some of them are even worse than the Reynolds-averaged Navier–Stokes (RANS) results [4]. This bad performance also happens in the nonequilibrium regions, even if the flow is free from boundaries [5–9]. Various attempts have been performed to repair this problem. The spirit of early works is to increase the near-wall resolution, for example the first grid point is usually located at y + < 1 with y + the normalized wall distance using the friction velocity. This method reduces the effect of subgrid-scale (SGS) model in the near-wall region with a direct numerical simulation (DNS)-like resolution. It can improve the LES performance, however, the calculation cost also increases obviously, which leads to difficulties for complex flows. For example, under the estimation of Spalart et al., a turbulent flow around an airplane or a car will require huge computational resources, which cannot be satisfied before the year 2045 [10]. This problem calls for later studies on the LES-RANS
*
Corresponding author. E-mail address: [email protected] (Y.W. Liu).
http://dx.doi.org/10.1016/j.physleta.2015.05.029 0375-9601/© 2015 Elsevier B.V. All rights reserved.
hybrid methods. Recently, in order to avoid the over-dense mesh, a detached-eddy simulation (DES) technique has been developed, which involves the RANS calculation in the regions of coherent structures [4,11]. The DES method has achieved a nice balance between accuracy and engineering efficiency, however, the problem of dynamically capture the coherent structure region prevents DES from being widely used in real applications. In particular, difficulties still exist in simulating the strongly-instationary flows as well as the complex flows. Therefore, we may conclude that the surgical operations on the numerical implementations of LES method have not really cured the underlying disease. Instead, in the present paper, we attempt to propose a different methodology for this problem based on the physical recognitions of the non-equilibrium property. In Section 2 we introduce the precise concept of spectral non-equilibrium and show its evident effect in LES. A statistical method to decompose the velocity field is then introduced in Section 3 for considering this non-equilibrium. Based on this decomposition, an improved Smagorinsky model is proposed in Section 4, which corrects the non-equilibrium behavior of the traditional Smagorinsky model. Finally the conclusions and discussions are presented in Section 5. 2. Definition of spectral non-equilibrium property in homogeneous isotropic turbulence Presently there exist different definitions of the non-equilibrium turbulence. Some researchers define non-equilibrium as the imbalance between the maximum non-linear energy transfer
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and the dissipation [8,12–14], for example the free-decaying turbulence is regarded as non-equilibrium under this definition. Another definition of non-equilibrium may be found from Townsend [15] and Bradshaw [16] in the studies of wall turbulence: in the governing equation of turbulence energy, if the productive terms and the dissipative terms are statistically equilibrium, the turbulent flow is defined to be “equilibrium”; otherwise it is “non-equilibrium”. Evidently, most realistic turbulent flows are non-equilibrium. Indeed, the non-equilibrium property is more obvious in the regions of coherent structures, such as transition, wake and separation [17–19]. Does the non-equilibrium property play an important role in the LES calculation and even prevent the simulations from accurately predicting the coherent structures? In order to clearly answer this question, a more clear recognition of the non-equilibrium property would be important. In fact, most traditional SGS models [20–25] were derived under the assumptions of high Reynolds homogeneous isotropic equilibrium turbulence, based on the equilibrium in the inertial range. It is then difficult to say that these SGS models can be naturally extended to complex flows. As pointed out by Frisch et al. [26], these assumptions are indeed far from the reality. If we carefully visit the regions of coherent structures, it can be found that none of these assumptions is satisfied: i) the Reynolds number may be not high (usually Reλ < 1000), which does not lead to obvious inertial range [27]; ii) the flow is usually inhomogeneous; iii) the local isotropy is not usually satisfied even in small scales; iv) the flow is in non-equilibrium. Existing studies have focused on and (partly) solved the first three problems [27–34], however, there is still few attempts on the fourth problem, i.e., the non-equilibrium fact, in the domain of SGS modeling [35,7]. In this case, in the following we would like to introduce our previous attempt of considering the non-equilibrium in a simplest situation, i.e., the homogeneous isotropic turbulence (HIT). In a previous study [36] we introduced the concept of timereversed turbulence, which means the change of all the velocity to − vectors u u in a full-developed turbulence field. (In the absence of viscosity, the dynamics of the Navier–Stokes equations are to − invariant under the simultaneous transformation u u , t to −t. This means that if at an instant t the velocity is reversed, the flow will evolve backward in time until the initial condition is reached. On the level of energy transfer between scales, this property implies that in the inviscid case the direction of the energy transfer reverses when the velocity is reversed or when time is reversed.) to the grid-scale (GS) In LES, we further divide the velocity field u < and the SGS part u > by employing a sharp cut-off filter part u in spectral space. The reversal of GS part then involves the various possibility of SGS part. For example, the “Reverse–Reverse” (RR) case, i.e., reversal at all scales (− u ), leads to a short-term increase of GS energy [36]; by contrast, the “Reverse–Normal” (RN) case, which corresponds to a reversed GS part and a normal SGS > ), does not yield the energy increase, but can cause part (− u< + u a gap-like energy spectrum in short time [37]. Both the Eddydamping-quasi-normal (EDQN) theory [37] and the DNS cases [36] showed that this reversal of GS velocity breaks the equilibrium of energy transfer in inertial range. Specifically, an inverse energy cascade appears at GS scales. It is then possible to define the “spectral non-equilibrium property” in HIT: in the inertial range (usually defined as the k−5/3 region) if the energy production from larger scales and the energy dissipation from smaller scales are statistically equilibrium, we call this “spectrally equilibrium”; otherwise it is “spectrally non-equilibrium”. This can also be understood as the balance between the forward and backward energy flux ε f and εb at an inertial scale (see Ref. [36] for details). It differs from the definition of Valente et al. [8], as we use an instantaneous spectrally-local balance instead of the overall energy balance. For example, in Fig. 1 of Ref. [8] that even in highly
Fig. 1. Time evaluation of GS energy. The time of reversal is denoted as t = 0. The turn-over time at t = 0 is used for normalization. DNS calculations under 2563 resolution. RR: Reverse–Reverse case; RN: Reverse–Normal case. (See Ref. [36] for the detailed definition.) The dashed lines present the differences between nonequilibrium cases and the equilibrium case, normalized by the equilibrium GS energy.
non-equilibrium flows, there are time instants where the energy transfer is balanced at an inertial scale, which in the present definition would be termed as “spectrally equilibrium”. In HIT the spectral equilibrium state usually corresponds to a fully developed turbulence, while the time-reversed turbulence is an extremely non-equilibrium state. Indeed, the practical complex flows are usually between the spectral-equilibrium turbulence and the timereversed turbulence. From DNS calculations it can be shown that the difference of GS energy between time-reversed turbulence and spectrally equilibrium turbulence can be the order of 150% (see Fig. 1), which illustrates the important role of the non-equilibrium behavior in HIT. This, more or less, answers the question in the beginning of this section and shows the importance of considering the spectral non-equilibrium property in LES. 3. Decomposition of non-equilibrium GS turbulence by using the skewness of velocity gradient Classical theories show that in full-developed HIT the skewness of longitudinal velocity gradient S k is a negative quasi-constant which is weakly dependent to Reynolds number [38]. It is also interesting to emphasize three facts: i) the Gaussian initial field → − leads to zero skewness; ii) the time-reversal of u u changes the signal of the skewness; iii) the skewness always tends to the quasi-constant state, i.e., an equilibrium state. From these facts, it is then natural to choose S k as the parameter which represents the spectral non-equilibrium property. In particular, in LES we only know the GS information, thus we choose the skewness of longitudinal GS velocity gradient S k< to present the non-equilibrium property of GS turbulence. This skewness S k< is defined as
S k< =
3 (∂1 u < 1) < 2 (∂1 u 1 ) 3/2
(1)
with the ensemble average. The equilibrium value of S k< is denoted as S k<,equi which is about −0.5 in HIT (the detailed discussion on this value can be found in Ref. [38]). It is then also possible to define the value of S k< in time-reversed turbulence as S k<,rev , with S k<,rev = − S k<,equi . Supposing that S k< is the skewness value
L. Fang et al. / Physics Letters A 379 (2015) 2331–2336
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in a real turbulent flow, in the following we introduce a statistical method to divide this field into a spectral-equilibrium part and a time-reversed part. In particular, the decomposition leads to the following equation: < < ∂1 u < 1 = ∂1 u 1,equi + ∂1 u 1,rev ,
(2)
where u < is the equilibrium GS velocity field, while u < is i ,rev i ,equi the time-reversed one. The idea is to consider the non-equilibrium turbulence as a superposition between two independent canonical velocity fields of equilibrium and time-reversed turbulence, respectively. Here we choose these two fields to be independent and assume that they will keep independent in short time. Note that this is indeed a crude short-time approximation, since these fields will definitively evolve to correlate with each other in long time due to the non-linear dynamics of Navier–Stokes equations. The advance of this simplification is to permit the corresponding SGS modeling, as will be discussed in the next section. The second- and third-order moments for these fields are then, respectively, defined as 2 D 2 = (∂1 u < 1 ) ,
3 D 3 = (∂1 u < 1 ) ,
2 D 2,equi = (∂1 u < 1,equi ) ,
3 D 3,equi = (∂1 u < 1,equi ) ,
2 D 2,rev = (∂1 u < 1,rev ) ,
3 D 3,rev = (∂1 u < 1,rev ) .
(3)
Note that in LES modeling, we are not really interested in the exact decomposition result of velocity fields (indeed, the decomposition is not unique). Instead, the second- and third-order moments would be important parameters in the modeling. In order to further determine the moments, statistical restrictions are necessary. Here we introduce the statistical independency between u < and i ,equi u< , which leads to i ,rev < ∂1 u < 1,equi ∂1 u 1,rev = 0, < 2 ∂1 u < 1,equi (∂1 u 1,rev ) = 0, 2 < (∂1 u < 1,equi ) ∂1 u 1,rev = 0.
(4)
From Eqs. (2)–(4) the final equations can be derived as 3/2 D 3,equi = S k<,equi D 2,equi ,
Fig. 2. Statistical results of the decomposition using the skewness of velocity gradient. The equilibrium skewness is chosen as S k,equi = −0.5. (a) Second-order moments; (b) third-order moments.
3/2 D 3,rev = S k<,rev D 2,rev ,
D 3,equi + D 3,rev = S k< ( D 2,equi + D 2,rev )3/2 , D 2,equi + D 2,rev = D 2 .
(5)
The four unknown variables D 2,equi , D 2,rev , D 3,equi and D 3,rev can then be solved. Although there is no analytical solution found, the numerical method is simple and leads to the results as shown in Fig. 2. From Fig. 2(a) we can also define a parameter
α ( S k< , S k<,equi ) = D 2,equi / D 2
(6)
which represents the proportion of the equilibrium part in the real field. From the figure we can find that the relation between α and S k is nearly linear, that is
α ( S k< , S k<,equi ) ≈
S k< − S k<,rev
− S k<,rev
S k<,equi
(7)
.
Note that S k<,equi = − S k<,rev is only valid when the sign of the velocity field is changed. From Ref. [38] the value of S k<,equi can be estimated as
√
S k<,equi ≈ −
15 30 35
ν
∞ 0
∞ 0
k4 E < (k)dk
k2 E < (k)dk
3/2 ,
(8)
with ν the viscosity, k the wave number and E < (k) the filtered energy spectrum.
4. An improved Smagorinsky model Smagorinsky model is one of the most famous eddy-viscosity SGS models in LES applications [20]. In HIT the standard Smagorinsky model writes
νt = (C s )2 | S < |, where | S < |
=
(S< S < )1/2 , S < ij ij ij
(9)
= (∂ j
u< i
+
∂i u
strain rate, is the filter size in physical space, and C s is the coefficient which can be dynamically determined or fixed depending on flow conditions [39]. Using the similar parameters as Ref. [36], in this study we use constant coefficient C s = 0.14. Also, a cusp is introduced to correct the dissipation spectrum near filter
νt< (k) = νt (1 + 34.6 exp(−3kc /k)).
(10)
Considering the non-equilibrium property, we suppose that the turbulence field can be divided to the equilibrium part and the time-reversed part, as was presented in the last section. It was noted that only the equilibrium part dissipates; the time-reversed part leads to different behaviors of GS energy evolution, depending on the SGS distribution. In the case without viscosity, the timereversed part can either increase or retain the GS energy in short
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time (see Fig. 7 in Ref. [36]). As already discussed in Ref. [36], time-reversible SGS models (e.g., the CZZS model [24]) can yield the increase of GS energy, but they are not always numerically stable, therefore are not appropriate in practical use; by contrast, time-nonreversible SGS models (e.g., the Smagorinsky model) are numerically stable, but they are over-dissipative in short time. As analyzed above, this over dissipation is because of the incorrect treatment on the time-reversed part. It is not appropriate to assume that this part can yield a short-time forward non-linear energy transfer flux, which is represented by the SGS dissipation in LES (see Refs. [40,7,8] for detailed discussions on the imbalance between the non-linear energy transfer and the viscous dissipation). Hence, a natural correction is to assume null dissipation for the time-reversed part. The improved Smagorinsky model is then written as
νt = α ( S k< , S k<,equi )(C s )2 | S < |,
(11)
where the additional coefficient α ( S k< , S k<,equi ) was defined in Eq. (6). Note that due to numerical discretization, the derivatives are usually calculated by finite difference method at the grid size, which indeed involves differences to the definition (1). However, as already discussed in Refs. [33] and [38], this difference is not evident and could be appropriately neglected in LES applications. Two groups of LES test cases under 643 resolution, with each containing six cases, are performed for validate the performance of this improved Smagorinsky model. The numerical details of the calculation can be found in Refs. [25] and [36]. In each group, three of the cases are normal decaying turbulence (denoted as N cases) using the Smagorinsky model, the CZZS model and the improved Smagorinsky model, respectively; the other three cases assume an instantaneous perturbation when the flow is fully developed (which is denoted as t = 0) which reverses all GS-scale informaare normalized by using tion (denoted as R cases). The time values√ √ the turn-over time before reversal T = L / E = 3/2( E /ε ), where L is the integral length-scale, the kinetic energy E appropriately equals to the GS energy E < , while the dissipation ε = ε < + ε > and the SGS dissipation ε > is evaluated by SGS models. Note that this calculation is only valid before reversal, as the normalized energy dissipation will change after reversal [12]. The Taylor-scale Reynolds numbers are 73.9 and 157 in these two groups respectively, corresponding to the equilibrium values of the skewness S k<,equi respectively −0.45 and −0.35 calculated with Eq. (8). The model performance is validated by comparing the Reλ = 73.9 cases with DNS results under the same Reynolds number. The detail about DNS calculations can be found in Ref. [36]. Fig. 3 shows the time evolution of GS energy. All the three N cases show almost the same trend, illustrating the correctness of the improved Smagorinsky model in free-decaying turbulence. Note that under some definitions the free-decaying turbulence may be considered as non-equilibrium [14], but using our definition in this paper, it is equilibrium due to the balance of energy transfer in the inertial range. For the performance of the R cases, as already addressed in Ref. [36], the CZZS model behaves similarly to the RR case of DNS, while the Smagorinsky model produces a wrong short-time evolution as shown in Fig. 3(c), since there is no immediate forward energy transfer after reversal. As expected, it is shown that the improved Smagorinsky model is distinct from the N cases in short time, corresponding to a energy decay with the molecular dissipation and without the forward energy cascade. From Fig. 3(c) this evolution of GS energy is found to be similar as the RZ and RN cases of DNS in short time. The improved Smagorinsky model is therefore here proposed to replace the traditional Smagorinsky model in practical applications, especially in the cases where the non-equilibrium property dominates. To illustrate, we also compare the evolution of energy flux which passes the cut-off wavenumber, as shown in Fig. 4. It is
Fig. 3. Comparison of LES and DNS cases on the time evolution of GS energy. At reversal Reλ = 73.9. (a) DNS results (see Ref. [36] for the detailed definition of NN, RR, RN and RZ); (b) LES results; (c) Comparison of typical cases in short time.
L. Fang et al. / Physics Letters A 379 (2015) 2331–2336
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Fig. 5. Comparison of LES and EDQN cases on the time evolution of GS energy. At reversal Reλ = 157. The subfigure shows the short-time evolution.
Fig. 4. Comparison of LES and DNS cases on the time evolution of energy flux. At reversal Reλ = 73.9. (a) DNS results; (b) LES results.
Fig. 6. LES time evolution of the skewness of velocity gradient. At reversal Reλ = 157.
shown that the original Smagorinsky model yields a wrong shorttime evolution, which does not agree to any DNS case. By contrast, the improved Smagorinsky model is, qualitatively, similar to the RN and RZ DNS cases, representing the null energy flux at reversal as well as a following short-time increase. In order to show the model performance under higher Reynolds numbers, we compare the Reλ = 157 cases to the EDQN theoretical estimation. As discussed in Ref. [37], in short time the EDQN theory can be considered as an appropriate approach for nonequilibrium turbulent flows. From Fig. 5 it is clearly shown that the original Smagorinsky model produces a wrong short-time GS energy evolution, while the improved Smagorinsky model agrees quite well to the RZ situation of the EDQN prediction. Here we do not show the RN case by EDQN method, as it is quite similar to the RZ one in short time [37]. Note that only the short-time evolution of EDQN method (the solid red line in Fig. 5) is considered as a reference, as in long time (less than T , i.e., dashed red line in Fig. 5) the EDQN method will not produce correct results [37]. It might be also interesting to show the evolution of the skewness S k< , which, as was discussed in Section 3, definitively plays a key role in the spectral non-equilibrium process. In Fig. 6, it is shown that the N cases behaves similar, while the R cases show difference in short time (less than T ). As shown in Fig. 6,
the improved Smagorinsky model leads to a slower evolution of S k< . Indeed, the explicit relation between S k< and νt in the improved Smagorinsky model dynamically takes into account the non-equilibrium property, and avoids the over-dissipation of the traditional Smagorinsky model. In Fig. 6, another interesting observation may be the over-equilibrium phenomenon in the region of about 0.7 < t /T < 3, where the skewness value is even smaller than the equilibrium value. It is currently difficult to give a convincing physical explanation, but phenomenologically we may imagine that it is related with the self-organization mechanism of turbulence. Future investigations on this over-equilibrium phenomenon might be therefore interesting. 5. Concluding remarks Natural phenomena are always complex. In order to scientifically recognize these phenomena, one usually needs some assumptions to simplify the investigation. These assumptions are necessary, however, the generalization studies towards complex situations are more important for extending people’s recognitions. Turbulence is such a complex phenomenon, and many assumptions have been involved in the past years, for instance homogeneity, isotropy, high Reynolds, and equilibrium. Based on those
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studies, more and more researchers are trying to step a little further, and are now concerned with the more general situations. The generalization steps, from homogeneity to non-homogeneity, from isotropy to anisotropy, and from high Reynolds to moderate Reynolds, have already achieved great success. However, due to complexity, there is still no systematic work on the generalization from equilibrium to non-equilibrium. We realize that there are different non-equilibrium processes in real turbulence, while a general description might be difficult to reach. Instead, simple non-equilibrium processes should be carefully visited. Therefore, we aim at defining and investigating a typical non-equilibrium property in HIT, expecting to push forward the generalization from equilibrium to non-equilibrium in the turbulence community. This non-equilibrium property is defined in the present letter as a spectral imbalance in inertial range. The concept of time-reversal is then introduced to permit a statistical decomposition. The skewness of GS velocity gradient, S k< , is proposed to be a key parameter in the non-equilibrium process. This decomposition then leads to an improvement of the traditional Smagorinsky model which does not include any non-equilibrium information. Numerical cases show that the improved Smagorinsky model produces the correct short-time GS energy evolution, which corresponds to the RN or RZ DNS cases. In addition, the improved Smagorinsky model is not a time-reversible model, as the backward energy cascade is naturally forbidden from the model assumptions. Compared with existing time-reversible models (e.g., the CZZS model), this property is important for the numerical stability in practical applications. We remark that this problem of short-time non-equilibrium process has already be observed in Ref. [40], but the physical mechanism was not analyzed. The spirit of the mixing model proposed in Ref. [40] is, indeed, similar as the present letter, but we would like to emphasize that our contribution is to rationally analyze this spectral non-equilibrium process. Therefore, the improved Smagorinsky model is a natural result in the present letter, based on the recognition on the spectral non-equilibrium property. We finally conclude that the investigation on the spectral nonequilibrium property in HIT is an attempting step for recognizing the natural complex turbulence and the related modeling problems, which may shed light on the future studies on the more generalized non-equilibrium turbulence. Acknowledgements We are grateful to Liang Shao, Jean-Pierre Bertoglio and Wouter Bos for the useful discussions, and to the anonymous reviews whose suggestions greatly improve the quality of the present paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11202013, 51376001, 51420105008) and the National Basic Research Program of China (Grant No. 2012CB720200). References [1] P. Sagaut, Large Eddy Simulation for Incompressible Flows, Springer, 2006. [2] P. Spalart, W. Jou, M. Strelets, S. Allmaras, Comments on the feasibility of LES for wings, and on a hybrid RANS, in: C. Liu, Z. Lin (Eds.), Advances in DNS/LES, Greyden Press, Columbus, 1997, p. 137. [3] L. Fang, L. Shao, J. Bertoglio, Recent understanding on the subgrid-scale modeling of large-eddy simulation in physical space, Sci. China, Phys. Mech. Astron. 57 (12) (2014) 2188–2193. [4] P. Spalart, Detached-eddy simulation, Annu. Rev. Fluid Mech. 41 (2009) 181–202. [5] J. Lumley, Some comments on turbulence, Phys. Fluids A, Fluid Dyn. 4 (1992) 203. [6] A. Yoshizawa, Nonequilibrium effect of the turbulent-energy-production process on the inertial-range energy spectrum, Phys. Rev. E 49 (1994) 4065.
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Physics Letters A www.elsevier.com/locate/pla
Spectral non-equilibrium property in homogeneous isotropic turbulence and its implication in subgrid-scale modeling Le Fang a , Ying Zhu a,b , Yangwei Liu b,∗ , Lipeng Lu b a
Laboratory of Mathematics and Physics, Ecole Centrale de Pékin, Beihang University, Beijing 100191, China National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, School of Energy and Power Engineering, Beihang University, Beijing 100191, China b
a r t i c l e
i n f o
Article history: Received 7 January 2015 Received in revised form 24 April 2015 Accepted 14 May 2015 Available online 18 May 2015 Communicated by F. Porcelli Keywords: Non-equilibrium turbulence Large-eddy simulation Subgrid-scale modeling
a b s t r a c t The non-equilibrium property in turbulence is a non-negligible problem in large-eddy simulation but has not yet been systematically considered. The generalization from equilibrium turbulence to nonequilibrium turbulence requires a clear recognition of the non-equilibrium property. As a preliminary step of this recognition, the present letter defines a typical non-equilibrium process, that is, the spectral non-equilibrium process, in homogeneous isotropic turbulence. It is then theoretically investigated by employing the skewness of grid-scale velocity gradient, which permits the decomposition of resolved velocity field into an equilibrium one and a time-reversed one. Based on this decomposition, an improved Smagorinsky model is proposed to correct the non-equilibrium behavior of the traditional Smagorinsky model. The present study is expected to shed light on the future studies of more generalized nonequilibrium turbulent flows. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The large-eddy simulation (LES) method has achieved great success in engineering applications and therefore is usually expected to be the main Computational Fluid Dynamics (CFD) technique in the future [1–3]. However, LES simulations for capturing largescale coherent structures (e.g., separation, transition and boundary layer) do not usually yield satisfactory results, while some of them are even worse than the Reynolds-averaged Navier–Stokes (RANS) results [4]. This bad performance also happens in the nonequilibrium regions, even if the flow is free from boundaries [5–9]. Various attempts have been performed to repair this problem. The spirit of early works is to increase the near-wall resolution, for example the first grid point is usually located at y + < 1 with y + the normalized wall distance using the friction velocity. This method reduces the effect of subgrid-scale (SGS) model in the near-wall region with a direct numerical simulation (DNS)-like resolution. It can improve the LES performance, however, the calculation cost also increases obviously, which leads to difficulties for complex flows. For example, under the estimation of Spalart et al., a turbulent flow around an airplane or a car will require huge computational resources, which cannot be satisfied before the year 2045 [10]. This problem calls for later studies on the LES-RANS
*
Corresponding author. E-mail address: [email protected] (Y.W. Liu).
http://dx.doi.org/10.1016/j.physleta.2015.05.029 0375-9601/© 2015 Elsevier B.V. All rights reserved.
hybrid methods. Recently, in order to avoid the over-dense mesh, a detached-eddy simulation (DES) technique has been developed, which involves the RANS calculation in the regions of coherent structures [4,11]. The DES method has achieved a nice balance between accuracy and engineering efficiency, however, the problem of dynamically capture the coherent structure region prevents DES from being widely used in real applications. In particular, difficulties still exist in simulating the strongly-instationary flows as well as the complex flows. Therefore, we may conclude that the surgical operations on the numerical implementations of LES method have not really cured the underlying disease. Instead, in the present paper, we attempt to propose a different methodology for this problem based on the physical recognitions of the non-equilibrium property. In Section 2 we introduce the precise concept of spectral non-equilibrium and show its evident effect in LES. A statistical method to decompose the velocity field is then introduced in Section 3 for considering this non-equilibrium. Based on this decomposition, an improved Smagorinsky model is proposed in Section 4, which corrects the non-equilibrium behavior of the traditional Smagorinsky model. Finally the conclusions and discussions are presented in Section 5. 2. Definition of spectral non-equilibrium property in homogeneous isotropic turbulence Presently there exist different definitions of the non-equilibrium turbulence. Some researchers define non-equilibrium as the imbalance between the maximum non-linear energy transfer
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and the dissipation [8,12–14], for example the free-decaying turbulence is regarded as non-equilibrium under this definition. Another definition of non-equilibrium may be found from Townsend [15] and Bradshaw [16] in the studies of wall turbulence: in the governing equation of turbulence energy, if the productive terms and the dissipative terms are statistically equilibrium, the turbulent flow is defined to be “equilibrium”; otherwise it is “non-equilibrium”. Evidently, most realistic turbulent flows are non-equilibrium. Indeed, the non-equilibrium property is more obvious in the regions of coherent structures, such as transition, wake and separation [17–19]. Does the non-equilibrium property play an important role in the LES calculation and even prevent the simulations from accurately predicting the coherent structures? In order to clearly answer this question, a more clear recognition of the non-equilibrium property would be important. In fact, most traditional SGS models [20–25] were derived under the assumptions of high Reynolds homogeneous isotropic equilibrium turbulence, based on the equilibrium in the inertial range. It is then difficult to say that these SGS models can be naturally extended to complex flows. As pointed out by Frisch et al. [26], these assumptions are indeed far from the reality. If we carefully visit the regions of coherent structures, it can be found that none of these assumptions is satisfied: i) the Reynolds number may be not high (usually Reλ < 1000), which does not lead to obvious inertial range [27]; ii) the flow is usually inhomogeneous; iii) the local isotropy is not usually satisfied even in small scales; iv) the flow is in non-equilibrium. Existing studies have focused on and (partly) solved the first three problems [27–34], however, there is still few attempts on the fourth problem, i.e., the non-equilibrium fact, in the domain of SGS modeling [35,7]. In this case, in the following we would like to introduce our previous attempt of considering the non-equilibrium in a simplest situation, i.e., the homogeneous isotropic turbulence (HIT). In a previous study [36] we introduced the concept of timereversed turbulence, which means the change of all the velocity to − vectors u u in a full-developed turbulence field. (In the absence of viscosity, the dynamics of the Navier–Stokes equations are to − invariant under the simultaneous transformation u u , t to −t. This means that if at an instant t the velocity is reversed, the flow will evolve backward in time until the initial condition is reached. On the level of energy transfer between scales, this property implies that in the inviscid case the direction of the energy transfer reverses when the velocity is reversed or when time is reversed.) to the grid-scale (GS) In LES, we further divide the velocity field u < and the SGS part u > by employing a sharp cut-off filter part u in spectral space. The reversal of GS part then involves the various possibility of SGS part. For example, the “Reverse–Reverse” (RR) case, i.e., reversal at all scales (− u ), leads to a short-term increase of GS energy [36]; by contrast, the “Reverse–Normal” (RN) case, which corresponds to a reversed GS part and a normal SGS > ), does not yield the energy increase, but can cause part (− u< + u a gap-like energy spectrum in short time [37]. Both the Eddydamping-quasi-normal (EDQN) theory [37] and the DNS cases [36] showed that this reversal of GS velocity breaks the equilibrium of energy transfer in inertial range. Specifically, an inverse energy cascade appears at GS scales. It is then possible to define the “spectral non-equilibrium property” in HIT: in the inertial range (usually defined as the k−5/3 region) if the energy production from larger scales and the energy dissipation from smaller scales are statistically equilibrium, we call this “spectrally equilibrium”; otherwise it is “spectrally non-equilibrium”. This can also be understood as the balance between the forward and backward energy flux ε f and εb at an inertial scale (see Ref. [36] for details). It differs from the definition of Valente et al. [8], as we use an instantaneous spectrally-local balance instead of the overall energy balance. For example, in Fig. 1 of Ref. [8] that even in highly
Fig. 1. Time evaluation of GS energy. The time of reversal is denoted as t = 0. The turn-over time at t = 0 is used for normalization. DNS calculations under 2563 resolution. RR: Reverse–Reverse case; RN: Reverse–Normal case. (See Ref. [36] for the detailed definition.) The dashed lines present the differences between nonequilibrium cases and the equilibrium case, normalized by the equilibrium GS energy.
non-equilibrium flows, there are time instants where the energy transfer is balanced at an inertial scale, which in the present definition would be termed as “spectrally equilibrium”. In HIT the spectral equilibrium state usually corresponds to a fully developed turbulence, while the time-reversed turbulence is an extremely non-equilibrium state. Indeed, the practical complex flows are usually between the spectral-equilibrium turbulence and the timereversed turbulence. From DNS calculations it can be shown that the difference of GS energy between time-reversed turbulence and spectrally equilibrium turbulence can be the order of 150% (see Fig. 1), which illustrates the important role of the non-equilibrium behavior in HIT. This, more or less, answers the question in the beginning of this section and shows the importance of considering the spectral non-equilibrium property in LES. 3. Decomposition of non-equilibrium GS turbulence by using the skewness of velocity gradient Classical theories show that in full-developed HIT the skewness of longitudinal velocity gradient S k is a negative quasi-constant which is weakly dependent to Reynolds number [38]. It is also interesting to emphasize three facts: i) the Gaussian initial field → − leads to zero skewness; ii) the time-reversal of u u changes the signal of the skewness; iii) the skewness always tends to the quasi-constant state, i.e., an equilibrium state. From these facts, it is then natural to choose S k as the parameter which represents the spectral non-equilibrium property. In particular, in LES we only know the GS information, thus we choose the skewness of longitudinal GS velocity gradient S k< to present the non-equilibrium property of GS turbulence. This skewness S k< is defined as
S k< =
3 (∂1 u < 1) < 2 (∂1 u 1 ) 3/2
(1)
with the ensemble average. The equilibrium value of S k< is denoted as S k<,equi which is about −0.5 in HIT (the detailed discussion on this value can be found in Ref. [38]). It is then also possible to define the value of S k< in time-reversed turbulence as S k<,rev , with S k<,rev = − S k<,equi . Supposing that S k< is the skewness value
L. Fang et al. / Physics Letters A 379 (2015) 2331–2336
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in a real turbulent flow, in the following we introduce a statistical method to divide this field into a spectral-equilibrium part and a time-reversed part. In particular, the decomposition leads to the following equation: < < ∂1 u < 1 = ∂1 u 1,equi + ∂1 u 1,rev ,
(2)
where u < is the equilibrium GS velocity field, while u < is i ,rev i ,equi the time-reversed one. The idea is to consider the non-equilibrium turbulence as a superposition between two independent canonical velocity fields of equilibrium and time-reversed turbulence, respectively. Here we choose these two fields to be independent and assume that they will keep independent in short time. Note that this is indeed a crude short-time approximation, since these fields will definitively evolve to correlate with each other in long time due to the non-linear dynamics of Navier–Stokes equations. The advance of this simplification is to permit the corresponding SGS modeling, as will be discussed in the next section. The second- and third-order moments for these fields are then, respectively, defined as 2 D 2 = (∂1 u < 1 ) ,
3 D 3 = (∂1 u < 1 ) ,
2 D 2,equi = (∂1 u < 1,equi ) ,
3 D 3,equi = (∂1 u < 1,equi ) ,
2 D 2,rev = (∂1 u < 1,rev ) ,
3 D 3,rev = (∂1 u < 1,rev ) .
(3)
Note that in LES modeling, we are not really interested in the exact decomposition result of velocity fields (indeed, the decomposition is not unique). Instead, the second- and third-order moments would be important parameters in the modeling. In order to further determine the moments, statistical restrictions are necessary. Here we introduce the statistical independency between u < and i ,equi u< , which leads to i ,rev < ∂1 u < 1,equi ∂1 u 1,rev = 0, < 2 ∂1 u < 1,equi (∂1 u 1,rev ) = 0, 2 < (∂1 u < 1,equi ) ∂1 u 1,rev = 0.
(4)
From Eqs. (2)–(4) the final equations can be derived as 3/2 D 3,equi = S k<,equi D 2,equi ,
Fig. 2. Statistical results of the decomposition using the skewness of velocity gradient. The equilibrium skewness is chosen as S k,equi = −0.5. (a) Second-order moments; (b) third-order moments.
3/2 D 3,rev = S k<,rev D 2,rev ,
D 3,equi + D 3,rev = S k< ( D 2,equi + D 2,rev )3/2 , D 2,equi + D 2,rev = D 2 .
(5)
The four unknown variables D 2,equi , D 2,rev , D 3,equi and D 3,rev can then be solved. Although there is no analytical solution found, the numerical method is simple and leads to the results as shown in Fig. 2. From Fig. 2(a) we can also define a parameter
α ( S k< , S k<,equi ) = D 2,equi / D 2
(6)
which represents the proportion of the equilibrium part in the real field. From the figure we can find that the relation between α and S k is nearly linear, that is
α ( S k< , S k<,equi ) ≈
S k< − S k<,rev
− S k<,rev
S k<,equi
(7)
.
Note that S k<,equi = − S k<,rev is only valid when the sign of the velocity field is changed. From Ref. [38] the value of S k<,equi can be estimated as
√
S k<,equi ≈ −
15 30 35
ν
∞ 0
∞ 0
k4 E < (k)dk
k2 E < (k)dk
3/2 ,
(8)
with ν the viscosity, k the wave number and E < (k) the filtered energy spectrum.
4. An improved Smagorinsky model Smagorinsky model is one of the most famous eddy-viscosity SGS models in LES applications [20]. In HIT the standard Smagorinsky model writes
νt = (C s )2 | S < |, where | S < |
=
(S< S < )1/2 , S < ij ij ij
(9)
= (∂ j
u< i
+
∂i u
strain rate, is the filter size in physical space, and C s is the coefficient which can be dynamically determined or fixed depending on flow conditions [39]. Using the similar parameters as Ref. [36], in this study we use constant coefficient C s = 0.14. Also, a cusp is introduced to correct the dissipation spectrum near filter
νt< (k) = νt (1 + 34.6 exp(−3kc /k)).
(10)
Considering the non-equilibrium property, we suppose that the turbulence field can be divided to the equilibrium part and the time-reversed part, as was presented in the last section. It was noted that only the equilibrium part dissipates; the time-reversed part leads to different behaviors of GS energy evolution, depending on the SGS distribution. In the case without viscosity, the timereversed part can either increase or retain the GS energy in short
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time (see Fig. 7 in Ref. [36]). As already discussed in Ref. [36], time-reversible SGS models (e.g., the CZZS model [24]) can yield the increase of GS energy, but they are not always numerically stable, therefore are not appropriate in practical use; by contrast, time-nonreversible SGS models (e.g., the Smagorinsky model) are numerically stable, but they are over-dissipative in short time. As analyzed above, this over dissipation is because of the incorrect treatment on the time-reversed part. It is not appropriate to assume that this part can yield a short-time forward non-linear energy transfer flux, which is represented by the SGS dissipation in LES (see Refs. [40,7,8] for detailed discussions on the imbalance between the non-linear energy transfer and the viscous dissipation). Hence, a natural correction is to assume null dissipation for the time-reversed part. The improved Smagorinsky model is then written as
νt = α ( S k< , S k<,equi )(C s )2 | S < |,
(11)
where the additional coefficient α ( S k< , S k<,equi ) was defined in Eq. (6). Note that due to numerical discretization, the derivatives are usually calculated by finite difference method at the grid size, which indeed involves differences to the definition (1). However, as already discussed in Refs. [33] and [38], this difference is not evident and could be appropriately neglected in LES applications. Two groups of LES test cases under 643 resolution, with each containing six cases, are performed for validate the performance of this improved Smagorinsky model. The numerical details of the calculation can be found in Refs. [25] and [36]. In each group, three of the cases are normal decaying turbulence (denoted as N cases) using the Smagorinsky model, the CZZS model and the improved Smagorinsky model, respectively; the other three cases assume an instantaneous perturbation when the flow is fully developed (which is denoted as t = 0) which reverses all GS-scale informaare normalized by using tion (denoted as R cases). The time values√ √ the turn-over time before reversal T = L / E = 3/2( E /ε ), where L is the integral length-scale, the kinetic energy E appropriately equals to the GS energy E < , while the dissipation ε = ε < + ε > and the SGS dissipation ε > is evaluated by SGS models. Note that this calculation is only valid before reversal, as the normalized energy dissipation will change after reversal [12]. The Taylor-scale Reynolds numbers are 73.9 and 157 in these two groups respectively, corresponding to the equilibrium values of the skewness S k<,equi respectively −0.45 and −0.35 calculated with Eq. (8). The model performance is validated by comparing the Reλ = 73.9 cases with DNS results under the same Reynolds number. The detail about DNS calculations can be found in Ref. [36]. Fig. 3 shows the time evolution of GS energy. All the three N cases show almost the same trend, illustrating the correctness of the improved Smagorinsky model in free-decaying turbulence. Note that under some definitions the free-decaying turbulence may be considered as non-equilibrium [14], but using our definition in this paper, it is equilibrium due to the balance of energy transfer in the inertial range. For the performance of the R cases, as already addressed in Ref. [36], the CZZS model behaves similarly to the RR case of DNS, while the Smagorinsky model produces a wrong short-time evolution as shown in Fig. 3(c), since there is no immediate forward energy transfer after reversal. As expected, it is shown that the improved Smagorinsky model is distinct from the N cases in short time, corresponding to a energy decay with the molecular dissipation and without the forward energy cascade. From Fig. 3(c) this evolution of GS energy is found to be similar as the RZ and RN cases of DNS in short time. The improved Smagorinsky model is therefore here proposed to replace the traditional Smagorinsky model in practical applications, especially in the cases where the non-equilibrium property dominates. To illustrate, we also compare the evolution of energy flux which passes the cut-off wavenumber, as shown in Fig. 4. It is
Fig. 3. Comparison of LES and DNS cases on the time evolution of GS energy. At reversal Reλ = 73.9. (a) DNS results (see Ref. [36] for the detailed definition of NN, RR, RN and RZ); (b) LES results; (c) Comparison of typical cases in short time.
L. Fang et al. / Physics Letters A 379 (2015) 2331–2336
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Fig. 5. Comparison of LES and EDQN cases on the time evolution of GS energy. At reversal Reλ = 157. The subfigure shows the short-time evolution.
Fig. 4. Comparison of LES and DNS cases on the time evolution of energy flux. At reversal Reλ = 73.9. (a) DNS results; (b) LES results.
Fig. 6. LES time evolution of the skewness of velocity gradient. At reversal Reλ = 157.
shown that the original Smagorinsky model yields a wrong shorttime evolution, which does not agree to any DNS case. By contrast, the improved Smagorinsky model is, qualitatively, similar to the RN and RZ DNS cases, representing the null energy flux at reversal as well as a following short-time increase. In order to show the model performance under higher Reynolds numbers, we compare the Reλ = 157 cases to the EDQN theoretical estimation. As discussed in Ref. [37], in short time the EDQN theory can be considered as an appropriate approach for nonequilibrium turbulent flows. From Fig. 5 it is clearly shown that the original Smagorinsky model produces a wrong short-time GS energy evolution, while the improved Smagorinsky model agrees quite well to the RZ situation of the EDQN prediction. Here we do not show the RN case by EDQN method, as it is quite similar to the RZ one in short time [37]. Note that only the short-time evolution of EDQN method (the solid red line in Fig. 5) is considered as a reference, as in long time (less than T , i.e., dashed red line in Fig. 5) the EDQN method will not produce correct results [37]. It might be also interesting to show the evolution of the skewness S k< , which, as was discussed in Section 3, definitively plays a key role in the spectral non-equilibrium process. In Fig. 6, it is shown that the N cases behaves similar, while the R cases show difference in short time (less than T ). As shown in Fig. 6,
the improved Smagorinsky model leads to a slower evolution of S k< . Indeed, the explicit relation between S k< and νt in the improved Smagorinsky model dynamically takes into account the non-equilibrium property, and avoids the over-dissipation of the traditional Smagorinsky model. In Fig. 6, another interesting observation may be the over-equilibrium phenomenon in the region of about 0.7 < t /T < 3, where the skewness value is even smaller than the equilibrium value. It is currently difficult to give a convincing physical explanation, but phenomenologically we may imagine that it is related with the self-organization mechanism of turbulence. Future investigations on this over-equilibrium phenomenon might be therefore interesting. 5. Concluding remarks Natural phenomena are always complex. In order to scientifically recognize these phenomena, one usually needs some assumptions to simplify the investigation. These assumptions are necessary, however, the generalization studies towards complex situations are more important for extending people’s recognitions. Turbulence is such a complex phenomenon, and many assumptions have been involved in the past years, for instance homogeneity, isotropy, high Reynolds, and equilibrium. Based on those
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studies, more and more researchers are trying to step a little further, and are now concerned with the more general situations. The generalization steps, from homogeneity to non-homogeneity, from isotropy to anisotropy, and from high Reynolds to moderate Reynolds, have already achieved great success. However, due to complexity, there is still no systematic work on the generalization from equilibrium to non-equilibrium. We realize that there are different non-equilibrium processes in real turbulence, while a general description might be difficult to reach. Instead, simple non-equilibrium processes should be carefully visited. Therefore, we aim at defining and investigating a typical non-equilibrium property in HIT, expecting to push forward the generalization from equilibrium to non-equilibrium in the turbulence community. This non-equilibrium property is defined in the present letter as a spectral imbalance in inertial range. The concept of time-reversal is then introduced to permit a statistical decomposition. The skewness of GS velocity gradient, S k< , is proposed to be a key parameter in the non-equilibrium process. This decomposition then leads to an improvement of the traditional Smagorinsky model which does not include any non-equilibrium information. Numerical cases show that the improved Smagorinsky model produces the correct short-time GS energy evolution, which corresponds to the RN or RZ DNS cases. In addition, the improved Smagorinsky model is not a time-reversible model, as the backward energy cascade is naturally forbidden from the model assumptions. Compared with existing time-reversible models (e.g., the CZZS model), this property is important for the numerical stability in practical applications. We remark that this problem of short-time non-equilibrium process has already be observed in Ref. [40], but the physical mechanism was not analyzed. The spirit of the mixing model proposed in Ref. [40] is, indeed, similar as the present letter, but we would like to emphasize that our contribution is to rationally analyze this spectral non-equilibrium process. Therefore, the improved Smagorinsky model is a natural result in the present letter, based on the recognition on the spectral non-equilibrium property. We finally conclude that the investigation on the spectral nonequilibrium property in HIT is an attempting step for recognizing the natural complex turbulence and the related modeling problems, which may shed light on the future studies on the more generalized non-equilibrium turbulence. Acknowledgements We are grateful to Liang Shao, Jean-Pierre Bertoglio and Wouter Bos for the useful discussions, and to the anonymous reviews whose suggestions greatly improve the quality of the present paper. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11202013, 51376001, 51420105008) and the National Basic Research Program of China (Grant No. 2012CB720200). References [1] P. Sagaut, Large Eddy Simulation for Incompressible Flows, Springer, 2006. [2] P. Spalart, W. Jou, M. Strelets, S. Allmaras, Comments on the feasibility of LES for wings, and on a hybrid RANS, in: C. Liu, Z. Lin (Eds.), Advances in DNS/LES, Greyden Press, Columbus, 1997, p. 137. [3] L. Fang, L. Shao, J. Bertoglio, Recent understanding on the subgrid-scale modeling of large-eddy simulation in physical space, Sci. China, Phys. Mech. Astron. 57 (12) (2014) 2188–2193. [4] P. Spalart, Detached-eddy simulation, Annu. Rev. Fluid Mech. 41 (2009) 181–202. [5] J. Lumley, Some comments on turbulence, Phys. Fluids A, Fluid Dyn. 4 (1992) 203. [6] A. Yoshizawa, Nonequilibrium effect of the turbulent-energy-production process on the inertial-range energy spectrum, Phys. Rev. E 49 (1994) 4065.
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