Journal of Turbulence
ISSN: (Print) 1468-5248 (Online) Journal homepage: http://www.tandfonline.com/loi/tjot20
Non-equilibrium turbulent phenomena in transitional channel flows F. Liu, L. P. Lu & L. Fang To cite this article: F. Liu, L. P. Lu & L. Fang (2018): Non-equilibrium turbulent phenomena in transitional channel flows, Journal of Turbulence, DOI: 10.1080/14685248.2018.1511906 To link to this article: https://doi.org/10.1080/14685248.2018.1511906
Published online: 22 Aug 2018.
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JOURNAL OF TURBULENCE https://doi.org/10.1080/14685248.2018.1511906
Non-equilibrium turbulent phenomena in transitional channel flows F. Liua,b , L. P. Lub and L. Fanga a Laboratory of Mathematics and Physics, Ecole Centrale de Pékin, Beihang University, Beijing, People’s
Republic of China; b National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, School of Energy and Power EngineeringBeihang University, Beijing, People’s Republic of China ABSTRACT
ARTICLE HISTORY
Non-equilibrium turbulent flows were considerably investigated in recent years, but most studies are restricted in grid-generated turbulence. In the present contribution we show by large-eddy simulation (LES), that there exists a weakly non-equilibrium turbulence region before the beginning of the full-developed turbulence. Both the dissipation coefficient C and the skewness of longitudinal velocity gradient Sk are used as characteristic non-equilibrium quantities, illustrating that the non-equilibrium turbulence region in channel flow is weakly similar to previous studies in grid-generated turbulence. The observation of non-equilibrium turbulence region in transitional channel flow is also expected to inspire future improvement of turbulence models.
Received 6 October 2017 Accepted 6 August 2018 KEYWORDS
Non-equilibrium turbulence; fully-developed turbulence; transitional channel flow; turbulence model
1. Introduction The traditional Kolmogorov 1 theory (K41) [1] leads to two conclusions in the inertial range of high-Reynolds turbulent flows: (i) −5/3 spectrum for the energy distribution; (ii) constant dissipation coefficient for the energy transfer. The latter one is also named as an equilibrium energy transfer. In recent 10 years, a series of experimental and numerical evidences have shown that non-equilibrium turbulent flows exist in reality and thus extended the traditional K41 theory. An early experiment on non-equilibrium turbulent phenomena was performed by Hurst and Vassilicos [2] in a free-decaying turbulent flow generated by fractal square grids. A non-equilibrium dissipation scaling was observed with non-constant dissipation coefficient, defined as C := L U 3 , with U root-meansquare (rms) of the velocity fluctuations, L integral length scale and turbulent energy dissipation rate. The non-equilibrium phenomena have also been found with other configurations, leading to various experimental and numerical studies [3–15]. One of the most important observations for non-equilibrium turbulence is the universal law C ∼ ReI m ReL n with m ≈ n ≈ 1 [8,16] (or a slightly different scaling 15/14 by the theory of Bos and Rubinstein [17]), where ReI denotes the inlet Reynolds number, ReL denotes CONTACT L. Fang
[email protected] Laboratory of Mathematics and Physics, Ecole Centrale de Pékin, Beihang University, Beijing 100191, People’s Republic of China © 2018 Informa UK Limited, trading as Taylor & Francis Group
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the local Reynolds number. The difference between these two Reynolds number is that ReL is directly related to the local position in the flow, while ReI is not. Usually the nonequilibrium region locates in the near-field behind a grid, while a sudden transition is observed in the far-field where C is constant [8,11,14]. However, there still exists many unclear points in both observing the non-equilibrium phenomena and understanding the underlying mechanisms. For example, besides the homogeneous isotropic turbulence (HIT), is it possible to observe the non-equilibrium phenomena in anisotropic turbulent flows? In the present paper we focus on a Poiseuille channel flow with spatial transition. Traditional understandings on the transition phenomena usually divide the flows into three spatial regions, i.e. receptivity, transition, and fully-developed turbulence [18–22]. The transition region can be further divided into many different regions (disturbance growth, formation and development of transitional flow structures, breakdown, . . . ) according to the different physical mechanisms [19,23–26]. The present work is rather interested to the region just after the main transition process completed. We will show that before the beginning of the fully-developed equilibrium turbulence, there are weak evidences showing non-equilibrium properties. We will describe the numerical details in Section 2 and present these observations in Section 3. We clarify that the term ‘non-equilibrium’ has been widely used in literature of wall turbulence, but the meanings are not the same. The definition of Townsend [27] describes a spatial imbalance between production and dissipation in boundary layers [28–30]; on another hand, ‘non-equilibrium’ has been used for describing the unstable boundary layer under temporally periodic pertubations [31–37]. According to previous studies, our definition of ‘non-equilibrium’ turbulence is on neither spatial nor temporal evolution, but aims at describing the imbalance of transfer among different scales, which is generally expressed as an imbalance between transfer and dissipation.
2. Description of the numerical simulation An LES calculation is performed to obtain the spatially transitional channel flow. The governing equations for resolved-scale incompressible flows are < ∂Ui < ∂ 1 ∂P< ∂ 2 Ui < + Ui Uj = − +ν , ∂t ∂xj ρ ∂xi ∂xi ∂xj
∂Ui < = 0, ∂xi
(1)
where the superscript < denotes a low-pass filter that Ui = Ui < + Ui > with Ui < the resolved velocity, P< is the resolved pressure, ρ is the density and ν is the kinematic viscosity. The operator of ensemble average is also defined here, which leads to Ui < = Ui < + ui < with ui < fluctuating velocity and ensemble average. The modelling of the nonlinear term (Ui Uj )< leads to various subgrid-scale (SGS) models, while in the present contribution we choose the commonly used Dynamic Smagorinsky Model (DSM) [38]. The sketch of the calculation is shown in Figure 1. The spatial discretisation is performed by employing the Fourier spectral method in both streamwise x and spanwise z directions, and the Chebyshev–Gauss–Lobatto collocation method in the normal y direction. The resolution is 1024 × 96 × 64 in x, y, z directions, respectively, corresponding to a 64πh × 2h × 2π h domain with h channel half-width. The Reynolds number is
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Figure 1. A sketch of the computational domains employed in transitional channel flows.
Rem = 7000, defined as Rem = Um h ν with Um bulk mean velocity, and ν kinematic viscosity, corresponding to the friction Reynolds number Reτ ≈ 395 in the fully-developed turbulence region, where Reτ = uτ h ν , uτ = τw ρ with τw wall shear stress, and ρ density. The spatial discretisation is homogeneous in both streamwise and spanwise direc+ tions, leading to mesh sizes (the superscript + denote normalisation by wall x ≈ 77 + + variables, e.g. x = uτ x ν ) and z ≈ 38, respectively. By contrast, the first-level normal mesh size from the wall is about y+ ≈ 0.2. Non-slip conditions are applied at wall (except for the PSB region, as will be explained below), while periodic conditions are used in both streamwise and span wise directions. Concerning the numerical method of the channel code, see Refs. [39,40] for more details. In order to generate a spatial transition, a PSB (periodic suction and blowing disturbance) region and a fringe region [41] are employed as presented in Figure 1. The PSB region locates in the streamwise interval [πh, 2πh]. The streamwise length of the fringe region is Lf = 8π h. Some of the numerical details of the PSB and fringe regions are introduced in the following parts (see Ref. [41] for more details). In the present calculation, the inflow condition is a laminar Poiseuille profile which will be subjected to a PSB through a slot located at some positions along the channel wall. The velocity profile in Poiseuille flow is constrained by U = Uc 1 − y2 ,
(2)
with U the streamwise velocity (U, V and W for velocity in the streamwise, normal and spanwise directions, respectively), Uc the centreline velocity and y the wall-normal location. In the PSB region, the non-slip wall boundary conditions are modified as U = 0, W = 0, 1 V = AV [cos (2x − 3π) + 1] sin (ωt) , 2
(3)
where ω = 0.5 is the disturbance frequency and AV = 0.00325Um the disturbance amplitude. The fringe region is employed to force the turbulent flow to return to a prescribed inflow profile with minimal reflection, in order to satisfy the periodic condition in the streamwise direction. The fringe functions are selected the same as in Ref. [41].
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3. Results 3.1. Location of the region of fully-developed turbulence As mentioned in Section 1, traditional understandings on the transition phenomena usually divide the flows into three spatial regions of receptivity, transition, and fully-developed turbulence. In the present contribution we aim at investigating the non-equilibrium phenomena in the later stage of turbulent transition, where the transitional flow structures decay completely before the turbulence to become fully-developed. Thus it is important to clarify these locations carefully. Figure 2(a–c) shows the streamwise evolution of the maximum value of the mean velocity Umax , the friction velocity uτ , and the streamwise rms velocity urms at several wall-normal locations, respectively. According to the criterions introduced by Refs. [19–21,42], the mean velocity profile changes from parabolic to inflectional in the breakdown region of transition, but basically remain unchanged in the earlier region of transition and in the region of fully-developed turbulence, respectively. In order to quantitatively represent this evolution, here we present the maximum value of the mean velocity concisely as shown in Figure 2(a), since it is one of the most important factors in the mean velocity profile. Umax settles to approximately 1.14 at Lx ≈ 23.4π confirming that the velocity profile have the typical shape of turbulent profile in the range Lx > 23.4π. Moser et al. show the maximum value of the mean velocity is approximately 1.13 in fully-developed channel flow [43], for results obtained from direct numerical simulation (DNS) data, the same Reynolds number with the present case. The streamwise
Figure 2. Streamwise evolution of (a) maximum of the streamwise mean velocity Umax in the wallnormal direction, (b) friction velocity uτ , (c) streamwise rms velocity urms at five wall-normal locations. (d) Total shear stress τ normalised by the friction velocity at different downstream positions around Lx = 23.4π .
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evolution of friction velocity also support this result, since uτ remains (nearly) constant in the range Lx > 23.4π , similar with the evolution of Umax . Although at Lx ≈ 23.4π, the main transition process has already been accomplished, but it is still far from the final fully-developed equilibrium state of turbulence. We confirmed this by studying the streamwise development of the total shear stress around Lx = 23.4π, as shown in Figure 2(d). The profile of the total shear stress is a straight line when the flow reaches an fully-developed equilibrium state. It is obvious that the flow has not yet been fully-developed until nearly Lx = 31π. Figure 2(c) approves our estimate of the beginning of fully-developed turbulence as streamwise rms velocity is approximately constant at different wall-normal locations for Lx > 31π . The description of the complete transition process from laminar to turbulence in current case is as follows [19,44]: (a) the disturbance growth and formation of transitional flow structures (Lx < 12π , roughly), in which the mean flow profile close to the parabolic profile as shown in Figure 2(a,b) the breakdown and the turbulent boundary layer development (12π < Lx < 23.4π); (c) the flow structures decay and approach to fully-developed state gradually (23.4π < Lx < 31π); and (d) the fully-developed turbulence (Lx > 31π). Note that turbulence statistics such as the rms velocity and the turbulent kinetic energy show a peak in the second stage, and the peak value is significantly higher than in the turbulent state [19]. Clearly, Figure 2(c) shows peaks for streamwise rms velocity at several wallnormal locations during the breakdown process. Compared with the wall-normal locations in the core region, streamwise rms velocity reach a peak earlier for wall-normal locations in the near-wall region. Moreover, the boundary layers meet at the centre of the channel at the end of this stage. Thus, after Lx ≈ 23.4π , the flow turn into turbulent. In general, these evolutions are in agreement with traditional recognition of the transition regions. The results for grid-generated turbulence reported by Hurst and Vassilicos reveal that the streamwise rms velocity increase along x on the centreline until they reach a peak point x = xpeak beyond which it decay. Thus they defined the production region for x < xpeak and the decay region for x > xpeak [2,4]. Enlightened by their studies and with the present transitional phenomenon taking into account, we then roughly divide the channel flow into three regions: (i) the transition region in the interval Lx < 23.4π (in the present contribution, we have merged the first and second stages mentioned above and not discussed in detail as they are not related to the non-equilibrium turbulence); (ii) we define the interval 23.4π < Lx < 31π specifically as statistics behave between the transition and the fully developed turbulence, and in the following parts we will refer to it as ‘non-equilibrium turbulence region’ for brevity although the non-equilibrium evidences are quite weak; (iii) the fully developed equilibrium turbulence is in the interval Lx > 31π. In brief, the explanations in this section verify the present region partitions, which will be important for the following parts which introduce the non-equilibrium phenomena. 3.2. Low-order flow structures In order to compare the flow structures among the three regions defined in the previous subsection, here we examine two typical low-order flow structures as follows. • At large scales, longitudinal streaks near wall are usually considered as typical coherent structures in channel, and are plotted in Figure 3. In both non-equilibrium
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Figure 3. Longitudinal fluctuating velocity magnitude at y+ ≈ 17 in (a) transition region, (b) starting of non-equilibrium turbulence region and (c) equilibrium turbulence region, respectively.
and equilibrium regions we can find similar longitudinal high-speed and low-speed streaks. • At small scales, a typical flow structure can be the helical velocity, i.e. the alignment between velocity and vorticity, corresponding the depression of nonlinearity and the enhancement of dissipation [45,46]. From Figure 4 it is clear that the distributions of the angle between fluctuating velocity and vorticity show no difference between non-equilibrium and equilibrium regions, by comparing to the transition region. In brief, both these flow structures show that the non-equilibrium region is similar to the equilibrium region, but differs from the transition region. This agrees with the region division in the previous subsection, where we consider the flows in the non-equilibrium region as turbulence according to traditional understandings. However, in the following subsection we will show that the relation between transfer and dissipation might be slightly non-equilibrium in this region. 3.3. Non-equilibrium phenomena In this subsection we will discuss the special non-equilibrium region in the later stage of transition, that is, a decaying region before the beginning of the fully developed turbulence, and the characteristic statistical quantities in which are similar to the non-equilibrium phenomena in grid-generated turbulence. In Section 3.3.1 the dissipation coefficient denotes the imbalance between transfer flux and dissipation; in Section 3.3.2 the skewness of longitudinal velocity derivative involves the third-order moments in physical space; in Section 3.3.3 a preliminary budget analysis of the skewness evolution is performed. 3.3.1. Dissipation coefficient As introduced in Section 1, the dissipation coefficient is the most commonly used statistical quantity for presenting the non-equilibrium phenomena in grid-generated turbulence. In
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Figure 4. PDFs of the angle between fluctuating velocity and vorticity, at (a) y+ ≈ 3, (b) y+ ≈ 17, (c) y+ ≈ 46, (d) y+ ≈ 124 and (e) y+ ≈ 380, respectively. Statistical points are selected at streamwise locations Lx = 14π (transition region), Lx = 24π (non-equilibrium region) and Lx = 40π (equilibrium region), respectively.
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anisotropic turbulence the extension is not evident, here we approximately use the spanwise components (since the flows is inhomogeneous in streamwise direction) to replace the longitudinal definitions in HIT: C = Lw W 3 , (4) with the turbulent energy dissipation rate. Because in the present case we only have one spatial homogeneous direction z, the integral scale has to be defined in this specified ∞ direction, yielding W the rms of spanwise fluctuating velocity, and Lw = 0 Rww (r)dr 2 the integral length scale with Rww (r) = w(z)w(z + r) w the autocorrelation of the spanwise velocity fluctuation. In the present contribution all statistics are obtained by the averaging in both time and spanwise direction. Figure 5 shows the streamwise evolution of C at y+ ≈ 17, 46, 124 and 380, respectively. Note that here we only focus on the region just after the main transition process completed, defined in the previous section as Lx > 23.4π, since in the transition region C does not have clear physics and can diverge. At each wall distance, we can find that the value of C increases weakly to constant. This phenomenon is slightly similar to the observations in grid-generated turbulence [2–4,16], which might imply that the turbulence is weakly non-equilibrium. Accordingly, following the definition in grid-generated turbulence, we define the ‘non-equilibrium’ region as 23.4π < Lx < 31π where C increases, and ‘equilibrium’ region as Lx > 31π with approximately constant C . In addition, the values of C near-wall are greater than those in the centre. The underlying reason is that at low Reynolds number C is approximately inversely proportional to the integral scale Reynolds number ReL = W Lw ν [47], which is lower at near-wall locations due to the narrower separation between energy-containing scale and dissipation scale. Also, the predictions in HIT [48] are in agreement with our results, as C is greater than 1 for a lower ReL (corresponding to y+ ≈ 17 and 46 in the present case) while is less than 1 for a moderately higher ReL (corresponding to y+ ≈ 124 and 380 in the present case). The relation between dissipation coefficient C and Reynolds number ReL at different wall distances is plotted in Figure 6. It is shown that the scalings between the
Figure 5. Streamwise evolution of the dissipation coefficient C .
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Figure 6. Dissipation coefficient C against Reynolds number ReL at (a) y+ ≈ 17, (b) y+ ≈ 46, (c) y+ ≈ 124, and (d) y+ ≈ 380, respectively.
non-equilibrium region (squares) and the equilibrium region (circles) differ at all wall distances, while the former is more similar to the transition region (triangles) where the turbulence is out of equilibrium. These results might be similar to the numerical cases of Goto and Vassilicos with large-scale periodic forcing [15,49], illustrating the nonequilibrium properties of the present case. Different to literature [8,11,12], at y+ ≈ 17, 46 and 124, in equilibrium region (circles) we observe that C is weakly dependent to ReL . This is perhaps due to the channel geometry, which resists the grow of boundary layer, involves additional energy transfer or diffusion along normal direction and affects the value of C which corresponds to the local balance between transfer and dissipation. In another study we are presently in process of performing a comparison between transitional channel and transitional boundary layer to obtain better understanding of this phenomenon. 3.3.2. Skewness of longitudinal velocity derivative Indeed, the extensions of dissipation coefficient to anisotropic flows are not evident, although results show similarity with grid-generated turbulence. In order to illustrate the
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non-equilibrium phenomena, the skewness of longitudinal velocity derivative is considered by following Refs. [12,14,50], defined along the streamwise direction as 3 ∂u ∂x , Sk = (5) 3 2 2 ∂u ∂x with u the streamwise fluctuating velocity and the averaging operators performed in both the spanwise direction and time. The reason for choosing the streamwise direction is that it is approximately consistent to the HIT definition in a-posteriori cases [51,52]. The evolutions of Sk at different wall distances y+ ≈ 3, 17, 46, 124 and 380 are shown in Figure 7. Here the statistics are performed in all the channel except the fringe region. According to the discussions, the whole process is divided into three regions, marked in the figure. In the beginning of the channel (Lx ∼ 0) Sk is approximately zero because all fluctuations can be considered as Gaussian noise. This value increases in the earlier region of transition, which, to our knowledge, has not been reported in literature, while one of the only related discussions might be Ayyalasomayajula and Warhaft [53], who pointed out that positive skewness corresponds to the fact that vortex compression dominates in an equilibrium flow. In Appendix 1 we attempt to give a simplified analytical analysis on this phenomenon, that increasing Sk stems from the break of symmetry of the velocity distribution along the normal direction. In the breakdown region and the non-equilibrium region, phenomena are more complex, while the skewness evolutes differently at different wall distances. We are also interested to the differences between the non-equilibrium and equilibrium regions of turbulence, which agree with the conclusion in Section 3.3.1: Sk slightly decreases in the non-equilibrium region, while is approximately constant in the equilibrium region. Note that the asymptotic values of Sk vary from −0.35 to −0.1 at different wall distances, this
Figure 7. Streamwise evolution of the skewness of longitudinal velocity derivative Sk .
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might be explained by using the conclusion in HIT, that Sk is dependent to local Reynolds number [54]. Following Hearst and Lavoie [14], the values of C and Sk in phase space is plotted in Figure 8 to illustrate their correlation. We select different points in the figure corresponds to the value of C and Sk at a specific streamwise location in the turbulence region, and calculate the joint PDF between C and Sk by statistical treatment in non-equilibrium and equilibrium regions, respectively. The distributions show differences between nonequilibrium and equilibrium regions: both C and Sk have a wider distribution in the non-equilibrium region (contour lines) by comparing with that in the equilibrium region (contour grayscales). Specifically, each equilibrium region in phase space is a corner contained by the corresponding non-equilibrium region. These results are in agreement with the observation of Hearst and Lavoie in grid-generated turbulence [14], that C and −Sk are correlated in the evolution from non-equilibrium turbulence to equilibrium turbulence. Moreover, although not exactly the same, this non-equilibrium behaviour in phase space is consistent to our previous study [55], in which a similar non-equilibrium process was observed in the (F3 /F1 , F3 /F1 ) phase space, with F1 , F2 and F3 fourth-order statistical invariants in HIT.
Figure 8. Relation between the dissipation coefficient C and the skewness of longitudinal velocity derivative Sk in phase space. Different points are selected as statistical samples in non-equilibrium region (contour lines) and equilibrium region (contour grayscales), respectively. Contour values are the joint PDF between C andSk . (a) y+ ≈ 17; (b) y+ ≈ 46; (c) y+ ≈ 124; (d) y+ ≈ 380.
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Non-zero values of Sk corresponds to a skewed probability density function (PDF) of the longitudinal fluctuating velocity gradient. We recall that in the case of turbulence experimentally generated by different grids, the PDFs are generally close to Gaussian in the decay region, but clearly non-Gaussian and skewed towards negative values in the production region [4,12,56]. We remark that the transition region of channel flows in the present case, including an earlier region and a breakdown region, is analogically similar to the production region in grid-generated turbulence, since the turbulence structures are self-organised in those regions. In Figure 9 the location Lx = 14π then corresponds to a typical state in the transition, showing highly non-Gaussian distribution and exhibit asymmetric tails, which are in agreement with the previous studies in grid-generated turbulence [4,12,56]. By contrast, the PDFs are almost the same between the non-equilibrium turbulence region (Lx = 24π as a typical location) and the equilibrium turbulence region (Lx = 40π as a typical location), this is also in agreement with grid-generated turbulence [4,12,56]. These evidences support the present observations and explanations on the non-equilibrium turbulence. Note that due to the intermittency of velocity field which is mostly evident in near-wall regions, the PDFs of both non-equilibrium and equilibrium turbulence regions are not exactly Gaussian. 3.3.3. Budget analysis of the skewness evolution As discussed in the previous part, the streamwise evolution of the skewness of velocity gradient is appropriate for describing the non-equilibrium properties of the channel flows. It is then interesting to investigate the budget of the skewness, written in a Lagrangian form to represent the spatial evolution: dSk = Q + + V + , (6) dt with Sk the skewness of longitudinal velocity derivative, d dt the Lagrangian derivative, and
∂u1 ∂u1 −(3/2) ∂u1 ∂u1 ∂Uj ∂u1 Q = −3 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂xj
∂u1 ∂u1 ∂u1 ∂u1 ∂u1 −(5/2) ∂u1 ∂Uj ∂u1 , +3 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂xj
3 ∂u1 ∂u1 −(3/2) ∂u1 ∂u1 ∂ 2 p =− ρ ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1
3 ∂u1 ∂u1 ∂u1 ∂u1 ∂u1 −(5/2) ∂u1 ∂ 2 p + , ρ ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1
2
∂u1 ∂u1 −(3/2) ∂ u1 ∂ 2 u1 ∂u1 1 ∂2 ∂u1 ∂u1 ∂u1 V = −3ν 2 − ∂x1 ∂x1 ∂x1 ∂xj ∂x1 ∂xj ∂x1 3 ∂xj ∂xj ∂x1 ∂x1 ∂x1
−(5/2) 2
∂ u1 ∂ 2 u1 1 ∂2 ∂u1 ∂u1 ∂u1 ∂u1 ∂u1 ∂u1 ∂u1 − , + 3ν ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂xj ∂x1 ∂xj 2 ∂xj ∂xj ∂x1 ∂x1
∂ 2 u1 ∂ 2 u1 ∂u1 ∂2 1 ∂u1 ∂u1 ∂u1 ∂u1 ∂u1 −(3/2) νt 2 νt − = −3 ∂x1 ∂x1 ∂x1 ∂xj ∂x1 ∂xj ∂x1 3 ∂xj ∂xj ∂x1 ∂x1 ∂x1
:=
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Figure 9. PDFs of the longitudinal velocity derivatives at (a) y+ ≈ 3, (b) y+ ≈ 17, (c) y+ ≈ 46, (d) y+ ≈ 1/2 124 and (e) y+ ≈ 380, respectively. g = (∂u/∂x) (∂u/∂x)2 is the normalised value of ∂u ∂x .
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Figure 10. Budget in the Lagrangian evolution equation of skewness of longitudinal velocity derivative. (a) y+ ≈ 17; (b) y+ ≈ 380.
∂ 2 u1 ∂ 2 u1 ∂u1 ∂u1 ∂νt ∂ 2 u1 ∂u1 ∂u1 ∂u1 ∂u1 ∂u1 −(5/2) − +3 νt ∂x1 ∂x1 ∂x1 ∂xj ∂xj ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂xj ∂x1 ∂xj
2 2 ∂ 1 ∂u1 ∂u1 ∂u1 ∂νt ∂ u1 − νt − (7) 2 ∂xj ∂xj ∂x1 ∂x1 ∂x1 ∂x1 ∂xj ∂xj are production, pressure, viscosity and SGS terms, respectively. The details on the derivation can be found in Appendix 1. Here we introduce the common-used eddy-viscosity assumption and use the SGS viscosity νt to describe the subgrid tensor, and will not discuss more about the inadequacy of this assumption [57,58]. Indeed, an LES case will always involve errors on these SGS quantities, neither using the values of νt to calculate nor assuming = −(Q + + V) with null would be convincing. We therefore do not present the values of in the present paper, and leave investigations in future DNS cases. This budget is shown in Figure 10 at two typical wall distances respectively. It can be observed that the production term Q is always less than 0, indicating that Q is dominant to the decrease of Sk ; by contrast, the role of viscosity term V is negligible. According to Figure 7, the decrease of Sk corresponds to a self-organising process in the velocity phase space to an equilibrium state. From the definition (7) we find that Q indicates the nonlinear velocity interactions, while V viscosity dissipation. We can therefore conclude that the underlying mechanism that non-equilibrium turbulence tends to the equilibrium state might be due to the nonlinear velocity interactions rather than the viscosity dissipation. This might support the modelling improvement on the non-equilibrium turbulent flows.
4. Conclusion The objective of the present paper is to look for non-equilibrium phenomena in transitional channel flows. Comparing to the grid-generated turbulence, we have not observed obvious non-equilibrium phenomena, but in the region before the beginning of the fully-developed turbulence, statistics weakly show the non-equilibrium properties. We summarise the conclusions as follows.
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(1) In a spatially transitional channel flow, according to traditional low-order statistical quantities and either the perspective of non-equilibrium turbulence, we divide the flow into three regions, with the transition, the non-equilibrium, and the fully-developed equilibrium region included. (2) Both the dissipation coefficient C and the skewness of longitudinal velocity gradient Sk are used as characteristic non-equilibrium quantities. Results are quantitatively in weak agreement with previous studies in grid-generated turbulence, implying that there might be weakly non-equilibrium turbulence in transitional channel flows. In the present study it is shown that no evident difference for the low-order structures (Section 3.2) between non-equilibrium and equilibrium regions, but the high-order statistics which are related to the transfer show differences (Section 3.3). This implies that besides the traditional understandings based on energy perturbation [17,59,60], the mechanism of non-equilibrium might be also due to the phase de-coherence [50,61]. This will possibly raise many related investigation topics. Whereas, we should admit that even if the phase de-coherence leads to non-equilibrium turbulence, it is very weak by comparing to the non-equilibrium phenomena caused by energy perturbation. Indeed, the reason that the non-equilibrium phenomena are very weak in the channel flow might be due to the presence of velocity gradient everywhere, which always acts as an energy production. By contrast, in grid turbulence the nonequilibrium region is observed in the part of the flow where the mean velocity gradients, and thereby the production diminishes in strength, leading to a strong non-equilibrium between production and dissipation in this region. The non-equilibrium phenomena are also expected to involve new insights in understanding and modelling turbulence. As an example, traditional RANS (Reynolds averaged Navier–Stokes) calculations on a transitional flow usually use turbulence models and transition models. In particular, in the turbulence region we only have turbulence models such as the k − model. However, many traditional turbulence models are based on the equilibrium assumption, which will be broken in the non-equilibrium region [16,61]. This then calls for future studies on improving these turbulence models to correctly simulation the non-equilibrium phenomena in transitional flows. Specifically, the following two facts might be useful: (1) From Figure 6 it is shown that the scaling of C in non-equilibrium region is more close to the transition region, by comparing to the equilibrium region. This is more evident when near wall. This then breaks the equilibrium assumption which was explicitly used in some classic turbulent models such as the k − model, inspiring further improvement by considering non-equilibrium; also, this calls for further investigations to estimate the performance of other turbulent models in the nonequilibrium region. (2) From Figure 10 it is illustrated that the self-organisation from non-equilibrium to equilibrium is caused by the nonlinear velocity interactions rather than by the viscosity dissipation. This indicates that simply changing the turbulence viscosity in a turbulence model might be inappropriate to represent the non-equilibrium turbulence; instead, a non-equilibrium turbulence model dealing with high-order moments might be a better choice.
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Acknowledgments We are grateful to Chong Pan, Jinjun Wang, Liang Shao, Wouter Bos and Robert Rubinstein for the inspiring discussions, and to the two anonymous referees for the many constructive suggestions as well as the guides on science.
Disclosure statement No potential conflict of interest was reported by the authors.
Funding This work is supported by the National Science Foundation of China [Grant numbers 11772032, 51420105008, 11572025] and the National Basic Research Program of China [Grant number 2014CB046405].
References [1] Kolmogorov AN. The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. Proc Math Phys Sci. 1941;30:301. [2] Hurst D, Vassilicos JC. Scalings and decay of fractal-generated turbulence. Phys Fluids. 2007;19(3):035103. [3] Seoud RE, Vassilicos JC. Idissipation and decay of fractal-generated turbulence. Phys Fluids. 2007;19:105108. [4] Mazellier N, Vassilicos JC. Turbulence without Richardson–Kolmogorov cascade. Phys Fluid. 2010;22(7):075101. [5] Laizet S, Lamballais E, Vassilicos JC. A numerical strategy to combine high-order schemes, complex geometry and parallel computing for high resolution DNS of fractal generated turbulence. Comput Fluids. 2010;39:471. [6] Valente PC, Vassilicos JC. The decay of turbulence generated by a class of multiscale grids. J Fluid Mech. 2011;687:300. [7] Krogstad PA, Davidson PA. Freely decaying, homogeneous turbulence generated by multi-scale grids. J Fluid Mech. 2011;680:417. [8] Valente PC, Vassilicos JC. Universal dissipation scaling for nonequilibrium turbulence. Phys Rev Lett. 2012;108:214503. [9] Nagata K, Sakai Y, Inaba T, et al. Turbulence structure and turbulence kinetic energy transport in multiscale/fractal-generated turbulence. Phys Fluids. 2013;25:065102. [10] Valente PC, Vassilicos JC. The non-equilibrium region of grid-generated decaying turbulence. J Fluid Mech. 2014;744:5–37. [11] Hearst RJ, Lavoie P. Decay of turbulence generated by a square-fractal-element grid. J Fluid Mech. 2014;741:567–584. [12] Isaza JC, Salazar R, Warhaft Z. On grid-generated turbulence in the near- and far field regions. J Fluid Mech. 2014;753:402–426. [13] Zhou Y, Nagata K, Sakai Y, et al. Relevance of turbulence behind the single square grid to turbulence generated by regular- and multiscale-grids. Phys Fluids. 2014;26:075105. [14] Hearst RJ, Lavoie P. Velocity derivative skewness in fractal-generated, non-equilibrium grid turbulence. Phys Fluids. 2015;27:071701. [15] Goto S, Vassilicos JC. Energy dissipation and flux laws for unsteady turbulence. Phys Lett A. 2015;379(16–17):1144–1148. [16] Vassilicos JC. Dissipation in turbulent flows. Annu Rev Fluid Mech. 2015;47:95–114. [17] Bos WJT, Rubinstein R. Dissipation in unsteady turbulence. Phys Rev Fluids. 2017;2:022601. [18] Kachanov YS. Physical mechanisms of laminar-boundary-layer transition. Annu Rev Fluid Mech. 1994;26:411–482.
JOURNAL OF TURBULENCE
17
[19] Klerser L. Numerical simulation of transition in wall-bounded shear flows. Annu Rev Fluid Mech. 1991;23:495–537. [20] Saiki EM, Biringen S, Danabasoglu G, Streett CL. Spatial simulation of secondary instability in plane channel flow: Comparison of k- and h-type disturbances. J Fluid Mech. 1993;253:485–507. [21] Matsubara M, Alfredsson PH. Disturbance growth in boundary layers subjected to freestream turbulence. J Fluid Mech. 2001;430:149–168. [22] Sano M, Tamai K. A universal transition to turbulence in channel flow. Nat Phys. 2015;12(3):249–253. [23] Saric WS, Reed HL, Kerschen EJ. Boundary-layer receptivity to freestream disturbances. Annu Rev Fluid Mech. 2002;34:291–319. [24] Klebanoff PS, Tidstrom KD, Sargent LM. The three-dimensional nature of boundary-layer instability. J Fluid Mech. 1962;12:1–34. [25] Bayly BJ, Orszag SA, Herber T. Instability mechanisms in shear-flow transition. Annu Rev Fluid Mech. 1988;20:359–391. [26] Henningson DS, Lundbladh A, Johansson AV. A mechanism for bypass transition from localized disturbances in wall-bounded shear flows. J Fluid Mech. 1993;250:169–207. [27] Townsend AA. Equilibrium layers and wall turbulence. J Fluid Mech. 1961;11:97. [28] Aubertine CD, Eaton JK. Turbulence development in a non-equilibrium turbulent boundary layer with mild adverse pressure gradient. J Fluid Mech. 2005;532:345–364. [29] Dixit SA, Ramesh ON. Determination of skin friction in strong pressuregradient equilibrium and near-equilibrium turbulent boundary layers. Exp Fluids. 2009;47(6):1045–1058. [30] Kawai S, Larsson J. Dynamic non-equilibrium wall-modeling for large eddy simulation at high Reynolds numbers. Phys Fluids. 2013;25:015105. [31] Scotti A, Piomelli U. Numerical simulation of pulsating turbulent channel flow. Phys Fluids. 2001;13:1367–1384. [32] Costamagna P, Vittori G, Blondeaux P. Coherent structures in oscillatory boundary layers. J Fluid Mech. 2003;474:1–33. [33] Spalart PR, Baldwin BS. Direct simulation of a turbulent oscillating boundary layerIn Turbulent shear flows Vol. 6. Berlin: Springer; 1987. p. 417–440. [34] Merkli P, Thomann H. Transition to turbulence in oscillating pipe flow. J Fluid Mech. 1975;68:567–575. [35] Mazzuoli M, Vittori G, Blondeaux P. Turbulent spots in oscillatory boundary layers. J Fluid Mech. 2011;685:365–376. [36] Jensen BL, Sumer BM, Fredsoe J. Turbulent oscillatory boundary layers at high Reynolds numbers. J Fluid Mech. 1989;206:265–297. [37] Tang YF, Akhavan R. Computations of equilibrium and non-equilibrium turbulent channel flows using a nested-les approach. J Fluid Mech. 2016;793:709–748. [38] Germano M, Piomelli U, Moin P, et al. A dynamic subgrid-scale eddy viscosity model. Phys Fluids A. 1991;3(7):1760–1765. [39] Xu CX, Zhang ZS, Nieuwstadt FTM. Origin of high kurtosis in viscous sublayer. Direct numerical simulation and experiment. Phys Fluids. 1996;8:1938–1942. [40] Yang ZX, Cui GX, Xu CX, et al. Large eddy simulation of rotating turbulent channel flow with a new dynamic global-coefficient nonlinear subgrid stress model. J Turbul. 2012; 13(1):48. [41] Ma B, Van Doorne CWH, Zhang Z, et al. On the spatial evolution of a wall-imposed periodic disturbance in pipe Poiseuille flow at Re = 3000. Part 1. Subcritical disturbance. J Fluid Mech. 1999;398:181–224. [42] Zang TA, Gilbert N, Kleiser L. Direct numerical simulation of the transitional zone. Instability Transit. 1990;2:283–299. [43] Moser RD, Kim J, Mansour NN. Direct numerical simulation of turbulent channel flow up to Reτ = 590. Phys Fluids. 1999;11:943. [44] Zagarola MV, Smits AJ. Mean-flow scaling of turbulent pipe flow. J Fluid Mech. 1998;373:33–79.
18
F. LIU ET AL.
[45] Kerr RM. Higher-order derivative correlations and the alignment of small-scale structure in isotropic numerical turbulence. J Fluid Mech. 1985;153:31–58. [46] Kraichnan R, Panda R. Depression of nonlinearity in decaying isotropic turbulence. Phys Fluids. 1988;31:2395. [47] Sreenivasan KR. On the scaling of the turbulence energy dissipation rate. Phys Fluids. 1984;27:1048. [48] Bos WJT, Shao L, Bertoglio JP. Spectral imbalance and the normalized dissipation rate of turbulence. Phys Fluids. 2007;19:045101. [49] Goto S, Vassilicos JC. The dissipation rate coefficient of turbulence is not universal and depends on the internal stagnation point structure. Phys Fluids. 2014;21(3):035104. [50] Fang L, Zhu Y, Liu YW, et al. Spectral non-equilibrium property in homogeneous isotropic turbulence and its implication in subgrid-scale modeling. Phys Lett A. 2015;379(38):2331–2336. [51] Cui GX, Zhou HB, Zhang ZS, et al. A new dynamic subgrid eddy viscosity model with application to turbulent channel flow. Phys Fluids. 2004;16(8):2835–2842. [52] Shao L, Zhang ZS, Cui GX, et al. Subgrid modeling of anisotropic rotating homogeneous turbulence. Phys Fluids. 2005;17(11):115106. [53] Ayyalasomayajula S, Warhaft Z. Nonlinear interactions in strained axisymmetric highReynolds-number turbulence. J Fluid Mech. 2006;566:273–307. [54] Bos WJT, Chevillard L, Scott J, et al. Reynolds number effects on the velocity increment skewness in isotropic turbulence. Phys Fluids. 2012;24:015108. [55] Fang L, Zhang YJ, Fang J, et al. Relation of the fourth-order statistical invariants of velocity gradient tensor in isotropic turbulence. Phys Rev E. 2016;94(2):023114. [56] Krogstad PA, Davidson PA. Near-field investigation of turbulence produced by multi-scale grids. Phys Fluids. 2012;24:035103. [57] Brun C, Friedrich R. Modelling the SGS tensor Tij: An issue in the dynamic approach. Phys Fluids. 2001;13(8):2373–2385. [58] Fang L, Shao L, Bertoglio JP, Cui G, et al. An improved velocity increment model based on Kolmogorov equation of filtered velocity. Phys Fluids. 2009;21(6):065108. [59] Rubinstein R, Clark TT. ‘Equilibrium’ and ‘non-equilibrium’ turbulence. Theor Appl Mech Lett. 2017;7(5):301. [60] Meldi M, Sagaut P. Investigation of anomalous very fast decay regimes in homogeneous isotropic turbulence. J Turbul. 2018;19(5):390–413. [61] Fang L, Zhao HK, Lu LP, et al. Quantitative description of non-equilibrium turbulent phenomena in compressors. Aerosp Sci Technol. 2017;71:78–89.
Appendices Appendix 1. A simplified analytical model on positive Sk in the initial stage of transition The present case is simplified as a two-dimensional flow with mean velocity v0 and PSB perturbations. It is assumed that the PSB perturbations yield small normal velocities, which are simply advected by mean velocity v0 . The continuous perturbations are further simplified to three instants at two streamwise points, as shown in Figure A1, respectively. These three instants are used to represent the symmetric distribution of PSB perturbations. Investigating the broken of this symmetry in the simplified case will then explain the positive values of Sk in the initial stage of transition. Here v1 and v2 are PSB disturbance velocities in the wall-normal direction with v1 < v2 . Due to constant mean velocity, the trajectories of advected micro-particles are simply straight lines, denoted in Figure A1. For incompressible flows, the continuous equation is ∂u ∂v + = 0, ∂x ∂y
(A1)
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Figure A1. The sketches of two-dimensional flows with PSB disturbance.
leading to ∂v ∂u =− . ∂x ∂y
(A2)
Thus the skewness of longitudinal velocity derivative is the opposite number of the skewness of normal velocity derivative, i.e.
3 3 ∂v ∂u ∂x
Sk = 3 ∂u 2 ∂x
/2
= −
∂y
∂v ∂y
2 3/2 .
(A3)
In the following the normal velocity derivatives are estimated by employing the difference method, g = ∂v/∂y ≈ v y . For the streamwise location A in Figure A1(a) (equivalent to location B in Figure A1(b) and location C in Figure A1(c)), the velocity gradient in normal direction is ga =
v0 (v1 − v2 ) < 0. v1 (x + l) − v2 x
(A4)
For streamwise location B in Figure A1(b) and streamwise location C in Figure A1(c), the velocity gradients in normal direction are, respectively, gb = 0,
gc =
v0 (v2 − v1 ) > 0. v2 (x + l) − v1 x
(A5)
G=
1 ga + gc , 3
(A6)
The average of ga , gb and gc is
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which then leads to the expression of Sk : 3 3 3 1 ga − G + gb − G + gc − G 3 Sk = − 2 2 2 3/2 . 1 g − G + g − G + g − G a c b 3
(A7)
Substituting Equations (A4), (A5) and (A6) into Equation (A7), we have Sk = −
2ga 3 + 2gc 3 − 3gc ga 2 − 3ga gc 2 3/2 . 2ga 2 + 2gc 3 − 2ga gc
(A8)
with a positive denominator. Dividing the numerator by gc 3 , and letting k = ga gc , one writes a function of k f (k) = 2k3 − 3k2 − 3k + 2, (A9) which have the same sign with the numerator in Equation (A8). Since k < −1, we always have f (k) < 0,
(A10)
which leads to positive skewness of longitudinal velocity gradient Sk > 0
(A11)
in the initial stage of transition. This simple model implies that the positive skewness might result from the advection of mean shear. Note that this 2D model does not means that the early stage of transition is similar to 2D turbulence, since in isotropic incompressible 2D turbulence there is always Sk = 0.
Appendix 2. Derivation of Equation (7) Taking Lagrangian derivative of skewness of longitudinal velocity gradient with respect to t, one obtains ∂u1 ∂u1 d ∂u1 ∂u1 ∂u1 d ∂u1 ∂u1 1 ∂u1 ∂u1 − 32 ∂u ∂x1 ∂x1 dt ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 dt ∂x1 ∂x1 dSk = . (A12) 5/2 dt ∂u1 ∂u1 ∂x1 ∂x1
In order to explicitly expand Equation (A12) in incompressible flows, in the following we will derive the expressions of (d/dt)((∂u1 /∂x1 )(∂u1 /∂x1 )(∂u1 /∂x1 )) and (d/dt)((∂u1 /∂x1 )(∂u1 /∂x1 )), respectively. We start from the Navier–Stokes equation for the fluctuating velocity of incompressible flow, ∂ua 1 ∂p ∂ 2 ua ∂ua =− +ν , (A13) + Uj ∂t ∂xj ρ ∂xa ∂xj ∂xj where capitalised velocities indicate the momentary values, while lowercase velocities and pressure indicate the fluctuating components. If the velocities and pressure are grid-scale quantities in an LES run, Equation (A13) is rewritten as (here we omit the filter operators) ∂ua ∂ 2 ua 1 ∂p ∂ 2 ua ∂ua =− +ν + νt , + Uj ∂t ∂xj ρ ∂xa ∂xj ∂xj ∂xj ∂xj
(A14)
where the SGS viscosity νt is provided by employing the eddy-viscosity assumption. Taking partial derivative of Equation (A14) with respect to xb and multiplying the equation by (∂uc /∂xd ), one obtains ∂uc ∂ 2 ua ∂uc ∂ 2 ua ∂uc ∂Uj ∂ua 1 ∂uc ∂ 2 p + =− + Uj ∂xd ∂xb ∂t ∂xd ∂xj ∂xb ∂xd ∂xb ∂xj ρ ∂xd ∂xa ∂xb +ν
∂uc ∂ 3 ua ∂uc ∂ 3 ua ∂νt ∂uc ∂ 2 ua + νt + . ∂xd ∂xj ∂xj ∂xb ∂xd ∂xj ∂xj ∂xb ∂xb ∂xd ∂xj ∂xj
(A15)
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Equation (A15) can be written in an alternative formula ∂ua ∂ 2 uc ∂ua ∂Uj ∂uc ∂ua ∂ 2 uc + + Uj ∂xb ∂xd ∂t ∂xb ∂xj ∂xd ∂xb ∂xd ∂xj =−
∂ua ∂ 3 uc 1 ∂ua ∂ 2 p ∂ua ∂ 3 uc ∂νt ∂ua ∂ 2 uc +ν + νt + . ρ ∂xb ∂xc ∂xd ∂xb ∂xj ∂xj ∂xd ∂xb ∂xj ∂xj ∂xd ∂xd ∂xb ∂xj ∂xj
(A16)
Adding Equation (A15) to (A16) yileds d dt
∂uc ∂Uj ∂ua ∂ua ∂Uj ∂uc 1 ∂ua ∂ 2 p ∂uc ∂ 2 p − − + ∂xd ∂xb ∂xj ∂xb ∂xd ∂xj ρ ∂xb ∂xc ∂xd ∂xd ∂xa ∂xb 2 2 2 2 ∂ ∂νt ∂ua ∂ 2 uc ∂ua ∂uc ∂ uc ∂ ua ∂νt ∂uc ∂ ua + . −2 + + (ν + νt ) ∂xj ∂xj ∂xb ∂xd ∂xj ∂xd ∂xj ∂xb ∂xb ∂xd ∂xj ∂xj ∂xd ∂xb ∂xj ∂xj (A17) ∂ua ∂uc ∂xb ∂xd
=−
Considering another LES equation ∂ui ∂ 2 ui 1 ∂p ∂ 2 ui ∂ui + Uj =− +ν + νt . ∂t ∂xj ρ ∂xi ∂xj ∂xj ∂xj ∂xj
(A18)
Taking partial derivative of Equation (A18) with respect to xf and multiplying the equation by (∂uk /∂xl )(∂um /∂xn ), we obtain ∂uk ∂um ∂ 2 ui ∂uk ∂um ∂Uj ∂ui ∂uk ∂um ∂ 2 ui + Uj + ∂xl ∂xn ∂xf ∂t ∂xl ∂xn ∂xj ∂xf ∂xl ∂xn ∂xf ∂xj
∂ 3 ui 1 ∂ 2p ∂uk ∂um ∂ 3 ui ∂νt ∂ 2 ui = − +ν + νt + . ∂xl ∂xn ρ ∂xi ∂xf ∂xj ∂xj ∂xf ∂xj ∂xj ∂xf ∂xf ∂xj ∂xj
(A19)
Similarly we write Equation (A19) in two alternative formulas: ∂ui ∂uk ∂ 2 um ∂ui ∂uk ∂Uj ∂um ∂ui ∂uk ∂ 2 um + + Uj ∂xf ∂xl ∂xn ∂t ∂xf ∂xl ∂xj ∂xn ∂xf ∂xl ∂xn ∂xj
∂ 3 um ∂ui ∂uk 1 ∂ 2p ∂ 3 um ∂νt ∂ 2 um = − +ν + νt + , ∂xf ∂xl ρ ∂xm ∂xn ∂xj ∂xj ∂xn ∂xj ∂xj ∂xn ∂xn ∂xj ∂xj
(A20)
and ∂ui ∂um ∂ 2 uk ∂ui ∂um ∂Uj ∂uk ∂ui ∂um ∂ 2 uk + + Uj ∂xf ∂xn ∂xl ∂t ∂xf ∂xn ∂xj ∂xl ∂xf ∂xn ∂xl ∂xj
2 3 ∂ 3 uk ∂ui ∂um 1 ∂ p ∂ uk ∂νt ∂ 2 uk = − +ν + νt + . ∂xf ∂xn ρ ∂xk ∂xl ∂xj ∂xj ∂xl ∂xj ∂xj ∂xl ∂xl ∂xj ∂xj
(A21)
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From Equations (A19), (A20) and (A21) we obtain d dt
∂ui ∂uk ∂um ∂ui ∂uk ∂Uj ∂um ∂ui ∂um ∂Uj ∂uk ∂uk ∂um ∂Uj ∂ui − − =− ∂xf ∂xl ∂xn ∂xl ∂xn ∂xf ∂xj ∂xf ∂xl ∂xn ∂xj ∂xf ∂xn ∂xl ∂xj
1 ∂uk ∂um ∂ 2 p ∂ui ∂uk ∂ 2 p ∂ui ∂um ∂ 2 p − + + ρ ∂xl ∂xn ∂xi ∂xf ∂xf ∂xl ∂xm ∂xn ∂xf ∂xn ∂xk ∂xl 2 ∂ ∂ui ∂uk ∂um ∂ 2 ui ∂ 2 uk ∂um −2 + (ν + νt ) ∂xj ∂xj ∂xf ∂xl ∂xn ∂xf ∂xj ∂xl ∂xj ∂xn 2 2 2 2 ∂ ui ∂ um ∂uk ∂ uk ∂ um ∂ui −2 −2 ∂xf ∂xj ∂xn ∂xj ∂xl ∂xl ∂xj ∂xn ∂xj ∂xf +
∂νt ∂ui ∂uk ∂ 2 um ∂νt ∂ui ∂um ∂ 2 uk ∂νt ∂uk ∂um ∂ 2 ui + + . ∂xf ∂xl ∂xn ∂xj ∂xj ∂xn ∂xf ∂xl ∂xj ∂xj ∂xl ∂xf ∂xn ∂xj ∂xj
(A22)
Employing ensemble averages on Equations (A17) and (A22), and considering the case that a = b = c = d = i = f = k = l = m = n = 1, one obtains
d ∂u1 ∂u1 dt ∂x1 ∂x1
2
2
∂u1 ∂Uj ∂u1 2 ∂u1 ∂ 2 p ∂ ∂u1 ∂u1 ∂ u1 ∂ 2 u1 = −2 − +ν − 2ν ∂x1 ∂x1 ∂xj ρ ∂x1 ∂x1 ∂x1 ∂xj ∂xj ∂x1 ∂x1 ∂xj ∂x1 ∂xj ∂x1
∂2 ∂ 2 u1 ∂ 2 u1 ∂u1 ∂u1 ∂νt ∂u1 ∂ 2 u1 − 2 νt +2 (A23) + νt ∂xj ∂xj ∂x1 ∂x1 ∂xj ∂x1 ∂xj ∂x1 ∂x1 ∂x1 ∂xj ∂xj and
d ∂u1 ∂u1 ∂u1 dt ∂x1 ∂x1 ∂x1
2
∂u1 ∂u1 ∂Uj ∂u1 3 ∂u1 ∂u1 ∂ 2 p ∂ ∂u1 ∂u1 ∂u1 = −3 − +ν ∂x1 ∂x1 ∂x1 ∂xj ρ ∂x1 ∂x1 ∂x1 ∂x1 ∂xj ∂xj ∂x1 ∂x1 ∂x1
2 ∂2 ∂u1 ∂u1 ∂u1 ∂ u1 ∂ 2 u1 ∂u1 + νt − 6ν ∂x1 ∂xj ∂x1 ∂xj ∂x1 ∂xj ∂xj ∂x1 ∂x1 ∂x1
∂ 2 u1 ∂ 2 u1 ∂u1 ∂νt ∂u1 ∂u1 ∂ 2 u1 +3 . (A24) − 6 νt ∂x1 ∂xj ∂x1 ∂xj ∂x1 ∂x1 ∂x1 ∂x1 ∂xj ∂xj Substituting Equations (A23) and (A24) to Equation (A12) then yields dSk dt
∂u1 ∂u1 ∂u1 ∂u1 ∂u1 −(5/2) ∂u1 ∂Uj ∂u1 ∂u1 ∂u1 −(3/2) ∂u1 ∂u1 ∂Uj ∂u1 +3 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂xj ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂xj
−(3/2)
−(5/2)
2 3 ∂u1 ∂u1 ∂u1 ∂u1 ∂ p 3 ∂u1 ∂u1 ∂u1 ∂u1 ∂u1 ∂u1 ∂ 2 p − + ρ ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ρ ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1
= −3
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∂u1 ∂u1 −(3/2) ∂ u1 ∂ 2 u1 ∂u1 1 ∂2 ∂u1 ∂u1 ∂u1 − 3ν 2 − ∂x1 ∂x1 ∂x1 ∂xj ∂x1 ∂xj ∂x1 3 ∂xj ∂xj ∂x1 ∂x1 ∂x1
−(5/2) 2
∂u1 ∂u1 ∂u1 ∂u1 ∂u1 ∂ u1 ∂ 2 u1 1 ∂2 ∂u1 ∂u1 + 3ν − ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂xj ∂x1 ∂xj 2 ∂xj ∂xj ∂x1 ∂x1
−(3/2)
∂u1 ∂u1 ∂ 2 u1 ∂ 2 u1 ∂u1 ∂2 1 ∂u1 ∂u1 ∂u1 νt −3 2 νt − ∂x1 ∂x1 ∂x1 ∂xj ∂x1 ∂xj ∂x1 3 ∂xj ∂xj ∂x1 ∂x1 ∂x1
∂u1 ∂u1 ∂νt ∂ 2 u1 ∂ 2 u1 ∂ 2 u1 ∂u1 ∂u1 ∂u1 ∂u1 ∂u1 −(5/2) − +3 νt ∂x1 ∂x1 ∂x1 ∂xj ∂xj ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂x1 ∂xj ∂x1 ∂xj
∂2 ∂u1 ∂u1 ∂u1 ∂νt ∂ 2 u1 1 − . − νt 2 ∂xj ∂xj ∂x1 ∂x1 ∂x1 ∂x1 ∂xj ∂xj
23
(A25)
