UUnder consideration for publication in J. Fluid Mech. An isolated logarithmic layer
Yongseok Kwon † and Javier Jim´enez School of Aeronautics, Universidad Polit´ecnica de Madrid, 28040 Madrid, Spain(Received xx; revised xx; accepted xx)
To isolate the multiscale dynamics of the logarithmic layer of wall-boundedturbulent flows, a novel numerical experiment is conducted in which the meantangential Reynolds stress is eliminated except in a subregion correspondingto the typical location of the logarithmic layer in channels. Various statisticalcomparisons against channel flow databases show that, despite some differences,this modified flow system reproduces the kinematics and dynamics of naturallogarithmic layers well, even in the absence of a buffer and an outer zone. Thissupports the previous idea that the logarithmic layer has its own autonomousdynamics. In particular, the results suggest that the mean velocity gradientand the wall-parallel scale of the largest eddies are determined by the heightof the tallest momentum-transferring motions, implying that the very large-scale motions of wall-bounded flows are not an intrinsic part of logarithmic-layer dynamics. Using a similar set-up, an isolated layer with a constant totalstress, representing the logarithmic layer without a driving force, is simulatedand examined.
1. Introduction
Due to their abundance in scientific and engineering applications, wall-boundedflows have been one of the key areas in turbulence research. Initially, the focuscentred on the near-wall region due to its direct relation to the generation of skinfriction. Over the last couple of decades, however, the focus has shifted towardsthe logarithmic layer, partly because advancement in experimental techniqueand numerical computing enabled access to flow databases with a sufficientlyresolved logarithmic layer. Yet, a more fundamental reason for the interestin the logarithmic layer is that it is of great importance in the large-scaleapplications of high Reynolds number wall-bounded turbulence (such as largetransportation devices). For instance, the amount of bulk turbulent kinetic energy(TKE) production and dissipation within the logarithmic layer (Marusic et al. et al. † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] F e b Y. Kwon and J. Jim´enez turbulence characteristics are well-replicated up to a wall-normal distance propor-tional to the spanwise domain size. They used them to study the characteristicsof a hierarchy of the minimal logarithmic layer structures without an influencefrom the large-scale outer layer motions. Later, Hwang (2015) combined thismethod with an overdamped large eddy simulation (LES) (use of intentionallyelevated value of eddy viscosity to damp out the small-scale motions; see Hwang& Cossu 2010) to isolate flow structures of a given step in the hierarchy. Basedon this experiment, he concluded that the flow structures at each hierarchy cansustain themselves. However, there is still a question of whether overdampingsimply filters out the small-scales (without affecting the large-scales) or modifiesthe dynamics of the whole flow by effectively reducing the Reynolds number(Feldmann & Avila 2018).While the above examples attempt to isolate structures of certain sizes orlocations, the study of statistically stationary homogeneous shear turbulence(SSHST) takes the different approach of isolating a particular element of thelogarithmic layer, namely the shear. Unlike experimental homogeneous shearflows, where the size of the structures grows indefinitely, an statistically stationarystate is achieved numerically by using a limited spanwise flow domain (e.g.Pumir 1996; Sekimoto et al. et al. (2017) conducted an extensive study of thecoherent structures in SSHST, and concluded that its structures are essen-tially symmetrised and unconstrained (by the wall) versions of the structuresin turbulent channels, suggesting that the shear is the main ingredient of thecoherent structure dynamics in the logarithmic layer. However, SSHST is stillnot equivalent to the logarithmic layer, because it cannot replicate the wall-normal dependence of characteristic length scale, or the inhomogeneity along thewall-normal direction.In this regard, a closer reproduction of the logarithmic layer is the numericalexperiment by Mizuno & Jim´enez (2013) where the buffer layer (as well as thewall itself) was removed and substituted by an off-wall boundary condition. Theyintroduced the scale variation along the wall-normal direction by using a rescaledinterior plane as the off-wall boundary (without the rescaling, the resulting flowwas very similar to SSHST). Their numerical experiment reproduced many char-acteristics of the natural logarithmic layer, although not perfectly. For example, aspurious ‘buffer layer’ formed near the off-wall boundary due to the formation ofsmall-scale vortices caused by the incoherence between the off-wall boundary andthe adjacent flow. Alternatively, Lozano-Dur´an & Bae (2019) achieved the sameobjective by utilizing slip and permeable boundary conditions. This experimentreproduces the outer layer dynamics of the no-slip channel well but only does soabove some adaptation height, which is of the order of the slip length applied forthe boundary conditions. In combination with that, Bae & Lozano-Dur´an (2019)used a minimal spanwise domain to remove the large-scale outer layer structuresto isolate the logarithmic layer.All the aforementioned studies were successful at replicating or isolating certainfeatures of the logarithmic layer, but also had some drawbacks which made themincompatible with the natural flow. In the previous attempts to isolate the loga-rithmic layer of turbulent channel flows, there have been numerous strategies forremoving the buffer layer dynamics. However, to our best knowledge, the removalof the outer layer large-scale structures has relied almost exclusively on the use n isolated logarithmic layer § § § §
2. Numerical experiment
For this investigation, turbulent flow between two parallel plates, separated bythe distance 2 h , is simulated at a nominal Re τ = hU τ /ν = 2000, where ν is the kinematic viscosity of the fluid and U τ is the friction velocity. Periodicboundary conditions are used along the wall-parallel directions and no-slip andimpermeable boundary conditions are applied at both walls. Throughout thepaper, the streamwise, wall-normal and spanwise coordinates are denoted by x , y and z , respectively, and the corresponding velocity components by U , V and W . The y -dependent ensemble-averaged quantities are represented by an Y. Kwon and J. Jim´enez
Case Type Re τ L x /h L z /h ∆x + ∆z + ∆y + tU τ /h Line styleLB LES 2002 2 π π π π π π ∞ π π π π Hoyas & Jim´enez (2006)Table 1: Simulation parameters for the numerical experiments, and for thereference DNS database. ∆ represents the grid spacing in each direction. Thegrid spacings in the wall-parallel directions are calculated after dropping 1/3 ofthe high wavenumber modes for de-aliasing. The LES cases include the basecase (LB), the main experiment with wider logarithmic layer (LW), thesupplementary experiment with narrower logarithmic layer (LN) and theexperiment with a constant stress profile (LWc). For LW, LWc and LN, U τ iscomputed by extrapolating the total shear stress to the wall. For LWc, h is thewall-normal simulation domain. The next-to-last column shows the total timeover which the statistics are gathered, in terms of the large-eddy turnover time( h/U τ ). overline, while lower-case velocity variables indicate fluctuations with respect tothis average (e.g. U = U + u ). A ‘+’ superscript indicates normalisation by theviscous scale ν/U τ for length, and by U τ for velocity. The domain length in x and z are L x = 2 πh and L z = πh , respectively, to make sure that the entireflow domain is not minimal in the wall-parallel directions (Flores & Jim´enez2010). The flow is simulated via LES with static Smagorinsky sub-grid scale(SGS) model (Smargorinsky 1963). The Smagorinsky constant is chosen to be C s = 0 .
1, and the statistics of the LES compare well with a direct numericalsimulation (DNS) database at the same Reynolds number. The computationalalgorithm and the numerical code are taken from those employed in Vela-Mart´ın et al. (2019), but adapted for LES. The code solves the wall-normal vorticityand the Laplacian of v formulation of the Navier-Stokes equations (Kim et al. et al. Re τ (Hoyas & Jim´enez 2006, hereafter referred to as HJ06) is examined as abenchmark, and it is found that more than 95% of the total turbulent kineticenergy, k = ( u + v + w ) / y + > x and z are chosen to be ∆x + = ∆z + (cid:39)
37, after de-aliasing. In the wall-normal direction, the grid is defined n isolated logarithmic layer y + U + (a) 10 −10123 y + u + , v + , w + , u v + (b) Figure 1: (a) Mean streamwise velocity. Solid line: LB, Dashed line: HJ06. (b)Second-order velocity statistics. (cid:3) : u (cid:48) + , (cid:52) : v (cid:48) + , (cid:35) : w (cid:48) + , (cid:50) : uv + . Solid lines withclosed symbols: LB, Dashed lines with open symbols: HJ06. by a hyperbolic tangent stretch function such that the n -th grid location is givenby y n /h = tanh[3( n − / − / / tanh(3 /
2) + 1 for n = 1 , , ..., y = 0 and the upper one at y = 2 h . The parameters of thesimulations are summarized in table 1. Here, Re τ for LW, LWc and LN are givenbased on U τ from the extrapolated total shear stress at the wall to highlight theagreement of the total stress profile within linear-stress layer. However, it doesnot carry the usual meaning of the ‘Reynolds number’ for the canonical channelflows because the scale separation and the wall-normal gradient of the total shearstress become independent parameters for our isolated layers (for the canonicalchannel flows, they are both related by Re τ ).In order to validate this numerical methodology, the base LES (case LB) isfirst conducted and compared against HJ06. For this case, a van Driest dampingfunction of the form D ( y + ) = [1 − exp( − y + / is used on the Smagorinskyeddy viscosity to enforce the zero SGS stress conditions at the wall. Figure 1shows the profiles of the mean streamwise velocity U , and of the second-ordervelocity statistics of LB. The primed velocity variables indicate the root meansquare (RMS) value. A good agreement is observed between the statistics of LBand HJ06. Although not shown here for brevity, the one-dimensional (1D) velocityspectra also show a good agreement. Throughout the paper, further statisticalcomparisons will be made where appropriate to demonstrate that our LES codereproduces well the logarithmic layer of turbulent channel flows.Several methods were tested to isolate the logarithmic layer by removing theturbulent fluctuations outside it. The initial strategy was to employ an elevatedvalue of C s outside the logarithmic layer (i.e. overdamped LES; see Hwang &Cossu 2010) and its effect on the flow was investigated by varying the value of C s in the usual buffer layer ( y + < C s near the wall,it is observed that the spectral signature of the near-wall cycle gradually movesoutwards instead of being eliminated at a fixed location (see Appendix A).Hence, instead of damping previously-created turbulent fluctuations, an alter-native approach is sought where the necessity of ‘active’ turbulent fluctuations iseliminated outside the logarithmic layer. This is achieved by setting a prescribedtotal mean shear stress (sum of viscous, Reynolds and SGS stresses) profile whichdrops to zero outside the nominal logarithmic layer. In practice, it is done byimposing a modified profile of the body force. This method is found to be effective Y. Kwon and J. Jim´enez
Case Equation Body force Linear layer δ a /hy l /h β l y u /h β u y bot /h y top /h LW (2.1) 0.025 120 0.35 20 0.045 0.235 0 . . . τ + xy = 1 − y/h forLW and LN, and τ + xy = 1 for LWc) is below 1%. δ a represents the height of theactive stress region where τ + xy > . y/h τ + x y Figure 2: Profiles of the mean shear stress. The lines are as indicated in table 1,except for the dashed-dotted line, τ + xy = 1 − y/h (LW, LWc and LN arepresented by red, green and blue colors, respectively). The linear-stress layer forLW and LWc is indicated by the grey shaded area and that for LN is indicatedby the vertical dotted lines. at eliminating the buffer layer, and is chosen as our preferred method for isolatingthe logarithmic layer. The method of modifying the stress profile also means thatthe elimination of the outer layer dynamics can be achieved without relying onthe restricted flow domain and hence allows us to investigate its effects on thelarge-scale structures, which is not possible in the case of the minimal logarithmiclayer experiments where the large-scales are, by construction, truncated. In fact,the idea is not new. For example, Tuerke & Jim´enez (2013) simulated turbulentchannel flows with a prescribed mean velocity profile to study its effects on thedynamics of energy containing eddies, and Borrell (2015) applied and extra bodyforce to model the effects of roughness near the wall. It is also known that amodified body force can lead to laminarization of the flow at transitional Reynoldsnumbers (He et al. et al. n isolated logarithmic layer τ xy = U τ − y/h ) (1 + tanh[ β l ( y/h − y l /h )]) (1 − tanh[ β u ( y/h − y u /h )]) . (2.1)This equation is defined for 0 (cid:54) y (cid:54) h , but the prescribed stress profile isextended to the opposite side of the channel using symmetry. The parameters y l and β l control the location and width of the region where the stress profiledecays from its natural value to zero between the nominal logarithmic layer andthe wall. Likewise, y u and β u control the location and width of the region wherethe stress profile decays smoothly to zero above the nominal logarithmic layer.The parameters for the stress profiles for the main experiment with isolatedlogarithmic layer (case LW) and for the supplementary experiment with narrowerlog layer (case LN) are given in table 2. The values of y l , β l , y u and β u are setempirically.In addition to cases LW and LN, another experiment is performed in which theprescribed stress profile is constant within the isolated layer, intended to mimicthe total stress profile in the logarithmic layer as the streamwise pressure gradientvanishes. For this experiment, referred to as LWc, the prescribed stress profile is τ xy = U τ β l ( y/h − y l /h )]) (1 − tanh[ β u ( y/h − y u /h )]) , (2.2)which only differs from (2.1) by the missing (1 − y/h ) factor. The parameters y l , β l , y u and β u are kept as in LW, to study the effect of changing the stress profileindependently from other factors. In conventional channel flows, the channelcentreline provides a natural symmetry plane for the total stress profile (i.e. τ xy = 0). This does not apply to LWc, where the extrapolated location ofzero mean shear stress would be infinitely far from the wall. Therefore, theupper wall of LWc is replaced by a free-slip impermeable boundary at y = h ( ∂u/∂y = ∂w/∂y = 0 and v = 0). Although physically not required, fine gridspacing is needed near the free-slip wall to numerically enforce the boundaryconditions (in particular, to numerically resolve exponential functions with largeexponents for high wavenumbers). Hence, a different wall-normal grid is used forLWc, such that the n -th grid location is given by y n /h = tanh[3( n − / − / / (2 tanh(3 / / n = 1 , , ...,
384 for 0 (cid:54) y n (cid:54) h . This grid is designedsuch that the wall-normal spacing is kept similar to that used for LB, LW andLN within the domain of interest (0 . (cid:46) y/h (cid:46) . h simply means the wall-normal domain size for LWc, not the channel half height.We define the ‘linear-stress’, or ‘active’, layer to be the region in which theprescribed shear-stress profile deviates by less than 1% from the natural stressprofile in channels with unmodified body force (i.e. τ + xy = 1 − y/h for LW andLN, and τ + xy = 1 for LWc). While eddies within this layer could be expected to be‘most natural’, taller eddies have to exist up to the level at which some tangentialstress has to be carried by the flow. We therefore also introduce a length scale, δ a , intended to be indicative of the height of the largest momentum-transferringeddies, defined as the point at which τ + xy = 0 .
01 (approximately 1% of the wallshear stress in the natural channel). The limits y bot < y < y top of the linear layer,and δ a of the active one, are given in table 2. Note that, although y top and δ a are related, they are independent parameters, whose ratio can be changed bymodifying the stress profile. Both will be used below to scale different quantities, Y. Kwon and J. Jim´enez y / h x/h (a) 0 2 4 6012 y / h z/h (c) 0 1 2 3012 z / h x/h (e) 0 2 4 60123 x/h (b)0 2 4 6 z/h (d)0 1 2 3 x/h (f) 0 2 4 6 Figure 3: Instantaneous fields of u for (a,c,e) LB; (b,d,f) LW. The wall-parallelplanes in (e,f) are at y/h = 0 .
15. The colors range from u = − U τ (dark blue)to 4 U τ (dark red). but, since y top /δ a ≈ . − . δ a that we can tentatively assign δ a = h in unmodifiedchannels.Within the linear-stress layer, the flow experiences a body force equivalent tothe one generated by the mean pressure gradient in a canonical channel, and mostof the stress is carried by the Reynolds stress. For example, the flow in LN andLW has to produce the same mean momentum flux as in a natural channel at Re τ = 2000, and could therefore be expected to have the same dynamics as thelogarithmic layer in such a channel. For LWc, no driving body force is present inthe linear region, and the flow is maintained by the localised body forces appliedabove and below the linear-stress layer. For the actual numerical computation,the stress profile is enforced by replacing the mean pressure gradient in the meanstreamwise momentum equation with − d τ xy / d y . No van Driest damping is used,since the buffer layer is outside the domain of validity of the experiments, andthere is no need to reproduce its behaviour. Figure 2 shows the actual stressprofiles for LW, LWc and LN, which follow the prescribed profiles well. For allthe numerical experiments in which the wall shear stress is intentionally modified,the effective U τ is estimated by extrapolating to the wall the stress in the linear-stress layer. This is the velocity scale used for normalization in table 1.As a preliminary result and a qualitative comparison, instantaneous snapshotsof the field of u for LB and LW are shown in figure 3. They demonstrate that the n isolated logarithmic layer y + U + (a) 10 y + l + m (b) 100 200 300 400 500050100150200 Figure 4: (a) Mean streamwise velocity (shifted by 15 U τ for LW, LWc and LN)(b) Mixing length. For LW, LWc and LN, only the linear shear stress region isshown. The lines are as indicated in table 1 (LB, LW, LWc and LN arepresented by black, red, green and blue colors, respectively). The linear-stresslayer for LW and LWc is indicated by gray shaded area and that for LN isindicated by vertical dotted lines. turbulent fluctuations in the centre of the channel are eliminated in LW. This isalso true in the buffer layer, although they are too small to be observed visually.Within the isolated layer, the turbulent structures are qualitatively similar inboth cases, although it is noteworthy that streaks whose streamwise length iscomparable to the streamwise domain are observed in LB but not in LW. Thisdifference will be further investigated by the spectral analysis in §
3. Assessment of the isolated logarithmic layer
One-point statistics
In this section, we examine whether the flow in an isolated linear-stress layercan replicate the characteristics of the natural logarithmic layer, by comparingthe statistics of the truncated cases, LN and LW, with those of the full channelLB. In addition to that, we examine the effects of changing the stress profile bycomparing case LW with LWc, which is intended to represent the limiting caseof channel flow without the driving force. Figure 4(a) shows the profiles of themean streamwise velocity. Note that, because the walls are outside the domainof validity of the three isolated cases, the no-slip condition does not providean absolute velocity reference, and a Galilean offset of the profile is required ingeneral (Mizuno & Jim´enez 2013). In fact, it is clear from the figure that the meanvelocity of the truncated layers vanishes below y + ≈
60 (and actually becomesslightly negative). Much of the effect of the no-slip condition is taken over bythe dragging effect of the body force, and the profiles for LW, LWc and LN needto be shifted by 15 U τ to be comparable to the canonical logarithmic layer. Theagreement of the mean velocity after this shift is fair within the active layer, butthe velocity gradient gets steeper as the linear-stress layer gets narrower. Thisis further examined in figure 4(b), which tests the mixing length, l m = U τ /S ,where S = d U / d y . For a logarithmic mean velocity, l + m ( y + ) is a linear functionwhose slope is the K´arm´an constant, but the mixing-length profile of LW, LWcand LN is not linear, even within the active layer. By comparing wall-boundedflows with different geometries (with an exception of Ekman layers), Johnstone et al. (2010) and Luchini (2017) found that in the logarithmic and outer layers,0 Y. Kwon and J. Jim´enez y/h S h / U τ (a) 0 0.1 0.2 0.3 0.4020406080100 y/ δ a S δ a / U τ (b) 0 0.5 1010203040 Figure 5: The mean shear profile scaled by (a) the channel half height; (b) thewidth of active stress layer. The lines are as indicated in table 1 (LB, LW, LWcand LN are presented by black, red, green and blue colors, respectively). Thelinear-stress layer for LW and LWc is indicated by gray shaded area and thatfor LN is indicated by vertical dotted lines. the negative streamwise pressure gradient induces a positive shift in the meanstreamwise velocity, and vice versa. In our experiments, the mean streamwisevelocity of LWc is higher than that of LW, which seems contradictory to thoseprevious results. However, a direct comparison is not possible here because thepositive shift in U profile of LWc with respect to LW is caused by the difference inthe wall-normal profile of the body force, rather than by the geometry or pressuregradient. Although the total integrated force must sum to zero in both LW andLWc, the magnitudes of integrated positive and negative forces (i.e. the differencebetween the minimum and maximum τ xy ) are about 5% larger in LWc, whichresults in the greater amount of mean shear and the positive shift in U .The difference in the mean velocity is further investigated by comparing themean shear profiles of the four LES cases. Figure 5(a) shows that they collapsepoorly when S is normalised by U τ and h . However, in our experiments, h does notconvey the usual meaning of an outer length scale, because the eddies of height h are purposely suppressed. Instead, we propose that the alternative length scale, δ a , is more relevant to the physics of the flow, since it represents the height ofthe tallest momentum-transferring eddies. Figure 5(b) shows that the profilescollapse well within the linear-stress layer when both S and y are normalisedwith δ a , at least up to y (cid:39) . δ a . It is particularly interesting that the profile ofthe mean shear scales with δ a even when the profile of total shear stress (whichalso represents the driving force) changes within the active layer, such as betweenLW and LWc. This suggests that the value of shear within the logarithmic layeris associated with, or possibly decided by, the size of the largest active eddies inthe flow, and that this size is controlled by δ a . Moreover, the fact that the profilesagree within the active layer, when properly scaled, suggests that the truncatedflows contain a self-similar eddy hierarchy, as in the natural logarithmic layer,although the range of sizes within the hierarchy may differ.Figure 6 compares the Reynolds stress profiles of the four LES cases. All thestresses decay for y (cid:38) δ a , and figure 6(d) shows that the shear stress agreeswell within the active layer for LB, LW and LN, as expected from the designof the experiment. An important observation is the absence of a buffer-layer u (cid:48) peak in LW, LWc and LN, which suggests that the buffer-layer dynamics n isolated logarithmic layer u + (a) 10 v + (b) 10 y + w + (c) 10 y + − u v + (d) 10 Figure 6: Profiles of (a) u (cid:48) + (b) v (cid:48) + (c) w (cid:48) + (d) − uv + . The lines are as indicatedin table 1 (LB, LW, LWc and LN are presented by black, red, green and bluecolors, respectively). The linear-stress layer for LW and LWc is indicated bygray shaded area and that for LN is indicated by vertical dotted lines. has been suppressed. There are some residual velocity fluctuations below thelinear-stress layer, but they are not involved in the net momentum transfer or inTKE production, because they only carry a negligible fraction of the tangentialReynolds stress (i.e. they are inactive, see figure 6d). The shape of the u (cid:48) profileswithin the linear-stress layer is similar for LB, LW and LN, but their amplitudedecreases as the width of the active layer does. The same decreasing trend isobserved for w (cid:48) when comparing LW and LN, and we will argue below that bothtrends are due to the attenuation of the large-scale fluctuations by the restrictedheight of the active layer. Here, the effects of changing the height of the activelayer can be solely attributed to the change in the scale separation within theeddy hierarchy, because LB, LW and LN share the same mean shear stress withinthe linear-stress layer. In contrast, the value of v (cid:48) is slightly higher for LW and LNthan for LB. The exact reason for this is not clear, but the likeliest explanation isthat an elevated v (cid:48) is required to compensate for the missing tangential Reynoldsstress that used to be contributed by the large-scale u -eddies that would otherwisehave originated above the active region (see figure 3). In the outer part of theflow, where y ∼ O ( δ a ), the profiles of the truncated simulations collapse wellwhen y is scaled with δ a (not shown), indicating that they have similar outerlayer dynamics.The RMS velocity fluctuations in LWc are stronger than in LW. This isexpected, because previous investigations of channel flows with altered stressprofiles (Tuerke & Jim´enez 2013; Lozano-Dur´an & Bae 2019) have concluded thatthe magnitude of the fluctuations within the logarithmic layer scales with the local2 Y. Kwon and J. Jim´enez u / u ∗ (a) 10 v / u ∗ (b) 10 y + w / u ∗ (c) 10 y + − u v / τ x y (d) 10 Figure 7: Profiles of (a) u (cid:48) (b) v (cid:48) (c) w (cid:48) (d) − uv . (a-c) Normalised by u ∗ = ( − uv ) / ; (d) Normalised by τ xy . The lines are as indicated in table 1(LB, LW, LWc and LN are presented by black, red, green and blue colors,respectively). In (a-c), solid lines with triangles are the Re τ = 934 channel bydel ´Alamo et al. (2004). The linear-stress layer for LW and LWc is indicated bygray shaded area and that for LN is indicated by vertical dotted lines. Thevertical scale is kept as in figure 6 to facilitate comparison. value of tangential Reynolds stress, and because the primary role of turbulentfluctuations in the logarithmic layer is to carry the tangential Reynolds stressrequired for the transfer of momentum. To check this, the RMS velocity profilesare shown in figure 7 scaled with the local velocity scale u ∗ = ( − uv ) / . Figure7(d) confirms that it is indeed true that most of the mean shear stress is carriedby − uv within the linear-stress layer, as in the logarithmic layer of natural flows.The profiles of LW and LWc now agree well, but the consistent decrease withdecreasing δ a remains, especially for u (cid:48) /u ∗ . Note that figure 7 includes profilesfrom the DNS channel by del ´Alamo et al. (2004), whose h + = 934 is comparableto the δ + a of LW and LWc. The three flows agree reasonably well.We therefore turn our attention to the effect of δ + a , and plot in figure 8 theaverage value in 100 < y + <
200 of the TKEs of the three velocity components, asfunctions of δ + a for different DNS databases and LES experiments. The averagingrange is chosen to be within the active or logarithmic layer in all the datasetsincluded, and we set δ a = h for the DNS databases. In all cases, the TKEs arenormalised with the local uv . For the DNS databases, u and w display a log-linear trend with respect to h + , while v stays roughly constant. This is consistentwith the predictions from the attached-eddy hypothesis (Perry & Abell 1977;Perry & Chong 1982), in which the main effect of increasing h + is considered tobe to extend the range of scales of the self-similar attached-eddy hierarchy. The n isolated logarithmic layer δ + a T K E Figure 8: TKE of each velocity component normalised by the local uv averagedover 100 < y + < (cid:3) : u / − uv , (cid:52) : v / − uv , (cid:35) : w / − uv . Open symbolsconnected with solid lines represent DNS databases at Re τ = 547 (del ´Alamo &Jim´enez 2003), 934 (del ´Alamo et al. results from the LES experiments (solid symbols) agree well with the trend of theDNS databases, except for a slight w excess for LW and LWc, which is due to themild hump in their w (cid:48) profile within the linear-stress layer (figure 7c). Besidesreinforcing the importance of δ a as a parameter, this agreement supports theequivalence of δ a and h in natural channels, suggesting that the active part of thelargest Townsend-type self-similar attached eddies reaches the channel centreline,even though they are obscured in that region by the presence of wake structures(see also del ´Alamo et al. et al. δ a acts as a control parameter that determines the scaleseparation as well as the mean shear as a function of y/δ a (figure 5b), and δ a isan independent parameter from the shear stress gradient, unlike natural channelflows. This is made especially clear by the agreement between LW, LWc and del´Alamo et al. (2004) despite having different mean Reynolds shear stress gradientand driving forces. Such comparisons are not possible in natural channel flowsbecause the scale separation within the self-similar attached eddies and the meanstress gradient both depend on the Reynolds number.3.2. Spectra
To examine the distribution of turbulent kinetic energy at different scales, one-dimensional premultiplied spectra are plotted in figure 9. All the spectra aresuppressed outside the linear-stress layer, but the most notable observation is theelimination of the near-wall spectral peak in the spectrum of u for LW and LN,which is especially clear in figure 9(b) and proves that our numerical experimenteffectively removes the dynamics of the buffer layer. Another important differenceis the attenuation, within the linear-stress layer of LW and LN, of the spectrum of u at very large λ x and λ z . The motions in this range of wavelengths are commonly4 Y. Kwon and J. Jim´enez y + (a) 10 y + (c) 10 y + (e) 10 y + (g) λ + x (b)(d)(f)(h) λ + z Figure 9: One-dimensional pre-multiplied spectral density of (a,b) u ; (c,d) v ;(e,f) w ; (g,h) − uv along the (a,c,e,g) streamwise; (b,d,f,h) spanwise direction.The gray shaded contours are for HJ06, solid contours are for LW and dashedcontours are for LN. Contour lines are drawn at multiples of 0 . U τ except for(a) 0.2; (b) 0.4; (g) 0.05. The horizontal dashed-dotted lines are y/h = 0 . .
11 and 0 . referred to as the very large-scale motions (VLSM) of the logarithmic layer (e.g.Jim´enez 1998; Kim & Adrian 1999) and some attention has been dedicated tothem, because they carry a substantial fraction of the TKE and of the Reynoldsstresses (Balakumar & Adrian 2007). However, the present result suggests that n isolated logarithmic layer y + (a) 100300500 y + (c) 100300500 y + (e) λ + x λ + z Figure 10: Difference between the one-dimensional pre-multiplied spectraldensity of (a,b) u ; (c,d) w ; (e,f) − uv along the (a,c,e) streamwise; (b,d,f)spanwise direction. The grey shaded contours are LB-LW and lines contours arefor LB-LN. In (a) the contours are separated by 0 . U τ . In (b), by 0 . U τ . In(c-f), by 0 . U τ . The horizontal lines are the limits for the two linear-stresslayers. the VLSMs are not part of the intrinsic dynamics of the logarithmic layer, but ofthe region above it, which has been suppressed by the body force in LN and LW.This idea is consistent with the concepts of ‘inertial waves’ in Jim´enez (2018)or of ‘global modes’ in del ´Alamo & Jim´enez (2003), introduced to describe theenergetic motions of u at very large wavelengths, and which occupy the majorityof the channel half width. Kwon (2016) tried a different way of eliminating theouter layer contributions to the velocity fluctuations, from the perspective of thequiescent core. He observed that, upon the removal of the velocity fluctuationsassociated with the quiescent core, most of the energy of u in the VLSM rangedisappears.The damping of the long and wide wavelengths in LN and LW is made explicitin figure 10, which shows the difference between their spectra and the full LEScase. The restricted layers exhibit an energy deficit with respect to LB, and itis restricted to the large scales. Moreover, the length of the region in which LNfalls below LB (e.g. λ + x ≈ . U τ level of k x φ uu in figure 10a) isapproximately twice shorter than for LW, proportionally to their respective δ a .The width of the spanwise defect follows a similar trend but the peak is locatedat λ z ≈ h in both cases, consistent with the known width of the VLSM (Jim´enez2018). Note that there are no plots for φ vv in figure 10. This velocity componenthas no VLSM, and the corresponding plots are almost empty.All these studies converge to the conclusion that the VLSMs do not belongto the self-similar wall-attached eddy hierarchy intrinsic to the logarithmic layer.This is not to say that they have no influence on its dynamics, but it suggests that6 Y. Kwon and J. Jim´enez λ z / l m (a) 10 (b) λ x /l m λ z / l m (c) 10 λ x /l m (d) 10 Figure 11: Contour plots of (a) k x k z φ uu ; (b) k x k z φ vv ; (c) k x k z φ ww ; (d)- k x k z φ uv against λ x /l m and λ z /l m , scaled by the mixing length at each height.The solid grey scale contours are for LW and computed at y/h (cid:39) .
1, 0.15 and0.2 (from light gray to black). The dashed color contours are for LB andcomputed at y/h (cid:39) . .
15 (red) and 0 . . . U τ ; (b,d) [0 .
03 0 . U τ ; (c) [0 .
05 0 . U τ . the origin and dynamics of the VLSMs are associated with the outer layer ratherthan with the logarithmic one. A similar attenuation of the large scales is observedfor w and, to a lesser degree, for uv , but not for v , consistent with Townsend’s(1976) idea that the u and w fluctuations are attached, in the sense that theyare created far from the wall and extend downwards to fill the space underneath,while the v fluctuations are local in y . This is also clear from the triangularspectral ‘skirts’ in figures 9(a,b) and 9(e,f). That these roots are ‘inactive’ withrespect to the tangential stress is shown by the lack of skirts in figures 9(c,d)and 9(g,h). It is interesting to observe the dependence of the large-scale energyattenuation on the thickness of the linear-stress layer, demonstrated by thegreater attenuation in LN compared to LW. This supports the idea that restrictingthe wall-normal dimension over which turbulent fluctuations can develop alsolimits their growth in the wall-parallel directions. Long structures at a given y arethe skirts of structures farther up, and truncating the top of the spectral triangle,also truncates the long wavelengths. In other words, the structures in the linear-stress layer are ‘minimal’ in the wall-normal direction, and the upper bound oflinear-stress layer acts as a ‘ceiling’ that limits the growth of the structures inthe wall-parallel directions as well. This also explains the decreasing trend of u (cid:48) and w (cid:48) with decreasing δ a in figure 8.Figure 11 presents two-dimensional velocity spectra at three wall-normal loca- n isolated logarithmic layer − uv / instead of U τ , but thedifference between the two scales is small for the range of wall-normal locationsin figure 11, and we have kept the traditional definition. The energetic cores ofthe spectra of LW at different wall-normal locations show an excellent collapse,supporting the conclusion that the mixing length is the correct length scale forthe energy-containing eddies in the logarithmic layer, even when the profile of themixing length is not linear. If the typical velocity scale within the logarithmiclayer is given by U τ , this implies that the time scale of the energy-containingeddies is dictated by the local mean shear, rather than by a local eddy turnoverbased on the distance from the wall and U τ . The core of the spectra of LWalso agree well with LB. The lack of collapse at the large-scale ends of k x k z φ uu and k x k z φ ww was already discussed in figure 9, and corresponds to the inactivestructures, which scale with h or with δ a . In particular, note the damping of thespectrum of LW in the upper-right corner of figure 11(a,d).Figure 12 examines the effect of changing the stress profile in layers of similarthickness by comparing cases LW and LWc. The one-dimensional spectra arenormalised with u ∗ , which was shown in the previous section to be the correctscale for the intensities, and shown only within the linear-stress layer. Theycollapse well, showing that the spectral distribution of the fluctuations, and notonly their TKE, is independent of the existence of a pressure gradient.To complement the observations on the trend of the large-scale energy attenua-tion, figure 13 compares two-dimensional energy spectra at y/h (cid:39) .
1, scaled with u ∗ . The spectra for the full LES (LB) agree well with HJ06, again demonstratingthe adequacy of the current LES simulations for the study of the logarithmic layer.There is some accumulation of energy at the scales close to the grid resolutionof LES, due to the slightly insufficient dissipation by the SGS model, but thiseffect does not extend to the energy-containing region. The spectra for LW andLWc agree well, reinforcing the conclusions from the one-dimensional data. Thecomparison between LB, LW and LN clearly shows the removal of large-scaleenergy as the width of the linear-stress layer decreases, especially for u and w .This also explains the previously observed decreasing trend of the u (cid:48) and w (cid:48) profiles with decreasing δ + a , discussed in figure 6.On the other hand, there is a TKE excess in the spectra of all the restricted-layer experiments with respect to LB and HJ06, at intermediate scales, whichcan be observed in figures 11 and 13. The wavelengths of the excess energy inLW scale with the mixing length above y/h = 0 .
075 ( y + = 150), and are centredaround ( λ x , λ z ) (cid:39) (15 l m , l m ) for v and w , and around ( λ x , λ z ) (cid:39) (30 l m , l m )for u . This energy excess also produces some extra Reynolds shear stress, whichappears as an upper ‘horn’ in figure 13(d), and which is needed to compensatefor the attenuated Reynolds shear stress at the large scales. The same energyexcess also appears in LN, where it is stronger because it has to compensate for8 Y. Kwon and J. Jim´enez y + (a) 100200300400500600 y + (c) 100200300400500600 y + (e) 100200300400500600 y + (g) λ + x λ + z Figure 12: One-dimensional pre-multiplied spectral density of (a,b) u ; (c,d) v ;(e,f) w ; (g,h) − uv along the (a,c,e,g) streamwise; (b,d,f,h) spanwise direction.The gray shaded contours are for HJ06, solid contours are for LW and dashedcontours are for LWc. Contour lines are drawn at multiples of 0 . u ∗ except for(a) 0.2; (b) 0.4; (g) 0.05. The horizontal dashed-dotted lines indicate y = 0 . h , and 0 . h , which mark the boundaries of linear-stress layers. an even larger attenuation. Except for these localised effects, the agreement inother regions of the two-dimensional spectra is very good.There is also an energy excess in the wide modes of v in LN, which can beobserved in figure 13(b), and which also appears in LW near the lower limit of thelinear-stress layer (not shown). Because its wavelengths are wide and relatively n isolated logarithmic layer λ z / l m (a)10 (b) λ z / l m (c) λ x /l m (d) λ x /l m Figure 13: Contour plots of (a) k x k z φ uu ; (b) k x k z φ vv ; (c) k x k z φ ww ; (d)- k x k z φ uv at y (cid:39) . h plotted against λ x /l m and λ z /l m . The shaded contoursare HJ06. The line contours are LES experiments as indicated in table 1 (LB,LW, LWc and LN are presented by black, red, green and blue colors,respectively). Contour levels are drawn at (a) [0 . . u ∗ ; (b,d) [0 .
03 0 . u ∗ ;(c) [0 .
05 0 . u ∗ . The black dashed diagonal lines represent λ x = λ z . short, it is tempting to attribute this extra energy to a spanwise instability ofthe shear layer that forms underneath the linear-stress region (see the peak at y + ≈
70 in figure 5). The extra energy in the near-wall region is also visible infigure 9(a,b) as a ‘stem’ at λ + x ≈
700 and λ + z ≈ − u at y + = 20, although weak overall, contains very wide structureswith λ + x ≈ et al. (2001) showed that any profile with an approximateinflexion point near the wall develops a Kelvin–Helmholtz like instability as soonas any amount of wall transpiration is allowed, and Garc´ıa-Mayoral & Jim´enez(2011) showed that, in ribbed surfaces modelled by a layer of retarding bodyforces, this effect results in transverse unstable rolls. This instability, like Kelvin–Helmholtz’s, is essentially inviscid and depends only on the mean u profile, andin the v velocity that separates the inflection point from the impermeable wall.Its typical wavelengths are of the order of 5 to 10 times the thickness of thedrag layer, which in the present case ( λ + x ≈ − Y. Kwon and J. Jim´enez of the present paper, but the direct influence of the possible instabilities does notseem to be significant, and the characteristics of the energy-containing eddies arewell-reproduced. 3.3.
Dynamical indicators of the flow
Examination of velocity statistics reveals that our numerical experiment canreplicate key kinematic properties of the natural logarithmic layer. In order tofurther assess the resemblance of the linear-stress layer to the natural logarithmiclayer, we also examine and compare some of the dynamical characteristics of theflow. Firstly, we will examine the ratio between the production and dissipationof TKE, since it is widely known that these two quantities are approximately inbalance in the logarithmic layer. For LES channel flows, the balance of the meanTKE of the filtered velocity fields is given by
DkDt = P + (cid:15) + Π p + Π t + Π v + Π r (3.1)where the terms in the right-hand side represent production, dissipation, pressuretransport, turbulent transport, viscous diffusion and diffusion by residual stress,respectively. They are defined as P = − Suv (3.2) (cid:15) = − τ v : E + τ r : E (3.3) Π p = − ∂vp∂y (3.4) Π t = − ∂vk∂y (3.5) Π v = ∇ · u · τ v (3.6) Π r = −∇ · u · τ r (3.7)(3.8)where u is the fluctuating velocity vector ( u, v, w ), p is the kinematic pressure, E is the strain rate tensor of the filtered velocity fields, τ v = 2 ν E is the viscousstress tensor and τ r = − ν r E , where ν r is the LES eddy viscosity given by theSGS model, is the residual stress tensor.Another key parameter for characterizing the dynamics of shear flow is theCorrsin parameter, which is defined as the ratio between the dissipative timescale of eddies (2 k/(cid:15) ) and the time scale of the mean shear (1 /S ). Therefore, itrepresents the relative importance of shear to the dynamics of turbulent eddies.For example, 2 kS/(cid:15) (cid:29) y + (cid:39) P/(cid:15) (cid:39)
P/(cid:15) profile with respect to LB remains less than 5%, located in 200 (cid:46) y + (cid:46)
390 (0 . (cid:46) y/h (cid:46) . P/(cid:15) ratio never reaches unity in LN, n isolated logarithmic layer y + P / (a) 10 y + k S / (b) 10 Figure 14: (a) Ratio between production and dissipation (b) Corrsin parameter.The lines are as indicated in table 1 (LB, LW, LWc and LN are presented byblack, red, green and blue colors, respectively). The linear-stress layer for LWand LWc is indicated by gray shaded area and that for LN is indicated byvertical dotted lines. y + y + P + , y + + (a) 10 −4−20246 y + l + m P + , l + m + (b) 10 −101 Figure 15: Profiles of production and dissipation. Lines on the positive andnegatives sides represent the production and dissipation, respectively. (a)Premultiplied by y (b) Premultiplied by l m . The lines are as indicated in table 1(LB, LW, LWc and LN are presented by black, red, green and blue colors,respectively). The linear-stress layer for LW and LWc is indicated by grayshaded area and that for LN is indicated by vertical dotted lines. In (b), profilesfor LW and LWc are plotted only near the vicinity of the linear-stress layer(70 < y + < presumably because the linear-stress layer is too narrow to fully recover the TKEbalance. In the middle of the linear-stress layer, the Corrsin parameter in figure14(b) agrees well among all the LES cases and HJ06, and stays approximatelyconstant at 2 kS/(cid:15) ≈ . This agreement suggests that similar dynamical processestake place in all these flows. These observations demonstrate that our numericalexperiments are able to produce a region whose dynamical characteristics aresimilar to those of the natural logarithmic layer.However, the profiles of
P/(cid:15) and of the Corrsin parameter for LW have peaksat y + (cid:39)
70 and y + (cid:39)
570 in figure 14, just outside of the linear-stress layer.This could potentially be worrying, because the excess
P/(cid:15) in these regions mayinfluence the dynamics of the linear-stress layer. To investigate this possibility,figure 15 compares the actual production and dissipation profiles. When P and (cid:15) are premultiplied by y in figure 15(a), to highlight the logarithmic and outer2 Y. Kwon and J. Jim´enez y + Θ + −1−0.500.511.5 Figure 16: Profiles of the wall-normal TKE flux. The lines are as indicated intable 1 (LB, LW, LWc and LN are presented by black, red, green and bluecolors, respectively). The linear-stress layer for LW and LWc is indicated bygray shaded area and that for LN is indicated by vertical dotted lines. regions, they do not agree well, even within the linear-stress layer. However,we saw in the previous section that the correct length scale for this region isthe mixing length, and when the production is premultiplied by l m in figure15(b), LW, LN and LB agree excellently, while LWc does not. This is essentiallyautomatic, because l + m P + = − uv + , which is set by the body force, whose profile isonly different for LWc. The behaviour of l + m (cid:15) + is more interesting. The agreementbetween LB and LW near y + (cid:39)
280 is consistent with the collapse of l + m P + andof P/(cid:15) in this region, but it is clear from figure 15(b) that the peaks of
P/(cid:15) nearthe edge of the linear layer are caused by a reduced level of dissipation, not by anincreased level of production. The balance of the two quantities is never reachedfor LN, because its linear-stress layer is too narrow (the scale separation betweenthe two edges is only a factor of 2). Tuerke & Jim´enez (2013) investigated aturbulent channel flow with a sharp change in the mean shear, and found thatthe production adapts to the change in the shear almost immediately, while thedissipation does so more gradually, in agreement with the behaviour near theedges of the linear-stress layer in figure 15(b). They attributed this phenomenonto the temporal delay between the production and dissipation mechanisms.To further inspect this behaviour, the wall-normal flux of the mean TKE isconsidered. The transport terms in (3.1) are in the form of a flux divergence, andthe wall-normal flux of the mean TKE, Θ , can be computed as Θ ( y ) = (cid:90) y ( Π p ( ξ ) + Π t ( ξ ) + Π v ( ξ ) + Π r ( ξ )) d ξ. (3.9)As per (3.1), regions with positive d Θ/ d y (i.e. positive transport) indicate netenergy sinks, (cid:15) > P , which draw energy from other wall-normal locations, andvice versa. Also, because (3.9) vanishes at the wall, a positive Θ indicates a netTKE flux towards the wall at that wall-normal location. Figure 16 shows Θ for theLES cases and HJ06 (Hoyas & Jim´enez 2008). LW, LWc and LN agree well belowthe linear-stress layer but LN does not exhibit a plateau region because it never n isolated logarithmic layer Θ in the logarithmicand linear-stress layers is negative, since P/(cid:15) is slightly greater than unity there(at least up to Re τ = 5200; see Bernardini et al. Θ has a positive slope at y + <
55 and y + >
645 for LW, indicatingthat the flow acts as an energy sink outside of the linear-stress layer, except inthe vicinity of the layer edges. In full channels, like LB and HJ06, the buffer layer(5 < y + <
40) acts as a strong energy source due to intense production activities.However, the removal of the buffer layer in LW turns this region into a net energysink, whose energy deficit is balanced by an energy flux coming from the linear-stress layer. The region above y + (cid:39)
645 also acts as a net energy sink and drawsenergy from the linear-stress layer. In LW and LWc, Θ crosses zero at y + (cid:39) y + <
125 and towardsthe centre at y + > < y + < Θ in LW approaches that of LB and HJ06,while the actual magnitude of TKE flux is smaller because there is less upwardTKE flux coming from below. This region coincides with the location where agood agreement is observed for P/(cid:15) and the Corrsin parameter between LW,LWc and LB. Overall, the shear production mechanism of the logarithmic layeris well reproduced in the linear-stress layer, although there are some differencesin how the TKE is transported and dissipated near the layer edges. Therefore,the current numerical experiment is an adequate reproduction of the naturallogarithmic layer as far as the energy producing and energy containing motionsare concerned.
4. Discussion
To isolate the logarithmic layer of wall-bounded turbulent flows, we have pre-sented a series of numerical experiments in which a body force is used to imposea prescribed total stress profile. The resulting flow has an ‘active’ layer in whichthe total stress follows a linear trend, as in natural channels, but the stress is madeto decay to zero elsewhere. As a result, the turbulent fluctuations are effectivelyeliminated outside the active layer, especially the ones carrying the tangentialReynolds stress. Various statistical comparisons demonstrate the kinematic anddynamic equivalence between the isolated active layer and the natural logarithmicone, and the experiments allow us to assess separately the effects of the range ofscales of the self-similar eddies (using cases LB, LW and LN), and of the profileof the shear stress within the active layer (using LW and LWc). These effectscannot be separated in natural channel flows, because both are controlled bythe Reynolds number. We show that the scale range of the self-similar eddies isrelated to the thickness, δ a , of the active layer, which controls the size of thelargest momentum-transferring eddies. This thickness determines the mean shearbelow y = 0 . δ a , and the largest wall-parallel scales of the flow. On the otherhand, the primary effect of the average shear stress within the linear-stress layeris to act as a scale for the velocity fluctuations, while the slope of the shear stress4 Y. Kwon and J. Jim´enez profile, which is equivalent in natural channels to the mean streamwise pressuregradient, does not significantly affect the dynamics.The characteristics of the energy-containing eddies are investigated using theirenergy spectrum. Within the linear-stress layer, our experiments agree withnatural channel flows when the wavelengths are scaled with the mixing length ofthe mean streamwise velocity profile. This agreement includes the spectrum atdifferent wall-normal locations, even if the mixing length profile is not strictlylinear in y . This suggests that the linear dependence of the length scale innatural channels is not a necessary condition for self-similarity, and that thelength scale of the self-similar eddies in the logarithmic layer is associated withthe local value of mean shear, not with the absolute distance from the wall, inline with the conclusions of Mizuno & Jim´enez (2011) and Lozano-Dur´an & Bae(2019). The implication is that the distance from the wall relative to the sizeof the largest active eddies determines the mean shear (figure 5), and the meanshear, together with the mean momentum flux (roughly U τ in the logarithmiclayer), determines the scale of the self-similar eddies in the logarithmic layer,rather than the absolute distance from the wall. In this regard, although theabsolute distance from the wall does not provide a scale for the self-similar activeeddies, the isolated layer is not truly independent of the wall because the valueof the mean shear depends on y/δ a . Another noteworthy difference between ournumerical experiments and the natural channel is the attenuation in the formerof the turbulent kinetic energy (TKE) of the very large-scale motions, suggestingthat these motions are not an intrinsic part of the dynamics of the logarithmiclayer.By comparing experiments having linear-stress layers of different thickness, weconfirm that the upper bound of the layer acts like a ‘ceiling’ for the structures,and that inhibiting the wall-normal growth of turbulent structures also limitstheir wall-parallel scales. This attenuation of the large-scale energy also explainsthe observed decrease of u (cid:48) and w (cid:48) as δ a decreases, or equivalently, as the scaleseparation within the self-similar eddy hierarchy gets narrower. The wavelengthsof the vertical energetic ridge in the spectrum of u in figures 9(a) and 9(b) areapproximately λ x ≈ δ a and λ z ≈ . δ a . It is interesting to compare this resultwith the aspect ratio of the vortex clusters (3:1:1.5 in x , y and z , in del ´Alamo et al. x , y and z , in Lozano-Dur´an et al. δ a should be relatedto the wall-normal dimension of those threshold-based structures.We finally compare the dynamical properties of the linear-stress layer withthose of the natural logarithmic layer. There is a central region within the activelayer in which the production and dissipation of the TKE match the ones of thenatural logarithmic layer, but the dissipation decays towards both ends of theactive layer because the TKE is transported away from it to compensate for theTKE deficit caused outside the active layer by the elimination of the buffer andouter layer dynamics, instead of being dissipated in place. On the other hand,the TKE production or, equivalently, the mean shear, compares well with thenatural logarithmic layer throughout the linear-stress layer when scaled with themixing length. The Corrsin parameter is approximately constant 2 kS/(cid:15) = 8, bothin the active layer and in the logarithmic layer, supporting the conclusion thatthe dynamics of the eddies is dominated by the effect of the mean shear in both n isolated logarithmic layer
5. Conclusions
In conclusion, we demonstrate that the linear- and constant-stress layer of thepresent experiments successfully reproduces the essential dynamics of the naturallogarithmic layer, even in the absence of a buffer and of an outer layer. Althoughthere are some differences between the two flows, such as a nonlinear mixing-length profile and the details of the TKE transport and dissipation, the essentialdynamics of the energy-producing and energy-containing motions in the naturallogarithmic layer is well-reproduced. Hence, the isolated system introduced hereshould be useful to identify other intrinsic features of the logarithmic layer aswell as the features that are not intrinsic to the logarithmic layer such as verylarge-scale structures. In the present paper, we use it to support the previousidea that the logarithmic layer has its own autonomous dynamics, which dependonly weakly on inputs from other parts of the flow. This is not to say that theother parts of the flow do not have an influence on the logarithmic layer. Inparticular, we show that the dimensions of the longest structures depend on theflow above. Moreover, the size of self-similar logarithmic layer eddies is related tothe height of the largest momentum-transferring eddies in the flow through theagency of the mean shear and momentum flux. However, the sustenance of thelogarithmic layer does not depend on the other parts of the flow. The key problemin simulating an isolated logarithmic layer is how to limit the tendency of the sizeof the turbulent structures to grow indefinitely as a result of the shear. To achievethis objective, most of the previous attempts (Flores & Jim´enez 2010; Hwang2015; Bae & Lozano-Dur´an 2019) take a ‘minimal box’ approach, which controlsthe wall-normal eddy size by limiting the spanwise domain dimension. However,the use of a minimal box inherently causes a significant portion of the TKE toremain outside the range of resolvable scales, and their aggregate dynamics isprojected onto the streamwise- or spanwise-uniform modes. Instead, the presentsystem represents the opposite approach of creating a non-uniform shear profileby directly limiting the wall-normal eddy size, which can accommodate the fullscale dynamics of the energy-containing eddies in the logarithmic layer.
Funding.
This work was supported by the European Research Council under Coturb GrantNo. ERC2014.AdG-669505.
Declaration of interests.
The authors report no conflict of interest.
Appendix A. Effects of overdamping on the near-wall turbulence
As a preliminary trial, overdamping is applied in the buffer layer to test itseffectiveness in suppressing the near-wall turbulence. Note that, because it was atest case, the simulation was conducted at a reduced spatial domain ( L x /h = π and L z /h = π/
2) and spatial resolution ( ∆x + = ∆z + (cid:39)
74) compared to LB.All other simulation parameters are identical to LB. The overdamping is appliedbelow y + = 70 and its degree is controlled by a parameter C s,w , which representsthe value of C s at the wall. The gradient of C s with respect to y is set to be zeroat the wall, and C s = 0 . y + = 70. For 0 < y + (cid:54)
70, a cubic polynomialis fitted such that C s is continuous and differentiable at y + = 70. No van Driest6 Y. Kwon and J. Jim´enez C s y + Figure 17: The profiles of C s for C s,w = 0 . y + = 70. C s,w = 0 . y / h −2 −1 C s,w = 0 . C s,w = 0 . λ z /h y / h −2 −1 −2 −1 C s,w = 0 . λ z /h −2 −1 Figure 18: Spanwise pre-multiplied spectra of u for different values of C s,w . Theblack (filled with gray scale colors in between) and red solid contours representHJ06 and the overdamped experiments, respectively. The contour lines aredrawn at multiples of 0 . U τ . The horizontal dashed lines indicates y + = 70. damping is applied close to the wall. The profiles of C s for C s,w = 0 .
1, 0.2, 0.4and 0.8 are shown in figure 17.As the most illustrative measure, the spanwise pre-multiplied spectra of u for different values of C s,w are shown in figure 18. Overdamping is effective atsuppressing velocity fluctuations below y + = 70. However, with increasing C s,w ,the spectral signature of the near-wall cycle simply moves away from the walland to the wider wavelengths instead of being eliminated at a fixed location. Forthe higher C s,w (especially for 0.8), it even protrudes into the non-overdampedregion. This is in-line with the observation by Feldmann & Avila (2018) that the n isolated logarithmic layer C s , which they interpret as an effectivereduction of the Reynolds number.This problem does not exist when the buffer layer is suppressed by a modifiedbody force. Figure 9(b) shows that the spectral signature of the near-wall cycleis eliminated without leaving a residual in the cases LW and LN. Therefore, amodification of the body force is chosen as the preferred method for suppressingthe buffer layer turbulence. REFERENCESdel ´Alamo, J. C. & Jim´enez, J.
Phys. Fluids , L41–L44. del ´Alamo, J. C., Jim´enez, J., Zandonade, P. & Moser, R. D. J. Fluid Mech. , 135–144. del ´Alamo, J. C., Jim´enez, J., Zandonade, P. & Moser, R. D.
J. Fluid Mech. , 329–358.
Bae, H. J. & Lozano-Dur´an, A.
CTR Annual Research Briefs , pp. 1–10.Stanford University.
Balakumar, B. J. & Adrian, R. J.
Phil. Trans. R. Soc. A , 665–681.
Barenblatt, G. I.
Scaling, Self-similarity, and Intermediate Asymptotics . CambridgeUniversity Press.
Bernardini, M., Pirozzoli, S. & Orlandi, P. Re τ = 4000. J. Fluid Mech. , 171–191.
Borrell, G.
Dong, S., Lozano-Dur´an, A., Sekimoto, A. & Jim´enez, J.
J. Fluid Mech. , 167–208.
Feldmann, D. & Avila, M. Re τ = 1500. J. Phys.: Conf. Ser. , 012016.
Flores, O. & Jim´enez, J.
Phys.Fluids , 071704. Garc´ıa-Mayoral, R. & Jim´enez, J.
J. Fluid Mech. , 317–347. de Giovanetti, M., Hwang, Y. & Choi, H.
J. Fluid Mech. , 511–538.
He, S., He, K. & Seddighi, M.
J. Fluid Mech. , 31–71.
Hoyas, S. & Jim´enez, J. Re τ = 2003. Phys. Fluids , 011702. Hoyas, S. & Jim´enez, J.
Phys. Fluids , 101511. Hutchins, N. & Marusic, I.
Phil. Trans.R. Soc. A , 647–664.
Hwang, Y.
J. Fluid Mech. , 254–289.
Hwang, Y. & Cossu, C.
Phys. Rev. Lett. , 1–4.
Jim´enez, J.
CTR Annual Research Briefs ,pp. 943–945. Stanford University.
Jim´enez, J.
J. Fluid Mech. , P1.
Jim´enez, J. & Moin, P.
J. Fluid Mech. , 213–240. Y. Kwon and J. Jim´enez
Jim´enez, J. & Moser, R. D.
AIAA J. ,605–612. Jim´enez, J. & Pinelli, A.
J. Fluid Mech. , 335–359.
Jim´enez, J., Uhlmann, M., Pinelli, A. & Kawahara, G.
J. Fluid Mech. , 89 – 117.
Johnstone, R., Coleman, G. N. & Spalart, P. R.
J. Fluid Mech. ,163–175.
Kim, J., Moin, P. & Moser, R. D.
J. Fluid Mech. , 133–166.
Kim, K. C. & Adrian, R. J.
Phys. Fluids ,417–422. K¨uhnen, J., Song, B., Scarselli, D., Budanur, N. B., Riedl, M., Willis, A. P., Avila,M. & Hof, B.
Nat. Phys. , 386–390. Kwon, Y. S.
Lee, M. & Moser, R. D. Re τ ≈ J. Fluid Mech. , 395–415.
Lele, S. K.
J. Comput.Phys. , 16–42.
Lozano-Dur´an, A. & Bae, H. J.
J. Fluid Mech. , 698–725.
Lozano-Dur´an, A., Flores, O. & Jim´enez, J.
J. Fluid Mech. , 100–130.
Lozano-Dur´an, A. & Jim´enez, J. Re τ = 4200. Phys. Fluids , 011702. Luchini, P.
Phys. Rev. Lett. , 224501.
Marusic, I., Mathis, R. & Hutchins, N.
Int. J. Heat and Fluid Flow , 418–428. Mizuno, Y. & Jim´enez, J.
Phys. Fluids , 085112. Mizuno, Y. & Jim´enez, J.
J. Fluid Mech. , 429–455.
Perry, A. E. & Abell, J.
J. Fluid Mech. , 785–799. Perry, A. E. & Chong, M. S.
J. Fluid Mech. , 137–217.
Pumir, A.
Phys. Fluids , 3112–3127. Russo, S. & Luchini, P.
J. Fluid Mech. , 104–127.
Sekimoto, A., Dong, S. & Jim´enez, J.
Phys.Fluids , 035101. Smargorinsky, J.
Mon. Weather Rev. , 99–164. Spalart, P. R., Moser, R. D. & Rogers, M. M.
J. Comput. Phys. , 297–324. Townsend, A. A.
The Structure of Turbulent Shear Flow , 2nd edn. Cambridge UniversityPress.
Tuerke, F. & Jim´enez, J.
J. Fluid Mech. , 587–603.