Toward a flow-structure-based wall-modeled large-eddy simulation paradigm
aa r X i v : . [ phy s i c s . f l u - dyn ] J a n Center for Turbulence ResearchAnnual Research Briefs 2020 Toward a flow-structure-based wall-modeledlarge-eddy simulation paradigm
By A. Elnahhas, A. Lozano-Dur´an
A N D
P. Moin1. Motivation and objectives
Large-eddy simulations (LES) fundamentally rely on a large separation of scales. Theyallow the large, energy-containing eddies to evolve spatiotemporally according to theirgoverning equations, while modeling the effect of the near-universal small scales of tur-bulence on those large eddies using subgrid-scale (SGS) models. When walls are present,the separation of scales between the eddies inhabiting the two ends of the energy cascadeceases to exist as the wall is approached. This is due to the scaling of both large- andsmall- eddying motions with distance from the wall (Tennekes & Lumley 1972; Townsend1976). Wall-resolved LES aims to capture the large, energy-containing eddies in the cas-cade at all wall-normal distances. The aforementioned behavior of the large- and small-eddying motions leads to the cost of wall-resolved LES to be only slightly more favorablethan DNS in terms of scaling with the Reynolds number based on the streamwise extent, L x , Re / L x and Re / L x , respectively, making both prohibitively expensive in simulatinghigh- Re flows (Choi & Moin 2012).Wall-modeled LES (WMLES) attempts to circumvent this problem by extending thenear-isotropic grid resolution in the outer region of the flow down to the wall whilemodeling the entire subgrid dynamics of the near-wall region using low-order models(Bose & Park 2018). This reduces the scaling of the cost of WMLES to Re L x , indicatingits practical importance (Choi & Moin 2012). The most commonly utilized wall modelsrely on the law of the wall or simplified RANS equations and make no use of the cur-rent vast knowledge on the structure of wall-bounded turbulent flows. For the currentapplication of interest, namely attached external boundary layer flows, it is importantto note that the first grid point away from the wall in WMLES typically lies within thelogarithmic region of the flow.The logarithmic region in high Reynolds number wall-bounded turbulent flows isempirically observed to be composed of different types of eddies. Most notably, theenergy-containing eddies are attached to the wall and geometrically self-similar with dis-tance from the wall (Townsend 1976; Marusic & Monty 2019). A synthetically generatedboundary layer composed of such attached eddies allows for the prediction of the meanturbulent intensities as a function of the distance from the wall (Perry & Chong 1982).Below the smallest eddies in this self-similar hierarchy, there is a near-wall self-sustainingcycle (Jim´enez & Pinelli 1999). Figure 1 shows an instantaneous streamwise velocitysnapshot in a turbulent channel at Re τ ≈ Re τ is the friction Reynolds num-ber, and depicts both the wall-attached eddies scaling with distance from the wall, aswell as the region where the near-wall self-sustaining cycle is active.Many possible reduced-order models of the near-wall cycle that make use of tur-bulent structures have been envisioned, but have not been used in WMLES. Thesemodels include some based on invariant solutions to the Navier-Stokes equations atlow Re τ (Kawahara et al. Elnahhas, Lozano-Dur´an & Moin (Farrell & Ioannou 2012), and a truncated Galerkin projection of the self-sustaining cy-cle (Waleffe 1997). Regardless of the form that a potential near-wall cycle wall modelcould take, describing the near-wall dynamics using a reduced-order dynamical systemmodel is currently limited to low- Re τ turbulent flows or, interpreted differently, the vis-cous and buffer layer dynamics in high- Re τ flows. As such, in developing a dynamicalstructure-based wall model that is valid for high- Re τ flows, any near-wall cycle reduced-order model would be valid in a region that is substantially smaller than the near-wallLES grid, and would only constitute a portion of the flow that needs modeling. Fur-thermore, coupling such reduced-order models to the LES flow would require boundaryconditions defined at the edge of the buffer layer, which would be subject to large model-ing errors. Instead, if the reduced-order model contains buffer layer dynamics as well asa portion of the self-similar hierarchy of eddies, a coupling can be more easily formulatedbetween the the reduced-order model and the LES flow, which captures the largest ofthese self-similar eddies away from the wall. However, since the LES flow field near thewall is under-resolved even with respect to the largest structures of the flow, a couplingbetween the near-wall reduced-order model and the LES flow field cannot directly de-pend on the flow variables computed on the LES grid point nearest to the wall. This isillustrated in Figure 2.We propose to use knowledge of the wall-normal self-similarity of high- Re τ wall-bounded turbulent flows as the bridge between the LES flow field and the model rep-resenting the near-wall cycle. As a surrogate for any potential reduced-order model ofthe near-wall cycle, a near-wall patch of DNS resolution is used. The domain extent ofthis DNS patch is fixed in inner units in all three directions and is independent of theLES grid size, capturing only the smallest eddies in the flow. This makes the scaling ofsuch a wall model independent of Re τ over the range of Re τ values of interest in ex-ternal aerodynamics applications. This physics-based multiscale modeling approach thatcouples DNS and LES should be contrasted with other similar patch ideas whose sizeis linked to the LES grid size, resulting in an unfavorable scaling of the number of gridpoints required with respect to Re τ , being Re ατ with 1 ≤ α ≤ et al. et al. a-priori results of applying the attached-eddy-based boundary conditions to the DNS near-wall patch. Finally, conclusions are drawnand future tasks are presented in Section 5.
2. Outer flow formulation in channel flow
Consider the continuity and the incompressible Navier-Stokes equations ∂u i ∂x i = 0 (2.1) ∂u i ∂t + u j ∂u i ∂x j = − ∂p∂x i + ν ∂ u i ∂x j ∂x j , (2.2) oward a flow-structure-based WMLES paradigm Figure 1 . Instantaneous streamwise velocity in a turbulent channel flow at Re τ ≈ Figure 2 . Schematic showing the interdependence of the DNS near-wall patch and the outer-flow LES flow field. The near-wall patch is fixed in inner units and its size is independent of theLES grid size. The outer flow centered around x AE is used to extract statistical constraints thatare enforced at the top boundary of the DNS near-wall patch. The inset shows the boundaryconditions of the DNS near-wall patch, where the dashed line indicates the internal plane whosestructure is copied and rescaled at the top boundary. where u i is the velocity component in the i th direction; x , x , and x are the streamwise,wall-normal, and spanwise directions, respectively; ν is the kinematic viscosity; and p isthe kinematic pressure. Defining c ( . ) to be the filtering operation, filtering Eqs. (2.1)-(2.2)leads to the LES governing equations ∂ b u i ∂x i = 0 (2.3) Elnahhas, Lozano-Dur´an & Moin ∂ b u i ∂t + b u j ∂ b u i ∂x j = − ∂ b p∂x i + ν ∂ b u i ∂x j ∂x j − ∂τ ij ∂x j , (2.4)where τ ij = d u i u j − b u i b u j is the SGS stress tensor. In the bulk of the channel, Eqs. (2.3)-(2.4) are solved on an isotropic grid. The channel is driven by a constant mass flux suchthat the wall-shear stress is not fixed a-priori . Since the LES equations are solved on agrid resolving only the large eddies in the logarithmic region of the flow, the appropriateboundary condition to apply at the wall for the tangential velocities is not the no-slipboundary condition. Instead, a boundary condition is placed on the wall-shear stress, τ LESw . Most commonly, a no-penetration boundary condition at the wall for the filteredvelocity is applied, forcing the wall-shear stress to be carried by resolved shear andSGS stress contributions. For eddy-viscosity-type SGS models, this makes the boundarycondition a Neumann-type boundary condition for the tangential velocities at the wall.A wall model, which is usually based on RANS formulations, is needed to compute τ LESw .Instead, we extract τ LESw from the DNS near-wall patch, which acts as the wall model.Due to the homogeneity of the channel in the wall-parallel directions, the same wall-shearstress is applied at every wall grid point in the LES, and a single DNS near-wall patchis simulated.
3. A near-wall patch wall model
A near-wall patch of DNS resolution is employed to model the near-wall dynamics.The main aim of this near-wall patch is to predict the wall-shear stress and one-pointstatistics such as the mean velocity profiles and turbulence intensities. Furthermore, weaim to accurately predict the velocity spectra. To make this model tractable, the costof the near-wall DNS patch should scale more favorably with Re τ than the cost of theLES to eventually make its cost marginal. To satisfy this constraint, the size of thenear-wall patch must be independent of the LES grid, and it is chosen to be fixed ininner units such that the physical size of the DNS patch shrinks as Re τ increases. Thenear-wall patch is designed to capture the near-wall self-sustaining cycle, along with aportion of the self-similar hierarchy of eddies such that a coupling with the LES flowfield that utilizes information from both flow domains is formulated. Figure 2 illustratesthe coupling between the DNS near-wall patch and the LES flow field.3.1. A near-wall patch in inner units
Consider a region near the wall that is homogeneous in both the streamwise and thespanwise directions. The wall-parallel averaged wall-shear stress τ w , can be used tofind a time-varying friction velocity u τ ( t ) = ( τ w , /ρ ) / , where ( . ) , denotes averagingalong the homogeneous directions, and ρ is the fluid density. Rescaling Eqs. (2.1)-(2.2)with this time-dependent friction velocity, and its corresponding time-dependent frictionlength scale δ v ( t ) = ν/u τ ( t ), leads to ∂u + i ∂x + i = 0 , (3.1) ∂u + i ∂t + + u + j ∂u + i ∂x + j = − ∂p + ∂x + i + ∂ u + i ∂x + j ∂x + j − u + i u τ du τ dt + , (3.2)where ( . ) + indicates time-dependent plus units. The linear forcing term on the right-handside of Eq. (3.2) resembles the linear forcing term found in forced homogeneous isotropic oward a flow-structure-based WMLES paradigm ∂u +1 /∂x +2 | w = 1 atall times, which follows directly from the definition of plus units, ensuring that this DNSnear-wall patch is fixed in inner units but varies in size in outer units as the flow develops.Note that as the near-wall patch reaches statistical equilibrium, this linear forcing termvanishes on average. The friction velocity u τ ( t + ) can be found as u τ ( t + ) = u τ (0) exp (cid:18) Z t + A ( t ′ + ) dt ′ + (cid:19) , (3.3)where A ≡ u τ du τ dt + = − U + b (cid:20) dU + b dt + + u +1 u +2 1 , δ + P (cid:12)(cid:12)(cid:12)(cid:12) δ + P + dP + dx +1 − δ + P (cid:18) ∂u +1 , ∂x +2 (cid:12)(cid:12)(cid:12)(cid:12) δ + P − (cid:19)(cid:21) , (3.4) U + b is the bulk velocity of the near-wall patch in inner units, δ + P is the wall-normalextent of the DNS near-wall patch in inner units, which is a parameter to be defined, dP + /dx +1 = d ˆ P + /dx +1 is the time-varying mean pressure gradient driving the constantmass flux LES channel, and u τ (0) is the friction velocity of the initial condition of thenear-wall patch. Given this formulation, τ LESw can be defined as τ LESw = ρu τ , completingthe formulation for the LES domain.The near-wall patch is homogeneous in the x and x directions, and periodic boundaryconditions are applied to all velocity components. In the wall-normal direction, the no-slipand no-penetration boundary conditions are applied at the bottom wall. However, thecorrect boundary condition to apply at the upper boundary is ambiguous, as it is placeda distance δ P ( t ) away from the wall, within a highly turbulent region. Specifying this topboundary condition for the three velocity components closes the system. In the followingsubsection, a physics-based approach utilizing the self-similar nature of the logarithmicregion with distance from the wall is pursued to obtain suitable top boundary conditionsfor the DNS patch. The DNS near-wall patch of Carney et al. (2020) was also fixed ininner units, but instead of being predictive of the wall-shear stress, the wall-shear stresswas a parameter defining the turbulent environment.3.2. Attached-eddy-based coupling between the LES and the near-wall patch
To specify the boundary conditions of the three velocity components at the top of thenear-wall patch, we separately specify the mean and fluctuating components of each ofthe three velocity components u i | δ P = u i | δ P + u ′ i | δ P . (3.5)To specify the mean component of the velocities u i , a logarithmic mean velocity profile isassumed to exist between the location in the outer LES flow x AE , from where statisticalinformation is extracted and to be defined in the following paragraphs, and the top ofthe DNS patch δ P u i , ( x AE ) − u i , ( δ P ) u τ = − κ log (cid:18) δ P x AE (cid:19) + B ∗ for i = 1 , , (3.6)where κ is the von K´arm´an constant and B ∗ is the intercept of the log-law. The parame-ters κ and B ∗ can be found dynamically from the outer LES velocity using a least-squaresmethod centered around x AE . The mean of the wall-normal velocity has to be zero dueto the symmetry of the flow. Elnahhas, Lozano-Dur´an & Moin
To fully define the fluctuating component of each of the velocities at the top boundary,a magnitude and a spatial structure need to be specified u ′ i | δ P = α i u ∗ i | δ P , (3.7)where α i and u ∗ i | δ P are the magnitude and spatial structure of the i th fluctuating velocitycomponent at the top boundary, respectively. To specify each of them, we appeal to theself-similar structure of the flow. To specify the magnitude, the statistical attributes of ageometrically self-similar flow are leveraged. For any wall-bounded flow, the wall-normalbehavior of the turbulent intensities can generally depend on the distance from the wall,the Reynolds number, the mean pressure gradients, and any other environment definingvariables u ′ u τ = f (cid:18) x δ , Re τ , dPdx , dPdx , . . . (cid:19) ,u ′ u τ = f (cid:18) x δ , Re τ , dPdx , dPdx , . . . (cid:19) ,u ′ u τ = f (cid:18) x δ , Re τ , dPdx , dPdx , . . . (cid:19) , (3.8)where δ is the channel half-height or boundary layer thickness, and ( . ) denotes averagingalong the homogeneous directions and time. At sufficiently high Re τ , and when there is aconstant stress layer in the flow, the logarithmic layer is hypothesized to be mostly com-posed of wall-attached eddies that are geometrically self-similar with distance from thewall (Townsend 1976; Perry & Chong 1982; Marusic & Monty 2019). These wall-attachededdies carry the bulk of the turbulent kinetic energy and momentum of the flow. In thispreliminary study, we focus on tackling the coupling problem between the DNS near-wallpatch and the LES domain in this limit of high- Re τ flows. Under these conditions, theturbulent intensity profiles in Eq. (3.8) reduce to Re τ -independent, logarithmic forms u ′ u τ = B − A log (cid:18) x δ (cid:19) ,u ′ u τ = B − A log (cid:18) x δ (cid:19) ,u ′ u τ = B , (3.9)where A , A , B , B , and B are coefficients that depend on the flow.Since wall-attached eddies are self-similar with distance from the wall, and they repre-sent the largest fully confined eddy below a given wall-normal height, a carefully chosenLES resolution can resolve them. Lozano-Dur´an & Bae (2019) showed that to resolve90% of the turbulent kinetic energy at some x while recovering two-dimensional energyspectra resembling those of DNS except at the smallest scales, the grid resolution needsto satisfy 2∆ min x ≈ . , min x ≈ . , (3.10)where ∆ and ∆ are the grid sizes in the streamwise and spanwise directions, respec-tively. Given that the wall-attached eddy scaling is valid at high Re τ in the range of2 . Re / τ ≤ x +2 ≤ . Re τ , as shown by Klewicki et al. (2009), then to resolve the scales oward a flow-structure-based WMLES paradigm x in this region, assuming an isotropic grid as is usually employed in WMLES,requires approximately 90 points in the wall-normal direction. This range of the validityof the attached-eddy scaling places the constraint on the value of δ + P ≥ . Re / τ . For the Re τ values of interest for external aerodynamics, Re τ ∼ O (10 − ) (Smits & Marusic2013), choosing δ + P ∼ Re τ .By resolving these wall-attached eddies in the LES flow field around x AE in the range2 . Re / τ ≤ x AE ≤ . Re τ , the coefficients u τ A , u τ A , u τ B , u τ B , and u τ B can beextracted dynamically from the LES solution and Eq. (3.9) using a least-squares methodcentered around x AE , where u τ is computed using the DNS near-wall patch. These pre-dicted coefficients provide the first piece of information, i.e. the statistics required tospecify the magnitude of the fluctuating component of the velocity boundary conditionsat the top of the DNS near-wall patch. To determine α i , any statistical quantities thatcan be predicted using Eq. (3.9) such as the turbulent intensity or the slope of the tur-bulent intensity of each velocity component can be utilized. In this study, we chose tohold the slopes of the turbulent intensities constant for the two wall-parallel componentsof the fluctuating velocities ∂u ′ i ∂x (cid:12)(cid:12)(cid:12)(cid:12) δ P = γ i for i = 1 , , (3.11)making α i = α i ( γ i ). Thus, Neumann boundary conditions are enforced for the slopeof the mean turbulent intensities in the wall-parallel directions. The magnitude of thewall-normal fluctuating velocity cannot be specified using this statistical constraint asexplained in the following paragraph.The aim is to find the boundary condition that least perturbs the flow, allowing for asmooth continuation in the internal solution up to the boundary. To do so, the spatialstructures of the fluctuating velocities in Eq. (3.7) are chosen such that the interiorof the domain is connected with the top boundary using wall-normal geometric self-similarity. Given that δ + P ≥ . Re / τ , the self-similar hierarchy of eddies extends beneaththe top boundary of the DNS domain. Therefore, copying internal planes of fluctuatingvelocities and scaling them upward in size, could serve as potential boundary conditions.For simplicity, we copy the first internal plane below the top boundary for each velocitycomponent u ∗ i | δ P ∝ u ′ i | δ P − ∆ x , (3.12)where ∆ x is the grid spacing at the top of the domain in the wall-normal direction,and the proportionality symbol indicates that only the spatial structure is extracted byrescaling the variance of the copied internal plane to unity. This choice of boundarycondition assumes that the DNS grid is sufficiently resolved in the wall-normal directionsuch that wall-parallel scaling of the size of the structures with distance from the wall isunnecessary. However, if an interior plane farther away from the top boundary is copied,the wall-parallel scaling of the structure of the flow will become necessary to maintain thegeometric self-similarity that is the basis for this boundary condition. The combinationof Eqs. (3.7),(3.9), and (3.12) imply that a zero Neumann boundary condition should beapplied to the instantaneous wall-normal velocity component, which is not physical. Inthis preliminary study, we explore the use of a Robin-type boundary condition for thewall-normal velocity component u = ℓ ∂u ∂x , (3.13) Elnahhas, Lozano-Dur´an & Moin where the slip length ℓ is a parameter to be specified.Given this two-sided enforcement of the wall-normal geometric self-similarity as aboundary condition, the system for the DNS near-wall patch is closed, and the couplingbetween the LES domain and the DNS domain is established in a dynamic fashion, rely-ing on no a-priori picked coefficients, except for the choice of ℓ . However, in principle, apurely parameter-free coupled system can be formulated which incorporates a dynamicboundary condition for the wall-normal velocity. Possible methods include supplement-ing the attached-eddy-based scalings in the wall-parallel directions with a statistical con-straint on the wall-normal turbulent intensity to allow the application of Eqs. (3.7) and(3.12) directly, or the internal rescaling of planes further inside the DNS patch domain.In future versions of this work, these dynamic boundary conditions for the wall-normalvelocity will be investigated. In the next section, we present a-priori results of applyingthe boundary conditions to a truncated channel flow.An alternative type of boundary condition that was tested at the initial stages ofthe investigation was a one-way enforcement of the wall-normal geometric self-similarity,where both the magnitude and the structure of the fluctuating velocities in Eq. (3.7)were extracted from the outer LES flow. An entire plane of fluctuating velocities fromthe LES solution at x AE was extracted, scaled linearly with distance from the wallin the wall-parallel directions, scaled in amplitude using Eq. (3.9), evaluated at x = δ + P , and translated in the wall-parallel directions by the logarithmic mean velocitiesin Eq. (3.6) to account for the difference in the advection velocity of the fluctuationsat different wall-normal heights. These modified velocity planes were then applied asDirichlet boundary conditions to the top of the DNS near-wall patch. However, due tothe disconnect between these planes and the internal flow field of the DNS near-wallpatch, spurious pressure fluctuations were introduced that modified the mean velocityprofile at distances proportional to δ P (Jim´enez & Vasco 1998; Mizuno & Jim´enez 2013).For this reason, this boundary condition was not pursued any further. A-priori testing of the attached-eddy-based boundary conditions
To test the ability of the boundary conditions in Eqs. (3.12)-(3.6) applied within thelogarithmic region of the flow in recovering realistic turbulent statistics inside the DNSnear-wall patch, an a-priori test is conducted at a low Re τ = 395. DNS of a full chan-nel flow at Re τ = 395 with ( L x , L x , L x ) = (2 π, , π ) is conducted with uniform gridresolutions in the wall-parallel directions of (∆ + x , ∆ + x ) = (9 . , .
87) and with a mini-mum and maximum grid resolution in the wall-normal direction of ∆ + x ,min = 0 .
51 and∆ + x ,max = 4 .
21, respectively. Once statistical equilibrium is reached, the channel is trun-cated at x +2 ≈ ℓ ∈ {− . , − . , − . , − . , − . , − . } and it wasfound that the first-order quantities of interest such as the mean velocity profile and u τ do not significantly change. As such, the results presented are for ℓ = − .
4. Furthermore,all results are presented in ( . ) + units normalized by the respective friction velocities of oward a flow-structure-based WMLES paradigm Figure 3 . Instantaneous streamwise velocity in a the DNS near-wall patch simulation at anominal Re τ = 395. Low velocities are indicated by darker colors and higher velocities withlighter colors. The flow in the x − x plane shows the near-wall streaky structure of the flow,the flow in the x − x plane shows ejections and sweeps, and the flow in the x − x plane showsthe low-speed streaks due to the lift-up effect of streamwise-aligned vortices. The two solid linesindicate the locations at which the cross-sections are taken.(a) (b) Figure 4 . First-order one-point statistics of the full channel and the surrogate DNS near-wallpatch. (a) Streamwise mean velocity profile. (b) Indicator function β + . The indicator function β + = x +2 ∂U +1 /∂x +2 is used to test for logarithmic regions in the mean velocity profile U +1 ( x +2 ). each simulation. Finally, note that because the initial condition was statistically close tothe final equilibrium solution, the linear forcing term in Eq. (3.2) was not included asit should be small. Figure 3 show instantaneous snapshots of the streamwise velocity inwall-parallel, cross-stream, and wall-normal streamwise-aligned planes.For the cases tested, u τ of the DNS patch was always predicted within 5% of that of thefull channel, showing initial success of the top boundary conditions. Figure 4(a) shows0 Elnahhas, Lozano-Dur´an & Moin the streamwise mean velocity profile from the full channel truncated to x +2 = 110 aswell as the mean velocity profile of the surrogate DNS near-wall patch. It is evident thatthe boundary conditions do not introduce significant artifacts, such as the emergenceof an internal adaptation layer at the top of the domain, and that the mean velocityprofile of the DNS patch tracks that of the full channel across the entirety of the domain.This is verified in Figure 4(b), which compares the logarithmic indicator function of theDNS patch with that of the full channel. The indicator function of the DNS patch tracksthat of the full channel across the entirety of the domain and captures the minimum inthe indicator function within the correct wall-normal region away from the wall. Thisminimum is underpredicted by about 7%. Note that there is no true logarithmic region inthe mean velocity profile at this low Re τ . However, as the magnitude of ℓ was increased,an artifical logarithmic region with an overpredicted von K´arm´an constant, where theindicator function attained a constant value with wall-normal distance, developed towardsthe edge of the domain.Figure 5 shows the turbulent intensities predicted by the DNS patch. Both the stream-wise and the spanwise turbulent intensities are predicted with reasonable accuracy acrossthe entire wall-normal extent of the DNS patch domain. However, there is a large over-prediction of the wall-normal turbulent intensity near the top boundary that extendstowards the wall. This overprediction in the wall-normal turbulent intensity could betied to the underprediction of the streamwise intensity through pressure-strain correla-tion and is most likely related to the choice of the Robin-type boundary condition forthe wall-normal velocity, as it was the quantity most sensitive to the value of ℓ . Further-more, even though no explicit constraints were placed on the Reynolds shear stress, it ispredicted accurately across the entire wall-normal distance. This high level of accuracyin the prediction of the Reynolds shear stress is explained through the mean momentumbalance, which ties the Reynolds shear stresses to the mean velocity profile, which waspredicted with accurately. The emergence of the artifical logarithmic region with theincreased magnitude of ℓ did not affect the Reynolds stress wall-normal profile signifi-cantly. Note that the lowest error in the prediction of κ and u τ was achieved when thechoice of ℓ made the wall-normal turbulent intensity closest to its correct value at theboundary. This suggests that the value of ℓ could be specified by enforcing the value ofthe wall-normal turbulent intensity at the boundary, which can be extracted from thewall-normal turbulent intensity from the LES flow at x AE in a fully coupled calculation.To compare the structure of the flow between the full channel and the surrogate DNSnear-wall patch, contours of premultiplied one-dimensional energy spectra in both thestreamwise and the spanwise directions are considered. Figures 6 and 7 show that thenear-wall peak in the streamwise spectra of the streamwise turbulent intensity is capturedaccurately, and that the considered contour levels match approximately all the way tothe top boundary of the patch. This is also approximately seen in the spanwise spectra ofthe streamwise turbulent intensity with any appreciable difference only observed towardthe very top of the domain. Qualitatively, both the streamwise and the spanwise one-dimensional spectra of both the wall-normal and the spanwise turbulent intensities inthe DNS patch match the full channel simulation, with two notable exceptions. First,the peak in the streamwise spectrum of the spanwise turbulent intensity is closer to thewall in the full channel than it is in the DNS patch, where it occurs at the boundary.Second, the same behavior can be observed in the spanwise spectrum of the wall-normalturbulent intensity, where the peak is closer to the top of the domain in the DNS patchthan it is in the full channel simulation. oward a flow-structure-based WMLES paradigm Figure 5 . A comparison of the wall-normal profiles of the four nonzero components of theReynolds stress tensor in the full channel with those in the DNS near-wall patch. The solid linesrepresent the DNS patch and the dash-dotted lines with matching symbols represent the fullchannel simulation at Re τ = 395. Figure 6 . Premultiplied one-dimensional energy spectra as a function of the wall-normal dis-tance and the streamwise wavelength. The solid lines represent the DNS near-wall patch and thedashed lines represent the full channel simulation. The contour levels are { . , . , . , . } × max( k x Φ + u ′ i u ′ i ).
5. Conclusions and future work
The current landscape of RANS-based wall models does not make use of any struc-tural information of the flow. In this preliminary study, we examine the feasibility of astructure-based wall model that aims to capture the near-wall self-sustaining cycle and aportion of the wall-normal self-similar hierarchy of eddies, such that a coupling is possiblewith the largest eddies in this hierarchy resolved by the LES grid. A framework for cou-2
Elnahhas, Lozano-Dur´an & Moin
Figure 7 . Premultiplied one-dimensional energy spectra as a function of the wall-normal dis-tance and the spanwise wavelength. The solid lines represent the DNS near-wall patch and thedashed lines represent the full channel simulation. The contour levels are { . , . , . , . } × max( k x Φ + u ′ i u ′ i ). pling a patch of DNS resolution fixed in inner units to the outer LES flow is presented.Dynamically computed statistical quantities from the LES are applied as constraints tothe self-similarity-based top boundary condition of this DNS patch. A-priori testing ofthis boundary condition was conducted. It is found that copying internal planes fromwithin the incipient logarithmic region of the flow does not introduce significant artifactsnear the top boundary and recovers quantitatively accurate first- and second-order one-point statistics. The predicted friction velocity u τ was within 5% of the full channel flow,and the mean velocity profile, streamwise turbulent intensity, and the Reynolds shearstress were predicted accurately across the entire domain. Two-point statistics in theform of the streamwise and spanwise premultiplied energy spectra for all three velocitycomponents compared quantitatively well with the full channel simulation and showedthat the structure of the flow was also realistic up to the top of the domain. The a-priori tests were conducted at the relatively low Re τ = 395. Since the validity of the assump-tions upon which the boundary conditions are based increases as Re τ increases, the a-priori tests are considered to be a stringent test of the proposed boundary conditions.It is hypothesized that the accuracy of the predictions would increase with Re τ .Even though a general framework was presented, some assumptions that were maderequire refinement. For example, the attached-eddy-based form of the turbulent inten-sities is valid for the dynamically computed coefficients only at extreme values of Re τ .To extend this coupling to moderate values of Re τ the form of the turbulent intensi-ties can be enhanced by accounting for the existence of other types of eddies, such aswall-detached self-similar eddies (Marusic & Monty 2019). Furthermore, enforcing otherstatistical quantities as constraints on the top boundary of the DNS patch, which wouldallow for the removal of the ad hoc choice of a Robin-type boundary condition for thewall-normal velocity, needs to be explored. Once all these refinements are made, a fullycoupled calculation between an outer-flow LES simulation and the DNS near-wall patchwill be conducted in both equilibrium and nonequilibrium settings. Acknowledgments
This study was funded by the Stanford Engineering Graduate Fellowship and by NASAgrant oward a flow-structure-based WMLES paradigm REFERENCES
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