Stochastic deconvolution of wall statistics in Reynolds‐averaged Navier–Stokes simulations based on one‐dimensional turbulence

Reynolds‐averaged Navier–Stokes simulation (RaNS) is state‐of‐the‐art for numerical analysis of complex flows at high Reynolds number. Standalone RaNS may yield a reasonable estimate of the wall‐shear stress and turbulent drag if a proper wall‐function is prescribed, but detailed turbulence statistics cannot be obtained, especially at the wall. This lack in modeling is addressed here by a stochastic deconvolution strategy based on a stochastic one‐dimensional turbulence (ODT) model. Here, a one‐way coupling strategy is proposed in which a forcing term is computed from the balanced RaNS solution that is in turn utilized in the ODT model. The temporally developing ODT solution exhibits turbulent perturbations but relaxes toward the local RaNS solution due to resolved molecular‐diffusive processes. It is demonstrated that the approach is able to recover the distribution of positive wall‐shear stress fluctuations in turbulent channel flow. When formulated as post‐processing tool, it is suggested that RaNS can be enhanced by ODT providing economical means for local high‐fidelity numerical modeling based on a low‐fidelity flow solution.


INTRODUCTION
Low-fidelity modeling based on the Reynolds-averaged Navier-Stokes (RaNS) equations is state-of-the-art for numerical simulation of complex flows at high Reynolds number, which is in particular the case for atmospheric boundary-layer flows [1,2], among other industrial applications.RaNS provides fast turn-around times at reasonable accuracy for applications that are well described by some calibrated set-up and dedicated modeling approach, especially for the wall [3,4].However, standalone RaNS exhibits a substantial amount of modeling as small-scale processes, that is, turbulent fluctuations, multiscale boundary-layer dynamics, and other transient processes, are not resolved.It is somewhat surprising that it is feasible to obtain a reasonable estimate of the mean flow in many situations with RaNS.Even the mean wall-shear stress can be reproduced rather well if a proper turbulence model and corresponding wall-functions are available [5][6][7].Nonetheless, standalone RaNS remains limited to the prediction of averaged flow variables.This limitation poses a challenge for applications in which small-scale fluctuations and backscatter from the direct turbulent cascade affect the mean state.Fluctuating boundary layer processes in combination with differential diffusion of heat and mass crucially F I G U R E 1 Sketch of the channel flow configuration.() (thick black profile) denotes the RaNS solution and () (thick red profile) the instantaneous ODT solution, both resolved as wall-normal profiles of the streamwise velocity.The boundary layer thickness ℎ extends over half of the channel height.Stochastic deconvolution of the bottom wall boundary layer is performed with the ODT model driven by the RaNS solution in a truncated domain (ODT line, thick red line).ODT, one-dimensional turbulence; RaNS, Reynolds-averaged Navier-Stokes simulations.
affect, for example, the yield of a catalytic through-flow reactor even if the mass flow rate and mean velocity field are barely affected by the near-wall reactions [8,9].Another example that requires an improved representation of small-scale processes is wind power forecasting on short time scales.Present approaches either infer the probability of power ramps from small ensembles of numerical weather prediction model simulations [10] or work with historical data [11].A physically more robust approach would provide and propagate also the spread in wind fluctuations consistent with the mean atmospheric flow state by deconvolution of the RaNS state if high-fidelity modeling by wall-resolved large-eddy simulation (LES) or direct numerical simulation (DNS) is constrained by costs.
The modeling error in the applications mentioned might be significantly improved at affordable cost by a stochastic enhancement of the RaNS results.The idea of utilizing a stochastic process for deconvolution of parameterized scales is not new, but usually made for a particle-based representation of probability density functions (PDFs) within a Langevin equation framework [12,13].Here, a fundamentally different approach is suggested that addresses the issues of feasibility, fidelity, and physical consistency in physical space without PDF approximations.Further technical aspects are the usability as post-processing tool and physical interpretability.All of the requirements mentioned are addressed by utilizing the map-based one-dimensional turbulence (ODT) model [14] for stochastic deconvolution of RaNS data.In ODT, turbulent advection is modeled by a stochastic process that is formulated with the aid of turbulent Baker's maps [15].Major cost reduction is achieved by formulating the model in a one-dimensional subspace that aims to resolve the wallnormal direction of the turbulent boundary layer flow.In this paper, it is demonstrated that the one-way coupling of ODT and RaNS is able to recover the probability density of positive wall-shear stress fluctuations in turbulent channel flow.The boundary layer structure is maintained since the formulation yields statistical relaxation of the ODT solution to the prescribed RaNS state.Nevertheless, some open issues remain with respect to the treatment of the outer layer cut-off that might have to be treated by an extension of the 1D domain to fully accommodate turbulent eddies up to the mixing length scale.Missing negative wall-shear stress fluctuations is not related to this issue but an artifact of the one-dimensional formulation [16].
The rest of this paper is organized as follows.Section 2 briefly summarizes channel flow configuration, the RaNS and ODT modeling approaches, and the one-way coupling in the present context.Section 3 contains the key results and a case study on the influence of the one-way coupling.Last, Section 4 closes with some concluding remarks.

Overview of the channel flow configuration
Figure 1 shows a sketch of the channel flow configuration investigated.The flow is driven by a prescribed constant mean pressure gradient (−∕ = const).In ODT, an additional force term is supplied in order to relax the standalone stochastic solution to the RaNS solution.At the walls that are located at  = 0 and  = 2ℎ, homogeneous no-slip boundary conditions are applied.The RaNS domain extends over the entire channel height, whereas the present ODT application as post-processing tool is performed for a truncated domain that extends over the boundary layer thickness ℎ.Further details of the RaNS and ODT approaches and the coupling strategy are discussed below.

Overview of the Reynolds-averaged Navier-Stokes model
Reynolds-averaged Navier-Stokes simulations exploit the assumption of small-scale universality of turbulent flows [17].Parameterization schemes are utilized in order to close the averaged conservation equations by expressing the statistical mean effect of notionally random turbulent motions based on physical relations with the large-scale flow.This approach is essentially based on the mean effect of the direct cascade phenomenology in three-dimensional turbulence [18] that neglects backscatter from the small-scale fluctuating processes.The key idea of using RaNS as prognostic flow model is to resolve the nonuniversal large-scale flow by modeling turbulence as a statistically diffusive and dissipative process.
The universality assumptions underlying the parameterizations, however, are not generally justified.This is in general deducible from the variety of available turbulence models, and specifically seen in the need to prescribe wall functions for statistical moments of the velocity field in order to capture the turbulent boundary layer [19].The RaNS equations (per unit mass) for an incompressible flow are given by where  is time, (  ) = (, , ) T the Cartesian coordinates of the configuration space, (  ) = (, , ) T the Cartesian components of the Reynolds-averaged velocity vector,  the Reynolds-averaged pressure,   the prescribed Reynolds-averaged momentum sources, like a prescribed streamwise specific mean pressure-gradient force,   = − −1 (∕)  1 , where   (,  = 1, 2, 3) is the Kronecker symbol,  and  is the fluid's constant mass density and kinematic viscosity, respectively, and   =     −     the unclosed Reynolds stress tensor.Einstein's summation convention is implied for indices that appear twice.By applying Boussinesq's [20] diffusive closure hypothesis, the Reynolds stress tensor is parameterized as   = 2 t   , where   = (  ∕  +   ∕  )∕2 denotes the RaNS-resolved mean rate-of-strain tensor and  t the turbulent eddy viscosity.Here, ANSYS Fluent [21] standard - model based on Wilcox [19] is used to obtain  t with boundary-layer modifications due to the ANSYS Fluent velocity wall function implementation based on Launder and Spalding [22].

Overview of the map-based stochastic one-dimensional turbulence model
Kerstein's [14] ODT model aims to resolve all relevant scales of turbulent flow.Molecular-diffusive transport processes are directly resolved along a single physical coordinate denoted as 'ODT line' in Figure 1, but distinguished from turbulentadvective processes.In contrast to previous 1D standalone ODT [14,16,23] and 3D ODTLES [24][25][26] applications, the ODT line here only extends over a fraction of the channel height.The dimensionally reduced, stochastic ODT equations are given by [16] where   =   (, ) denotes the instantaneous Cartesian velocity vector components and   =   (, ) the momentum sources resolved along a truncated local wall-normal coordinate .The effects of turbulent advection and fluctuating pressure gradients are modeled by a stochastically sampled sequence of eddy events, symbolically represented as   occurring at discrete times  e , where ( −  e ) is the Dirac delta distribution.These events punctuate the deterministic advancement and are formulated with the aid of a measure preserving map that addresses fundamental conservation principles within the dimensionally reduced setting [14,15].Temporal integration regularizes a 'delta kick' to a finite 'Heaviside jump' contribution to the piecewise continuous ODT solution.Note that, in contrast to RaNS, there is no closure and no closure modeling in ODT.
ODT eddy events are sampled from an unknown PDF that depends on the current flow state.In practice, the construction of this PDF is avoided and a more economical thinning-and-rejection algorithm [14] is used.A size- eddy event is probabilistically accepted with local rate  −1 =  √ 2∕ 2 , where  =  kin −  vp is the current specific available eddy energy that for incompressible shear flow only takes kinetic energy ( kin ) and viscous 'penalty' energy ( vp ) contributions.In the present application, the free ODT model parameters are fixed at  = 6.5 and  = 300 (following Glawe [27]), which is a reasonable selection for low-Reynolds number channel and boundary-layer type flows [28][29][30].For further details on the model formulation, the reader is directed to [14,23,31]

2.4
One-dimensional turbulence-enhanced Reynolds-averaged Navier-Stokes simulation: a weak one-way coupling strategy for post-processing applications The forcing term   in the ODT momentum equation (3) provides a generic interface for the investigation of the laminar and turbulent response to a prescribed forcing which can be exploited for local stochastic deconvolution of a RaNS solution.A weak coupling strategy could be to assume that the RaNS solution is a reasonable approximation that should be reproduced on average by the ODT model.It is practical to define a residual forcing that makes the ODT solution relax toward a prescribed mean state.This is feasible as demonstrated by a conceptually related approach in [32], where a variable momentum source was derived for the investigation of the transient mixing in a confined jet.Starting from standalone ODT applications to channel flow [14,16], the forcing term should minimally contain the driving pressure-gradient force of the RaNS and the advective terms not represented by eddies.
The starting point for the present application is the RaNS-based balance equation, In ODT, the stochastically sampled eddy events have to collectively contribute to the turbulent fluxes yielding   () as conditional average of the map-induced changes [14,16].Correspondingly, to temporal average Equation (3) yields essentially the same expression as in the case of RaNS discussed above.This holds at least for a prescribed constant mean pressure gradient as often utilized in standalone ODT channel flow simulations [16,23].Thus, the Reynolds stresses   are represented by ODT eddy events and not considered within the forcing.In order to generalize the approach, it is important to note that the streamwise mean pressure gradient ∕ is not known when ODT is applied locally as postprocessing tool.
The proposed weak one-way coupling strategy for the purpose of stochastic deconvolution of a RaNS solution allows for flow profile variability between the ODT and RaNS realizations.Differences in the turbulent flux representation between RaNS and ODT must therefore be compensated by a corresponding difference in the mean velocity profiles and vice versa.This is not a contradiction between RaNS and ODT, but provides the necessary freedom for model application to an inverse problem that would be overdetermined if ODT is required to reproduce the RaNS exactly.
Equation ( 4) thus yields the necessary expression for the ODT forcing term that is now fully determined by a RaNS velocity solution, where the spatial average yields an effective, spatially homogeneous, specific pressure-gradient force by application of the boundary-layer approximation [33].Note that the ODT domain extends only over the lower half of the channel (height ℎ) and not the full height (2ℎ) as in previous ODT channel flow simulations by various authors, see, for example, [14,16,27].Note further that the advection terms in Equation ( 4) vanish for the fully developed channel flow and   recovers the constant mean pressure gradient in this case.Expression ( 5) is used straightforwardly in the ODT momentum equation ( 3) without further assumptions or simplifications.Except for the circumstance that stochastic eddy sampling is constrained to a truncated, one-sided ODT line as sketched in Figure 1.This spatial truncation is not a problem since it is assumed that the RaNS solution can be evaluated for all required locations and all fluxes at the upper boundary condition can be computed straightforwardly.
F I G U R E 2 Mean streamwise velocity profiles over the bottom half of the channel for   = 395.ODT simulations were driven by the RaNS and performed for the truncated domain.Permissively sampling and back-shifting ODT eddy events that cross the midplane only has a weak effect on the velocity overshoot.ODT, one-dimensional turbulence; RaNS, Reynolds-averaged Navier-Stokes simulations.

RESULTS
Preliminary ODT-enhanced RaNS results are presented below.First, it is demonstrated that the boundary layer structure and mean velocity field are captured by the approach.After that, second order but also detailed turbulence statistics are presented in order to highlight strengths and shortcomings of the approach.

Mean streamwise velocity
Figure 2 shows the mean velocity profile in bulk units for a simulation with friction Reynolds number   =   ℎ∕ = 395, where   = √ (∕) w is evaluated at the bottom wall located at  = 0 for the fully resolved boundary layer in ODT or obtained from the wall function prescribed in RaNS.The mean velocity profile ū() is a prognostic result of the RaNS approach, but a diagnostic result of the ODT.For the latter, it is obtained by temporal averaging of the instantaneous solution such that () = (, ).Two ODT solutions are shown in Figure 2 in order to demonstrate the influence of the domain truncation.Naturally, eddy events that would cross the midplane at  = ℎ are completely neglected.This approach leads to an artificial velocity overshoot which can be explained by domain-truncation-induced reduction of the momentum mixing.It is noteworthy that the symmetry of the flow is not broken since the domain truncation respects the statistical symmetry in the flow.Second, a run with backshifting of eddy events that would cross the midplane has been performed by sampling the lower edge location of eddy events and the size.This approach reduces the velocity overshoot, but momentum mixing is only partially restored.The model has no information of the opposing wall boundary condition, so that the bulk velocity profile in the ODT-enhanced RaNS is in fact closer to a flat-plate boundary layer [28,29] than channel flow [16,23].
Figure 3 shows the law-of-the-wall plot showing the normalized variables  + = ∕  and  + =   ∕ corresponding to the cases discussed above.The application of ODT aims to fully resolve the turbulent boundary layer, so that there is no wall function evaluated.It can be seen that ODT accurately reproduces the boundary-layer structure of available reference DNS [34].In particular, the turbulent and viscous inner layer with  + < 100 is well captured, considerably improving the limited fidelity of the wall-modeled RaNS.This observation is supported by standalone ODT applications to boundarylayer-type flows [16,29] and hits at a generic property of ODT that is exploited for the stochastic deconvolution of the RaNS solution.are shown here as ODT, in its present formulation, that is unable to fully represent the anisotropy of the boundary-layer turbulence.This manifests itself by identical wall-normal and spanwise velocity fluctuations [16,29,31].An acceptable large statistics gathering interval has been used for the results shown in Figures 2-4 due to which the r.m.s.velocity profiles and the mean velocity profiles are sufficiently converged.ODT exhibits about 50% less fluctuations than the reference DNS and a spatial artifact in the vicinity of the wall at around  + ≃ 20.Both effects are well-known modeling errors documented previously for various applications of the standalone formulation [14,29,35].Nevertheless, it is remarkable that ODT is able to provide reasonable estimates of the velocity fluctuations at the wall and throughout the boundary layer, which is relevant if, for example, the detailed heat and mass transfer to a wall [16] shall be analyzed based on a RaNS solution.

3.3
Detailed statistics of the local wall-shear stress low-order flow and detailed wall statistics.However, due to the dimensionally reduced formulation, the current modeling of turbulent pressure-velocity couplings [31] is unable to reproduce the occurrence of negative streamwise wall shear stress events.This might be a significant limitation of the model if such data would be crucial.This could be the detection of flow separation events, but for such cases surrogate analysis of ODT eddy events would seem more reasonable [30].Hence, it is perhaps most important that the ODT model provides economical means to statistically evaluate the magnitude and frequency of some relevant diversions from the mean state, enabled solely by post-processing of a RaNS solution.

CONCLUSION
Utilization of stochastic methods for deconvolution of parameterized scales and processes in RaNS simulations is desirable for various industrial applications, ranging from risk assessment over noise abatement to catalysis and wind power predictions.Such applications require a physics-based representation of small-scale processes and consistency of the deconvolution with the underlying RaNS solution, assuming that the RaNS solution is a reasonable approximation to the ground truth.Here, a map-based stochastic modeling approach based on an ODT model has been suggested that utilizes random sampling based on the stochastic nature of turbulent fluctuations in high-Reynolds-number flows but introduces memory effects by forward propagation of momentary flow profiles.Motivated by an assessment of local boundary layer properties, the RaNS solution in only half of the channel flow domain was supplied to the ODT.It was demonstrated that ODT is able to provide detailed wall-shear stress fluctuation statistics for turbulent channel flow without degeneration of the boundary layer structure.While the method works rather well for the positive wall-shear stress variations and maintained inner and log layer regions, negative wall-shear stress fluctuations and outer layer properties were not as well captured.The former is a known limitation of the 1D formulation [16] that can be mitigated in more costly 3D model extensions [27], whereas the latter is a profile truncation error.Especially the large detached eddies contributing the most to momentum mixing in the outer layer and bulk are implicitly filtered.Less momentum mixing is hence consistent with the excess velocity that has been observed for the truncated domain size.An extension of the ODT domain made did not cure the issue since the boundary layer from the opposing wall is not captured at all.The results obtained suggest that a generic application of the one-way coupled ODT-enhanced RaNS is generally possible and promising for the intended estimation of detailed fluctuation statistics.However, care has to be taken where flow profiles are truncated, which is presumably less of an issue for boundary layers than internal flows due to the extended free-stream region of the former.For the latter, statistical symmetry or vanishing shear detection in RaNS in conjunction with uniform flow velocity extension in ODT may be used in the future to mimic free-stream behavior.

A C K N O W L E D G M E N T S
Open access funding enabled and organized by Projekt DEAL.

Figure 4
Figure 4 shows wall-normal profiles of root-mean-square (r.m.s.) velocity fluctuations  ,rms = √  2  −  2 for ODT and available reference DNS[34] for the same simulations shown in Figure3.Only streamwise and spanwise fluctuations

Figure 5
Figure5shows the probability distribution function (PDF) of the streamwise component of the momentary wall-shear stress  w = (∕) w evaluated at the bottom wall located at  = 0. Deconvoluted RaNS data from the ODT solution in Figures2-4are shown together with available reference DNS[36].It is remarkable that ODT captures the positive streamwise wall-shear stresses very well.Even short stochastic simulations are sufficient to get reasonable estimates of