Systematic Study of Accuracy of Wall-Modeled Large Eddy Simulation using Uncertainty Quantification Techniques

The predictive accuracy of wall-modeled large eddy simulation is studied by systematic simulation campaigns of turbulent channel flow. The effect of wall model, grid resolution and anisotropy, numerical convective scheme and subgrid-scale modeling is investigated. All of these factors affect the resulting accuracy, and their action is to a large extent intertwined. The wall model is of the wall-stress type, and its sensitivity to location of velocity sampling, as well as law of the wall's parameters is assessed. For efficient exploration of the model parameter space (anisotropic grid resolution and wall model parameter values), generalized polynomial chaos expansions are used to construct metamodels for the responses which are taken to be measures of the predictive error in quantities of interest (QoIs). The QoIs include the mean wall shear stress and profiles of the mean velocity, the turbulent kinetic energy, and the Reynolds shear stress. DNS data is used as reference. Within the tested framework, a particular second-order accurate CFD code (OpenFOAM), the results provide ample support for grid and method parameters recommendations which are proposed in the present paper, and which provide good results for the QoIs. Notably, good results are obtained with a grid with isotropic (cubic) hexahedral cells, with $15\, 000$ cells per $\delta^3$, where $\delta$ is the channel half-height (or thickness of the turbulent boundary layer). The importance of providing enough numerical dissipation to obtain accurate QoIs is demonstrated. The main channel flow case investigated is ${\rm Re}_\tau=5200$, but extension to a wide range of ${\rm Re}$-numbers is considered. Use of other numerical methods and software would likely modify these recommendations, at least slightly, but the proposed framework is fully applicable to investigate this as well

A Library for Wall-Modelled Large-Eddy Simulation Based on OpenFOAM Technology

This work presents a feature-rich open-source library for wall-modelled large-eddy simulation (WMLES), which is a turbulence modelling approach that reduces the computational cost of traditional (wall-resolved) LES by introducing special treatment of the inner region of turbulent boundary layers (TBLs). The library is based on OpenFOAM and enhances the general-purpose LES solvers provided by this software with state-of-the-art wall modelling capability. In particular, the included wall models belong to the class of wall-stress models that account for the under-resolved turbulent structures by predicting and enforcing the correct local value of the wall shear stress. A review of this approach is given, followed by a detailed description of the library, discussing its functionality and extensible design. The included wall-stress models are presented, based on both algebraic and ordinary differential equations. To demonstrate the capabilities of the library, it was used for WMLES of turbulent channel flow and the flow over a backward-facing step (BFS). For each flow, a systematic simulation campaign was performed, in order to find a combination of numerical schemes, grid resolution and wall model type that would yield a good predictive accuracy for both the mean velocity field in the outer layer of the TBLs and the mean wall shear stress. The best result was achieved using a mildly dissipative second-order accurate scheme for the convective fluxes applied on an isotropic grid with 27000 cells per $\delta^3$-cube, where $\delta$ is the thickness of the TBL or the half-height of the channel. An algebraic model based on Spalding's law of the wall was found to perform well for both flows. On the other hand, the tested more complicated models, which incorporate the pressure gradient in the wall shear stress prediction, led to less accurate results

Assessment of uncertainties in hot-wire anemometry and oil-film interferometry measurements for wall-bounded turbulent flows

In this study, the sources of uncertainty of hot-wire anemometry (HWA) and oil-film interferometry (OFI) measurements are assessed. Both statistical and classical methods are used for the forward and inverse problems, so that the contributions to the overall uncertainty of the measured quantities can be evaluated. The correlations between the parameters are taken into account through the Bayesian inference with error-in-variable (EiV) model. In the forward problem, very small differences were found when using Monte Carlo (MC), Polynomial Chaos Expansion (PCE) and linear perturbation methods. In flow velocity measurements with HWA, the results indicate that the estimated uncertainty is lower when the correlations among parameters are considered, than when they are not taken into account. Moreover, global sensitivity analyses with Sobol indices showed that the HWA measurements are most sensitive to the wire voltage, and in the case of OFI the most sensitive factor is the calculation of fringe velocity. The relative errors in wall-shear stress, friction velocity and viscous length are 0.44%, 0.23% and 0.22%, respectively. Note that these values are lower than the ones reported in other wall-bounded turbulence studies. Note that in most studies of wall-bounded turbulence the correlations among parameters are not considered, and the uncertainties from the various parameters are directly added when determining the overall uncertainty of the measured quantity. In the present analysis we account for these correlations, which may lead to a lower overall uncertainty estimate due to error cancellation. Furthermore, our results also indicate that the crucial aspect when obtaining accurate inner-scaled velocity measurements is the wind-tunnel flow quality, which is more critical than the accuracy in wall-shear stress measurements

Effect of grid resolution on large eddy simulation of wall-bounded turbulence

The effect of grid resolution on large eddy simulation (LES) of wall-bounded turbulent flow is investigated. A channel flow simulation campaign involving systematic variation of the streamwise ($\Delta x$) and spanwise ($\Delta z$) grid resolution is used for this purpose. The main friction-velocity based Reynolds number investigated is 300. Near the walls, the grid cell size is determined by the frictional scaling, $\Delta x^+$ and $\Delta z^+$, and strongly anisotropic cells, with first $\Delta y^+ \sim 1$, thus aiming for wall-resolving LES. Results are compared to direct numerical simulations (DNS) and several quality measures are investigated, including the error in the predicted mean friction velocity and the error in cross-channel profiles of flow statistics. To reduce the total number of channel flow simulations, techniques from the framework of uncertainty quantification (UQ) are employed. In particular, generalized polynomial chaos expansion (gPCE) is used to create meta models for the errors over the allowed parameter ranges. The differing behavior of the different quality measures is demonstrated and analyzed. It is shown that friction velocity, and profiles of velocity and the Reynolds stress tensor, are most sensitive to $\Delta z^+$, while the error in the turbulent kinetic energy is mostly influenced by $\Delta x^+$. Recommendations for grid resolution requirements are given, together with quantification of the resulting predictive accuracy. The sensitivity of the results to subgrid-scale (SGS) model and varying Reynolds number is also investigated. All simulations are carried out with second-order accurate finite-volume based solver. The choice of numerical methods and SGS model is expected to influence the conclusions, but it is emphasized that the proposed methodology, involving gPCE, can be applied to other modeling approaches as well.Comment: 27 pages, The following article has been accepted by Physics of Fluids. After it is published, it will be found at https://aip.scitation.org/journal/phf. Copyright 2018 Saleh Rezaeiravesh and Mattias Liefvendahl. This article is distributed under a Creative Commons Attribution (CC-BY-NC-ND 4.0) Licens

Direct numerical simulation of turbulent pipe flow up to Reτ=5200Re_\tau=5200

Well-resolved direct numerical simulations (DNSs) have been performed of the flow in a smooth circular pipe of radius $R$ and axial length $10\pi R$ at friction Reynolds numbers up to $Re_\tau=5200$. Various turbulence statistics are documented and compared with other DNS and experimental data in pipes as well as channels.Small but distinct differences between various datasets are identified. The friction factor $\lambda$ overshoots by $2\%$ and undershoots by $0.6\%$ of the Prandtl friction law at low and high $Re$ ranges, respectively. In addition, $\lambda$ in our results is slightly higher than that in Pirozzoli et al. (J. Fluid. Mech., 926, A28, 2021), but matches well with the experiments in Furuichi et al. (Phys. Fluids, 27, 095108, 2015). The log-law indicator function, which is nearly indistinguishable between the pipe and channel flows up to $y^+=250$, has not yet developed a plateau further away from the wall in the pipes even for the $Re_\tau=5200$ cases. The wall shear stress fluctuations and the inner peak of the axial velocity intensity -- which grow monotonically with $Re_\tau$ -- are lower in the pipe than in the channel, but the difference decreases with increasing $Re_\tau$. While the wall values are slightly lower in channel than pipe flows at the same $Re_\tau$, the inner peaks of the pressure fluctuations show negligible differences between them. The Reynolds number scaling of all these quantities agrees with both the logarithmic and defect power laws if the coefficients are properly chosen. The one-dimensional spectrum of the axial velocity fluctuation exhibits a $k^{-1}$ dependence at an intermediate distance from the wall -- as also seen in the channel flow. In summary, this high-fidelity data enable us to provide better insights into the flow physics in the pipes and the similarity/difference among different types of wall turbulence.Comment: 22 pages, 15 figure

In-situ Estimation of Time-averaging Uncertainties in Turbulent Flow Simulations

The statistics obtained from turbulent flow simulations are generally uncertain due to finite time averaging. The techniques available in the literature to accurately estimate these uncertainties typically only work in an offline mode, that is, they require access to all available samples of a time series at once. In addition to the impossibility of online monitoring of uncertainties during the course of simulations, such an offline approach can lead to input/output (I/O) deficiencies and large storage/memory requirements, which can be problematic for large-scale simulations of turbulent flows. Here, we designed, implemented and tested a framework for estimating time-averaging uncertainties in turbulence statistics in an in-situ (online/streaming/updating) manner. The proposed algorithm relies on a novel low-memory update formula for computing the sample-estimated autocorrelation functions (ACFs). Based on this, smooth modeled ACFs of turbulence quantities can be generated to accurately estimate the time-averaging uncertainties in the corresponding sample mean estimators. The resulting uncertainty estimates are highly robust, accurate, and quantitatively the same as those obtained by standard offline estimators. Moreover, the computational overhead added by the in-situ algorithm is found to be negligible. The framework is completely general and can be used with any flow solver and also integrated into the simulations over conformal and complex meshes created by adopting adaptive mesh refinement techniques. The results of the study are encouraging for the further development of the in-situ framework for other uncertainty quantification and data-driven analyses relevant not only to large-scale turbulent flow simulations, but also to the simulation of other dynamical systems leading to time-varying quantities with autocorrelated samples

Applying Bayesian Optimization with Gaussian Process Regression to Computational Fluid Dynamics Problems

Bayesian optimization (BO) based on Gaussian process regression (GPR) is applied to different CFD (computational fluid dynamics) problems which can be of practical relevance. The problems are i) shape optimization in a lid-driven cavity to minimize or maximize the energy dissipation, ii) shape optimization of the wall of a channel flow in order to obtain a desired pressure-gradient distribution along the edge of the turbulent boundary layer formed on the other wall, and finally, iii) optimization of the controlling parameters of a spoiler-ice model to attain the aerodynamic characteristics of the airfoil with an actual surface ice. The diversity of the optimization problems, independence of the optimization approach from any adjoint information, the ease of employing different CFD solvers in the optimization loop, and more importantly, the relatively small number of the required flow simulations reveal the flexibility, efficiency, and versatility of the BO-GPR approach in CFD applications. It is shown that to ensure finding the global optimum of the design parameters of the size up to 8, less than 90 executions of the CFD solvers are needed. Furthermore, it is observed that the number of flow simulations does not significantly increase with the number of design parameters. The associated computational cost of these simulations can be affordable for many optimization cases with practical relevance

Quantification of time-averaging uncertainties in turbulence simulations

An automatic method is proposed for the removal of the initialization bias that is intrinsic to the output of any statistically stationary simulation. The general techniques based on optimization approaches such as Beyhaghi et al. [1] following the Marginal Standard Error Rules (MSER) method of White et al. [16] were observed to be highly sensitive to the fluctuations in a time series and resulted in frequent overprediction of the length of the initial truncation. As fluctuations are an innate part of turbulence data, these techniques performed poorly on turbulence quantities, meaning that the local minima was often wrongly interpreted as the minimum variance in the time series and resulted in different transient point predictions for any increments to the sample size. This limitation was overcome by considering the finite difference of the slope of the variance computed in the optimization algorithm. The start of the zero slope region was considered as the initial transient truncation point. This modification to the standard approach eliminated the sensitivity of the scheme, and led to consistent estimates of the transient truncation point, provided that the finite difference time interval was chosen large enough to cover the fluctuations in the time series. Therefore, the step size for the finite difference slope was computed using both visual inspection of the time series and trial and error. We propose the Augmented Dickey­Fuller test as an automatic and reliable method to determine the truncation point, from which the time series is considered stationary and without an initialization bias