Stochastic turbulence modeling in RANS simulations via Multilevel Monte Carlo

A multilevel Monte Carlo (MLMC) method for quantifying model-form uncertainties associated with the Reynolds-Averaged Navier-Stokes (RANS) simulations is presented. Two, high-dimensional, stochastic extensions of the RANS equations are considered to demonstrate the applicability of the MLMC method. The first approach is based on global perturbation of the baseline eddy viscosity field using a lognormal random field. A more general second extension is considered based on the work of [Xiao et al.(2017)], where the entire Reynolds Stress Tensor (RST) is perturbed while maintaining realizability. For two fundamental flows, we show that the MLMC method based on a hierarchy of meshes is asymptotically faster than plain Monte Carlo. Additionally, we demonstrate that for some flows an optimal multilevel estimator can be obtained for which the cost scales with the same order as a single CFD solve on the finest grid level.Comment: 40 page

Stochastic turbulence modeling in RANS simulations via multilevel Monte Carlo

A multilevel Monte Carlo (MLMC) method for quantifying model-form uncertainties associated with the Reynolds-Averaged Navier-Stokes (RANS) simulations is presented. Two, high-dimensional, stochastic extensions of the RANS equations are considered to demonstrate the applicability of the MLMC method. The first approach is based on global perturbation of the baseline eddy viscosity field using a lognormal random field. A more general second extension is considered based on the work of [Xiao et al. (2017)], where the entire Reynolds Stress Tensor (RST) is perturbed while maintaining realizability. For two fundamental flows, we show that the MLMC method based on a hierarchy of meshes is asymptotically faster than plain Monte Carlo. Additionally, we demonstrate that for some flows an optimal multilevel estimator can be obtained for which the cost scales with the same order as a single CFD solve on the finest grid level

Estimation of Model Error Using Bayesian Model-Scenario Averaging with Maximum a Posterori-Estimates

International audienceThe lack of an universal modelling approach for turbulence in Reynolds-Averaged Navier–Stokes simulations creates the need for quantifying the modelling error without additional validation data. Bayesian Model-Scenario Averaging (BMSA), which exploits the variability on model closure coefficients across several flow scenarios and multiple models, gives a stochastic, a posteriori estimate of a quantity of interest. The full BMSA requires the propagation of the posterior probability distribution of the closure coefficients through a CFD code, which makes the approach infeasible for industrial relevant flow cases. By using maximum a posteriori (MAP) estimates on the posterior distribution, we drastically reduce the computational costs. The approach is applied to turbulent flow in a pipe at Re= 44,000 over 2D periodic hills at Re=5600, and finally over a generic falcon jet test case (Industrial challenge IC-03 of the UMRIDA project)

Updating the QR decomposition of block tridiagonal and block Hessenberg matrices

Abstract We present an efficient block-wise update scheme for the QR decomposition of block tridiagonal and block Hessenberg matrices. For example, such matrices come up in generalizations of the Krylov space solvers MinRes, SymmLQ, GMRes, and QMR to block methods for linear systems of equations with multiple right-hand sides. In the non-block case it is very efficient (and, in fact, standard) to use Givens rotations for these QR decompositions. Normally, the same approach is also used with column-wise updates in the block case. However, we show that, even for small block sizes, block-wise updates using (in general, complex) Householder reflections instead of Givens rotations are far more efficient in this case, in particular if the unitary transformations that incorporate the reflections determined by a whole block are computed explicitly. Naturally, the bigger the block size the bigger the savings. We discuss the somewhat complicated algorithmic details of this block-wise update, and present numerical experiments on accuracy and timing for the various options (Givens vs. Householder, block-wise vs. column-wise update, explicit vs. implicit computation of unitary transformations). Our treatment allows variable block sizes and can be adapted to block Hessenberg matrices that do not have the special structure encountered in the above mentioned block Krylov space solvers

Bayesian Predictions of Reynolds-Averaged Navier–Stokes Uncertainties Using Maximum a Posteriori Estimates

Computational fluid dynamics analyses of high-Reynolds-number flows mostly rely on the Reynolds-averaged Navier–Stokes equations. The associated closure models are based on multiple simplifying assumptions and involve numerous empirical closure coefficients, which are calibrated on a set of simple reference flows. Predicting new flows using a single closure model with nominal values for the closure coefficients may lead to biased predictions. Bayesian model-scenario averaging is a statistical technique providing an optimal way to combine the predictions of several competing models calibrated on various sets of data (scenarios). The method allows a stochastic estimate of a quantity of interest in an unmeasured prediction scenario to be obtain by 1) propagating posterior probability distributions of the parameters obtained for multiple calibration scenarios, and 2) computing a weighted posterior predictive distribution. Although step 2 has a negligible computational cost, step 1 requires a large number of samples of the solver. To enable the application of the proposed approach to computationally expensive flow configurations, a modified formulation is used where a maximum posterior probability approximation is used to drastically reduce the computational burden. The predictive capability of the proposed simplified approach is assessed for two-dimensional separated and three-dimensional compressible flows

دور المشروعات الصغیرة والمتوسطة فى التنمیة الصناعیة لمصر

Micro, small and medium sized enterprises (M/SMEs) are a dynamic force for sustained economic growth and job creation. They are a valid, crucial component of a vibrant industrial society.M/SMEs stimulate private ownership and entrepreneurial skills; they are flexible and can adapt quickly to changing market demand and supply conditions; they generate employment, help diversify economic activities and make significant contribution to export and trade.Many small projects stalled, especially in light of a great revolution of January 25, 2011. The role of small and medium-sized enterprises in the industrial development of Egypt

Effect of metal ions on the physical properties of multilayers from hyaluronan and chitosan, and the adhesion, growth and adipogenic differentiation of multipotent mouse fibroblasts

[EN] Polyelectrolyte multilayers (PEMs) consisting of the polysaccharides hyaluronic acid (HA) as the polyanion and chitosan (Chi) as the polycation were prepared with layer-by-layer technique (LbL). The [Chi/HA](5) multilayers were exposed to solutions of metal ions (Ca2+, Co2+, Cu2+ and Fe3+). Binding of metal ions to [Chi/HA](5) multilayers by the formation of complexes with functional groups of polysaccharides modulates their physical properties and the bioactivity of PEMs with regard to the adhesion and function of multipotent murine C3H10T1/2 embryonic fibroblasts. Characterization of multilayer formation and surface properties using different analytical methods demonstrates changes in the wetting, surface potential and mechanical properties of multilayers depending on the concentration and type of metal ion. Most interestingly, it is observed that Fe3+ metal ions greatly promote adhesion and spreading of C3H10T1/2 cells on the low adhesive [Chi/HA](5) PEM system. The application of intermediate concentrations of Cu2+, Ca2+ and Co2+ as well as low concentrations of Fe3+ to PEMs also results in increased cell spreading. Moreover, it can be shown that complex formation of PEMs with Cu2+ and Fe3+ ions leads to increased metabolic activity in cells after 24 h and induces cell differentiation towards adipocytes in the absence of any additional adipogenic media supplements. Overall, complex formation of [Chi/HA](5) PEM with metal ions like Cu2+ and Fe3+ represents an interesting and cheap alternative to the use of growth factors for making cell-adhesive coatings and guiding stem cell differentiation on implants and scaffolds to regenerate connective-type of tissues.This work was part of the High-Performance Center Chemical and Biosystems Technology Halle/Leipzig, supported by the European Regional Development Fund (ERDF), and a grant to HK from the Martin Luther University Halle-Wittenberg for female PhD students. The work was further supported by the Fraunhofer Internal Programs under Grant No. Attract 069-608203 (CEHS). TG acknowledges the kind support by the Ministry of Science and Higher Education of the Russian Federation within the framework of state support for the creation and development of World-Class Research Centers ``Digital biodesign and personalized healthcare'' 075-15-2020926. GGF acknowledges funding by the State Research Agency. Ministry of Science and Innovation of Spain, grant PID2019106000RB-C21/AEI/10.13039/501100011033 project. We are grateful for the kind support by Christian Willems for the help in formatting and proof reading the manuscript.Kindi, H.; Menzel, M.; Heilmann, A.; Schmelzer, CEH.; Herzberg, M.; Fuhrmann, B.; Gallego-Ferrer, G.... (2021). Effect of metal ions on the physical properties of multilayers from hyaluronan and chitosan, and the adhesion, growth and adipogenic differentiation of multipotent mouse fibroblasts. Soft Matter. 17(36):8394-8410. https://doi.org/10.1039/d1sm00405k83948410173