Invariant subspaces on multiply connected domains

The lattice of invariant subspaces of several Banach spaces of analytic functions on the unit disk, for example the Bergman spaces and the Dirichlet spaces, have been studied recently. A natural question is to what extent these investigations carry over to analogously defined spaces on an annulus. We consider this question in the context of general Banach spaces of analytic functions on finitely connected domains Ω. The main result reads as follows: Assume that B is a Banach space of analytic functions satisfying some conditions on the domain Ω. Assume further that M(B) is the set of all multipliers of B. Let Ω1 be a domain obtained from Ω by adding some of the bounded connectivity components of C\Ω. Also, let B1 be the closed subspace of B of all functions that extend analytically to Ω1. Then the mapping I 7→ clos(I · M(B)) gives a one-to-one correspondence between a class of multiplier invariant subspaces I of B1, and a class of multiplier invariant subspaces J of B. The inverse mapping is given by J 7→ J ∩ B1

A survey of machine learning wall models for large eddy simulation

This survey investigates wall modeling in large eddy simulations (LES) using data-driven machine learning (ML) techniques. To this end, we implement three ML wall models in an open-source code and compare their performances with the equilibrium wall model in LES of half-channel flow at eleven friction Reynolds numbers between $180$ and $10^{10}$. The three models have ''seen'' flows at only a few Reynolds numbers. We test if these ML wall models can extrapolate to unseen Reynolds numbers. Among the three models, two are supervised ML models, and one is a reinforcement learning ML model. The two supervised ML models are trained against direct numerical simulation (DNS) data, whereas the reinforcement learning ML model is trained in the context of a wall-modeled LES with no access to high-fidelity data. The two supervised ML models capture the law of the wall at both seen and unseen Reynolds numbers--although one model requires re-training and predicts a smaller von K\'arm\'an constant. The reinforcement learning model captures the law of the wall reasonably well but has errors at both low ($Re_\tau<10^3$) and high Reynolds numbers ($Re_\tau>10^6$). In addition to documenting the results, we try to ''understand'' why the ML models behave the way they behave. Analysis shows that the errors of the supervised ML model is a result of the network design and the errors in the reinforcement learning model arise due to the present choice of the ''states'' and the mismatch between the neutral line and the line separating the action map. In all, we see promises in data-driven machine learning models

Bathymetric modelin from satellite imagery via Single Band Algorithm (SBA) and Principal Components Analysis (PCA) in southern Caspian Sea

Remotely sensed imagery is proving to be a useful tool to estimate water depths in coastal zones. Bathymetric algorithms attempt to isolate water attenuation and hence depth from other factors by using different combinations of spectral bands. In this research, images of absolute bathymetry using two different but related methods in a region in the southern Caspian Sea coasts has been produced. The first method used a Single Band Algorithm (SBA) and assumed a constant water attenuation coefficient throughout the blue band. The second method used Principal Components Analysis (PCA) to adjust for varying water attenuation coefficients without additional ground truth data. PCA method (r=-0.672394) appears to match our control points slightly better than single band algorithm (r=-0.645404). It is clear that both methods can be used as rough estimates of bathymetry for many coastal zone studies in the southern Caspian Sea such as near shore fisheries, coastal erosion, water quality, recreation siting and so forth. The presented methodology can be considered as the first step toward mapping bathymetry in the southern Caspian Sea. Further research must investigate the determination of the nonlinear optimization techniques as well as the assessment of these models’ performance in the study area

Log-law recovery through reinforcement-learning wall model for large-eddy simulation

This paper focuses on the use of reinforcement learning (RL) as a machine-learning (ML) modeling tool for near-wall turbulence. RL has demonstrated its effectiveness in solving high-dimensional problems, especially in domains such as games. Despite its potential, RL is still not widely used for turbulence modeling and is primarily used for flow control and optimization purposes. A new RL wall model (WM) called VYBA23 is developed in this work, which uses agents dispersed in the flow near the wall. The model is trained on a single Reynolds number ($Re_\tau = 10^4$) and does not rely on high-fidelity data, as the back-propagation process is based on a reward rather than output error. The states of the RLWM, which are the representation of the environment by the agents, are normalized to remove dependence on the Reynolds number. The model is tested and compared to another RLWM (BK22) and to an equilibrium wall model, in a half-channel flow at eleven different Reynolds numbers ($Re_\tau \in [180;10^{10}]$). The effects of varying agents' parameters such as actions range, time-step, and spacing are also studied. The results are promising, showing little effect on the average flow field but some effect on wall-shear stress fluctuations and velocity fluctuations. This work offers positive prospects for developing RLWMs that can recover physical laws, and for extending this type of ML models to more complex flows in the future.Comment: arXiv admin note: text overlap with arXiv:2211.0361

Influence of vertical distribution of phytoplankton on remote sensing signal of Case II waters : southern Caspian Sea case study

Reliable monitoring of coastal waters is not possible without using remote sensing data. On the other hand, it is quite difficult to develop remote sensing algorithms that allow one to retrieve water characteristics (like chlorophyll-a concentration) in optically complex coastal and inland waters (called also Case II waters) as the concentrations of optically active substances (phytoplankton, suspended matter, and colored dissolved organic matter) vary independently from each other and the range of variability is often high. Another problem related to developing remote sensing algorithms for retrieving concentrations of optically active substances in such complex waters is vertical distribution of these substances. For example, phytoplankton distribution in the water column is often characterized with maxima just below the surface mixed layer, and some phytoplankton species even have the capability to migrate in the water column and tend to form layers at depths optimal for their growth. Twenty-three field campaigns were performed during the spring-summer period in the coastal waters of the southern Caspian Sea where vertical distribution of phytoplankton was measured by means of chlorophyll-a fluorometer. There results showed that there is usually a chlorophyll-a maximum between 10 and 20 m where the concentration is about one order of magnitude higher than in the top mixed layer. The Hydrolight 5.0 radiative transfer model used to estimate if the vertical distribution of biomass have detectable impact on remote sensing signal in these waters. For that purpose, several stations with distinctly different chlorophyll-a profiles were selected and two simulations for each of those measuring stations was carried out. First the Hydrolight was run with the actual chlorophyll-a vertical distribution profile and second a constant chlorophyll-a value (taken as an average of measured chlorophyll-a in the surface layer) was used in the model simulation. The modelling results show that the “deep” chlorophyll maximum has negligible effect on the remote sensing reflectance spectra. Consequently, there is no need to take into account the vertical distribution of phytoplankton while developing remote sensing algorithms for the Caspian Sea coastal water

Progressive augmentation of Reynolds stress tensor models for secondary flow prediction by computational fluid dynamics driven surrogate optimisation

Generalisability and the consistency of the a posteriori results are the most critical points of view regarding data-driven turbulence models. This study presents a progressive improvement of turbulence models using simulation-driven surrogate optimisation based on Kriging. We aim for the augmentation of secondary-flow reconstruction capability in a linear eddy-viscosity model without violating its original performance on canonical cases e.g. channel flow. Explicit algebraic Reynolds stress correction models (EARSCMs) for $k-\omega$ SST turbulence model are obtained to predict the secondary flow which the standard model fails to capture. The optimisation of the models is achieved by a multi-objective approach based on duct flow quantities, and numerical verification of the developed models is performed for various test cases. The results of testing new models on channel flow cases guarantee that new models preserve the performance of the original $k-\omega$ SST model. Regarding the generalisability of the new models, results of unseen test cases demonstrate a significant improvement in the prediction of secondary flows and streamwise velocity. These results highlight the potential of the progressive approach to enhance the performance of data-driven turbulence models for fluid flow simulation while preserving the robustness and stability of the solver.Comment: 23 pages, 20 figure

Extension of the law of the wall exploiting the concepts of strong and weak universality of velocity fluctuations in a channel

This paper explores the universality of the instantaneous velocity profiles in a channel, which, despite the equilibrium nature of a channel flow, are subjected to strong non-equilibrium effects. In the analysis, we employ a one-dimensional scalar variant of the proper orthogonal decomposition (POD) and exploit the concepts of strong and weak universalities. Strong universality requires that all POD modes are universal with respect to the Reynolds number, while weak universality only requires that the first few POD modes are universal. As POD analysis concerns information at more than one location, these universalities are more general than various similarities and universality in the literature concerning single-point flow statistics, e.g., outer layer similarity or universality of the log law. We examine flows at $Re_\tau=$180, 540, 1000, and 5200. Strong universality is observed in the outer layer, and weak universality is found in both the inner layer and the outer part of the logarithmic layer. The presence of weak universality suggests the existence of an extension to the law of the wall (LoW). Here, we propose such an extension based on the results from one-dimensional POD analysis. The usefulness of the LoW extension is assessed by comparing flow reconstructions according to the extended LoW and the equilibrium LoW. We show that the extended LoW provides strikingly accurate flow reconstruction in the wall layer across a wide range of Reynolds numbers, capturing fine-scale motions that are entirely missed by the equilibrium LoW.Comment: 20 pages, 16 figure