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Register a new userA turbulence model based on deep neural network considering the near-wall effect
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There exists continuous demand of improved turbulence models for the closure of Reynolds Averaged Navier-Stokes (RANS) simulations. Machine Learning (ML) offers effective tools for establishing advanced empirical Reynolds stress closures on the basis of high fidelity simulation data. This paper presents a turbulence model based on the Deep Neural Network(DNN) which takes into account the non-linear relationship between the Reynolds stress anisotropy tensor and the local mean velocity gradient as well as the near-wall effect. The construction and the tuning of the DNN-turbulence model are detailed. We show that the DNN-turbulence model trained on data from direct numerical simulations yields an accurate prediction of the Reynolds stresses for plane channel flow. In particular, we propose including the local turbulence Reynolds number in the model input.
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Modelling the near-wall region of wall-bounded turbulent flows is a widespread practice to reduce the computational cost of large-eddy simulations (LESs) at high Reynolds number. As a first step towards a data-driven wall-model, a neural-network-based approach to predict the near-wall behaviour in a turbulent open channel flow is investigated. The fully-convolutional network (FCN) proposed by Guastoni et al. [preprint, arXiv:2006.12483] is trained to predict the two-dimensional velocity-fluctuation fields at $y^{+}_{rm target}$, using the sampled fluctuations in wall-parallel planes located farther from the wall, at $y^{+}_{rm input}$. The data for training and testing is obtained from a direct numerical simulation (DNS) at friction Reynolds numbers $Re_{tau} = 180$ and $550$. The turbulent velocity-fluctuation fields are sampled at various wall-normal locations, i.e. $y^{+} = {15, 30, 50, 80, 100, 120, 150}$. At $Re_{tau}=550$, the FCN can take advantage of the self-similarity in the logarithmic region of the flow and predict the velocity-fluctuation fields at $y^{+} = 50$ using the velocity-fluctuation fields at $y^{+} = 100$ as input with less than 20% error in prediction of streamwise-fluctuations intensity. These results are an encouraging starting point to develop a neural-network based approach for modelling turbulence at the wall in numerical simulations.
A new velocity scale is derived that yields a Reynolds number independent profile for the streamwise turbulent fluctuations in the near-wall region of wall bounded flows for $y^+<25$. The scaling demonstrates the important role played by the wall shear stress fluctuations and how the large eddies determine the Reynolds number dependence of the near-wall turbulence distribution.
Despite the nonlinear nature of turbulence, there is evidence that part of the energy-transfer mechanisms sustaining wall turbulence can be ascribed to linear processes. The different scenarios stem from linear stability theory and comprise exponential instabilities, neutral modes, transient growth from non-normal operators, and parametric instabilities from temporal mean-flow variations, among others. These mechanisms, each potentially capable of leading to the observed turbulence structure, are rooted in theoretical and conceptual arguments. Whether the flow follows any or a combination of them remains elusive. Here, we evaluate the linear mechanisms responsible for the energy transfer from the streamwise-averaged mean-flow ($bf U$) to the fluctuating velocities ($bf u$). We use cause-and-effect analysis based on interventions. This is achieved by direct numerical simulation of turbulent channel flows at low Reynolds number, in which the energy transfer from $bf U$ to $bf u$ is constrained to preclude a targeted linear mechanism. We show that transient growth is sufficient for sustaining realistic wall turbulence. Self-sustaining turbulence persists when exponential instabilities, neutral modes, and parametric instabilities of the mean flow are suppressed. We further show that a key component of transient growth is the Orr/push-over mechanism induced by spanwise variations of the base flow. Finally, we demonstrate that an ensemble of simulations with various frozen-in-time $bf U$ arranged so that only transient growth is active, can faithfully represent the energy transfer from $bf U$ to $bf u$ as in realistic turbulence. Our approach provides direct cause-and-effect evaluation of the linear energy-injection mechanisms from $bf U$ to $bf u$ in the fully nonlinear system and simplifies the conceptual model of self-sustaining wall turbulence.
Deformation-induced lateral migration of a bubble slowly rising near a vertical plane wall in a stagnant liquid is numerically and theoretically investigated. In particular, our focus is set on a situation with a small clearance $c$ between the bubble interface and the wall. Motivated by the fact that experimentally measured migration velocity (Takemura et al. (2002, J. Fluid Mech. {bf 461}, 277)) is higher than the velocity estimated by the available analytical solution (Magnaudet et al. (2003, J. Fluid Mech. {bf 476}, 115)) using the Fax{e}n mirror image technique for $kappa(=a/(a+c))ll 1$ (here $a$ is the bubble radius), when the clearance parameter $epsilon(=c/a)$ is comparable to or smaller than unit, the numerical analysis based on the boundary-fitted finite-difference approach by solving the Stokes equation is performed to complement the experiment. To improve the understandings of a role of the squeezing flow within the bubble-wall gap, the theoretical analysis based on a soft-lubrication approach (Skotheim & Mahadevan (2004, Phys. Rev. Lett. {bf 92}, 245509)) is also performed. The present analyses demonstrate the migration velocity scales $propto{rm Ca} epsilon^{-1}V_{B1}$ (here, $V_{B1}$ and ${rm Ca}$ denote the rising velocity and the capillary number, respectively) in the limit of $epsilonto 0$.
This paper reviews results from the study of wall-bounded turbulent flows using statistical state dynamics (SSD) that demonstrate the benefits of adopting this perspective for understanding turbulence in wall-bounded shear flows. The SSD approach used in this work employs a second-order closure which isolates the interaction between the streamwise mean and the equivalent of the perturbation covariance. This closure restricts nonlinearity in the SSD to that explicitly retained in the streamwise constant mean together with nonlinear interactions between the mean and the perturbation covariance. This dynamical restriction, in which explicit perturbation-perturbation nonlinearity is removed from the perturbation equation, results in a simplified dynamics referred to as the restricted nonlinear (RNL) dynamics. RNL systems in which an ensemble of a finite number of realizations of the perturbation equation share the same mean flow provide tractable approximations to the equivalently infinite ensemble RNL system. The infinite ensemble system, referred to as the S3T, introduces new analysis tools for studying turbulence. The RNL with a single ensemble member can be alternatively viewed as a realization of RNL dynamics. RNL systems provide computationally efficient means to approximate the SSD, producing self-sustaining turbulence exhibiting qualitative features similar to those observed in direct numerical simulations (DNS) despite its greatly simplified dynamics. Finally, we show that RNL turbulence can be supported by as few as a single streamwise varying component interacting with the streamwise constant mean flow and that judicious selection of this truncated support, or band-limiting, can be used to improve quantitative accuracy of RNL turbulence. The results suggest that the SSD approach provides new analytical and computational tools allowing new insights into wall-turbulence.
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