scholarly journals A Hybrid Analytics Paradigm Combining Physics-Based Modeling and Data-Driven Modeling to Accelerate Incompressible Flow Solvers

Fluids ◽  
2018 ◽  
Vol 3 (3) ◽  
pp. 50 ◽  
Author(s):  
Sk. Rahman ◽  
Adil Rasheed ◽  
Omer San
Keyword(s):  
Poisson Equation ◽  
Incompressible Flow ◽  
Data Exchange ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Time Integration ◽  
Data Driven ◽  
Integration Algorithm ◽  
Reduced Order ◽  
Advection Diffusion Equation

Numerical solution of the incompressible Navier–Stokes equations poses a significant computational challenge due to the solenoidal velocity field constraint. In most computational modeling frameworks, this divergence-free constraint requires the solution of a Poisson equation at every step of the underlying time integration algorithm, which constitutes the major component of the computational expense. In this study, we propose a hybrid analytics procedure combining a data-driven approach with a physics-based simulation technique to accelerate the computation of incompressible flows. In our approach, proper orthogonal basis functions are generated to be used in solving the Poisson equation in a reduced order space. Since the time integration of the advection–diffusion equation part of the physics-based model is computationally inexpensive in a typical incompressible flow solver, it is retained in the full order space to represent the dynamics more accurately. Encoder and decoder interface conditions are provided by incorporating the elliptic constraint along with the data exchange between the full order and reduced order spaces. We investigate the feasibility of the proposed method by solving the Taylor–Green vortex decaying problem, and it is found that a remarkable speed-up can be achieved while retaining a similar accuracy with respect to the full order model.

Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow

1969 ◽  
Vol 37 (4) ◽  
pp. 727-750 ◽  
Author(s):  
Gareth P. Williams
Keyword(s):  
Poisson Equation ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Wave Flow ◽  
Trigonometric Interpolation ◽  
Time Step ◽  
Navier Stokes Equations ◽  
The Poisson Equation

A method of numerically integrating the Navier-Stokes equations for certain three-dimensional incompressible flows is described. The technique is presented through application to the particular problem of describing thermal convection in a rotating annulus. The equations, in cylindrical polar co-ordinate form, are integrated with respect to time by a marching process, together with the solving of a Poisson equation for the pressure. A suitable form of the finite difference equations gives a computationally-stable long-term integration with reasonably faithful representation of the spatial and temporal characteristics of the flow.Trigonometric interpolation techniques provide accurate (discretely exact) solutions to the Poisson equation. By using an auxiliary algorithm for rapid evaluation of trigonometric transforms, the proportion of computation needed to solve the Poisson equation can be reduced to less than 25% of the total time needed to’ advance one time step. Computing on a UNIVAC 1108 machine, the flow can be advanced one time-step in 2 sec for a 14 × 14 × 14 grid upward to 96 sec for a 60 × 34 × 34 grid.As an example of the method, some features of a solution for steady wave flow in annulus convection are presented. The resemblance of this flow to the classical Eady wave is noted.

Data-driven construction of a reduced-order model for supersonic boundary layer transition

2019 ◽  
Vol 874 ◽  
pp. 1096-1114 ◽  
Author(s):  
Ming Yu ◽  
Wei-Xi Huang ◽  
Chun-Xiao Xu
Keyword(s):  
Compressible Flows ◽  
Incompressible Flows ◽  
Temporal Evolution ◽  
Reduced Order Model ◽  
Boundary Layer Transition ◽  
Galerkin Projection ◽  
Data Driven ◽  
Dynamic Mode ◽  
Order Model ◽  
Reduced Order

In this study, a data-driven method for the construction of a reduced-order model (ROM) for complex flows is proposed. The method uses the proper orthogonal decomposition (POD) modes as the orthogonal basis and the dynamic mode decomposition method to obtain linear equations for the temporal evolution coefficients of the modes. This method eliminates the need for the governing equations of the flows involved, and therefore saves the effort of deriving the projected equations and proving their consistency, convergence and stability, as required by the conventional Galerkin projection method, which has been successfully applied to incompressible flows but is hard to extend to compressible flows. Using a sparsity-promoting algorithm, the dimensionality of the ROM is further reduced to a minimum. The ROMs of the natural and bypass transitions of supersonic boundary layers at $Ma=2.25$ are constructed by the proposed data-driven method. The temporal evolution of the POD modes shows good agreement with that obtained by direct numerical simulations in both cases.

Discrete Energy-Conservation Properties in the Numerical Simulation of the Navier–Stokes Equations

2019 ◽  
Vol 71 (1) ◽  
Author(s):  
Gennaro Coppola ◽  
Francesco Capuano ◽  
Luigi de Luca
Keyword(s):  
Energy Conservation ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Time Integration ◽  
Navier Stokes ◽  
Discrete Energy ◽  
Navier Stokes Equations ◽  
Fundamental Symmetry ◽  
Symmetry Properties ◽  
High Reynolds

Nonlinear convective terms pose the most critical issues when a numerical discretization of the Navier–Stokes equations is performed, especially at high Reynolds numbers. They are indeed responsible for a nonlinear instability arising from the amplification of aliasing errors that come from the evaluation of the products of two or more variables on a finite grid. The classical remedy to this difficulty has been the construction of difference schemes able to reproduce at a discrete level some of the fundamental symmetry properties of the Navier–Stokes equations. The invariant character of quadratic quantities such as global kinetic energy in inviscid incompressible flows is a particular symmetry, whose enforcement typically guarantees a sufficient control of aliasing errors that allows the fulfillment of long-time integration. In this paper, a survey of the most successful approaches developed in this field is presented. The incompressible and compressible cases are both covered, and treated separately, and the topics of spatial and temporal energy conservation are discussed. The theory and the ideas are exposed with full details in classical simplified numerical settings, and the extensions to more complex situations are also reviewed. The effectiveness of the illustrated approaches is documented by numerical simulations of canonical flows and by industrial flow computations taken from the literature.

Numerical solution of the reduced Navier-Stokes equations for internal incompressible flows

AIAA Journal ◽  
2000 ◽  
Vol 38 ◽  
pp. 1603-1614
Author(s):  
Martin Scholtysik ◽  
Bernhard Mueller ◽  
Torstein K. Fannelop
Keyword(s):  
Numerical Solution ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations

Lattice-BGK Methods for Simulating Incompressible Fluid Flows

1997 ◽  
Vol 08 (04) ◽  
pp. 793-803 ◽  
Author(s):  
Yu Chen ◽  
Hirotada Ohashi
Keyword(s):  
Fluid Flow ◽  
Incompressible Fluid ◽  
Poisson Equation ◽  
General Model ◽  
Incompressible Flows ◽  
Fluid Flows ◽  
Incompressible Limit ◽  
Numerical Example ◽  
Incompressible Fluid Flows

The lattice-Bhatnagar-Gross-Krook (BGK) method has been used to simulate fluid flow in the nearly incompressible limit. But for the completely incompressible flows, two special approaches should be applied to the general model, for the steady and unsteady cases, respectively. Introduced by Zou et al.,1 the method for steady incompressible flows will be described briefly in this paper. For the unsteady case, we will show, using a simple numerical example, the need to solve a Poisson equation for pressure.

Machine learning–based reduced-order modeling of hydrodynamic forces using pressure mode decomposition

2021 ◽  
pp. 095441002199986
Author(s):  
Hassan F Ahmed ◽  
Hamayun Farooq ◽  
Imran Akhtar ◽  
Zafar Bangash
Keyword(s):  
Machine Learning ◽  
Short Term Memory ◽  
Stokes Equations ◽  
Decomposition Analysis ◽  
Hydrodynamic Forces ◽  
Reduced Order Modeling ◽  
Navier Stokes ◽  
Reduced Order ◽  
Mode Decomposition ◽  
Lift And Drag

In this article, we introduce a machine learning–based reduced-order modeling (ML-ROM) framework through the integration of proper orthogonal decomposition (POD) and deep neural networks (DNNs), in addition to long short-term memory (LSTM) networks. The DNN is utilized to upscale POD temporal coefficients and their respective spatial modes to account for the dynamics represented by the truncated modes. In the second part of the algorithm, temporal evolution of the POD coefficients is obtained by recursively predicting their future states using an LSTM network. The proposed model (ML-ROM) is tested for flow past a circular cylinder characterized by the Navier–Stokes equations. We perform pressure mode decomposition analysis on the flow data using both POD and ML-ROM to predict hydrodynamic forces and demonstrate the accuracy of the proposed strategy for modeling lift and drag coefficients.

Data-driven reduced order modeling based on tensor decompositions and its application to air-wall heat transfer in buildings

SeMA Journal ◽  
2021 ◽  
Author(s):  
M. Azaïez ◽  
T. Chacón Rebollo ◽  
M. Gómez Mármol ◽  
E. Perracchione ◽  
A. Rincón Casado ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Reduced Order Modeling ◽  
Data Driven ◽  
Wall Heat Transfer ◽  
Reduced Order ◽  
Tensor Decompositions

scholarly journals Data-Driven Reduced-Order Modeling of Convective Heat Transfer in Porous Media

Fluids ◽  
2021 ◽  
Vol 6 (8) ◽  
pp. 266
Author(s):  
Péter German ◽  
Mauricio E. Tano ◽  
Carlo Fiorina ◽  
Jean C. Ragusa
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Porous Media ◽  
Steady State ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Data Driven ◽  
Reduced Order ◽  
Medium Formulation ◽  
Flow Resistances

This work presents a data-driven Reduced-Order Model (ROM) for parametric convective heat transfer problems in porous media. The intrusive Proper Orthogonal Decomposition aided Reduced-Basis (POD-RB) technique is employed to reduce the porous medium formulation of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations coupled with heat transfer. Instead of resolving the exact flow configuration with high fidelity, the porous medium formulation solves a homogenized flow in which the fluid-structure interactions are captured via volumetric flow resistances with nonlinear, semi-empirical friction correlations. A supremizer approach is implemented for the stabilization of the reduced fluid dynamics equations. The reduced nonlinear flow resistances are treated using the Discrete Empirical Interpolation Method (DEIM), while the turbulent eddy viscosity and diffusivity are approximated by adopting a Radial Basis Function (RBF) interpolation-based approach. The proposed method is tested using a 2D numerical model of the Molten Salt Fast Reactor (MSFR), which involves the simulation of both clean and porous medium regions in the same domain. For the steady-state example, five model parameters are considered to be uncertain: the magnitude of the pumping force, the external coolant temperature, the heat transfer coefficient, the thermal expansion coefficient, and the Prandtl number. For transient scenarios, on the other hand, the coastdown-time of the pump is the only uncertain parameter. The results indicate that the POD-RB-ROMs are suitable for the reduction of similar problems. The relative L2 errors are below 3.34% for every field of interest for all cases analyzed, while the speedup factors vary between 54 (transient) and 40,000 (steady-state).

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