A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

1. Introduction.- 1 Fundamentals of the Theory of Digital Signal Processing.- 2. Elements of Signal Theory.- 2.1 Signals as Mathematical Functions.- 2.2 Signal Space.- 2.3 The Most Common Systems of Basis Functions.- 2.3.1 Impulse Basis Functions.- 2.3.2 Harmonic Functions.- 2.3.3 Walsh Functions.- 2.3.4 Haar Functions.- 2.3.5 Sampling Functions.- 2.4 Continuous Representations of Signals.- 2.5 Description of Signal Transformations.- 2.5.1 Linear Transformations.- 2.5.2 Nonlinear Element-by-Element Transforms.- 2.6 Representation of Linear Transformations with Respect to Discrete Bases.- 2.6.1 Representation Using Vector Responses.- 2.6.2 Matrix Representations.- 2.6.3 Representation of Operators by Means of Their Eigenfunctions and Eigenvalues.- 2.7 Representing Operator with Respect to Continuous Bases.- 2.7.1 Operator Kernel.- 2.7.2 Description in Terms of Impulse Responses.- 2.7.3 Description Using Frequency Transfer Functions.- 2.7.4 Description with Input and Output Signals Referred to Different Bases.- 2.7.5 Description Using Eigenfunctions.- 2.8 Examples of Linear Operators.- 2.8.1 Shift-Invariant Filters.- 2.8.2 The Identity Operator.- 2.8.3 Shi ft Operator.- 2.8.4 Sampling Oparator.- 2.8.5 Gating Operator (Multiplier).- 3. Discretization and Quantization of Signals.- 3.1 Generalized Quantization.- 3.2 Concepts of Discretization and Element-by-Element Quantization.- 3.2.1 Discretization.- 3.2.2 Element-by-Element Quantization.- 3.3 The Sampling Theorem.- 3.4 Sampling Theory for Two-Dimensional Signals.- 3.5 Errors of Discretization and Restoration of Signals in Sampling Theory.- 3.6 Other Approaches to Discretization.- 3.7 Optimal Discrete Representation and Dimensionality of Signals.- 3.8 Element-by-Element Quantization.- 3.9 Examples of Optimum Quantization.- 3.9.1 Example: The Threshold Metric.- 3.9.2 Example: Power Criteria for the Absolute Value of the Quantization Error.- 3.9.3 Example: Power Criteria for the Relative Quantization Error.- 3.10 Quantization in the Presence of Noise. Quantization and Representation of Numbers in Digital Processors.- 3.11 Review of Picture-Coding Methods.- 4. Discrete Representations of Linear Transforms.- 4.1 Problem Formulation and General Approach.- 4.2 Discrete Representation of Shift-Invariant Filters for Band-Limited Signals.- 4.3 Digital Filters.- 4.4 Transfer Functions and Impulse Responses of Digital Filters.- 4.5 Boundary Effects in Digital Filtering.- 4.6 The Discrete Fourier Transform (DFT).- 4.7 Shifted, Odd and Even DFTs.- 4.8 Using Discrete Fourier Transforms.- 4.8.1 Calculating Convolutions.- 4.8.2 Signal Interpolation.- 4.9 Walsh and Similar Transforms.- 4.10 The Haar Tansform. Addition Elements of Matrix Calculus.- 4.11 Other Orthogonal Tansforms. General Representations. Review of Applications.- 5. Linear Transform Algorithms.- 5.1 Fast Algorithms of Discrete Orthogonal Transforms.- 5.2 Fast Haar Transform (FHT) Algorithms.- 5.3 Fast Walsh Transform (FWT) Algorithms.- 5.4 Fast Discrete Fourier Transform (FFT) Algorithms.- 5.5 Review of Other Fast Algorithms. Features of Two-Dimensional Transforms.- 5.5.1 Truncated FFT and FWT Algorithms.- 5.5.2 Transition Matrices Between Various Transforms.- 5.5.3 Calculation of Two-Dimensional Transforms.- 5.6 Combined DFT Algorithms.- 5.6.1 Combined DFT Algorithms of Real Sequences.- 5.6.2 Combined SDFT (1/2, 0) Algorithms of Even and Real Even Sequences.- 5.7 Recursive DFT Algorithms.- 5.8 Fast Algorithms for Calculating the DFT and Signal Convolution with Decreased Multiplication.- 6. Digital Statistical Methods.- 6.1 Principles of the Statistical Description of Pictures.- 6.2 Measuring the Grey-Level Distribution.- 6.2.1 Step Smoothing.- 6.2.2 Smoothing by Sliding Summation.- 6.2.3 Smoothing with Orthogonal Transforms.- 6.3 The Estimation of Correlation Functions and Spectra.- 6.3.1 Averaging Local Spectra.- 6.3.2 Masking (Windowing) the Process by Smooth Functions.- 6.3.3 Direct Smoothing of Spectra.- 6.4 Generating Pseudorandom Numbers.- 6.5 Measuring Picture Noise.- 6.5.1 The Prediction Method.- 6.5.2 The Voting Method.- 6.5.3 Measuring the Variance and the Auto-Correlation Function of Additive Wideband Noise.- 6.5.4 Evaluation of the Intensity and Frequency of the Harmonic Components of Periodic Interference and Other Types of Interference with Narrow Spectra.- 6.5.5 Evaluation of the Parameters of Pulse Noise, Quantization Noise and Strip-Like Noise.- 2 Picture Processing.- 7. Correcting Imaging Systems.- 7.1 Problem Formulation.- 7.2 Suppression of Additive Noise by Linear Filtering.- 7.3 Filtering of Pulse Interference.- 7.4 Correction of Linear Distortion.- 7.5 Correction of Amplitude Characteristics.- 8. Picture Enhancement and Preparation.- 8.1 Preparation Problems and Visual Analysis of Pictures.- 8.1.1 Feature Processing.- 8.1.2 Geometric Transformations.- 8.2 Adaptive Quantization of Modes.- 8.3 Preparation by Nonlinear Transformation of the Video Signal Scale.- 8.4 Linear Preparation Methods.- 8.5 Methods of Constructing Graphical Representation: Computer Graphics.- 8.6 Geometric Picture Transformation.- 8.6.1 Bilinear Interpolation.- 8.6.2 Interpolation Using DFT and SDFT.- 9. Measuring the Coordinates of Objects in Pictures.- 9.1 Problem Formulation.- 9.2 Localizing a Precisely Known Object in a Spatially Homogeneous Picture.- 9.3 Uncertainty in the Object and Picture Inhomogeneity. Localization in "Blurred" Pictures.- 9.3.1 "Exhaustive" Estimator.- 9.3.2 Estimator Seeking an Averaged Object.- 9.3.3 Adjustable Estimator with Fragment-by Fraament Optimal Fi1tering.- 9.3.4 Non-Adjustable Estimator.- 9.3.5 Localization in Blurred and Noisy Pictures.- 9.4 Optimal Localization and Picture Contours. Choice of Reference Objects.- 9.5 Algorithm for the Automatic Detection and Extraction of Bench-Marks in Aerial and Space Photographs.- 10-Conclusion.- References.

Digital picture processing : an introduction

In this paper we introduce and analyze an algorithm for continuous data assimilation for a three-dimensional Brinkman-Forchheimer-extended Darcy (3D BFeD) model of porous media. This model is believed to be accurate when the flow velocity is too large for Darcy's law to be valid, and additionally the porosity is not too small. The algorithm is inspired by ideas developed for designing finite-parameters feedback control for dissipative systems. It aims to obtaining improved estimates of the state of the physical system by incorporating deterministic or noisy measurements and observations. Specifically, the algorithm involves a feedback control that nudges the large scales of the approximate solution toward those of the reference solution associated with the spatial measurements. In the first part of the paper, we present few results of existence and uniqueness of weak and strong solutions of the 3D BFeD system. The second part is devoted to the setting and convergence analysis of the data assimilation algorithm.

/pdf/continuous-data-assimilation-for-the-three-dimensional-2si7au2c7a.pdf

Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model

https://escholarship.org/content/qt7138b5ns/qt7138b5ns.pdf

Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements

In this paper we propose a continuous data assimilation (downscaling) algorithm for a two-dimensional Benard convection problem. Specifically we consider the two-dimensional Boussinesq system of a layer of incompressible fluid between two solid horizontal walls, with no-normal flow and stress-free boundary conditions on the walls, and the fluid is heated from the bottom and cooled from the top. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the horizontal velocity. We show that under an appropriate choice of the nudging parameter and the size of the spatial coarse mesh observables, and under the assumption that the observed data are error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed data on the horizontal component of the velocity.

/pdf/continuous-data-assimilation-for-a-2d-benard-convection-56ncwempkc.pdf

Continuous Data Assimilation for a 2D Bénard Convection System Through Horizontal Velocity Measurements Alone

We study the numerical performance of a continuous data assimilation (downscaling) algorithm, based on ideas from feedback control theory, in the context of the two-dimensional incompressible Navier– Stokes equations. Our model problem is to recover an unknown reference solution, asymptotically in time, by using continuous-in-time coarse-mesh nodal-point observational measurements of the velocity field of this reference solution (subsampling), as might be measured by an array of weather vane anemometers. Our calculations show that the required nodal observation density is remarkably less that what is suggested by the analytical study; and is in fact comparable to the number of numerically determining Fourier modes, which was reported in an earlier computational study by the authors. Thus, this method is computationally efficient and performs far better t the analytical estimates suggest.

/pdf/a-computational-study-of-a-data-assimilation-algorithm-for-jvc2beu907.pdf

A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier-Stokes Equations

Impact of mineralogy and wettability on pore-scale displacement of NAPLs in heterogeneous porous media

We study the numerical performance of a continuous data assimilation (downscaling) algorithm, based on ideas from feedback control theory, in the context of the two-dimensional incompressible Navier--Stokes equations. Our model problem is to recover an unknown reference solution, asymptotically in time, by using continuous-in-time coarse-mesh nodal-point observational measurements of the velocity field of this reference solution (subsampling), as might be measured by an array of weather vane anemometers. Our calculations show that the required nodal observation density is remarkably less that what is suggested by the analytical study; and is in fact comparable to the {\it number of numerically determining Fourier modes}, which was reported in an earlier computational study by the authors. Thus, this method is computationally efficient and performs far better than the analytical estimates suggest.

A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier--Stokes Equations

This study proposes a novel approach to investigate the diffusion-limited deposition of asphaltenes in flow lines to better understand their nanoscale behavior at interfaces and aid in the developm...

Masakazu Gesho

Papers

A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier-Stokes Equations

Impact of mineralogy and wettability on pore-scale displacement of NAPLs in heterogeneous porous media

A Computational Study of a Data Assimilation Algorithm for the Two-dimensional Navier--Stokes Equations

Nanoscale Investigation of Asphaltene Deposition under Capillary Flow Conditions

Auto-segmentation technique for SEM images using machine learning: Asphaltene deposition case study.