A Triangle Based Finite Volume Method for the Integration of Lubrication’s Incompressible Bulk Flow Equations

2000 ◽  
Vol 123 (1) ◽  
pp. 118-124 ◽  
Author(s):  
Mihai Arghir ◽  
Jean Fre^ne
Keyword(s):  
Flow System ◽  
A Priori ◽  
Simple Algorithm ◽  
Bulk Flow ◽  
System Of Equations ◽  
Flow Equations ◽  
Application Range ◽  
Lubrication Equation ◽  
Laminar And Turbulent Flow ◽  
Control Volumes

It is well known that for a reduced Reynolds number Re*=ρVH/μs˙H/L greater than unity, inertia forces have a dominant effect in the transport equations, thus rendering the classical lubrication equation inapplicable. The so called “bulk flow” system of equations is then the appropriate mathematical model for describing the flow in bearing and seals operating at Re*⩾1. The difficulty in integrating this system of equations is that one has to deal with coupled pressure and velocity fields. Analytic methods have a very narrow application range so a numerical method has been proposed by Launder and Leschziner in 1978. It represents a natural extrapolation of the successful SIMPLE algorithm applied to the bulk flow system of equations. The algorithm used rectangular, staggered control volumes and represented the state of the art at that moment. In the present work we introduced a method using triangular control volumes. The basic advantage of triangles versus rectangles is that non rectangular domains can be dealt without any a priori limitation. The present paper is focused on the description of the discretized equations and of the solution algorithm. Validations for bearings and seals operating in incompressible, laminar and turbulent flow regime are finally proving the accuracy of the method.

scholarly journals An SNR Estimation Technique Based on Deep Learning

Electronics ◽  
2019 ◽  
Vol 8 (10) ◽  
pp. 1139 ◽  
Author(s):  
Kai Yang ◽  
Zhitao Huang ◽  
Xiang Wang ◽  
Fenghua Wang
Keyword(s):  
Deep Learning ◽  
Signal To Noise Ratio ◽  
A Priori ◽  
Intermediate Frequency ◽  
Estimation Technique ◽  
Signal To Noise ◽  
Snr Estimation ◽  
Application Range ◽  
Noise Ratio ◽  
Priori Information

Signal-to-noise ratio (SNR) is a priori information necessary for many signal processing algorithms or techniques. However, there are many problems exsisting in conventional SNR estimation techniques, such as limited application range of modulation types, narrow effective estimation range of signal-to-noise ratio, and poor ability to accommodate non-zero timing offsets and frequency offsets. In this paper, an SNR estimation technique based on deep learning (DL) is proposed, which is a non-data-aid (NDA) technique. Second and forth moment (M2M4) estimator is used as a benchmark, and experimental results show that the performance and robustness of the proposed method are better, and the applied ranges of modulation types is wider. At the same time, the proposed method is not only applicable to the baseband signal and the incoherent signal, but can also estimate the SNR of the intermediate frequency signal.

scholarly journals X. On Hamilton's numbers

1887 ◽  
Vol 178 ◽  
pp. 285-312
Keyword(s):  
General Equation ◽  
Minimum Degree ◽  
A Priori ◽  
Senior Author ◽  
System Of Equations ◽  
Absolute Minimum ◽  
Doctor Of Philosophy ◽  
The University ◽  
The Given ◽  
Trinomial Equation

In the year 1786 Erland Samuel Bring, Professor at the University of Lund in Sweden, showed how by an extension of the method of Tschirnhausen it was possible to deprive the general algebraical equation of the 5th degree of three of its terms without solving an equation higher than the 3rd degree. By a well-understood, however singular, academical fiction, this discovery was ascribed by him to one of his own pupils, a certain Sven Gustaf Sommelius, and embodied in a thesis humbly submitted to himself for approval by that pupil, as a preliminary to his obtaining his degree of Doctor of Philosophy in the University. The process for effecting this reduction seems to have been overlooked or forgotten, and was subsequently re-discovered many years later by Mr. Jerrard. In a report contained in the ‘Proceedings of the British Association’ for 1836, Sir William Hamilton showed that Mr. Jerrard was mistaken in supposing that the method was adequate to taking away more than three terms of the equation of the 5th degree, but supplemented this somewhat unnecessary refutation of a result, known à priori to be impossible, by an extremely valuable discussion of a question raised by Mr. Jerrard as to the number of variables required in order that any system of equations of given degrees in those variables shall admit of being satisfied without solving any equation of a degree higher than the highest of the given degrees. In the year 1886 the senior author of this memoir showed in a paper in Kronecker'e (better known as Crelle’s ) ‘Journal that the trinomial equation of the 5th degree, upon which by Bring’s method the general equation of that degree can be made to depend, has necessarily imagmaiy coefficients except in the case where four of the roots of the original equation are imaginary, and also pointed out method of obtaining the absolute minimum degree M of an equation from which an given number of specified terms can be taken away subject to the condition of no having to solve any equation of a degree higher than M. The numbers furnished be Hamilton’s method, it is to be observed, are not minima unless a more stringer condition than this is substituted, viz., that the system of equations which have to be resolved in order to take away the proposed terms shall be the simplest possible i. e ., of the lowest possible weight and not merely of the lowest order; in the memo: in ‘Crelle,’ above referred to, he has explained in what sense the words weight an order are here employed. He has given the name of Hamilton’s Numbers to these relative minima (minima, i. e ., in regard to weight) for the case where the terms to be taken away from the equation occupy consecutive places in it, beginning with the second.

scholarly journals Computation of the Jet-Wake Flow Structure in a Low Speed Centrifugal Impeller

1988 ◽  
Author(s):  
B. L. Lapworth ◽  
R. L. Elder
Keyword(s):  
Three Dimensional ◽  
Simple Algorithm ◽  
Design Flow ◽  
Wake Flow ◽  
Staggered Grid ◽  
Centrifugal Impeller ◽  
Low Speed ◽  
Flow Equations ◽  
System A ◽  
Speed Flow

The low speed flow through the shrouded de-Havilland Ghost centrifugal impeller is computed using an incompressible elliptic calculation procedure. The three dimensional viscous flow equations are solved using the SIMPLE algorithm in an arbitrary generalised coordinate system. A non-staggered grid arrangement is implemented in which pressure oscillations are eliminated using an amended pressure correction scheme. Flow computations are performed at ‘nominal’ low speed design and above design flow rates, and (on the coarse grids used in the calculations) good agreement is obtained with the experimentally observed jet-wake structure of the flow.

A Bulk-Flow Model of Angled Injection Lomakin Bearings

2002 ◽  
Author(s):  
Luis San Andre´s ◽  
Thomas Soulas ◽  
Florence Challier ◽  
Patrice Fayolle
Keyword(s):  
Flow Model ◽  
Control Volume ◽  
Dynamic Performance ◽  
Bulk Flow ◽  
Computational Method ◽  
Design Parameters ◽  
Dynamic Force ◽  
Injection Angle ◽  
Flow Equations ◽  
Force Coefficients

The paper introduces a bulk-flow model for prediction of the static and dynamic force coefficients of angled injection Lomakin bearings. The analysis accounts for the flow interaction between the injection orifices, the supply circumferential groove, and the thin film lands. A one control-volume model in the groove is coupled to a bulk-flow model within the film lands of the bearing. Bernoulli-type relationships provide closure at the flow interfaces. Flow turbulence is accounted for with shear stress parameters and Moody’s friction factors. The flow equations are solved numerically using a robust computational method. Comparisons between predictions and experimental results for a tangential-against-rotation injection water Lomakin bearing show the novel model predicts well the leakage and direct stiffness and damping coefficients. Computed cross-coupled stiffness coefficients follow the experimental trends for increasing rotor speeds and supply pressures, but quantitative agreement remains poor. A parameter investigation evidences the effects of the groove and land geometries on the Lomakin bearing flowrate and force coefficients. The orifice injection angle does not influence the bearing static performance, although it largely affects its stability characteristics through the evolution of the cross-coupled stiffnesses. The predictions confirm the promising stabilizing effect of the tangential-against-rotation injection configuration. Two design parameters, comprising the feed orifices area and groove geometry, define the static and dynamic performance of Lomakin bearing. The analysis also shows that the film land clearance and length have a larger impact on the Lomakin bearing rotordynamic behavior than its groove depth and length.

Design of Hybrid Hydrostatic/Hydrodynamic Journal Bearings for Optimum Self-Compensation Under Misaligning External Loads

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
L. F. Martinez Esparza ◽  
J. G. Cervantes de Gortari ◽  
E. J. Chicurel Uziel
Keyword(s):  
Calculated Data ◽  
Journal Bearings ◽  
Radial Clearance ◽  
Lubrication Equation ◽  
Exact Function ◽  
Laminar And Turbulent Flow ◽  
Bearing Design ◽  
Integral Differential Equations ◽  
The Mathematical Model ◽  
External Loads

A method to design hybrid hydrostatic/hydrodynamic journal bearings, with the criterion of optimized self-compensation under misaligning loads, is presented. An analysis considering laminar and turbulent flow of a Newtonian incompressible lubricant between the bearing and a misaligned shaft, with restricted lubricant supply to each recess, is discussed. The mathematical model considers the modified steady-state Reynolds lubrication equation, an exact function for the local bearing radial clearance with a misaligned shaft, the continuity integral–differential equations at the recess limits, and boundary conditions at the cavitation zone and outer limits. The finite-difference method was used, and a modular computer program was developed. The procedure follows a univariate search to determine the optimum size and position of recesses and therefore obtain the design with the maximum reactive moment under misaligning loads. A validation of the model was obtained comparing the results with experimental and calculated data from the literature. Results for a 4 + 4 LBP hybrid bearing design are presented.

Estimation of density, magnetization, and depth to source: A nonlinear and noniterative 3-D potential‐field method

Geophysics ◽  
1997 ◽  
Vol 62 (3) ◽  
pp. 814-830 ◽  
Author(s):  
Maurizio Fedi
Keyword(s):  
A Priori ◽  
Real Data ◽  
Field Method ◽  
Magnetic Data ◽  
System Of Equations ◽  
Source Depth ◽  
A Priori Information ◽  
Magnetization Direction ◽  
Priori Information ◽  
Magnetic Case

The depth to the top, or bottom, and the density of a 3-D homogeneous source can be estimated from its gravity or magnetic anomalies by using a priori information on the maximum and minimum source depths. For the magnetic case, the magnetization direction is assumed to be constant and known. The source is assumed to be within a layer of known depth to the top h and thickness t. A depth model, satisfying both the data and the a priori information is found, together with its associated density/magnetization contrast. The methodology first derives, from the measured data, a set of apparent densities [Formula: see text] (or magnetizations), which do not depend on the layer parameters h and t, but only on source thickness. A nonlinear system of equations based on [Formula: see text], with source thicknesses as unknowns, is constructed. To simplify the solution, a more practical system of equations is formed. Each equation depends on only one value of thickness. Solving for the thicknesses, taking into account the above a priori information, the source depth to the top (or to the bottom) is determined uniquely. Finally, the depth solutions allow a unit‐density gravity model to be computed, which is compared to the observed gravity to determine the density contrast. A similar procedure can be used for magnetic data. Tests on synthetic anomalies and on real data demonstrate the good performance of this method.

FREE-BOUNDARY PROBLEM OF THE ONE-DIMENSIONAL EQUATIONS FOR A VISCOUS AND HEAT-CONDUCTIVE GASEOUS FLOW UNDER THE SELF-GRAVITATION

2013 ◽  
Vol 23 (08) ◽  
pp. 1377-1419 ◽  
Author(s):  
MORIMICHI UMEHARA ◽  
ATUSI TANI
Keyword(s):  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Global Solvability ◽  
The Self ◽  
Fundamental Result ◽  
System Of Equations ◽  
One Dimensional ◽  
Unique Existence ◽  
The One

In this paper we consider a system of equations describing the one-dimensional motion of a viscous and heat-conductive gas bounded by the free-surface. The motion is driven by the self-gravitation of the gas. This system of equations, originally formulated in the Eulerian coordinate, is reduced to the one in a fixed domain by the Lagrangian-mass transformation. For smooth initial data we first establish the temporally global solvability of the problem based on both the fundamental result for local in time and unique existence of the classical solution and a priori estimates of its solution. Second it is proved that some estimates of the global solution are independent of time under a certain restricted, but physically plausible situation. This gives the fact that the solution does not blow up even if time goes to infinity under such a situation. Simultaneously, a temporally asymptotic behavior of the solution is established.

Comparison Between Numerical and Experimental Dynamic Coefficients of a Hybrid Aerostatic Bearing

2012 ◽  
Vol 134 (12) ◽  
Author(s):  
Mohamed Amine Hassini ◽  
Mihai Arghir ◽  
Manuel Frocot
Keyword(s):  
Experimental Data ◽  
Theoretical Models ◽  
Bulk Flow ◽  
Major Influence ◽  
Aerostatic Bearing ◽  
Complex Flow ◽  
Flow Equations ◽  
Dynamic Coefficients ◽  
Equivalent Area ◽  
Feeding Pressure

Hybrid journal bearings have been considered for many years as a possible replacement for ball bearings in turbopumps used by the aerospace industry. Due to flow regimes dominated by inertia and due to the nature of the lubricant (cryogenic fluids), the prediction of the linearized dynamic coefficients in these bearings must be based on the compressible bulk-flow equations. Theoretical models based on these equations were validated for hybrid bearings working with water or for liquid or gas annular seals. Validations for hybrid compressible bearings are missing. Experimental data obtained for an air lubricated hybrid aerostatic bearing designed with shallow pockets were recently presented; the data consist of linearized dynamic coefficients obtained for rotation speeds up to 50 krpm and up to 7 bars feeding pressure. The present work introduces a consolidated numerical approach for predicting static and linearized dynamic characteristics. Theoretical predictions are based on bulk flow equations in conjunction with CFD analysis. It was found that, for a given feeding pressure, the value of the pressure downstream the orifice has a major influence on all results. Special care was then taken to describe the complex flow in the feeding system and the orifice. Three dimensional CFD was employed because the bulk-flow equations are inappropriate in this part of the bearing. The pressure downstream the orifice stemming from CFD results and the feeding pressure were next imposed in the bulk flow model and the equivalent area of the orifice was obtained from the numerical solution of the steady flow in the bearing. Since the pockets of the hybrid bearing are shallow, this equivalent area is considered as being the harmonic average of the orifice cross section area and of the cylindrical curtain area located between the orifice and the rotor. The comparisons between theoretical dynamic coefficients and experimental data validated this approach of the equivalent area of the orifice.

scholarly journals A priori error estimates for a discretized poro-elastic–elastic system solved by a fixed-stress algorithm

2019 ◽  
Vol 74 ◽  
pp. 24 ◽  
Author(s):  
Vivette Girault ◽  
Mary F. Wheeler ◽  
Tameem Almani ◽  
Saumik Dana
Keyword(s):  
Error Estimates ◽  
A Priori ◽  
Elastic Region ◽  
Elastic Displacement ◽  
Complete Analysis ◽  
Time Step ◽  
Galerkin Scheme ◽  
Flow Equations ◽  
Mixed Scheme ◽  
Continuous Galerkin

We consider a poro-elastic region embedded into an elastic non-porous region. The elastic displacement equations are discretized by a continuous Galerkin scheme, while the flow equations for the pressure in the poro-elastic medium are discretized by either a continuous Galerkin scheme or a mixed scheme. Since the overall system of equations is very large, a fixed-stress algorithm is used at each time step to decouple the displacement from the flow equations in the poro-elastic region. We prove a priori error estimates for the resulting Galerkin scheme as well as the mixed scheme, with the expected order of accuracy, provided the algorithm is sufficiently iterated at each time step. These theoretical results are confirmed by a numerical experiment performed with the mixed scheme. A complete analysis including a posteriori estimates for both the Galerkin and the mixed scheme has been done but is too long to appear here.

On the Modeling Approach of DC Arc Plasma Flows by Implementation of the CFD Phoenics Code

1996 ◽  
Author(s):  
P. Eichert ◽  
C. Coddet ◽  
M. Imbed
Keyword(s):  
Gas Flow ◽  
Simple Algorithm ◽  
Elliptic Type ◽  
Spray Parameters ◽  
Arc Plasma ◽  
Algebraic Equations ◽  
Plasma Flows ◽  
Modeling Approach ◽  
Control Volumes ◽  
Dc Arc

Abstract This paper is devoted to the presentation of a modeling approach of DC arc plasma flows by implementation of the CFD PHOENICS code. The equations of mass, momentum and energy, of elliptic type, are discretized using a finite/control volumes method. The closure of the equations set is obtained with the standard k-ε turbulence model. Finally, the algebraic equations set is solved by means of the SIMPLE algorithm. 3D calculations are performed from the point of the cold gas injection into the torch to the plasma ejection into the surrounding atmosphere, taking into account some currently used spray parameters (i.e., plasma gas flow rate, power, etc.). Results are discussed and compared with experimental values taken from the literature.

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