Non-isothermal Laminar Flow of Gases through Cooled Tubes

1973 ◽  
Vol 95 (1) ◽  
pp. 85-92 ◽  
Author(s):  
Lloyd H. Back
Keyword(s):  
Laminar Flow ◽  
Numerical Solutions ◽  
Isothermal Flow ◽  
Wall Friction ◽  
Gas Flows ◽  
Average Heat Transfer Coefficient ◽  
Average Heat ◽  
Flow Equations ◽  
Flow Through ◽  
Good Agreement

Numerical solutions of the laminar-flow equations in differential form are presented for gas flows through cooled tubes. For nearly isothermal flow there is good agreement with available experimental data, as is also found for the case of a large amount of wall cooling. This correspondence along with a check on the satisfaction of the global momentum and energy constraints allowed an appraisal of the effect of wall cooling on flow through tubes. In general, the effect of wall cooling was to decrease the wall friction and the change in pressure along tubes, but the average heat-transfer coefficient did not vary much.

scholarly journals Groundwater flow equation, overview, derivation, and solution

2021 ◽  
Vol 314 ◽  
pp. 04007
Author(s):  
Lhoussaine El Mezouary ◽  
Bouabid El Mansouri
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Groundwater Flow ◽  
Numerical Solutions ◽  
Flow Equation ◽  
Heat Transfer Equation ◽  
Flow Equations ◽  
Basic Law ◽  
Flow Through ◽  
Groundwater Flow Equation

Darcy’s law is the basic law of flow, and it produces a partial differential equation is similar to the heat transfer equation when coupled with an equation of continuity that explains the conservation of fluid mass during flow through a porous media. This article, titled the groundwater flow equation, covers the derivation of the groundwater flow equations in both the steady and transient states. We look at some of the most common approaches and methods for developing analytical or numerical solutions. The flaws and limits of these solutions in reproducing the behavior of water flow on the aquifer are also discussed in the article.

The stability and transition of a two-dimensional jet

1960 ◽  
Vol 7 (1) ◽  
pp. 53-80 ◽  
Author(s):  
Hiroshi Sato
Keyword(s):  
Laminar Flow ◽  
Numerical Solutions ◽  
Two Dimensional ◽  
Eigenvalues And Eigenfunctions ◽  
Response Characteristics ◽  
Laminar Jets ◽  
The Stability ◽  
Linear Disturbance ◽  
Good Agreement ◽  
Small Disturbances

A study was made of the transition of a two-dimensional jet. In the region where laminar flow becomes unstable, two kinds of sinusoidal velocity fluctuation have been found; one is symmetrical and the other is anti-symmetrical with respect to the centre line of the jet. The fluctuations grow exponentially at first and develop into turbulence without being accompanied by abrupt bursts or turbulent spots.The response characteristics of laminar jets to artificial external excitation were investigated in detail by using sound as an exciting agent. The effect of excitation was seen to be most remarkable when the frequency of excitation coincides with that of self-excited sinusoidal fluctuations.Numerical solutions of equation of small disturbances superposed on laminar flow were obtained assuming the Reynolds number as infinity. Theoretical eigenvalues and eigenfunctions are in good agreement with experimental results, thus verifying the existence of a region of linear disturbance in the two-dimensional jet.

Loss Coefficients for Periodically Unsteady Flows in Conduit Components: Illustrated for Laminar Flow in a Circular Duct and a 90 Degree Bend

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
Bastian Schmandt ◽  
Heinz Herwig
Keyword(s):  
Laminar Flow ◽  
Flow Field ◽  
Numerical Solutions ◽  
Steady Flows ◽  
Circular Duct ◽  
Pipe System ◽  
Second Law Analysis ◽  
Flow Through ◽  
Loss Coefficients ◽  
Laminar Pipe Flow

Losses in a flow through conduit components of a pipe system can be accounted for by head loss coefficients K. They can either be determined experimentally or from numerical solutions of the flow field. The physical interpretation is straight forward when these losses are related to the entropy generation in the flow field. This can be done based on the numerical solutions by the second law analysis (SLA) successfully applied for steady flows in the past. This analysis here is extended to unsteady laminar flow, exemplified by a periodic pulsating mass flow rate with the pulsation amplitude and the frequency as crucial parameters. First the numerical model is validated by comparing it to results for unsteady laminar pipe flow with analytical solutions for this case. Then K-values are determined for the benchmark case of a 90 deg bend with a square cross section which is well-documented for the steady case already. It turns out that time averaged values of K may significantly deviate from the corresponding steady values. The K-values determined for steady flow are a good approximation for the time-averaged values in the unsteady case only for small frequencies and small amplitudes.

A numerical comparison between the multiple-scales and finite-element solution for sound propagation in lined flow ducts

2001 ◽  
Vol 437 ◽  
pp. 367-384 ◽  
Author(s):  
SJOERD W. RIENSTRA ◽  
WALTER EVERSMAN
Keyword(s):  
Finite Element ◽  
Numerical Solutions ◽  
Sound Propagation ◽  
Multiple Scales ◽  
Finite Element Solution ◽  
Numerical Comparison ◽  
Element Solution ◽  
Mean Flow ◽  
Flow Equations ◽  
Good Agreement

An explicit, analytical, multiple-scales solution for modal sound transmission through slowly varying ducts with mean flow and acoustic lining is tested against a numerical finite-element solution solving the same potential flow equations. The test geometry taken is representative of a high-bypass turbofan aircraft engine, with typical Mach numbers of 0.5–0.7, circumferential mode numbers m of 10–40, dimensionless wavenumbers of 10–50, and both hard and acoustically treated inlet walls of impedance Z = 2 − i. Of special interest is the presence of the spinner, which incorporates a geometrical complexity which could previously only be handled by fully numerical solutions. The results for predicted power attenuation loss show in general a very good agreement. The results for iso-pressure contour plots compare quite well in the cases where scattering into many higher radial modes can occur easily (high frequency, low angular mode), and again a very good agreement in the other cases.

The Influence of Exhaust Poppet Valve Gas Flow on Reciprocating Engine Noise

1975 ◽  
Vol 189 (1) ◽  
pp. 461-469 ◽  
Author(s):  
T. J. Williams ◽  
J. B. Cox
Keyword(s):  
Gas Flow ◽  
Combustion Engine ◽  
Flow Characteristics ◽  
Exhaust System ◽  
Gas Flows ◽  
Frequency Spectra ◽  
Reciprocating Engine ◽  
Flow Through ◽  
Field Noise ◽  
Good Agreement

The far field noise generated by cold air flowing through stationary and reciprocating exhaust poppet valves into a cylindrical duct or pipe has been investigated. A method of predicting the intensity and frequency spectrum of the noise generated in such circumstances in terms of the known or assumed geometry and flow characteristics of the valves is presented. Comparisons of the predicted frequency spectra with measured values show good agreement for steady gas flows through stationary valves and for unsteady flow through a simplified exhaust system of a motored single cylinder internal combustion engine.

Laminar Flow in a Uniformly Porous Channel

1964 ◽  
Vol 15 (3) ◽  
pp. 299-310 ◽  
Author(s):  
Thein Wah
Keyword(s):  
Differential Equation ◽  
Reynolds Number ◽  
Laminar Flow ◽  
Numerical Solutions ◽  
Large Reynolds Number ◽  
Two Dimensional ◽  
Series Solutions ◽  
Porous Walls ◽  
Linear Differential ◽  
Good Agreement

SummaryThe flow in a two-dimensional channel with porous walls through which fluid is uniformly injected or extracted is considered. Following Berman, a solution is obtained giving a fourth-order non-linear differential equation which depends on a suction Reynolds number R. Numerical solutions of this equation have been obtained. Series solutions of this equation for small and large Reynolds number are given and are shown to give good agreement with the numerical solutions.

Analysis of Two-Phase Homogeneous Bubbly Flows Including Friction and Mass Addition

2004 ◽  
Vol 126 (1) ◽  
pp. 102-109 ◽  
Author(s):  
Marat Mor ◽  
Alon Gany
Keyword(s):  
Single Phase ◽  
Mathematical Formulation ◽  
Bubbly Flows ◽  
Wall Friction ◽  
Gas Flows ◽  
Propulsion Systems ◽  
Two Phase ◽  
Jet Propulsion ◽  
Flow Equations ◽  
Phase Velocities

The present work is a theoretical investigation of two-phase bubbly flows. The main objective is to get a better insight of the basic phenomena associated with such flows through nozzles via physical modeling and mathematical formulation. Introducing Mach number into the flow equations, we find novel, closed-form analytical solutions and expressions for homogeneous bubbly flows including the influence of wall friction and mass addition. The expressions obtained demonstrate an analogy to those of classical, single-phase gas flows. The study deals with homogeneous flows, however, its approach and results can also be applied to investigate flows with unequal phase velocities, to study instability phenomena, as well as to design and analyze water jet propulsion systems.

Numerical Solutions of Viscous Flow through a Pipe Orifice at Low Reynolds Numbers

1968 ◽  
Vol 10 (2) ◽  
pp. 133-140 ◽  
Author(s):  
R. D. Mills
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Viscous Incompressible Fluid ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Axially Symmetric ◽  
Navier Stokes Equations ◽  
Low Reynolds Numbers ◽  
Flow Through ◽  
Good Agreement

Numerical solutions of the Navier-Stokes equations have been obtained in the low range of Reynolds numbers for steady, axially symmetric, viscous, incompressible fluid flow through an orifice in a circular pipe with a fixed orifice/pipe diameter ratio. Streamline patterns and vorticity contours are presented as functions of Reynolds number. The theoretically determined discharge coefficients are in good agreement with experimental results of Johansen (2).

Steady flow through a channel with a symmetrical constriction in the form of a step

1980 ◽  
Vol 372 (1750) ◽  
pp. 393-414 ◽  
Keyword(s):  
Asymptotic Theory ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Volumetric Flow Rate ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Uniform Grids ◽  
Flow Through ◽  
Good Agreement

Numerical solutions of the Navier-Stokes equations are given for the steady, two-dimensional, laminar flow of an incompressible fluid through a channel with a symmetric constriction in the form of a semi-infinite step change in width. The flow proceeds from a steady Poiseuille velocity distribution far enough upstream of the step in the wider part of the channel to a corresponding distribution downstream in the narrower part and is assumed to remain symmetrical about the centre line of the channel. The numerical scheme involves an accurate and efficient centred difference treatment developed by Dennis & Hudson (1978) and results are obtained for Reynolds numbers, based on half the volumetric flow rate, up to 2000. For a step that halves the width of the channel it is found that very fine uniform grids, with 80 intervals spaced across half of the wider channel upstream, are necessary for resolution of the solution for the flow downstream of the onset of the step. Slightly less refined grids are adequate to resolve the flow upstream. The calculated flow ahead of the step exhibits very good agreement with the asymptotic theory of Smith (1979 b)for Reynolds numbers greater than about 100; indeed, comparisons of the upstream separation position and of the wall vorticity nearby are believed to yield the best agreement between numerical and asymptotic solutions yet found. Downstream there is also qualitative agreement regarding separation and reattachment as the grid size is refined in the numerical results.

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