scholarly journals A RAYLEIGH–RITZ METHOD FOR NAVIER–STOKES FLOW THROUGH CURVED DUCTS

2019 ◽  
Vol 61 (1) ◽  
pp. 1-22 ◽  
Author(s):  
BRENDAN HARDING
Keyword(s):  
Cross Sections ◽  
Cell Separation ◽  
Ritz Method ◽  
Cross Flow ◽  
Axial Flow ◽  
Navier Stokes ◽  
Cross Sectional ◽  
Straightforward Method ◽  
Flow Through ◽  
Curved Ducts

We present a Rayleigh–Ritz method for the approximation of fluid flow in a curved duct, including the secondary cross-flow, which is well known to develop for nonzero Dean numbers. Having a straightforward method to estimate the cross-flow for ducts with a variety of cross-sectional shapes is important for many applications. One particular example is in microfluidics where curved ducts with low aspect ratio are common, and there is an increasing interest in nonrectangular duct shapes for the purpose of size-based cell separation. We describe functionals which are minimized by the axial flow velocity and cross-flow stream function which solve an expansion of the Navier–Stokes model of the flow. A Rayleigh–Ritz method is then obtained by computing the coefficients of an appropriate polynomial basis, taking into account the duct shape, such that the corresponding functionals are stationary. Whilst the method itself is quite general, we describe an implementation for a particular family of duct shapes in which the top and bottom walls are described by a polynomial with respect to the lateral coordinate. Solutions for a rectangular duct and two nonstandard duct shapes are examined in detail. A comparison with solutions obtained using a finite-element method demonstrates the rate of convergence with respect to the size of the basis. An implementation for circular cross-sections is also described, and results are found to be consistent with previous studies.

Study of Fully Developed Incompressible Flow in Curved Ducts, Using a Multi-Grid Technique

1987 ◽  
Vol 109 (3) ◽  
pp. 226-236 ◽  
Author(s):  
K. N. Ghia ◽  
U. Ghia ◽  
C. T. Shin
Keyword(s):  
Cross Sections ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Cross Flow ◽  
Internal Flow ◽  
Navier Stokes ◽  
Dean Number ◽  
Similarity Parameter ◽  
Fine Grid ◽  
Curved Ducts

Fully developed flows inside curved ducts of rectangular as well as polar cross sections have been analyzed using the Navier-Stokes equations in terms of the axial velocity and vorticity and the cross-flow stream function. Numerical solutions of the three second-order coupled elliptic partial differential equations governing this flow have been obtained using efficient numerical schemes. For curved-duct flows, the similarity parameter of significance is the Dean number K, rather than the Reynolds number Re. Results have been obtained for curved ducts with square cross sections for K up to 900 which, in the present study, corresponds to Re = 9,000 for this internal flow configuration. The fine-grid calculations show that, for square cross-section ducts, Dean’s instability occurs at K ≈ 125 and, further, that this phenomenon does not disappear even for K = 900. In ducts of polar cross sections, which are geometrically more representative of turbomachinery cascade passages, the phenomenon of Dean’s instability is not seen to occur for K up to 600.

A 3-D Thin Layer Navier-Stokes Code for Supersonic Laminar and Turbulent Flows

2002 ◽  
Author(s):  
Omid Abouali ◽  
Mohammad M. Alishahi ◽  
Homayoon Emdad ◽  
Goodarz Ahmadi
Keyword(s):  
Experimental Data ◽  
Shock Wave ◽  
Mach Number ◽  
Turbulent Flows ◽  
Turbulence Modeling ◽  
Cross Flow ◽  
Navier Stokes ◽  
Cross Sectional ◽  
Laminar And Turbulent Flows ◽  
Circumferential Pressure

A 3-D Thin Layer Navier-Stokes (TLNS) code for solving viscous supersonic flows is developed. The new code uses several numerical algorithms for space and time discretization together with appropriate turbulence modeling. Roe’s method is used for discretizing the convective terms and the central differencing scheme is employed for the viscous terms. An explicit time marching technique and a finite volume space discretization are used. The developed computational model can handle both laminar and turbulent flows. The Baldwin-Lomax model and Degani-Schiff modifications are used for turbulence modeling. The computational model is applied to a hypersonic laminar flow at Mach 7.95 around a cone at different incidence angles. The circumferential pressure distribution is compared with the experimental data. The cross-sectional Mach number contours are also presented. It is shown that in addition to the outer shock, a cross-flow shock wave is also present in the flow field. The cases of supersonic turbulent flows with Mach number 3 around a tangent-ogive with incidence angles of 6° and a secant-ogive with incidence angles of 10° are also studied. The circumferential pressure distributions are compared with the experimental data and the Euler code results and good agreement is obtained. The cross-sectional Mach number contours are also presented. It is shown that in this case also in addition to the outer shock, a cross-flow shock wave is also present at the incidence angle of 10°.

scholarly journals The Computation of Adjacent Blade-Row Effects in a 1.5 Stage Axial Flow Turbine

1997 ◽  
Author(s):  
Rolf Emunds ◽  
Ian K. Jennions ◽  
Dieter Bohn ◽  
Jochen Gier
Keyword(s):  
Experimental Data ◽  
Numerical Simulation ◽  
Boundary Conditions ◽  
Axial Flow ◽  
The Other ◽  
Navier Stokes ◽  
Blade Row ◽  
Flow Through ◽  
The University

This paper deals with the numerical simulation of flow through a 1.5 stage axial flow turbine. The 3-row configuration has been experimentally investigated at the University of Aachen where measurements behind the first vane, the first stage and the full configuration were taken. These measurements allow single blade row computations, to the measured boundary conditions taken from complete engine experiments, or full multistage simulations. The results are openly available inside the framework of ERCOFTAC 1996. There are two separate but interrelated parts to the paper. Firstly, two significantly different Navier-Stokes codes are used to predict the flow around the first vane and the first rotor, both running in isolation. This is used to engender confidence in the code that is subsequently used to model the multiple bladerow tests, the other code is currently only suitable for a single blade row. Secondly, the 1.5 stage results are compared to the experimental data and promote discussion of surrounding blade row effects on multistage solutions.

Interference in a three-dimensional array of jets

2015 ◽  
Vol 26 (5) ◽  
pp. 795-819
Author(s):  
P. E. WESTWOOD ◽  
F. T. SMITH
Keyword(s):  
Cross Sections ◽  
Flow Direction ◽  
Three Dimensional ◽  
Cross Flow ◽  
Longitudinal Vortex ◽  
Cross Sectional ◽  
Dimensional Array ◽  
Unsteady Jets ◽  
The Cross ◽  
Jet Cross Sections

The theoretical investigation here of a three-dimensional array of jets of fluid (air guns) and their interference is motivated by applications to the food sorting industry especially. Three-dimensional motion without symmetry is addressed for arbitrary jet cross-sections and incident velocity profiles. Asymptotic analysis based on the comparatively long axial length scale of the configuration leads to a reduced longitudinal vortex system providing a slender flow model for the complete array response. Analytical and numerical studies, along with comparisons and asymptotic limits or checks, are presented for various cross-sectional shapes of nozzle and velocity inputs. The influences of swirl and of unsteady jets are examined. Substantial cross-flows are found to occur due to the interference. The flow solution is non-periodic in the cross-plane even if the nozzle array itself is periodic. The analysis shows that in general the bulk of the three-dimensional motion can be described simply in a cross-plane problem but the induced flow in the cross-plane is sensitively controlled by edge effects and incident conditions, a feature which applies to any of the array configurations examined. Interference readily alters the cross-flow direction and misdirects the jets. Design considerations centre on target positioning and jet swirling.

Importance of Added Mass for the Interaction Between IL and CF Vibrations of Free Spanning Pipelines

2011 ◽  
Author(s):  
Ida M. Aglen ◽  
Carl M. Larsen
Keyword(s):  
Cross Sections ◽  
Amplitude Ratio ◽  
Large Diameter ◽  
Cross Flow ◽  
Added Mass ◽  
Flexible Beam ◽  
Hydrodynamic Coefficients ◽  
Cross Sectional ◽  
Mass Coefficient ◽  
The Stability

The importance of cross-flow (CF) response generated by vortex induced vibrations (VIV) of free spanning pipelines has long been recognised. The significance of in-line (IL) vibrations has recently been understood and hence also been subjected to research. The combined effect of CF and IL vibrations is, however, still not fully described. This paper highlights the CF-IL interaction with focus on the transition zone from pure IL to CF dominated response, giving special attention to how the added mass influences the interaction. Results from extensive flexible beam tests connected to the Ormen Lange (OL) development have been used as a basis for this study. Trajectories for cross sectional motions from the flexible beam test were identified, and then used as forced motions of a large diameter rigid cylinder exposed to uniform flow. Non-dimensional parameters like Reynolds number (Re), amplitude ratio and reduced frequency were identical for the two tests. Hence, forces found from the forced motion test could be used to find hydrodynamic coefficients valid for the flexible beam case. This paper discusses the results from the flexible beam tests with a relatively short length to diameter ratio (L/D) of 145. Modal analyses by Nielsen et al. (2002) show that the first mode dominates in both directions for these particular tests, even though the IL response frequency is twice the CF frequency. In this paper the added mass variations along the OL flexible beam is studied. Forces acting on 4 different cross sections along the beam are measured for 7 different prototype velocities. For each test the hydrodynamic coefficients are calculated, and the results show how the added mass changes along the beam for increasing velocities, and thereby creates resonance for both IL and CF response. The stability of the added mass coefficient throughout the time series is also evaluated.

scholarly journals Numerical simulation of turbulent flow through Schiller’s wavy pipe

2014 ◽  
Vol 761 ◽  
pp. 241-260 ◽  
Author(s):  
G. Daschiel ◽  
V. Krieger ◽  
J. Jovanović ◽  
A. Delgado
Keyword(s):  
Turbulent Flow ◽  
Structural Changes ◽  
Stokes Equations ◽  
Resistance Coefficient ◽  
Navier Stokes ◽  
Cross Sectional ◽  
Navier Stokes Equations ◽  
Flow Through ◽  
Pressure Resistance ◽  
Almost All

AbstractThe development of incompressible turbulent flow through a pipe of wavy cross-section was studied numerically by direct integration of the Navier–Stokes equations. Simulations were performed at Reynolds numbers of $4.5\times 10^{3}$ and $10^{4}$ based on the hydraulic diameter and the bulk velocity. Results for the pressure resistance coefficient ${\it\lambda}$ were found to be in excellent agreement with experimental data of Schiller (Z. Angew. Math. Mech., vol. 3, 1922, pp. 2–13). Of particular interest is the decrease in ${\it\lambda}$ below the level predicted from the Blasius correlation, which fits almost all experimental results for pipes and ducts of complex cross-sectional geometries. Simulation databases were used to evaluate turbulence anisotropy and provide insights into structural changes of turbulence leading to flow relaminarization. Anisotropy-invariant mapping of turbulence confirmed that suppression of turbulence is due to statistical axisymmetry in the turbulent stresses.

Introduction to Engineering Fluid Mechanics

2018 ◽  
Author(s):  
Marcel Escudier
Keyword(s):  
Fluid Mechanics ◽  
Dimensional Analysis ◽  
Gas Flow ◽  
Stokes Equations ◽  
Reaction Force ◽  
Axial Flow ◽  
Hydraulic Turbine ◽  
Navier Stokes ◽  
Pipe Bend ◽  
Flow Through

Turbojet and turbofan engines, rocket motors, road vehicles, aircraft, pumps, compressors, and turbines are examples of machines which require a knowledge of fluid mechanics for their design. The aim of this undergraduate-level textbook is to introduce the physical concepts and conservation laws which underlie the subject of fluid mechanics and show how they can be applied to practical engineering problems. The first ten chapters are concerned with fluid properties, dimensional analysis, the pressure variation in a fluid at rest (hydrostatics) and the associated forces on submerged surfaces, the relationship between pressure and velocity in the absence of viscosity, and fluid flow through straight pipes and bends. The examples used to illustrate the application of this introductory material include the calculation of rocket-motor thrust, jet-engine thrust, the reaction force required to restrain a pipe bend or junction, and the power generated by a hydraulic turbine. Compressible-gas flow is then dealt with, including flow through nozzles, normal and oblique shock waves, centred expansion fans, pipe flow with friction or wall heating, and flow through axial-flow turbomachinery blading. The fundamental Navier-Stokes equations are then derived from first principles, and examples given of their application to pipe and channel flows and to boundary layers. The final chapter is concerned with turbulent flow. Throughout the book the importance of dimensions and dimensional analysis is stressed. A historical perspective is provided by an appendix which gives brief biographical information about those engineers and scientists whose names are associated with key developments in fluid mechanics.

scholarly journals Effects of Bridge-Shaped Microchannel Geometry on the Performance of a Micro Laminar Flow Fuel Cell

Micromachines ◽  
2019 ◽  
Vol 10 (12) ◽  
pp. 822
Author(s):  
Muhammad Tanveer ◽  
Kwang-Yong Kim
Keyword(s):  
Fuel Cell ◽  
Numerical Model ◽  
Laminar Flow ◽  
Cross Section ◽  
Cross Sections ◽  
Three Dimensional ◽  
Narrow Channel ◽  
Navier Stokes ◽  
Cross Sectional ◽  
Square Channel

A laminar flow micro fuel cell comprising of bridge-shaped microchannel is investigated to find out the effects of the cross-section shape of the microchannel on the performance. A parametric study is performed by varying the heights and widths of the channel and bridge shape. Nine different microchannel cross-section shapes are evaluated to find effective microchannel cross-sections by combining three bridge shapes with three channel shapes. A three-dimensional fully coupled numerical model is used to calculate the fuel cell’s performance. Navier-Stokes, convection and diffusion, and Butler-Volmer equations are implemented using the numerical model. A narrow channel with a wide bridge shape shows the best performance among the tested nine cross-sectional shapes, which is increased by about 78% compared to the square channel with the square bridge shape.

The developing tangential velocity profile for axial flow in an annulus with a rotating inner cylinder

1968 ◽  
Vol 307 (1488) ◽  
pp. 55-69 ◽  
Keyword(s):  
Velocity Profile ◽  
Stokes Equations ◽  
Tangential Velocity ◽  
Axial Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Taylor Vortices ◽  
Flow Through ◽  
Momentum Integral ◽  
Tangential Velocity Profile

A method is described of predicting the growth of a tangential velocity profile in fully developed laminar axial flow through a concentric annulus when the inner surface is rotated at speeds which are insufficient to generate Taylor vortices. The treatment, which is based on simplification and subsequent solution of the Navier-Stokes equations, as Fourier-Bessel series, appears preferable to momentum-integral techniques through greater simplicity of expression and in requiring fewer assumptions about the developing tangential profile. The validity of the predictions is best at high axial Reynolds number.

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