scholarly journals The steady propagation of a semi-infinite bubble into a tube of elliptical or rectangular cross-section

2002 ◽  
Vol 470 ◽  
pp. 91-114 ◽  
Author(s):  
ANDREW L. HAZEL ◽  
MATTHIAS HEIL
Keyword(s):  
Aspect Ratio ◽  
Cross Section ◽  
Capillary Number ◽  
Stokes Equations ◽  
Rectangular Cross Section ◽  
Scaling Law ◽  
Propagation Speed ◽  
Cross Sectional ◽  
Steady Propagation ◽  
Rectangular Tubes

This paper investigates the propagation of an air finger into a fluid-filled, axially uniform tube of elliptical or rectangular cross-section with transverse length scale a and aspect ratio α. Gravity is assumed to act parallel to the tube's axis. The problem is studied numerically by a finite-element-based direct solution of the free-surface Stokes equations.In rectangular tubes, our results for the pressure drop across the bubble tip, Δp, are in good agreement with the asymptotic predictions of Wong et al. (1995b) at low values of the capillary number, Ca (ratio of viscous to surface-tension forces). At larger Ca, Wong et al.'s (1995b) predictions are found to underestimate Δp. In both elliptical and rectangular tubes, the ratio Δp(α)/Δp(α = 1) is approximately independent of Ca and thus equal to the ratio of the static meniscus curvatures.In non-axisymmetric tubes, the air-liquid interface develops a noticeable asymmetry near the bubble tip at all values of the capillary number. The tip asymmetry decays with increasing distance from the bubble tip, but the decay rate becomes very small as Ca increases. For example, in a rectangular tube with α = 1.5, when Ca = 10, the maximum and minimum finger radii still differ by more than 10% at a distance 100a behind the finger tip. At large Ca the air finger ultimately becomes axisymmetric with radius r∞. In this regime, we find that r∞ in elliptical and rectangular tubes is related to r∞ in circular and square tubes, respectively, by a simple, empirical scaling law. The scaling has the physical interpretation that for rectangular and elliptical tubes of a given cross-sectional area, the propagation speed of an air finger, which is driven by the injection of air at a constant volumetric rate, is independent of the tube's aspect ratio.For smaller Ca (Ca < Ca), the air finger is always non-axisymmetric and the persisting draining flows in the thin film regions far behind the bubble tip ultimately lead to dry regions on the tube wall. Ca increases with increasing α and for α > αˆ dry spots will develop on the tube walls at all values of Ca.

Saint-Venant Decay Rates for the Rectangular Cross Section Rod

2004 ◽  
Vol 71 (3) ◽  
pp. 429-433 ◽  
Author(s):  
N. G. Stephen ◽  
P. J. Wang
Keyword(s):  
Finite Element ◽  
Aspect Ratio ◽  
Transfer Matrix ◽  
Cross Section ◽  
Cross Sections ◽  
Rectangular Cross Section ◽  
Decay Rates ◽  
Elastic Rod ◽  
Cross Sectional ◽  
Element Transfer

A finite element-transfer matrix procedure developed for determination of Saint-Venant decay rates of self-equilibrated loading at one end of a semi-infinite prismatic elastic rod of general cross section, which are the eigenvalues of a single repeating cell transfer matrix, is applied to the case of a rectangular cross section. First, a characteristic length of the rod is modelled within a finite element code; a superelement stiffness matrix relating force and displacement components at the master nodes at the ends of the length is then constructed, and its manipulation provides the transfer matrix, from which the eigenvalues and eigenvectors are determined. Over the range from plane stress to plane strain, which are the extremes of aspect ratio, there are always eigenmodes which decay slower than the generalized Papkovitch-Fadle modes, the latter being largely insensitive to aspect ratio. For compact cross sections, close to square, the slowest decay is for a mode having a distribution of axial displacement reminiscent of that associated with warping during torsion; for less compact cross sections, slowest decay is for a mode characterized by cross-sectional bending, caused by self-equilibrated twisting moment.

The Exact One-Dimensional Theory for End-Loaded Fully Anisotropic Beams of Narrow Rectangular Cross Section

2001 ◽  
Vol 68 (6) ◽  
pp. 865-868 ◽  
Author(s):  
P. Ladeve`ze ◽  
J. G. Simmonds
Keyword(s):  
Cross Section ◽  
Plane Stress ◽  
Rectangular Cross Section ◽  
Dimensional Theory ◽  
Explicit Formulas ◽  
Cross Sectional ◽  
Rectangular Shape ◽  
One Dimensional ◽  
Anisotropic Elastic ◽  
Derivatives Of

The exact theory of linearly elastic beams developed by Ladeve`ze and Ladeve`ze and Simmonds is illustrated using the equations of plane stress for a fully anisotropic elastic body of rectangular shape. Explicit formulas are given for the cross-sectional material operators that appear in the special Saint-Venant solutions of Ladeve`ze and Simmonds and in the overall beamlike stress-strain relations between forces and a moment (the generalized stress) and derivatives of certain one-dimensional displacements and a rotation (the generalized displacement). A new definition is proposed for built-in boundary conditions in which the generalized displacement vanishes rather than pointwise displacements or geometric averages.

Dependence of AC losses on aspect ratio in superconducting wires with rectangular cross section

2004 ◽  
Vol 412-414 ◽  
pp. 1045-1049 ◽  
Author(s):  
K. Kajikawa ◽  
T. Hayashi ◽  
K. Funaki ◽  
E.S. Otabe ◽  
T. Matsushita
Keyword(s):  
Aspect Ratio ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Ac Losses ◽  
Superconducting Wires

Numerical Investigation on Diffusion Slot Hole With Various Cross-Sectional End Shapes

2017 ◽  
Vol 139 (9) ◽  
Author(s):  
Bai-Tao An ◽  
Jian-Jun Liu
Keyword(s):  
Cross Section ◽  
Film Cooling ◽  
Separation Bubble ◽  
Rectangular Cross Section ◽  
Pressure Ratio ◽  
Smooth Transition ◽  
Shape Variation ◽  
Cross Sectional ◽  
Cooling Effectiveness ◽  
Shaped Hole

The diffusion hole constructed on a slot-type cross section has the potential to obtain high film cooling performance. However, the end shape of the cross section can greatly affect film cooling characteristics. This study examined eight cases of diffusion slot holes with various cross-sectional end shapes. The comparison of the eight diffusion slot holes and a typical fan-shaped hole was performed with a flat plate model using a three-dimensional (3D) steady computational fluid dynamics (CFD) method. The rectangular cross section had an aspect ratio of about 3.4. The end shape variation can be described based on sidewall contraction location, size, and form. The simulations were performed under an engine-representative condition of mainstream inlet Mach number 0.3 and turbulence intensity 5.2%. The simulated results showed that a strip separation bubble caused by inlet “jetting effect” occurs near the downstream wall of the diffusion slot hole and interacts with the diffusion flow. The different end shape of the rectangular cross section leads to different sidewall static pressure and exit velocity profiles, thereby produces three cooling effectiveness patterns, single-peak, bipeak, and tripeak patterns. The tripeak pattern produces higher cooling effectiveness and relatively uniform film coverage. The structure with moderate contraction and smooth transition on two sides of the downstream wall favors creation of a tripeak pattern. Compared with the fan-shaped hole, the discharge coefficient of diffusion slot hole is slightly small in low pressure ratio range, the pressure loss ratio has little difference.

scholarly journals Flow through a helical pipe with rectangular cross-section

1970 ◽  
Vol 4 (2) ◽  
pp. 99-110
Author(s):  
Md Mahmud Alam ◽  
Delowara Begum ◽  
K Yamamoto
Keyword(s):  
Aspect Ratio ◽  
Iterative Method ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Main Tool ◽  
Numerical Approach ◽  
Dean Number ◽  
Helical Pipe ◽  
Flow Through ◽  
Marine Engineering

The effects of torsion, aspect ratio and curvature on the flow in a helical pipe of rectangular cross- section are studied by introducing a non-orthogonal helical coordinate system. Spectral method is applied as main tool for numerical approach where Chebyshev polynomial is used. The numerical calculations are obtained by the iterative method. The calculations are carried out for 0≤ δ ≤0.02, 1≤ λ ≤ 2.85, 1≤ γ ≤2.4, at Dn = 50 & 100 respectively, where d is the non-dimensional curvature, l the torsion parameter, g the aspect ratio and  Dn the pressure driven parameter (Dean number).DOI: http://dx.doi.org/10.3329/jname.v4i2.991 Journal of Naval Architecture and Marine Engineering Vol.4(2) 2007 p.99-110

Film Cooling Investigation of a Slot-Based Diffusion Hole

2016 ◽  
Author(s):  
Bai-Tao An ◽  
Jian-Jun Liu ◽  
Si-Jing Zhou ◽  
Xiao-Dong Zhang ◽  
Chao Zhang
Keyword(s):  
Cross Section ◽  
Film Cooling ◽  
Rectangular Cross Section ◽  
Density Ratio ◽  
Vortex Structures ◽  
Cross Sectional ◽  
Cooling Effectiveness ◽  
Film Cooling Effectiveness ◽  
Blowing Ratio ◽  
Shaped Hole

This paper presents a new configuration of discrete film hole, i.e., the slot-based diffusion hole. Retaining the similar diffusion features to a traditional diffusion hole, the slot-based diffusion hole transforms the cross section of circle for the traditional diffusion hole to a flattened rectangle with respect to the equivalent cross-sectional area. Consequently, the exit width of the new hole is effectively enlarged. To verify the film cooling effectiveness, a low speed flat plate experimental facility incorporated with Pressure Sensitive Paint (PSP) measurement technique was employed to obtain the adiabatic film cooling effectiveness. The experiments were performed with hole pitch to diameter ratio p/D=6 and density ratio DR=1.38. The blowing ratio was varied from M=0.5 to M=2.5. A fan-shaped hole and two slot-based diffusion holes were tested and compared. Three-dimensional numerical simulation was employed to analyze the flow field in detail. The experimental results showed that the area averaged effectiveness of two slot-based diffusion holes is significantly higher than that of the fan-shaped hole when the blowing ratio exceeds 1.0. The slot-based diffusion hole demonstrates the great advantage over the fan-shaped hole at hole exit and maintains this to far downstream. The numerical results showed that the ends shape of the flattened rectangular cross section has large influences on film distribution patterns and downstream vortex structures. The semi-circle and straight line ends shapes lead to a bi-peak and a single-peak effectiveness pattern, respectively. The optimal ends shape can regulate the vortex structures and improve the film cooling effectiveness further.

Exact Solution of the Navier-Stokes Equations for the Fully Developed, Pulsating Flow in a Rectangular Duct With a Constant Cross-Sectional Velocity1

2003 ◽  
Vol 125 (2) ◽  
pp. 382-385 ◽  
Author(s):  
S. Tsangaris ◽  
N. W. Vlachakis
Keyword(s):  
Pressure Gradient ◽  
Stokes Equations ◽  
Rectangular Cross Section ◽  
Gradient Flow ◽  
Pulsating Flow ◽  
Artificial Kidney ◽  
Navier Stokes ◽  
Rectangular Duct ◽  
Cross Sectional ◽  
Navier Stokes Equations

The Navier-Stokes equations have been solved in order to obtain an analytical solution of the fully developed laminar flow in a duct having a rectangular cross section with two opposite equally porous walls. We obtained solutions both for the case of steady flow as well as for the case of oscillating pressure gradient flow. The pulsating flow is obtained by the superposition of the steady and oscillating pressure gradient solutions. The solution has applications for blood flow in fiber membranes used for the artificial kidney.

Influence of the 180° Bend Geometry on the Pressure Loss and Heat Transfer in a High Aspect Ratio Rectangular Smooth Channel

2004 ◽  
Author(s):  
Detlef Pape ◽  
Herve´ Jeanmart ◽  
Jens von Wolfersdorf ◽  
Bernhard Weigand
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Aspect Ratio ◽  
Cross Section ◽  
High Aspect Ratio ◽  
Pressure Loss ◽  
Rectangular Cross Section ◽  
Flow Structures ◽  
Cooling Channel ◽  
Smooth Channel

An experimental and numerical investigation of the pressure loss and the heat transfer in the bend region of a smooth two-pass cooling channel with a 180°-turn has been performed. The channels have a rectangular cross-section with a high aspect ratio of H/W = 4. The heat transfer has been measured using the transient liquid crystal method. For the investigations the Reynolds-number as well as the distance between the tip and the divider wall (tip distance) are varied. While the Reynolds number varies from 50’000 to 200’000 and its influence on the normalized pressure loss and heat transfer is found to be small, the variations of the tip distance from 0.5 up to 3.65 W produce quite different flow structures in the bend. The pressure loss over the bend thus shows a strong dependency on these variations.

The shape of a Möbius band

1993 ◽  
Vol 440 (1908) ◽  
pp. 149-162 ◽  
Keyword(s):  
Aspect Ratio ◽  
Asymptotic Solution ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Elastic Rods ◽  
Large Deflections ◽  
Mechanical Equilibrium ◽  
Elastic Rod ◽  
Flexible Material ◽  
Möbius Band

The shape of a Möbius band made of a flexible material, such as paper, is determined. The band is represented as a bent, twisted elastic rod with a rectangular cross-section. Its mechanical equilibrium is governed by the Kirchhoff–Love equations for the large deflections of elastic rods. These are solved numerically for various values of the aspect ratio of the cross-section, and an asymptotic solution is found for large values of this ratio. The resulting shape is shown to agree well with that of a band made from a strip of plastic.

A Numerical Study of Dean Instability in Non-Newtonian Fluids

2005 ◽  
Vol 128 (1) ◽  
pp. 34-41 ◽  
Author(s):  
H. Fellouah ◽  
C. Castelain ◽  
A. Ould El Moctar ◽  
H. Peerhossaini
Keyword(s):  
Aspect Ratio ◽  
Cross Section ◽  
Power Law ◽  
Numerical Study ◽  
Rectangular Cross Section ◽  
Newtonian Fluids ◽  
Dean Number ◽  
Bingham Number ◽  
Curved Channels ◽  
Power Law Index

We present a numerical study of Dean instability for non-Newtonian fluids in a laminar 180deg curved-channel flow of rectangular cross section. A methodology based on the Papanastasiou model (Papanastasiou, T. C., 1987, J. Rheol., 31(5), pp. 385–404) was developed to take into account the Bingham-type rheological behavior. After validation of the numerical methodology, simulations were carried out (using FLUENT CFD code) for Newtonian and non-Newtonian fluids in curved channels of square or rectangular cross section and for a large aspect and curvature ratios. A criterion based on the axial velocity gradient was defined to detect the instability threshold. This criterion was used to optimize the grid geometry. The effects of curvature and aspect ratio on the Dean instability are studied for all fluids, Newtonian and non-Newtonian. In particular, we show that the critical value of the Dean number decreases with increasing curvature ratio. The variation of the critical Dean number with aspect ratio is less regular. The results are compared to those for Newtonian fluids to emphasize the effect of the power-law index and the Bingham number. The onset of Dean instability is delayed with increasing power-law index. The same delay is observed in Bingham fluids when the Bingham number is increased.

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