Conjugate-flow theory for heterogeneous compressible fluids, with application to non-uniform suspensions of gas bubbles in liquids

1972 ◽  
Vol 54 (3) ◽  
pp. 545-563 ◽  
Author(s):  
T. Brooke Benjamin
Keyword(s):  
Cross Section ◽  
Gas Bubbles ◽  
Arbitrary Cross Section ◽  
Compressible Fluids ◽  
Lagrangian Description ◽  
Pressure Force ◽  
Primary Flow ◽  
Cross Sectional ◽  
Dissipative Flow ◽  
Conjugate Flows

Conjugate flows have been defined generally as flows uniform in the direction of streaming that separately satisfy the relevant hydrodynamical equations, so allowing a transition from one flow to its conjugate to be consistent with mass and energy conservation. In previous studies of various examples, certain general principles have been found to apply to conjugate flows: in particular, one in a pair of such flows is subcritical (subsonic) and the other supercritical (supersonic), the former having greater flow force (i.e. momentum flux plus pressure force). In this paper these principles are confirmed in another field of application, for which the theory of conjugate flows takes a novel course.The theoretical model defined in § 2 consists of a straight duct of arbitrary cross-section filled with a perfect fluid whose constitutive properties vary with cross-sectional position, and whose primary, prescribed flow is axial with a velocity distribution that may be non-uniform. In § 3 the possibility of a conjugate flow in the same duct is investigated, and its principal properties relative to those of the primary flow are deduced from certain simple inequalities between integrals over the cross-section. A Lagrangian description of the conjugate flow is essential, but the properties in question are established without the necessity of determining this flow explicitly. At the end of § 3, a modification of the model is discussed accounting for dissipative, flow-force conserving transitions (shocks). The application of the theory to flows of non-uniform suspensions of gas bubbles is considered in § 4.

Numerical Simulating Open-Channel Flows with Regular and Irregular Cross-Section Shapes Based on Finite Volume Godunov-Type Scheme

2020 ◽  
pp. 2050047
Author(s):  
Xiaokang Xin ◽  
Fengpeng Bai ◽  
Kefeng Li
Keyword(s):  
Finite Volume ◽  
Cross Section ◽  
Cross Sections ◽  
Open Channel ◽  
Arbitrary Cross Section ◽  
Second Order ◽  
Cross Sectional ◽  
Shallow Flows ◽  
Natural Streams ◽  
Saint Venant Equations

A numerical model based on the Saint-Venant equations (one-dimensional shallow water equations) is proposed to simulate shallow flows in an open channel with regular and irregular cross-section shapes. The Saint-Venant equations are solved by the finite-volume method based on Godunov-type framework with a modified Harten, Lax, and van Leer (HLL) approximate Riemann solver. Cross-sectional area is replaced by water surface level as one of primitive variables. Two numerical integral algorithms, compound trapezoidal and Gauss–Legendre integrations, are used to compute the hydrostatic pressure thrust term for natural streams with arbitrary and irregular cross-sections. The Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and second-order Runge–Kutta methods is adopted to achieve second-order accuracy in space and time, respectively. The performance of the resulting scheme is evaluated by application in rectangular channels, trapezoidal channels, and a natural mountain river. The results are compared with analytical solutions and experimental or measured data. It is demonstrated that the numerical scheme can simulate shallow flows with arbitrary cross-section shapes in practical conditions.

Estimation of Nusselt Number in Microchannels of Arbitrary Cross-Section With Constant Axial Heat Flux

2008 ◽  
Author(s):  
Ehsan Sadeghi ◽  
Majid Bahrami ◽  
Ned Djilali
Keyword(s):  
Heat Flux ◽  
Nusselt Number ◽  
Cross Section ◽  
Cross Sections ◽  
Numerical Solutions ◽  
Arbitrary Cross Section ◽  
Geometrical Parameters ◽  
Thermal Systems ◽  
Cross Sectional ◽  
Axial Heat

In many practical instances such as basic design, parametric study, and optimization analysis of thermal systems, it is often very convenient to have closed form relations to obtain the trends and a reasonable estimate of the Nusselt number. However, finding exact solutions for many practical singly-connected cross-sections, such as trapezoidal microchannels, is complex. In the present study, the square root of cross-sectional area is proposed as the characteristic length scale for Nusselt number. Using analytical solutions of rectangular, elliptical, and triangular ducts, a compact model for estimation of Nusselt number of fully-developed, laminar flow in microchannels of arbitrary cross-sections with “H1” boundary condition (constant axial wall heat flux with constant peripheral wall temperature) is developed. The proposed model is only a function of geometrical parameters of the cross-section, i.e., area, perimeter, and polar moment of inertia. The present model is verified against analytical and numerical solutions for a wide variety of cross-sections with a maximum difference on the order of 9%.

scholarly journals LONG WAVES IN CHANNELS Of ARBITRARY CROSS-SECTION

1968 ◽  
Vol 1 (11) ◽  
pp. 12
Author(s):  
D.H. Peregrine
Keyword(s):  
Cross Section ◽  
Gravity Waves ◽  
Reasonable Agreement ◽  
Arbitrary Constant ◽  
Arbitrary Cross Section ◽  
Cross Sectional ◽  
Curved Channels ◽  
Channel Curvature ◽  
M Channels ◽  
Theoretical Results

This warier summarises some recent work on lone gravity waves on still water m channels of arbitrary constant cross-section. Theoretical results have been obtained for both straight and curved channels. Some experimental work has been performed m straight trapezoidal channels and shows reasonable agreement with theory. For straight channels some details of the second approximation are given, and the cases where the approximation breaks down are indicated. For curved channels it is found that the effect of channel curvature is more pronounced when the cross-sectional shane of the channel is not symmetric with resnect to its centre-line.

Slip-Flow Pressure Drop in Microchannels of General Cross Section

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
M. Bahrami ◽  
A. Tamayol ◽  
P. Taheri
Keyword(s):  
Boundary Condition ◽  
Pressure Drop ◽  
Cross Section ◽  
Slip Flow ◽  
Homogeneous Boundary Condition ◽  
Slip Boundary Condition ◽  
Arbitrary Cross Section ◽  
Geometrical Parameters ◽  
Cross Sectional ◽  
Slip Boundary

In the present study, a compact analytical model is developed to determine the pressure drop of fully-developed, incompressible, and constant properties slip-flow through arbitrary cross section microchannels. An averaged first-order Maxwell slip boundary condition is considered. Introducing a relative velocity, the difference between the bulk flow and the boundary velocities, the axial momentum reduces to Poisson’s equation with homogeneous boundary condition. Square root of area is selected as the characteristic length scale. The model of Bahrami et al. (2006, “Pressure Drop of Laminar, Fully Developed Flow in Microchannels of Arbitrary Cross Section,” ASME J. Fluids Eng., 128, pp. 1036–1044), which was developed for no-slip boundary condition, is extended to cover the slip-flow regime in this study. The proposed model for pressure drop is a function of geometrical parameters of the channel: cross sectional area, perimeter, polar moment of inertia, and the Knudsen number. The model is successfully validated against existing numerical and experimental data collected from different sources in literature for several shapes, including circular, rectangular, trapezoidal, and double-trapezoidal cross sections and a variety of gases such as nitrogen, argon, and helium.

Vibration of Stress-Free Hollow Cylinders of Arbitrary Cross Section

1995 ◽  
Vol 62 (3) ◽  
pp. 718-724 ◽  
Author(s):  
K. M. Liew ◽  
K. C. Hung ◽  
M. K. Lim
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Three Dimensional ◽  
Arbitrary Cross Section ◽  
Mode Shapes ◽  
Width Ratio ◽  
Cross Sectional ◽  
Hollow Cylinders ◽  
Displacement Based ◽  
Bending Modes

A three-dimensional elasticity solution to the vibrations of stress-free hollow cylinders of arbitrary cross section is presented. The natural frequencies and deformed mode shapes of these cylinders are obtained via a three-dimensional displacement-based energy formulation. The technique is applied specifically to the parametric investigation of hollow cylinders of different cross sections and sizes. It is found that the cross-sectional property of the cylinder has significant effects on the normal mode responses, particularly, on the transverse bending modes. By varying the length-to-width ratio of these elastic cylinders, interesting results demonstrating the dependence of frequencies on the length of the cylinder have been concluded.

Anisotropic Elastic Tubes of Arbitrary Cross Section Under Arbitrary End Loads: Separation of Beamlike and Decaying Solutions

2004 ◽  
Vol 72 (4) ◽  
pp. 500-510
Author(s):  
P. Ladevèze ◽  
J. G. Simmonds
Keyword(s):  
Cross Section ◽  
Arbitrary Cross Section ◽  
Constant Thickness ◽  
Cross Sectional ◽  
Edge Zone ◽  
One Dimensional ◽  
Elastic Tubes ◽  
Anisotropic Elastic ◽  
Relative Errors ◽  
Complex Valued

First approximation analytical solutions are constructed for finite and semi-infinite, fully anisotropic elastic tubes of constant thickness h and arbitrary cross section, subject to purely kinetic or purely kinematic boundary conditions. Final results contain relative errors of O(h∕R), where R is some equivalent cross sectional radius. Solutions are decomposed into the sum of an exact beamlike or Saint-Venant solution, treated in Ladevèze et al. (Int. J. Solids Struct., 41, pp. 1925–1944, 2004) and extended in an appendix; a rapidly decaying edge-zone solution; and a slowly decaying semi-membrane-inextensional-bending (MB) solution. Explicit conditions on the boundary data are given that guarantee decaying solutions. The MB solutions are expressed as an infinite series of complex-valued exponential functions times real-valued one-dimensional eigenfunctions which satisfy a fourth-order differential equation in the circumferential coordinate and depend on the pointwise cross sectional curvature only.

Slip-Flow Pressure Drop in Microchannels of General Cross-Section

2008 ◽  
Author(s):  
A. Tamayol ◽  
M. Bahrami ◽  
P. Taheri
Keyword(s):  
Boundary Condition ◽  
Pressure Drop ◽  
Cross Section ◽  
Slip Flow ◽  
Homogeneous Boundary Condition ◽  
Slip Boundary Condition ◽  
Arbitrary Cross Section ◽  
Geometrical Parameters ◽  
Cross Sectional ◽  
Slip Boundary

In the present study, a compact analytical model is developed to determine the pressure drop of fully-developed, incompressible, and constant properties slip-flow through arbitrary cross-section microchannels. An averaged first-order Maxwell slip boundary condition is considered. Introducing a relative velocity, the difference between the bulk flow and the boundary velocities, the axial momentum reduces to the Poisson’s equation with homogeneous boundary condition. Square root of area is selected as the characteristic length scale. Bahrami et al.’s model, which was developed no-slip boundary condition, is extended to cover the slip-flow regime in this study. The proposed model is a function of geometrical parameters of the channel: cross-sectional area, perimeter, polar moment of inertia and the Knudsen number. The model is successfully validated against existing numerical and experimental data from different sources in the literature for several shapes, including: circular, rectangular, trapezoidal, and double-trapezoidal cross-sections and a variety of gases such as: nitrogen, argon, and helium.

The effect of compressibility on the speed of propagation of a vortex ring

1985 ◽  
Vol 397 (1812) ◽  
pp. 87-97 ◽  
Keyword(s):  
Vortex Ring ◽  
Cross Section ◽  
Sound Speed ◽  
Rigid Rotation ◽  
Compressible Fluids ◽  
Perfect Gas ◽  
Cross Sectional ◽  
The Core ◽  
Speed Of Propagation ◽  
Compressibility Effects

A circular vortex filament of radius R , cross sectional area π a 2 and circulation Г propagates steadily in an inviscid, calorically perfect gas. The flow outside the filament is assumed to be irrotational and isentropic. If it is further assumed that a/R ≪ 1, the cross section is approximately circular and the speed of propagation of the filament is shown to depend on the distribution of circulatory velocity v 0 and entropy s 0 within the core. If s 0 is constant and equal to its value in the isentropic exterior of the filament, the vortex ring is slowed down by compressibility effects, whatever the distribution of circulatory velocity. If the circulatory velocity corresponds to rigid rotation in the core cross section, the speed, U , of propagation is given by U = Γ/4π R [ln 8R/ a - ¼ - 5/12 M 2 + O ( M 4 )], where M is the Mach number Γ /2π ac ∞ and c ∞ is the sound speed far from the vortex ring. Numerical results for finite M are also given in this case. These results enable the cut-off theory of filament motion to be extended to compressible fluids.

scholarly journals Evolution of long water waves in variable channels

1994 ◽  
Vol 266 ◽  
pp. 303-317 ◽  
Author(s):  
Michelle H. Teng ◽  
Theodore Y. Wu
Keyword(s):  
Cross Section ◽  
Wave Energy ◽  
Water Waves ◽  
Mass Conservation ◽  
Arbitrary Cross Section ◽  
Channel Cross Section ◽  
Cross Sectional ◽  
Reflected Waves ◽  
Conservation Property ◽  
Divergent Channel

This paper applies two theoretical wave models, namely the generalized channel Boussinesq (gcB) and the channel Korteweg–de Vries (cKdV) models (Teng & Wu 1992) to investigate the evolution, transmission and reflection of long water waves propagating in a convergent–divergent channel of arbitrary cross-section. A new simplified version of the gcB model is introduced based on neglecting the higher-order derivatives of channel variations. This simplification preserves the mass conservation property of the original gcB model, yet greatly facilitates applications and clarifies the effect of channel cross-section. A critical comparative study between the gcB and cKdV models is then pursued for predicting the evolution of long waves in variable channels. Regarding the integral properties, the gcB model is shown to conserve mass exactly whereas the cKdV model, being limited to unidirectional waves only, violates the mass conservation law by a significant margin and bears no waves which are reflected due to changes in channel cross-sectional area. Although theoretically both models imply adiabatic invariance for the wave energy, the gcB model exhibits numerically a greater accuracy than the cKdV model in conserving wave energy. In general, the gcB model is found to have excellent conservation properties and can be applied to predict both transmitted and reflected waves simultaneously. It also broadly agrees well with the experiments. A result of basic interest is that in spite of the weakness in conserving total mass and energy, the cKdV model is found to predict the transmitted waves in good agreement with the gcB model and with the experimental data available.

scholarly journals Shear Lag Analysis due to Flexure of Prismatic Beams with Arbitrary Cross-Sections by FEM

2021 ◽  
Author(s):  
Dang-Bao Tran ◽  
Jaroslav Navrátil
Keyword(s):  
Numerical Method ◽  
Cross Section ◽  
Cross Sections ◽  
Arbitrary Cross Section ◽  
Local Analysis ◽  
3D Analysis ◽  
Shear Lag ◽  
Isoparametric Element ◽  
Cross Sectional ◽  
The Cross

This paper presents the use of a finite element method (FEM) to analyze the shear lag effect due to the flexure of beams with an arbitrary cross-section and homogeneous elastic material. Beams are constrained by the most common types of supports, such as fixed, pinned, and roller. The transverse, concentrated, or distributed loads act on the beams through the shear center of the cross-section. The presented FEM transforms the 3D analysis of the shear lag phenomenon into separated 2D cross-sectional and 1D beam modeling. The characteristics of the cross-section are firstly derived from 2D FEM, which uses a 9-node isoparametric element. Then, a 1D FEM, which uses a linear isoparametric element, is developed to compute the deflection, rotation angle, bending warping parameter, and stress resultants. Finally, the stress field is obtained from the local analysis on the 2D-cross section. A MATLAB program is executed to validate the numerical method. The validation examples have proven the efficiency and reliability of the numerical method for analyzing shear lag flexure, which is a common problem in structural design.

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