Dimensionless Radii Of Curvature Research Articles

A Numerical Computation of Heat and Fluid Flow in L-Shaped Curved Channels

A numerical analysis of steady, hydrodynamically and thermally fully developed, incompressible laminar flow with constant physical properties is presented. Two different geometrical cases of L-shaped channels are considered: right-curved and left-curved. The solution of the discretized continuity, momentum, and energy equations was obtained by using an elliptic Fortran program based on the SIMPLE algorithm. Solutions are obtained for Dean number ranges from 4.1 to 210.6 with dimensionless radius of curvature of 100, and Prandtl number of 0.7. The secondary flow streamlines, the isotherms, average Nusselt numbers, and friction factors are presented depending on Dean number and L-shaped channel orientation. As a result, it is observed that the secondary flows resulting from centrifugal forces change the distribution of the velocity and the temperature fields. In addition, it is determined that channel orientation has a profound effect on the flow and temperature fields.

Determination of the distribution density of specular points on the sea surface: Formulation of the inverse problem

With reference to the model of a random Gaussian homogeneous cylindrical (one-dimensional) sea surface z = ζ(x), the inverse problem is formulated in the form of the integral Fredholm equation of the first kind to determine the distribution density of the number of specular points on the sea surface. The kernel of the equation is determined in terms of the Fourier transform of the distribution density of radii of surface curvature at the points of specular reflection. The equation derived by the author earlier for the distribution density and written for the dimensionless radius of curvature contains no parameters, a result that is indicative of the universal character of this distribution for an arbitrary Gaussian surface. The validity of the original formulas obtained in this paper was verified by simulations.

The Influence of Finite Three-Dimensional Multiple Axial Erosions on the Fatigue Life of Partially Autofrettaged Pressurized Cylinders

Erosion geometry effects on the mode I stress intensity factor (SIF) for a crack emanating from the farthest erosion’s deepest point in a multiply, finite-length or full-length eroded, partially autofrettaged, pressurized, thick-walled cylinder is investigated. The problem is solved via the FEM method. Autofrettage, based on von Mises’ yield criterion, is simulated by thermal loading and SIFs are determined by the nodal displacement method. SIFs were evaluated for a variety of relative crack depths, a/t=0.01-0.30 and crack ellipticities, a/c=0.5-1.5 emanating from the tip of the erosion of various geometries, namely, (a) semi-circular erosions of relative depths of 1–10% of the cylinder’s wall thickness, t; (b) arc erosions for several dimensionless radii of curvature, r′/t=0.05-0.3; and (c) semi-elliptical erosions with ellipticities of d/h=0.5-1.5. In the cases of finite erosions, the semi-erosion length to the semi-crack length, Le/c, was between two and ten, erosion angular spacing, α, was between 7 and 120 degrees, whereas percent autofrettage investigated included 30%, 60%, and 100%. The normalized SIFs and the normalized effective SIFs of a crack emanating from the farthest finite erosion are found to rise sharply for values of Le/c<3. Both the normalized SIF and normalized effective SIF values are mitigated as the amount of partial autofrettage increases with the most rapid decrease occurring between 0–60% autofrettage. The purpose of this study is to detail these findings.

Erosions and Their Effect on the Fatigue Life of Thick Walled, Autofrettaged, Pressurized Vessels

This paper summarizes the results that have been found in evaluating the effect of erosions on thick walled, autofrettaged, pressurized, cracked vessels. The problem is solved numerically via the FEM method. Autofrettage, based on von Mises yield criterion, is simulated by thermal loading and stress intensity factors (SIF’s) are determined by the nodal displacement method. SIF’s were evaluated for a variety of relative crack depths a/t and crack ellipticities a/c emanating from the tip of the erosion of various geometries, namely, (a) semi-circular erosions of small relative depths of the cylinder’s wall thickness t; (b) arc erosions for several dimensionless radii of curvature r′/t; and (c) semi-elliptical erosions with ellipticities of d/h. Other parameters evaluated were, in the cases of finite erosions, the semi-erosion length to the semicrack length Le/c, the erosion angular spacing α, and the autofrettage level. First, we summarize the differences found between a vessel with one erosion and one with multiple erosions. We show that for full cylinder length erosions, the erosions tend to make smaller cracks more dangerous than larger cracks in fully autofrettaged vessels and that as the crack grows the stress intensity factor initially decreases. We then show that as the crack grows further, the effect is to increase the effective stress intensity factor (SIF) but also to practically void the existence of the erosion. We show further that lower levels of autofrettage will lead to higher effective SIF’s but that partially eroded cylinders (cylinders where erosions are a fraction of the cylinder length) lead to lower SIF’s. Affecting these values in all cases, of course, are the erosion geometry and depth as well as the crack geometry and depth.

AXIAL FLOW BETWEEN ECCENTRIC CYLINDERS

Plastic pipe production must use annular dies with eccentricity to compensate for gravity flow in the cooling chamber. Here an algebraic analytic solution is obtained in bipolar cylindrical coordinates for Newtonian and power-law liquids. Dimensionless average velocity profiles around eccentric dies are provided to help plastic pipe engineers determine the needed eccentricity. In curved hose manufacture, the die centerpiece is made eccentric to cause curvature. Dimensionless radii of curvature are plotted to help curved hose manufacturers select proper die eccentricity.

The Influence of Multiple Axial Erosions on a Three-Dimensional Crack in Determining the Fatigue Life of Autofrettaged Pressurized Cylinders

Erosion geometry effects on the mode I stress intensity factor (SIF) for a crack emanating from an erosion’s deepest point in a multiply eroded, autofrettaged, pressurized, thick-walled cylinder are investigated. The problem is solved via the finite element method (FEM). Autofrettage, based on von Mises yield criterion, is simulated by thermal loading and SIFs are determined by the nodal displacement method. SIFs are evaluated for a variety of relative crack depths, a0/t=0.01-0.40, and crack ellipticities, a0/c=0.5-1.5, emanating from the tip of erosions of different geometry, namely: (a) semi-circular erosions of relative depths of 1–10 percent of the cylinder’s wall thickness, t; (b) arc erosions for several dimensionless radii of curvature, r′/t=0.05-0.4; and (c) semi-elliptical erosions with ellipticities of d/h=0.3-2.0. The erosion separation angle, α, is taken from 7 to 360 deg. Deep cracks are found to be almost unaffected by the erosion. The effective SIF for relatively short cracks is enhanced by the presence, separation distance and geometry of the erosion, as well as the crack geometry, and may result in a significant decrease in the vessel’s fatigue life of up to an order of magnitude.

The Influence of Multiple Axial Erosions on the Fatigue Life of Autofrettaged Pressurized Cylinders

Erosion geometry effects on the mode I stress intensity factor (SIF) for a crack emanating from an erosion’s deepest point in a multiply eroded, autofrettaged, pressurized, thick-walled cylinder are investigated. The problem is simulated as a two-dimensional problem and is solved via the finite element method. Autofrettage, based on von Mises yield criterion, is simulated by thermal loading and SIFs are determined by the nodal displacement method. SIFs are evaluated for a variety of relative crack lengths, a0/t=0.01-0.45 emanating from the tip of erosions of different geometries, namely, (a) semi-circular erosions of relative depths of 1–10 percent of the cylinder’s wall thickness, t; (b) arc erosions for several dimensionless radii of curvature, r′/t=0.05-0.4; and (c) semi-elliptical erosions with ellipticities of d/h=0.5-1.5, and erosion span angle, α, from 6 deg to 360 deg. The effective SIF for relatively short cracks is found to be increased by the presence of the erosion, which in turn may result in a significant decrease in the vessel’s fatigue life of up to an order of magnitude. Deep cracks are found to be almost unaffected by the erosion.

Cracks Emanating From an Erosion in a Pressurized Autofrettaged Thick-Walled Cylinder—Part I: Semi-Circular and Arc Erosions

Erosion geometry effects on the mode I stress intensity factor (SIF) for a crack emanating from the erosion’s deepest point in an autofrettaged, pressurized, thick-walled cylinder are investigated. The problem is solved via the FEM method and knowledge of the asymptotic behavior of short cracks. Autofrettage, based on von Mises yield criterion, is simulated by thermal loading and SIFs are determined by the nodal displacement method. SIFs are evaluated for a variety of relative crack lengths, a0/W = 0.01 – 0.45, emanating from the tip of erosions of different geometries. In Part I of this paper, two configurations are considered: (a) semi-circular erosions of relative depths of 5 percent of the cylinder’s wall thickness, W; and (b) arc erosions for several dimensionless radii of curvature, r′/W = 0.05 – 0.4. While deep cracks are almost unaffected by the erosion, the effective SIF for relatively short cracks is found to be significantly enhanced by the presence and geometry of the erosion and might reduce the vessel’s fatigue life.

Forced Laminar Convection in a Curved Isothermal Square Duct

A numerical study is made to investigate the forced laminar convection in the hydrodynamically and thermally fully developed region of a curved isothermal square duct. Solutions with one and two pairs of vortices superimposed on the main flow are obtained. In the thermally fully developed region, a three-dimensional energy equation of elliptic type is reduced to a two-dimensional one with an eigenvalue, and the axial diffusion term is considered for a small value of the Peclet number. Flow characteristics for cases of dimensionless radius of curvature β = ∞, 50, 10, and 5 with the square of the Dean number ranging from 0 to 106 in a square duct are studied. In addition, heat transfer characteristics for large dimensionless radii of curvature, Pr=0.7 and 7.0, and Peclet number Pe=∞, 10, 5, and 1 are also examined.

Mass Transfer of a Submerged Jet Impinging on a Cylindrical Surface

The mass transfer rate and wall shear stress for a submerged two‐dimensional slot jet impinging on a convex surface were obtained by measuring the limiting current of the electrochemical reduction of ferricyanide ion. The local mass transfer rate was measured over a range of nozzle Reynolds number (Re) from 2,500 to 13,000, dimensionless nozzle height from 0.2 to 3 and dimensionless radius of curvature from 2 to 6. A relation of was obtained. The value of β was found to be in the impingement region and in the wall jet region.