Symmetric Shear Research Articles

A Transformation for the Numerical Solution of Two-Dimensional Free Mixing Flow Problems

A coordinate transformation which removes the variation of the mixing layer width is introduced. The boundary layer equations are then solved in the new coordinate plane using a multilevel difference scheme. The calculated results for two-dimensional symmetric mixing and free shear layer flows for turbulent flow are compared with experimental data and with other solutions.

Transverse surface wave as high-frequency limit of a shear-horizontal piezoelectric plate wave

It is hown that the lowest symmetric shear horizontal plate mode in a width-polarized ferroelectric ceramic plate becomes a double transverse-surface (Bleustein) wave when the wavelength is small compared with the plate thickness. The frequency equations for the above plate wave are derived for different boundary conditions, as well as their surface-wave approximations. Numerical solutions are used to determine the behavior of the wave at all frequencies.

Modified Beam Method for Analyzing Symmetrical Interconnected Shear Walls

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Two-dimensional turbulent wakes

The equations of mean motion indicate that two-dimensional turbulent wakes, when subjected to appropriately tailored adverse pressure gradients, can be self-preserving. An experimental examination of two nearly self-preserving wakes is reported here. Mean velocity, longitudinal and lateral turbulence intensity, inter-mittency and shear stress distributions have been measured and are compared with Townsend's data from the small-deficit undistorted wake. In comparison with the undistorted case, the present wakes have slightly lower turbulent intensities and significantly lower shear stresses, all quantities being non-dimensionalized by a local velocity scale taken as the maximum mean velocity deficit. A consideration of the reasons for the shear stress reduction leads to an expression from which the shear stresses in any symmetrical free equilibrium shear flow can be found. This relationship is used to calculate the rate of growth in the measured wakes, with reasonable success.

Axisymmetric Torsional Wave Propagation in Circular Viscoelastic Rods

A semi-infinite viscoelastic circular cylindrical rod is subjected at its end surface to an arbitrary radial, axially symmetric tangential shear stress distribution, harmonic in time. Expressions are derived for the tangential and radial shear stress distributions, as well as for the tangential displacement. Near the forced end of the rod, the tangential displacement takes on a complicated form which consists of the superposition of a simple rotational oscillation of the undeformed cross section and motions deforming the cross section in its plane. The last type of motion permits infinitely many nodal circles. Owing to the large magnitude of the space attenuation coefficients, this second type of motion dies out rapidly and only the simple rotation, which is less attenuated, remains. Numerical computations were performed for two polymethyl methacrylate rods of 1 and 10 cm diam, subjected to a shear stress distribution varying periodically with time and cubic in the radial distance from the center. The results are presented in graphical form.

Stress Concentration in Nonlinear Creep of a Simple Shell

A long, thin circular cylindrical shell is loaded at one edge by symmetrical radial shear Qx and bending moment Mx. (No interior pressure.) The shell is made of material which under applied stress creeps with a strain rate which is proportional to the rth power of the stress. Previous results are used to derive, approximately, the greatest stress in the shell for any Qx, Mx, and r. It is shown that for any load the greatest stress decreases as r increases, and is approximately a linear function of 1/r. The case r = 1 is exactly analogous to a linear elastic problem, and the case r → = ∞ corresponds exactly to a perfectly plastic problem. Results for any exponent r may thus be found approximately by simple interpolation between results obtained in linearelastic analysis and perfectly plastic analysis.

Some aspects of friction and wear of tyres arising from deformations, slip and stresses at the ground contact

Beginning with the subject of friction of rubber on a dry surface which has been theoretically and experimentally studied at the Vehicle Research Laboratory, the important role of tread patterns in adhesion on wet roads is emphasized. The tread patterns cause the stress distribution in the area of contact to be wavy and variable both in time and space. The deformation of an inflated tyre against a hard, flat surface induces a symmetrical and inwardly-directed shear stress distribution at the ground contact. The shear stresses have been calculated from frictional consideration under certain simplifying assumptions. Apart from the importance of cornering and stability of automobiles, the study of a tyre set at a slip angle (the angle between the direction of travel, and the plane of the tyre) is also important from the point of view of wear, especially because of the high temperatures prevailing at the tread surface under these conditions. The forces developed on wet road surfaces are considered. The results obtained show that cornering forces on well-lubricated surfaces are very low. This leads to the relatively new concept of employing a high hysteresis tread rubber to give additional friction on such surfaces. The tests carried out elsewhere at high speeds on roads sprayed with water have been of limited value in evaluating the hysteresis component of friction resulting from deformations since the change in tread properties resulting from unknown contact temperatures due to surface heating was not considered. Measurements on the glass plate apparatus with speed of only 4 1 2 cm/sec on dry and lubricated surfaces indicate that the contribution due to hysteresis can be appreciable. The cornering forces developed by two tyres with different resilience and hardness were measured on dry and well-lubricated surfaces with different roughness. Finally, a short note is added describing some of the theories explaining the mechanism of wear occurring at the contact between rubber and a hard surface.

A linearized theory for rotational supercavitating flow

In this paper methods are given for establishing qualitative and quantitative measures of the effects of rotation in supercavitating flows past slender bodies. A linearized theory is developed for steady, two-dimensional flow under the assumption that the flow has a constant rotation throughout. The stream function of the rotational flow satisfies Poisson's equation. By using a particular solution of this equation, the rotational problem is reduced to a problem involving Laplace's equation and harmonic perturbation velocities. The resulting boundary-value problem is solved by use of conformal mapping and singularities from thinairfoil theory. The theory is then applied to asymmetric shear flow past wedges and hydrofoils and to symmetric shear flow past wedges. The presence of rotation is shown to create significant changes in the forces acting on the slender bodies and in the shape and size of the trailing cavities.

The End Problem of Cylinders

Abstract When self-equilibrating rotationally symmetric normal and shear tractions sk(r), tk(r) act on an end face of a cylinder, stresses ensue which decay with distance from the loaded end as e−αk2, where αk is the real part of an eigenvalue parameter γk. The analysis is carried out in a variational approximation; the method is based on Sadowsky-Sternberg stress functions Φk(z, r) which are so modified as to identically satisfy the equilibrium equations. The factors Fk(r) of the functions Φk = Gk(z) Fk(r) or Φk = Hk(z)Fk(r) (k = 2, 4, 6, …) are polynomials of degree k + 2, which are orthogonal in a “generalized sense”; their appropriate derivatives constitute sk(r) and tk(r). The polynomials sk(r), tk(r), of degrees k and k + 1, respectively, form two complete sets of tractions in terms of which arbitrary rotationally symmetric self-equilibrating end tractions may be expanded. The factors Gk(z), Hk(z) of the stress functions Φk are exponentially decaying sinusoidals; Gk(z) is appropriate for the normal load problem, Hk(z) for the shear-load problem. The function Φ2 leads, for ν = 0.3, to the fundamental variational eigenvalue γ2 = 2.69 + i 1.34, as contrasted with the rigorous value γ2 = 2.72 + i 1.35.

Statistical Theory of Turbulence, III Extension of Similarity Theory of Turbulence

As the second stage of developments of our statistical theory of turbulence, extenston of the initial-period similarity-law is attempted in order to give general explanations to distributions of turbulent quantity across the mean direction of flow, particularly in decaying and nondecaying shear turbulence. In (A), as the statistical hypothesis in our theory, we first give the Gaussian form to P -function. Then, as a simple theoretical result of the above extended similarity law, it can be proved that the distribution of an intermittency factor γ, introduced by A. A.Townsend in his study of turbulent wake, takes the form of Gaussian integral function, which is surveyed further by experimental observations. In (B), by introducing the above result to our fundamental expressions of turbulent intensity, it is interpreted that in all the cases of shear turbulence \(\textbf{\itshape u}/\sqrt{\textbf{\itshape u}^{2}+\textbf{\itshape v}^{2}}\), \(\overline{\textbf{\itshape uv}}/\textbf{\itshape uv}\) and w / u have respective nearly constant values in the y -direction. These characters are investigated by experimental measurements. In (C), as the first step to study \(\overline{V}_{*}\)-function in shear turbulent state, the problem of distortion of isotropic turbulence in a contracting stream is taken up. By rather qualitative hydrodynamical discussion on \(\overline{V}_{*}\)-function in the shearless turbulence with a non-uniform mean velocity, our fundamental formula in this case gives a result that \(\textbf{\itshape v}/\textbf{\itshape u}{=}\overline{U}(x)/\overline{U}_{0}\), which is also checked in our measurements. In (D), by using all the results in (A) and (C) we derive theoretically the Reynolds stress distribution across a symmetrical shear turbulence, which is the simplest shear turbulence and corresponds nearly to a free mixing flow. Mean velocity distribution is also determined by the Reynolds equations. They are compared with our experimental measurements.