Method For Calculation Of Integrals Research Articles

Two-dimensional turbulent separated flow

TkJTANY different flow cases with nominally twolYJLdimensional turbulent separated flow regions are discussed in Refs. 1 and 2. Because intermittent flow reversal and backflow occur in the near-wall region, directionally sensitive measurement techniques must be used to examine the flow structure. The experimentally-observed structure of detached flows on streamlined surfaces and around sharp-edged corners is discussed in detail for steady and unsteady incompressible and compressible cases. In all cases, large-scale structures dominate the flow behavior, producing large shearing stresses in the middle of the detached shear flow and strongly influencing the local intermittent backflow. The turbulence structure strongly lags the mean flow behavior in the detachment and reattachment processes. For unsteady periodic separation, there is considerable hysteresis of the flow during a cycle. A number of differential and integral calculation methods are discussed. Traditional attached flow turbulence models and correlations do not describe detached flows well. Methods that include experimentally-observed features and/or correlations of detached flow parameters seem to perform best, although further improvements are still needed.

A UHF probe for NMR micro-imaging experiments.

A long solenoid is capable of producing, at least theoretically, a B1 field that is relatively more homogeneous than other known structures. This paper describes the design of a UHF probe incorporating a one-turn solenoid associated with a pair of parallel-plate transmission line feeders and presents a practical example used in a 360-MHz spectrometer for micro-imaging experiments. Numerical evaluation of the solenoid inductance using King's method for direct calculation of elliptic integrals is also included.

Flow analysis in two-dimensional impeller of centrifugal pump by the surface singularity method combined with a flow-model of three-dimensional boundary layer.

An iterative solution method has been developed for calculating flows in two-dimensional centrifugal pump impellers. The method was formed by combining an integral calculation method for a three-dimensional boundary layer on the whole channel wall with a surface singularity method for a two-dimensional inviscid core-flow. In the inviscid analysis, the blade shape and the area of flow passage were modified by calculated displacement thickness of boundary layer, and then a finite number of vortices were distributed on the modified blade surfaces with the Kutta condition at the outlet edge on the suction surface. This method was applied to flows in two impellers with different blade thicknesses. Comparing the results of calculated and measured flows, both were qualitatively in good agreement, including an effect of the outlet blade surface with constant radius, which showed that relative velocity near the pressure surface was faster than one near the suction surface.

Contribution to the theory of laminar momentumless wakes

The self-similar laws of decay of the velocity field in a plane momentumless viscous incompressible wake of a hydrodynamic propulsion system were first analyzed in [1]. Turbulent wakes in the near and far flow regions were investigated in [2] on the basis of an integral calculation method. In [3] the asymptotic laws of degeneration of the passive admixture concentration, temperature and velocity fields were obtained, making it possible to estimate the effect of purely molecular diffusion and convective transfer o the passive impurity concentration distribution in the wake. In the present paper the limiting self-similar solutions of the problem of a swirled momentumless viscous incompressible wake are obtained, together with the self-similar solutions of the problem of the development of a plane momentumless wake in a medium nonuniform with respect to temperature (passive admixture concentration).

A classical fluid-like approach to the density-functional formalism of many-electron systems

Within the framework of a new local thermodynamic transcription [Ghosh, Berkowitz, and Parr, Proc. Natl. Acad. Sci. U.S.A. 81, 8028 (1984)] of ground state density-functional theory, the electron cloud is interpreted as an inhomogeneous fluid at an effective local temperature. The correlation functions are defined and integral equations are introduced for this electron fluid along the lines of the theory of inhomogeneous classical liquids. An equation of state and a local compressibility equation are also developed. Extension to a mixture of two fluid components is explored as a means for investigating spin-polarized cases. The formalisms broaden the scope of the hydrodynamic analogy to quantum mechanics and provide an alternative integral equation method for calculation of the electron density of many-electron systems.

An unsteady momentum integral calculation for the turbulent spot

Velocity and wall shear stress measurements were made along the centerline of turbulent spots generated in a laminar boundary layer on a flat plate. A momentum integral calculation method based on two-dimensional boundary layer theory and the assumption of potential flow above the spot was developed by transforming the unsteady momentum equation into nondimensional similarity coordinates. The centerline wall shear stress, calculated using ensemble-averaged data for the velocity field, shows fairly good agreement with the measured values.

Direct and inverse integral calculation methods for three-dimensional turbulent boundary layers

It is both a privilege and a pleasure to be asked to contribute to an edition of the Aeronautical Journal to celebrate Professor Alec Young's 70th birthday. He has been my mentor for nearly a quarter of a century.The subject of integral prediction methods for boundary layers is one with which Alec Young has long been associated. Starting with a method for laminar compressible flow he has guided the development of, amongst others, integral methods for compressible laminar flow with heat transfer, three-dimensional laminar flow, three-dimensional turbulent flow and flows with pressure gradients normal to the surface.This paper attempts to describe the integral methods for compressible turbulent boundary layers which are most widely used in aerodynamic design and to discuss the application of inverse integral methods to three-dimensional separated flows.

A Computation and Comparison With Measurements of Transonic Flow in an Axial Compressor Stage With Shock and Boundary Layer Interaction

The flow field within a transonic axial flow compressor stage has been computed using a three-dimensional time-marching technique. Limited viscous effects are considered by including a calculation of the blade surface boundary layers. The boundary layer calculation forms an integral part of the whole computation scheme, which consists of, respectively: (i) inviscid Mach number calculations, (ii) blade surface boundary layer displacement thickness calculations, (iii) inviscid Mach number calculations with mass flow adjustment (based on the calculated displacement thicknesses) on the blade surfaces. The boundary layer computation is done by using integral calculation methods and has specifically been developed to account for a shock and boundary layer interaction (should one exist). Comparisons are made with measured results obtained with an advanced laser velocimeter. The calculated Mach number contours are in extremely good agreement with the experimental results. It is concluded that the calculation technique is a useful tool in the design of transonic axial flow turbomachines.

A Single Formula for the Complete Velocity Profile in a Turbulent Boundary Layer

An expression for the wake deficit g(Π, y/δ) in a turbulent boundary layer is combined with a function f(U+) for the inner layer to yield a single formula for the complete velocity profile in the form y+ exp {Kg(Π, y/δ)} = f(U+). Close agreement is shown with data from boundary layers in a wide range of pressure gradients and beneath a turbulent free stream. The formula will be particularly useful for integral and differential calculation methods.

An integral equation method for the numerical calculation of ion drift and diffusion in evaporated dielectrics

A method is presented to solve the partial differential equation for a time dependent drift — diffusion process in a one dimensional finite space. The method is based on an integral equation technique, which is easily solved numerically on a digital computer. This technique has been used to study the transport phenomenon of mobile ions in evaporated dielectrics under influence of an electric field.

Turbulent Compressible Three-DimensionaI Mean Flow Profiles

A problem facing theoretical fluid dynamicists today is the prediction of the growth and development of compressible three-dimensional transitional/turbulent boundary layers. Cross-flow pressure gradients which maintain flow components in the lateral direction are found on virtually all proposed aeronautical flight vehicles. Several numerical' and integral' three-dimensional compressible turbulent boundary-layer calculation methods have been developed; however, there is a very limited amount of compressible threedimensional turbulent boundary-layer data available for use as test cases (e.g., Refs. 5, 6), in contrast to incompressible flow. This paper presents the first detailed developing threedimensional compressible transitional/turbulent boundary-layer profile test case to check out, develop, and improve the accuracy of the numerical and integral calculation methods.