Showing papers on "Streamlines, streaklines, and pathlines published in 1975"

Some remarks on the initiation of inertial Taylor columns

Abstract: The quasi-geostrophic flow over an obstacle placed on the lower of two horizontal planes in rapid rotation about the vertical z axis is considered. The flow field is calculated in the limit of vanishing viscosity after assuming the flow far upstream of the obstacle to be uniform, of magnitude V. The effects that occur in homogeneous flow are compared with those that occur in stratified flow.If the flow is homogeneous, there is a region of closed streamlines if \[ h_0R^{-1} > \min_r\left[r\left/\int_0^r xh(x)dx\right. \right], \] where the obstacle is assumed to be cylindrically symmetric and given by z = Hh0h(r/L), H is the distance between the planes and R is the Rossby number V/(fL). For any obstacle the right-hand side of (1) is greater than zero and hence h0 must be positive for a closed-streamline region to occur. It is argued, and illustrated by a particular example, that because (1) involves an integral of h(x) a representative flow pattern can be obtained for obstacles of less than critical height by considering the special case of a flat-topped obstacle, as is done by Ingersoll (1969).If the flow is stratified with constant Brunt-Vaisala frequency N, the condition for the existence of closed streamlines is shown to be \[ h_0R^{-1} > \min_r \left[B\int_0^{\infty}\int_0^{\infty}xt\cot h(Bt)h(x)J_0(tx)J_1(tr)\,dt\,dx \right]^{-1}, \] where B = NH/fL. In contrast to the homogeneous situation, the right-hand side of (2) can be zero and is so if the obstacle is somewhere vertical. Such obstacles will produce a closed-streamline region no matter now small their height and will hence not lead to patterns representative of smooth obstacles. This is because a stratified column of fluid cannot be stretched or compressed over an infinitesimal distance. Instead, the column bends markedly and the fluid flows around the obstacle. The critical conditions (1) and (2) for a number of specific obstacles are calculated and discussed.

The creeping motion of liquid drops through a circular tube of comparable diameter

Abstract: The creeping motion through a circular tube of neutrally buoyant Newtonian drops which have an undeformed radius comparable to that of the tube was studied experimentally. Both a Newtonian and a viscoelastic suspending fluid were used in order to determine the influence of viscoelasticity. The extra pressure drop owing to the presence of the suspended drops, the shape and velocity of the drops, and the streamlines of the flow are reported for various viscosity ratios, total flow rates and drop sizes.

The injection molding behavior of thermoplastics in thin rectangular cavities

Abstract: Two models are presented for simulating the injection molding of thermoplastics in thin, rectangular cavities. One is a much simplified bounded-radial-flow model that utilizes an existing numerical model for semi-circular cavities to adjust for the non-radial flow region bounded by the lateral walls. The other is a two-dimensional analysis which assumes that both viscosity and temperature change strongly across the narrow gap but vary weakly in the directions of flow. The latter analysis allows application of the potential theory and the determination of streamlines and progressing front shapes. Both models deal with a non-Newtonian viscosity with temperature variation. Comparisons between experimental and computational results are shown.

Pressure- and buoyancy-driven thermal convection in a rectangular enclosure

Abstract: Results are presented for unsteady laminar thermal convection in compressible fluids at various reduced levels of gravity in a rectangular enclosure which is heated on one side and cooled on the opposite side. The results were obtained by solving numerically the equations of conservation for a viscous, compressible, heat-conducting, ideal gas in the presence of a gravitational body force. The formulation differs from the Boussinesq simplification in that the effects of variable density are completely retained. A conservative, explicit, time-dependent, finite-difference technique was used and good agreement was found for the limited cases where direct comparison with previous investigations was possible. The solutions show that the thermally induced motion is acoustic in nature at low levels of gravity and that the unsteady-state rate of heat transfer is thereby greatly enhanced relative to pure conduction. The nonlinear variable density profile skews the streamlines towards the cooler walls but is shown to have little effect on the steady-state isotherms.

Aerodynamic Heating on 3-D Bodies Including the Effects of Entropy-Layer Swallowing

Abstract: A relatively simple method is presented to include the effects of entropy-layer swallowing in a method developed previously for calculating laminar, transitional, and turbulent heating rates on three-dimensional bodies in hypersonic flows. The boundary layer swallows the entropy layer when the boundary layer downstream of the nose region has entrained those streamlines which passed through the nearly normal part of the bow shock wave. The entropy at the edge of the boundary layer is then determined by equating the mass flow inside the boundary layer to that entering part of the bow shock wave. A new inviscid flowfield solution, which is an extension of Maslen's axisymmetric method, is developed to calculate the three-dimensional shock shape and couple the inviscid solution with the viscous solution. The calculated heating rates compare favorably with Mayne's theory and experimental data for blunted circular and elliptical cones at angles of attack. The effects of entropy layer swallowing on the calculated heating rates were small for laminar heating but large increases were noted for the turbulent heating rates. The computer program developed to calculate the results presented herein required only about 7 sec per streamline on the IM 370/165 computer.

The Interference of Two-Dimensional Parallel Jets : 3rd Report, The Region near the Nozzles in Triple Jets

Abstract: Flow patterns of three two-dimensional parallel jets were investigated experimentally. In order to examine the effect of varying the ratio of the velocities in the central and outside jets on flow pattern, detailed measurements of the flow by a hot-wire anemometer and flow visualization by oil film were carried out. The results obtained are as follows. (1) The flow fields near the nozzles can be classified in three patterns of the flow according to the velocity ratio. 82) There is a specific velocity ratio at which the flow is unstable. (3) The flow patterns by means of visualization agree well with those of the streamlines, qualitatively.

Behavior of Severely Asymmetric Flow in a Vaneless Diffuser

Abstract: There is controversy about the mechanism of decay of the asymmetric flow in the vaneless diffuser of centrifugal blowers. In order to clearly observe the behavior of asymmetric flow, every other flow passage of a centrifugal impeller is blocked with a punched plate so that a severely asymmetric flow is induced. The flow behavior in the vaneless diffuser is measured with unsteady flow measuring instruments as well as with conventional instruments for steady flow. The experimental results indicate that there are some flow phenomena which cannot be explained by the conception of mixing process. Such flow phenomena can be explained quantitatively as the isentropic energy exchange between relative streamlines due to the circumferential pressure variation. In addition, the wall roughness of the vaneless diffuser is changed, and the influence of the wall friction on the total pressure loss is compared with the predictions based on the two controversial hypothesis. While these data are handled, it is recognized that the time average total pressure is considerably larger than the mass average total pressure for a severely asymmetric flow. Therefore, if instruments for steady flow measurement are used near the impeller exit to measure the total pressure, the impeller efficiency may be overestimated and the diffuser efficiency may be underestimated.

A note on the creeping motion of a viscoelastic fluid past a sphere

Abstract: In this paper we consider the creeping motion of a viscoelastic fluid past a sphere. Detailed visualization of the flow was carried out using small suspended tracer particles, forReynolds numbers Re, less than 0.1 andWeissenberg numbers, We, ranging from 10−2 to 22. Although some upstream shifting of the streamlines was found for ReWe = O(1), comparison shows that the streamline shift is very much smaller than predicted by the theoretical analysis ofUltman andDenn (1971). This discrepancy has led us to re-examine the validity of the basic assumptions of the analysis, as reported in the last section of the present communication.

Developing flow in circular conduits: transition from plug flow to tube flow

Abstract: A numerical solution of the complete Navier–Stokes equations of motion by means of an implicit finite-difference method is presented for the following developing-flow problem: a piston forced with constant speed through an infinitely long tube of circular cross-section. The transition of the velocity profile of an incompressible isothermal Newtonian fluid from the plug-flow profile in front of the piston to the parabolic profile of developed flow is analysed. Streamlines, vorticity distributions, velocity profiles, the excess pressure drop and the entrance length are given for Reynolds numbers from 0 to 800.

Numerical solution of the Navier-Stokes equations for arbitrary two-dimensional airfoils

Abstract: A method for numerical solution of the Navier-Stokes equations for the flow about arbitrary airfoils or other bodies is presented. This method utilizes a numerically generated curvilinear coordinate system having a coordinate line coincident with the body contour. Streamlines, velocity profiles, and pressure and force coefficients for several airfoils and an arbitrary rock are given. Potential flow solutions are also presented. The procedure capable of treating multiple-element airfoils, and potential flow results are presented.

The development of magnetohydrodynamic flow due to an electric current discharge

Abstract: The development of the magnetohydrodynamic flow field due to the discharge of an electric current J0 from a point on a plate bounding a semi-infinite viscous incompressible conducting fluid is considered The flow field is the response of the fluid to the Lorentz force set up by the electric current and the associated magnetic field The problem is formulated in terms of the dimensionless variable (vt)½/r and solved numerically Here ν is the coefficient of kinematic viscosity, t the time from the application of the electric current and r the distance from the discharge It is shown that the streamlines of the developing flow field in a cross-section through the axis of the discharge are closed loops about a stagnation point As the flow field develops, the stagnation point moves to infinity along a ray emanating from the discharge with a speed proportional to t−½ The steady state, within a distance r from the discharge, is practically established when t = r2/ν

Analysis. of a Cross Head Die With the Flow Analysis Network (FAN) Method

Abstract: The Flow Analysis Network (FAN) method was adapted for solving the isothermal flow problem in a cross head die. Given the polymer rheology and the die geometry, the flow streamlines in the die and the flow rate uniformity at the exit can be calculated for any given head pressure. The optimum geometrical configuration of the die can be computed by repeated simulations. Results of the computations with the present method are shown to be in good agreement with previously published computations.

Semi-similar solutions to the three-dimensional laminar boundary layer

Abstract: The theory of semi-similar solutions is developed for and applied to the problem of three-dimensional laminar boundary layer flow. A number of specific examples are calculated. Particular attention is given to certain flows in which separation is approached and the nature of three-dimensional laminar boundary layer separation is inferred from the behavior of these solutions close to separation. Two types of separation are observed: “singular” separation characterized by the vanishing of the total shear along the line of separation and “ordinary” separation characterized by limiting streamlines which become parallel to the line of separation.

A generalized hyperbolic marching technique for three-dimensional supersonic flow with shocks

Abstract: The current paper introduces a generalized, nonorthogonal coordinate system for application to supersonic flows with such large angles between streamlines of the initial Cauchy data and the marching direction that the velocity component in the marching direction is actually subsonic. Finite-difference operators are developed to solve the three-dimensional, steady, inviscid conservation equations of fluid flow referenced to this general frame. Moreover, a new method of aligning the difference mesh to the bow shock wave is presented which eliminates both the necessity for differencing the free-stream flow properties and the spurious fluctuations that often arise in conservative variables when they are differenced across the discontinuity. Results are given for the case of flow past a conical surface.

Fluid mechanical investigations of ciliary propulsion

Abstract: Fluid mechanical investigations of ciliary propulsion are carried out from two points of view. In Part I, using a planar geometry, a model is developed for the fluid flow created by an array of metachronally coordinated cilia. The central concept of this model is to replace the discrete forces of the cilia ensemble by an equivalent continuum distribution of an unsteady body force within the cilia layer. This approach facilitates the calculation for the case of finite amplitude movement of cilia and takes into account the oscillatory component of the flow. Expressions for the flow velocity, pressure, and the energy expended by a cilium are obtained for small oscillatory Reynolds numbers. Calculations are carried out with the data obtained for the two ciliates Opalina ranarurn and Paramecium multimicronucleatum. The results are compared with those of previous theoretical models and some experimental observations. In Part II a model is developed for representing the mechanism of propulsion of a finite ciliated micro-organism having a prolate spheroidal shape. The basic concept of the model is to replace the micro-organism by a prolate spheroidal control surface at which certain boundary conditions on the fluid velocity are prescribed. These boundary conditions, which admit specific tangential and normal components of the flow velocity relative to the control surface, are proposed as a reasonable representation of the overall features of the flow field generated by the motion of the cilia system. Expressions are obtained for the velocity of propulsion, the rate of energy dissipation in the fluid exterior to the cilia layer, and the stream function of the motion. The effect of the shape of the organism upon its locomotion is explored. Experimental streak photographs of the flow around both freely swimming and inert sedimenting Paramecia are presented and compared with the theoretical prediction of the streamlines.

Experimental Investigation on Supersonic Plasma Flow through Diverging lines of Magnetic Force

Abstract: A supersonic plasma flow through the diverging lines of magnetic force is investigated experimentally. The Hall parameter of electrons is greater than unity, but that of ions is much smaller than unity. Measurements are made of the floating potential, electron temperature, ion density and ion velocity. The expansion of the plasma flow is suppressed by the external magnetic field. When the stream lines and the lines of magnetic force intersect, the electron temperature is higher at the downstream; when they are parallel to each other, a large potential difference occurs along the streamline, and equi-potential lines are crossed by the streamline. An extraordinary speed of ions is caused by the electrothermal acceleration. The strong electric field intensity along the streamlines agrees with the values estimated by the generalized Ohm's law.

Plane MHD Steady Flows with Constantly Inclined Magnetic and Velocity Fields

Abstract: Plane steady state viscous fluid flows, in which the magnetic field and velocity field are constantly inclined to one another, are considered. Necessary and sufficient physical conditions have been derived for flows with zero current density and the general solutions for these flows are obtained. Irrotational flows and flows with straight streamlines are also studied.

A note on the geostrophic flow past a crater in a rotating fluid

Abstract: Laboratory experiments are described on the flow past a solid obstacle in a rotating, homogeneous fluid. Specifically, the obstacle has the form of a walled crater specially constructed so that the volume of the depression is identically equal to the volume of the walls. The results show that closed streamlines occur rather more easily above such topography than above other obstacle types of the same scale but that the conditions for closure are determined essentially by the detailed geometry of the crater, the value of the Rossby number, and the depth of the fluid. The observed flow patterns are analysed and classified and attempts to quantify the most common flow type are made.

Effect of Uniform Rotation on Streamlines in Benard Convection Cells Induced by Surface Tension and Buoyancy

Abstract: The work of Nield [1] is extended to include the effect of uniform rotation about a vertical axis. As the rate of rotation increases the velocity perturbation vector becomes more and more symmetrical with the midlayer for buoyancy driven convection, and asymmetrical for surface tension driven convection. The result which shows that the rotation weakens the coupling of the buoyancy and surface tension forces is confirmed.

A note on co-current laminar flows of two immiscible elastico-viscous liquids

Abstract: An analysis is presented of steady (isothermal) co-current laminar flows of two immiscible elasticoviscous liquids in cylindrical channels to include (i) unidirectional stratified flow with ripple-free, plane liquid interface, and (ii) concentric-layered swirling flow with ripple-free cylindrical liquid interface. The general conditions are derived for such two-phase channel flows to be physically realizable. It is shown that, whereas (under certain circumstances)single-phase laminar flows are physically possible,two-phase flows, on the other hand, of liquids of the same class may not be. But liquids of theRoberts type (Roberts 1953), with a normal stress difference equivalent to an extra simple tension along the streamlines in simple shearing, are capable of steady unidirectional flowin all circumstances (whether in single or two-phase flow), though they are not in a privileged position so far astwo-phase swirling flows are concerned.

Analysis of three-dimensional potential flow around a ship hull

Abstract: The method of singularities is applied for the solution of three-dimensi onal potential flow around arbitrarily shaped bodies. The three-dimensional bodies are represented by systems of singular functions with singularities placed inside the body. An optimization procedure was applied to determine the optimal locations of the singular functions. A rigorous treatment of the problem is presented to establish the validity of the mathematical formulation and the optimization procedure. The method developed here can be applied for analyzing flow around obstacles for which the singularities at the boundaries could not be treated efficiently using presently used techniques. Presented are computational considerations and numerical results showing the velocity distribution around a regularly shaped obstacle and for the bow of an icebreaker. URING recent years there have been various applica- tions of numerical methods for solution of potential flow problems around arbitrarily shaped bodies.l'5 Many of these methods have been essentially based on Prager's concept6 of representing the surface of a three-dimensional body in a flow by a surface distribution of vorticity. Such a vorticity field is approximated at a series of discrete points on the body. Hess and Smith 1»2 have demonstrated the applicability of surface distributions of singularities to the calculation of potential flows around ship hulls and aircraft fuselage-wing com- binations. This method has produced interesting numerical results. The procedure given here was developed during an in- vestigation of potential flow around an icebreaker ship. A considerable portion of the resistance encountered by a moving ice-breaker is due to motion of the broken ice pieces depending on the flow characteristics in the bow region. The influence of the geometric configuration of the bow is com- plex, and produces several near singularities in the flow. In addition to the consideration of the bow shape, nowadays it is desirable to include the effects of pitching motion of the ship, and sometimes to include even the effect of a bubbler system on the flow patterns. Inclusion of such features in an analysis requires an elaborate and accurate discretized representation of the hull geometry. Difficulties of such a nature can hinder the application of previously developed methods, particularly from the standpoint of computational efficiency. In the following, details of the mathematical analysis of the potential flow are discussed. Following the method of von Karman,7 the three-dimensional body is represented by a system of singular functions with singularities placed inside the body. However, compared to previous analyses, an op- timum configuration of such functions was sought with the intent to increase the efficiency of the method. The presen- tation of the mathematical derivation of the analysis is essen- tial for understanding both the method and its advantages. Numerical results were obtained which show the velocity distributions for a regularly shaped obstacle, a sphere, and for an irregularly shaped body, the bow of the USSC Mackinaw,

Inviscid interpenetration of two streams with unequal total pressures

Abstract: A theory is proposed for analyzing the inviscid interpretation of two streams in the case when the difference in total pressure between the streams is relatively small. A stream is considered which discharges from a nozzle or reservoir into a partially moving and partially stationary environment in such a way that the flows leave the solid boundaries in a tangential direction where the two streams first interact. The problem is solved by expanding in a small parameter related to the difference in total pressure between the streams, the zeroth-order solution is obtained by classical methods, and a technique similar to that employed in thin-airfoil theory is used to transfer the first-order boundary conditions to the zeroth-order boundary. A procedure is developed to transform the problem into one that can be solved by standard techniques of the theory of sectionally analytic functions. Solutions are obtained for flows with and without free streamlines, and the general theory is applied to several specific flow configurations.

Computation of Steady Compressible Swirl Flows with Closed Streamlines at High Reynolds Numbers.

Abstract: : Flows with closed streamlines in a swirl chamber can be computed with equations in which viscosity and heat conduction are neglected provided that the entropy, the swirl, and the Bernoulli 'constant', in their dependence upon the streamfunction, are chosen in such a manner that the cumulative effects of viscosity and heat conduction (computed from the Navier Stokes equations and the energy equation) vanish The basic theory is found in a previous report The present report describes the computational aspects This includes the computation of the flow field and an iterative modification of the three functions mentioned above The resulting flow fields have been analyzed from the point of view of thermodynamics The particles undergo a thermodynamic cycle with heat input from the dissipation and heat conduction of the primary flow and heat output through heat conduction of the secondary flow The Prandtl number is very important Under present conditions where the prevalent heat input occurs at low temperatures, it hinders the secondary motion Physically, this can be explained as a buoyancy effect (Author)

A finite element analysis of the exact nonlinear formulation of a lifting surface in steady incompressible flow, with the evaluation of the correct wake geometry

Abstract: The problem of steady incompressible flow for lifting surfaces is considered. An integral equation is solved relating the values of the potential discontinuity on the lifting surface and its wake to the values of the normal derivative of the potential which are known from the boundary conditions. The lifting surface and the wake are divided into small quadrilateral surface elements. The values of the potential discontinuity and the normal derivative of the potential are assumed to be constant within each lifting surface element and equal to their values at the centroids of the lifting surface elements. This yields a set of linear algebraic equations. An iteration procedure is used to obtain the wake geometry: the velocities at the corner points of the wake elements are calculated and the wake streamlines are aligned to be parallel to the velocity vector. The procedure is repeated until convergence is attained.

Theory of Inviscid Shear Flow over Bluff Bodies Attached to a Plane Wall : Part 1, Theoretical Formulation and Applications to Normal Plate and Semicircular Projection

Abstract: A theoretical investigation is attempted on a separated flow past two-dimensional bluff bodies attached to a plane wall on which a turbulent boundary layer develops. The velocity profile in the turbulent boundary layer which would have been measured if the bluff body had been absent was replaced by a hypothetical inviscid shear flow of constant vorticity. This model admits analytical solutions and generates closed streamlines in front of the bluff bodies which are geometrically akin to the observed separation bubbles. Because of the lack of knowledge about the turbulent mixing along the edge of the separation bubbles, the present theory includes three or four constants which should be determined on the basis of the experimental information, the number of constants depending upon the shape of the body. For bluff bodies for which these constants can properly be estimated, theoretical pressure distributions agree well with experimental results. Flows around two typical shapes of bluff bodies, i.e. the normal plate and the semicircular projection are worked out in detail.

Axisymmetric rotating flow past a prolate spheroid

Abstract: The steady, inviscid, axisymmetric, rotating flow past a prolate spheroid in an unbounded liquid is determined on the hypothesis that all streamlines originate in a uniform flow far upstream of the body. The similarity parameters for the flow are κ = 2Ωa/U and δ = a/b, where 2a and 2b are the minor and major axes and Ω and U are the angular and axial velocities of the basic flow. Solutions are obtained both by separation of variables in prolate spheroidal co-ordinates and through the slender-body limit δ ↓ 0 with κ = O(1). Forward separation is found to occur for κ > κ*, where κ* lies between 2·2 and 2·3 for 0 < δ [les ] 1. The velocity on the body, the upstream axial velocity and the wave drag are calculated for κ < κ*.

Swirling flows of a gas with tangent equation of state

Abstract: An example is given of plane steady swirling flow of a gas with tangent equation of state and streamlines which are the involutes of an arbitrary closed convex sectionally smooth curve.

Two-dimensional transonic vortex flows of an ideal gas

Abstract: Equations are obtained for two-dimensional transonic adiabatic (nonisoenergetic and nonisoentropic) vortex flows of an ideal gas, using the natural coordinatesϕψ (ψ=const is the family of streamlines, and ϕ=const is the family of lines orthogonal to them). It is not required that the transonic gas flow be close to a uniform sonic flow (the derivation is given without e estimates). Solutions are found for equations describing vortex flows inside a Laval nozzle and near the sonic boundary of a free stream.