Two-dimensional Wake Research Articles

Coupled radiation and conduction in free mixing.

Abstract : Heat transfer by both grey radiation and conduction is examined in the case of Oseen-like free-mixing flow which is formally analogous to a one-dimensional unsteady transfer problem. The equation describing the transport of energy by non-convective means is obtained by linearization of the conservation of energy relation. The geometry of the time-dependent problem may be envisaged as a slab of finite width separating two infinite half-spaces; this configuration is analogous to the steady, free-mixing flow of a two-dimensional jet or wake. The initial temperature profile of the jet with respect to the surrounding gas is taken to be a step function. By means of the transform calculus the determination of the temperature for the entire region of the jet and the fluid adjacent to it may be given as an inversion integral. This approach does not require a kernel substitution in the solution of the integro-differential equation for the temperature and thus avoids some of the limitations imposed thereby. Analytic solutions in the limiting cases of 'thin' and 'thick' gas behavior are readily recovered from the general solution. From the bounded variation of the integrand the temperature distribution may be computed for all possible combinations of the heat conduction and radiation parameters. Exact numerical evaluation of the inversion integral is employed to provide a complete description of the temperature variations in the flow. (Author)

On the singular points in the laminar two-dimensional near wake flow field

It is shown that useful information concerning the flow in the neighbourhood of the various separation and stagnation points in the laminar near wake of a blunt-based two-dimensional wedge can be learned from the locally valid Stokes type series solutions to the incompressible Navier-Stokes vorticity equation derived previously by Dean & Montagnon (1949) and Moffatt (1964). This theory, which is in qualitative agreement with the experiments of Hama (1967) and Donaldson (1967), shows that the flow separates from the base of a blunt-based body and not from its trailing edge. The series solution for the two-dimensional stagnation point is treated in detail and compared with Howarth's (1934) numerical solution in order to study the convergence and conditions for completeness of the Stokes type series solution. Finally, the wake rear stagnation point is examined to provide insight into the problem of wake closure.

The development of three-dimensional wave-packets in unbounded parallel flows

In this paper we examine the stability of a two-dimensional wake profile of the form u(y) = U∞(1 – r e-sy2) with respect to a pulsed disturbance at a point in the fluid. The disturbed flow forms an expanding wave packet which is convected downstream. Far downstream, where asymptotic expansions are valid, the motion at any point in the wave packet is described by a particular three-dimensional wave having complex wave-numbers. In the special case of very unstable flows, where viscosity does not have a significant influence, it is possible to evaluate the three-dimensional eigenvalues in terms of two-dimensional ones using the inviscid form of Squire's transformation. In this way each point in the physical plane can be linked to a particular two-dimensional wave growing in both space and time by simple algebraic expressions which are independent of the mean flow velocity profile. Computed eigenvalues for the wake profile are used in these relations to find the behaviour of the wave packet in the physical plane.

Role of intermittency in free turbulent flows.

A theoretical analysis of the effects of intermittency on turbulent transport processes in free turbulent flows is presented. Turbulent fluctuations, in addition to those in the fully turbulent fluid, are produced by the intermittency effects. It is shown that these additional fluctuations are significant for scalar quantities introduced in the turbulent fluid, but are insignificant for the velocity field based on Townsend's measurements. Dynamic equations governing the mean quantities and the fluctuations for both the velocity and temperature fields, including the intermittency effects, are derived. Based on these equations, the turbulent transport processes for momentum and heat are analyzed. It is found that 1) the turbulent Prandtl number is proportional to the intermittency factor, and 2) the mean velocity (defect) and temperature profiles are not the same, and the spread of the temperature profile is wider. A two-dimensional wake flow is worked out as a numerical example. The relevance of the present results to the hypersonic wake problem is discussed.

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.

A note on the expansion of turbulent wakes

An analysis has been made of a two-dimensional turbulent wake which was subjected to a homogeneous strain. This was applied in such a way that the rate of growth of the wake tended to be accelerated but the main stream convective velocity remained constant. An attempt to describe the flow by classical self-preservation techniques was made. However, the results suggested that a detrainment of wake fluid would result. In lieu of this a more realistic approach was sought and it was found that in certain situations where the flow has had an opportunity to develop before the strain is applied, the growth of the wake in the strain field can be described successfully by a simple geometric distortion. It appears likely that in these cases the interaction between the external strain and the turbulent structure within the wake is negligible, the wake remaining essentially passive. Suggestions as to the mechanism of the turbulent mixing and transport under these conditions are made.

An experimental examination of the large-eddy equilibrium hypothesis

The large-eddy energy equilibrium hypothesis states that the largest eddies of a turbulent shear flow are in approximate energy equilibrium throughout a significant part of their lives. This hypothesis leads to a relationship between the mean rate of shear strain and the Reynolds shear stress which involves the scale of the large eddies. By assuming that the large-eddy scale is proportional to the standard deviation of the free turbulent boundary, or laminar superlayer, the validity of this hypothesis may be checked experimentally. Intermittency and mean velocity measurements made in five different two-dimensional shear flows are presented and these results, together with values calculated from Townsend's measurements in a two-dimensional wake, support the form of relationship suggested by the energy equilibrium hypothesis.

Note on instability of compressible jets and wakes to long-wave disturbances

A two-dimensional jet or wake is observed in a frame of reference moving with the fluid at infinity, so that the velocity w(y) in the x-direction tends to zero as y → ± ∞. The fluid is assumed to be an inviscid perfect gas, to undergo-adiabatic changes, and the local speed of sound to be a function of y such that a(y)→ a∞, as y → ±∞. If the disturbance pressure has the form

The Stability of Two-Dimensional Laminar Wakes at Low Reynolds Numbers

This paper is concerned with the stability of a two-dimensional laminer wake. The development of small disturbances excited artificially is investigated both experimentally and theoretically. Experiments are made in an experimental water-tank at Reynolds numbers between 10 and 600 for a flat plate and between 0.8 and 60 for a circular cylinder. The amplification rate, the wave length and the wave velocity of the disturbance are measured from the photographs of the flow. The neutral stability curve is determined both for the fiat plate and for the circular cylinder. The minimum critical Reynolds number below which all disturbances are damped is found to be about 1.0 for the circular cylinder. Theoretical investigations are made by the method of small oscillations. The agreement between the theory and the experiment is fairly good.

Impuls- und wärmeübertragung in turbulenten windschatten hinter rotationskörpern

The development of the width of a turbulent axi-symmetric wake depends on the shape of the body of revolution generating the wake. Within certain limits, the shape of the momentum loss distribution is also influenced by the shape of the body. Behind slender bodies of revolution bell-shaped distributions of velocity and temperature are found. The coefficient of heat exchange is slightly greater than the coefficient of momentum exchange. The outer potential flow restricts the turbulent exchange in axi-symmetric wakes more than in two-dimensional wakes and in jets.

An Investigation of the End-Wall Boundary Layer of a Turbine-Nozzle Cascade

Abstract As a part of a research program aimed at an explanation of the low efficiencies measured near the casings of turbines, an experimental investigation of the flow through a rectilinear cascade of turbine nozzles was conducted to study the effects of the end-wall boundary layer on the flow pattern through the nozzles. Secondary flows in the boundary-layer fluid produce an accumulation of low-energy fluid in the cascade exit plane that is considerably different from the flow field of the two-dimensional wakes. To describe the effects of the secondary flows, detailed data are included showing blade-pressure-distribution variations through the end-wall boundary layer, flow directions, and total and static pressure at cascade discharge, both with and without an inlet boundary layer on the end wall. From these data, qualitative conclusions have been drawn concerning the nature and origin of these low-energy accumulations. The most significant of these conclusions are (a) that the end-wall boundary-layer behavior inside the nozzle passage is essentially independent of the boundary layer entering the cascade; (b) the low-energy fluid which appears in the suction-surface end-wall corner at cascade discharge has its origin only in the end-wall and suction-surface boundary layers.

Note on Free Turbulent Flows, with Special Reference to the Two-Dimensional Wake

Note on Free Turbulent Flows, with Special Reference to the Two-Dimensional Wake