Acoustically generated vorticity in an internal flow

A mathematical model is formulated to describe the initiation and evolution of intense unsteady vorticity in a low Mach number (M), weakly viscous internal flow sustained by mass addition through the sidewall of a long, narrow cylinder. An O(M) axial acoustic velocity disturbance, generated by a prescribed harmonic transient endwall velocity, interacts with the basically inviscid rotational steady injected flow to generate time-dependent vorticity at the sidewall. The steady radial velocity component convects the vorticity into the flow. The axial velocity associated with the vorticity field varies across the cylinder radius and in particular has an instantaneous oscillatory spatial distribution with a characteristic wavelength O(M) smaller than the radius. Weak viscous effects cause the vorticity to diffuse on the small radial length scale as it is convected from the wall toward the axis. The magnitude of the transient vorticity field is larger by O(M−1) than that in the steady flow.An initial-boundary-value formulation is employed to find nonlinear unsteady solutions when a pressure node exists at the downstream exit of the cylinder. The complete velocity consists of a superposition of the steady flow, an acoustic (irrotational) field and the rotational component, all of the same magnitude.

A study of turbulent boundary-layer separation and reattachment

An experimental and theoretical study is made of a suddenly separating and reattaching two-dimensional turbulent boundary layer on a flat surface. A separation bubble is formed on the floor of a wide parallel-sided wind-tunnel duct with the pressure field causing the bubble formation produced by fixing the shape of the flexible roof of the duct. Boundary layers on the roof are controlled and remain attached. It is found that a very satisfactory model for the flow is an inviscid one.The boundary layer on the floor of the duct is represented by a region of constant vorticity with slip at the boundary, and it is assumed that the separation process is dominated by the interaction between this ‘vortical’ region and the irrotational field between the vortical region and the roof (of prescribed shape). The interface between the rotational and irrotational regions is a free boundary and may be located when all necessary boundary conditions are given. These conditions include two characteristic parameters for the adverse-pressure-gradient turbulent boundary layer which is developing upstream of the region of interest.The problem is solved by an electrical analog method. The theoretical size and shape of the bubble and positions of separation and reattachment are in agreement with observations. The advantage of the model over most previous attempts to predict separation is that the governing equations are elliptic rather than parabolic or hyperbolic and therefore the interaction between the boundary-layer flow and the irrotational free stream is included in the calculations.

Irrotational fluctuations near a turbulent boundary layer

A verification of some of the predictions of the theory of Phillips (1955) is presented, and the relation between one-dimensional and two-dimensional wavenumber spectra is discussed. The convection velocity of the irrotational field deduced from measurements of the frequency spectrum alone appears to be about 0.9U1in the frequency range carrying most of the energy. It follows that the pressure-fluctuation spectrum is proportional to the velocity-fluctuation spectrum and varies as ω2at low frequency. The discrepancy between this result and measurements of wall-pressure spectra is plausibly attributed to extraneous sound.