scholarly journals Some similarity solutions for three-dimensional boundary layers

2019 ◽  
Vol 475 (2229) ◽  
pp. 20190267 ◽  
Author(s):  
R. H. Vaz ◽  
A. J. Mestel
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Wall Jet ◽  
Dimensional Solution ◽  
Similarity Reduction ◽  
Outer Flow ◽  
Dimensional Boundary ◽  
Constant Potential

A similarity solution of a three-dimensional boundary layer is investigated. The outer flow is given by U  = ( −  xz , −  yz , z 2 ), corresponding to an axisymmetric poloidal circulation with constant potential vorticity. This flow is an exact solution of the Navier–Stokes. A wall is introduced at y  = 0 along which a boundary layer develops towards x  = 0. We show that a similarity reduction to a system of ODEs is possible. Two distinct solutions are found, one of them through numerical path-continuation, and their stability is investigated. A second three-dimensional solution is also identified for two-dimensional outer flow. The solutions are generalized for outer flows scaling with different powers of z and similar results are found. This behaviour is related to the non-uniqueness of the Falkner–Skan flows in a three-dimensional sense, with a transverse wall-jet.

On vortex boundary layers

1985 ◽  
Vol 400 (1819) ◽  
pp. 253-261 ◽  
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Three Dimensional ◽  
Radial Pressure ◽  
Outer Flow ◽  
The Core ◽  
Dimensional Boundary ◽  
Radial Pressure Gradient ◽  
Potential Vortex ◽  
Vortex Axis

Flow visualization is used to study the flow that results when a potential vortex rotates normal to a stationary horizontal disc. Viscosity is seen to remove the singularity on the vortex axis and lead to the development of a three-dimensional boundary layer. The flow remains laminar below a Reynolds number, Re , of about 10 4 , where Re is based on radius and velocity at the disc edge. With further increases in Re the boundary layer becomes turbulent but relaminarizes as it is advected radially inwards by the highly favourable radial pressure gradient associated with the outer flow. The radius of the zone of relaminarized fluid decreases with increasing Re . Close to the axis the flow effuses vertically to form the core of the vortex which, for Re < 10 4 , is observed to undergo a massive disruption, either of the axisymmetric or helical form. The sense of the helix was observed on some occasions to be with that of the outer flow and on others to be opposite that of the outer flow.

Effect of Adverse Pressure Gradients on Turbulent Boundary Layers in Axisymmetric Conduits

1957 ◽  
Vol 24 (2) ◽  
pp. 191-196
Author(s):  
J. M. Robertson ◽  
J. W. Holl
Keyword(s):  
Boundary Layer ◽  
Flow Parameter ◽  
Three Dimensional ◽  
Turbulent Boundary Layers ◽  
Pressure Gradients ◽  
Two Dimensional ◽  
Flow Conditions ◽  
Straight Pipe ◽  
Outer Flow ◽  
Dimensional Boundary

Abstract The development of the three-dimensional boundary layer in a diffuser with several discharge arrangements has been studied for air flow, in continuation of the work of Uram (1). The flow conditions in a diffuser when followed by a straight pipe, an additional length of the diffuser, or a jet, are compared. Extension of the method of analysis developed by Ross for two-dimensional layers is presented. In some cases the use of three-dimensionally defined parameters leads to different results. Ross’s (2) unique outer-flow parameter is found to be no longer satisfactory. Other outer parameters are presented as possible substitutes.

A three-dimensional boundary-layer separation

1980 ◽  
Vol 99 (1) ◽  
pp. 185-224 ◽  
Author(s):  
F. T. Smith
Keyword(s):  
Boundary Layer ◽  
Flow Field ◽  
Three Dimensional ◽  
Vortex Sheet ◽  
Boundary Layer Separation ◽  
Similarity Solutions ◽  
Structural Investigations ◽  
Boundary Layer Problem ◽  
Dimensional Boundary ◽  
Free Interaction

A nonlinear three-dimensional boundary-layer problem governing the flow upstream of a particular disturbance (e.g. a shallow obstacle) at the wall is considered. The upstream response, a free interaction, takes place under zero displacement of the boundary layer, and the solution is found numerically using Fourier series truncation and varying the number of terms kept in the series. In one part of the flow field regular separation is encountered, beyond which the motion becomes strongly attached to the wall elsewhere in the flow field. Analytically, local structural investigations then suggest that the attached part of the upstream response terminates at a line singularity, while the separated part can continue indefinitely far downstream. The former structure leads to a new set of similarity solutions of the three-dimensional boundary-layer equations, while the latter develops a vortex sheet formation. The three-dimensional flow problem has most relevance to pipe flows, but some connexion also with external flows, and the implications for these are discussed.

The boundary layer beneath a Rankine-like vortex

1987 ◽  
Vol 411 (1840) ◽  
pp. 177-192 ◽  
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Radial Pressure ◽  
Reynolds Numbers ◽  
Tangential Component ◽  
Plane Boundary ◽  
Outer Flow ◽  
Law Of The Wall ◽  
Dimensional Boundary ◽  
Transitory State

An experimental investigation is made of the three-dimensional boundary layer that results when a Rankine-like vortex is bounded by a fixed plane boundary, in particular by a horizontal disc coaxial with, and perpendicular to, the axis of rotation of the vortex. A laser-Doppler anemometer is used to make velocity traverses through both the vortex and the boundary layer, for Reynolds numbers, Re , ranging from 5000 to 30000, where Re is based on velocity and radius at the disc edge. The boundary layer is laminar at Re = 5000 and the data agree well with the theory of Belcher et al . ( J . Fluid Mech . 52, 753-780 (1972)); at Re = 10000 the layer is in a transitory state, while for Re ≽ 15000 it is turbulent over some of the disc. The radial pressure gradient associated with the outer flow has a stabilizing effect on the boundary layer and, for 10000 ≼ Re ≼ 30000, acts to revert it to a laminar state, but with diminishing effect as Re increases. In spite of the high threedimensionality of the layer, the tangential component of velocity conforms to the same law-of-the-wall as its streamwise counterpart in two-dimensional turbulent boundary layers.

Effects of Streamwise Vortices on Laminar Boundary-Layer Flow

1968 ◽  
Vol 35 (2) ◽  
pp. 424-426 ◽  
Author(s):  
T. K. Fannelop
Keyword(s):  
Boundary Layer ◽  
Boundary Layer Flow ◽  
Stokes Equations ◽  
Transverse Velocity ◽  
Three Dimensional ◽  
Perturbation Expansion ◽  
Navier Stokes ◽  
Boundary Layer Equations ◽  
Dimensional Boundary ◽  
Layer Flow

The effects of periodic transverse velocity fluctuations are investigated for boundary-layer flow over a flat plate. The method used is a perturbation expansion of the three-dimensional boundary-layer equations in terms of the small transverse velocity component. The equations are reduced to similarity form by means of suitable transformations. The second-order terms are expressed in terms of the first-order (Blasius) variables and are found to increase linearly with the streamwise coordinate. The present heat-transfer solution agrees with the more qualitative results of Persen. The derived velocity profiles are in exact agreement with the results of Crow’s more elaborate analysis based on the Navier-Stokes equations.

scholarly journals Application of Advanced Computational Codes in the Design of an Experiment for a Supersonic Throughflow Fan Rotor

1988 ◽  
Vol 110 (2) ◽  
pp. 270-279
Author(s):  
J. R. Wood ◽  
J. F. Schmidt ◽  
R. J. Steinke ◽  
R. V. Chima ◽  
W. G. Kunik
Keyword(s):  
Boundary Layer ◽  
Mach Number ◽  
Pressure Ratio ◽  
Three Dimensional ◽  
Axial Flow ◽  
Navier Stokes ◽  
Nasa Lewis Research ◽  
Two Dimensional ◽  
Final Design ◽  
Dimensional Boundary

Increased emphasis on sustained supersonic or hypersonic cruise has revived interest in the supersonic throughflow fan as a possible component in advanced propulsion systems. Use of a fan that can operate with a supersonic inlet axial Mach number is attractive from the standpoint of reducing the inlet losses incurred in diffusing the flow from a supersonic flight Mach number to a subsonic one at the fan face. The data base for components of this type is practically nonexistent; therefore, in order to furnish the required information for assessment of this type fan, a program has been initiated at the NASA Lewis Research Center to design, build, and test a fan rotor that operates with supersonic axial velocities from inlet to exit. This paper describes the design of the experiment using advanced computational codes to calculate the unique components required. The fan rotor has constant hub and tip radii and was designed for a pressure ratio of 2.7 with a tip speed of 457 m/s. The rotor was designed using existing turbomachinery design and analysis codes modified to handle fully supersonic axial flow through the rotor. A two-dimensional axisymmetric throughflow design code plus a blade element code were used to generate fan rotor velocity diagrams and blade shapes. A quasi-three-dimensional, thin shear layer Navier–Stokes code was used to assess the performance of the fan rotor blade shapes. The final design was stacked and checked for three-dimensional effects using a three-dimensional Euler code interactively coupled with a two-dimensional boundary layer code. A translating nozzle was designed to produce a uniform flow parallel to the fan up to the design axial Mach number of 2.0. The nozzle was designed with the three-dimensional Euler/interactive boundary layer code. The nozzle design in the expansion region was analyzed with a three-dimensional parabolized viscous code, which corroborated the results from the Euler code. A translating supersonic diffuser was designed using these same codes.

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