A two-dimensional boundary layer encountering a three-dimensional hump

1977 ◽  
Vol 83 (1) ◽  
pp. 163-176 ◽  
Author(s):  
F. T. Smith ◽  
R. I. Sykes ◽  
P. W. M. Brighton
Keyword(s):  
Boundary Layer ◽  
Constant Width ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Flow Properties ◽  
Nonlinear Motion ◽  
Dimensional Boundary ◽  
Incompressible Boundary ◽  
Insight Into ◽  
Steady Laminar

A shallow three-dimensional hump disturbs the two-dimensional incompressible boundary layer developed on an otherwise flat surface. The steady laminar flow is studied by means of a three-dimensional extension of triple-deck theory, so that there is the prospect of separation in the nonlinear motion. As a first step, however, a linearized analysis valid for certain shallow obstacles gives some insight into the flow properties. The most striking features then are the reversal of the secondary vortex motions and the emergence of a ‘corridor’ in the wake of the hump. The corridor stays of constant width downstream and within it the boundary-layer displacement and skin-friction perturbation are much greater than outside. Extending outside the corridor, there is a zone where the surface fluid is accelerated, in contrast with the deceleration near the centre of the corridor. The downstream decay (e.g. of displacement) here is much slower than in two-dimensional flows.

Marginal separation of a three-dimensional boundary layer on a line of symmetry

1985 ◽  
Vol 158 ◽  
pp. 95-111 ◽  
Author(s):  
S. N. Brown
Keyword(s):  
Boundary Layer ◽  
Asymptotic Form ◽  
Separation Bubble ◽  
Three Dimensional ◽  
Additional Parameter ◽  
Analytical Prediction ◽  
Two Dimensional ◽  
Boundary Layer Equations ◽  
Dimensional Boundary ◽  
Incompressible Boundary

The marginal separation of a laminar incompressible boundary layer on the line of symmetry of a three-dimensional body is discussed. The interaction itself is taken to be quasi-two-dimensional but the results differ from those for a two-dimensional boundary layer in that the effect of the gradient of the crossflow is included. Solutions of the resulting integral equation are computed for two values of the additional parameter, and comparisons made with an analytical prediction of the asymptotic form as the length of the separation bubble tends to infinity. The occurrence of the phenomenon is confirmed by an examination of the results of an existing numerical integration of the boundary-layer equations for the line of symmetry of a paraboloid.

Singularities in solutions of the three-dimensional laminar-boundary-layer equations

1985 ◽  
Vol 160 ◽  
pp. 257-279 ◽  
Author(s):  
James C. Williams
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Unsteady Boundary Layer ◽  
Boundary Layer Flows ◽  
Boundary Layer Equations ◽  
New Type ◽  
Steady Laminar

The three-dimensional steady laminar-boundary-layer equations have been cast in the appropriate form for semisimilar solutions, and it is shown that in this form they have the same structure as the semisimilar form of the two-dimensional unsteady laminar-boundary-layer equations. This similarity suggests that there may be a new type of singularity in solutions to the three-dimensional equations: a singularity that is the counterpart of the Stewartson singularity in certain solutions to the unsteady boundary-layer equations.A family of simple three-dimensional laminar boundary-layer flows has been devised and numerical solutions for the development of these flows have been obtained in an effort to discover and investigate the new singularity. The numerical results do indeed indicate the existence of such a singularity. A study of the flow approaching the singularity indicates that the singularity is associated with the domain of influence of the flow for given initial (upstream) conditions as is prescribed by the Raetz influence principle.

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.

Modelling the calmed region behind a spot

2005 ◽  
Vol 363 (1830) ◽  
pp. 1069-1078 ◽  
Author(s):  
S.N Brown ◽  
F.T Smith
Keyword(s):  
Boundary Layer ◽  
Theoretical Model ◽  
Slip Velocity ◽  
Three Dimensional ◽  
Cross Flow ◽  
Two Dimensional ◽  
Turbulent Spot ◽  
Mean Flow ◽  
Flow Surface ◽  
Dimensional Boundary

A theoretical model of the laminar ‘calmed region’ following a three-dimensional turbulent spot within a transitioning two-dimensional boundary layer is formulated and discussed. The flow is taken to be inviscid, and the perturbation mean flow surface streamlines calculated represent disturbances to the basic slip velocity. Available experimental evidence shows a fuller, more stable, streamwise profile in a considerable region trailing the spot, with cross-flow ‘inwash’ towards the line of symmetry. Present results are in qualitative agreement with this evidence.

Reynolds stress development in pressure-driven three-dimensional turbulent boundary layers

1989 ◽  
Vol 202 ◽  
pp. 263-294 ◽  
Author(s):  
Shawn D. Anderson ◽  
John K. Eaton
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Reynolds Stress ◽  
Free Stream ◽  
Three Dimensional ◽  
Radius Of Curvature ◽  
Two Dimensional ◽  
Stress Vector ◽  
Dimensional Boundary ◽  
Cross Wire

The development of the Reynolds stress field was studied for flows in which an initially two-dimensional boundary layer was skewed sideways by a spanwise pressure gradient ahead of an upstream-facing wedge. Two different wedges were used, providing a variation in the boundary-layer skewing. Measurements of all components of the Reynolds stress tensor and all ten triple products were measured using a rotatable cross-wire anemometer. The results show the expected lag of the shear stress vector behind the strain rate. Comparison of the two present experiments with previous data suggests that the lag can be estimated if the radius of curvature of the free-stream streamline is known. The magnitude of the shear stress vector in the plane of the wall is seen to decrease rapidly as the boundary-layer skewing increases. The amount of decrease is apparently related to the skewing angle between the wall and the free stream. The triple products evolve rapidly and profiles in the three-dimensional boundary layer are considerably different than two-dimensional profiles, leaving little hope for gradient transport models for the Reynolds stresses. The simplified model presented by Rotta (1979) performs reasonably well providing that an appropriate value of the T-parameter is chosen.

scholarly journals Cancellation of Tollmien–Schlichting waves with surface heating

2021 ◽  
Vol 128 (1) ◽  
Author(s):  
Georgia S. Brennan ◽  
Jitesh S. B. Gajjar ◽  
Richard E. Hewitt
Keyword(s):  
Boundary Layer ◽  
Asymptotic Methods ◽  
Three Dimensional ◽  
Exact Formula ◽  
Surface Heating ◽  
Two Dimensional ◽  
Boundary Layer Flows ◽  
Dimensional Boundary ◽  
Tollmien Schlichting ◽  
Layer Flow

AbstractTwo-dimensional boundary layer flows in quiet disturbance environments are known to become unstable to Tollmien–Schlichting waves. The experimental work of Liepmann et al. (J Fluid Mech 118:187–200, 1982), Liepmann and Nosenchuck (J Fluid Mech 118:201–204, 1982) showed how it is possible to control and reduce unstable Tollmien–Schlichting wave amplitudes using unsteady surface heating. We consider the problem of an oncoming planar compressible subsonic boundary layer flow with a three-dimensional vibrator mounted on a flat plate, and with surface heating present. It is shown using asymptotic methods based on triple-deck theory that it is possible to choose an unsteady surface heating distribution to cancel out the response due to the vibrator. An approximation based on the exact formula is used successfully in numerical computations to confirm the findings. The results presented here are a generalisation of the analogous results for the two-dimensional problem in Brennan et al. (J Fluid Mech 909:A16-1, 2020).

The Görtler vortex instability mechanism in three-dimensional boundary layers

1985 ◽  
Vol 399 (1816) ◽  
pp. 135-152 ◽  
Keyword(s):  
Boundary Layer ◽  
Relative Size ◽  
Three Dimensional ◽  
Practical Interest ◽  
Basic Flow ◽  
Two Dimensional ◽  
Instability Mechanism ◽  
Order Unity ◽  
Concave Wall ◽  
Dimensional Boundary

It is well known that the two-dimensional boundary layer on a concave wall is centrifugally unstable with respect to vortices aligned with the basic flow for sufficiently high values of the Görtler number. However, in most situations of practical interest the basic flow is three-dimensional and previous theoretical investigations do not apply. In this paper the linear stability of the flow over an infinitely long swept wall of vari­able curvature is considered. If there is no pressure gradient in the boundary layer it is shown that the instability problem can always be related to an equivalent two-dimensional calculation. However, in general, this is not the case and even for small values of the crossflow velocity field dramatic differences between the two- and three-dimensional problems emerge. In particular, it is shown that when the relative size of the crossflow and chordwise flow is O ( Re –½ ),where Re is the Reynolds number of the flow, the most unstable mode is time-dependent. When the size of the crossflow is further increased, the vortices in the neutral location have their axes locally perpendicular to the vortex lines of the basic flow. In this régime the eigenfunctions associated with the instability become essentially 'centre modes’ of the Orr–Sommerfeld equation destabilized by centrifugal effects. The critical Görtler number for such modes can be predicted by a large wavenumber asymptotic analysis; the results suggest that for order unity values of the ratio of the crossflow and chordwise velocity fields, the Görtler instability mechanism is almost certainly not operational.

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