Stability of a Laminar Boundary Layer Flowing Along a Concave Surface

1989 ◽  
Vol 111 (4) ◽  
pp. 376-386 ◽  
Author(s):  
M. V. Finnis ◽  
A. Brown
Keyword(s):  
Boundary Layer ◽  
Laminar Boundary Layer ◽  
Streamwise Velocity ◽  
Wave Number ◽  
Radius Of Curvature ◽  
Normal Velocity ◽  
Mean Flow ◽  
Boundary Layer Flows ◽  
The Stability ◽  
The Galerkin Method

Go¨rtler instability for incompressible laminar boundary-layer flows over constant curvature concave surfaces is considered. The full linearized disturbance equations are solved by the Galerkin method using Chebyshev polynomials to represent the disturbance functions. Stability curves relating Go¨rtler number, wave number, and vortex amplification for a Blasius mean flow are presented. The effect of streamwise pressure variation is investigated using the Falkner–Skan boundary-layer solutions for the mean flow. The importance of including the normal velocity terms for these flows is shown by their effect on the stability curves. The streamwise velocity distribution in the boundary layer on a 3-m radius of curvature plate was investigated experimentally. The results are compared with the stability curves and predicted disturbance functions.

A new inviscid mode of instability in compressible boundary-layer flows

2015 ◽  
Vol 785 ◽  
pp. 301-323 ◽  
Author(s):  
Adam P. Tunney ◽  
James P. Denier ◽  
Trent W. Mattner ◽  
John E. Cater
Keyword(s):  
Boundary Layer ◽  
Heated Surface ◽  
Direct Consequence ◽  
Streamwise Velocity ◽  
Stream Velocity ◽  
Boundary Layer Flows ◽  
New Class ◽  
The Stability ◽  
The Impact ◽  
Inviscid Compressible Fluid

The stability of an almost inviscid compressible fluid flowing over a rigid heated surface is considered. We focus on the boundary layer that arises. The effect of surface heating is known to induce a streamwise acceleration in the boundary layer near the surface. This manifests in a streamwise velocity which exhibits a maximum larger than the free-stream velocity (i.e. the streamwise velocity exhibits an ‘overshoot’ region). We explore the impact of this overshoot on the stability of the boundary layer, demonstrating that the compressible form of the classical Rayleigh equation (which governs the development of short wavelength instabilities) possesses a new unstable mode that is a direct consequence of this overshoot. The structure of this new class of modes in the small wavenumber limit is detailed, providing a valuable confirmation of our numerical results obtained from the full inviscid eigenvalue problem.

scholarly journals Analysis of axisymmetric boundary layers

2018 ◽  
Vol 849 ◽  
pp. 927-941 ◽  
Author(s):  
Praveen Kumar ◽  
Krishnan Mahesh
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Boundary Layers ◽  
Axial Flow ◽  
Streamwise Velocity ◽  
Skin Friction Coefficient ◽  
Radius Of Curvature ◽  
Normal Velocity ◽  
Displacement Thickness ◽  
Analytical Relations

Axisymmetric boundary layers are studied using integral analysis of the governing equations for axial flow over a circular cylinder. The analysis includes the effect of pressure gradient and focuses on the effect of transverse curvature on boundary layer parameters such as shape factor ($H$) and skin-friction coefficient ($C_{f}$), defined as $H=\unicode[STIX]{x1D6FF}^{\ast }/\unicode[STIX]{x1D703}$ and $C_{f}=\unicode[STIX]{x1D70F}_{w}/(0.5\unicode[STIX]{x1D70C}U_{e}^{2})$ respectively, where $\unicode[STIX]{x1D6FF}^{\ast }$ is displacement thickness, $\unicode[STIX]{x1D703}$ is momentum thickness, $\unicode[STIX]{x1D70F}_{w}$ is the shear stress at the wall, $\unicode[STIX]{x1D70C}$ is density and $U_{e}$ is the streamwise velocity at the edge of the boundary layer. Relations are obtained relating the mean wall-normal velocity at the edge of the boundary layer ($V_{e}$) and $C_{f}$ to the boundary layer and pressure gradient parameters. The analytical relations reduce to established results for planar boundary layers in the limit of infinite radius of curvature. The relations are used to obtain $C_{f}$ which shows good agreement with the data reported in the literature. The analytical results are used to discuss different flow regimes of axisymmetric boundary layers in the presence of pressure gradients.

Stability of spatially developing boundary layers in pressure gradients

1995 ◽  
Vol 300 ◽  
pp. 117-147 ◽  
Author(s):  
Rama Govindarajan ◽  
R. Narasimha
Keyword(s):  
Boundary Layer ◽  
Stability Boundary ◽  
Streamwise Velocity ◽  
Present Approach ◽  
Pressure Gradients ◽  
Mean Flow ◽  
Rational Analysis ◽  
Displacement Effect ◽  
The Stability ◽  
New Formulation

A new formulation of the stability of boundary-layer flows in pressure gradients is presented, taking into account the spatial development of the flow and utilizing a special coordinate transformation. The formulation assumes that disturbance wavelength and eigenfunction vary downstream no more rapidly than the boundary-layer thickness, and includes all terms nominally of orderR−1in the boundary-layer Reynolds numberR. In Blasius flow, the present approach is consistent with that of Bertolottiet al.(1992) toO(R−1) but simpler (i.e. has fewer terms), and may best be seen as providing a parametric differential equation which can be solved without having to march in space. The computed neutral boundaries depend strongly on distance from the surface, but the one corresponding to the inner maximum of the streamwise velocity perturbation happens to be close to the parallel flow (Orr-Sommerfeld) boundary. For this quantity, solutions for the Falkner-Skan flows show the effects of spatial growth to be striking only in the presence of strong adverse pressure gradients. As a rational analysis toO(R−1) demands inclusion of higher-order corrections on the mean flow, an illustrative calculation of one such correction, due to the displacement effect of the boundary layer, is made, and shown to have a significant destabilizing influence on the stability boundary in strong adverse pressure gradients. The effect of non-parallelism on the growth of relatively high frequencies can be significant at low Reynolds numbers, but is marginal in other cases. As an extension of the present approach, a method of dealing with non-similar flows is also presented and illustrated.However, inherent in the transformation underlying the present approach is a lower-order non-parallel theory, which is obtained by dropping all terms of nominal orderR−1except those required for obtaining the lowest-order solution in the critical and wall layers. It is shown that a reduced Orr-Sommerfeld equation (in transformed coordinates) already contains the major effects of non-parallelism.

scholarly journals Analysis of Unsteady Laminar Boundary Layer Flow by an Integral Method

1973 ◽  
Vol 95 (2) ◽  
pp. 237-247 ◽  
Author(s):  
R. W. Miller ◽  
L. S. Han
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Laminar Boundary Layer ◽  
Integral Method ◽  
Momentum Equation ◽  
Asymptotic Solutions ◽  
Mean Flow ◽  
Boundary Layer Flows ◽  
Approximate Integral ◽  
Layer Flow

An approximate integral method of analysis is developed for unsteady laminar boundary layer flows. The case of a flat plate in a free stream with small harmonic velocity oscillations about a steady mean is used to formulate the method. Results are compared with the available experimental data. The essence of the method is to: First obtain asymptotic solutions for limiting cases of the flow under consideration. Then, assume velocity profiles with sufficient generality to include the asymptotic solutions. The profile form functions are determined by applying integral relations (velocity weighted averages of the momentum equation) and compatibility conditions (normal derivatives of the momentum equation evaluated at the boundary). As an example of the method, the second-order mean flow correction is determined for oscillating flow over a flat plate.

Experimental and Theoretical Investigation of the Supersonic Boundary Layer Stability on the Swept Wing

2008 ◽  
Vol 3 (3) ◽  
pp. 34-38
Author(s):  
Sergey A. Gaponov ◽  
Yuri G. Yermolaev ◽  
Aleksandr D. Kosinov ◽  
Nikolay V. Semionov ◽  
Boris V. Smorodsky
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Leading Edge ◽  
Supersonic Boundary Layer ◽  
Mean Flow ◽  
Swept Wing ◽  
Wing Model ◽  
Dimensional Boundary ◽  
The Stability ◽  
Good Agreement

Theoretical and an experimental research results of the disturbances development in a swept wing boundary layer are presented at Mach number М = 2. In experiments development of natural and small amplitude controllable disturbances downstream was studied. Experiments were carried out on a swept wing model with a lenticular profile at a zero attack angle. The swept angle of a leading edge was 40°. Wave parameters of moving disturbances were determined. In frames of the linear theory and an approach of the local self-similar mean flow the stability of a compressible three-dimensional boundary layer is studied. Good agreement of the theory with experimental results for transversal scales of unstable vertices of the secondary flow was obtained. However the calculated amplification rates differ from measured values considerably. This disagreement is explained by the nonlinear processes observed in experiment

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