A Simplified Theory of Oscillating Aerofoils in Transonic Flow: Review and Extension

1980 ◽  
Vol 31 (4) ◽  
pp. 252-284
Author(s):  
E.H. Dowell
Keyword(s):  
Boundary Condition ◽  
Unsteady Flow ◽  
Flow Separation ◽  
Transonic Flow ◽  
Three Dimensional ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Bernoulli’S Equation ◽  
Surface Boundary Condition

SummarySignificant new results are presented to show to what extent a simplified theory for transonic flow may be used. Solutions are obtained by classical techniques and compared with experiment. Results are given for two-dimensional and three-dimensional, steady and unsteady flow. The effects of flow separation and improvements in Bernoulli’s equation and the surface boundary condition are also briefly discussed.

scholarly journals On the time to tracer equilibrium in the global ocean

2008 ◽  
Vol 5 (3) ◽  
pp. 471-506
Author(s):  
F. Primeau ◽  
E. Deleersnijder
Keyword(s):  
Boundary Condition ◽  
Ocean Circulation ◽  
Dirichlet Boundary Condition ◽  
Deep Sea ◽  
Dirichlet Boundary ◽  
Three Dimensional ◽  
Sea Surface ◽  
Global Ocean ◽  
Surface Boundary ◽  
Surface Boundary Condition

Abstract. An important issue for the interpretation of data from deep-sea cores is the time for tracers to be transported from the sea surface to the deep ocean. Global ocean circulation models can help shed light on the timescales over which a tracer comes to equilibrium in different regions of the ocean. In this note, we discuss how the most slowly decaying eigenmode of a model can be used to obtain a relevant timescale for a tracer that enters through the sea surface to become well mixed in the ocean interior. We show how this timescale depends critically on the choice between a Neumann surface boundary condition in which the flux of tracer is prescribed or a Dirichlet surface boundary condition in which the concentration is prescribed. Explicit calculations with a 3-box model and a three-dimensional ocean circulation model show that the Dirichlet boundary condition when applied to only part of the surface ocean greatly overestimate the time needed to reach equilibrium. As a result regional-"injection" calculations which prescribe the surface concentration instead of the surface flux are not relevant for interpreting the regional disequilibrium between the Atlantic and Pacific found in paleo-tracer records from deep-sea cores. For tracers such as δ18O that enter the ocean from melt water, a Neumann boundary condition is more relevant. For tracers that enter the ocean through air-sea gas exchange such as 14C, a prescribed concentration boundary condition can be used to infer relevant timescales, but the Dirichlet Boundary condition must be applied over the entire ocean surface and not only to a patch of limited area. Our three-dimensional model results based on a steady-state modern circulation suggest that the relative disequilibrium between the deep Atlantic and Pacific is on the order of "only" 1200 years or less and does not depend on the size and location of the patch where the tracer is injected.

scholarly journals On the time to tracer equilibrium in the global ocean

Ocean Science ◽  
2009 ◽  
Vol 5 (1) ◽  
pp. 13-28 ◽  
Author(s):  
F. Primeau ◽  
E. Deleersnijder
Keyword(s):  
Boundary Condition ◽  
Ocean Circulation ◽  
Deep Sea ◽  
Neumann Boundary Condition ◽  
Three Dimensional ◽  
Neumann Boundary ◽  
Sea Surface ◽  
Global Ocean ◽  
Surface Boundary ◽  
Surface Boundary Condition

Abstract. An important issue for the interpretation of data from deep-sea cores is the time for tracers to be transported from the sea surface to the deep ocean. Global ocean circulation models can help shed light on the timescales over which a tracer comes to equilibrium in different regions of the ocean. In this note, we discuss how the most slowly decaying eigenmode of a model can be used to obtain a relevant timescale for a tracer that enters through the sea surface to become well mixed in the ocean interior. We show how this timescale depends critically on the choice between a Neumann surface boundary condition in which the flux of tracer is prescribed, a Robin surface boundary condition in which a combination of the flux and tracer concentration is prescribed or a Dirichlet surface boundary condition in which the concentration is prescribed. Explicit calculations with a 3-box model and a three-dimensional ocean circulation model show that the Dirichlet boundary condition when applied to only part of the surface ocean greatly overestimate the time needed to reach equilibrium. As a result regional-"injection" calculations which prescribe the surface concentration instead of the surface flux are not relevant for interpreting the regional disequilibrium between the Atlantic and Pacific found in paleo-tracer records from deep-sea cores. For tracers that enter the ocean through air-sea gas exchange a prescribed concentration boundary condition can be used to infer relevant timescales if the air-sea gas exchange rate is sufficiently fast, but the boundary condition must be applied over the entire ocean surface and not only to a patch of limited area. For tracers with a slow air-sea exchange rate such as 14C a Robin-type boundary condition is more relevant and for tracers such as δ18O that enter the ocean from melt water, a Neumann boundary condition is presumably more relevant. Our three-dimensional model results based on a steady-state modern circulation suggest that the relative disequilibrium between the deep Atlantic and Pacific is on the order of "only" 1200 years or less for a Neumann boundary condition and does not depend on the size and location of the patch where the tracer is injected.

Three-Dimensional Non-Reflecting Boundary Condition for Linearized Flow Solvers

2010 ◽  
Author(s):  
Paul J. Petrie-Repar
Keyword(s):  
Boundary Condition ◽  
Unsteady Flow ◽  
Steady Flow ◽  
Three Dimensional ◽  
Far Field ◽  
Two Dimensional ◽  
Flow Equations ◽  
Flow Simulations ◽  
Unsteady Aerodynamic ◽  
2D And 3D

A three-dimensional (3D) non-reflecting boundary condition for linearized flow solvers is presented. The unsteady aerodynamic modes at the inlet and outlet (far-field) are numerically determined by solving an eigen problem for the semi-discretized flow equations on a two-dimensional mesh. Unlike previous methods the shape of the far-field can be general and the non-uniformity of the steady flow across the far-field is considered. The calculated unsteady modes are used to decompose the unsteady flow at the far-field into modes. The direction of each mode is determined, and incoming modes are prescribed and outgoing modes are extrapolated. The results of 2D and 3D inviscid linearised flow simulations using the new boundary condition are presented.

Theory of Blade Design for Large Deflections: Part I—Two-Dimensional Cascade

1984 ◽  
Vol 106 (2) ◽  
pp. 346-353 ◽  
Author(s):  
W. R. Hawthorne ◽  
C. Wang ◽  
C. S. Tan ◽  
J. E. McCune
Keyword(s):  
Three Dimensional ◽  
Tangential Velocity ◽  
Mean Value ◽  
Three Dimensions ◽  
Surface Boundary ◽  
Large Deflections ◽  
Two Dimensional ◽  
Axial Compressors ◽  
A Value ◽  
Surface Boundary Condition

As a step in the development of an analytical method for designing highly loaded, three-dimensional blade profiles for axial compressors and turbines, a simple two-dimensional method was first investigated. The fluid is assumed to be incompressible and inviscid, the blades of negligible thickness, and the mean tangential velocity is prescribed. The blades are represented by a distributed bound vorticity whose strength is determined by the prescribed tangential velocity. The velocity induced by the bound vortices is obtained by a conventional Biot-Savart method assuming a first approximation to the blade profile. Using the blade surface boundary condition, the profile is then obtained by iteration. It is shown that this procedure is successful even for large pitch-chord ratios and large deflections. In order to develop a method for use in three dimensions, the velocity is divided into a pitchwise mean value and a value varying periodically in the pitchwise direction. By using generalized functions to represent the bound vorticity and a Clebsch formulation for the periodic velocity, series expressions are obtained which can be adapted to three-dimensional problems. Several numerical results were obtained using both approaches.

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