Closure to “Discussions of ‘A Three-Dimensional Integral Method for Calculating Incompressible Turbulent Skin Friction’” (1975, ASME J. Fluids Eng., 97, pp. 555–556)

1975 ◽  
Vol 97 (4) ◽  
pp. 556-557
Author(s):  
F. M. White ◽  
R. C. Lessmann ◽  
G. H. Christoph
Keyword(s):  
Skin Friction ◽  
Integral Method ◽  
Three Dimensional ◽  
Dimensional Integral

A Three-Dimensional Integral Method for Calculating Incompressible Turbulent Skin Friction

1975 ◽  
Vol 97 (4) ◽  
pp. 550-555 ◽  
Author(s):  
F. M. White ◽  
R. C. Lessmann ◽  
G. H. Christoph
Keyword(s):  
Skin Friction ◽  
Integral Method ◽  
Three Dimensional ◽  
Separated Flow ◽  
Turbulent Boundary Layers ◽  
Shape Factors ◽  
Coupled Partial Differential Equations ◽  
Separation Line ◽  
Curved Duct ◽  
Dimensional Integral

A new integral method is proposed for the analysis of three-dimensional incompressible turbulent boundary layers. The method utilizes velocity profile expressions in wall-law form to derive two coupled partial differential equations for the two components of surface skin friction. No shape factors or empirical shear stress correlations are needed in the method. The only requirements are a knowledge of the external velocity and streamline distribution and initial values of skin friction along a starting crossflow line of the flow. The method is insensitive to sidewall conditions and may be continued downstream until the complete three-dimensional separation line of the flow has been computed. Two comparisons with experiment are shown: a curved-duct unseparated flow and a T-shaped-box separated flow. The calculations are very straightforward and agree reasonably well with the data for friction, crossflow angle, and separation line.

A three-dimensional field-integral method for the calculation of transonic flow on complex configurations — theory and preliminary results

1988 ◽  
Vol 92 (916) ◽  
pp. 235-241 ◽  
Author(s):  
P. M. Sinclair
Keyword(s):  
Transonic Flow ◽  
Integral Method ◽  
Three Dimensional ◽  
Full Potential ◽  
Potential Equation ◽  
Direct Extension ◽  
Body Shapes ◽  
Dimensional Integral ◽  
Field Integral ◽  
Good Agreement

Summary A three-dimensional integral formulation for the solution of the full potential equation and the associated numerical algorithm, the field-integral method, are presented. The method is a direct extension of a two-dimensional method and in particular retains the simple grid generation requirements noted in that method. Results are presented for the flow over body shapes and a complex winglet configuration, and are compared with existing transonic methods and experiments with good agreement. The further work necessary to provide a fast, robust method for use in design is outlined.

scholarly journals The on-axis magnetic well and Mercier's criterion for arbitrary stellarator geometries

2021 ◽  
Vol 87 (2) ◽  
Author(s):  
P. Kim ◽  
R. Jorge ◽  
W. Dorland
Keyword(s):  
Magnetic Field ◽  
Original Work ◽  
Three Dimensional ◽  
Radial Distance ◽  
Analytical Form ◽  
One Dimensional ◽  
Modern Notation ◽  
Magnetic Well ◽  
First Time ◽  
Dimensional Integral

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.

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