Application of Kraichnan's direct interaction approximation to kinematic dynamo theory. I. Incompressible isotropic turbulence and a singular integral equation

1973 ◽  
Vol 14 (10) ◽  
pp. 1381-1388 ◽  
Author(s):  
I. Lerche
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Isotropic Turbulence ◽  
Direct Interaction ◽  
Dynamo Theory ◽  
Kinematic Dynamo ◽  
Direct Interaction Approximation ◽  
Interaction Approximation

Velocity-derivative skewness and two-time velocity correlations of isotropic turbulence as predicted by the LET theory

1989 ◽  
Vol 208 ◽  
pp. 91-114 ◽  
Author(s):  
W. D. Mccomb ◽  
V. Shanmugasundaram ◽  
P. Hutchinson
Keyword(s):  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Direct Interaction ◽  
Reynolds Numbers ◽  
Time Correlation ◽  
General Belief ◽  
Direct Interaction Approximation ◽  
The Difference ◽  
Decaying Turbulence ◽  
Interaction Approximation

The local-energy-transfer (LET) theory was used to calculate freely decaying turbulence for arbitrary initial conditions over a range of microscale-based Reynolds numbers 0.5 [les ] Rλ(tf) [les ] 1009, where tf is the final time of computation. The predicted skewness factor S(Rλ) agreed closely with the results of numerical simulations at low-to-moderate Reynolds numbers and followed the same general trend at larger values of Rλ. It was also found that, for Rλ(tf) [les ] 5, the LET calculation was almost indistinguishable from that of the direct-interaction approximation (DIA), with the difference between the two theories tending to zero as Rλ(tf)∞ 0.Two-time correlation and propagator (or response) functions were also obtained. Tests of their scaling behaviour suggest that, contrary to general belief, the convective sweeping of the energy-containing range is much less important than the Kolmogorov timescale in determining inertial-range behaviour. This result raises questions about the accepted explanation for the failure of the direct-interaction approximation, thus motivating a discussion about the relevance of random Galilean invariance (RGI). It is argued that, for a properly constructed ensemble of transformations to inertial frames, invariance in every realization necessarily implies RGI. It is suggested that the defects of the direct-interaction approximation can be understood in terms of a failure to renormalize the stirring forces.

Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

2008 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
P. KARCZMAREK
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Trigonometric Polynomials ◽  
Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

A semianalytical and finite-element solution to the unbonded contact between a frictionless layer and an FGM-coated half-plane

2017 ◽  
Vol 24 (2) ◽  
pp. 448-464 ◽  
Author(s):  
Jie Yan ◽  
Changwen Mi ◽  
Zhixin Liu
Keyword(s):  
Integral Equation ◽  
Finite Element ◽  
Singular Integral Equation ◽  
Contact Problem ◽  
Singular Integral ◽  
Material Properties ◽  
Elastic Layer ◽  
Coated Substrate ◽  
Receding Contact ◽  
Contact Size

In this work, we examine the receding contact between a homogeneous elastic layer and a half-plane substrate reinforced by a functionally graded coating. The material properties of the coating are allowed to vary exponentially along its thickness. A distributed traction load applied over a finite segment of the layer surface presses the layer and the coated substrate against each other. It is further assumed that the receding contact between the layer and the coated substrate is frictionless. In the absence of body forces, Fourier integral transforms are used to convert the governing equations and boundary conditions of the plane receding contact problem into a singular integral equation with the contact pressure and contact size as unknowns. Gauss–Chebyshev quadrature is subsequently employed to discretize both the singular integral equation and the force equilibrium condition at the contact interface. An iterative algorithm based on the method of steepest descent has been proposed to numerically solve the system of algebraic equations, which is linear for the contact pressure but nonlinear for the contact size. Extensive case studies are performed with respect to the coating inhomogeneity parameter, geometric parameters, material properties, and the extent of the indentation load. As a result of the indentation, the elastic layer remains in contact with the coated substrate over only a finite interval. Exterior to this region, the layer and the coated substrate lose contact. Nonetheless, the receding contact size is always larger than that of the indentation traction. To validate the theoretical solution, we have also developed a finite-element model to solve the same receding contact problem. Numerical results of finite-element modeling and theoretical development are compared in detail for a number of parametric studies and are found to agree very well with each other.

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