Navier Stokes Derivative Estimates in Three Dimensions with Boundary Values and Body Forces

1991 ◽  
Vol 43 (6) ◽  
pp. 1161-1212 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Stokes Equations ◽  
Finite Energy ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Boundary Values ◽  
Vector Solution ◽  
Navier Stokes Equations ◽  
Body Forces ◽  
Derivatives Of

AbstractFor a vector solution u(x, t) with finite energy of the Navier Stokes equations with body forces and boundary values on a region Ω ⊆ R3 for t > 0, conditions are established on the L6/5(Ω) and L2(Ω) norms of derivatives of the data that ensure the estimates and max , up to any given integer value of the weighted order 2r+s, where r or s = s1 + s2 + s3 > 0 and 0 < T < ∞.

Singularities

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Stokes Equations ◽  
Negative Answer ◽  
Three Dimensions ◽  
Picard Iteration ◽  
Navier Stokes ◽  
Scale Invariant ◽  
Navier Stokes Equations ◽  
L2 Norm ◽  
Two Dimensional Flow ◽  
Physical Consequences

The initial value problem of the incompressible Navier–Stokes equations is explained. Leray’s classic study of it (using Picard iteration) is simplified and described in the language of physics. The ideas of Lebesgue and Sobolev norms are explained. The L2 norm being the energy, cannot increase. This gives sufficient control to establish existence, regularity and uniqueness in two-dimensional flow. The L3 norm is not guaranteed to decrease, so this strategy fails in three dimensions. Leray’s proof of regularity for a finite time is outlined. His attempts to construct a scale-invariant singular solution, and modern work showing this is impossible, are then explained. The physical consequences of a negative answer to the regularity of Navier–Stokes solutions are explained. This chapter is meant as an introduction, for physicists, to a difficult field of analysis.

scholarly journals 3D Hydrodynamic Modelling Enhances the Design of Tendaho Dam Spillway, Ethiopia

Water ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 82
Author(s):  
Getnet Kebede Demeke ◽  
Dereje Hailu Asfaw ◽  
Yilma Seleshi Shiferaw
Keyword(s):  
Stokes Equations ◽  
Flow Behavior ◽  
Fluid Flows ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Hydraulic Structures ◽  
Hydrodynamic Modelling ◽  
Navier Stokes Equations ◽  
Pressure Distributions ◽  
Computational Fluid Dynamics Cfd

Hydraulic structures are often complex and in many cases their designs require attention so that the flow behavior around hydraulic structures and their influence on the environment can be predicted accurately. Currently, more efficient computational fluid dynamics (CFD) codes can solve the Navier–Stokes equations in three-dimensions and free surface computation in a significantly improved manner. CFD has evolved into a powerful tool in simulating fluid flows. In addition, CFD with its advantages of lower cost and greater flexibility can reasonably predict the mean characteristics of flows such as velocity distributions, pressure distributions, and water surface profiles of complex problems in hydraulic engineering. In Ethiopia, Tendaho Dam Spillway was constructed recently, and one flood passed over the spillway. Although the flood was below the designed capacity, there was an overflow due to superelevation at the bend. Therefore, design of complex hydraulic structures using the state-of- art of 3D hydrodynamic modelling enhances the safety of the structures. 3D hydrodynamic modelling was used to verify the safety of the spillway using designed data and the result showed that the constructed hydraulic section is not safe unless it is modified.

Computational Investigation of Tower Shadow Effects on Wind Turbines

2011 ◽  
Author(s):  
Adrian Sescu ◽  
Brett Andersen ◽  
Abdollah A. Afjeh
Keyword(s):  
Wind Turbines ◽  
Stokes Equations ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Turbulent Regime ◽  
Navier Stokes Equations ◽  
Shadow Effect ◽  
Horizontal Axis Wind Turbines ◽  
Computational Fluid Dynamics Cfd ◽  
Tower Shadow

In this paper, a Computational Fluid Dynamics (CFD) analysis of tower shadow effect associated with downwind configuration horizontal axis wind turbines (HAWT) is presented. This work is part of a large project focused on the design and installation of a multi-megawatt, downwind configuration, offshore HAWT in Lake Erie. The numerical tool is a dedicated commercial CFD solver, capable of performing accurate simulations in three dimensions, by solving the Navier-Stokes equations in turbulent regime. The tower shadow effect is analyzed by comparing results from the downwind configuration to results from the corresponding upwind configuration and free rotor.

Sign in / Sign up

Export Citation Format

Share Document