Evolution of wavelike disturbances in shear flows : a class of exact solutions of the Navier-Stokes equations

1986 ◽  
Vol 406 (1830) ◽  
pp. 13-26 ◽  
Keyword(s):  
Exact Solutions ◽  
Shear Flows ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Periodic Disturbances ◽  
Method Of Solution ◽  
Spatially Periodic ◽  
Basic Shear

New classes of exact solutions of the incompressible Navier-Stokes equations are presented. The method of solution has its origins in that first used by Kelvin ( Phil. Mag . 24 (5), 188-196 (1887)) to solve the linearized equations governing small disturbances in unbounded plane Couette flow. The new solutions found describe arbitrarily large, spatially periodic disturbances within certain two- and three-dimensional ‘ basic ’ shear flows of unbounded extent. The admissible classes of basic flow possess spatially uniform strain rates; they include two- and three- dimensional stagnation point flows and two-dimensional flows with uniform vorticity. The disturbances, though spatially periodic, have time-dependent wavenumber and velocity components. It is found that solutions for the disturbance do not always decay to zero ; but in some instances grow continuously in spite of viscous dissipation. This behaviour is explained in terms of vorticity dynamics.

scholarly journals On the Navier-Stokes equations with temperature-dependent transport coefficients

2006 ◽  
Vol 2006 ◽  
pp. 1-14 ◽  
Author(s):  
Eduard Feireisl ◽  
Josef Málek
Keyword(s):  
Stokes Equations ◽  
Transport Coefficients ◽  
Three Dimensional ◽  
Periodic Problem ◽  
Large Data ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Long Time ◽  
Spatially Periodic ◽  
Dependent Transport

We establish long-time and large-data existence of a weak solution to the problem describing three-dimensional unsteady flows of an incompressible fluid, where the viscosity and heat-conductivity coefficients vary with the temperature. The approach reposes on considering the equation for the total energy rather than the equation for the temperature. We consider the spatially periodic problem.

scholarly journals Exact Solutions to the Three-Dimensional Navier–Stokes Equations Using the Extended Beltrami Method

2019 ◽  
Vol 87 (1) ◽  
Author(s):  
Nolan J. Dyck ◽  
Anthony G. Straatman
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Natural Extension ◽  
Three Dimensional ◽  
Linear Partial Differential Equation ◽  
Auxiliary Function ◽  
Curvilinear Coordinates ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Auxiliary Functions

Abstract In a 1966 publication, Chi-Yi Wang used the streamfunction in concert with the vorticity equations to develop a methodology for obtaining exact solutions to the incompressible Navier–Stokes equations, now known as the extended Beltrami method. In Wang's approach, the vorticity is represented by the sum of a linear function of the streamfunction and an assumed auxiliary function, such that the vorticity equation can be reduced to a quasi-linear partial differential equation, and exact solutions are obtainable for many choices of the auxiliary function. In the present work, a natural extension of Wang's formulation to three-dimensional flows in arbitrary orthogonal curvilinear coordinates has been derived, wherein two auxiliary functions are formed at the outset, with the caveat that the pressure and velocity components may vary in two spatial dimensions. As is the case with two-dimensional extended Beltrami flows, exact solutions are only obtainable when the forms of the auxiliary functions are “simple enough” to render the governing equations solvable. To demonstrate the solutions which may be obtained using the extended formulation, the well-known Kovasznay flow is generalized to a three-dimensional flow. A unique solution in plane polar coordinates is found. An extension to the solution to Burgers vortex has been derived and discussed in the context of existing literature. Finally, a new 3D swirling flow solution which is the angular analogue to Kovasznay flow has been developed.

scholarly journals A class of exact solutions to the three-dimensional incompressible Navier–Stokes equations

2010 ◽  
Vol 23 (11) ◽  
pp. 1388-1396 ◽  
Author(s):  
Gunawan Nugroho ◽  
Ahmed M.S. Ali ◽  
Zainal A. Abdul Karim
Keyword(s):  
Exact Solutions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations

Eddy viscosity of three-dimensional flow

1995 ◽  
Vol 288 ◽  
pp. 249-264 ◽  
Author(s):  
A. Wirth ◽  
S. Gama ◽  
U. Frisch
Keyword(s):  
Large Scale ◽  
Eddy Viscosity ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Cubic Symmetry ◽  
A Value ◽  
Spatially Periodic

Detailed theoretical and numerical results are presented for the eddy viscosity of three-dimensional forced spatially periodic incompressible flow.As shown by Dubrulle & Frisch (1991), the eddy viscosity, which is in general a fourth-order anisotropic tensor, is expressible in terms of the solution of auxiliary problems. These are, essentially, three-dimensional linearized Navier–Stokes equations which must be solved numerically.The dynamics of weak large-scale perturbations of wavevector k is determined by the eigenvalues – called here ‘eddy viscosities’ – of a two by two matrix, obtained by contracting the eddy viscosity tensor with two k-vectors and projecting onto the plane transverse to k to ensure incompressibility. As a consequence, eddy viscosities in three dimensions, but not in two, can become complex. It is shown that this is ruled out for flow with cubic symmetry, the eddy viscosities of which may, however, become negative.An instance is the equilateral ABC-flow (A = B = C = 1). When the wavevector k is in any of the three coordinate planes, at least one of the eddy viscosities becomes negative for R = 1/v > Rc [bsime ] 1.92. This leads to a large-scale instability occurring for a value of the Reynolds number about seven times smaller than instabilities having the same spatial periodicity as the basic flow.

Applications of exact solutions to the Navier–Stokes equations: free shear layers

1994 ◽  
Vol 274 ◽  
pp. 267-291 ◽  
Author(s):  
Eric Varley ◽  
Brian R. Seymour
Keyword(s):  
Exact Solutions ◽  
Shear Layer ◽  
Stokes Equations ◽  
Mathematical Problem ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Plane Boundary ◽  
Turbulent Shear ◽  
Navier Stokes Equations ◽  
Linear Ordinary Differential Equations

A family of exact solutions to the Navier—Stokes equations is used to analyse unsteady three-dimensional viscometric flows that occur in the vicinity of a plane boundary that translates and rotates with time-varying velocities. Such flows are important in the study of flows that are produced by rotating machinery. They are also useful in describing local behaviour in more complex global flows, such as that produced in a shear layer by the passage of a disturbance in the mainstream. An example is the flow produced in a turbulent shear layer by the passage of the core of a Rankine vortex. When the effect of viscosity is unimportant, the use of Lagrangian coordinates reduces the mathematical problem to that of solving a set of linear ordinary differential equations.

Effects of stator splitter blades on aerodynamic performance of a single-stage transonic axial compressor

2020 ◽  
Vol 14 (4) ◽  
pp. 7369-7378
Author(s):  
Ky-Quang Pham ◽  
Xuan-Truong Le ◽  
Cong-Truong Dinh
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Aerodynamic Performance ◽  
Numerical Results ◽  
Single Stage ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Operating Stability ◽  
Length Coverage

Splitter blades located between stator blades in a single-stage axial compressor were proposed and investigated in this work to find their effects on aerodynamic performance and operating stability. Aerodynamic performance of the compressor was evaluated using three-dimensional Reynolds-averaged Navier-Stokes equations using the k-e turbulence model with a scalable wall function. The numerical results for the typical performance parameters without stator splitter blades were validated in comparison with experimental data. The numerical results of a parametric study using four geometric parameters (chord length, coverage angle, height and position) of the stator splitter blades showed that the operational stability of the single-stage axial compressor enhances remarkably using the stator splitter blades. The splitters were effective in suppressing flow separation in the stator domain of the compressor at near-stall condition which affects considerably the aerodynamic performance of the compressor.

scholarly journals Dynamics of Single Droplet Splashing on Liquid Film by Coupling FVM with VOF

Processes ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 841
Author(s):  
Yuzhen Jin ◽  
Huang Zhou ◽  
Linhang Zhu ◽  
Zeqing Li
Keyword(s):  
Liquid Film ◽  
Weber Number ◽  
Stokes Equations ◽  
Numerical Study ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Single Droplet ◽  
Navier Stokes Equations ◽  
Volume Method ◽  
The Relationship

A three-dimensional numerical study of a single droplet splashing vertically on a liquid film is presented. The numerical method is based on the finite volume method (FVM) of Navier–Stokes equations coupled with the volume of fluid (VOF) method, and the adaptive local mesh refinement technology is adopted. It enables the liquid–gas interface to be tracked more accurately, and to be less computationally expensive. The relationship between the diameter of the free rim, the height of the crown with different numbers of collision Weber, and the thickness of the liquid film is explored. The results indicate that the crown height increases as the Weber number increases, and the diameter of the crown rim is inversely proportional to the collision Weber number. It can also be concluded that the dimensionless height of the crown decreases with the increase in the thickness of the dimensionless liquid film, which has little effect on the diameter of the crown rim during its growth.

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