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.

Simulation of Three-Dimensional Shear Flow Around a Nozzle-Afterbody at High Speeds

1992 ◽  
Vol 114 (2) ◽  
pp. 178-185 ◽  
Author(s):  
Oktay Baysal ◽  
Wendy B. Hoffman
Keyword(s):  
Shear Layer ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Surface Flow ◽  
Navier Stokes ◽  
Turbulent Shear ◽  
Duct Flow ◽  
Navier Stokes Equations ◽  
Eddy Viscosity Model ◽  
Pressure Distributions

Turbulent shear flows at supersonic and hypersonic speeds around a nozzle-afterbody are simulated. The three-dimensional, Reynolds-averaged Navier-Stokes equations are solved by a finite-volume and implicit method. The convective and the pressure terms are differenced by an upwind-biased algorithm. The effect of turbulence is incorporated by a modified Baldwin-Lomax eddy viscosity model. The success of the standard Baldwin-Lomax model for this flow type is shown by comparing it to a laminar case. These modifications made to the model are also shown to improve flow prediction when compared to the standard Baldwin-Lomax model. These modifications to the model reflect the effects of high compressibility, multiple walls, vortices near walls, and turbulent memory effects in the shear layer. This numerically simulated complex flowfield includes a supersonic duct flow, a hypersonic flow over an external double corner, a flow through a non-axisymmetric, internal-external nozzle, and a three-dimensional shear layer. The specific application is for the flow around the nozzle-afterbody of a generic hypersonic vehicle powered by a scramjet engine. The computed pressure distributions compared favorably with the experimentally obtained surface and off-surface flow surveys.

A Numerical Examination of the Inkjet Refilling

2004 ◽  
Author(s):  
Kai-Shing Yang ◽  
Ing-Young Chen ◽  
Chi-Chuan Wang
Keyword(s):  
Dynamic Viscosity ◽  
Bubble Growth ◽  
Stokes Equations ◽  
Numerical Study ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Navier Stokes ◽  
Plane Boundary ◽  
Navier Stokes Equations ◽  
Numerical Examination

A numerical study is conducted to examine the flow characteristics of the inkjet print-head with special attentions on the refilling process. By solving the full set of three-dimensional transient Navier-Stokes equations and considering the process of bubble growth and collapse as a movable membrane, it is found that the double refilling channels can reduce the flow surge phenomenon considerably due to the imposed friction. However, for the additional cylinder obstacle placed at the filling channel, the flow surge phenomenon is still present. This is because of the jet-like flow along the cylinder leading to a collision and eruption of fluid angled towards the plane boundary with the presence of cylinder. The calculated results also indicated the flow surge can be moderately suppressed for fluid having larger dynamic viscosity.

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 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

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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