scholarly journals A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equations

2021 ◽  
Author(s):  
Taofiq Amoloye
Keyword(s):  
Reynolds Number ◽  
Numerical Simulations ◽  
Potential Theory ◽  
Stream Function ◽  
Continuity Equation ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Consistent Solution

Abstract The three main approaches in fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. Numerical simulations and theoretical solutions are based on the continuity equation and Navier-Stokes equations (NSE) that govern experimental observations of fluid dynamics.Theoretical solutions can offer huge advantages over numerical solutions and experiments in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, theoretical solutions have been limited by the prized NSE problem that seeks a physically consistent solution than what classical potential theory (CPT) offers. Therefore, the current author refined CPT. He introduced refined potential theory (RPT) that provides a viscous potential/stream function as a physically consistent solution to the NSE problem. This function captures observable unsteady flow features including separation, wake, vortex shedding, compressibility, turbulence, and Reynolds-number-dependence. It appropriately combines the properties of a three-dimensional potential function that satisfy the inertia terms of NSE and the features of a stream function that satisfy the continuity equation, the viscous vorticity equation, and the viscous terms of NSE. RPT has been verified and validated against experimental and numerical results of incompressible unsteady sub-critical Reynolds number flows on stationary finite circular cylinder, sphere, and spheroid.

scholarly journals A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equation

2021 ◽  
Author(s):  
Taofiq Omoniyi Amoloye
Keyword(s):  
Fluid Dynamics ◽  
Reynolds Number ◽  
Analytical Solution ◽  
Numerical Simulations ◽  
Closed Form ◽  
Continuity Equation ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Consistent Solution ◽  
Closed Form Analytical Solution

Abstract The three main approaches to exploring fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. Numerical simulations and theoretical solutions are based on the continuity equation and Navier-Stokes equations (NSE) that govern experimental observations of fluid dynamics. Theoretical solutions can offer huge advantages over numerical solutions and experiments in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, theoretical solutions have been limited by the prized NSE problem that seeks a physically consistent solution than what classical potential theory (CPT) offers. Therefore, the current author embarked on a doctoral research on the refinement of CPT. He introduced the Refined Potential Theory (RPT) that provides the Kwasu function as a physically consistent solution to the NSE problem. The Kwasu function is a viscous scalar potential function that captures known and observable unsteady features of experimentally observed wall bounded flows including flow separation, wake formation, vortex shedding, compressibility effects, turbulence and Reynolds-number-dependence. It is appropriately defined to combine the properties of a three-dimensional potential function to satisfy the inertia terms of the NSE and the features of a stream function to satisfy the continuity equation, the viscous vorticity equation and the viscous terms of the NSE. RPT has been verified and validated against experimental and numerical results of incompressible unsteady sub-critical Reynolds number flows on stationary finite circular cylinder, sphere and spheroid. It is concluded that the Kwasu function is a physically consistent and closed-form analytical solution to the NSE problem.

scholarly journals Nonlinear amplification in hydrodynamic turbulence

2021 ◽  
Vol 930 ◽  
Author(s):  
Kartik P. Iyer ◽  
Katepalli R. Sreenivasan ◽  
P.K. Yeung
Keyword(s):  
Reynolds Number ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Developed Turbulence ◽  
Advection Term ◽  
Nonlinear Amplification

Using direct numerical simulations performed on periodic cubes of various sizes, the largest being $8192^3$ , we examine the nonlinear advection term in the Navier–Stokes equations generating fully developed turbulence. We find significant dissipation even in flow regions where nonlinearity is locally absent. With increasing Reynolds number, the Navier–Stokes dynamics amplifies the nonlinearity in a global sense. This nonlinear amplification with increasing Reynolds number renders the vortex stretching mechanism more intermittent, with the global suppression of nonlinearity, reported previously, restricted to low Reynolds numbers. In regions where vortex stretching is absent, the angle and the ratio between the convective vorticity and solenoidal advection in three-dimensional isotropic turbulence are statistically similar to those in the two-dimensional case, despite the fundamental differences between them.

scholarly journals Numerical Simulation of Slip effect on Lid-Driven Cavity Flow Problem for High Reynolds Number: Vorticity – Stream Function Approach

2021 ◽  
Vol 8 (3) ◽  
pp. 418-424
Author(s):  
Syed Fazuruddin ◽  
Seelam Sreekanth ◽  
G. Sankara Sekhar Raju
Keyword(s):  
Reynolds Number ◽  
Stream Function ◽  
Stokes Equations ◽  
Cavity Flow ◽  
Partial Slip ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Driven Cavity ◽  
Slip Conditions ◽  
Driven Cavity Flow

Incompressible 2-D Navier-stokes equations for various values of Reynolds number with and without partial slip conditions are studied numerically. The Lid-Driven cavity (LDC) with uniform driven lid problem is employed with vorticity - Stream function (VSF) approach. The uniform mesh grid is used in finite difference approximation for solving the governing Navier-stokes equations and developed MATLAB code. The numerical method is validated with benchmark results. The present work is focused on the analysis of lid driven cavity flow of incompressible fluid with partial slip conditions (imposed on side walls of the cavity). The fluid flow patterns are studied with wide range of Reynolds number and slip parameters.

Finite-Reynolds-Number Effects in Steady, Three-Dimensional Airway Reopening

2006 ◽  
Vol 128 (4) ◽  
pp. 573-578 ◽  
Author(s):  
Andrew L. Hazel ◽  
Matthias Heil
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Elastic Tube ◽  
Navier Stokes ◽  
Fluid Inertia ◽  
Navier Stokes Equations ◽  
Airway Reopening ◽  
Reynolds Number Effects ◽  
Steady Propagation

Motivated by the physiological problem of pulmonary airway reopening, we study the steady propagation of an air finger into a buckled elastic tube, initially filled with viscous fluid. The system is modeled using geometrically non-linear, Kirchhoff-Love shell theory, coupled to the free-surface Navier-Stokes equations. The resulting three-dimensional, fluid-structure-interaction problem is solved numerically by a fully coupled finite element method. Our study focuses on the effects of fluid inertia, which has been neglected in most previous studies. The importance of inertial forces is characterized by the ratio of the Reynolds and capillary numbers, Re∕Ca, a material parameter. Fluid inertia has a significant effect on the system’s behavior, even at relatively small values of Re∕Ca. In particular, compared to the case of zero Reynolds number, fluid inertia causes a significant increase in the pressure required to drive the air finger at a given speed.

Solution of three-dimensional incompressible flow problems

1977 ◽  
Vol 82 (2) ◽  
pp. 309-319 ◽  
Author(s):  
S. M. Richardson ◽  
A. R. H. Cornish
Keyword(s):  
Stream Function ◽  
Incompressible Flow ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
Flow Problems ◽  
Vector Potentials

A method for solving quite general three-dimensional incompressible flow problems, in particular those described by the Navier–Stokes equations, is presented. The essence of the method is the expression of the velocity in terms of scalar and vector potentials, which are the three-dimensional generalizations of the two-dimensional stream function, and which ensure that the equation of continuity is satisfied automatically. Although the method is not new, a correct but simple and unambiguous procedure for using it has not been presented before.

scholarly journals Estimating intermittency in three-dimensional Navier–Stokes turbulence

2009 ◽  
Vol 625 ◽  
pp. 125-133 ◽  
Author(s):  
J. D. GIBBON
Keyword(s):  
Reynolds Number ◽  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Space Time ◽  
Navier Stokes ◽  
Local Velocity ◽  
Vortex Sheets ◽  
Navier Stokes Equations ◽  
Local Number

The issue of why computational resolution in Navier–Stokes turbulence is hard to achieve is addressed. Under the assumption that the three-dimensional Navier–Stokes equations have a global attractor it is nevertheless shown that solutions can potentially behave differently in two distinct regions of space–time $\mathbb{S}$± where $\mathbb{S}$− is comprised of a union of disjoint space–time ‘anomalies’. If $\mathbb{S}$− is non-empty it is dominated by large values of |∇ω|, which is consistent with the formation of vortex sheets or tightly coiled filaments. The local number of degrees of freedom ± needed to resolve the regions in $\mathbb{S}$± satisfies $\mathcal{N}^{\pm}(\bx,\,t)\lessgtr 3\sqrt{2}\,\mathcal{R}_{u}^{3},$, where u = uL/ν is a Reynolds number dependent on the local velocity field u(x, t).

scholarly journals Numerical Implementations for 2D Lid-Driven Cavity Flow in Stream Function Formulation

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
K. Poochinapan
Keyword(s):  
Reynolds Number ◽  
Stream Function ◽  
Stokes Equations ◽  
Operator Splitting ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Driven Cavity ◽  
Function Formulation ◽  
High Reynolds ◽  
Stream Function Formulation

The aim of this paper is to study the properties of approximations to nonlinear terms of the 2D incompressible Navier-Stokes equations in the stream function formulation (time-dependent biharmonic equation). The nonlinear convective terms are numerically solved by using the method with internal iterations, compared to the ones which are solved by using explicit and implicit schemes (operator splitting scheme Christov and Marinova; (2001)). Using schemes and algorithms, the steady 2D incompressible flow in a lid-driven cavity is solved up to Reynolds number Re =5000 with second-order spatial accuracy. The schemes are thoroughly validated on grids with different resolutions. The result of numerical experiments shows that the finite difference scheme with internal iterations on nonlinearity is more efficient for the high Reynolds number.

The Magnetohydrodynamic (MHD) Effects on the Performance of a Hydrostatic Thrust Bearing With Hybrid Raleigh Step

2010 ◽  
Author(s):  
Frank E. Horvat ◽  
Minel J. Braun
Keyword(s):  
Reynolds Number ◽  
Thrust Bearing ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Angular Speed ◽  
Navier Stokes ◽  
Operational Parameters ◽  
Navier Stokes Equations ◽  
Pressure Profiles ◽  
Ansys Cfx

This paper studies the numerical development of flow patterns and pressure profiles inside a hybrid Rayleigh step thrust bearing (HRSB) where the working magnetohydrodynamic (MHD) fluid is subject to an imposed magnetic field. This hybrid type bearing stems from integrating two classical component: the modified Rayleigh step (variable depth) and the hydrostatic feed entering at the center of the circular thrust bearing. The parameters used in this study consist of one geometric parameter, the Rayleigh step aspect ratio (depth to length ratio) and two dimensionless operational parameters, (i) the Reynolds number based on the hydrostatic fluid jet velocity entering the restrictor (Rejet) and the Reynolds number based on the smooth upper plate angular speed (Replate). The numerical results are obtained using the commercially available package ANSYS (CFX) [4], which utilizes the full three-dimensional Navier-Stokes equations for the steady-state incompressible MHD fluid with constant properties. Results to be presented will will contain both vector and pressure fields within the Rayleigh step profile and on the smooth lands.

scholarly journals Exact coherent structures at extreme Reynolds number

2016 ◽  
Vol 794 ◽  
pp. 1-4 ◽  
Author(s):  
G. P. Chini
Keyword(s):  
Reynolds Number ◽  
Coherent Structures ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Large Reynolds Number ◽  
Navier Stokes Equations ◽  
Wall Flows ◽  
Tollmien Schlichting ◽  
Turbulent Wall

Exact coherent structures (ECS), unstable three-dimensional solutions of the Navier–Stokes equations, play a fundamental role in transitional and turbulent wall flows. Dempsey et al. (J. Fluid Mech., vol. 791, 2016, pp. 97–121) demonstrate that at large Reynolds number reduced equations can be derived that simplify the computation and facilitate mechanistic understanding of these solutions. Their analysis shows that ECS in plane Poiseuille flow can be sustained by a novel inner–outer interaction between oblique near-wall Tollmien–Schlichting waves and interior streamwise vortices.

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