scholarly journals Asymptotic solution of the low Reynolds-number flow between two co-axial cones of common apex

1984 ◽  
Vol 7 (4) ◽  
pp. 765-784 ◽  
Author(s):  
M. A. Serag-Eldin ◽  
Y. K. Gayed
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Asymptotic Series ◽  
Creeping Flow ◽  
Navier Stokes ◽  
Cross Sectional ◽  
Navier Stokes Equations ◽  
Spherical Coordinate ◽  
Flow Solution ◽  
Reynolds Number Flow

The paper is concerned with the axi-symmetrlc, incompressible, steady, laminar and Newtonian flow between two, stationary, conical-boundaries, which exhibit a common apex but may include arbitrary angles. The flow pattern and pressure field are obtained by solving the pertinent Navier-Stokes' equations in the spherical coordinate system. The solution is presented in the form of an asymptotic series, which converges towards the creeping flow solution as a cross-sectional Reynolds-number tends to zero. The first term in the series, namely the creeping flow solution, is given in closed form; whereas, higher order terms contain functions which generally could only be expressed in infinite series form, or else evaluated numerically. Some of the results obtained for converging and diverging flows are displayed and they are demonstrated to be plausible and informative.

Stokes flow between two intersecting circular arcs

1965 ◽  
Vol 61 (1) ◽  
pp. 271-274 ◽  
Author(s):  
K. B Ranger
Keyword(s):  
Reynolds Number ◽  
Stokes Flow ◽  
Stokes Equations ◽  
Flow Region ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reynolds Number Flow ◽  
Circular Arcs ◽  
Converging Flow ◽  
Point Of Intersection

This paper considers a family of viscous flows closely related to the exact Jeffery-Hamel solution ((l), (2)) of the two-dimensional Navier-Stokes equations, for diverging or converging flow in a channel. It is known that if the walls of the channel intersect at an angle less than π then there is a unique solution of the Navier-Stokes equations in which the streamlines are straight lines issuing from the point of intersection of the walls and the flow is everywhere diverging or everywhere converging. The flow parameters depend on the total fluid mass M emitted at the point of intersection and the angle 2α between the walls. By taking the Reynolds number R = M/ν, where v is the kinematic viscosity, the stream function can be expanded in a power series in R in which the leading term is a Stokes flow. Alternatively the solution can be developed by perturbing the Stokes flow and is one of very few examples known in which a Stokes flow can be regarded as a uniformly valid first approximation everywhere in an infinite fluid region. The class of flows to be considered is a generalization of the Jeffery–Hamel flow by taking the flow region to be finite and bounded by two circular arcs which intersect at an angle less than π At one point of intersection fluid is forced into the region and an equal amount is absorbed out at the other point. It is found to the first order that the flow at the two points of intersection corresponds to the zero Reynolds number limit for diverging and converging flow, respectively. Now since the flow at these points can be developed by perturbing the Stokes flow solution it is reasonable to assume that the zero Reynolds number flow in the entire finite region bounded by the arcs is a Stokes flow since the most likely region in which this approximation becomes invalid is locally at the points of intersection but here the validity of the approximation is ensured. A comparison of the convection terms with the viscous terms verifies that this conclusion is borne out.

Incompressible Flow Between Finite Disks

1982 ◽  
Vol 49 (1) ◽  
pp. 1-9 ◽  
Author(s):  
M. L. Adams ◽  
A. Z. Szeri
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reynolds Number Flow ◽  
Parallel Disks ◽  
Rotationally Symmetric ◽  
Number Flow ◽  
High Reynolds ◽  
Multiple Cells

Solutions were developed and are shown here for the primary laminar steady flow field that occurs in an incompressible, isoviscous, Newtonian fluid which is contained between two finite parallel disks. One of the disks is made to rotate at constant velocity and the other is held stationary, and either a source or a sink is located concentric to the axis of rotation. The analysis is general, containing all terms of the Navier-Stokes equations for rotationally symmetric flows, and produces a four-parameter family of solutions. The high Reynolds number flow contains multiple cells, arranged along the radius, and the flow appears to be uniquely defined by the boundary condition and the Reynolds number.

Drag Force on a Sphere Moving in Low-Reynolds-Number Pipe Flows

2007 ◽  
Vol 23 (4) ◽  
pp. 423-432 ◽  
Author(s):  
S.-H. Lee ◽  
Tzuyin Wu
Keyword(s):  
Reynolds Number ◽  
Drag Force ◽  
Stokes Equations ◽  
Low Reynolds Number ◽  
Time Derivative ◽  
Creeping Flow ◽  
Navier Stokes ◽  
Sphere Diameter ◽  
Navier Stokes Equations ◽  
Low Reynolds

AbstractIn this paper, the drag force on a sphere moving constantly along the centerline of a circular pipe filled with viscous fluid (the falling-sphere problem) under low Reynolds number condition is investigated via numerical calculation. The incompressible Navier-Stokes equations are formulated in a pseudocompressibility form. The numerical scheme makes use of finite-volume method and the numerical flux terms are evaluated using the Total-Variation Diminishing (TVD) strategy commonly applied to the compressible flow. Steady solution is obtained by marching (iterating) in time until the artificial time derivative of pressure term in the continuity equation drops to zero.In the calculation, six different Reynolds number (Re) ranging from 0.1 to 1 and seven different pipe-to-sphere diameter ratios (D/d) ranging from 5 to 40 are selected to study the pipe-wall effect. In each case, the drag force on the sphere is evaluated and the results are compared with the existing approximate theoretical values derived from correcting the Stokes' formula. Both results agree in trend, but with noticeable deviation in values, particularly for cases with large pipe-to-sphere diameter ratios. The deviation is due to the fact that theoretical values were based on the solution to the linearized Navier-Stokes equations (Stokes' creeping-flow equations), while the fully nonlinear form of the Navier-Stokes equations are adopted in the present calculations. Finally, a least-square regression technique is applied to collapse the calculated results into a single expression exhibiting the functional relationship between the drag force, Reynolds number (Re), and the pipe-to-sphere diameter ratio (D/d).

Low Reynolds-Number Flow in the Vicinity of Axisymmetric Constrictions

1979 ◽  
Vol 21 (2) ◽  
pp. 73-84 ◽  
Author(s):  
N. S. Vlachos ◽  
J. H. Whitelaw
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reynolds Number Flow ◽  
Local Shear ◽  
Number Flow ◽  
Recirculation Length

Numerical solutions of the two-dimensional, Navier-Stokes equations are presented for boundary conditions corresponding to the laminar flow of Newtonian and non-Newtonian fluids in a round tube with axisymmetric constrictions. The influence of Reynolds number, blockage diameter ratio and length on the velocity components, streamlines, local shear stress and pressure drop are quantified and, in the case of the first two, shown to be large. The non-Newtonian stress-strain relationship corresponds to that for blood flowing in venules and results in an increased recirculation length and larger regions of high shear stress.

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

Free-stream coherent structures in parallel boundary-layer flows

2014 ◽  
Vol 752 ◽  
pp. 602-625 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Coherent Structures ◽  
Free Stream ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Homotopy Continuation ◽  
Boundary Layer Flows ◽  
Navier Stokes Equations ◽  
High Reynolds

AbstractOur concern in this paper is with high-Reynolds-number nonlinear equilibrium solutions of the Navier–Stokes equations for boundary-layer flows. Here we consider the asymptotic suction boundary layer (ASBL) which we take as a prototype parallel boundary layer. Solutions of the equations of motion are obtained using a homotopy continuation from two known types of solutions for plane Couette flow. At high Reynolds numbers, it is shown that the first type of solution takes the form of a vortex–wave interaction (VWI) state, see Hall & Smith (J. Fluid Mech., vol. 227, 1991, pp. 641–666), and is located in the main part of the boundary layer. On the other hand, here the second type is found to support an equilibrium solution of the unit-Reynolds-number Navier–Stokes equations in a layer located a distance of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}O(\ln \mathit{Re})$ from the wall. Here $\mathit{Re}$ is the Reynolds number based on the free-stream speed and the unperturbed boundary-layer thickness. The streaky field produced by the interaction grows exponentially below the layer and takes its maximum size within the unperturbed boundary layer. The results suggest the possibility of two distinct types of streaky coherent structures existing, possibly simultaneously, in disturbed boundary layers.

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.

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