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.

Numerical Simulations of Riser Vortex-Induced Vibrations

2005 ◽  
Author(s):  
Juan P. Pontaza ◽  
Hamn-Ching Chen ◽  
Chia-Rong Chen
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reynolds Number Flow ◽  
Vortex Induced Vibrations ◽  
Number Flow ◽  
Low Mass ◽  
High Reynolds ◽  
Low Mass Ratio ◽  
Important Design

Vortex-induced vibrations (VIV) is an important design consideration for marine risers in offshore drilling and production. In an effort to better understand the VIV phenomena, we present numerical simulation results for two-dimensional incompressible flow past freely vibrating multi-cylinder configurations found in offshore engineering. Of interest is the response of the structure for low mass ratio, low damping, and high Reynolds number flow conditions. The governing incompressible Navier-Stokes equations are numerically solved and time-integrated using a local-analytic-based discretization procedure, implemented in conjunction with overset (Chimera) grid capabilities for zonal-based resolution of the flow field.

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.

Trailing-Edge Flow

2018 ◽  
Author(s):  
Anatoly I. Ruban
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Trailing Edge ◽  
Displacement Effect ◽  
Navier Stokes Equations ◽  
Reynolds Number Flow ◽  
Number Flow ◽  
High Reynolds ◽  
Interaction Problem ◽  
Edge Flow

Chapter 3 focuses on the high-Reynolds number flow of an incompressible fluid near the trailing edge of a flat plate. It begins with Goldstein’s (1930) solution for a viscous wake behind the plate, and shows that the displacement effect of the wake produces a singular pressure gradient near the trailing edge. It further shows that this singularity leads to a formation triple-deck viscous-inviscid interaction region that occupies a small vicinity of the trailing edge. A detailed analysis of the flow in each tier of the triple-deck structure is conducted based on the asymptotic analysis of the Navier–Stokes equations. As a result, the so-called ‘interaction problem’ is formulated. It concludes with the numerical solution of so-called ‘interaction problem’.

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.

Dynamics of a macroscopic elastic fibre in a polymeric cellular flow

2017 ◽  
Vol 817 ◽  
pp. 388-405 ◽  
Author(s):  
Qiang Yang ◽  
Lisa Fauci
Keyword(s):  
Reynolds Number ◽  
Viscoelastic Fluid ◽  
Relaxation Times ◽  
Elastic Fibre ◽  
Weissenberg Number ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Reynolds Number Flow ◽  
Number Flow ◽  
Multiple Cells

We study the dynamics and transport of an elastic fibre in a polymeric cellular flow. The macroscopic fibre is much larger than the infinitesimal immersed polymer coils distributed in the surrounding viscoelastic fluid. Here we consider low-Reynolds-number flow using the Navier–Stokes/Fene-P equations in a two-dimensional, doubly periodic domain. The macroscopic fibre supports both tensile and bending forces, and is fully coupled to the viscoelastic fluid using an immersed boundary framework. We examine the effects of fibre flexibility and polymeric relaxation times on fibre buckling and transport as well as the evolution of polymer stress. Non-dimensional control parameters include the Reynolds number, the Weissenberg number, and the elasto-viscous number of the macroscopic fibre. We find that large polymer stresses occur in the fluid near the ends of the fibre when it is compressed. In addition, we find that viscoelasticity hinders a fibre’s ability to traverse multiple cells in the domain.

scholarly journals The emergence of localized vortex–wave interaction states in plane Couette flow

2013 ◽  
Vol 721 ◽  
pp. 58-85 ◽  
Author(s):  
Kengo Deguchi ◽  
Philip Hall ◽  
Andrew Walton
Keyword(s):  
Reynolds Number ◽  
Turbulent Flows ◽  
High Reynolds Number ◽  
Stokes Equations ◽  
Wave Interaction ◽  
Critical Layer ◽  
Navier Stokes ◽  
Turbulent Spots ◽  
Navier Stokes Equations ◽  
High Reynolds

AbstractThe recently understood relationship between high-Reynolds-number vortex–wave interaction theory and computationally generated self-sustaining processes provides a possible route to an understanding of some of the underlying structures of fully turbulent flows. Here vortex–wave interaction (VWI) theory is used in the long streamwise wavelength limit to continue the development found at order-one wavelengths by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). The asymptotic description given reduces the Navier–Stokes equations to the so-called boundary-region equations, for which we find equilibrium states describing the change in the VWI as the wavelength of the wave increases from $O(h)$ to $O(Rh)$, where $R$ is the Reynolds number and $2h$ is the depth of the channel. The reduced equations do not include the streamwise pressure gradient of the perturbation or the effect of streamwise diffusion of the wave–vortex states. The solutions we calculate have an asymptotic error proportional to ${R}^{- 2} $ when compared to the full Navier–Stokes equations. The results found correspond to the minimum drag configuration for VWI states and might therefore be of relevance to the control of turbulent flows. The key feature of the new states discussed here is the thickening of the critical layer structure associated with the wave part of the flow to completely fill the channel, so that the roll part of the flow is driven throughout the flow rather than as in Hall & Sherwin as a stress discontinuity across the critical layer. We identify a critical streamwise wavenumber scaling, which, when approached, causes the flow to localize and take on similarities with computationally generated or experimentally observed turbulent spots. In effect, the identification of this critical wavenumber for a given value of the assumed high Reynolds number fixes a minimum box length necessary for the emergence of localized structures. Whereas nonlinear equilibrium states of the Navier–Stokes equations are thought to form a backbone on which turbulent flows hang, our results suggest that the localized states found here might play a related role for turbulent spots.

scholarly journals Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels

2013 ◽  
Vol 734 ◽  
pp. 275-316 ◽  
Author(s):  
Rashad Moarref ◽  
Ati S. Sharma ◽  
Joel A. Tropp ◽  
Beverley J. McKeon
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Low Rank ◽  
Navier Stokes ◽  
Mean Velocity ◽  
Navier Stokes Equations ◽  
Self Similar ◽  
The Mean ◽  
High Reynolds

AbstractWe study the Reynolds-number scaling and the geometric self-similarity of a gain-based, low-rank approximation to turbulent channel flows, determined by the resolvent formulation of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), in order to obtain a description of the streamwise turbulence intensity from direct consideration of the Navier–Stokes equations. Under this formulation, the velocity field is decomposed into propagating waves (with single streamwise and spanwise wavelengths and wave speed) whose wall-normal shapes are determined from the principal singular function of the corresponding resolvent operator. Using the accepted scalings of the mean velocity in wall-bounded turbulent flows, we establish that the resolvent operator admits three classes of wave parameters that induce universal behaviour with Reynolds number in the low-rank model, and which are consistent with scalings proposed throughout the wall turbulence literature. In addition, it is shown that a necessary condition for geometrically self-similar resolvent modes is the presence of a logarithmic turbulent mean velocity. Under the practical assumption that the mean velocity consists of a logarithmic region, we identify the scalings that constitute hierarchies of self-similar modes that are parameterized by the critical wall-normal location where the speed of the mode equals the local turbulent mean velocity. For the rank-1 model subject to broadband forcing, the integrated streamwise energy density takes a universal form which is consistent with the dominant near-wall turbulent motions. When the shape of the forcing is optimized to enforce matching with results from direct numerical simulations at low turbulent Reynolds numbers, further similarity appears. Representation of these weight functions using similarity laws enables prediction of the Reynolds number and wall-normal variations of the streamwise energy intensity at high Reynolds numbers (${Re}_{\tau } \approx 1{0}^{3} {\unicode{x2013}} 1{0}^{10} $). Results from this low-rank model of the Navier–Stokes equations compare favourably with experimental results in the literature.

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