scholarly journals Confined axisymmetric laminar jets with large expansion ratios

2002 ◽  
Vol 456 ◽  
pp. 319-352 ◽  
Author(s):  
A. REVUELTA ◽  
A. L. SÁNCHEZ ◽  
A. LIÑÁN
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Outer Region ◽  
Outer Wall ◽  
Relevant Parameter ◽  
Order Unity ◽  
Laminar Jet ◽  
Recirculating Flow ◽  
Large Expansion ◽  
Downstream Flow

This paper investigates the steady round laminar jet discharging into a coaxial duct when the jet Reynolds number, Rej, is large and the ratio of the jet radius to the duct radius, ε, is small. The analysis considers the distinguished double limit in which the Reynolds number Rea = Rejε for the final downstream flow is of order unity, when four different regions can be identified in the flow field. Near the entrance, the outer confinement exerts a negligible influence on the incoming jet, which develops as a slender unconfined jet with constant momentum flux. The jet entrains outer fluid, inducing a slow back flow motion of the surrounding fluid near the backstep. Further downstream, the jet grows to fill the duct, exchanging momentum with the surrounding recirculating flow in a slender region where the Reynolds number is still of the order of Rej. The streamsurface bounding the toroidal vortex eventually intersects the outer wall, in a non-slender transition zone to the final downstream region of parallel streamlines. In the region of jet development, and also in the main region of recirculating flow, the boundary-layer approximation can be used to describe the flow, while the full Navier–Stokes equations are needed to describe the outer region surrounding the jet and the final transition region, with Rea = Rejε entering as the relevant parameter to characterize the resulting non-slender flows.

Flow induced by jets and plumes

1981 ◽  
Vol 108 ◽  
pp. 55-65 ◽  
Author(s):  
W. Schneider
Keyword(s):  
Stokes Equations ◽  
Outer Region ◽  
Reynolds Numbers ◽  
Slip Condition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Laminar Jet ◽  
Order Of Magnitude ◽  
Valid Solution ◽  
Solid Walls

The order of magnitude of the flow velocity due to the entrainment into an axisymmetric, laminar or turbulent jet and an axisymmetric laminar plume, respectively, indicates that viscosity and non-slip of the fluid at solid walls are essential effects even for large Reynolds numbers of the jet or plume. An exact similarity solution of the Navier-Stokes equations is determined such that both the non-slip condition at circular-conical walls (including a plane wall) and the entrainment condition at the jet (or plume) axis are satisfied. A uniformly valid solution for large Reynolds numbers, describing the flow in the laminar jet region as well as in the outer region, is also given. Comparisons show that neither potential flow theory (Taylor 1958) nor viscous flow theories that disregard the non-slip condition (Squire 1952; Morgan 1956) provide correct results if the flow is bounded by solid walls.

scholarly journals On the decay of dispersive motions in the outer region of rough-wall boundary layers

2019 ◽  
Vol 862 ◽  
Author(s):  
Johan Meyers ◽  
Bharathram Ganapathisubramani ◽  
Raúl Bayoán Cal
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Boundary Layers ◽  
Outer Layer ◽  
Stokes Equations ◽  
Outer Region ◽  
Rough Wall ◽  
Rough Boundary ◽  
Mean Flow ◽  
Wall Boundary

In rough-wall boundary layers, wall-parallel non-homogeneous mean-flow solutions exist that lead to so-called dispersive velocity components and dispersive stresses. They play a significant role in the mean-flow momentum balance near the wall, but typically disappear in the outer layer. A theoretical framework is presented to study the decay of dispersive motions in the outer layer. To this end, the problem is formulated in Fourier space, and a set of governing ordinary differential equations per mode in wavenumber space is derived by linearizing the Reynolds-averaged Navier–Stokes equations around a constant background velocity. With further simplifications, analytically tractable solutions are found consisting of linear combinations of $\exp (-kz)$ and $\exp (-Kz)$, with $z$ the wall distance, $k$ the magnitude of the horizontal wavevector $\boldsymbol{k}$, and where $K(\boldsymbol{k},Re)$ is a function of $\boldsymbol{k}$ and the Reynolds number $Re$. Moreover, for $k\rightarrow \infty$ or $k_{1}\rightarrow 0$ (with $k_{1}$ the stream-wise wavenumber), $K\rightarrow k$ is found, in which case solutions consist of a linear combination of $\exp (-kz)$ and $z\exp (-kz)$, and are independent of the Reynolds number. These analytical relations are compared in the limit of $k_{1}=0$ to the rough boundary layer experiments by Vanderwel & Ganapathisubramani (J. Fluid Mech., vol. 774, 2015, R2) and are in reasonable agreement for $\ell _{k}/\unicode[STIX]{x1D6FF}\leqslant 0.5$, with $\unicode[STIX]{x1D6FF}$ the boundary-layer thickness and $\ell _{k}=2\unicode[STIX]{x03C0}/k$.

scholarly journals On the uniqueness of steady flow past a rotating cylinder with suction

2000 ◽  
Vol 411 ◽  
pp. 213-232 ◽  
Author(s):  
E. V. BULDAKOV ◽  
S. I. CHERNYSHENKO ◽  
A. I. RUBAN
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Stokes Equations ◽  
Cross Flow ◽  
Rotation Velocity ◽  
Rotating Cylinder ◽  
Large Reynolds Number ◽  
Order Unity ◽  
Flow Past ◽  
Exponentially Small

The subject of this study is a steady two-dimensional incompressible flow past a rapidly rotating cylinder with suction. The rotation velocity is assumed to be large enough compared with the cross-flow velocity at infinity to ensure that there is no separation. High-Reynolds-number asymptotic analysis of incompressible Navier–Stokes equations is performed. Prandtl's classical approach of subdividing the flow field into two regions, the outer inviscid region and the boundary layer, was used earlier by Glauert (1957) for analysis of a similar flow without suction. Glauert found that the periodicity of the boundary layer allows the velocity circulation around the cylinder to be found uniquely. In the present study it is shown that the periodicity condition does not give a unique solution for suction velocity much greater than 1/Re. It is found that these non-unique solutions correspond to different exponentially small upstream vorticity levels, which cannot be distinguished from zero when considering terms of only a few powers in a large Reynolds number asymptotic expansion. Unique solutions are constructed for suction of order unity, 1/Re, and 1/√Re. In the last case an explicit analysis of the distribution of exponentially small vorticity outside the boundary layer was carried out.

scholarly journals Characterization of the Behavior of Confined Laminar Round Jets

2012 ◽  
Author(s):  
D. Tyler Landfried ◽  
A. Jana ◽  
M. L. Kimber
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Stokes Equations ◽  
Flow Characteristics ◽  
Expansion Ratio ◽  
Downstream Location ◽  
Practical Applications ◽  
Free Jet ◽  
Laminar Jet ◽  
Jet Behavior

Confined laminar fluid jets have many practical applications in industry. Several examples include expansions in pipes and flow of gas into a large plenum. While much consideration has been given experimentally to heat transfer and pressure gradients within the confinement, little attention has been paid to quantify the velocity profiles and transitions between various flow behaviours. Using a finite volume CFD code, OpenFOAM ®, the Navier-Stokes equations were solved for varying expansion ratio, 1/ε = renclosure/rj, and varying Reynolds numbers. In the present analysis, Reynolds number based on the inlet jet diameter is varied from 30 to 70, well within the accepted range for laminar jet behavior. The expansion ratio, 1/ε is varied from 20–200. Of primary focus in the current study are compact correlations for the jet centreline velocity as a function of jet Reynolds number, Rej and expansion ratio. Similar functional dependences for the “linear” decay region of the jet, and the location of the stagnation point on the enclosure wall, are also investigated. These are all important features of the global flow field for the confined jet. Results suggest that initially, the flow characteristics are identical to a free jet. At some downstream location, the presence of the enclosure is felt by the jet and deviations begin to be seen from free jet behavior. This transition region continues until at a sufficiently large downstream location, the flow becomes fully developed, internal Poiseuille flow. In this paper, we analyse these transition regions and offer explanations and practical correlations to successfully predict the important flow physics that occur between free jet behavior and Poiseuille flow. Key dimensionless parameters are identified, the magnitude of which can be used to classify the flow conditions.

Laminar flow through slots

1988 ◽  
Vol 190 ◽  
pp. 179-200 ◽  
Author(s):  
E. G. Tulapurkara ◽  
B. H. Lakshmana Gowda ◽  
N. Balachandran
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Flow Visualization ◽  
Vortex Shedding ◽  
Stokes Equations ◽  
Computer Code ◽  
Channel Height ◽  
Visualization Technique ◽  
Recirculating Flow ◽  
Flow Through

Laminar flow through slots is investigated using a flow-visualization technique and the numerical solution of the Navier-Stokes equations for steady flow. In the flow situation studied here, the fluid enters an upper channel blocked at the rear end and leaves through a lower channel blocked at the front end. The two channels are interconnected by one, two and three slots. The flow-visualization technique effectively brings out the various features of the flow through slot(s). The ratio of the slot width to the channel height w/h is varied between 0.5 to 4.0 and the Reynolds number Re, based on the velocity at the entry to the channel and the height of the channel, is varied between 300 and 2000. Both w/h and Re influence the flow in general and the extent of the regions of recirculating flow in particular. The Reynolds number at which the vortex shedding begins depends on w/h. Computations are carried out using the computer code 2/E/FIX of Pun & Spalding (1977). The computed flow patterns closely resemble the observed patterns at various Reynolds numbers investigated except around the Reynolds number where the vortex shedding begins.

A Modified Low-Reynolds-Number k-ω Model for Recirculating Flows

1997 ◽  
Vol 119 (4) ◽  
pp. 867-875 ◽  
Author(s):  
Shia-Hui Peng ◽  
Lars Davidson ◽  
Sture Holmberg
Keyword(s):  
Reynolds Number ◽  
Low Reynolds Number ◽  
Expansion Ratio ◽  
Diffusion Term ◽  
Step Flow ◽  
Cross Diffusion ◽  
Recirculating Flow ◽  
Large Expansion ◽  
Recirculating Flows ◽  
Low Reynolds

A modified form of Wilcox’s low-Reynolds-number k-ω model (Wilcox, 1994) is proposed for predicting recirculating flows. The turbulent diffusion for the specific dissipation rate, ω, is modeled with two parts: a second-order diffusion term and a first-order cross-diffusion term. The model constants are re-established. The damping functions are redevised, which reproduce correct near-wall asymptotic behaviors, and retain the mechanism describing transition as in the original model. The new model is applied to channel flow, backward-facing step flow with a large expansion ratio (H/h = 6), and recirculating flow in a ventilation enclosure. The predictions are considerably improved.

Scaling of small vortices in stably stratified shear flows

2017 ◽  
Vol 821 ◽  
pp. 582-594 ◽  
Author(s):  
Kengo Deguchi
Keyword(s):  
Reynolds Number ◽  
Richardson Number ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Wave Interaction ◽  
Maximum Size ◽  
Navier Stokes ◽  
Large Reynolds Number ◽  
Order Unity ◽  
Steady Solutions

The present paper treats the large Reynolds number scaling of coherent structures in stably stratified flows sheared between two horizontally placed walls. Three-dimensional steady solutions are used to confirm the theoretical scaling. For small values of the Richardson number, the previously known scaling based on the vortex–wave interaction/self-sustaining process is found to give excellent predictions of the numerical results. When the Richardson number is increased, the maximum size of the vortices is limited by the Ozmidov scale. The largest possible Richardson number to sustain the vortices is predicted to be of order unity when the typical length scale of the vortices reaches the Kolmogorov scale. The minimum-scale vortices are governed by unit Reynolds number Navier–Stokes equations.

Three-Dimensional Steady Flow Through A Bifurcation

1990 ◽  
Vol 112 (2) ◽  
pp. 189-197 ◽  
Author(s):  
Chain-Nan Yung ◽  
Kenneth J. De Witt ◽  
Theo G. Keith
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Steady Flow ◽  
Stokes Equations ◽  
Control Volume ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Simple Algorithm ◽  
Outer Wall ◽  
Navier Stokes

Steady flow of an incompressible, Newtonian fluid through a symmetric bifurcated rigid channel was numerically analyzed by solving the three-dimensional Navier-Stokes equations. The upstream Reynolds number ranged from 100 to 1500. The bifurcation was symmetrical with a branch angle of 60 deg and the area ratio of the daughter to the mother vessel was 2.0. The numerical procedure utilized a coordinate transformation and a control volume approach to discretize the equations to finite difference form and incorporated the SIMPLE algorithm in performing the calculation. The predicted velocity pattern was in qualitative agreement with experimental measurements available in the literature. The results also showed the effect of secondary flow which can not be predicted using previous two-dimensional simulations. A region of reversed flow was observed near the outer wall of the branch except for the case of the lowest Reynolds number. Particle trajectory was examined and it was found that no fluid particles remained within the recirculation zone. The shear stress was calculated on both the inner and the outer wall of the branch. The largest wall shear stress, located in the vicinity of the apex of the branch, was of the same order of magnitude as the level that can cause damage to the vessel wall as reported in a recent study.

The influence of the inertially dominated outer region on the rheology of a dilute dispersion of low-Reynolds-number drops or rigid particles

2011 ◽  
Vol 674 ◽  
pp. 307-358 ◽  
Author(s):  
GANESH SUBRAMANIAN ◽  
DONALD L. KOCH ◽  
JINGSHENG ZHANG ◽  
CHAO YANG
Keyword(s):  
Reynolds Number ◽  
Shear Flow ◽  
Simple Shear ◽  
Stokes Equations ◽  
Outer Region ◽  
Simple Shear Flow ◽  
Dispersed Phase ◽  
Leading Order ◽  
Normal Stress Differences ◽  
Neutrally Buoyant

We calculate the rheological properties of a dilute emulsion of neutrally buoyant nearly spherical drops at O(φRe3/2) in a simple shear flow(u∞ = x211, being the shear rate) as a function of the ratio of the dispersed- and continuous-phase viscosities (λ = /μ). Here, φ is the volume fraction of the dispersed phase and Re is the micro-scale Reynolds number. The latter parameter is a dimensionless measure of inertial effects on the scale of the dispersed-phase constituents and is defined as Re = a2ρ/μ, a being the drop radius and ρ the common density of the two phases. The analysis is restricted to the limit φ, Re ≪ 1, when hydrodynamic interactions between drops may be neglected, and the velocity field in a region around the drop of the order of its own size is governed by the Stokes equations at leading order. The dominant contribution to the rheology at O(φRe3/2), however, arises from the so-called outer region where the leading-order Stokes approximation ceases to be valid. The relevant length scale in this outer region, the inertial screening length, results from a balance of convection and viscous diffusion, and is O(aRe−1/2) for simple shear flow in the limit Re ≪ 1. The neutrally buoyant drop appears as a point-force dipole on this scale. The rheological calculation at O(φRe3/2) is therefore based on a solution of the linearized Navier–Stokes equations forced by a point dipole. The principal contributions to the bulk rheological properties at this order arise from inertial corrections to the drop stresslet and Reynolds stress integrals. The theoretical calculations for the stresslet components are validated via finite volume simulations of a spherical drop at finite Re; the latter extend up to Re ≈ 10.Combining the results of our O(φRe3/2) analysis with the known rheology of a dilute emulsion to O(φRe) leads to the following expressions for the relative viscosity (μe), and the non-dimensional first (N1) and second normal stress differences (N2) to O(φRe3/2): μe = 1 + φ[(5λ+2)/(2(λ+1))+0.024Re3/2(5λ+2)2/(λ+1)2]; N1=φ[−Re4(3λ2+3λ+1)/(9(λ+1)2)+0.066Re3/2(5λ+2)2/(λ+1)2] and N2 = φ[Re2(105λ2+96λ+35)/(315(λ+1)2)−0.085Re3/2(5λ+2)2/(λ+1)2].Thus, for small but finite Re, inertia endows an emulsion with a non-Newtonian rheology even in the infinitely dilute limit, and in particular, our calculations show that, aside from normal stress differences, such an emulsion also exhibits a shear-thickening behaviour. The results for a suspension of rigid spherical particles are obtained in the limit λ → ∞.

Low Reynolds-Number Pipe Flow in the Vicinity of Three-Dimensional Obstacles

1979 ◽  
Vol 21 (5) ◽  
pp. 335-343 ◽  
Author(s):  
A. D. Gosman ◽  
N. S. Vlachos ◽  
J. H. Whitelaw
Keyword(s):  
Reynolds Number ◽  
Turbulent Flows ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Recirculating Flow ◽  
Local Shear ◽  
Sector Angle

Numerical solutions of the three-dimensional Navier-Stokes equations are presented for boundary conditions corresponding to the laminar flow of Newtonian and non-Newtonian fluids in a round pipe with truncated sector-shaped obstacles. The influences of Reynolds number and sector angle on the velocity distributions, local shear stress and pressure drop are quantified and shown to be large. The results are complementary to those previously reported by Vlachos and Whitelaw (1)§ for axisymmetric obstacles, where related two-dimensional effects were quantified. They provide new information on three-dimensional, recirculating flow in ducts and form a basis for future calculations of corresponding turbulent flows.

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