Pulsatile Blood Flow in a Channel of Small Exponential Divergence—I. The Linear Approximation for Low Mean Reynolds Number

1975 ◽  
Vol 97 (3) ◽  
pp. 353-360 ◽  
Author(s):  
D. J. Schneck ◽  
Simon Ostrach
Keyword(s):  
Reynolds Number ◽  
Vascular Endothelium ◽  
Coronary Circulation ◽  
Viscous Incompressible Fluid ◽  
Axial Direction ◽  
Reynolds Numbers ◽  
Pulsatile Blood Flow ◽  
Viscous Effects ◽  
Radial Velocity Component ◽  
Phase Lags

The pulsating flow of a viscous, incompressible fluid through rigid circular channels having walls which diverge at a slow exponential rate is examined analytically. Linearized solutions for low mean Reynolds numbers reveal that viscous effects lead to radially dependent phase shifts between different layers of fluid oscillating in the axial direction, and characteristic phase lags between flow and pressure curves. When the Reynolds number and channel divergence are each small, the flow does not separate, but there is a downstream attenuation of both flow and pressure, together with the appearance of a finite radial velocity component. Utilizing data relevant to basal conditions existing in the major blood vessels of the human coronary circulation, it is found (in the absence of any persistent flow anomalies) that the shear stress at the wall is at least one to two orders of magnitude lower than values reported to be damaging to vascular endothelium.

scholarly journals Viscous effects in the absolute–convective instability of the Batchelor vortex

2002 ◽  
Vol 459 ◽  
pp. 371-396 ◽  
Author(s):  
C. OLENDRARU ◽  
A. SELLIER
Keyword(s):  
Reynolds Number ◽  
Convective Instability ◽  
Axial Flow ◽  
Reynolds Numbers ◽  
Absolute Instability ◽  
Viscous Effects ◽  
Swirling Jets ◽  
Bending Mode ◽  
Transitional Mode ◽  
The Absolute

The effects of viscosity on the instability properties of the Batchelor vortex are investigated. The characteristics of spatially amplified branches are first documented in the convectively unstable regime for different values of the swirl parameter q and the co-flow parameter a at several Reynolds numbers Re. The absolute–convective instability transition curves, determined by the Briggs–Bers zero-group velocity criterion, are delineated in the (a, q)-parameter plane as a function of Re. The azimuthal wavenumber m of the critical transitional mode is found to depend on the magnitude of the swirl q and on the jet (a > −0.5) or wake (a < −0.5) nature of the axial flow. At large Reynolds numbers, the inviscid results of Olendraru et al. (1999) are recovered. As the Reynolds number decreases, the pocket of absolute instability in the (a, q)-plane is found to shrink gradually. At Re = 667; the critical transitional modes for swirling jets are m = −2 or m = −3 and absolute instability prevails at moderate swirl values even in the absence of counterflow. For higher swirl levels, the bending mode m = −1 becomes critical. The results are in good overall agreement with those obtained by Delbende et al. (1998) at the same Reynolds number. However, a bending (m = +1) viscous mode is found to partake in the outer absolute–convective instability transition for jets at very low positive levels of swirl. This asymmetric branch is the spatial counterpart of the temporal viscous mode isolated by Khorrami (1991) and Mayer & Powell (1992). At Re = 100, the critical transitional mode for swirling jets is m = −2 at moderate and high swirl values and, in order to trigger an absolute instability, a slight counterflow is always required. A bending (m = +1) viscous mode again becomes critical at very low swirl values. For wakes (a < −0.5) the critical transitional mode is always found to be the bending mode m = −1, whatever the Reynolds number. However, above q = 1.5, near-neutral centre modes are found to define a tongue of weak absolute instability in the (a, q)-plane. Such modes had been analytically predicted by Stewartson & Brown (1985) in a strictly temporal inviscid framework.

An almost-inviscid geostrophic flow

1962 ◽  
Vol 12 (1) ◽  
pp. 129-134 ◽  
Author(s):  
L. M. Hocking
Keyword(s):  
Reynolds Number ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Axial Direction ◽  
Reynolds Numbers ◽  
Rigid Rotation ◽  
Velocity Variation ◽  
Geostrophic Flow ◽  
The Difference ◽  
Streaming Motion

An almost rigid rotation of a viscous fluid is produced by dividing the containing cylinder into two sections and rotating them at slightly different speeds. The fluid velocity can be separated into two parts, a swirl about the axis and a streaming motion in the axial planes. When the difference in the speeds of rotation of the two sections is small, the equations of motion can be linearized. The solution is found for large Reynolds numbers and provides an illustration of the way in which the conditions of geostrophic flow (no velocity variation in the axial direction and an inability to insist on undistrubed flow at infinity) are approached as the Reynolds number tends to infinity.

Effect of Upstream Flow Processes on Hydrodynamic Development in a Duct

1977 ◽  
Vol 99 (3) ◽  
pp. 556-560 ◽  
Author(s):  
E. M. Sparrow ◽  
C. E. Anderson
Keyword(s):  
Reynolds Number ◽  
Reynolds Numbers ◽  
Center Line ◽  
Viscous Effects ◽  
Mean Velocity ◽  
Inertia Effects ◽  
Reynolds Number Flow ◽  
Compact Size ◽  
Number Flow ◽  
The Mean

Consideration is given to the developing laminar flow in a parallel plate channel, with the fluid being drawn from a large upstream space. The flow fields upstream and downstream of the channel inlet were solved simultaneously. A finite-difference technique was employed which was facilitated by a coordinate transformation that telescoped the broadly extended flow domain into a more compact size. For the solutions, the Reynolds number was assigned values from 1 to 1000, covering the range from viscous-dominated flows to those where both viscous and inertia effects are relevant. Streamline maps indicate that whereas a low Reynolds number flow glides smoothly into the channel, a high Reynolds number flow has to turn sharply to enter the channel, with the result that the sharply turning fluid tends to overshoot at first and then readjust. A significant amount of upstream predevelopment occurs at low and intermediate Reynolds numbers. Thus, for example, at Re = 1 and 100, the center-line velocities at inlet are, respectively, 1.37 and 1.13 times the mean velocity (the fully developed center-line velocity is 1.5 times the mean). The upstream pressure drop, measured in terms of the velocity head, is substantially increased by viscous effects at low and intermediate Reynolds numbers.

scholarly journals Laminar-to-turbulent transition of pipe flows through puffs and slugs

2008 ◽  
Vol 614 ◽  
pp. 425-446 ◽  
Author(s):  
MINA NISHI ◽  
BÜLENT ÜNSAL ◽  
FRANZ DURST ◽  
GAUTAM BISWAS
Keyword(s):  
Reynolds Number ◽  
Pipe Flow ◽  
Axial Direction ◽  
Reynolds Numbers ◽  
Turbulent Pipe Flow ◽  
Test Facility ◽  
Pipe Flows ◽  
Laminar To Turbulent Transition ◽  
Turbulent Transition ◽  
Slug Flows

Laminar-to-turbulent transition of pipe flows occurs, for sufficiently high Reynolds numbers, in the form of slugs. These are initiated by disturbances in the entrance region of a pipe flow, and grow in length in the axial direction as they move downstream. Sequences of slugs merge at some distance from the pipe inlet to finally form the state of fully developed turbulent pipe flow. This formation process is generally known, but the randomness in time of naturally occurring slug formation does not permit detailed study of slug flows. For this reason, a special test facility was developed and built for detailed investigation of deterministically generated slugs in pipe flows. It is also employed to generate the puff flows at lower Reynolds numbers. The results reveal a high degree of reproducibility with which the triggering device is able to produce puffs. With increasing Reynolds number, ‘puff splitting’ is observed and the split puffs develop into slugs. Thereafter, the laminar-to-turbulent transition occurs in the same way as found for slug flows. The ring-type obstacle height, h, required to trigger fully developed laminar flows to form first slugs or puffs is determined to show its dependence on the Reynolds number, Re = DU/ν (where D is the pipe diameter, U is the mean velocity in the axial direction and ν is the kinematic viscosity of the fluid). When correctly normalized, h+ turns out to be independent of Reτ (where h+ = hUτ/ν, Reτ = DUτ/ν and $U_{\tau}\,{=}\,\sqrt{\tau_{w}/ \rho}$; τw is the wall shear stress and ρ is the density of the fluid).

Pulsatile Blood Flow in a Channel of Small Exponential Divergence—III. Unsteady Flow Separation

1977 ◽  
Vol 99 (2) ◽  
pp. 333-338 ◽  
Author(s):  
Daniel J. Schneck
Keyword(s):  
Reynolds Number ◽  
Blood Flow ◽  
Steady State ◽  
Flow Separation ◽  
Reynolds Numbers ◽  
Critical Value ◽  
Flow Oscillation ◽  
Pulsatile Blood Flow ◽  
Steady Streaming ◽  
Flow Through

Analysis of pulsatile flow through exponentially diverging channels reveals the existence of critical mean Reynolds numbers for which the flow separates at a downstream axial station. These Reynolds numbers vary directly with the frequency of flow oscillation and inversely with the rate of channel divergence. Increasing the Reynolds number above its critical value results in a rapid upstream displacement of the point of separation. For a tube of fixed geometry, periodic unsteadiness causes flow separation to occur at lower Reynolds numbers and upstream of a corresponding steady-state situation. The point of separation moves progressively downstream, however, towards its steady-state location, as the frequency of oscillation increases. These results are discussed as consequences of the nonlinear steady streaming phenomenon described in an earlier paper.

Numerical Analysis of Sub-Millimeter-Scale Microturbomachinery Aerothermodynamics

2008 ◽  
Author(s):  
Philippe B. Martel ◽  
Luc G. Fre´chette
Keyword(s):  
Reynolds Number ◽  
Large Scale ◽  
Microelectromechanical Systems ◽  
Numerical Study ◽  
Reynolds Numbers ◽  
Viscous Effects ◽  
Analysis And Design ◽  
Moderate Reynolds Number ◽  
Throat Width ◽  
Solid Foundation

This paper presents a complete numerical study of the aerothermodynamics of subsonic moderate Reynolds number microturbomachinery using 2D computational fluid dynamics (CFD) on 24 cascade geometries and covering over 2000 conditions. Profile and mixing losses, as well as deviation and heat transfer correlations are developed for use in mean-line analysis and design. Both losses and thermal transfer tend to increase with decreasing Reynolds number, Mach number, and throat width. Deviation follows large scale turbomachinery behavior but tends to increase with viscous effects. A slender cascade geometry using a modified profile is suggested, potentially increasing isentropic efficiency by as much as 15%. This work defines a solid foundation for the design of microturbines used in power microelectromechanical systems (MEMS), such as gas and steam microturbines with sub-millimeter-scale blade chords operating at moderate Reynolds numbers (100 &lt; Rec &lt; 2000).

Pulsatile Blood Flow in a Channel of Small Exponential Divergence—Part II: Steady Streaming Due to the Interaction of Viscous Effects With Convected Inertia

1976 ◽  
Vol 98 (4) ◽  
pp. 707-713 ◽  
Author(s):  
D. J. Schneck ◽  
F. J. Walburn
Keyword(s):  
Viscous Incompressible Fluid ◽  
Exponential Rate ◽  
Inertial Effects ◽  
Pulsatile Blood Flow ◽  
Viscous Effects ◽  
Flow Phenomenon ◽  
Steady Streaming ◽  
Self Consistent ◽  
Downstream Flow ◽  
Streaming Motion

This paper describes a secondary streaming motion that appears during the pulsatile flow of a viscous, incompressible fluid through rigid circular channels having walls which diverge at a slow exponential rate. Arising primarily from the interaction of viscous effects with convected inertial effects, this steady streaming motion acts to continuously retard downstream flow near the wall surface and enhance such flow nearer midstream. The secondary flow phenomenon is shown to be directly proportional to mean Reynolds Number, inversely proportional to the unsteadiness parameter of the flow, and to attenuate with decreasing rates of channel divergence. These effects are all self-consistent and interdependent.

scholarly journals Performance of Overset Mesh in Modeling the Wake of Sharp-Edge Bodies

Computation ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 66
Author(s):  
Suyash Verma ◽  
Arman Hemmati
Keyword(s):  
Reynolds Number ◽  
Reynolds Numbers ◽  
Sharp Edge ◽  
Solution Technique ◽  
Viscous Effects ◽  
Surface Pressure Distribution ◽  
Flow Features ◽  
Background Grid ◽  
High Reynolds ◽  
Flow Variables

The wake dynamics of sharp-edge rigid panels is examined using Overset Grid Assembly (OGA) utilized in OpenFOAM, an open-source platform. The OGA method is an efficient solution technique based on overlap of a single or multiple moving grids on a stationary background grid. Five test cases for a stationary panel at different angle of attack are compared with available computational data, which show a good agreement in predicting global flow variables, such as mean drag. The models also provided accurate results in predicting the main flow features and structures. The flow past a pitching square panel is also investigated at two Reynolds numbers. The study of surface pressure distribution and shear forces acting on the panel suggests that a higher streamwise pressure gradient exists for the high Reynolds number case, which leads to an increase in lift, whereas the highly viscous effects at low Reynolds number lead to an increased drag production. The wake visualizations for the stationary and pitching motion cases show that the vortex shedding and wake characteristics are captured accurately using the OGA method.

A Numerical Comparison of 2D and 3D Unsteady Confined Impinging Air Jets and Associated Microelectronics Cooling Impact

2007 ◽  
Author(s):  
Victor Adrian Chiriac ◽  
Jorge Luis Rosales
Keyword(s):  
Reynolds Number ◽  
Steady Flow ◽  
Reynolds Numbers ◽  
Spatial Behavior ◽  
Temperature Fields ◽  
Numerical Comparison ◽  
Viscous Effects ◽  
Reynolds Number Flow ◽  
Number Flow ◽  
2D And 3D

A numerical investigation to compare the flow and heat transfer characteristics of 2D and 3D single impinging slot jets was performed at various Reynolds numbers. The present study is a continuation of the authors’ earlier work [1], and identifies the main similarities and differences arising from the expansion to the third dimension. For comparison purposes, a single slot jet impinges on an adiabatic lower wall with a flush mounted heat source. Two Reynolds numbers have been selected, 300 and 600, such that the jets are steady at the lower Reynolds number flow for both 2D and 3D models and unsteady for the higher Reynolds number flow. Both simulations are in the laminar regime. The steady cases at a Reynolds number of 300 show that the 3D slot jet produces the same values as the 2D case. The flow produces a symmetrical, steady flow hydrodynamic pattern with the jet being deflected laterally by the wall. By further increasing the Reynolds number to 600, a complex and highly unsteady flow develops for both 2D and 3D simulations. The complex flow patterns reveal the vortex pairing effects observed before for the 2D flows, leading to the jet “buckling and sweeping” behavior. However, the 3D unsteady jet produces results that deviate from the 2D unsteady case due to 3D viscous effects that are more pronounced than for the steady flow. The relevant difference between the 3D spatial behavior versus the 2D planar behavior occurring for the unsteady flows are documented by comparing the Nusselt numbers on the target wall for the cases under evaluation. Plots of the velocity, vorticity and temperature fields for both 2D and 3D cases are provided together with detailed discussion of the results.

Heat transfer from a sphere at low Reynolds numbers

1973 ◽  
Vol 60 (2) ◽  
pp. 273-283 ◽  
Author(s):  
S. C. R. Dennis ◽  
J. D. A. Walker ◽  
J. D. Hudson
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Viscous Incompressible Fluid ◽  
Reynolds Numbers ◽  
Experimental Measurements ◽  
Low Reynolds Numbers ◽  
Nusselt Numbers ◽  
The Mean ◽  
Low Reynolds ◽  
Prandtl Numbers

The heat transfer due to forced convection from an isothermal sphere in a steady stream of viscous incompressible fluid is calculated for low values of the Reynolds number and Prandtl numbers ofO(1). The mean Nusselt number is compared with the results of experimental measurements. At very low Reynolds numbers, both the local and mean Nusselt numbers are compared with the results obtained from the theory of matched asymptotic expansions.

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