Laminar Inward Flow of an Incompressible Fluid Between Rotating Disks, With Full Peripheral Admission

1968 ◽  
Vol 35 (2) ◽  
pp. 229-237 ◽  
Author(s):  
K. E. Boyd ◽  
W. Rice
Keyword(s):  
Reynolds Number ◽  
Flow Rate ◽  
Stokes Equations ◽  
Complete Solution ◽  
Applied Pressure ◽  
Tangential Velocity ◽  
Tangential Component ◽  
Rotating Disks ◽  
Outer Periphery ◽  
Inlet Conditions

The laminar flow of an incompressible Newtonian fluid, radially inward between parallel co-rotating disks is considered. The through-flow is supported by an externally applied pressure difference between the outer periphery and a circular fluid exhaust hole at an inner radius. The fluid supplied at the outer periphery is considered with arbitrary velocity components, such that the tangential component may be greater or less than the disk peripheral velocity. A sufficiently complete problem statement is formulated from the Navier-Stokes’ equations. The problem has three parameters: a Reynolds number, a flow-rate parameter, and a peripheral tangential velocity component parameter. A numerical method of solution is detailed and typical numerical results are given illustrating the phenomena that occur in the inlet region for various inlet conditions. It is shown that the solution becomes the asymptotic solution given by previous investigators at interior radii following the inlet. Correspondence between the complete solution given herein and the earlier asymptotic solutions is established as dependent on corresponding values of Reynolds number and flow rate only. The results are discussed from the point of view of application of the solution in the development of multiple-disk turbines.

Inward Flow Between Stationary and Rotating Disks

2014 ◽  
Vol 136 (10) ◽  
Author(s):  
Achhaibar Singh
Keyword(s):  
Laminar Flow ◽  
Flow Field ◽  
Velocity Distribution ◽  
Stokes Equations ◽  
Tangential Velocity ◽  
Rotating Disk ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Rotating Disks ◽  
Navier Stokes Equations

The present study predicts the flow field and the pressure distribution for a laminar flow in the gap between a stationary and a rotating disk. The fluid enters through the peripheral gap between two concentric disks and converges to the center where it discharges axially through a hole in one of the disks. Closed form expressions have been derived by simplifying the Navier– Stokes equations. The expressions predict the backflow near the rotating disk due to the effect of centrifugal force. A convection effect has been observed in the tangential velocity distribution at high throughflow Reynolds numbers.

Vortex Pump as Turbine for Energy Recovery in Viscous Fluid Flows with Reynolds Number Effect

2021 ◽  
Author(s):  
Wen-Guang LI
Keyword(s):  
Reynolds Number ◽  
Viscous Fluid ◽  
Flow Rate ◽  
Power Efficiency ◽  
Stokes Equations ◽  
Entropy Generation Rate ◽  
Hydraulic Loss ◽  
Centrifugal Pumps ◽  
Vortex Pump ◽  
Turbine Mode

Abstract A vortex pump with a specific speed of 76 was studied in its turbine mode by using Fluent 6.3 based on the steady, three-dimensional, incompressible, Reynolds time-averaged Navier-Stokes equations, standard k-? turbulence model and non-equilibrium wall function in multiple reference system. The performance and flow structure of six liquids with different densities and viscosities were characterized, and the hydraulic, volumetric, and mechanical losses were discomposed. The correction factors of flow rate, head, shaft-power, efficiency, and disc friction power in turbine mode were correlated with impeller Reynolds number at three operational points. The conversion factors of flow rate, head, efficiency from the pump mode to the turbine mode were expressed with Reynolds number and compared with the counterparts of centrifugal pumps in the literature. It was indicated that the vortex pump can produce power as a turbine but becomes inefficient with increasing viscosity or decreasing impeller Reynolds number, especially as the number is smaller than 104 due to increased hydraulic, volumetric, and mechanical power losses. A vortex structure with radial, axial, and meridian vortices occurs in the impeller at different flow rates and viscosities. The incidence at blade leading edge and deviation angle at blade trailing edge depend largely on flow rate and viscosity. The impeller should be modified to improve its hydraulic performance under highly viscous fluid flow conditions. The entropy generation rate method cannot demonstrate the change in hydraulic loss with viscosity when the Reynolds number is below 104.

The axially symmetric flow of a viscous fluid between two infinite rotating disks

1962 ◽  
Vol 266 (1324) ◽  
pp. 109-121 ◽  
Keyword(s):  
Stokes Equations ◽  
Transverse Velocity ◽  
Complete Solution ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Axially Symmetric ◽  
Rotating Disks ◽  
Symmetric Flow ◽  
Navier Stokes Equations ◽  
High Reynolds

This is a numerical investigation of the similarity solutions of the Navier-Stokes equations describing the steady axially symmetric flow of a viscous incompressible fluid between two infinite rotating disks. Several cases have been examined in detail and the radial and transverse velocity profiles are displayed; value of the torque experienced in these cases are also given. It is found that at high Reynolds numbers, the main core of the fluid is in a state of solid rotation for practically all values of the ratio of angular velocity of the two disks. When the disks are rotating in the same sense, and when one is at rest and the other is rotating, the results show that edge effects must be taken into account in any complete solution to the problem. However, when the disks rotate in opposite directions, the solutions exhibit features which appear unlikely to occur in practice.

Secondary Flows in Centrifugal Pump Impellers: PIV Measurements and CFD Computations

2009 ◽  
Author(s):  
R. W. Westra ◽  
L. Broersma ◽  
K. van Andel ◽  
N. P. Kruyt
Keyword(s):  
Reynolds Number ◽  
Centrifugal Pump ◽  
Flow Rate ◽  
Stokes Equations ◽  
Suction Side ◽  
Velocity Profiles ◽  
Secondary Flows ◽  
Design Flow ◽  
Low Velocity ◽  
Flow Rates

Two-dimensional Particle Image Velocimetry measurements and three-dimensional Computational Fluid Dynamics (CFD) analyses have been performed of the flow field inside the impeller of a low specific-speed centrifugal pump operating with a vaneless diffuser. Flow rates ranging from 80% to 120% of the design flow rate are considered in detail. It is observed from the velocity measurements that secondary flows occur. These flows result in the formation of regions of low velocity near the intersection of blade suction side and shroud. The extent of this jet-wake structure decreases with increasing flow rate. Velocity profiles have also been computed from Reynolds-averaged Navier-Stokes equations with the Spalart-Allmaras turbulence model, using a commercial CFD-code. For the considered flow rates the qualitative agreement between measured and computed velocity profiles is very good. Overall, the average relative difference between these velocity profiles is around 7%. Additional CFD computations have been performed to assess the influence of Reynolds number and shape of the inlet velocity profile on the computed velocity profiles. It is found that the influence of Reynolds number is mild. The shape of the inlet profile only has a weak effect at the shroud.

Numerical Study of the Steady-State Tip Vortex Flow Over a Finite-Span Hydrofoil

1998 ◽  
Vol 120 (2) ◽  
pp. 345-353 ◽  
Author(s):  
Chao-Tsung Hsiao ◽  
Laura L. Pauley
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Vortex Core ◽  
Axial Velocity ◽  
Stokes Equations ◽  
Numerical Study ◽  
Angle Of Attack ◽  
Tangential Velocity ◽  
Tip Vortex ◽  
Finite Span

The flow over a finite-span hydrofoil creating a tip vortex was numerically studied by computing the full Navier-Stokes equations. A good agreement in pressure distribution and oil flow pattern was achieved between the numerical solution and available experimental data. The steady-state roll-up process of the tip vortex was described in detail from the numerical results. The effect of the angle of attack, the Reynolds number, and the hydrofoil planform on the tip vortex was investigated. The axial and tangential velocities within the tip-vortex core in the near-field wake region were greatly influenced by the angle of attack. A jet-like profile in the axial velocity was found within the tip-vortex core at high angle of attack, while a wake-like profile in the axial velocity was found at low angle of attack. Increasing the Reynolds number was found to increase the maximum axial velocity, but only had a slight impact on the tangential velocity. Finally, a swept hydrofoil planform was found to attenuate the strength of the tip vortex due to the low-momentum boundary layer traveling into the tip vortex on the suction side.

scholarly journals Effect of Lower Surface Roughness on Nonlinear Hydraulic Properties of Fractures

Geofluids ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Jinglong Li ◽  
Xianghui Li ◽  
Bo Zhang ◽  
Bin Sui ◽  
Pengcheng Wang ◽  
...  
Keyword(s):  
Surface Roughness ◽  
Reynolds Number ◽  
Fluid Flow ◽  
Pressure Drop ◽  
Flow Rate ◽  
Stokes Equations ◽  
Flow Regimes ◽  
Lower Surface ◽  
Nonlinear Flow ◽  
Hydraulic Pressure

This study investigates the effect of fracture lower surface roughness on the nonlinear flow behaviors of fluids through fractures when the aperture fields are fixed. The flow is modeled with hydraulic pressure drop = 10 − 4 ~ 10 5   Pa / m by solving the Navier-Stokes equations based on rough fracture models with lower surface roughness varying from JRC = 1 to JRC = 19 . Here, JRC represents joint roughness coefficient. The results show that the proposed numerical method is valid by comparisons between numerically calculated results with theoretical values of three parallel-plate models. With the increment of hydraulic pressure drop from 10-4 to 105 Pa/m spanning ten orders of magnitude, the flow rate increases with an increasing rate. The nonlinear relationships between flow rate and hydraulic pressure drop follow Forchheimer’s law. With increasing the JRC of lower surfaces from 1 to 19, the linear Forchheimer coefficient decreases, whereas the nonlinear Forchheimer coefficient increases, both following exponential functions. However, the nonlinear Forchheimer coefficient is approximately three orders of magnitude larger than the linear Forchheimer coefficient. With the increase in Reynolds number, the normalized transmissivity changes from constant values to decreasing values, indicating that fluid flow transits from linear flow regimes to nonlinear flow regimes. The critical Reynolds number that quantifies the onset of nonlinear fluid flow ranges from 21.79 to 185.19.

Computational Investigation of Heat Transfer Characteristics of Counter-Rotating Turbine Disks With Annulus

2015 ◽  
Author(s):  
Dongdong Liu ◽  
Xiang Luo ◽  
Zhi Tao
Keyword(s):  
Heat Transfer ◽  
Reynolds Number ◽  
Vortex Pair ◽  
Rotating Disks ◽  
Heat Transfer Characteristics ◽  
Complex Flow ◽  
Transfer Characteristics ◽  
Air Inlet ◽  
Inlet Conditions ◽  
Cooling Air

The heat transfer characteristics of counter-rotating disks with annulus flow were investigated numerically in this paper, in which the effects of coolant inlet conditions (inlet geometry, turbulent parameter and inlet preswirl) of the cavity were emphasized. The axial Reynolds number of annulus Rew was set to be 5.99×105 and rotating Reynolds number Reϕ was set to be 2.35×105. Two kinds of cooling air inlet, radial inlet from the middle of cavity and axial inlet from one side of cavity, were adopted and investigated. The turbulent parameter and preswil ratio varied from 0.028 to 0.197 and −1.5 to 1.5, respectively. According to the calculation results:a vortex pair that generated by the radial inlet and non-preswirling cooling air is somehow unstable which rendered a complex flow field and wall heat transfer pattern in the cavity. For the influence of pumping effect of rotating disks, preswirling of cooling air led to uneven division of cooling air for two counter-rotating disks in radial cooling air inlet type. More cooling air flows towards the disk with same rotating direction of preswirl cooling air and leads to higher Nu in that wall which was enhanced by the increase of preswirl ratio. Axial cooling air inlet results in a much more stable flow than that with radial inlet. The Nu of the wall that cooling air flow towards was higher than that of the other one and this difference increased with the increase of turbulent parameter λT.

An Asymptotic Solution for Laminar Flow of an Incompressible Fluid Between Rotating Disks

1968 ◽  
Vol 35 (1) ◽  
pp. 155-159 ◽  
Author(s):  
L. Matsch ◽  
W. Rice
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Radial Velocity ◽  
Velocity Component ◽  
Tangential Velocity ◽  
Outer Radius ◽  
Creeping Flow ◽  
Arbitrary Distribution ◽  
Rotating Disks ◽  
Inner Radius

Laminar flow is considered between parallel rotating disks having a circular exhaust hole at an inner radius and supplied with fluid at the outer radius with pressure higher than the available sink pressure. The problem statement for asymptotic (fully developed) flow is formulated. A procedure for perturbing a creeping flow solution and an iteration scheme are developed to produce a solution for higher Reynolds numbers. The solution depends on two parameters, a Reynolds number and a mass flow parameter, and is asymptotic in the sense that a third parameter would be necessary for a solution with an arbitrary tangential velocity component specified at the outer radius of the disks and/or an arbitrary distribution of the radial velocity component between the disks. From computations conducted by digital computer, the region having uninflected radial velocity profiles is delineated. Typical results are presented for the velocity components as functions of Reynolds number, the average radial component of velocity at the entrance, and the inner radius of the disks.

scholarly journals Laser Velocimeter Measurements in a Centrifugal Pump With a Synchronously Orbiting Impeller

1990 ◽  
Author(s):  
Ronald J. Beaudoin ◽  
Steven M. Miner ◽  
Ronald D. Flack
Keyword(s):  
Centrifugal Pump ◽  
Flow Rate ◽  
Reverse Flow ◽  
Tangential Velocity ◽  
Velocity Profiles ◽  
Radial Component ◽  
Tangential Component ◽  
Design Flow ◽  
Recirculation Region ◽  
Design Flow Rate

Velocity profiles were measured in the impeller of a centrifugal pump with a two directional laser velocimeter. Blade to blade profiles were measured at four circumferential positions and four radii within and one outside the four bladed impeller. Data is presented herein at two circumferential and three radial locations. The pump was tested in two configurations; with the impeller running centered within the pump, and with the impeller orbiting with a synchronous motion (ϵ/r2). Variation in velocity profiles among the individual passages in the orbiting impeller were found. At design flow rate, these variations ranged from 30 to 60 percent for the radial component, and 15 to 25 percent for the tangential component. Tangential velocity profiles near the impeller exit (r/r2 = 0.973) were near uniform across each individual passage. Differences in the magnitude of the exit tangential velocities among the passages, however, were detected. Systematic differences in the velocity profile shapes of the centered and orbiting impellers were in general not measured, the only exception being at r/r2 = 0.973 at 40% of the design flow rate. At this condition, two distinct radial velocity profiles were measured. Two of the impeller passages of the orbiting impeller contained a recirculation region covering 20–30% of the blade passage while the other two passages contained no recirculation region. The centered impeller also contained this region of reverse flow. Finally, velocity data was numerically integrated to find the forces and stiffnesses due to momentum fluxes on the impeller for the orbiting condition.

Laser Velocimeter Measurements in a Centrifugal Pump With a Synchronously Orbiting Impeller

1992 ◽  
Vol 114 (2) ◽  
pp. 340-349 ◽  
Author(s):  
R. J. Beaudoin ◽  
S. M. Miner ◽  
R. D. Flack
Keyword(s):  
Centrifugal Pump ◽  
Flow Rate ◽  
Reverse Flow ◽  
Tangential Velocity ◽  
Velocity Profiles ◽  
Radial Component ◽  
Tangential Component ◽  
Design Flow ◽  
Recirculation Region ◽  
Design Flow Rate

Velocity profiles were measured in the impeller of a centrifugal pump with a two-directional laser velocimeter. Blade-to-blade profiles were measured at four circumferential positions and four radii within and one outside the four-bladed impeller. Data are presented herein at two circumferential and three radial locations. The pump was tested in two configurations; with the impeller running centered within the pump, and with the impeller orbiting with a synchronous motion (ε/r2 = 0.016). Variation in velocity profiles among the individual passages in the orbiting impeller were found. At design flow rate, these variations ranged from 30 to 60 percent for the radial component, and 15 to 25 percent for the tangential component. Tangential velocity profiles near the impeller exit (r/r2 = 0.973) were near uniform across each individual passage. Differences in the magnitude of the exit tangential velocities among the passages however, were detected. Systematic differences in the velocity profile shapes of the centered and orbiting impellers were in general not measured, the only exception being at r/r2 = 0.973 at 40 percent of the design flow rate. At this condition, two distinct radial velocity profiles were measured. Two of the impeller passages of the orbiting impeller contained a recirculation region covering 20-30 percent of the blade passage while the other two passages contained no recirculation region. The centered impeller also contained this region of reverse flow. Finally, velocity data were numerically integrated to find the forces and stiffnesses due to momentum fluxes on the impeller for the orbiting condition.

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