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2014,26(2):165-177 DOI: 10.1016/S1001-6058(14)60019-6

Direct numerical simulation of Open Von Krmn Swirling Flow* XING Tao Department of Mechanical Engineering, College of Engineering, University of Idaho, Idaho 83844-0902, USA, E-mail: xing@uidaho.edu (Received January 15, 2014, Revised April 10, 2014) Abstract: Direct numerical simulations are used to investigate the Open Von Krmn Swirling Flow, a new type of unsteady three-dimensional flow that is formed between two counter-rotating coaxial disks with an axial extraction enclosed by a cylinder chamber. Solution verification shows that monotonic convergence is achieved on three systematically refined grids for average pressure at the disk periphery with a small grid uncertainty at 3.5%. Effects of the rotational speeds and flow rates on the flow field are examined. When the disks are rotating at the lowest speed, 100 RPM, only circular vortices are formed regardless of the flow rates. When the disks are rotating at 300 RPM and 500 RPM, negative spiral vortex network is formed. The radial counterflow concept for such spiral vortex network is verified by examining various horizontal cuts and radial velocity component, which show radial outflows in two bands near the two disks and radial inflow in one band between them. Overall, the flow is similar to the Stewartson type flow but with significant differences for all three velocity components due to the axial suction at the upper disk center and gap between the disk periphery and chamber wall. Key words: direct numerical simulation, Open Von Krmn flow, swirling, radial counterflow Introduction

Fluid motion between two coaxial disks/plates has been studied extensively for decades due to their importance to industrial applications. Applications of such flows in practice include heat and mass excha- ngers[1], disk reactor for intensified synthesis of bio- diesel[2], open clutch system[3,4], lubrication[5], rotating packed beds[6], and internal cooling-air systems of most gas turbines[7], etc. The two disks/plates may co- rotate, counter-rotate, or one disk is stationary and the other rotates (rotor-stator system), which creates dra- matically different flow patterns.

Limited number of studies used analytical metho- ds, likely due to the strong viscous effect within the boundary layers near the disk surfaces and strong three-dimensional features of the flow. Batchelor[8] so- lved the steady rotationally-symmetric viscous lami- nar flow between two infinite disks. When the two disks are exactly counter-rotating, the distribution of

Biography: XING Tao (1973-), Male, Ph. D., Assistant Professor

tangential velocity is symmetrical about the mid-plane and exhibits five distinct regions: two disk boundary layers, a transition shear layer at mid-plane, and two rotating cores on either side of the transition layer. Stewartson[9] draw controversial conclusions on the flow structure as he found that the flow is divided into only three zones when the Reynolds number =HRe

2 / 100H , where H is the disk spacing, is the rotational speed of the disk and is the fluid ki- nematic viscosity. The three regions are two boundary layers on the two disk surfaces separated by a region that has zero tangential velocity and uniform radial in- flow. The work by Wilson and Schryer[10] numerically solved the steady viscous flow between two coaxial infinite disks with one stationary and the other rota- ting. The effects of applying a uniform suction throu- gh the rotating disk are determined. At large Reynolds numbers, the equilibrium flow approaches an asymp- totic state in which thin boundary layers exist near both disks and an interior core rotates with nearly con- stant angular velocity. The flow field is assumed to be axisymmetric. This study also demonstrates that more than one steady (equilibrium) solution exist for the

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time-dependent equations of motion. Witkowski et al.[11] studied the first bifurcation in the axisymmetric flow between two exactly counter-rotating disks with very large aspect ratio /R H , where R is the disk radius and 2H is the inter-disk spacing. By ne- glecting the effect of curvature, they were able to re- duce the order of Navier-Stokes equations and axisy- mmetric flow to parallel flow. They found that a cen- trifugal instability will always occur no matter how large the local radius considered, which is different from the plane Couette flow. As a result of the assu- mptions for axisymmetric flow, infinite disk sizes, and very large aspect ratios, conclusions drawn by these analytical studies may not be applicable for most in- dustrial applications.

The first systematic experimental study on the flow between two rotating co-axial disks at a relative- ly wide range of rotational conditions was conducted by Soong et al.[12]. Three different modes of disk rota- tions, i.e., co-rotation, rotor-stator, and counter-rota- tion, were considered. When there is no shroud near the disk rim, co-rotating disk flows are characterized by an inboard core region of solid-body rotation, out- board vortical flow region, and Ekman layers over disks with the presence of the vortex chains in gap- view of the two-cell flow structure. Flow between counter-rotating disks encounters large tangential shear stemming from opposite tangential Coriolis force near two disks, which enhance the fluid mixing characteristics. The size of the disk gap plays an im- portant role in formation of the flow structures. In ge- neral, smaller gap size reduces the size of the vortices, weakens the external fluid ingestion in the gap region, and suppresses the flow instability or turbulence. Gauthier et al.[13] experimentally investigated the flow between two rotating disks (aspect ratio of 20.9) en- closed by a cylinder in the cases of both co- and cou- nter-rotation. It was found that the co-rotation case and the weak counter-rotation cases are very similar to the rotor-stator case in that the basic flow consists of two boundary layers near each disk and the instability patterns are the axisymmetric vortices and the positive spirals. When the two disks are counter-rotating with a higher rotation ratio, a new kind of instability pattern is observed, called negative spirals. The recirculation flow becomes organized into a two-cell structure with the appearance of a stagnation circle on the slower disk. Moisy et al.[14] conducted experimental and nu- merical study of the shear layer instability for the same geometry but with various aspect ratios between 2 and 21. It was shown that the instability can be de- scribed in terms of a classical Kelvin-Helmholtz insta- bility, where curvature has only a weak effect. The ob- served surrounding spiral arms result from the intera- ction of this unstable shear layer with the Ekman

boundary layers over the faster disk. Poncet et al.[15,16] used computational fluid dynamics (CFD) to investi- gate the turbulent Von Krmn flow generated by two counter-rotating smooth flat (viscous stirring) or bla- ded (inertial stirring) disks enclosed by a cylinder. For viscous stirring, the flow close to the rotation axis is of Stewartson type and shows three distinct regions: two boundary layers and one shear layer at mid-plane. For regions close to the periphery of the cavity, flow is of Batchelor type with five distinct zones: two boundary layers on the disks, a shear layer at mid- plane and two zones enclosed between the two.

Few studies investigated two-phase flows bet- ween two rotating coaxial disks. Yuan et al.[17] studied aeration for disengaged wet clutches using a gas-li- quid two-phase CFD model with experimental measu- rements of drag torque for validation. When the sepa- rator disk is stationary, air enters the clearance from the outer periphery of it and oil flies off from the rota- ting disk, which reduce the drag torque. When the two disks are counter-rotating, the two-phase flow pattern depends on the difference between the two angular ve- locities. If the difference is large, air enters from the low speed side of the plates. Otherwise, air enters from the middle of the clearance at the outer radius and both sides keep a thin oil film.

The objective of this study is to investigate a new type of flow, Open Von Krmn Swirling Flow, which features radial counterflow between two cou- nterrotating disks enclosed in a chamber. It differs from the well-known Von Krmn Flow as it has an axial suction (outlet) at the center of the upper disk and a gap between the two disks and chamber wall. This serves as a simplified model of the McCutchen Processors developed by Vorsana Inc. that centrifuga- lly separates a fluid mixture using vortices created in high shear between axially fed counter-rotating disk impellers. The processor makes use of the radial cou- nterflow concept. As the fluid mixture is spun at high speed in the vortices, centrifugal force moves heavy fractions toward the outside of the vortices and away from the axis of rotation, while light fractions remain inside the vortices, and are sucked inward by an axial pump. This radial counterflow concept has not been experimentally or numerically verified in previous works. This study uses CFD to examine this concept and other flow physics for Open Von Krmn Swirling Flow. Parametric studies are performed to elucidate the effect of flow rates and rotational speeds on the formation of vortical structures. Quantitative solution verification is performed on three systemati- cally refined grids to estimate the grid uncertainties. Validation of the CFD model is achieved by compa- ring with available experimental data for similar geo- metries and flow conditions.

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1. Computational methods The commercial CFD software, ANSYS/

FLUENT version 14.0[18] is used for all the simu- lations. FLUENT is a finite volume solver which pro- vides a suite of numerical schemes and transition and turbulence modeling options. For this study, transient single-phase simulations are conducted using the pre- ssure-based solver option, which is the typical predi- ctor-corrector method with solution of pressure via a pressure Poisson equation to enforce mass conserva- tion. Pressure-velocity coupling is performed using the SIMPLE scheme. The convective terms in the mo- mentum equation are discretized using the third-order MUSL scheme. Unsteady terms are discretized using a second order implicit scheme. The time step is cho- sen to be 0.005 s for all simulations with large maxi- mum iterations per time step to ensure that the mini- mum residuals are lower than 105 for the continuity and three momentum equations for all simulations. Due to the small disk radius and low rotational speed of the disks, no turbulence model is applied. All simu- lations are conducted using ANSYS Academic Resea- rch CFD with high performance computing on a Dell Precision T7500 that has 12 cores and 48 GB RAM. 1.1 Governing equations

Since direct numerical simulations are performed, no turbulence models are used. For Cartesian coordi- nates, the incompressible continuity and momentum equations are:

( ) = 0 V (1)

2 T( ) + ( ) = + [ ( + )] +pt V V V V g

(2) where V is the velocity vector, is the dynamic vis- cosity, is the density, p is the pressure, and g is the gravitational acceleration.

Fig.1 Geometry and grid 1.2 Geometry and grid

The geometry is shown in Fig.1. Two co-axial disks are counter-rotating at the same angular velocity

enclosed by a stationary cylindrical chamber. The upper disk rotates counter-clockwise viewed from the top and the lower disk rotates in the opposite direction at the same rotational speed . The disk diameter is 0.28 m. The gap size between the two disks is 0.0033 m. The chamber diameter is 0.32 m. The two disks are away from the chamber wall by 0.0133 m in the vertical direction ( )Z . Fluid enters the chamber through a circular hole with diameter 0.01965 m at the bottom surface of the chamber. Two additional holes with the same diameter are drilled on the upper disk and top chamber wall that are connected by a short circular pipe, which serves as the outlet of the fluid. An O-type grid is created to model the flow. The fine grid has a total of 798 675 points. For solution verifi- cation, additional two coarser grids are created syste- matically using a constant grid refinement ratio 2 in all three spatial directions. 1.3 Simulation design and flow parameters

A total of nine simulations are performed as sum- marized in Table 1. The simulations cover three flow rates (48 GPM, 72 GPM and 96 GPM) and three rota- tional speeds (100 RPM, 300 RPM and 500 RPM). Velocity inlet and pressure outlet are specified as boundary conditions for the fluid. Rotational wall boundaries are enforced using the prescribed rotatio- nal speeds. The fluid is water liquid with density 998.2 kg/m3 and kinematic viscosity 0.001003 kg/ms.

To facilitate the discussion and also generalize the conclusions such that they are independent of spe- cific geometry and flow properties, non-dimensional parameters are used. The rotational Reynolds number is 20 /Re R , where 0R is the outer radius of the disks and is the kinematic viscosity of the fluid. The aspect ratio is defined as the ratio of disk spacing S and 0R , 0= /G S R . For the cylindrical coordina- tes, the non-dimensional radial and axial coordinates are defined as 0= /r r R

and 0= /Z Z R , respective-

ly. Thus, =r 0 is the rotational axis located at the center of the disks and =r 1 is located at the disk periphery. The upper disk is located at =Z 0 and the lower disk is located at =Z 0.023. For the Cartesian coordinates, Z is the same as in cylindrical coordina- tes and X and Y are non-dimensionalized using 0R . Velocities are non-dimensionalized using * = /V V

0( )R , where V can be u , v , w in the Cartesian Coordinates or ru , u , and zu in the cylindrical coo- rdinates. For simplicity, asterisks for all dimensionless variables are dropped and units for dimensional varia- bles are specified.

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Table 1 Simulation matrix

Simulations 1 2 3 4 5 6 7 8 9

Q (GPM) 48 72 96 48 72 96 48 72 96

(RPM) 100 100 100 300 300 300 500 500 500 Re 3.25104 3.25104 3.25104 9.75104 9.75104 9.75104 1.63105 1.63105 1.63105

Fig.2 Gap-view flow structures using streamlines and pressure contour near the disk rim for =Re 7.910

3 2. Validation of the CFD model

The CFD model is first validated against experi- mental data reported in previous literature for flows between two counter-rotating disks without an axial suction, either qualitatively for the flow pattern or quantitatively for the disk drag torque. 2.1 Flow pattern between two counter-rotating disks

This validation case follows Soong et al.[12]. The disk peripheries are open to atmosphere. A 100 (in gap direction)600 (in radial direction) grid is used. Simulation is conducted for =Re 7.910

3. Two as-

pect ratios are examined. As shown by Fig.2, the fluid structures inside the gap are very sensitive to the gap spacing when it is reduced to a certain value. In this study, =G 0.08 (Fig.2(a)) creates staggered vortex chains and a wavy interface. The vortices close to the upper disk are rotating clockwise whereas those close to the lower disk are rotating counter-clockwise. This is the same as what observed in experiment (Fig.2(b)). When the gap size increases to 0.1, the vortex chain suddenly disappears (Fig.2(c)) and further increase of the rotational speed will not change the flow pattern.

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Table 2 Validation of drag torque for a disengaged wet clutch pack

Friction plate speed (RPM) Drag torque on fixed separator plate (Nm)

Experiment ( )D CFD ( % )E D Yuan et al.[17] ( % )E D

200 0.3295 0.3583 (8.7%) 0.3798 (15.3%)

316 0.5596 0.5634 (0.7%) 0.5682 (1.5%)

368 0.6402 0.6578 (2.7%) 0.6549 (2.3%)

2.2 Drag torque

Accurate prediction of the drag torque using the CFD model is important to estimate the power consu- mption. The drag torque also indicates the accuracy of the CFD model to predict the shear stresses near the plate surfaces. The drag torque predicted by the curre- nt CFD model is validated for a disengaged wet clutch pack. The simulation is compared to the experimental data and CFD simulation by Yuan et al.[17]. To ensure that the CFD results are independent of the grid reso- lution, a very fine grid (800 in radial100 in axial = 80 000 nodes) is used. This is about nine times finer than the grid used by Yuan et al.[17] in their CFD simulations, which had 300 in radial30 in axial = 9 000 nodes. The CFD model is built in axisymmetric flow with swirl.

The experiment measures the drag torque on the fixed separator plate when the friction plate is rotating at different speeds. It was shown that at low rotational speed, the drag torque increases almost linearly versus speed to a peak value (Phase I). After the peak value that corresponds to a critical friction plate speed, the torque is reduced rapidly to nearly zero (Phase II). By examining the flow field, it was found that Phase I shows single-phase flow whereas Phase II shows two- phase flow between the two plates. In other words, air starts to enter the clearance at the critical speed and the aeration causes the oil film to shrink. The greater the speed, the more the air enters the clearance. Be- cause the air viscosity is much smaller than oil visco- sity, the drag torque rapidly decreases. Since this study only investigates single-phase flow, validation is conducted for three rotational speeds in Phase I and results are summarized in Table 2. Compared to the CFD by Yuan et al.[17], the current CFD has similar re- lative error for the 368 RPM but much lower relative error for 200 RPM and 316 RPM. Since the CFD solver used is the same, this improvement was likely due to the much finer grid used in the current study. Table 3 Solution verification for average pressure (Pa) at

disk periphery

Mesh number R P 1(% )GU S

1 2 3

1 850 1 881 2 303 0.07 2.51 3.5%

3. Results and discussion 3.1 Solution verification

Solution verification is a process for assessing simulation numerical errors and associated uncertai- nties. In this study, the discretization error due to limi- ted number of grid points is the main source of nume- rical errors. In this study, solution verification is per- formed for the average pressure at disk periphery on three systematically refined grids that are generated using a constant grid refinement ratio = 2r in all three spatial directions. The factor of safety me- thod[19,20] is used to estimate the grid uncertainties and results are summarized in Table 3. The fine grid (mesh 1) has 770 788 grid points. Meshes 2 and 3 re- present the medium and coarse grids, respectively. Simulation 9 that has the highest disk rotational speed and largest inlet flow rate is selected for solution veri- fication. The solutions on the fine, medium, and coa- rse grids are 1S , 2S and 3S , respectively. Solution changes for medium-fine and coarse-medium solu- tions and the convergence ratio R are defined by

21 2 1= S S , 32 3 2= S S , 2132

=R (3)

When 0 1R , monotonic convergence is achieved. Then the three grid solutions can be used to compute the estimated order of accuracy REp , error RE , and grid uncertainty 1(% )GU S .

32

21

ln=

ln( )REp

r

(4)

21=1RERE pr

(5) When solutions are in the asymptotic range, then

=RE thp p , however, in many circumstances, especia- lly for industrial applications, solutions are far from the asymptotic range such that REp is greater or sma- ller than thp

[21]. The ratio of REp to thp is used as the distance metric

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Fig.3 Three-dimensional vortical structures (Iso-surface of =Q 200 is colored by pressure in Pa) of single-phase water between the

two counter-rotating disks view from the top (vortices above the upper disk and below the lower disk have been blanked out): (a)-(i) correspond to Simulations 1 to 9 in Table 1, respectively, and (j) averaged pressure of fluids at disk periphery for the nine simulations

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Fig.4 Streamlines and contour of the velocity component v in the slice at =Y 0 for Simulation 5 (length ratio of X and Z is

0.25)

= REth

pPp

(6)

and the grid uncertainty is estimated by

= (2.45 0.85 )G REU P , 0 1P (7a)

= (16.4 14.8 )G REU P P , 1P (7b) As shown in Table 3, monotonic convergence is ac- hieved with a low grid uncertainty of 13.5%S . This suggests that the current fine grid resolution is suffi- cient and this fine grid is used for all simulations. 3.2 Flow physics

Three-dimensional top view of the vortical stru- ctures within the processor is shown in Fig.3 for the nine simulations in Table 1. The vortical structures are identified by the Q -criterion[22] and colored by pre- ssure. To focus on the flow between the two disks, the vortical structures above the upper disk and below the lower disk have been blanked out.

For all the nine simulations, the highest and lo- west pressures are located near the chamber wall and axial suction (outlet), respectively. For the same ro- tational speed, the range of pressure values increases with the increase of the inlet flow rate. When the disks are rotating at the lowest speed, 100 RPM (Figs.3(a)-3(c)), only circular vortices are formed re- gardless of the flow rates. With the increase of the in- let flow rate, more circular vortices move toward the axial suction. For the two higher rotational speeds ( 300 RPM and 500 RPM ), negative spiral vortex network is formed, which is similar to what was obse- rved in the experiments by Gauthier et al.[13]. It also shows for these two higher rotational speeds that in- crease of the flow rate creates larger size vortices but the number of vortices decreases near the disk center.

To examine quantitatively the effect of flow rates

at the three different rotational speeds, average pressu- res of fluids at disk periphery are plot for the nine simulations, as shown in Fig.3(j). Overall, the pressure increases almost linearly with the increase of rotatio- nal speed for the two lower rotational speeds 100 RPM, 300 RPM and 300 RPM shows a larger slope. For rotational speed 500 RPM, the pressure in- creases non-linearly (quadratically) as the increase of flow rates.

Figure 4 shows streamlines and contour of the velocity component v in the slice at =Y 0 for Simu- lation 5. In order to clearly show the flow field, the length ratio of X and Z has been reduced from 1 to 0.25. Two vortex streets staggered to each other are formed near the upper and lower disk surfaces, respe- ctively, which is similar to the flow pattern shown in Fig.2(a) for the study by Soong et al.[12]. The upper vortices are rotating counter-clockwise whereas the lower vortices are rotating clockwise. This results in a shear layer between the two vortex streets where fluid flows from the disk periphery to the center of the disk. Fluids very close to the two disk surfaces are swept out by the rotating of the disks, regardless of their ro- tating direction. However, due to the opposite rotatio- nal directions of the two disks, the upper and lower vortex streets show negative and positive v velocities, respectively, which is consistent with the rotational di- rection of the adjacent disk.

Figure 5 shows different views by examining flows in various Z cuts for Simulation 5. All the sub- figures in Fig.5 are colored by the Z velocity. By comparing with Fig.3(e) and streamlines in Fig.4, the interface between the positive and negative Z velo- cities in Fig.5 is corresponding to the local core of the spiral vortices. For fluid inside the boundary layer of the upper disk as shown in Fig.5(a), it has two veloci- ty components. The first component is caused by the local disk rotation and no-slip boundary condition en- forced on the disk surface, r , where r is the radius of the local point on the disk surface with respect to the Z axis. The other component is the velocity in the

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Fig.5 Streamlines and contour of Z velocity component in various Z cuts (Simulation 5): (a) inside the boundary layer of the

upper disk 8( = 1 10 )Z , (b) inside the boundary layer of the lower disk ( = 0.99)Z , (c) plane cross the upper vortex street center ( = 0.22)Z , (d) plane cut through the lower vortex street center ( = 0.61)Z , and (e) plane cut through the shear layer between the two vortex streets ( = 0.45)Z

radial direction caused by the centrifugal force. As a result, fluid flows radially outward from the rotational Z axis following negative spiral paths. However, there is a small circular region with diameter 0.048 m near the center where fluid flows toward the rotational axis through a positive spiral. This is called a spiral eye that is larger than the outlet diameter 0.01965 m, which is caused by the strong suction at the outlet lo- cated at the center of the upper disk. Figure 5(b)

shows the flow inside the boundary layer of the lower disk. Similar to the flow inside the boundary layer of the upper disk, the fluid flows radially outward from the rotational axis. But it follows positive spiral paths as it rotates in the opposite direction to the upper disk. There is also a spiral eye near the center with dia- meter 0 =r 0.034 m, which is smaller than observed inside the upper disk boundary layer. Inside the spiral eye, fluid flows towards the Z axis through a nega-

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tive spiral. For the plane cutting through the upper vortex street center (Fig.5(c)), the streamlines are very curvy with overall flow direction from the disk peri- phery to the center. The spiral vortex network is clea- rly shown and agrees well with the vortical structures observed in Fig.3(e). The flow patterns in the plane cutting through the lower vortex street center (Fig.5(d)) and plan across the shear layer between the two vortex streets (Fig.5(e)) are similar to the flow pattern in the plane cut across the upper vortex street center but with a much smoother negative spiral.

Fig.6 Conversion of velocity vectors from Cartesian coordinate

to cylindrical polar coordinate

To better examine the radial counterflow concept, various annular control surfaces are extracted within the flow field. These annual control surfaces are cross- sections of the flow between the disks at a constant ra- dius from the rotational Z axis. To facilitate the ana- lysis, the three Cartesian velocity components ( , ,u v

)w are converted to be the components in the cylin- drical coordinates ( , , )r zu u u using

= cos + sinru u v , = sin + cosu u v ,

=zu w , tan =yx

(8)

The correlation between these velocity compone- nts is shown in Fig.6. Figure 7 shows instantaneous pressure (contour flood) and ru (contour line) in three annular control surfaces for X 0 that are projected to the vertical ( , )Y Z plane ( X 0 exhibits similar features and thus not shown). There are two bands with positive ru that are located inside boundary layers of the two disks. This suggests that the net mo- mentum for fluid close to the disk surface is radially outward. Between the two bands, there is a shear layer where ru is negative and the net momentum is radia- lly inward. As the annual control surface move closer

to the rotational Z axis (smaller r ), the size of the shear layer band increases and the band near the upper disk is significantly suppressed. This is due to the axial suction at the center of the upper disk. The lar- gest negative radial velocity is located near the mid- plane between the two disks. Overall the magnitude of the radial velocity in the shear layer band decreases when r decreases, likely due to the increase of the shear layer band size. The pressure variation in the vertical Z direction is minor. Low and high pressures are corresponding to the largest negative radial velo- city regions and regions between them, respectively, which can be explained using the Bernoulli effect.

The interfaces between the three bands can be visualized using Iso-surface of the radial velocity

=ru 0, where the radial velocity changes direction. Figures 8(a) and 8(b) show the instantaneous and time averaged Iso-surface of =ru 0, respectively. Overall the band near the lower disk is much thicker than the band near the upper disk. The arrows show the flow direction for each band. The time-averaging process smooths the wavy interface observed for the insta- ntaneous flow field. The averaged interface between the two lower bands shows a smooth circle whereas the averaged interface between the two upper bands shows a shape similar to a volcano due to the axial suction near the upper disk center.

Ravelet et al.[23] found that the structure of the mean Von Krmn flow in the exact counter-rotating regime can be decomposed into two poloidal recircu- lations in the ( , )r z plane. Similar flow pattern is ob- served in the current study as shown in Fig.9. The fluid near the upper and lower disks are moved by two opposite rotation speeds ( )u , and then swiped radia- lly outward. As a result of mass conservation, a shear layer develops between the two disks with radially in- ward velocity. However, unlike the Von Krmn flow where the shear layer is located in the equatorial plane (mid-plane between the two disks), the shear layer in this study is closer to the upper disk and moves farther away from the lower disk when flow approaches the Z axis. The difference is caused by the axial suction at the center of the upper disk. Axial profiles of the tangential velocity component u of the mean flow at four radial locations are show in Fig.10(a). The tange- ntial velocity is small except inside the boundary laye- rs of the upper and lower disks. When r increases, the magnitude of the tangential velocity increases in- side the two disk boundary layers but remains almost constant near the shear layer at Z 0.0075. Similar to the Stewartson flow structure observed by Poncet et al.[16] for Von Krmn flow when r is small, three zones are observed: an almost constant tangential ve- locity zone enclosed by two boundary layers on each disk. However, the tangential velocity constant is

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Fig.7 Instantaneous pressure (flood, Pa) and ru (line) in various annular control surfaces for X 0 that are projected to the

vertical ( , )Y Z plane (Simulation 5) negative, not zero as observed for Von Krmn flow. This indicates that most regions of the flow are impa- cted more by the lower disk than by the upper disk, which is consistent with the much thicker lower bands shown in Fig.8. The boundary layers of the disks are also much thicker than those observed by Poncet et al. who examined turbulent Von Krmn flows that have much larger Re . While the Von Krmn flow shows almost zero radial and axial velocity components, the current open Von Krmn swirling flow exhibits sig- nificant magnitude of ru and weak zu (still non-zero), as shown in Fig.10(b) and Fig.10(c), respectively. The radial velocity component is about 25% of the magni- tude of the tangential velocity component and reaches

maximum positive value and maximum negative value inside the boundary layers of the disks and near the shear layer between them, respectively. When r increases, the magnitudes of both maximum positive and maximum negative values also increase. The axial velocity component is the smallest among the three velocity components. It is zero on the two disk surfa- ces and in the region near the shear layer. It is positive and negative in the regions between the upper disk and the shear layer and between the shear layer and lower disk, respectively. 4. Conclusions and future work

For the first time, unsteady three-dimensional

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Fig.8 Iso-surface of =ru 0 that separates the three annular bands for Simulation 5 (length ratio of X and Z is 0.05)

Fig.9 Streamlines of the mean flow between the two disks colored by u for Simulation 5 direct numerical simulations are conducted to investi- gate flows between two counter-rotating coaxial disks with an axial extraction enclosed by a cylinder cha- mber, which is called the Open Von Krmn Swirling Flow. The CFD model is built on top of the commercial CFD software, ANSYS FLUENT 14.0, and validated by comparing against experimental data published in previous literatures, either qualitatively for the flow pattern or quantitatively for the drag tor- que. Quantitative solution verification is performed on three systematically refined grids. Monotonic conve- rgence is achieved for the average pressure at disk pe- riphery with a small grid uncertainty at 3.5%. The fine grid is then used for all the nine simulations that cover three rotational speeds (100 RPM, 300 RPM, and 500 RPM) and three flow rates (48 GPM, 72 GPM, and 96 GPM).

This study reveals strong three-dimensional flow structures, which undermines the use of axisymmetric model with a two-dimensional grid to approximate the

flow field in most previous studies for similar geome- try and flow conditions. The highest and lowest pre- ssures are located near the chamber wall and axial su- ction, respectively. For the same rotational speed, the range of pressure values increases with the increase of the inlet flow rate. When the disks are rotating at the lowest speed, 100 RPM , only circular vortices are formed regardless of the flow rates. For the two higher rotational speeds ( 300 RPM and 500 RPM ), nega- tive spiral vortex network is formed. The slice cutting through the spiral vortices at =Y 0 and X 0 shows two staggered vortex streets that rotates counter-clo- ckwise and clockwise near the upper and lower disks, respectively.

The radial counterflow concept is verified by examining various Z cuts and radial velocity com- ponent ru in the cylindrical coordinate. Two bands with positive ru are located in regions very close to the two disk surfaces where the net momentum of

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fluid is radially outward. Between the upper and lower bands, there is a shear layer where ru is negative and the net momentum is radially inward. Overall the lower band near the lower disk is much thicker than the upper band near the upper disk. As the location moves closer to the rotational Z axis, the size of the shear layer band increases and the upper band is signi- ficantly suppressed. This is due to the axial suction at the center of the upper disk. No significant change of the lower band thickness is observed. Further analysis of the two poloidal recirculations and the three veloci- ty components in the ( , )r z plane show features simi- lar to Stewartson flow but with significant differences on the location of the shear layer and non-zero radial and axial velocity components.

Fig.10 Axial profiles of the three velocity components of the

mean flow in =Y 0 at four radial locations for Simu- lation 5

Future work includes extension of the current geometry from model-scale to full-scale and validate CFD simulations using full-scale experimental data upon available. The smooth flat disks (viscous stirring) may be replaced by bladed disks (inertial stirring) to increase the efficiency of the disks in forcing the flow. The current single phase simulations need to be exte- nded to two- and multi-phase simulations to investiga- te the effect of the spiral vortex network on separation of various phases. Preliminary results of the air-water mixture flows show that the lighter-phase air tends to be locked in the spiral vortex cores. Acknowledgement

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