Direct numerical simulation of Open Von Kármán Swirling Flow*

XING Tao

Department of Mechanical Engineering, College of Engineering, University of Idaho, Idaho 83844-0902, USA, E-mail: xing@uidaho.edu

Direct numerical simulation of Open Von Kármán 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)

Direct numerical simulations are used to investigate the Open Von Kármán 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.

direct numerical simulation, Open Von Kármán 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 exchangers[1], disk reactor for intensified synthesis of biodiesel[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 corotate, counter-rotate, or one disk is stationary and the other rotates (rotor-stator system), which creates dramatically different flow patterns.

Limited number of studies used analytical methods, 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]solved the steady rotationally-symmetric viscous laminar flow between two infinite disks. When the two disks are exactly counter-rotating, the distribution of 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 ReH= ΩH2/ν＞100, where H is the disk spacing, Ω is the rotational speed of the disk and ν is the fluid kinematic 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 inflow. The work by Wilson and Schryer[10]numerically solved the steady viscous flow between two coaxial infinite disks with one stationary and the other rotating. The effects of applying a uniform suction through the rotating disk are determined. At large Reynolds numbers, the equilibrium flow approaches an asymptotic state in which thin boundary layers exist near both disks and an interior core rotates with nearly constant angular velocity. The flow field is assumed to be axisymmetric. This study also demonstrates that more than one steady (equilibrium) solution exist for thetime-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 Γ≡R/ H , where R is the disk radius and 2H is the inter-disk spacing. By neglecting the effect of curvature, they were able to reduce the order of Navier-Stokes equations and axisymmetric flow to parallel flow. They found that a centrifugal 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 assumptions for axisymmetric flow, infinite disk sizes, and very large aspect ratios, conclusions drawn by these analytical studies may not be applicable for most industrial applications.

The first systematic experimental study on the flow between two rotating co-axial disks at a relatively wide range of rotational conditions was conducted by Soong et al.[12]. Three different modes of disk rotations, i.e., co-rotation, rotor-stator, and counter-rotation, 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, outboard vortical flow region, and Ekman layers over disks with the presence of the vortex chains in gapview 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 important role in formation of the flow structures. In general, 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) enclosed by a cylinder in the cases of both co- and counter-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 numerical 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 described in terms of a classical Kelvin-Helmholtz instability, where curvature has only a weak effect. The observed surrounding spiral arms result from the interaction 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 investigate the turbulent Von Kármán flow generated by two counter-rotating smooth flat (viscous stirring) or bladed (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 midplane and two zones enclosed between the two.

Few studies investigated two-phase flows between two rotating coaxial disks. Yuan et al.[17]studied aeration for disengaged wet clutches using a gas-liquid two-phase CFD model with experimental measurements of drag torque for validation. When the separator disk is stationary, air enters the clearance from the outer periphery of it and oil flies off from the rotating 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 velocities. 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 Kármán Swirling Flow, which features radial counterflow between two counterrotating disks enclosed in a chamber. It differs from the well-known Von Kármán 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 centrifugally separates a fluid mixture using vortices created in high shear between axially fed counter-rotating disk impellers. The processor makes use of the “radial counterflow” 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 Kármán 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 systematically refined grids to estimate the grid uncertainties. Validation of the CFD model is achieved by comparing with available experimental data for similar geometries and flow conditions.

1. Computational methods

The commercial CFD software, ANSYS/ FLUENT? version 14.0[18]is used for all the simulations. FLUENT is a finite volume solver which provides a suite of numerical schemes and transition and turbulence modeling options. For this study, transient single-phase simulations are conducted using the pressure-based solver option, which is the typical predictor-corrector method with solution of pressure via a pressure Poisson equation to enforce mass conservation. Pressure-velocity coupling is performed using the SIMPLE scheme. The convective terms in the momentum equation are discretized using the third-order MUSL scheme. Unsteady terms are discretized using a second order implicit scheme. The time step is chosen to be 0.005 s for all simulations with large maximum iterations per time step to ensure that the minimum residuals are lower than 10–5for 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 simulations are conducted using ANSYS Academic Research 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 coordinates, the incompressible continuity and momentum equations are:

where V is the velocity vector, μ is the dynamic viscosity, ρ 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 Ω. 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 verification, additional two coarser grids are created systematically 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 summarized in Table 1. The simulations cover three flow rates (48 GPM, 72 GPM and 96 GPM) and three rotational 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 rotational speeds. The fluid is water liquid with density 998.2 kg/m3and kinematic viscosity 0.001003 kg/m·s.

To facilitate the discussion and also generalize the conclusions such that they are independent of specific geometry and flow properties, non-dimensional parameters are used. The rotational Reynolds number is ReΩ≡/ν, where R0is the outer radius of the disks and ν is the kinematic viscosity of the fluid. The aspect ratio is defined as the ratio of disk spacing S and R0, G=S/ R0. For the cylindrical coordinates, the non-dimensional radial and axial coordinates are defined as r*=r/ Rand Z*=Z/ R, respective-0ly. 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 coordinates and X*and Y*are non-dimensionalized using R0. Velocities are non-dimensionalized using V*=V/ (ΩR0), where V can be u, v, w in the Cartesian Coordinates or ur, uθ, and uzin the cylindrical coordinates. For simplicity, asterisks for all dimensionless variables are dropped and units for dimensional variables are specified.

Table 1 Simulation matrix

Fig.2 Gap-view flow structures using streamlines and pressure contour near the disk rim for ReΩ=7.9×103

2. Validation of the CFD model

The CFD model is first validated against experimental 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 ReΩ=7.9×103. Two aspect 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.

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

2.2 Drag torque

Accurate prediction of the drag torque using the CFD model is important to estimate the power consumption. 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 current 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 resolution, a very fine grid (800 in radial×100 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 radial×30 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 twophase 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. Because the air viscosity is much smaller than oil viscosity, 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 relative 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

3. Results and discussion

3.1 Solution verification

Solution verification is a process for assessing simulation numerical errors and associated uncertainties. In this study, the discretization error due to limited number of grid points is the main source of numerical errors. In this study, solution verification is performed for the average pressure at disk periphery on three systematically refined grids that are generated using a constant grid refinement ratio r=2 in all three spatial directions. The factor of safety method[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 represent the medium and coarse grids, respectively. Simulation 9 that has the highest disk rotational speed and largest inlet flow rate is selected for solution verification. The solutions on the fine, medium, and coarse grids are S1, S2and S3, respectively. Solution changes ε for medium-fine and coarse-medium solutions and the convergence ratio R are defined by

When 0＜R＜1, monotonic convergence is achieved. Then the three grid solutions can be used to compute the estimated order of accuracy pRE, error δRE, and grid uncertainty UG(%S1).

When solutions are in the asymptotic range, then pRE=pth, however, in many circumstances, especially for industrial applications, solutions are far from the asymptotic range such that pREis greater or smaller than. The ratio of pREto pthis used as the distance metric

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

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)

and the grid uncertainty is estimated by

As shown in Table 3, monotonic convergence is achieved with a low grid uncertainty of 3.5%S1. This suggests that the current fine grid resolution is sufficient and this fine grid is used for all simulations.

3.2 Flow physics

Three-dimensional top view of the vortical structures 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 pressure. 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 lowest pressures are located near the chamber wall and axial suction (outlet), 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 (Figs.3(a)-3(c)), only circular vortices are formed regardless of the flow rates. With the increase of the inlet 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 observed in the experiments by Gauthier et al.[13]. It also shows for these two higher rotational speeds that increase 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 pressures 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 rotational 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 increases 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 Simulation 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, respectively, 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 rotating direction. However, due to the opposite rotational directions of the two disks, the upper and lower vortex streets show negative and positive v velocities, respectively, which is consistent with the rotational direction of the adjacent disk.

Figure 5 shows different views by examining flows in various Z cuts for Simulation 5. All the subfigures 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 velocities 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 velocity components. The first component is caused by the local disk rotation and no-slip boundary condition enforced on the disk surface, rω, 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 theradial 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 located 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 diameter r0=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-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 periphery to the center. The spiral vortex network is clearly 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.5 Streamlines and contour of Z velocity component in various Z cuts (Simulation 5): (a) inside the boundary layer of the upper disk (Z =-1× 10-8), (b) inside the boundary layer of the lower disk (Z=-0.99), (c) plane cross the upper vortex street center (Z=-0.22), (d) plane cut through the lower vortex street center (Z=-0.61), and (e) plane cut through the shear layer between the two vortex streets (Z=-0.45)

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 crosssections of the flow between the disks at a constant radius from the rotational Z axis. To facilitate the analysis, the three Cartesian velocity components (u, v, w) are converted to be the components in the cylindrical coordinates (ur, uθ, uz) using

The correlation between these velocity components is shown in Fig.6. Figure 7 shows instantaneous pressure (contour flood) and ur(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 urthat are located inside boundary layers of the two disks. This suggests that the net momentum for fluid close to the disk surface is radially outward. Between the two bands, there is a shear layer where uris negative and the net momentum is radially 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 largest negative radial velocity is located near the midplane 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 velocity 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 ur=0, where the radial velocity changes direction. Figures 8(a) and 8(b) show the instantaneous and time averaged Iso-surface of ur=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 instantaneous 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 Kármán flow in the exact counter-rotating regime can be decomposed into two poloidal recirculations in the (r, z) plane. Similar flow pattern is observed in the current study as shown in Fig.9. The fluid near the upper and lower disks are moved by two opposite rotation speeds (uθ), and then swiped radially outward. As a result of mass conservation, a shear layer develops between the two disks with radially inward velocity. However, unlike the Von Kármán 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 uθof the mean flow at four radial locations are show in Fig.10(a). The tangential velocity is small except inside the boundary layers of the upper and lower disks. When r increases, the magnitude of the tangential velocity increases inside 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 Kármán flow when r is small, three zones are observed: an almost constant tangential velocity zone enclosed by two boundary layers on each disk. However, the tangential velocity constant isnegative, not zero as observed for Von Kármán flow. This indicates that most regions of the flow are impacted 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 Kármán flows that have much larger ReΩ. While the Von Kármán flow shows almost zero radial and axial velocity components, the current open Von Kármán swirling flow exhibits significant magnitude of urand weak uz(still non-zero), as shown in Fig.10(b) and Fig.10(c), respectively. The radial velocity component is about 25% of the magnitude 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 surfaces 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.

Fig.7 Instantaneous pressure (flood, Pa) and ur(line) in various annular control surfaces for X＞0 that are projected to the vertical (Y, Z) plane (Simulation 5)

4. Conclusions and future work

For the first time, unsteady three-dimensionaldirect numerical simulations are conducted to investigate flows between two counter-rotating coaxial disks with an axial extraction enclosed by a cylinder chamber, which is called the “Open Von Kármán 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 torque. Quantitative solution verification is performed on three systematically refined grids. Monotonic convergence is achieved for the average pressure at disk periphery 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).

Fig.8 Iso-surface of ur=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 uθfor Simulation 5

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 geometry and flow conditions. The highest and lowest pressures are located near the chamber wall and axial suction, 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), negative 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-clockwise and clockwise near the upper and lower disks, respectively.

The radial counterflow concept is verified by examining various Z cuts and radial velocity component urin the cylindrical coordinate. Two bands with positive urare located in regions very close to the two disk surfaces where the net momentum offluid is radially outward. Between the upper and lower bands, there is a shear layer where uris 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 significantly 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 velocity components in the (r, z) plane show features similar 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 Simulation 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 extended to two- and multi-phase simulations to investigate 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

The author deeply appreciates the sponsorship from Vorsana Inc. on this research.

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10.1016/S1001-6058(14)60019-6

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